Stefano Mossa Publications

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1 Stefano Mossa Publications E. Marinari, S. Mossa, and G. Parisi, Glassy Potts model: a disordered Potts model without a ferromagnetic phase, Phys. Rev. B 59, 8401 (1999). 2. S. Mossa, R. Di Leonardo, G. Ruocco, and M. Sampoli, Molecular dynamics simulation of the fragile glass former ortho-terphenyl: a flexible molecule model, Phys. Rev. E 62, 612 (2000). 3. S. Mossa, G. Ruocco, and M. Sampoli, Molecular dynamics simulation of the fragile glass former ortho-terphenyl: a flexible molecule model. II. Collective dynamics, Phys. Rev. E 64, (2001). 4. S. Mossa, G. Monaco, G. Ruocco, M. Sampoli, and F. Sette, Molecular dynamics simulation study of the high frequency sound waves in the fragile glass former orthoterphenyl, J. Chem. Phys. 116, 1077 (2002). 5. S. Mossa, M. Barthélémy, H. E. Stanley, and L. A. N. Amaral, Truncation of power law behavior in scale-free network models due to information filtering, Phys. Rev. Lett. 88, (2002). 6. S. Mossa, E. La Nave, H. E. Stanley, C. Donati, F. Sciortino, and P. Tartaglia, Dynamics and configurational entropy in the Lewis-Wahnström model for supercooled orthoterphenyl, Phys. Rev. E 65, (2002). 7. S. Mossa, G. Ruocco, F. Sciortino, and P. Tartaglia, Quenches and crunches: Does the system explore in ageing the same part of the configuration space explored in equilibrium?, Philos. Mag. B 82, 695 (2002). 8. M. Yamada, S. Mossa, H. E. Stanley, and F. Sciortino, Interplay between timetemperature-transformation and the liquid-liquid phase transition in water, Phys. Rev. Lett. 88, (2002). 9. E. La Nave, S. Mossa, and F. Sciortino, Potential energy landscape equation of state, Phys. Rev. Lett. 88, (2002). 10. S. Mossa, G. Ruocco, and M. Sampoli, Orientational and induced contributions to the depolarized Rayleigh spectra of liquid and supercooled ortho-terphenyl, J. Chem. Phys. 117, 3289 (2002). 11. S. Mossa, G. Monaco, and G. Ruocco, Vibrational origin of the fast relaxation processes in molecular glass-formers, Europhys. Lett. 60, 92 (2002). 12. H. E. Stanley, M. C. Barbosa, S. Mossa, P. A. Netz, F. Sciortino, F. W. Starr, and M. Yamada, Water at Positive and Negative Pressures, in Proc. NATO Advanced Research Workshop Liquids Under Negative Pressure, February 23-25, 2002, A. Imre, Ed. (Kluwer, Dordrecht, 2002), preprint cond-mat/

2 13. H. E. Stanley, M. C. Barbosa, S. Mossa, P. A. Netz, F. Sciortino, F. W. Starr, and M. Yamada, Statistical physics and liquid water at negative pressures, Physica A 315, 281 (2002). 14. S. Mossa, E. La Nave, F. Sciortino, and P. Tartaglia, Aging and energy landscapes: application to liquids and glasses, Eur. Phys. J. B 30, 351 (2002). 15. S. Mossa, E. La Nave, P. Tartaglia, and F. Sciortino, Equilibrium and out-ofequilibrium thermodynamics in supercooled liquids and glasses, J. Phys.: Condens. Matter 15, S351 (2003). 16. H. E. Stanley, S. V. Buldyrev, N. Giovanbattista, E. La Nave, S. Mossa, A. Scala, F. Sciortino, F. W. Starr, and M. Yamada, Application of statistical physics to understand static and dynamic anomalies in liquid water, Journal of Statistical Physics 110, 1039 (2003). 17. E. La Nave, F. Sciortino, P. Tartaglia, C. De Michele, and S. Mossa, Numerical evaluation of the statistical properties of a potential energy landscape, J. Phys.: Condens. Matter 15, S1085 (2003). 18. C. A. Angell, Y. Yue, L.-M. Wang, J. R. D. Copley, S. Borick, and S. Mossa, Potential energy, relaxation, vibrational dynamics and the boson peak, of hyperquenched glasses, J. Phys.: Condens. Matter 15, S1051 (2003). 19. S. Mossa and G. Tarjus, Locally preferred structure in simple atomic liquids, J. Chem. Phys. 119, 8069 (2003).

3 PHYSICAL REVIEW B VOLUME 59, NUMBER 13 1 APRIL 1999-I Glassy Potts model: A disordered Potts model without a ferromagnetic phase Enzo Marinari Dipartimento di Fisica and Istituto Nationale di Fisica Nucleare, Università di Cagliari, via Ospedale 72, Cagliari, Italy Stefano Mossa Dipartimento di Fisica and Istituto Nazionale di Fisica della Materia, Università di L Aquila, Località Coppito, L Aquila, Italy Giorgio Parisi Dipartimento di Fisica and Istituto Nationale di Fisica Nucleare, Università di Roma La Sapienza, P. A. Moro 2, Roma, Italy Received 30 July 1998 We introduce a Potts model with quenched, frustrated disorder, which enjoys a gauge symmetry that forbids spontaneous magnetization and allows the glassy phase to extend from T c down to T 0. We study numerically the four-dimensional model with q 4 states. We show the existence of a glassy phase, and we characterize it by studying the probability distributions of an order parameter, the binder cumulant, and the divergence of the overlap susceptibility. We show that the dynamical behavior of the system is characterized by aging. S The generalization of the Ising model to a frustrated model containing quenched disorder, the Ising spin glass, has provided us with a large amount of new physics. 1 Replica symmetry breaking has been found in the mean field theory, 2 and mainly numerical simulations strongly hint at its validity in finite-dimensional disordered Ising spin glasses. 3 The need for a generalization of such systems to the Potts models has lately been clear: technical motivations are obvious, while physical motivations include the need to describe systems where the Z 2 symmetry of the Ising model is not relevant real glasses being potentially among them 4. The most straightforward construction of a Potts spin glass, where the spin variables can be in q states and are randomly connected by a positive or a negative coupling, has been analyzed in detail, 5 13 but it has the unappealing feature which we will justify in the following of acquiring a spontaneous magnetization i.e., of entering a phase with the usual ferromagnetic ordering at low T values. The glassy regime is only present in a small-t region, making it impractical to be studied numerically and unplausible for a faithful description of real glasses which do not order at low temperatures. Here we will define and study what we consider to be the naturally glassy generalization of the Ising spin glasses to the Potts model. We will study the finite-dimensional version of the model, and we will show that these systems do indeed undergo a phase transition that leads them to a glassy phase, different from the usual Sherrington-Kirkpatrick spin glass phase. We regard here as crucial the exact gauge invariance that is found in the usual Ising spin glasses both in the finite number of dimensions and in the mean field approximation. The Hamiltonian of the Ising spin glass is i J i, j j, where the sum runs over couples of first-neighboring sites of the simple hypercubic d-dimensional lattice, the J are 1 with uniform probability or Gaussian variables, and the spin takes the values 1. Let us consider the site i and transform i i. If at the same time we flip all the 2d couplings J i, j involving the site i and one of its firstneighboring sites j), the energy of the system does not change. Because of this symmetry, the expectation value of the magnetization is zero, and a ferromagnetic phase is not allowed. The generalization to a disordered model of Refs is not protected by such a local symmetry, and is allowed to magnetize as it indeed does. We propose instead a generalization to a quenched, frustrated spin model, where the gauge invariance is preserved. The Hamiltonian of our model is H i, j i, i, j j, 1 where the sum runs over first-neighboring sites on a simple cubic hyperlattice or over all site couples in the mean field model, the spin variables can take q values (0,1,...,q 1), and the i, j are link-attached, quenched, random permutations of (0,...,q 1) there are q! of them. Inthe following we will denote this model by M q!. It is clear that, when written as a sum of functions, the Ising spin glass has exactly this form: in this sense this is a very natural generalization of the model, where we just increase the number of allowed states. The link i, j will give a nonzero contribution to the action, not, as in the usual Potts model, if i j, but if i is equal to i, j ( j ). In this model the same gauge symmetry we have described before protects us against magnetizing: if we transform i from 0 to 1, we will interchange, in the quenched random permutations involving the site i, the state 0 and the state 1. This feature makes this model the glassy Potts model a good candidate for a description of the glassy state. The model we have just defined, M q!, has a drawback: it is very difficult to check if it has reached thermal equilibrium. The Z 2 symmetry of the Ising model is indeed precious at this end: in a spin glass checking the symmetry of the probability distribution of the overlap is crucial for establish /99/59 13 / /$15.00 PRB The American Physical Society

4 8402 BRIEF REPORTS PRB 59 ing thermalization. 3 Our first numerical simulations of the model M q! have confirmed how difficult it is to establish on firm grounds thermalization without being able to count on a slow mode that has to exhibit a symmetry. It is possible to solve this problem at least for even values of the number of allowed states, q. One considers a permutation R such that R 2 1 for example, we can change the state 2k with the state 2k 1 for k 0,...,q/2 1), and allows in the function only permutations that commute with R. We have introduced the model M co where co stands for commutative where the Hamiltonian of Eq. 1 contains only the permutation that commutes with the permutation R (0,1,2,3) (2,3,0,1), i.e., R R. This model is symmetric under R, and invariance under R can be tested in order to check if thermal equilibrium has been reached. In order to do that we have defined a modified overlap that is 1 if two spins are in the same state, 1 for the couples (0,2),(1,3),(2,0), and (3,1), and 0 otherwise (q will be the usual overlap, where we sum 1 if two spins are equal and 0 otherwise. Because of the symmetry we have introduced by selecting only R commuting permutations, the probability distribution P( ) is symmetric at equilibrium under. The two models are expected and turn out to be equivalent, as we will show in the following. When using M co it is easy to check thermalization, and the coincidence of the results with the ones obtained when studying M q! sheds light on their physical meaning. We have studied M q! and M co with q 4 states in four spatial dimensions d. We have used a normal Monte Carlo method. We will present thermalized data in the broken phase for lattices of volume L and 5 4, and data in the warm phase for a 8 4 lattice. The data on the two smaller lattice volumes have been obtained from a slow annealing with full sweeps of the lattice at each temperature point; the one with L 8 used sweeps per T point. We have averaged over ten realizations of the disorder. We have preferred to have long thermal runs, since thermalization of the samples needs to be completely sure to make the results reliable and to keep the number of samples quite small. The numerical simulations have taken on the order of 2 months of CPU time of medium size workstations. We only report results for which we are sure of having reached full thermalization the main criteria used being the symmetry of the probability distribution of M co and the request of a good stability in time of the observables. Working on lattices of linear size 4 and 5 we have succeeded in getting some control over the finite-size behavior. A large amount of evidence, which we will describe in the following, makes clear the existence of a phase transition to a low-t glassy phase. The Binder parameters of the modified overlap, 2 g T , 2 show the clear signature of a phase transition. For M co, g 0 with good accuracy in the warm phase because of the symmetry we have implemented. It becomes different from zero at T 1.4, and grows basically linearly in our statistical accuracy at low T. AtT 1.2 the lowest T 2 FIG. 1. (T) vst.atlowt, L 4 is the lower curve and L 5 is the upper curve. Notice the vertical logarithmic scale. value we are sure we have thermalized both at L 4 and L 5)g 1 2. In the precision given by our statistical errors of the average over disorder, evaluated directly for the Binder parameter by a jackknife method, the L 4 and L 5 results coincide at T 1.2 the relative error in the Binder parameter is of the order of 10%: for example, for L 5 we have g ). For the model M q!, g has the same pattern but it becomes slightly negative at T 1.7 but close to zero : atlowtit increases like for the other model. The Binder parameter of the straightforward overlap does not behave in a interesting way for both models. From the analysis of the Binder parameters we deduce as a first guess that T c 1.5. In Fig. 1 we plot the replica symmetry breaking parameter introduced in Ref. 14, , for M co where we have assumed that 0). seems to give an even clearer signature of the phase transition than the Binder parameter qualifying it without ambiguities as a replica symmetry breaking transition. In the infinite-volume limit is zero if the overlap distribution is self-averaging, and becomes nonzero if broken replica symmetry makes it non-self-averaging. Reference 14 shows that while the Binder parameter is very effective in detecting symmetry breaking accompanied by the breaking of spin reversal symmetry, but when spin reversal symmetry is absent tends to be a better estimator. Figure 1 shows a sharp transition notice that the vertical scale is logarithmic. Again a good estimate for T c is 1.5. The evidence for a phase transition is clear. The fact we can detect it by using makes it clear it is a replica symmetry breaking transition. Again, the situation in M q! is very similar: there is a change of regime close to T 1.5, where does not go to zero with increasing L. Maybe the most interesting evidence about the behavior of the system comes from the analysis of the probability

5 PRB 59 BRIEF REPORTS 8403 FIG. 2. P( ) symmetrized vs at T Annealed runs with sweeps per T point. For 0, L 4 is the lower curve and L 5 is the upper curve. FIG. 3. N J ( ) the non-normalized P J ( ) for four typical samples, vs at T Annealed runs with sweeps per T point. distributions of the overlap averaged over the different disorder samples. At high T, P( ) inm q! tends to a Gaussian when increasing the lattice size. We show in Fig. 2 P( ) at T 1.25 for L 4 and 5. There is a clear nontrivial behavior notice that the support in is very extended; i.e., P( ) is very flat. On larger lattices a peak in 0 is emerging, separated by a flat minimum from a second peak the minimum becomes sharper at lower T values, but there we know we did not reach full thermal equilibrium and we cannot safely attach a firm significance to the data : this behavior is reminiscent of the one of random energy models REM s, and fits with what we expect for a glassy state. 4 The important point here is that we have been able to thermalize the L 5 system even in the cold phase (T 1.25 is the lowest temperature value where we are sure about an adequate thermalization. It is useful and necessary to look at the individual, single sample probability distributions P J ( ) in order to qualify the behavior of the individual systems. In Fig. 3 we plot N J ( ) the non-normalized P J ( ) for four typical samples, versus at T The level of the asymmetry of the histograms is a measure of our statistical error and the fact that the functions look symmetric the sign of a good thermalization. The N J ( ) are nontrivial: some samples have double peaks, some have their support close to 0, and some have support at zero overlap and peaks at finite overlap. On a qualitative level we remark that the system looks harder than the usual spin glass: the dynamics is more jumpy, visiting in a quite discontinuous manner different parts of the phase space that is why checking thermalization has been difficult and crucial. Our evidence seems to suggest that deep minima do not have large basins of attraction in the free energy phase space. The probability distribution for the modified overlap in the model M q! is similar to the one we have shown. The detailed shape is not exactly the same, but it also becomes nontrivial at T On the L 5 lattice a two-peak structure starts to emerge from T 1.25 down one in 0 and one at a finite value. The nonmodified overlap distributions for both models enjoy the same main features even if they are nonsymmetric and, as we have already discussed, the thermalization is better checked by using the symmetry of P( ) inm co : it tends to a Gaussian when increasing L at T 1/0.6, it is nontrivial at T 1/0.7, and it develops a twopeak structure at lower T values. We have used the T data in the warm phase from the large, L 8 lattice to fit the divergence of the overlap susceptibility V 2, which T T c behaves as (T T c ). From our data we can only give a preliminary estimate that puts T c among 1.4 and 1.5 compatible with the value we have deduced from the direct analysis of P( ) and the exponent among 1.3 and 1.5. This value is differ- FIG. 4. Dynamical correlation function C(t,t w )vstfor different values of t w. Curves from top to bottom for decreasing waiting times (t , 3000, 1000, and 300.

6 8404 BRIEF REPORTS PRB 59 ent from the one quoted for the Ising four-dimensional 4D spin glass, The dynamical behavior of the system shows all the typical features of the complex dynamics. In Fig. 4 we show the aging behavior: the spin-spin correlation function C(t,t w ), depending on the waiting time t w and on the time t, versus t for different waiting times t w. The rate of the time decay depends on t w. Let us notice at last that the energies of the two models are very similar: on the L 5 lattice they are equal in our statistical precision, while on the L 8 lattice they are of the order of 1 per The two models seem to present the same kind of critical and off-critical behavior, and our best guess is that they do belong to the same universality class. We have introduced a disordered generalization of the Potts model that we regard as a very hopeful candidate to describe the glassy state. Our numerical simulations of the four-state, four-dimensional model show clearly the existence of a glassy phase, and they stress large differences with the usual Sherrington-Kirkpatrick spin glass phase, exhibiting a more discontinuous behavior, reminiscent of the random energy model one-step replica symmetry breaking. 1 We thank Renate Loll for the gift of a script computing commuting permutations. 1 M. Mezard, G. Parisi, and M. A. Virasoro, Spin Glass Theory and Beyond World Scientific, Singapore, G. Parisi, Phys. Rev. Lett. 43, ; J. Phys. A 13, ; 13, ; 13, L ; Phys. Rev. Lett. 50, E. Marinari, G. Parisi, and J. J. Ruiz-Lorenzo, in Spin Glasses and Random Fields, edited by P. Young World Scientific, Singapore See, for example, G. Parisi, cond-mat/ unpublished. 5 D. Elderfield and D. Sherrington, J. Phys. C 16, L ; 16, L ; 16, L A. Erzan and E. J. S. Lage, J. Phys. C 16, L D. J. Gross, I. Kanter, and H. Sompolinsky, Phys. Rev. Lett. 55, T. R. Kirkpatrick and P. G. Wolynes, Phys. Rev. B 36, ; 38, T. R. Kirkpatrick and D. Thirumalai, Phys. Rev. B 37, G. Cwilich and T. R. Kirkpatrick, J. Phys. A 22, ; 23, M. Scheucher, J. D. Reger, K. Binder, and A. P. Young, Phys. Rev. B 42, E. De Santis, G. Parisi, and F. Ritort, J. Phys. A 28, K. Binder, in Spin Glasses and Random Fields, edited by P. Young World Scientific, Singapore, E. Marinari, C. Naitza, G. Parisi, M. Picco, F. Ritort, and F. Zuliani, Phys. Rev. Lett. 81, G. Parisi, F. Ricci Tersenghi, and J. J. Ruiz-Lorenzo, J. Phys. A 29,

7 PHYSICAL REVIEW E VOLUME 62, NUMBER 1 JULY 2000 Molecular dynamics simulation of the fragile glass-former orthoterphenyl: A flexible molecule model S. Mossa, 1 R. Di Leonardo, 1 G. Ruocco, 1 and M. Sampoli 2 1 Dipartimento di Fisica and INFM, Università di L Aquila, Via Vetoio, Coppito, L Aquila, I-67100, Italy 2 Dipartimento di Energetica and INFM, Università di Firenze, Via Santa Marta 3, Firenze, I-50139, Italy Received 16 December 1999 We present a realistic model of the fragile glass-former orthoterphenyl and the results of extensive molecular dynamics simulations in which we investigated its basic static and dynamic properties. In this model the internal molecular interactions between the three rigid phenyl rings are described by a set of force constants, including harmonic and anharmonic terms; the interactions among different molecules are described by Lennard-Jones site-site potentials. Self-diffusion properties are discussed in detail together with the temperature and momentum dependencies of the self-intermediate scattering function. The simulation data are compared with existing experimental results and with the main predictions of the mode-coupling theory. PACS number s : Pf, Pd, Em, p I. INTRODUCTION In recent years a renewed interest on the glass transition phenomenon has motivated extensive experimental and theoretical works see 1 and reference therein. On the theoretical side, new descriptions of the glass transition have been developed: they emphasized either the dynamic as the mode-coupling theory MCT of Götze 2 the reader may also consult Ref. 3 and Schilling and Kob 3 or the coupled oscillators model of Ngai and Tsang 4 or the thermodynamic as the first principle computation based on a replica formulation of 5 or the inherent structure formalism computation of 6 aspects of the transition itself. A common feature of all these theories is they have been developed for model systems, often monoatomic models. The comparison of the theoretical results with the real experiment are, therefore, complicated by the trivial observation that in the real world the glass-forming systems are made out of molecular systems. As a consequence in the current literature there is a large debate on the applicability of the theoretical predictions to the experimental outcome and the more stringent tests of the theories come from molecular dynamics MD works. As an example, it is highly debated in literature the origin, in a molecular glass former, of secondary relaxations that can be observed by several experimental techniques beside the well-known microscopic and structural dynamics see, among others, Ref. 7 and references therein. For instance, fast relaxations i.e., in the s range have been observed in several glasses and it is not yet clear if their origin is related to the molecular center of mass motion, as MCT would explain in terms of the process, or rather to rotational or intramolecular dynamics. It is clear the crucial role that MD simulations can play to solve this specific problem. If it is possible to build a realistic model able to take into account the internal degrees of freedom, as well as the translational dynamics, computer simulations allow one to access any observable quantity of the system, and also those not directly measurable by present experimental techniques. Such possibility, together with the physical intuition, could allow the identification of the microscopic mechanisms underlying the different observed relaxation processes. In this paper we want to address the problem to set up a realistic potential for a glass model system capable of accounting for the internal molecular degrees of freedom. Among the glass-forming molecular liquids characterized by an extremely rich dynamical behavior, the organic fragile 1 glass former orthoterphenyl OTP (T m 329 K, T c 290 K, T G 243 K) has received much attention from both experimental and numerical simulation points of view. The structure of the OTP molecule, shown in Fig. 1, is known from neutron 8 and x-ray 9 diffraction studies; in condensed state the OTP molecules are bound together only by van der Waals forces, which resemble the Lennard-Jones ones often used by most theories and computer simulations aiming to study the glass transition problem. Due to its structural complexity, if we would like to obtain reliable results to be compared with experimental data we need to take into account not only the translations of the molecular center of masses and the rotations of the molecules as a whole, but also intramolecular motions like stretching along the molecular bonds, tilt of the bonds, rotation of the side rings with respect to the central one, and so on. In other words, we need to describe the dynamics of the liquid at the atomic level. On the contrary, in order to set up FIG. 1. Molecular structure of OTP (C 18 H 14 ); it is constituted by three phenyl rings, the two side rings being attached to the parent i.e., central ring by covalent bonds X/2000/62 1 / /$15.00 PRE The American Physical Society

8 PRE 62 MOLECULAR DYNAMICS SIMULATION OF THE a computational scheme which is affordable in not-too-long time with the nowadays computer capability, we need the simplest model potential able to capture the relevant features of the dynamical behavior of the real system. In the literature numerical studies of OTP have been proposed making use of different techniques ranging from harmonic lattice dynamics 10 to molecular dynamics simulations on the atomic level based on a general force field provided by the standard program Alchemy III 11. Nevertheless, to our knowledge, only two studies based on molecular dynamics simulations of molecular models of OTP have been proposed so far. They are as follows. i Lewis and co-workers 12,13 represent the molecule like a three-sites complex, each site playing the role of a whole phenyl ring, without internal dynamics and an intermolecular interaction of the Lennard-Jones LJ type. This model takes into account only the dynamical behavior associated with the translations of the molecular center of masses and with the rotations of the molecules as a whole. ii Kudchadkar and Wiest 14 propose a more realistic model with the true structure of the molecule. The intermolecular interaction is of LJ type and, as internal degrees of freedom, only the rotational dynamics of the side rings with respect to the central one is taken into account. These internal degrees of freedom are effectively the most relevant; nevertheless, the authors parametrize the potential in such a way that the side rings are, at equilibrium, in a configuration that corresponds to a saddle point in the molecular energy surface. The model potential we are going to introduce is much more efficient in mimicking the complexity of the dynamical behavior of the real system. The paper is organized as follows: in Sec. II we introduce the intramolecular model potential; in Sec. III we explain how we calculated the force constants in order to reproduce a realistic isolated molecule vibrational spectrum. In Sec. IV we present some computational details and in Sec. V we note some of the main wellestablished predictions of the ideal mode-coupling theory used in Sec. VI to test the center of mass dynamical behavior. In Sec. VI we discuss our MD simulation results mainly with regard to the study of the diffusion and self-dynamic properties. Section VII contains an overall discussion and the conclusions. II. THE MOLECULAR MODEL In our model the OTP molecule is constituted by three rigid hexagons phenyl rings of side L a nm connected as shown in Fig. 1, i.e., two adjacent vertices of the parent central ring are bonded to one vertex of the two side rings by bonds whose length, at equilibrium, is L b 0.15 nm. In our scheme, each vertex of the hexagons is thought to be occupied by a fictious atom of mass M CH 13 amu representing a carbon-hydrogen pair C-H. The choice of such a fictious atom, with its renormalized mass, greatly simplifies the computer simulation but presents some drawbacks: i in the real molecule the two couples of carbon atoms connecting the rings are not bonded to hydrogen atoms, while in our model we consider all the 18 vertices having the same mass M CH so that the total molecular mass is overestimated 234 rather than 230 amu and ii the moments of inertia of the model rings are smaller than the real ones the hydrogen atoms being too close to the ring center. Nevertheless, we expect only minor effects on the overall dynamics from the previous simplification. The three rings of a given molecule interact among themselves by an intramolecular potential, such potential being chosen i to preserve the molecule from dissociation; ii to give the correct relative equilibrium positions for the three rings; and iii to represent the real intramolecular vibrational spectrum as close as possible. The interaction among different molecules, actually among the rings pertaining to different molecules, is accounted for by a site-site pairwise additive potential energy of the 6-12 Lennard-Jones type, each site being one of the hexagons vertices. To sum up, the total interaction potential energy is written as the sum of an intermolecular and an intramolecular term, V tot V inter V intra. The first term can be written explicitly as V inter 1 2 V LJ r i l r j l, i j ll where r i l is the position of the l th atom (l 1,...,6) in the th ring ( 1,...,3, hereafter 1 indicates the parent ring belonging to the ith molecule (i 1,...,N), and V LJ R 4 R 12 R 6. The optimal choice of the two intermolecular force parameters, and, will be discussed later. In principle, the intramolecular interaction potential can be expressed in terms of the degrees of freedom describing the center of mass positions (R, 2,3) and orientations e.g., the set of Eulerian angles of the side rings with respect to the parent ring. However, for computational purposes, it is simpler to express the intramolecular potential in terms of orthonormal unit vectors attached to each ring or better in terms of quantities built from these vectors. With reference to Fig. 2 the sets of unit vectors for each ring lˆ,mˆ,nˆ are defined as lˆ 1,0,0, mˆ 0,1,0, nˆ 0,0,1, i.e., lˆ and mˆ are orthogonal unit vectors in the ring plane, while nˆ is the normal to that plane. The unit vectors that are parallel to the ring-ring bonds at equilibrium are given by ū lˆ2 3mˆ 2, ū lˆ3 3mˆ 3, ū 1(2) 1 2 lˆ1 3mˆ 1, 1 2 3

9 614 S. MOSSA, R. Di LEONARDO, G. RUOCCO, AND M. SAMPOLI PRE 62 On the other hand, no direct chemical bond is present between the two side rings, and we expect a less stiff spring for the fluctuation of the distance between the side ring centers. We model this interaction by V B c 2 R 2 R 3 2L c 2. 5 The determination of the force constants c 1 and c 2, as well as the others we are going to introduce, will be discussed later. B. Tilt of the ring-ring bond FIG. 2. Model geometry: each phenyl ring is represented by a rigid hexagon of side L a nm and the equilibrium bond length is L b nm. C 1,C 2,C 3 represent the origins of the reference frames fixed with the rings; ū 2,ū 3,ū 1(2),ū 1(3) are the vectors parallel to the ring bonds; lˆ1 and mˆ 1 are two versors identifying the parent ring plane; P 2,P 3,P 1(2),P 1(3) are the positions of the carbon atoms bonding together the rings; S 2,S 3,S 1(2),S 1(3) are four interaction sites introduced to force the rings towards the coplanar equilibrium condition; the symbols P A and P B have been introduced to identify the angles P 2 P 1(2) P A and P 3 P 1(3) P B. ū 1(3) 1 2 lˆ1 3mˆ 1. The positions of the four carbon atoms that link the three rings, i.e., P 1(2) and P 1(3) in the parent ring and P 2 and P 3 in the side rings, are given by, with respect to their ring centers R, P 1(2) R 1 L a ū 1(2), P 1(3) R 1 L a ū 1(3), P 2 R 2 L a ū 2, and P 3 R 3 L a ū 3. Finally, it is useful to define four further interaction sites, two pertaining to the parent ring, S 1(2) R 1 L c ū 1(2) and S 1(3) R 1 L c ū 1(3) and two pertaining to the side rings 2 and 3, respectively, S 2 R 2 L c ū 2 and S 3 R 3 L c ū 3. Here L c L a L b /2, so that at the equilibrium position S 1(2) S 2 and S 1(3) S 3. The variable we have introduced will be used to simply express the different contributions to the intramolecular potential in Secs. II A II C. A. Stretching along the ring-ring bonds and between the side rings The fluctuations of the distances among the centers of mass of the three rings are accounted for by introducing three springs. The parent-side ring stretching implies the elongation of a C-C bond, which is expected to have a high stiffness. The corresponding potential in harmonic approximation is written as V S c 1 P 1(2) P 2 L b 2 c 1 P 1(3) P 3 L b 2. 4 In the OTP crystal structure 15, the bond angles P 2 P 1(2) P 1(3) and P 3 P 1(3) P 1(2) are and 123.0, respectively, while the angles P 2 P 1(2) P A and P 3 P 1(3) P B are and see Fig. 2 Further, in the isolated molecule, the ring-ring bonds are forced out of the plane of the parent ring so that the dihedral angle P 2 P 1(2) P 1(3) P 3 is 5.2. This lack of planarity is due to the little asymmetry introduced by the difference between a carbon bonded to a hydrogen and a carbon bonded to a carbon of another ring. In our model, all these angles at the equilibrium are set equal to 120 and equal to 0. We model the restoring forces for these angles by using the scalar product of the unit vectors ū 2 and ū 1(2) as well as that of ū 3 and ū 1(3) ). Since ū 2 ū 1(2) ū 3 ū 1(3) 1at equilibrium, the quadratic term in the small oscillation approximation is given by V T1 c 3 1 ū 2 ū 1(2) c 3 1 ū 3 ū 1(3). However, this term is not enough to ensure the coplanarity of the vectors ū 2 and ū 3 with the parent ring; to force the rings towards the coplanar equilibrium condition we make use of the sites S 2 and S 1(2) as well as S 3 and S 13 ) introducing between them a spring of vanishing equilibrium length: V T2 c 4 S 1(2) S 2 2 c 4 S 1(3) S 3 2. C. Rotation of the side rings along the ring-ring bond Inside the intramolecular dynamics we expect that a crucial role is played by the rotation of the side rings planes around the ring-ring bonds 15,16 in interfering or modifying the intermolecular relaxation processes. The relevant variables describing this motion are the two angles 2, 3 between the normals to the side ring and parent ring planes. In the crystalline structure, the two side rings have slightly different angles, 42.4 and However, we model the disordered condensed phases liquid and glass by using as equilibrium angle values those of the isolated molecule, i.e., 54.0, as discussed below. We have to remark that the isolated molecule symmetry implies two isoenergetic configurations separated by a finite barrier as is qualitatively illustrated in Fig. 3. We performed an ab initio calculation of the single molecule potential energy surface as a function of 2 and 3 with all the other internal degrees of freedom fixed to their equilibrium values. Such calculation consists in the minimization of the Hartree-Fock energy over the Gauss- 6 7

10 PRE 62 MOLECULAR DYNAMICS SIMULATION OF THE n 1 n 2 n 1 n 3. 2 The final form of the internal rotation potential will be V R V R1 V R2, 9 with V R1 b 1 2 b 2 4 b 3 6, V R2 c FIG. 3. Rotational energy: each point has been determined by an ab initio Hartree-Fock calculation of the potential energy surface as a function of the rotational angles 1 and 2 expressed in degrees with all the other degrees of freedom fixed to their equilibrium values; the energy is expressed in Hartree units 1 Hartree 27.2 ev. We indicate by A the isolines ranging from to mhartree with an increment mhartree, by B the ones from to mhartree, with mhartree, and by C those from 0.5 to 1 mhartree with mhartree. This figure has to be considered only from a semiquantitative point of view as explained in the text. ian basis set 3-21G. For each atomic species the inner shell is made up of three Gaussians while the valence shell is a linear combination of two Gaussian orbitals plus one Gaussian orbital; the minimization was carried out by the standard package Gaussian 94. In Fig. 3 a contour plot of the Hartree-Fock potential energy is shown as a function of the rotation angles. It is important to quote that this map has only a semiquantitative meaning; the structure of the whole molecule is not reoptimized during the scan of 2 and 3 angles. However, a careful study, performed reoptimizing the whole molecular structure, has been done around the saddle point and around the two equivalent minima that turn out to be at and From this calculation the barrier height has been estimated to be V s /k B 580 K and, from this value, it is possible to envisage the nature of the rotational motion at the temperatures of interest: the two side rings can pivot in phase around the bonds crossing from one minimum to the other degenerate one. Moreover, they can perform librational out-of-phase motions of approximatively harmonic type. In order to represent this potential surface we express the in-phase rotation of the two side rings with a high-order 6th polynomium and the out-of-phase rotation in the harmonic approximation. For this purpose we use as primary variables the scalar products n 1 n 2 and n 1 n 3 their equilibrium value being (n 1 n 2) eq (n 1 n 2) eq and to disentangle the in-phase from the out-of-phase motion, we introduce the two variables n 1 n 2 n 1 n 3, 2 8 The parameters b 1, b 2, and b 3 describing the in-phase rotation potential are derived according to the following procedure. In the harmonic approximation, in proximity of the equilibrium position o for the scalar products, it must hold V R1 c 5 o 2, 12 and taking into account that the barrier height must be equal to V s, we have the following conditions: implying V R1 o V s, V R1 o 0, V R1 o 2c 5, b 1 V s o 2 c 5 4 2b 3 o 4, b 2 c 5 4 o 2 3b 3 o 2, b 3 1 o 4 V s o 2 c III. FORCE CONSTANTS We can finally write our internal model potential like V intra V S V B V T1 V T2 V R1 V R2, 14 this potential is parametrized by the set of six free coefficients c k whose actual value can be tuned in order to obtain a realistic free molecule vibrational spectrum. Indeed, in the small oscillations approximation, we can determine the values of the coefficients c k by diagonalizing the dynamical matrix and fitting the resulting eigenfrequencies DIAG to the lowest frequencies HF obtained by a Hartree-Fock calculation of the vibrational frequencies in the electronic ground state of the isolated molecule. We have 18 eigenvalues for the dynamics of three rings ( 1,...,18) butactually only N e 12 eigenvalues DIAG are nonvanishing, the other being associated with the translations and rotations of the molecule as a whole.

11 616 S. MOSSA, R. Di LEONARDO, G. RUOCCO, AND M. SAMPOLI PRE 62 TABLE I. Values of the internal potential coefficients determined by a least square minimization of the error function, Eq. 15. The coefficients c k are the force constants associated with the different intramolecular potential terms; the b k, derived from the c k via the Eq. 13, describe the form of the in-phase rotational potential according to Eq. 10. c J/nm 2 c J/nm 2 c J c J/nm 2 c J c J b J b J b J More explicitly, the set c k is obtained by minimizing, by the standard Levenberg-Marquardt algorithm 17, the error function N e 1 DIAG HF 2 15 where, for a given set of c k, the quantities DIAG solution of the eigenvalues problem 18 Vˆ 2 Tˆ 0. are the 16 Here Vˆ and Tˆ are the Hessian matrixes of the second partial derivatives of the potential and kinetic energy, respectively, with respect to the translational and rotational degrees of freedom of the three rings. In Table I are reported the values of the coefficients c k determined by the minimization together with the values for TABLE II. Free molecule vibrational frequencies: in the second column are reported the values HF determined by a Hartree-Fock calculation of the ground state of the isolated molecule we consider only the 12 lowest eigenvalues ; in the third column are shown the frequencies MD corresponding to the different peaks of the spectrum determined from the atomic velocity autocorrelation function calculated by means of a preliminary molecular dynamics simulation of the isolated molecule ( 0) at T 1 K. HF (cm 1 ) MD (cm 1 ) b k derived from Eqs. 13. The corresponding set of frequencies is shown in Table II where MD are the frequencies derived from a computer simulation on a system of isolated molecules at low temperature. These frequencies have been identified via the peaks of the spectrum of the velocity autocorrelation function. Other model details are reported in Ref. 19. IV. COMPUTATIONAL DETAILS We have studied a system composed of 108 molecules 324 rings, 1944 interaction centers ; the sample is large enough to neglect finite size effects on the investigated properties with reasonable computation times. The values of the parameters entering the site-site Lennard-Jones interactions were determined by preliminary simulations: the value of were firstly determined by comparing the computed and the experimental self-diffusion coefficients versus temperature 34 in the range 380 K T 440 K; the value of was estimated by tuning the static structure factor so to place the first maximum in the right position 32. Successive iterations led to /k B 14 K and 0.4 nm. To speed up the calculation of the intermolecular potential and to assure all the torques to be estimated in a consistent way, the cutoff for LJ interactions is applied between rings centers, i.e., the sites are considered not interacting when they pertain to rings whose center distance is larger than r c 3( L a ) 1.6 nm. At this stage a remark about the rotational potential expressed in Eq. 9 is needed: such potential presents a serious drawback since the intrinsic ambiguity due to the parity of the scalar products involved can force the side rings in the wrong positions. This problem can be bypassed introducing a new term in the intramolecular potential that gives zero contribution to the energy when the molecule is close to the equilibrium position. If we define the product w (lˆ1 nˆ 2)(lˆ1 nˆ 3), and we set c 7 10c 6 the actual value of c 7 is irrelevant, it must only be able to force the rings in the right way, we can write V A c 7w 2 if w 0 0 otherwise. 17 This term has been activated only during the preparation of the initial configuration i.e., at high temperature ; during the thermalized evolution of the system we expect molecules do not drive too much away from their equilibrium configuration and then this term to be always equal to zero. To integrate the equations of motion we have treated each ring as a separate rigid body, identified by the position of its center of mass R i and by its orientation expressed in terms of quaternions q i 20. The standard Verlet leap-frog algorithm 20 has been used to integrate the translational motion while, for the most difficult orientational part, the refined algorithm due to Ruocco and Sampoli 21 has been employed; such choices allow a very stable integration with a relatively long time step. The rotational dynamical problem can be written as

12 PRE 62 MOLECULAR DYNAMICS SIMULATION OF THE dj i i dt R j, q j, dq i Mˆ dt q i J i 18 TABLE III. Simulation runs details: at each temperature T, the system is coupled for a time t resc to a stochastic heat bath, then it evolves freely, at constant energy, for a time t term. At the end of this process, we consider the final configuration to be in a good equilibrium state and we start a trajectory t prod ps long, saving a system configuration every t save ps. where J i is the angular momentum of ring in the molecule i, i is the torque acting on it, and Mˆ i is the inverse of the inertia tensor in quaternion coordinates. The expression of the torques is simplified as the rotational part V rot of the intramolecular potential energy has been expressed in terms of suitable scalar products written in the general form s v 1 / v 1 v 2 / v 2. Therefore the value of the torques v1 and v2 associated with the degree of freedom corresponding to the angle arccos(s) can then be evaluated by 18 v1 v2 V rot v 1 v 2. s 19 The leap-frog algorithm for the rotational motion reported can be written in the form 21 J i t t J i t t 2 i with 0 t t, t t 2 t 20 q i t q i t 0 dt Mˆ q i t t J i t t 21 with 0 t, J i t t 2 J i t t 2 i t t. 22 The crucial point is that the dependence of the matrix Mˆ i on the angular variables implies the need of using a time step t to perform the numerical integration appearing in Eq. 21, smaller then t used for the center of mass cm integration. In turn, the value of t is limited by the highest vibrational frequency of about 450 cm 1. In our case, the chosen values of t 2 fs and t t/5 are found to be sufficiently small to reduce the fluctuations of the total energy to a negligible fraction of the kinetic energy. It is important to note that the use of the smaller t does not increase significantly the CPU computational time since the time-consuming part in each MD step is the calculation of the forces and torques which are kept fixed during the integration of Eq. 20. We considered a wide temperature range spanning the liquid phase and reaching the region close to T c as shown in Table III in which the whole set of simulation times is reported. During the different temperature runs, the size of the cubic box has been rescaled in order to keep the system at the experimental density which, for T T g, can be fitted by the polynomium 22 T T T 2, 23 T t resc t term t prod t save K ps ps ps ps where is in g/cm 3 and T in K. At each temperature T we have organized the simulations following this scheme: The system was coupled for a time t resc, chosen in a somehow arbitrary way, to a stochastic heat bath, i.e., the velocities of the rings were replaced following a logarithmic pattern with the velocities drawn from a Boltzmann distribution corresponding to such temperature. At this point the system was at the desired temperature and we let it perform a microcanonical time evolution constant energy for a period t term comparable with the experimental structural relaxation time at the same temperature; in such a way we expect every slow degree of freedom of the system to be correctly thermalized and we control that there was no drift in temperature and the degree of energy conservation fluctuations are always less than 1% of the kinetic energy. We considered the final system configuration obtained in this way as a good equilibrium starting state for a molecular dynamics trajectory. At each temperature we perform three different runs: the first one 20 ps long with a ps configuration saving time has been used to compute the small-time dynamical behavior of the system; a second one 640 ps long with a ps saving time has been used for some intermediate frequency analysis. The last one, with length t prod and saving time t save dependent on temperature, has been used to calculate static quantities and the long-time behavior of the system. All the calculations have been performed on a cluster of four -CPU with a frequency of 500 MHz; every nanosecond of simulated dynamics needed approximately 24 h of CPU time. V. RECALL OF MAIN RESULTS FROM THE MODE-COUPLING THEORY A great improvement to our understanding of the glassy state of matter has come from the extension of the theoretical

13 618 S. MOSSA, R. Di LEONARDO, G. RUOCCO, AND M. SAMPOLI PRE 62 building of the mode-coupling theory MCT 2,3 developed for the equilibrium description of the dynamics of simple, i.e., monatomic, liquid to the study of the glassy state. Although such theory is the only microscopic approach to the glass transition leading to many predictions on the experimental data, it is still at the center of a strong debate and some questions stay open. In fact, even if the real range of validity of MCT for the study of molecular liquids has been cleared in recent years see, among others, Refs ,44, some experimental results seem to contradict fundamental predictions of the idealized version of MCT, such as the presence of the so-called knee characterizing the lowfrequency behavior of the light scattering susceptibility 26,27 or the presence of a cusp in the nonergodicity parameter 28. In this section we sum up the main predictions of the so-called ideal MCT where it is hypothesized a complete dynamical freezing and the so-called thermally activated hopping processes are neglected; such predictions will be compared with our simulation data. In the ideal MCT the glass formation is interpreted as a dynamical transition from an ergodic to a nonergodic behavior at a crossover temperature T c. MCT provides a self-consistent dynamical treatment 2 for the density correlation function of an isotropic system F q,t 1 N q * t q * 0, 24 where N is the number of the particles,, N q (t) i 1 exp iq r i(t), and r i(t) is the position of particle i at time t. MCT proposes a particular ansatz for the memory kernel in the related integrodifferential generalized Langevin equation, such kernel is coupling nonlinearly the density fluctuations with one another. If the coupling increases upon lowering the temperature, the resulting dynamical feedback leads to a progressive slowing down of the density fluctuations until they become completely frozen at the critical temperature T c. The ideal MCT describes the behavior as much as the temperature approaches T c, i.e., the parameter (T c T)/T c is small however, real comparisons have to be made for not too small, in contrast to the case of scaling laws in phase transitions. For temperature T T c, F(q,t) is characterized by two step decays taking place at different time scales and the theory gives specific predictions for such different time regions. The first one, the -process region, is centered around a time which is predicted to scale like T T c 1/2a with 0 a 0.5 and to be bounded in the interval 0, where 0 is the time scale of the microscopic dynamics and is the structural rearrangement time scale. In this region the factorization property holds, in the sense that the density correlation function can be written as F q,t f q h q G t/, 25 where f (q) is the nonergodicity parameter i.e., Debye- Waller factor for collective correlators or Mössbauer-Lamb factor for single-particle correlators, h(q) is an amplitude independent of temperature and time, and the in G corresponds to time larger or smaller with respect to. So, the time dependence of the correlation functions is all embedded in the q-independent function G, namely spatial and temporal correlations result to be completely independent. G (t) is asymptotically expressed by two power laws, respectively the critical decay and the von Schweidler law 2, characterized by the temperature and momentum independent exponents a and b, G t t/ a, 0 t t/ b, t. 26 Here a is the same exponent of the power divergence of at T c and it is related to the exponent b (0 b 1) via the equation 2 1 a 1 2a 2 1 b 1 2b, 27 where is the gamma function. The second time region is the so-called region where the second decaying step takes place. This region is connected to the collective structural relaxations and asymptotically the theory predicts the validity of the well-known timetemperature superposition principle; it states that, on time scales of the same order of magnitude as, the following scaling law holds at every temperature T: t F q,t F 28 T. In other words, the correlation functions of any observables at different temperatures can be collapsed into a master curve when the time is scaled with t/. Moreover, MCT predicts that this master curve can be fitted by a Kolrausch-William- Watts function stretched exponential F q,t f q exp t. 29 The time scale depends on temperature through a power law of the form T T c, 30 where the q-independent exponent is related to the power exponents a and b of the region by the relation 1 2a 1 2b. 31 The inverse of the diffusion constant D 1 (T) is predicted to scale like 2 and consequently it follows Eq. 30. Up to now, all dynamical results reviewed are universal in the sense that they are predicted to hold for the correlators of every observable with nonzero overlap with density; in particular this is true for both the one-particle and the collective density correlation functions. Nevertheless important differences are predicted to hold for the q dependence in these two cases: in the former case f (q) and h(q) depend smoothly on q, in the latter one they oscillate, respectively, in phase and out of phase with the static structure factor S(q). Moreover, is predicted to be a smooth function of q in the one-

14 PRE 62 MOLECULAR DYNAMICS SIMULATION OF THE TABLE IV. Some thermodynamical results: temperatures T effectively measured, total energy E tot, total potential energy V tot, and kinetic energy T. T E tot V tot T K kj/mol kj/mol kj/mol particle case; at variance, it shows pronounced oscillations in phase with S(q) in the collective case. VI. RESULTS A. Thermodynamics In this section some thermodynamical time-independent results are shown such as potential, kinetic, and total energy see Table IV and Fig. 4. The interest in these results is clarified by the following argument: in computer simulations dealing with the glass transition it is possible to define a temperature often named T g sim 29 at which one-time quantities show some sort of discontinuity. Such discontinuity, whose position depends on the thermal history of the system, represents the thermodynamical point at which the system undergoes a glass transition on the time scale of the computer simulation, falling out of equilibrium. It is clear from Fig. 4 that no discontinuity is present, i.e., T g sim of FIG. 5. Pair static distribution functions at T 300 K calculated on atoms A, ring centers of mass B, and molecular centers of mass C ; full lines represent the total contribution of both intramolecular and intermolecular distances, dashed lines only the intermolecular contribution. our simulations is less than the lowest temperature studied and then we have a good chance to have well-thermalized results. Nevertheless, whether or not the system is in equilibrium can be checked only a posteriori by comparing the total simulation time with the measured relaxation time. From the linearity of E(T) we deduce a specific heat c(t) constant in the temperature range investigated and equal to 140 J K 1 mol 1 ; such value must be compared with the experimental value of J K 1 mol It is possible to explain the inconsistency between the two values keeping in mind that our MD value is a classical nonquantic result and, more important, we are neglecting many ( 78) degrees of freedom concerning the deformations of the phenyl rings. B. Structure In general the static structure of a fluid is well described by the pair distribution function 31, g r V N 2 i j i r r ij ). 32 FIG. 4. Temperature dependence of the energies tabulated in Table IV, E tot circles, V tot squares, and T triangles up together with the internal V intra triangles down and the intermolecular Lennard-Jones V inter diamonds contributions to V tot. In computer simulations 20, we can identify the distances r ij with different quantities. In Fig. 5 we report some g(r) s at T 300 K where we have considered as r ij the distances between the carbon atoms belonging to different rings A and between the center of masses of rings B and molecules C ; both total solid line and intermolecular dashed line contributions are shown in order to separate the internal molecular structure and the mean structural organization of the whole bulk sample. In Fig. 5 B a two peak

15 620 S. MOSSA, R. Di LEONARDO, G. RUOCCO, AND M. SAMPOLI PRE 62 FIG. 6. Static distribution functions of the scalar products nˆ 2 nˆ 3 triangles, lˆ1 nˆ 2,3 circles, and nˆ 1 nˆ 2,3 squares evaluated at T 280 K. structure is present: the first sharp peak is placed at r 0.42 nm corresponding to the mean distance between rings belonging to the same molecule; the second one, of intermolecular origin, is placed at r 0.6 nm. It is worth noting that such distance is less than the greatest intramolecular C-C distance ( 0.7 nm). Moreover, the molecular centers of mass g(r) also show a large value on distances less than 0.7 nm, giving the evidence of a strong packing of the molecules. All these features appear to be approximately temperature independent. Such packing depends strongly on the orientational internal configuration of the molecules, namely on the positions of the two side rings with respect to the parent one; the computation of the probability distribution of the scalar products among the versors lˆ, mˆ, and nˆ, introduced in Sec. II, is somehow instructive in this sense. In Fig. 6 the distribution functions for the quantities lˆ1 nˆ 2,3, nˆ 2 nˆ 3, nˆ 1 nˆ 2,3 are shown. The first two distributions are practically temperature independent and give us only informations on the correctness of the simulated geometry: they are sharply peaked on the the correct equilibrium positions of about 0.71 and 0.69, respectively. At variance with the distribution of x nˆ 2 nˆ 3 and of nˆ 1 nˆ 2,3 that are symmetric around x 0, the distribution of lˆ1 nˆ 2,3 does not present the symmetric peak on negative values so that we can argue that the auxiliary term V A worked correctly. The most interesting distribution is the third one in which the peak intensity the peak is correctly placed at ) is higher the lower the temperature, as shown in Fig. 7, indicating therefore that the correspondent degree of freedom the in-phase motion of nˆ 1 nˆ 2,3 ) is more and more frozen on its equilibrium value with decreasing the temperature. We have seen that in the isolated molecule the rotational motion of the two side rings can be separated in two contributions: an out-of-phase harmonic libration and an in-phase pivoting around the bonds which permits rings to cross from one equilibrium position to the other degenerate one. It is clear from the structure of the distribution function in proximity of nˆ 1 nˆ 2,3 0 shown in Fig. 8 that the time needed for FIG. 7. Intensity of the peaks of the static distribution function of nˆ 1 nˆ 2,3 at T 440,390,330,280 K from bottom to top. Anenlargement of this figure around x 0 is shown in Fig. 8. the transition from a minimum to the other one will be longer lowering the temperature; moreover, Fig. 8 can be considered as a restatement of the energy map shown in Fig. 3, since the intensity of the maximum in zero is a measure of the transition probability between the two minima. Such fenomenology will be clarified in future communications where we will study the relaxation processes associated with the angular degrees of freedom. The space Fourier transform of g(r) is the static structure factor. In a poliatomic system this quantity is defined as S q 1 N i, j b i b j e iq (r i r j ), 33 where the coefficients b i are the scattering lengths in principle different for each species involved. The S(q) has been determined experimentally for OTP by neutron scattering 32,33, and the following main features have been observed. FIG. 8. Temperature dependence of the static distribution function of nˆ 1 nˆ 2,3 near the saddle point position nˆ 1 nˆ 2,3 0 at T 440,420,390,370,350,330,320,300,280 K from top to bottom.

16 PRE 62 MOLECULAR DYNAMICS SIMULATION OF THE FIG. 9. Static structure factors at T 300 K calculated on the molecular solid line and ring centers of mass dashed line ; each q point is the average over all the independent Miller indices corresponding to it. i In contrast to atomic systems its main peak is split in two subpeaks placed around 14 and 19 nm 1. ii In the q 0 region, by lowering the temperature a reduction of scattering intensity is observed due to the decrease of the isothermal compressibility T T S(q 0). iii By increasing the density, a slight shift of the peak position to higher q values is observed. iv By decreasing temperature, the height of the peak around 19 nm 1 increases while the intensity of the peak at 14 nm 1 remains nearly unaffected, except for a slight reduction mostly connected to the decrease in T. In Fig. 9 we show our results for the structure factors calculated assuming as scattering centers the molecules and the rings centers of mass with b i 1; every point is an average on all the independent Miller indices corresponding to the given q. It is much more interesting to make a comparison among the MD results and the experimental data obtained by neutron Fig. 10 A and to test what is expected for x-ray scattering Fig. 10 B. In evaluating S(q) by computer simulation for a comparison with neutron data, we have to take into account the contribution due to both carbon and hydrogen atoms; H atoms are not considered in our dynamics; nevertheless, it is possible to place them in fixed positions on the line extending from the center of the ring through a carbon atom at a fixed C-H distance computed to be d C-H nm. In this case we would have to consider different scattering lengths for the two species, b H and b C ; nevertheless, they are both positive and about the same magnitude so that the product b i b j in Eq. 33 is an ineffective constant. In Fig. 10 A the calculated S(q) att 300 K is shown and compared with the data of Ref. 32 at T 324 K; in this paper the authors show their results in terms of the coherent scattering cross section (d /d ) measured in m 2 which is proportional to our S(q). In order to compare the two results we renormalized the experimental data in such a way the values of the two curves coincide at large q. The high-q region of the calculated S(q) appears to be in excellent agreement with the experiment but no double peak structure FIG. 10. Top: Comparison among molecular dynamics structure factor solid line calculated taking into account both carbon and hydrogen atoms as scattering centers and experimental structure factor dashed line measured by neutron scattering from 32. Bottom: Molecular dynamics structure factor solid line calculated taking into account only carbon atoms; this should be the correct result to be compared with the experimental structure factor measured by x-ray scattering. is present at low momenta. In particular the MD calculated first peak presents a small bump at about 18 nm 1 ; this is better seen in Fig. 11 where we show the small-q part of S(q) calculated at T 280 K together with the error bars estimated by means of the statistical fluctuation of the data. The noise cannot allow us to determine the correct structure FIG. 11. Enlargement of the low-momenta region of S(q) calculated at T 280 K: the error bars are estimated from the fluctuation of the single configuration S(q) s.

17 622 S. MOSSA, R. Di LEONARDO, G. RUOCCO, AND M. SAMPOLI PRE 62 TABLE V. Temperature dependence of the molecular dynamics self-diffusion coefficient D. T (K) 10 7 D (cm 2 /s) FIG. 12. Temperature dependence of the mean-square displacement r 2 (t) calculated on ring centers of mass at all temperature investigated except T 440,420 K higher temperature on top. Inset: linear scale plot of the mean-square displacement at some selected temperatures open symbols together with the long-time linear behavior dashed lines. of the main peak. It is worth noting, moreover, that at this low temperature the characteristic relaxation time is of order 1 ns, so that, considering a simulation run 10 ns long, we have only about ten really independent system configurations. In order to calculate the simulated S(q) as is expected by x-ray scattering, we consider only the carbon atoms; also in this case no double-peak structure is observed in the data but a clear prepeak appears at a q value less than the q of the first maximum, since the high-q behavior is similar to the neutron case, as shown in Fig. 10 B. C. Self-diffusion coefficient An important quantity to consider in the study of the dynamics of our system at a microscopic level is the meansquared displacement MSD defined as The temperature dependence of the MSD is shown in Fig. 12; each curve follows the usual cage-effect scenario. At small time less than 0.2 ps they present the t 2 behavior corresponding to the ballistic motion; at long time the diffusive linear time dependence of Eq. 35 is found. At intermediate times a small region is present where MSD stays almost constant and whose duration increases with decreasing temperature; on these time scales molecules are trapped in cages built up by their neighbors, and they can only vibrate in these limited regions, the length of the plateau being a measure of the mean lifetime of the cages. The calculated values of the self-diffusion coefficient are shown in Table V and plotted in Fig. 13 open circles as a function of temperature, together with the power-law temperature dependence solid line predicted by the MCT D 1 T T T c. 36 A three parameters fit to these data has been performed obtaining the following values: T c (D) 278 3, (D) , In the same figure we also show the experimental data full squares 34,35 that are well represented by Eq. 36 dashed line with the values and T c K. r 2 t 1 N i R i t R i 0 2, 34 where R i (t) is the position of the center of mass of the ring in the molecule i at time t; from the MSD is possible to determine the self-diffusion coefficient D(T) via the Einstein relation 1 D lim 6t r2 t. t 35 FIG. 13. Temperature dependence of molecular dynamics open circles and experimental full triangles diffusion coefficients together with the power-law fits in the form of Eq. 36 solid and dashed lines, respectively ; we show also the MD data shifted of 20 K open squares as explained in the text.

18 PRE 62 MOLECULAR DYNAMICS SIMULATION OF THE FIG. 14. Time dependence of the non-gaussian parameter 2 (t) for all temperature investigated lower temperatures on top. It is clear from these values and from Fig. 13 that a discrepancy is present among the lower temperatures diffusive behavior of the simulated and real system, respectively; this is most likely due to the fact that we have tuned the value of the LJ potential parameters and in order to reproduce the high-t diffusion properties of the real system. However, it is worth noting that it is possible to reproduce quite well the experimental results on the whole investigated temperature range shifting the molecular dynamics points at temperatures 20 K above their true values. In other words, we have to assume that our actual thermodynamic point is shifted with respect to the real one; from now on, whenever we will compare our molecular dynamics results with the experimental ones, our calculated points will be shifted 20 K above the measured temperature and the competing temperatures will be indicated as T c. On these grounds from the previous study of the self-diffusion properties of our model we obtain T c(d) to be compared with the experimental value T c A different way to determine the parameters entering in the power law reported in Eq. 36 is possible, even if not independent from the previous one; it is based on the study of the non-gaussian parameter 2 (t) defined as 36,37 2 t 9 5 r 4 t r 2 t 2 1, 39 where the mean-square displacement r 2 (t) and r 4 (t) are, respectively, the second and forth momenta of the Van Hove self-correlation function G s r,t 1 N i r R i t R i The parameter 2 (t) quantifies the degree of non- Gaussianity of G s (r,t) in space as a function of time and it is normalized in such a way that, if G s (r, t ) was a Gaussian function in space at a given time t, we would have 2 ( t ) 0. The time dependence of 2 (t) at all temperatures investigated is shown in Fig. 14. We are not interested here in the FIG. 15. Power-law fit of the temperature dependence of the position t max of the maximum of the non-gaussian parameter. specific time dependence of such function but only in the fact that t max, the position of the maximum of 2 (t), has the power dependence on T similar to that of Eq see Fig. 15. A fit to this quantity performed in the same way as before gives us the values t , T c t , compatible with the values determined by the temperature dependence of D, even if the error bars are larger in this case. D. Single particle dynamics Comparisons of the coherent collective and incoherent self- density fluctuations dynamics data measured by different techniques neutron time-of-flight and backscattering spectroscopy, photon correlation spectroscopy, depolarized Raman and Rayleigh-Brillouin light scattering with the main predictions of MCT have been reported in literature with great details In this section we will study the single particle density fluctuation dynamics of our model and we will compare our results with the experimental results mainly contained in Refs. 38,39 and with the MCT predictions. The single particle dynamics of the model is embedded in the incoherent self-intermediate scattering function defined as F s q,t 1 N i, e iq [R i (t) R i (0)] 43 where, again, R i (t) is the position of the center of mass of the ring in the molecule i at time t. At every temperature considered two sets of configurations, produced with two different storing times as described in Sec. IV, have been used to reconstruct the whole curve. We considered the T dependence of F s (q,t) at the two momentum values q 14,19 nm 1 corresponding to the first and second peaks of the static structure factor, averaging on values of q falling in the interval q q with q 0.2 nm 1. Finally, we

19 624 S. MOSSA, R. Di LEONARDO, G. RUOCCO, AND M. SAMPOLI PRE 62 FIG. 16. Temperature dependence of F s (q,t) calculated at q 14 nm 1 for all temperatures investigated except T 410,430 K lower temperatures on top. spanned at T 300 K the whole interesting q space in the interval q 2 30 nm 1 averaging on the values of q falling in the same interval 2 q wide. In Fig. 16 we show F s (q,t) for nearly all temperatures investigated at q q max 14 nm 1 ; all the curves decay to zero, i.e., the length of all the simulations allows the fluctuations to become completely uncorrelated. We are in a good thermodynamical equilibrium at every temperature, at least on the space scales corresponding to the inverse of q max. At temperatures lower then T 330 K the relaxation follows clearly the predicted two step pattern: on microscopic time scales the correlation is quadratic in time, this timescale being the one on which the intramolecular vibrations happen; on intermediate time scales we observe the formation of a plateau, whose height is the nonergodicity parameter f (q) and whose length in time is comparable to the one of the plateau in the MSD r 2 (t). On long-time scales we observe the structural relaxation in the form of a stretched FIG. 18. Temperature dependence of at q 14,19 nm 1 circles and squares, respectively together with the power-law fits with 296 K and T c 2.0 solid and dashed lines, respectively ; also the experimental shear viscosity s data full triangles are reported see 22 and reference therein multiplied by a factor 1.5 ps/poise. Molecular dynamics results have been shifted 20 K with respect to the measured temperatures, as explained in text. exponential. At the highest temperatures no double pattern is visible anymore and only a nearly exponential relaxation can be recognized. A stretched exponential fit see Eq. 29 on the structural time scale ( process gives us the temperature dependence of the three free parameters,, and f (q). The parameter circles is shown in Fig. 17; it appears to be nearly T independent for temperature lower than T 400 K and its mean value 0.8 dashed line has to be compared with the experimental value 0.6. For temperature in the higher region it tends toward the value 1 errors are clearly much more greater ; such behavior is due to the fact that in this temperature region it is no longer possible to sharply separate the long-time relaxation region from the microscopic short-time one. The study of the temperature dependence of the nonergodicity parameter f (q) in the interesting region is not possible due to our limited tem- FIG. 17. Temperature dependence of the stretching parameter circles and of the nonergodicity parameter f q squares ; the horizontal lines indicate the mean values of dashed line and f q dot-dashed line. FIG. 19. F s (q,t) at q 14 nm 1 rescaled to t/ ; all the curves verify the time-temperature superposition principle.

20 PRE 62 MOLECULAR DYNAMICS SIMULATION OF THE FIG. 20. Q-dependence of F s (q,t) at T 300 K for q 2n nm 1 with n 3,...,15 from top to bottom. perature range which do not permit to observe the expected low-temperature (T T g ) harmonic Debye-like behavior, and the onset of the anomalous decrease of f (q) with increasing T for T g T T c. Our data suggest us T c 283 K and the mean value f (q) 0.7 dot-dashed line agrees with the experimental value determined at T 290 K shown later in Fig. 21. It is worth testing the power-law temperature dependence Eq. 30 for the relaxation time ; the calculated relaxation times circles shifted 20 K with respect to the measured temperatures, as explained above, are plotted in Fig. 18 together with the experimental full triangles shear viscosity s data ( s is expected to be proportional to ) of Ref. 22 and the theoretical fitted curve solid line of parameters 296 7, T c S to be compared with the experimental results of Ref. 38, T c K and These values are compatible, FIG. 22. Q-dependence of the inverse relaxation time 1 ;molecular dynamics data open circles have been multiplied by a factor 6.5 in order to overlap the experimental 38 data full circles, as explained in the text. The solid line is the correct small-q behavior 1 (q) 6Dq 2, where 6D nm 2 /ps. within the statistical error, with the values calculated from the diffusion data, so we can conclude that the diffusive behavior and the self-dynamics of our model follow the same critical power law with T c 297 K and 1.8. Also the values of for q 19 nm 1 squares corresponding to the second peak of the static structure function are reported with the theoretical curve. A fit has been performed dashed line only on the prefactor keeping fixed the values of the other two parameters in order to show that these data also are compatible with the same power law. The crucial observation here is that the values of the two parameters T c and are effectively q independent and they can be considered universal for our model, as predicted by the MCT. The relaxation time can be also used to test the timetemperature superposition principle Eq. 28. In Fig. 19 the curves are shown in function of the rescaled time t/ and it is clearly seen that all the curves tend to collapse on the same master curve as predicted by the theory. FIG. 21. Q dependence of the stretching and non-ergodicity parameters: Left: molecular dynamics circles and experimental filled circles, from 38 values of the coefficient as determined by the stretched exponential fits. Right: experimental values filled squares of the nonergodicity parameter 38 together with the molecular dynamics results as determined by a MCT analysis of both squares and triangles regions and the Gaussian fit solid line to -region results with nm 2.

21 626 S. MOSSA, R. Di LEONARDO, G. RUOCCO, AND M. SAMPOLI PRE 62 FIG. 23. Left: Temperature dependence of the product D at q 1.4 nm 1. Right: q dependence at T 300 K of Dq 2. These quantity are expected to be constant. We now quantify the q dependence of the self-dynamics long-time behavior of the system at T 300 K. In Fig. 20 are reported the curves F s (q,t) for values of q 2n nm 1 with n 3,...,15; the choice of the temperature value T 300 K has been due to the need of well-thermalized results in a large range of q. Also in these data are the welldefined two-step behavior and we can calculate the long-time stretched exponential fit parameters; the resulting values are shown in Figs. 21 and 22. In Fig. 21 the values of left side and f (q) right side are shown. open circles appears to be a smooth function of q and it tends, for large values of q, to the experimental full circles evaluated value of 0.6. Such behavior is quite general see, for instance, Ref. 37 and can be easily explained by the following argument 37 : for large values of q, corresponding to length scales of the same order of magnitude of the cages dimension, the dynamics becomes slower and slower approaching the cage dynamics described through the von Schweidler exponent b. At variance, in the opposite limit of small q, we consider a diffusive dynamics on large distances; at such length scales the decay of the self-density fluctuations is of the usual purely exponential form exp ( Dq 2 t) corresponding to 1 see Fig. 27. At this stage, however, we have not a reasonable explanation of the disagreement with the experimental data. In the right side of Fig. 21 the q dependence of the nonergodicity parameter f (q) is also shown as calculated from the short-time limit of the process squares, from the long-time part triangles of the region as we will see below, and from the experimental data full squares 38 ; it seems clear a good agreement between our values and the experimental results. The data appear to be monotonic decreasing as increasing q and this dependence is expected to be approximately Gaussian; in Fig. 21 a Gaussian fit solid line in the form exp( q 2 /2 2 ) with 19 nm 1 to the molecular dynamics region data is also shown. It is clear that the q range considered here is too limited to really decide on the validity of this functional form a linear approximation would work well too, a good estimate of the error bars lacking in this case. In Fig. 22 the q dependence of 1 is shown circles together with the experimental data 38 full circles. Molecular dynamics points have been rescaled by a factor MD (T 280 K)/ MD (T 300 K) 6.5 to take into account the fact, as discussed above, that our system temperature is 20 K higher than the real one. The correct square-law behavior at low-q 1 (q) 6Dq 2 see Eq. 47 is also shown as a solid line; here D is the self-diffusion coefficient and 6D nm 2 /ps. Finally, in Fig. 23, two products of some calculated quantity, expected to be constant, are shown; on the left-hand side the statement D 1 (T) (T) is proven for q q max good at highest temperature. On the right-hand side we show the product q 2 D evaluated at T 300 K, which is nearly constant up to q 18 as expected, this value being approximately the crossover point among the correct quadratic behavior and the asymptotically linear regime as we found above. Let us now probe the MCT conclusions about the region which is predicted to follow the power laws of Eq. 26 : FIG. 24. Power laws in the region for q 8,10,12,14 nm 1 ; critical decay of exponent a dot-dashed lines and the von Scweidler law of exponent b dashed lines have been reported with simulation points.

22 PRE 62 MOLECULAR DYNAMICS SIMULATION OF THE FIG. 25. Q dependence of the power-laws exponents. Left: the experimental values filled circles for b(q) are plotted together with the molecular dynamics values for b(q) open circles and a(q) open squares ; the mean values for our results are shown by the dashed and dot-dashed lines, respectively. Right: our values for the q dependence of the exponent determined by the relation 1/2a 1/2b. we show here the results at fixed temperature T 300 K for values of q 6 n nm 1 with n 0,...,17. We fit all the curves by two power functions on the time ranges 1 and 2, F s q,t f q c 1 t a, t 1 f q c 2 t b, t 2 46 where :2 (ps) and 2 3:20 (ps); some selected fits are shown in Fig. 24 and they seem to work quite well. Some observations are needed on the following analysis: a great uncertainty stems from the choice of the fit range i.e., from the choice of 1 and 2 ) due to the consideration of a crossover region between two processes not sharply separated in time; moreover, an analogous problem cannot be excluded between the microscopic region and the critical decay region characterized by the exponent a. Such difficulty implies a great uncertainty on the determination of f (q) which is supposed to be considered as the long-time limit of the process and the short-time behavior of the relaxation. At this point it is clear that, lacking a careful error analysis on the data points, such fits can only state that a parametrization of the data in the form of Eq. 46 is possible which is consistent with the theory predictions 39, the current values of the determined parameters being considered only from a qualitative point of view. Nevertheless the values of such fitting parameters, shown in Figs. 25 and 26, appear to be in good agreement with the values obtained from the experimental data. In Fig. 25 are shown left the values of the power exponents a squares and b circles and of the exponent triangles calculated by means of Eq. 31 right side ; the mean values are 0.3, 0.5, 2.6, respectively, to be compared with the experimental determined values a 0.31 dot-dashed line, b 0.52 dashed line, 2.55 solid line 38. It is clear from these results that one of the main predictions of MCT, namely the q independence of a and b is verified in the limit of the error fluctuations. Moreover, it is important to note that the parameter, given by Eq. 31, remains constant, as expected, being its mean value 2.6 is clearly compatible with the value 2.55 determined from experimental data 38 ; nevertheless, this value overestimates the value at q 14 nm 1 previously determined by the fit to the region. Furthermore, b(q) is always less than the determined value of the large-q value of the stretching parameter 0.6 see Fig. 21 verifying another MCT prediction, namely 0 b. From Eqs. 25, 26, and 46 the two parameters c 1 (q) and c 2 (q) result to be proportional to h(q), the proportionality constants being dependent on,, a, and b. From T c 280 K we have 0.3 while we choose as a good estimate of the intersection point of the two power laws of Fig. 24, obtaining 2 ps; if we put a 0.31 and b 0.52 we finally obtain the factors 2.7 and 2.3 for c 1 and c 2, respectively. Unfortunately these value are not able to correctly rescale our data on the experimental results, the correct values being 0.7 and 4 as shown in Fig. 26; this result could be expected considering the great uncertainties on the parameter values used for the estimate. Nevertheless, simulated data agree quite well with the experimental points at low-q presenting a strong bending toward a constant value in the region q 16 nm 1. To complete the picture of the self-motion in our model, we test the validity of the Gaussian approximation to F s (q,t) in the limit of small momentum q. The first-order term of the expansion of F s (q,t) in powers of q 2 gives F s q,t exp q2 6 r2 t In Fig. 27 some curves at T 330 K and different values of q are shown together with the corresponding approximations; such approximation seems to work quite good and it becomes worse on increasing q as expected. At the end, we show in Fig. 28 all the time scales related to centers of mass motion investigated up to now as a function of temperature. Full circles and squares indicate, respectively, the experimental structural relaxation time V F expt (T),

23 628 S. MOSSA, R. Di LEONARDO, G. RUOCCO, AND M. SAMPOLI PRE 62 FIG. 26. Q dependence of the experimental full circles coefficient h of Eq. 25 and of our fitting parameters c 1 triangles, c 2 squares ; note that our results have been rescaled by a factor 0.7 and 5, respectively in order to superimpose to experimental data. following the Vogel-Fulcher law, and the experimental inverse self-diffusion coefficient D 1 expt (T) multiplied by a factor cm 2 in order to superimpose to V F expt (T). The open symbols are used to represent the molecular dynamics s results: MD diamonds is the relaxation time of the oneparticle dynamics at q 14 nm 1 multiplied by a scale factor 1.5 and D 1 MD (T) triangles up is the inverse of the diffusion coefficient rescaled by the same factor cm 2 we used for the experimental points. All the molecular dynamics points have been shifted of 20 K with respect to the measured temperatures; it is quite clear that both experimental and molecular dynamics data points collapse on a well-defined master curve. Our model is, at least, a good model for centers of mass dynamics of OTP. VII. CONCLUSIONS In this paper we have introduced an interaction potential model capable of describing the intramolecular dynamics of FIG. 27. Gaussian approximation to F s (q,t) att 330 K for q 2,3,4,6,8,1,1.2 nm 1 from top to bottom. FIG. 28. Master plot of the temperature dependence of all the centers of mass time scales discussed: we have used full symbols for experimental results and open symbols for molecular dynamics data. Molecular dynamics points collapse exactly on the master curve identified by the experimental data if they are multiplied by a scale factor taking into account the momentum dependencies of the s relaxation time MD and the correct dimensionality of the diffusion coefficients and the corresponding temperatures are shifted 20 K above the measured ones, as discussed in the text. In particular, s MD open diamonds is multiplied by a factor 1.5, and the selfdiffusion coefficients D expt full squares and D 1 MD open triangles 1 are rescaled by a factor cm 2. the fragile glass-former OTP; such a model appears to be much more efficient with respect to the ones introduced so far in the sense that it represents a much better compromise between the resulting computing needs and its capability to mimic all the complexities of the dynamical behavior of the real system. It takes into account not only the translational and rotational dynamics of the molecules as a whole but also the stretching along the molecular bonds, and the tilt of the bonds, the rotations of the side rings with respect to the parent ring. It is tuned in such a way to reproduce the isolated molecule vibrational spectrum. In this way, most likely, we have introduced the degrees of freedom whose interplay causes the complex dynamical behavior of the real system. We have, then, presented the results of molecular dynamics computer simulations of such a model; we mainly studied the static structure of a bulk sample, the self-diffusion properties, and the self-part of the density-density correlation functions. The static structure factor simulated in such a way is compared with the experimental measures and shows a good agreement with the neutron scattering data, except for the very low momenta behavior due, probably, to the finite size of our system. Moreover, we have no evidence of the splitting of the main peak in two subpeaks placed at q 14,19 nm 1, only the first one being clearly visible. This luck may be due mainly to the temperature range investigated the intensity of the second subpeak increases with lowering temperature. The self-diffusion properties of the system have been investigated through the mean-squared displacement and the self-diffusion coefficient temperature behavior; comparisons

24 PRE 62 MOLECULAR DYNAMICS SIMULATION OF THE with experimental self-diffusion data give a very good agreement, showing the evidence of compatible critical dynamics behavior in approaching the instability temperature T c of MCT which is here found to be T c to be compared with the experimental value T c determined by a MCT analysis of the dynamics of the density fluctuations, a discrepancy which is most to likely ascribed to the intermolecular LJ potential parameters (, ) that have been tuned in the temperature region close to T 300 K. Moreover, we considered the critical temperature dependence of the socalled non-gaussianity parameter 2 (t) obtaining compatible values for the power-law parameters. The self-dynamics of the density fluctuations has been studied in great detail on the whole accessible time window, spanning the range from a time scale of the order of few femtoseconds to times of order of some nanoseconds; moreover, its dependence on temperature and momenta has been investigated. All the correlation curves calculated show the typical two step behavior predicted by MCT, the first one on short time being associated with the so-called microscopic processes, i.e., the vibrational motion of molecules in the cage built up by their neighbors; the second one being associated with the process which controls the structural rearrangements of molecules on a long-time scale. The critical dynamics on an time scale approaching the correspondent T c is in good agreement with experimental findings; indeed the estimate values for our model of the exponent must be compared with the experimental value 2.5. The q dependence at T 300 K in the momentum range q 6 30 nm 1 has been analyzed in terms of a stretched exponential fit; the values of the determined parameters are in good agreement with the ones calculated by fitting the experimental points and with the MCT expected behavior. To summarize, in the present work we have shown that our model for OTP fluid is mimicking rather well the center of mass dynamical features of the real system, giving results in most cases fully compatible with the experimental findings. It is clear that its ability to help us in the understanding of the most exotic dynamic features of the real systems and, in particular, of the relevance of the internal degrees of freedom on the translational dynamics, has not been fully displayed in this paper; problems such as the collective dynamics density fluctuation behavior, the rotational dynamics, the origin of the unusual fast relaxational dynamics, and many others, will be addressed in future works 19. ACNOWLEDGMENTS The authors wish to thank W. Götze for a critical reading of the manuscript, L. Bernardini and G. Giuliani of the Parco Tecnologico d Abruzzo for technical support, G. Monaco and F. Sciortino for some useful discussions. 1 C. A. Angell, Science 267, W. Götze, in Liquids, Freezing and the Glass Transition, edited by J. P. Hansen, D. Levesque, and J. Zinn-Justin North- Holland, Amsterdam, 1991 ; W.Götze and L. Sjörgen, Rep. Prog. Phys. 55, ; W.Götze, J. Phys.: Condens. Matter 11, A R. Schilling, in Disorder Effects on Relaxational Processes, edited by A. Richert and A. Blumen Springer-Verlag, Berlin, 1994 ; W. Kob, in Experimental and Theoretical Approaches to Supercooled Liquids: Advances and Novel Applications, edited by J. Fourkas et al. ACS Books, Washington, K. L. Ngai and K. Y. Tsang, Phys. Rev. E 60, M. Mézard and G. Parisi, Phys. Rev. Lett. 82, F. Sciortino, W. Kob, and P. Tartaglia, J. Phys.: Condens. Matter 12, G. Monaco, D. Fioretto, C. Masciovecchio, G. 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25 630 S. MOSSA, R. Di LEONARDO, G. RUOCCO, AND M. SAMPOLI PRE W. Kob and H. C. Andersen, Phys. Rev. E 51, S. S. Chang and A. B. Bestful, J. Chem. Phys. 56, J. P. Hansen and J. R. Mc Donald, Theory of Simple Liquids Academic Press Limited, London, E. Bartsch, H. Bertagnolli, P. Chieux, A. David, and H. Sillescu, Chem. Phys. 169, A. Tölle, H. Schober, J. Wuttke, and F. Fujara, Phys. Rev. E 56, D. W. McCall, D. C. Douglass, and D. R. Falcone, J. Chem. Phys. 50, F. Fujara, B. Geil, H. Sillescu, and G. Fleischer, Z. Phys. B: Condens. Matter 88, T. Odagaki and Y. Hiwatari, Phys. Rev. A 43, F. Sciortino, P. Gallo, P. Tartaglia, and S. H. Chen, Phys. Rev. E 54, W. Petry, E. Bartsch, F. Fujara, M. Kiebel, H. Sillescu, and B. Farago, Z. Phys. B: Condens. Matter 83, M. Kiebel, E. Bartsch, O. Debus, F. Fujara, W. Petry, and H. Sillescu, Phys. Rev. B 45, E. Bartsch, F. Fujara, J. F. Legrand, W. Petry, H. Sillescu, and J. Wuttke, Phys. Rev. E 52, A. Tölle, J. Wuttke, H. Schober, O. G. Randl, and F. Fujara, Eur. Phys. J. B 5, Y. Hwang and G. Q. Shen, J. Phys.: Condens. Matter 11, W. Steffen, A. Patkowski, H. Gläser, G. Meier, and E. W. Fischer, Phys. Rev. E 49, R. Schilling and T. Scheidsteger, Phys. Rev. E 56,

26 PHYSICAL REVIEW E, VOLUME 64, Molecular dynamics simulation of the fragile glass former orthoterphenyl: A flexible molecule model. II. Collective dynamics S. Mossa, 1,2 G. Ruocco, 2 and M. Sampoli 3 1 Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts Dipartimento di Fisica and Istituto Nazionale di Fisica per la Materia, Università di Roma La Sapienza, Piazzale Aldo Moro 2, Roma, I-00185, Italy 3 Dipartimento di Energetica and INFM, Università di Firenze, Via Santa Marta 3, Firenze I-50139, Italy Received 27 December 2000; revised manuscript received 27 March 2001; published 24 July 2001 We present a molecular dynamics study of the collective dynamics of a model for the fragile glass former orthoterphenyl. In this model, introduced by Mossa, Di Leonardo, Ruocco, and Sampoli Phys. Rev. E 62, , the intramolecular interaction among the three rigid phenyl rings is described by a set of force constants whose value has been fixed in order to obtain a realistic isolated molecule spectrum. The interaction between different molecules is described by a Lennard Jones site-site potential. We study the behavior of the coherent scattering functions F t (q,t), considering the density fluctuations of both molecular and phenyl-ring centers of mass; moreover we directly simulate the neutron scattering spectra taking into account both the contributions due to carbon and hydrogens atoms. We compare our results with the main predictions of the mode-coupling theory and with the available coherent neutron scattering experimental data. DOI: /PhysRevE PACS number s : Pf, Pd, Em, p I. INTRODUCTION The understanding of the supercooled liquid and glassy phases in molecular systems is, nowadays, one of the main tasks of the physics of disordered materials see 1,2 and references therein for a general review. Two main theories provide us a description of the glass transition, respectively, from a thermodynamical and dynamical point of view. The first one see 3 and references therein is based on first-principles computation of the equilibrium thermodynamics of glasses and considers the glass transition as a true thermodynamic transition. In this context, the onset of the glassy state is associated with an entropy crisis, i.e., the vanishing of the configurational entropy of the thermodynamically relevant states. The second approach is the mode-coupling theory MCT 4,5, which studies the long-time structural dynamics and its relation with the glass transition; in this context this transition has to be considered not as a regular thermodynamical phase transition involving singularities of some observables, but as a kinetically induced transition from an ergodic to a nonergodic behavior. The structural dynamics becomes so slow that the system of interest appeared frozen on the experimental time scales. On the experimental side, the collective dynamics of a huge variety of molecular systems has been investigated by means of several experimental techniques; colloids 6,7 and ortho-terphenyl OTP 8 11 are the most widely studied fragile 1 supercooled liquids. They are only examples of an enormous experimental work the interested reader is referred to Ref. 2 for an accurate comprehensive review. Moreover, in the last ten years the analysis of experimental results has been flanked by extensive use of numerical techniques, mainly molecular dynamics MD and Monte Carlo simulations. The almost exponential growth of computational capabilities allows us to reach simulation times of the order of microseconds for simple monoatomics systems and of several tens of nanoseconds in the case of complex molecular systems; such performances definitely permit comparisons of numerical results with the structural very-longtime properties of the real systems. We remember, among many others, the results concerning the structural dynamics of two molecular liquids like simple point charge extended SPC/E water 12 and OTP 13,14. We emphasize here the fact that the models involved in these studies are molecular and rigid in the sense that they have a structure and take into account the orientational properties of the molecules, but they disregard the role played by internal degrees of freedom on the overall dynamical behavior of the system. In a recent paper 15 we have introduced a new model for the intramolecular dynamics of orthoterphenyl (T m 329 K, T c 290 K, T g 243 K), one of the most deeply studied substances in the liquid, supercooled, and glassy state. The introduction of such a flexible model allows us to understand the role of internal degrees of freedom in the short-time fast dynamics 16 taking place on the time scale of a few picoseconds and to study their possible coupling with the long-time center of mass dynamics 17. Moreover, it would allow us to emphasize once more the universality of the MCT approach for supercooled liquids and, in particular, of its molecular version in the case of complex molecular liquids. In Ref. 15 we introduced in detail the intermolecular model and we showed some results based on MD calculations; the self- one-particle dynamical properties of a system composed of 108 molecules have been studied in details and we have found good agreement with the main predictions of the MCT and with the experimental results related to the self-intermediate-scattering function 18,19 and the selfdiffusion 20,21. In the present paper we complete the picture considering the collective highly cooperative structural dynamics controlling the rearrangement of big portions of the system. We describe the dynamics of the molecules at the level of the molecular center of mass and at the level of the X/2001/64 2 / /$ The American Physical Society

27 S. MOSSA, G. RUOCCO, AND M. SAMPOLI PHYSICAL REVIEW E phenyl-ring center of mass. Moreover, we calculate the neutron coherent scattering function taking into account both the contributions due to carbon and hydrogen atoms in order to make a direct comparison with the experimental results. The paper is organized as follows. In Sec. II we summarize the main predictions of the MCT and we give a schematic introduction to the molecular mode-coupling theory MMCT. In Sec. III we briefly describe the model and we recall some computational details. In Sec. IV we study the temperature and momentum dependence of the collective dynamics of the molecular and of the phenyl-ring centers of mass. In Sec. V we make a deep comparison between the experimental results of neutron scattering and the simulated neutron spectra calculated taking into account the interactions with both carbon and hydrogen atoms. Finally, Sec. VI contains a discussion of the results obtained and some conclusions. II. MODE-COUPLING AND MOLECULAR MODE- COUPLING THEORY As we have already stressed in Sec. I, two main theories, one intrinsically thermodynamical 3 and the other purely dynamical 4,5, have been developed up to now to describe the phenomenology observed in the glass transition. In this paper we will make a comparison of our numerical results with the main predictions of MCT about the center of mass structural relaxation dynamics. The reason for this is twofold: first of all this is, indeed, the theory taking into account the states actually accessible by the system on the time scale of a typical simulation. Moreover, even if some experimental results, like the presence of the so-called knee characterizing the low-frequency behavior of the light scattering susceptibility 22,23 or the presence of a cusp in the nonergodicity parameter 24, seem to contradict some of its predictions, such a theory has been verified to hold for different experimental data on a very wide time region. Finally, in the last two years successful efforts have been made in order to generalize the theory to liquids of rigid molecules of arbitrary shape 25 see also Ref. 26 for the particular case of a liquid of linear rigid molecules taking into account both translational and rotational dynamics. In Ref. 15 we have summarized the main predictions of the so-called ideal MCT so that here we recall only the fundamental equations of the theory, concerning the collective intermediate scattering function that we will use in the following sections. MCT interprets the glass formation as a dynamical transition from an ergodic to nonergodic behavior at a crossover temperature T c ; the theory is written as a self-consistent dynamical treatment 4 of the intermediate scattering function, i.e., the time correlation of the density fluctuations of momentum q. This theoretical scheme can be considered as the mathematical description of the physical picture of the cage effect. Following the dynamics of a tagged particle it is possible to recognize two main dynamical regions. On a small time scale of the order of some picoseconds the region, the dynamics of the particle is confined into a limited region the cage built up by the nearest neighbors. In this regime it is possible to write the intermediate scattering function as F q,t f q h q T T c T c G t/ 1 where f (q) is the nonergodicity parameter also referred to as the Debye-Waller factor, h(q) is an amplitude independent of temperature and time and the in G corresponds to time larger or smaller with respect to, a parameter that fixes the time scale of the process. At this stage, the time dependence of the correlation functions is all embedded in the q-independent function G. G (t) is asymptotically expressed by two power laws, respectively, the critical decay G t/ t/ a, 0 t 2 and the von Schweidler law, G t/ t/ b, t ; 3 characterized by the temperature- and momentumindependent exponents a and b; here is the structural relaxation time. At time scales of order of the cages start to break down and the particle starts to diffuse approaching pure brownian motion. This long-time part of the dynamics is the so-called region and is well described by a stretched exponential function F q,t f q exp t, 4 which verifies the time-temperature superposition principle TTSP. The time scale depends on temperature trough a power law of the form T T c. The momentum dependence of the dynamical parameters in the collective dynamics case is not trivial as in the singleparticle case. When looking at the structural collective dynamics, studying different values of the momentum q means studying the highly cooperative time evolution of cages of average dimension 2 /q; it is clear that such time evolution is strongly coupled to the static topological structure of the system. More precisely, MCT predicts that the parameters f (q), h 1 (q),, and (q) oscillate in phase with the static structure factor S(q). The result concerning the momentum dependence of the collective relaxation time is quite general and is well known as de Gennes narrowing 27. A general relation 28 holds among the one-particle s and the collective c relaxation times, namely, c (q) S(q) s (q); if the diffusion limit is appropriate for F s (q,t), i.e., for values of q close to the first peak of the static structure factor, we obtain c q,t 1 S q D T q

28 MOLECULAR DYNAMICS....II... PHYSICAL REVIEW E where D(T) is the diffusion coefficient at temperature T and S(q) is supposed to be nearly temperature independent. In summary, the momentum dependence of the collective dynamics is nontrivially driven by the static structure of the system. In particular, for molecular systems, the small length scale structure is determined by orientational properties of the single molecules; a pure molecular translational dynamics will, obviously, loose all the dynamical features controlled by the high momentum part of the static structure factor. An important step toward a correct explanation of the dynamics of molecular systems is to write down a MMCT 25,26 taking into account both translational and rotational degrees of freedom. If we consider a system of N identical rigid molecules of arbitrary shape described by their center of mass positions r j(t) and by the Euler angles j (t) j (t), j (t), j (t) we can write the time-dependent microscopic one-particle density as N r,,t r r n t, n t. n 1 Expanding with respect to the complete set of functions l given by the plane waves and the Wigner matrices D mn ( ), we have the tensorial density modes lmn q,t i l 2l 1 1/2 n 1 N e iq r n (t) D l mn * n t. Then, the generalization of the intermediate scattering function to the molecular case is the tensorial quantity 7 8 S lmn;l m n q,t 1 N lmn * q,t l * q. m n 9 These correlators are directly related to experimental quantities 26 ; for l l 0, they describe the dynamics of translational degrees of freedom that can be measured by neutron scattering when looking at the center of mass low-frequency part of the spectrum; if the molecules possess a permanent dipolar moment, the correlators with l l 1 give information related to dielectric measurements and l l 2 is finally related to the orientational contribution to light scattering. At this stage, provided the static angular correlators S lmn;l m n (q,0) and the number density, it is possible to give a closed set of equations for the matrix S that completely solve the problem of a liquid of rigid molecules. The problem is that MMCT seems to be not enough for a high structured molecular system like OTP; we will see that it is not possible to explain some features of the momentum dependence of the structural dynamics without taking into account the internal degrees of freedom i.e., rotations of the side rings with respect to the central one that turn out to be strongly coupled to the long-time behavior of the density fluctuations. III. MODEL AND COMPUTATIONAL DETAILS In this section we give a brief description of the model and we refer the reader to Ref. 15 for details. In our model, the OTP molecule is constituted by three rigid hexagonal rings of side L a nm representing the phenyl rings; two adjacent vertices of the central ring are bonded to one vertex of the two lateral rings by bonds of equilibrium length L b 0.15 nm. In the isolated molecule equilibrium position, the two lateral rings lie in planes that form an angle of about 54 with respect to the central ring s plane. In the model the two lateral rings are free to rotate along the molecular bonds, to stretch along the bonds, and to tilt out of the plane identified by the central ring. The intramolecular potential is then written as a sum of harmonic and anharmonic terms, each one controlling one of these features. Every term is multiplied by a coupling constant whose actual value is determined in order to have a realistic isolated molecule vibrational spectrum. The intermolecular interaction is of the site-site Lennard-Jones type; each of these sites corresponds to a vertex of a hexagon and is occupied by a fictious atom of mass M CH 13 amu representing a carbon-hydrogen pair. The actual values of the parameters LJ and LJ have been fixed in order to have the first maximum of the static structure factor S(q) in the experimentally determined position 29 and to obtain the correct diffusional properties 20,21 ; the cutoff has been fixed to the value r c 1.6 nm 1. It is worth noting here that obviously the parametrization of the potential cannot be perfect; in our case it is possible to reproduce quite well the experimental results on the whole investigated temperature range shifting the MD thermodynamical points at temperatures of 20 K above their true values. The MD simulated system is composed of 108 molecules 324 phenyl rings for a total of 1944 Lennard- Jones interaction sites ; at each time step the intramolecular and intermolecular interaction forces are calculated and the equation of motion for the rings are solved for the translational and rotational parts separately. Wide temperature and momentum ranges have been investigated for values of temperature 380 T 440 K and momentum 2 q 30 nm 1 the runs details are shown in Table III of Ref. 15 and the total simulation time is of almost a hundred nanoseconds. IV. MOLECULES AND PHENYL RINGS The collective density fluctuations dynamics is embedded in the coherent intermediate scattering function in general defined as F t q,t 1 N NS q i 1 N j 1 exp iq x i t x j 0 10 where N is the number of molecules involved and S(q) the static structure factor. In the present case the position variables x k (t) can be identified with different quantities; here

29 S. MOSSA, G. RUOCCO, AND M. SAMPOLI PHYSICAL REVIEW E we are interested in the dynamics of the molecular and phenyl-ring centers of mass so that we will consider the following scattering functions F (M) 1 t q,t N M S (M) q exp iq M t M 0, 11 F (R) 1 t q,t N R S (R) q ij exp iq R i t R j FIG. 1. Intermediate coherent scattering functions F (R) t (q,t) calculated on phenyl rings at q 14 nm 1 for the temperatures T 280, 300, 320, 350, 370, 390, 410, and 430 K from top to bottom ; in the inset we show the same curves rescaled as a function of t/ (R). Here M (t) is the position of the center of mass of the molecule at time t ( 1,...,N M ), R i (t) is the position of the center of mass of the phenyl ring i (i 1,...,3) pertaining to the molecule ; the functions are renormalized to the corresponding static structure factors. From now on, the superscripts (R) and (M) will refer to rings and molecular quantities, respectively. As in the case of the incoherent scattering function, at every temperature investigated we have reconstructed the whole curve, even on very-short-time scales, by means of two sets of system configurations campionated with different frequencies see Table III of Ref. 15. At every investigated temperature, we considered the momentum values q 1 14 nm 1 and q 2 19 nm 1 corresponding to the first and second peak of the experimental static structure factor, averaging on the values of q falling in the interval q q with q 0.2 nm 1. Moreover, the momentum dependence of the principal dynamical parameters has been investigated at T 280, 300, and 330 K for values of momenta ranging from 2to30nm 1. We made a long-time analysis in terms of the usual stretched exponential form of Eq. 4 determining the temperature and momentum dependence of the fitting parameters,, and f q and verifying the TTSP. In Fig. 1 we show F (R) t (q,t) calculated at q 1 for the temperatures T 280, 300, 320, 350, 370, 390, 410, and 430 K from top to bottom ; as in the case of the self-dynamics, every curve decays to zero in the considered time window and the two-step decaying pattern is clearly visible. The long-time part of these F (R) t have been fitted to Eq. 4 and the parameters,, and f q are determined by a leastsquares fitting routine. In the inset we plot the same curves as a function of the rescaled time t t/ (R) ; all the curves collapse pretty well on a single master curve as predicted by the TTSP. The temperature dependence of the nonergodicity (R) parameter f q top panel and of the stretching parameter (R) bottom panel are shown in Fig. 2; they are temperature independent in the limit of our error bars as predicted by MCT. The mean values f (R) (R) q 0.78 and 0.83 dashed line have to be compared with the values determined in the case of the self-dynamics f q 0.7 and The two values of are equal in the limit of the (R) error bars; at variance, the value for f q in the collective case is greater than the value found in the one-particle case. In Fig. 3 we plot the structural relaxation times at q 1 and q 2 for molecules triangles and diamonds and phenyl rings (M) (R) circles and squares in order to test if both and follow the same power law, which is supposed to be momentum independent. Both sets of data have found to be consistent with a power law of the form of Eq. 5 with parameters FIG. 2. Temperature dependence of the stretched exponential parameters calculated from F t (R) (q,t) together with the corresponding mean values dashed lines. Top: nonergodicity parameter f q (R) (T). Bottom: stretching parameter (R) (T)

30 MOLECULAR DYNAMICS....II... PHYSICAL REVIEW E FIG. 3. Temperature dependence of the structural relaxation times at q 1 14 nm 1 and q 2 19 nm 1 calculated both on ring circles and squares, respectively and molecule triangles and diamonds, respectively centers of mass. In the inset the date are shown in a double-log scale as a function of the rescaled temperature (T T c ); the points have been shifted in order to maximize the mutual overlap and to stress the power law behavior. The power law of exponent 2.3 is also shown dashed line ; the value for T c is 268 K. FIG. 4. Intermediate coherent scattering functions F t (R) (q,t) calculated at fixed temperature T 300 K for different values of momentum q; the corresponding long-time stretched exponential fits are also shown solid lines. T c 268, 2.3; these values have to be compared with the results concerning the one-particle dynamics T c K and The inset show the data plotted as a function of T T T c, in order to stress the power law dependence, and rescaled by an arbitrary factor in order to maximize the overlap. We now consider the momentum dependence of the collective dynamics at few selected temperatures T 280, 300, and 330 K, spanning the momentum region in the interval 2 30 nm 1. We test the long-time dynamics in terms of the stretched exponential function and we verify the MCT predictions on the von Schweidler time region, characterized by the power exponent b. In Fig. 4 we show the stretched exponential fits solid lines to F t (R) at some selected values of q; they work pretty well at least for time values greater than 5 ps. From this figure, it is qualitatively clear that the relaxation time depends non trivially on the momentum values. As we reminded in Sec. II, it is a general property that in the collective case, at fixed temperature, the relaxation times oscillate in phase with the static structure factor, i.e., they are strongly coupled to the static structure of the system. In Fig. 5 we plot the momentum dependence of the collective relaxation times for molecules (A) and rings (B) at T 280, 300, and 330 K circles, squares, and triangles, respectively multiplied by the corresponding diffusion coefficients from Ref. 15 ; Eq. 6, which is valid for values of q close to the first maximum of the static structure factor, predicts that these products are temperature independent. In our case the data collapse is not perfect, showing a systematic shift of the data with the temperature. This result is not unexpected due to the fact that Eq. 6 is verified for monoatomic liquids 28, while for molecular systems some discrepancies have been observed 14. In the two lower panels of Fig. 5 we plot the corresponding static structure factors divided by q 2 c and d for FIG. 5. a Structural relaxation times (M) (q) for molecules at temperatures T 280, 300, and 330 K circles, squares, and triangles, respectively multiplied by the correspondent diffusion coefficients D(T); this product is supposed to be temperature independent. b As panel a but for calculated on phenyl-ring (R) centers of masses. c Structure factor calculated on molecular centers of masses divided by q 2. d Structure factor calculated on ring centers of masses divided by q

31 S. MOSSA, G. RUOCCO, AND M. SAMPOLI PHYSICAL REVIEW E molecules and phenyl rings, respectively. The correlation among the relaxation times and the respective structure factors of molecules and phenyl rings is evident; it is also evident that, at variance with the temperature dependence, the momentum dependence of the relaxation times is completely different in the two cases. In the molecular case Fig. 5 a only a maximum at q 9 nm 1 corresponding to intermolecular correlations is present a small shoulder at q 14 nm 1, related to correlations between rings pertaining to different molecules, can be also identified. In the case of the phenyl rings Fig. 5 b the momentum dependence is much more structured and three main features are present: i a maximum at q 9 nm 1 related to correlations between molecular centers of mass; ii a shoulder at q 14 nm 1 well developed in a maximum at the lowest temperature, T 280 K) related to correlations between rings belonging to different molecules; iii a maximum at q 22 nm 1 related to correlations between rings pertaining to the same molecules; this is the most important features related mainly to the orientation of the lateral rings with respect to the central ring. It is clear from this result that in complex molecular glass formers there are intramolecular rotational and vibrational degrees of freedom that couple to the translational long-time modes as already stressed in Ref. 10 ; a theory not taking into account these degrees of freedom cannot explain the whole momentum dependence of the centers of mass dynamics. In Fig. 6 a we plot the momentum dependence of the stretching parameter (R) at the three selected temperatures T 280, 300, and 330 K circles, squares, and triangles, respectively while in panel b we show the nonergodicity parameter f (R) q. Also in this case a correlation between the oscillations of these parameters and those of S (R) (q)/q 2 Fig. 6 e is somehow clear. We have seen in Sec. II that the long-time limit of the region can be described by the von Schweidler power law equations 1, 2, and 3. The exponent b is expected to be momentum independent and to assume the same value of the self-dynamics case, namely, b ; on the contrary, the amplitude h(q) is expected to be momentum dependent and to oscillate out of phase with the static structure factor. We then calculated a power law fit in the form F (R) t (q,t) f (R) q c (R) 2 (q)t b(r) for all values of momentum considered, in a time window depending on the particular q value but always included in the interval 2 30 ps; moreover, we considered the three parameters free as in the case of the selfdynamics. All the observations done in the previous work concerning the great uncertainties on the estimated values of the fitting parameters hold in the present case. In Fig. 6 c we plot the power exponent b (R) (q) that is supposed to be momentum independent; some smooth oscillations are nevertheless present but this can be due to the interplay during the fitting procedure with the other oscillating parameters. In Fig. 6 d we finally plot the quantity 1/c (R) 2 (q); in this case some oscillations can be recognized but the noise prevents us from reaching any conclusion. It is worth noting that the values of the nonergodicity parameter f (R) q calculated by FIG. 6. Momentum dependence of the stretching parameters (R) a and f q (R) b as calculated by the stretching exponential fit. Also shown are the momentum dependencies of the fitting parameters for the region b (R) (q) c and 1/c 2 (R) (q) d. Oscillations in phase with the structure factor e are somehow evident. the von Schweidler law is consistent with the one reported in panel b. The value of the plateau, indeed, must be the same if determined as the small-time limit of the region or the long-time limit of the process. V. NEUTRON SCATTERING Neutron scattering is one of the most powerful tools used in the study of supercooled liquids and glasses in the q region covered by the MD simulations. Experimentally the scattering function F t (q,t) of Eq. 10 can be determined 9 by neutron scattering experiments either directly on neutron spin echo instruments, or by Fourier transforming the dynamical structure factor S(q, ), S q, S q 2 1 dte i t F t q,t 13 calculated by means of triple axis backscattering or time-of

32 MOLECULAR DYNAMICS....II... PHYSICAL REVIEW E flight spectroscopy. The experimental neutron scattering cross section (d /d de) is generally composed of a coherent and an incoherent part d /d de b 2 S coh q, b 2 b 2 S incoh q, 14 where b is the scattering length and the symbol denotes an average over the distribution of nuclear spins and isotopes. The isotopic composition of the sample allows us to study selectively the collective motion via coherent scattering from deuterated samples the scattering lengths of D and C atoms are basically coincident and the one-particle motion via incoherent scattering from protonated samples. The interaction of neutrons with a bulk sample of OTP can be simulated numerically taking into account the interactions of neutrons with both carbon C and deuterium D atoms. H atoms are not considered in our dynamics but it is a reasonable approximation to put them in fixed positions on the line extending from the center of the ring through a carbon atom at the fixed C-H distance d C - H nm; so that, knowing the coordinates of the rings, it is trivial to reconstruct their own positions. We then define a neutron N coherent scattering function F t (N) (q,t) as F (N) 1 t q,t N A S (N) q b b ij exp iq r i t r j 0 15 FIG. 7. Comparison among the temperature dependencies of the stretched exponential parameters as calculated by MD simulated neutron spectra open symbols and experimental neutron scattering full symbols at q 14 nm 1. a Relaxation time (N) ; in the inset the MD data have been shifted by 20 K as explained in the text. b Nonergodicity parameter f q (N). C stretching parameter (N). where N A N C N H 3456 in this case is the total number of atoms, r i (t) is the position of the atom pertaining to the ring i in the molecule, and S (N) (q) is the static structure factor of Fig. 11 of Ref. 15. The number of hydrogen atoms is four for each central ring and five for each lateral ring. The scattering lengths b are, in principle, different for the carbon and deuterium atoms but, as we observed in Ref. 15, they are both positive and of the same magnitude, so that is a good approximation to consider the product b b an ineffective positive constant. The function F (N) t (q,t) is the quantity directly comparable with the experimental data. In the present section we present a comparison between the temperature and momentum dependencies of the MD and experimental spectra of Ref. 11 calculated from perdeuterated C 18 D 14 by means of coherent neutron time-of-flight and backscattering spectroscopy. At this stage few observations must be made on the momentum dependence of the MD and experimental sets of data. All these data are supposed to depend on the structure of the systems so that, in general, some differences are expected experimental and MD structures are slightly different 15. Anyway, two observations about the experimental results must be made. First of all, as reported in Ref. 11, some reservation is necessary for the experimental data at the smallest momenta, q 6 nm 1, where f q tends toward 1. Indeed, in this region one expects significant background from incoherent scattering, which contributes about 15% of the total cross section, and from multiple scattering. Moreover the technique used to determine the values of the dynamical parameters from the experimental data are quite different with respect to the MD computation. Indeed, due to the limited dynamical window of the available spectrometers, in the experimental case a direct fit of the data to the stretched exponential function with three independent parameters is not possible. Based on the observations that (q) (T)/T (T) is the viscosity at temperature T and that the line shape is independent of temperature, at fixed momentum q the spectra at different temperatures are rescaled in time to t t/t s where the scaling time is given by t s (T)/ (T 290K) and (T)/T. In this way the data converge toward a temperature-independent long-time asymptote; this is the curve actually fitted to the stretched exponential. In Fig. 7 we show the temperature dependence of the

33 S. MOSSA, G. RUOCCO, AND M. SAMPOLI PHYSICAL REVIEW E FIG. 8. Top: Simulated neutron coherent scattering function F t (N) (q,t) at T 290 K as a function of momentum. Bottom: F t (N) (q,t)/f q (N) plotted as a function of the rescaled time t/ (N).All the curves collapse on a master curve as expected in this case f q (N) depends on q). stretched exponential fitting parameters for the region of both MD open circles and experimental solid circles data determined at q 14 nm 1. In panel a we show the structural relaxation times; the inset shows the same data but the MD points have been shifted by 20 K as already explained in Ref. 15. We stress, indeed, that a nonperfect parametrization of the diffusive behavior of the model system controlled by the value of LJ, has shifted the MD thermodynamical point about 20 K above the corresponding experimental temperature. From the figure it is clear that the agreement among the two sets of data is very good on a very wide time region. In panel b we show our results for the nonergodicity parameter f q and also in this case the agreement is very good among the two sets of data; in panel c we plot the stretching parameters. In this case the MD points are systematically above the experimentally determined data. This effect can be due to the observations made before; moreover, for the determination of the parameter, the very long-time points are crucial and they seem to lack in the experimental data analysis. With Fig. 8 we start the comparison of the momentum dependence of the two sets of data. In the top panel of Fig. 8 we plot the simulated scattering functions F t (N) (q,t) symbols at T 290 K at the indicated q together with the stretched exponential long-time fits solid lines. In this case f q (N) is expected to be momentum dependent so that, in order to have a master plot bottom panel of Fig. 8, we have to FIG. 9. Comparison among the values of the dynamical parameters calculated by the MD simulations left panels and the experimental ones right panels together with the appropriate structure factors divided by q 2 respectively, panels g and h. Structural relaxation times (N) a and b, nonergodicity parameters f q (N) c and d, and stretching factors (N) e and f. show F t(n) (q,t) F t (N) (q,t)/f q (N) as a function of the rescaled time t t/ (N). The rescaled curves collapse quite well on a master curve as expected. In Fig. 9 we show the comparison for the momentum dependence of the MD data at the three temperatures T 280, 290, and 300 K and the experimental data at T 313, 320, and 330 K; the two sets of temperatures should be comparable due to the 20-K shift of the MD data. In Fig. 9 we show the MD left panels and experimental right panels results together with the corresponding static structure factors renormalized to q 2 panel g for MD and h for experimental results. Striking similarities are clear in the case of the structural relaxation times panels a and b even if maxima on MD results correspond to bumps of experimental results. The clearest difference is the decrease of (N) at small values of q that appears to be completely absent in the experimental case. This could be due to the incoherent background on the experimental data stressed above; moreover, it is well known that sometimes MD is not able to determine the correct values of at small value of q but usually tends to overestimate its correct value, at variance with this case. Anyway, the agreement among the two sets of data at the higher values of q corresponding to the intramolecular correlations is surprising; rigid model could have done that. In Figs. 9 c and 9 d we show the results concerning the nonergodicity parameter f q (N). The different curves are temperature independent as expected; both sets of data show a

34 MOLECULAR DYNAMICS....II... PHYSICAL REVIEW E maximum at q 14 nm 1, while the maximum at q 9 nm 1 for the MD data correspond to a little shoulder in the experimental case. Also in this case a decreasing part at small q is present in the MD data at variance with the experimental results. But according to Fig. 7 of Ref. 10 the calculation of the nonergodicity parameter fitting the region gives a plateau at q 6 nm 1 at variance with the increasing behavior of the region analysis. However, as already stressed in Sec. IV, these data are supposed to coincide. Concluding, also in this case, the experimental data at small q seem to be not reliable. In Figs. 9 e and 9 f we finally show the stretching parameters (N). MD data are very noisy, nevertheless it is possible to recognize an oscillatory behavior; moreover, the points at the smallest available value of q seem to catch the decreasing behavior of the experimental data. VI. SUMMARY AND CONCLUSIONS In this paper we have concluded the analysis of the longtime center of mass dynamics of the intramolecular model for OTP introduced in Ref. 15 ; there we found good agreement among the diffusion properties of the simulated and the real system as well as among the two single-particle dynamics. Moreover in Ref. 15 we found a very good agreement with the main predictions of MCT. In the present paper we definitely confirm such an agreement. We have studied the collective density fluctuations on a large temperature and momentum range, considering both the fluctuations related to molecular and phenyl-ring centers of mass. With respect to the temperature dependence we found the usual double-step decaying pattern and we confirmed the main predictions of the MCT about the behavior of the stretched exponential parameters; in particular, the relaxation times obey the same power law. The momentum dependence of the stretching parameters appears really interesting; MCT predicts that the behavior in the momentum space of the structural relaxation time, at variance with the trivial square law of the one-particle case, is driven by the structure, namely, it is proportional to S(q)/q 2. In both cases, molecules and phenyl rings, this prediction is completely fulfilled. The phenyl-ring behavior is particularly interesting; every feature of the quantity S(q)/q 2 is mirrored on the (R) (q) curve at the three investigated temperatures T 280, 300, and 330 K up to a value of momentum q 30 nm 1. In particular, the maximum at q 22 nm 1 is related to fluctuations taking place on molecular length scales; the long-time structural dynamics appears, indeed, coupled to the dynamics of internal molecular degrees of freedom. Similar oscillations are present for the other stretched parameters, and also the correct behavior for the short-time region is found. The next step has been a comparison among the experimental and simulated neutron scattering spectra calculated considering both the scattering from carbon and deuterium atoms. The temperature dependences of the relaxation times in both cases are well described by the same power law for almost three decades; a good agreement is also found for the other stretching parameters. The momentum dependence is FIG. 10. Master plot of all the time scales related to the center of mass dynamics of the system, calculated by means of molecular dynamics and measured experimentally: experimental viscosity circles, neutron scattering collective experimental relaxation time left triangles, and inverse of the experimental diffusion coefficient squares, MD one particle diamonds and structural relaxation time triangles down calculated on phenyl rings, MD neutron spectra relaxation time right triangles, and inverse of the MD diffusion coefficients triangles up. The data have been rescaled by arbitrary constants in order to maximize the overlap with the viscosity experimental results. All these time scales follow the same MCT power law with exponents T c 290 K and qualitatively very similar in both cases although some different features not clear are present. At this point some conclusions can be drawn about the capability of our model to describe the long-time dynamics of the real system. All the center of mass time scales calculated by molecular dynamics have been found to be consistent with a power law, although some discrepancies were present for the actual value of the power exponent mainly due to uncertainties on the fitting procedure. Moreover it has been shown that our actual MD thermodynamical point is shifted by about 20 K with respect to the corresponding experimental point. Taking into account all these informations, we plot in Fig. 10 all the time scales related to the centers of mass dynamics considered up to now, both experimental and numerical, as a function of the rescaled temperature T T T c where T c 290 K 18 for the experimental points and T c 270 K for the MD results. In particular, V-F (T) circles is the time scale related to the shear viscosity s of Ref. 30, (T) left triangles is the neutron scattering structural relaxation time of Ref. 11 at q 14 nm 1, and the inverse of the experimental diffusion coefficient squares is from Refs. 20,21. The MD one-particle relaxation time calculated on phenyl rings s (R) (T) diamonds and self-diffusion coefficient up triangles are from Ref. 15 ; finally, (R) (T) down triangles is the structural relaxation time of Fig. 3 calculated at q 14 nm 1 on phenyl rings and (N) (T) is the simulated neutron scattering relaxation time of Fig. 7. All the data collapse pretty well although we consider only one decade in temperature on the same straight line corresponding, on a double-log scale, to a power law of the form of Eq. 5 of exponent

35 S. MOSSA, G. RUOCCO, AND M. SAMPOLI PHYSICAL REVIEW E Concluding, our model has been found to be a very successful model for the centers of mass dynamics of the real system, showing a critical behavior consistent with the experimental results in a wide time window. The implementation of an intramolecular dynamics is relevant to such an extent; in particular, the internal degrees of freedom appear to be strongly coupled to the long-time structural dynamics as is also clear from the study of the rotational properties of the system 31. The next step will be the understanding of the role played by the internal degrees of freedom in the fast relaxations observed experimentally 17. ACKNOWLEDGMENTS The authors wish to thank F. Sciortino for very useful discussions, J. Wuttke for the raw experimental data of Ref. 11, and L. A. N. Amaral for very useful suggestions. 1 C. A. Angell, Science 267, C. A. Angell, K. L. Ngai, G. B. McKenna, P. F. McMillan, and S. W. Martin, J. Appl. Phys. 88, M. Mézard and G. Parisi, Phys. Rev. Lett. 82, ; J. Phys.: Condens. Matter 12, W. Götze, in Liquids, Freezing and the Glass Transition, edited by J. P. Hansen, D. Levesque, and J. Zinn-Justin North- Holland, Amsterdam, 1991 ; W.Götze and L. Sjörgen, Rep. Prog. Phys. 55, ; W.Götze, J. Phys.: Condens. Matter 11, A R. Schilling, in Disorder Effects on Relaxational Processes, edited by A. Richert and A. Blumen Springer-Verlag, Berlin, 1994 ; W. Kob, in Experimental and Theoretical Approaches to Supercooled Liquids: Advances and Novel Applications, edited by J. Fourkas et al. ACS Books, Washington, W. van Megen and S. M. Underwood, Phys. Rev. Lett. 70, ; Phys. Rev. E 49, P. N. Segrè, S. P. Meeker, P. N. Pusey, and W. C. K. Poon, Phys. Rev. Lett. 75, ; P. N. Segrè and P. N. Pusey, ibid. 77, Y. Hwang and G. Q. Schen, J. Phys.: Condens. Matter 11, E. Bartsch, F. Fujara, J. F. Legrand, W. Petry, H. Sillescu, and J. Wuttke, Phys. Rev. E 52, ; 53, A. Tölle, H. Schober, J. Wuttke, and F. Fujara, Phys. Rev. E 56, A. Tölle, J. Wuttke, H. Schober, O. G. Randl, and F. Fujara, Eur. Phys. J. B 5, P. Gallo, F. Sciortino, P. Tartaglia, and S. H. Chen, Phys. Rev. Lett. 76, ; F. Sciortino, P. Gallo, P. Tartaglia, and S. H. Chen, Phys. Rev. E 54, ; F. Sciortino, L. Fabbian, S. H. Chen, and P. Tartaglia, ibid. 56, ; F.W. Starr, F. Sciortino, and H. E. Stanley, ibid. 60, ; C. Y. Liao, F. Sciortino, and S. H. Chen, ibid. 60, ; F. Sciortino, Chem. Phys. 258, ; S.-H Chen, P. Gallo, F. Sciortino, and P. Tartaglia, Phys. Rev. E 56, L. J. Lewis and G. Wahnström, Phys. Rev. E 50, ; F. Sciortino and P. Tartaglia, J. Phys.: Condens. Matter 11, A A. Rinaldi, F. Sciortino, and P. Tartaglia, Phys. Rev. E S. Mossa, R. Di Leonardo, G. Ruocco, and M. Sampoli, Phys. Rev. E 62, G. Monaco, D. Fioretto, C. Masciovecchio, G. Ruocco, and F. Sette, Phys. Rev. Lett. 82, ; G. Monaco, S. Caponi, R. Di Leonardo, D. Fioretto, and G. Ruocco, Phys. Rev. E 62, R S. Mossa, G. Monaco, G. Ruocco, M. Sampoli, and F. Sette, e-print cond-mat/ ; S. Mossa, G. Monaco, and G. Ruocco, e-print cond-mat/ W. Petry, E. Bartsch, F. Fujara, M. Kiebel, H. Sillescu, and B. Farago, Z. Phys. B: Condens. Matter 83, M. Kiebel, E. Bartsch, O. Debus, F. Fujara, W. Petry, and H. Sillescu, Phys. Rev. B 45, D. W. McCall, D. C. Douglass, and D. R. Falcone, J. Chem. Phys. 50, F. Fujara, B. Geil, H. Sillescu, and G. Fleischer, Z. Phys. B: Condens. Matter 88, J. Wiedersich, T. Blochowicz, S. Benkhof, A. Kudlik, N. V. Surovtsev, C. Tschirwitz, V. N. Novikov, and E. Rössler, J. Phys.: Condens. Matter 11, A J. Gapinski, W. Steffen, A. Patkowski, A. P. Sokolov, A. Kisliuk, U. Buchenau, M. Russina, F. Mezei, and H. Schober, J. Chem. Phys. 110, F. Mezei and M. Russina, J. Phys.: Condens. Matter 11, A L. Fabbian, A. Latz, R. Schilling, F. Sciortino, P. Tartaglia, and C. Theis, Phys. Rev. E 60, ; C. Theis, F. Sciortino, A. Latz, R. Schilling, and P. Tartaglia, ibid. 62, ; L. Fabbian, A. Latz, R. Schilling, F. Sciortino, P. Tartaglia, and C. Theis, ibid. 62, R. Schilling and T. Scheidsteger, Phys. Rev. E 56, P. G. degennes, Physica Utrecht 25, P. A. Madden, in Liquids, Freezing and the Glass Transition Ref E. Bartsch, H. Bertagnolli, P. Chieux, A. David, and H. Sillescu, Chem. Phys. 169, G. Monaco, Ph.D. thesis, Università di L Aquila, S. Mossa, Ph.D. thesis, Università di L Aquila,

36 JOURNAL OF CHEMICAL PHYSICS VOLUME 116, NUMBER 3 15 JANUARY 2002 Molecular dynamics simulation study of the high frequency sound waves in the fragile glass former orthoterphenyl S. Mossa Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts and Dipartimento di Fisica and INFM, Università di Roma La Sapienza, P.zza Aldo Moro 2, Roma I-00185, Italy G. Monaco European Synchrotron Radiation Facility, BP220, Grenoble Cedex F-38043, France G. Ruocco Dipartimento di Fisica and INFM, Università di Roma La Sapienza, P.zza Aldo Moro 2, Roma I-00185, Italy M. Sampoli Dipartimento di Energetica and INFM, Università di Firenze, Via Santa Marta 3, Firenze I-50139, Italy F. Sette European Synchrotron Radiation Facility, BP220, Grenoble Cedex F-38043, France Received 18 April 2001; accepted 17 October 2001 Using a realistic flexible molecule model of the fragile glass former orthoterphenyl, we calculate via molecular dynamics simulation the collective dynamic structure factor S(Q, ), recently measured in this system by inelastic x-ray scattering. The comparison of the simulated and measured dynamic structure factor, and the study of the S(Q, ) in an extended momentum Q, frequency, and temperature T range allows us i to conclude that the utilized molecular model gives rise to S(Q, ) in agreement with the experimental data, for those thermodynamic states and Q values where the latter are available; ii to confirm the existence of a slope discontinuity on the T dependence of the sound velocity that, at finite Q s, takes place at a temperature T x higher than the calorimetric glass transition temperature T g ; iii to find that the value of T x is Q-dependent and that its Q 0 limit is consistent with T g. The latter finding is interpreted within the framework of the current description of the dynamics of supercooled liquids in terms of exploration of the potential energy landscape American Institute of Physics. DOI: / I. INTRODUCTION The understanding of the dynamics of the supercooled liquids 1,2 and of its relation with the glass transition has received great attention in the last few years. Different details of the dynamics of supercooled liquids and glasses seem to be on the way to be settled, while other are still obscure. As an example, considering the dynamics of the density fluctuations which are the main issue for this paper, on one side the connection between the glass transition phenomenology and the long time dynamics structural rearrangement has been almost clarified, on the other side the effect of the structural arrest on the high frequency collective vibrational motion sound waves is much less clear. From the experimental point of view the study of the time behavior of the density fluctuations is made via the determination of the dynamic structure factor S(Q, ), i.e., the power spectrum of the Q component of the number density Q (t). In the hydrodynamic regime this quantity is measurable by laser light scattering, a well established technique Q 0.03 nm 1 corresponding to frequency in the GHz range. Thanks to this technique much is known about the collective dynamics in the nanosecond time scale. The limit of high frequencies picosecond time scales is traditionally the realm of the inelastic neutron scattering technique. This technique, however, suffers of strong kinematics limitations, that prevent the possibility to investigate the mesoscopic Q region (Q 1 10nm 1 ) that is particularly interesting as it is the region where the transition from a genuine collective behavior of the density fluctuation fades in the single particle dynamics. Such kinematics limitations have been recently overcome by the inelastic x-ray scattering IXS technique. Using this technique it has been possible to firmly establish few common features of the collective dynamics in glasses in the mesoscopic region. 3 In particular, beside specific quantitative differences among different systems, all the investigated glasses show some qualitative common features that can be summarized as follows: i there exist propagating acousticlike excitations up to a max Q value (Q m ) given by Q m a 1 3 showing up as more or less defined Brillouin peaks at (Q) in the S(Q, ) here a is the average interparticle distance. The specific value of Q m a result to be correlated with the fragility of the glass; ii the slope of the almost linear (Q) vsq dispersion relation in the Q 0 limit extrapolates to the macroscopic sound velocity; iii the broadening of the Brillouin peaks, (Q), follows a power law, (Q) DQ, with 2 within the statistical uncertainties; iv the value of D does not depend significantly on /2002/116(3)/1077/8/$ American Institute of Physics Downloaded 17 Oct 2002 to Redistribution subject to AIP license or copyright, see

37 1078 J. Chem. Phys., Vol. 116, No. 3, 15 January 2002 Mossa et al. temperature, indicating that the broadening i.e., the sound attenuation in the high frequency region does not have a dynamic origin, but rather that it is due to the topological disorder. 4 From a theoretical point of view, two main theories have been developed for the study of supercooled and glassy systems, one based on first principle computations of the equilibrium thermodynamics 5 and a second one, the mode coupling theory MCT, 6,7 dynamical in nature. Very recently they have been complemented with the interpretation of the thermodynamics and of the slow structural dynamics in terms of the topological properties of the underlying potential energy surface PES Such interpretation is based on the concept of inherent structures, introduced many years ago by Stillinger and Weber; 11 upon cooling, the system populates basins of the PES whose local minima have depth increasing on lowering the temperature. Moreover, in this framework, the dynamics of the systems has been thought to be decomposed in a fast -high frequencyvibrational dynamics describing the exploration of a specific basin and a slow -diffusive-one, associated with the exploration of different basins. In these studies a key role has been played by computer simulations that, considering the nowadays technology, permit a significant sampling of the PES also at low temperature and, in principle, the study of the dynamics of the model systems at every time scale. Among other, one not yet fully explained issue in the high frequency collective dynamics of supercooled liquids and glasses is the recent finding that the temperature dependence of the excitations frequency at fixed Q value shows a slope discontinuity at a temperature, T x, that is larger than the calorimetric glass transition temperature T g. 12,13 This observation seems to connect to each other the shape of the PES basins which determines the vibrational frequency and hence also the excitations frequency at a given Q s with the depth of the minima explored at that specific temperature. It is our aim to further investigate this issue. In this paper we show results, obtained by means of molecular dynamics simulations, concerning the high frequency dynamics of a realistic flexible model for the fragile glass former orthoterphenyl OTP. Our aim is then twofold. First, we want to study the capability of the utilized molecular model to reproduce the high frequency dynamics of OTP by comparing the calculated dynamics structure factor with the analogous experimental quantity. Once the model has been validate, our aim is i to extend the experimental data to other Q s and other thermodynamic points and ii to provide an interpretation of the experimental finding of the existence of a slope discontinuity in the temperature dependence of the sound velocity. 12,13 The paper is organized as follows: in Sec. II we briefly describe the model and we define the dynamic structure factor S(Q, ) that will be compared with the experimental results. In Sec. III we recall some of the theoretical model used in the analysis of the dynamic structure factor in liquids; we compare our results with the experimental sets of data 12,13 in a large temperature and momentum range finding a very good agreement; we discuss the connection between our findings and the PES approach to the dynamics. Finally in Sec. IV we discuss the results obtained and we draw some conclusions. II. COMPUTATIONAL DETAILS In our model 14 the OTP molecule is constituted by three rigid hexagons phenyl rings of side L a nm. Two adjacent vertices of the parent central ring are bonded to one vertex of the two side rings by bonds whose length, at equilibrium, is L b 0.15 nm. In this scheme, each vertex of the hexagons is thought to be occupied by a fictitious atom of mass M CH 13 amu representing a carbon hydrogen pair C H. In the isolated molecule, at equilibrium, the two lateral rings lie in planes that form an angle of about 54 with respect to the central ring s plane. The three rings of a given molecule interact among themselves by an intramolecular interaction potential; such potential is chosen in such a way to give the correct relative equilibrium positions for the three phenyl rings, to preserve the molecule from dissociation, and to represent at best the isolated molecule vibrational spectrum. The interaction among the rings pertaining to different molecules is described by a site site pairwise additive Lennard-Jones 6 12 potential, each site corresponding to one of the six hexagons vertices. The details of the intramolecular and intermolecular interaction potentials, together with the values of the involved constants, are reported in Ref. 14. Previous studies of the temperature dependence of the self-diffusion coefficient 14 and of the structural relaxation times 14,15 indicate that the introduced model is capable of quantitatively reproducing the dynamical behavior of the real system, but the actual simulated thermodynamic temperature has to be shifted by 20 K upward. In the following, as is our aim to compare the simulation results with the experiments, the reported MD temperatures are always shifted by such an amount. We have studied a microcanonical constant energy system composed by 108 molecules 324 rings, 1944 Lennard Jones interaction sites enclosed in a constant volume cubic box with periodic boundary conditions. To integrate the equations of motion we have treated each ring as a separate rigid body, identified by the position of its center-of-mass R i and by its orientation expressed in terms of quaternions q i. 17 The standard Verlet leapfrog algorithm 17 has been used to integrate the translational motion while, for the orientational part, a refined algorithm has been used. 18 The integration time step is t 2 fs, which gives rise to an overall energy conservation better than 0.01% of the kinetic energy. We studied a wide temperature range spanning the liquid phase and reaching the glass region. In the different temperature runs the size of the cubic box has been chosen in order to keep the system at the coexistence curve experimental density. 19,20 At each temperature, after an equilibration run 15 ns long we start the calculation of the molecular dynamics trajectory. It is worth noting that the system in the low temperature side of the supercooled region and in the glassy phase is not equilibrated, as the diverging structural relaxation times do not allow simulation long enough to reach the equilibrium condition. Nevertheless, in the glassy phase a nonthermalized intrabasin dynamics is not expected to effect Downloaded 17 Oct 2002 to Redistribution subject to AIP license or copyright, see

38 J. Chem. Phys., Vol. 116, No. 3, 15 January 2002 Sound waves in orthoterphenyl 1079 the fast, intrabasin vibrational dynamics. Obviously the characteristics of this fast dynamics are determined by the curvature of the potential energy at the specific minima, a quantity that is known to be dependent on the energy of the minima itself. Therefore, as shown in Refs , strong aging and nonequilibrium effects are expected for the vibrational frequency distribution. Actually, also in the real experiments, whose outcome we are going to compare with the simulation, one investigates the glassy phase in a similar nonequilibrium situation; the relaxation times in the glassy phase are very long, even in comparison to the experimental measuring times. During the MD evolution, lasting for t M 640 ps, the configurations of the system are stored for subsequent analysis every time step for the runs at T ( )280 K. All the calculations have been performed on a cluster of four -CPU with a frequency of 500 MHz; every nanosecond of simulated dynamics needed approximately 24 hours of CPU time. From the stored configuration we calculate the dynamic structure factor as measured in an inelastic x-ray scattering IXS experiment. 12,13 The IXS technique measures the electron charge density fluctuation correlation function. As we consider the phenyl ring as a rigid body i.e., we are not interested in the intraring vibrational dynamics that takes place at frequencies much higher than those investigated here we can consider as fixed the phenyl ring charge distribution. Within this approximation, the scattering center can be considered as the ring s center-of-mass, and the effect of the spatial distribution of the electron charge is summarized in a Q dependent phenyl ring form factor not effecting the frequency shape of S(Q, ). Moreover, as we are not comparing neither the calculated and measured absolute scattering intensities nor their Q-dependence, we neglect hereafter the presence of such a form factor. Therefore, the appropriate dynamic structure factor 25 to be compared with the IXS experimental results is given by the power spectrum of the phenyl ring center number density fluctuation Q (t), Q t 1 N j e iq R j t, 1 S Q, 1 t t M Mdt Q t e i t FIG. 1. Selected examples of the Q dependence of the S(Q, ) calculated from the MD runs open circles at T 50 K and at the indicated Q values. The data are shown in linear and logarithmic scale in the main figures and in the insets, respectively. The full lines represent the fit of the data to Eq. 7, while the dashed dotted dashed lines is the individual elastic inelastic contributions to Eq. 7. In this equation t M is the observation time for the variable Q (t). The dynamic structure factors have been evaluated at the Q values allowed by periodic boundary conditions of the simulation box, Q 2 /L (n,m,l), with n, m, l integers and has been sampled at circular frequency between 0 and 22 ps 1 with a step of ps 1. The smallest accessible Q value is 1.8 nm 1. To increase the statistics, the S(Q, ) at different Q values have been binned in channels 0.2 nm 1 wide, a value comparable to the experimental Q resolution. The frequency resolution, dictated by the time extension of the runs, is ps 1 for T ( )280 K. The choice of different frequency resolutions below and above 280 K is due to the fact that below this temperature the width of the central line proportional to the inverse of the structural relaxation time becomes so small that is no longer measurable. It is therefore preferable to relax the energy resolution in order to increase the statistics of the data. Above 280 K, the width of the central line is measurable, but a good energy resolution is needed; this procedure, obviously, produces data with much poorer statistics. III. RESULTS Selected S(Q, ) Ref. 26 at T 50 K and at Q 1.8, 2.6, 3.2, and 3.7 nm 1 are reported in Fig. 1 open circles. In all the figures, to be consistent with the experimental data, the S(Q, ) have been reported as a function of the energy E measured in mev rather than as a function of the circular frequency. For completeness we recall that (ps 1 ) 1.61E mev. As the inelastic features appear as weak shoulder of an intense central peak, for each Q values the S(Q, ) are shown both in linear main figures and in logarithmic insets intensity scale. We are interested in extracting the relevant parameters from the calculated S(Q, ). A formally exact way to treat these functions goes through the description of the density fluctuation correlation function F(Q,t) Q (t) Q *(0), i.e., the frequency Fourier transform of S(Q, ), in terms of a generalized Langevin equation, 27 F 2 t Q,t 0 F Q,t m Q,t t Ḟ Q,t dt 0, 0 3 where 2 0 k B TQ 2 /MS(Q), m(q,t) is the memory function, and S(Q) is the static structure factor. This equation is Downloaded 17 Oct 2002 to Redistribution subject to AIP license or copyright, see

39 1080 J. Chem. Phys., Vol. 116, No. 3, 15 January 2002 Mossa et al. exact, but all the difficulties associated to the calculation of the S(Q, ) have been transferred to the determination of m(q,t). The advantage of this formulation stays in the fact that independently from the choice of m(q,t) the first two sum rules for S(Q, ) are automatically satisfied, d S Q, F Q,t 0 S Q, 4 K d 2 B T S Q, F Q,t 0 M Q2. By Fourier transform of Eq. 3, it is easy to show that 1 S Q 2 0 m Q, S Q, m Q, 2 m Q, 2, 5 where m (Q, ) and m (Q, ) are the real and imaginary part of the time Fourier transform of the memory function. In the 1 limit, a limit that is valid in all the investigated temperature range, 20 the memory function can be considered as the sum of two contributions: a constant which reflects the -process frozen on the time scale of the sound waves plus a very fast decay at short time. The latter contribution to the memory function often referred to as microscopic or instantaneous is usually represented as a delta-function. Introducing the two constants representing the area of the instantaneous process and the long time limit of the memory function, 2 (Q) and 2 (Q), the memory function is approximated by m Q,t 2 Q t 2 Q, and therefore, using Eq. 5, the S(Q, ) reduces to S Q, S Q f Q 1 f q 1 2 Q Q 2 2 Q Q, where (Q) 2 (Q) 2 0 and f Q / 2 (Q). This expression is the sum of an elastic line the frozen process and of an inelastic feature which is formally identical to a damped harmonic oscillator DHO function; the central line accounts for a fraction f Q the Debye Waller or nonergodicity factor of the total intensity. Recently it has been shown both via MD Ref. 28 and via IXS Ref. 29 that the microscopic contribution to the memory function cannot be represented by a delta function. There are, indeed, clear indications that this microscopic part is responsible for a positive dispersion of the sound velocity in harmonic model glasses, 28 and is responsible for the majority of the positive dispersion of the sound velocity in liquid lithium. 29 The simplest generalization of the delta function appearing in Eq. 6 considers a simple Debye-type microscopic contribution to the memory function, m Q,t 2 Q e t/ 2 Q. 6 8 FIG. 2. Selected examples of the T dependence of the S(Q, ) at the fixed Q value of 2.6 nm 1. Similarly to Fig. 1 the data are shown in linear and logarithmic scale in the main figures and in the insets respectively. The full lines represent the fit of the data to Eq. 7, while the dashed dotted-dashed lines is the individual elastic inelastic contributions to Eq. 7. The quantity entering in this equation is the characteristic time of the microscopic contribution to the memory function. Obviously, whenever 1, Eq. 8 produces for S(Q, ) a DHO-like line shape with (Q) 2 (Q) 0 2 and (Q) 2 (Q). In analyzing the IXS spectra of OTP, the author of Ref. 12 used the simplified DHO expression for the S(Q, ) without noticing any systematic deviation of the experimental data from the fitting line shape. Here, in order to better compare the MD results with the IXS one, we will follow the same procedure and analyze the calculated S(Q, ) to the line shape reported in Eq. 7. It is worth noting, however, that the MD data, less noisy and less effected by the resolution function than the experimental data, actually allow us to establish that the S(Q, ) obtained by the use of the memory function in Eq. 8 produces a better fit of the data. The fits to the MD S(Q, ) using Eq. 7 has been made via the minimization of the standard 2 function, and the results are reported in Fig. 1 as full lines. The individual contributions elastic and inelastic are also shown dashed and dotted dashed lines. The agreement between the MD data and the fitting function is satisfactory for all the investigated T and for all the investigated Q. An example of the T dependence of the S(Q, ) atq 2.6 nm 1 is reported in Fig. 2, again in linear main figure and logarithmic inset scale and together with the best fit to a DHO line shape. The Q-dependencies of the DHO parameters (Q) and (Q) together with the experimental results of Refs. 12, 13 at selected temperatures are reported in Figs. 3 and 4, respectively. 30 These figures confirm the experimental findings see Fig. 1 in Ref. 12 and Fig. 2 in Ref. 13 that: i The excitations frequencies (Q) show a clear Q dependence, which approach a linear behavior at small Q s and bent down Downloaded 17 Oct 2002 to Redistribution subject to AIP license or copyright, see

40 J. Chem. Phys., Vol. 116, No. 3, 15 January 2002 Sound waves in orthoterphenyl 1081 FIG. 3. Q dependence of the excitations frequency (Q) at the indicated selected temperatures open symbols for MD data and closed symbols for IXS data as derived from the fit of the S(Q, ) toeq. 7. The dashed lines are guide to the eye. at increasing Q values; ii The slope at small Q of (Q) shows a marked T dependence together with its overall shape; iii The parameters (Q) follow a Q 2 behavior; and iv The temperature dependence of (Q) is much less pronounced than that of (Q). The latter T dependence is better emphasized in Fig. 5, where the parameter (Q) is reported for the indicated selected Q values at all the investigated temperatures. A decrease line narrowing of (Q) of about 20% is observed between the glassy phase at 50 K and the liquid phase 450 K. This decreasing behavior can be explained remembering that, in the (Q) 1 limit, the parameter (Q) is given by Q 2 Q 2 Q 2 Q 9 and considering that both (Q) and (Q) are linear in Q and proportional to the sound velocities appropriate to frequencies always much larger than 1/ but respectively smaller and larger than 1/. The decrease of (Q) with FIG. 5. T dependence of the excitations broadening (Q) at the indicated selected Q values as derived from the fit of the S(Q, ) toeq. 7. increasing T reflects therefore the decrease of the sound velocities with increasing T. It is also important to point out that Eq. 9 gives also an explanation for the observed Q 2 behavior of (Q), when the reasonable hypotheses of a weak Q dependence of and of the sound velocities are made. Overall the MD results reported in Figs. 2 5 are in agreement with the IXS experimental findings and give further support to the picture where propagating high frequency sound modes exist in glasses and liquids. These excitations are characterized in the small Q limit by a linear and a quadratic Q dependence of the excitations frequency and sound absorption coefficient respectively. To be more quantitative, we report in Fig. 6 the comparison of the T dependence of the (Q) parameter at fixed Q value of 2.5 nm 1 as derived from the MD Ref. 31 and IXS data. The two sets of data are in satisfactory agreement. The MD data lie slightly above the experimental ones, a feature that we ascribe to the nonperfect tuning of the molecular potential model. It is important to underline that, similarly to FIG. 4. Q dependence of the excitations broadening (Q) at the indicated selected temperatures open symbols for MD data and closed symbols for IXS data as derived from the fit of the S(Q, ) to Eq. 7. The full line is a quadratic (Q) dq 2 fit to the whole set of data. FIG. 6. T dependence of the excitations frequency (Q) atq 2.6 nm 1 scaled at Q 2.5 nm 1 as explained in Ref. 31 as derived from the fit of the calculated S(Q, ) toeq. 7 open circles. Also reported are the values of (Q) atq 2.5 nm 1 determined from the IXS experiment full squares Ref. 12. The full lines are the best fit to the experimental points in the lowand high-temperature regions; they cross at the temperature T x. The dashed lines indicate the limit of predictions band. Downloaded 17 Oct 2002 to Redistribution subject to AIP license or copyright, see

41 1082 J. Chem. Phys., Vol. 116, No. 3, 15 January 2002 Mossa et al. FIG. 7. T dependence of the excitations frequencies (Q) at the indicated Q values derived from the fit of the calculated S(Q, ) to Eq. 7 open symbols. The full lines are the best fit to the points in the low- and hightemperature regions; they cross at the temperature T x (Q), indicated by stars. The dashed lines indicate the limit of predictions band; their lowest and highest T crossing are used to define the error bar for T x (Q). The dotted dashed line is a guide to the eye and connects the different discontinuity temperatures. the experimental data, also the MD derived (Q 2.5 nm 1 ) show a discontinuity of slope at a temperature around T g. A discontinuity that also in the MD data appears to be located at a temperature slightly above the calorimetric glass transition temperature. However, the single set of data at Q 2.5 nm 1 is not conclusive. A more complete picture can be derived from the data reported in Fig. 7, where the T dependence of (Q) is reported for four different Q values. Here one can clearly see that the temperature T x (Q) where is observed the slope discontinuity shows a well defined Q dependence, and seems to approach T g decreasing Q. Obviously the main origin of the variation of the sound velocity with the temperature comes from the temperature dependence of the mass density both the IXS experiment in Ref. 12 and the present MD simulation have been performed along the coexistence curve. The well known fact that the temperature dependence of the density is much steeper on the liquid side than on the glass side is reflected in the faster T-dependence of the sound velocity for temperatures larger than the calorimetric glass transition than for T T g. However, beside the trivial density dependence of the sound velocity, there are in the IXS data and even more evident in the MD ones clear indications that nontrivial effects associated with the properties of the PES are also present. Indeed, i the temperature T x at which we observe the change of slope for (q 2.5 nm 1 ) in agreement with the IXS results is definitely different from T g ; ii there is a clear Q-dependence of this temperature. These observations indicate that the shape of the dispersion relation (Q) is modified by changing T. Therefore it is not possible to scale all the (Q) vsq curves on top of each others by a factor, and it is important to study the Q 0 limit of these dispersion relations. This extrapolation is illustrated in Fig. 8 where, at four selected temperatures, the apparent sound velocity c(q) (Q)/Q is reported vs Q. Afitof FIG. 8. The apparent sound velocities c(q) (Q)/Q are reported as a function of Q at the four indicated temperatures. The dashed lines are the best fits of c(q) to a quadratic function, c(q) c(q 0) aq 2, and are used to determine the Q 0 limit of the sound velocity. these apparent velocities to a quadratic function, c(q) c(q 0) aq 2 is then performed dashed lines in Fig. 8. The Q 0 sound velocity, obtained by such an extrapolation are reported in Fig. 9, together with the similar quantity derived from the IXS data. 13 Again we found a good agreement between simulated and experimental data, and again the two sets of data show a slope discontinuity at a comparable temperature, that now coincide within the statistic uncertainties with T g. The overall picture is summarized in Fig. 10, where the discontinuity temperature T x (Q) is reported as a function of Q. Despite the large statistical error bars, it is evident that T x has a marked Q dependence and, in particular, it is clear that it extrapolates to a temperature compatible with T g as Q goes to zero. This result can be interpreted in the framework of the current understanding of the dynamics of supercooled liquids. Numerous MD simulations study 8 10 have recently FIG. 9. T dependence of the sound velocity at Q 0 derived from the fits as in Fig. 8 open circles. Also reported are the corresponding values determined from the IXS experiment full squares Ref. 13. The full lines are the best fit to the MD points in the low- and high-temperature regions: they cross at the temperature T x (Q 0). The dashed lines indicate the limit of predictions band. Downloaded 17 Oct 2002 to Redistribution subject to AIP license or copyright, see

42 J. Chem. Phys., Vol. 116, No. 3, 15 January 2002 Sound waves in orthoterphenyl 1083 FIG. 10. Q dependence of the discontinuity temperature T x (Q) derived from Fig. 7 open circles and from Fig. 9 open square. The error bars are determined by the crossing of the prediction bands. The dashed line is a guide to eye. The horizontal arrow indicates the calorimetric glass transition temperature. pointed out the validity of the description of the supercooled liquid state in terms of the formalism initially introduced by Stillinger and Weber; 11 the point that represents the system in the 3N-dimensions configurational space moves, in this space, performing a fast vibrational dynamics around a given minima of the potential energy hypersurface and occasionally with a rate dictated by the structural relaxation time performs jumps between basins pertaining to different minima. 8 This naive description acquires more physical meaning with the demonstration 21,22,32,33 that the two dynamical regimes are almost decoupled, to the extent that the aging processes in glasses can be quantitatively described supposing that the two subsystem are equilibrated at different temperatures. In this framework, it has been demonstrated 21,22 that, approaching the glass transition temperature, the representative point visits minima of lower and lower potential energy, and exists a strict relation between the temperature and the energy of the minima visited. As demonstrated in the case of binary mixture Lennard-Jones BMLJ Refs. 21, 22 systems, different minima have also on average different metric properties, as for example the curvatures at the minimum point and therefore the distribution of vibrational eigenfrequencies. In particular in Ref. 21 the authors show a deformation not a simple scaling of the equilibrium density of states at different temperatures. On lowering temperature the first momentum of the density of state decreases considerably. At the same time the effect of the temperature change on the overall shape of the density of state is striking. On lowering temperature the modes at high frequency reach their ideal equilibrium value much faster than the lower frequencies do. Concluding, there is an overall deformation of the shape of the density of state on lowering the temperature. The same features have been recently found in Refs. 23, 24 for the case of the rigid OTP model. All these findings agree with our results and the observed Q-dependence of the discontinuity temperature T x can be interpreted within this framework. Indeed, probing different values of the momentum Q is equivalent to consider normal modes in different zones of the density of states; these zones, in turn, behave in different ways on lowering temperature. Concluding, similarly to Lennard-Jones systems, our data suggest that there is a deformation of the shape of the density of vibrational states in changing the energy of the minima. In particular, upon cooling, i.e., on going to lower and lower energy minima, the highest curvatures highest frequency, highest Q reach their limiting values those pertaining to the ideal glassy minima before than the lowest curvatures do. Due to this deformation the modification of the eigenfrequencies local curvatures does not follow the simple scaling expected by a simple change of the elastic constants. In other words, the curve (Q,T) does not depend on T according to a law of the form (Q,T) (Q,T 0 )c(t)/c(t 0 ), this result following a rescaling of the elastic constants with T. At variance is the Q-dependence of (Q) which changes with T. IV. CONCLUSIONS In this paper we have reported results about Molecular Dynamics calculation of the high frequency dynamics on the flexible molecule model of orthoterphenyl introduced in Ref. 14. The collective dynamics structure factor has been determined in a wide exchanged momentum and temperature range and has been compared to its experimental measurements performed by means of inelastic x-ray scattering. 12,13,20 The relevant parameters that describe the overall spectral shape have been determined using the same model already utilized in the analysis of the experimental data, i.e., the sum of an elastic contribution and a DHO line shape. The resulting parameters (Q) and (Q) the position and the broadening of the inelastic peaks, respectively are in good agreement with the corresponding experimentally determined quantities. Moreover the present simulation confirms the existence in liquids and glasses of propagating sound waves at high frequency and shows that: i The excitation frequencies (Q) show a clear Q dependence, which approach a linear behavior at small Q s and bent down at increasing Q values; ii The slope at small Q of (Q) shows a marked T dependence together with its overall shape; iii The parameters (Q) follow a Q 2 behavior; and iv The temperature dependence of the high frequency sound attenuation coefficient, related to (Q), is only weakly dependent on T. The latter result further supports the findings of a sound attenuation mechanism not related to anharmonicity. 4 More important, we find that similarly to the experimental findings of Ref. 12 the temperature dependence of the excitation frequency at constant Q value shows a slope discontinuity at a temperature T x close to, but definitively larger than, the calorimetric glass transition temperature. By extending the Q values where this analysis is performed, we found clear evidence of the Q-dependence of T x. In particular, we find that T x (Q) approaches T g in the Q 0 limit. The existence of a T dependence of the S(Q, ) peaks position can be explained by the results of recent MD studies of Lennard-Jones systems, where the T-dependence of the curvatures of the PES at the inherent structures were clearly evidentiated. The finding of a Q-dependent transition temperature is an indication of the fact that the density of vibra- Downloaded 17 Oct 2002 to Redistribution subject to AIP license or copyright, see

43 1084 J. Chem. Phys., Vol. 116, No. 3, 15 January 2002 Mossa et al. tional states does not change according to a frequency scaling law. Rather it is deformed in such a way that the highest frequencies reach the limiting value pertaining to the ideal glassy minimum before than the lowest frequencies do. Finally, our results indicate a possible experimental way to validate or disprove the PES-based interpretation of the glass transition phenomenology. ACKNOWLEDGMENTS We thank R. Di Leonardo and T. Scopigno for intensive discussions during the data analysis and F. Sciortino for useful discussions. 1 C. A. Angell, Science 267, C. A. Angell, K. L. Ngai, G. B. McKenna, P. F. McMillan, and S. W. Martin, J. Appl. Phys. 88, F. Sette, M. Krisch, C. Masciovecchio, G. Ruocco, and G. Monaco, Science 280, G. Ruocco, F. Sette, R. Di Leonardo, D. Fioretto, M. Lorentzen, M. Krisch, C. Masciovecchio, G. Monaco, F. Pignon, and T. Scopigno, Phys. Rev. Lett. 83, M. Mézard and G. Parisi, Phys. Rev. Lett. 82, ; J. Phys.: Condens. Matter 12, W. Götze, in Liquids, Freezing, and the Glass Transition, edited by J. P. Hansen, D. Levesque, and J. Zinn-Justin North Holland, Amsterdam, 1991 ; W.Götze and L. Sjörgen, Rep. Prog. Phys. 55, ; W. Götze, J. Phys.: Condens. Matter 11, A R. Schilling, in Disorder Effects on Relaxational Processes, edited by A. Richert and A. Blumen Springer-Verlag, Berlin, 1994 ; W. Kob, in Experimental and Theoretical Approaches to Supercooled Liquids: Advances and Novel Applications, edited by J. Fourkas et al. ACS Books, Washington, F. Sciortino, W. Kob, and P. Tartaglia, Phys. Rev. Lett. 83, L. Angelani, R. Di Leonardo, G. Ruocco, A. Scala, and F. Sciortino, Phys. Rev. Lett. 85, ; E. La Nave, A. Scala, F. W. Starr, F. Sciortino, and H. E. Stanley, ibid. 84, ; L. Angelani, R. Di Leonardo, G. Parisi, and G. Ruocco, ibid. 87, A. Heuer, Phys. Rev. Lett. 78, ; S. Buechner and A. Heuer, Phys. Rev. E 60, ; S. Sastry, P. G. Debenedetti, and F. H. Stillinger, Nature London 393, ; B. Coluzzi, P. Verrocchio, and G. Parisi, Phys. Rev. Lett. 84, ; T. B. Schrøder, S. Sastry, J. Dyre, and S. C. Glotzer, J. Chem. Phys. 112, ; S. Sastry, Nature London 409, F. H. Stillinger and T. A. Weber, Phys. Rev. A 25, ; Science 225, ; F. H. Stillinger, ibid. 267, C. Masciovecchio, G. Monaco, F. Sette, A. Cunsolo, M. Krish, A. Mermet, M. Solwisch, and R. Verbeni, Phys. Rev. Lett. 80, G. Monaco, C. Masciovecchio, G. Ruocco, and F. Sette, Phys. Rev. Lett. 80, S. Mossa, R. Di Leonardo, G. Ruocco, and M. Sampoli, Phys. Rev. E 62, S. Mossa, G. Ruocco, and M. Sampoli, Phys. Rev. E 64, S. Mossa, G. Monaco, and G. Ruocco, preprint cond-mat/ M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids Clarendon, Oxford, G. Ruocco and M. Sampoli, Mol. Phys. 82, J. N. Andrews and A. R. Ubbelohde, Proc. R. Soc. London, Ser. A 228, ; R. J. Greet and D. Turnbull, J. Chem. Phys. 46, ; G. Friz, G. Kuhlbörsch, R. Nehren, and F. Reiter, Atomkernenergie 13, ; W. H. Hedley, M. V. Milnes, and W. H. Yanko, J. Chem. Eng. Data 15, ; M. Naoki and S. Koeda, J. Phys. Chem. 93, G. Monaco, Ph.D. thesis, Università di L Aquila, 1997 unpublished. 21 W. Kob, F. Sciortino, and P. Tartaglia, Europhys. Lett. 49, F. Sciortino and P. Tartaglia, Phys. Rev. Lett. 86, S. Mossa, G. Ruocco, F. Sciortino, and P. Tartaglia, preprint cond-mat/ S. Mossa, E. La Nave, H. E. Stanley,C. Donati, F. Sciortino, and P. Tartaglia, preprint cond-mat/ J. P. Hansen and J. R. Mc Donald, Theory of Simple Liquids Academic, London, It is worth noting that the following general discussion of the models for the dynamic structure factor will be given in terms of the frequency, at variance with all the available experimental data which express the energy in mev. In view of the comparison between MD and the experimental results all the data points in the pictures are expressed in mev. 27 U. Balucani and M. Zoppi, Dynamics of the Liquid State Clarendon, Oxford, G. Ruocco, F. Sette, R. Di Leonardo, G. Monaco, M. Sampoli, T. Scopigno, and G. Viliani, Phys. Rev. Lett. 84, T. Scopigno, U. Balucani, G. Ruocco, and F. Sette, Phys. Rev. Lett. 85, ; J. Phys. C 12, It is worth noting that sometimes in the literature in Eq. 6 appears the term instead of 2 ; this is actually the case of Refs. 3 and 4. In order to preserve the consistency with Eq. 6 we then show both the actual experimental and MD values for multiplied by The experimental data were collected at Q 2.5 nm 1, while the present MD data are calculated at Q 2.6 nm 1. To account for the small differences in the Q values and therefore in the values we have scaled the MD calculated (Q) by the factor 2.5/2.6, appropriate for a linear Q dependencies of the excitation frequency. 32 G. Parisi, Phys. Rev. Lett. 79, ; J. L. Barrat and W. Kob, Europhys. Lett. 46, R. Di Leonardo, L. Angelani, G. Parisi, and G. Ruocco, Phys. Rev. Lett. 84, Downloaded 17 Oct 2002 to Redistribution subject to AIP license or copyright, see

44 VOLUME 88, NUMBER 13 P H Y S I C A L R E V I E W L E T T E R S 1 APRIL 2002 Truncation of Power Law Behavior in Scale-Free Network Models due to Information Filtering Stefano Mossa, 1,2 Marc Barthélémy, 3 H. Eugene Stanley, 1 and Luís A. Nunes Amaral 1 1 Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts Dipartimento di Fisica, INFM UdR, and INFM Center for Statistical Mechanics and Complexity, Università di Roma La Sapienza, Piazzale Aldo Moro 2, I-00185, Roma, Italy 3 CEA-Service de Physique de la Matière Condensée, BP 12, Bruyères-le-Châtel, France (Received 18 October 2001; published 14 March 2002) We formulate a general model for the growth of scale-free networks under filtering information conditions that is, when the nodes can process information about only a subset of the existing nodes in the network. We find that the distribution of the number of incoming links to a node follows a universal scaling form, i.e., that it decays as a power law with an exponential truncation controlled not only by the system size but also by a feature not previously considered, the subset of the network accessible to the node. We test our model with empirical data for the World Wide Web and find agreement. DOI: /PhysRevLett There is a great deal of current interest in understanding the structure and growth mechanisms of global networks [1 3], such as the World Wide Web (WWW) [4,5] and the Internet [6]. Network structure is critical in many contexts such as Internet attacks [2], spread of an virus [7], or dynamics of human epidemics [8]. In all these problems, the nodes with the largest number of links play an important role on the dynamics of the system. It is therefore important to know the global structure of the network as well as its precise distribution of the number of links. Recent empirical studies report that both the Internet and the WWW have scale-free properties; that is, the number of incoming links and the number of outgoing links at a given node have distributions that decay with power law tails [4 6]. It has been proposed [9] that the scale-free structure of the Internet and the WWW may be explained by a mechanism referred to as preferential attachment [10] in which new nodes link to existing nodes with a probability proportional to the number of existing links to these nodes. Here we focus on the stochastic character of the preferential attachment mechanism, which we understand in the following way: New nodes want to connect to the existing nodes with the largest number of links i.e., with the largest degree because of the advantages offered by being linked to a well-connected node. For a large network it is not plausible that a new node will know the degrees of all existing nodes, so a new node must make a decision on which node to connect with based on what information it has about the state of the network. The preferential attachment mechanism then comes into play as nodes with a larger degree are more likely to become known. This picture has one underlying and unstated assumption, that the new nodes will process (i.e., gather, store, retrieve, and analyze) information concerning the state of the entire network. For very large networks, such as the WWW or the scientific literature, this would correspond to the unrealistic situation in which new nodes can process PACS numbers: Hh, i, Da, Hc an extremely large amount of information i.e., have unlimited information-processing capabilities. Indeed, it is likely that nodes have limited information-processing capabilities and so must filter incoming information according to their particular interests. Thus, new nodes of a large growing network will process only information concerning a subset of existing nodes, since there is a cost associated with processing information. The new nodes will then make decisions on with whom to link, based on filtered information. From the standpoint proposed here, most models studied in the literature work under the unrealistic assumption of unfiltered information i.e., a new node processes information about all the existing nodes in the network. Here we consider for the first time the effect on network growth of filtering information due to limited information-processing capabilities. First, we calculate the in-degree distributions of web pages using two databases. The first database, which comprises pages [9], surveys a very significant fraction of the entire WWW, while the second, which comprises pages, lists the University of Notre Dame domain [4] i.e., the set of URLs containing the string nd.edu. For the first database, we calculate the cumulative in-degree distributions P k P k 0.k p k 0, where p k is the probability distribution. We confirm that the in-degree distribution decays as a power law [9] of the form P k k 2gin, (1) with an exponent g in (Fig. 1). Further, we find an exponential truncation of the scale-free behavior for k. k , in contrast with the plateau reported in other studies [2,11]. For the second database, we also find a power law regime with the same exponent, but the exponential truncation appears to be absent, suggesting that the truncation is not due to the finite size of the databases (13) (4)$ The American Physical Society

45 VOLUME 88, NUMBER 13 P H Y S I C A L R E V I E W L E T T E R S 1 APRIL Cumulative distribution WWW Notre-Dame Number of incoming links, k FIG. 1. Distribution of the number of incoming links for the WWW. Cumulative in-degree distribution from two databases, the entire Web [9], and the University of Notre Dame domain [4]. We also plot a power law function with exponent g in 1.25 (dashed line) and a Yule function [10] of the form k 2gin exp 2k k x (solid line). A cutoff degree k is visible in the data. To explain these empirical results, we hypothesize that the authors of new web pages filter some of the information regarding existing web pages, that is, the new nodes make linking decisions under information-filtering conditions. To investigate this process, we consider network growth models in which new nodes process information from only a fraction of existing nodes which one may view as matching the interests of the new nodes. If the fraction f of interesting nodes in the network is much less than one, then the attachment of new links is a random process, so the generated network will be a random graph with an exponentially decaying in-degree distribution. In contrast, if f 1, then preferential attachment is recovered and the in-degree distribution is scale-free. We first define the network growth rule: At time t 0, one creates n o nodes with n o 2 1 links each. At each time step, one adds to the network a new node with n o 2 1 outgoing links. These n o links can connect to a randomly selected subset C containing n t t 1 n o f nodes. The links to the nodes in the subset are selected according to the preferential attachment rule, i.e., the probability that node i belonging to C is selected is proportional to the number of incoming links k i to it p i, t k x k i PjeC k j. (2) In Fig. 2(a), we show our numerical results for the indegree cumulative distributions for networks with S nodes and n o 1, for a sequence of f values. For f 1, we reproduce the results reported for the scale-free model [9] i.e., we observe an in-degree distribution that decays as a power law with an exponent g in 2. For f, 10 22, we observe a crossover at k k 3 from power law behavior to exponential behavior. P (k) P (k) γ k in P (k) 10 0 f k (b) f = 10 3 S 3,900 15,625 62, ,000 1,000, (c) (a) S=500,000 k k / (S f) θ 1 n (d) S=500, , (f) k (e) S=15,625 62, ,000 n=2 n=1,000 n= k k / n θ 2 FIG. 2. In-degree cumulative probability distributions P k under information filtering. Constant f case: (a) Results for S and different values of f. (b) Results for f and different values of S. (a) and (b) show that k 3 decreases with f and increases with S. (c) Data collapse of the numerical results according to Eq. (3) with g in and u Constant n case: (d) Results for S and different values of n showing the decrease in the cutoff degree k 3 with decreasing n. (e) Results for n 2, 10, and 1000 for different values of S showing that P k does not depend on S. (f) Data collapse according to Eq. (4) with g in and u To further investigate the effect of changes in f on the cutoff degree k 3, we plot in Fig. 2(b) the in-degree distributions for different network sizes S and a fixed value of f. Wefind that k 3 increases as a power law with S. All of our numerical results can be expressed compactly by the scaling form µ k P k, f, S ~ k 2gin F 1, (3) with k 3 Sf u 1. We find g in , u and F 1 x const for x ø 1, F 1 x e 2x for x 1. As a test of the scaling form Eq. (3), we plot in Fig. 1(c) the scaled cumulative distribution versus the scaled in-degree. The figure confirms our scaling ansatz, since all data collapse onto a single curve, the scaling function F 1 x. We consider next a situation in which new nodes are not processing information from a constant fraction f of nodes but from a constant number n of nodes. That is, as the network grows, the new nodes are able to process information about a smaller fraction of existing nodes. This model may be more plausible for networks that have grown to a very k 3

46 VOLUME 88, NUMBER 13 P H Y S I C A L R E V I E W L E T T E R S 1 APRIL 2002 large size, since the fraction f of all nodes represents a very large number. In the case of the scientific literature, this effect leads to the fragmentation of a scientific field as it grows [12]. For the constant n case, the fraction of known nodes at time t is f t n t 1 n o, implying that as the network grows there are two antagonistic trends affecting k 3. The first is a tendency to increase due to the growing size of the network, and the second is a tendency to decrease due to the decreasing value of f. Hence, one may hypothesize that there will be a characteristic network size S c above which k 3 will no longer depend on S. We now test these arguments with numerical simulations. In Figs. 2(d) and 2(e), we show our results for growing networks for which new nodes process information only from n randomly selected existing nodes. We find, in agreement with our scaling arguments, that for S S c the in-degree distribution obeys the µ scaling relation P k, n, S ~ k 2g in k F 2, (4) k 3 with k 3 n u 2, g in , u , and where the scaling function F 2 x has the same limiting behavior as F 1 x. To test the scaling form Eq. (4), we plot in Fig. 2(f) the scaled cumulative distribution versus the scaled in-degree. This confirms our scaling ansatz since the data collapse onto a single curve, the scaling function F 2 x. Comparison of the two scaling relations Eqs. (3) and (4) reveals an unexpected result. By replacing Sf by n in (3) one would naively expect to obtain (4) with u 1 u 2 and F 1 x F 2 x. Surprisingly, we find that u 1 is significantly different from u 2 and that F 1 x is significantly different from F 2 x. In order to understand this result, consider two growing networks that have reached size S. For the first, new nodes process information from a fraction f of existing nodes, while, for the second, new nodes process information from n fs existing nodes. At a time t, prior to the network having reached its final size S, there are t 1 n o, S sites, and the preferential attachment is acting for the first network on a number of nodes t 1 n o f, Sf n. The preferential attachment mechanism can operate effectively only when it acts on a number of nodes comparable to S, so the fact that for the first network new nodes have always processed information from fewer existing nodes suggests the first network will not develop nodes with as large a degree as the second network. Thus, we expect that (i) the two resulting networks have different in-degree distributions, and (ii) the in-degree distribution for f fixed has a sharper truncation and a smaller cutoff than for n fixed, which is indeed what we find. Our numerical results are in qualitative agreement with empirical data. However, the value of the power law exponent g in 1.25 found for the WWW is significantly smaller than the value g in 2 predicted by the model. This fact prompts the question of the effect of the cost of information filtering on models generating an in-degree distribution closer to the empirical results. To answer this question, we investigate two possible explanations for the observed value g in (i) Effect of out-degree distribution on g in. The scalefree model [9] is missing an important ingredient: a heterogeneous distribution of number of outgoing links. Indeed, the out-degree distribution considered so far is restricted to a single value m n o 2 1, i.e., p out m d m,no 21, while for the empirical data of the WWW it decays as a power law of the form p out m m 2g out with g out We show in Fig. 3 the computed value of the exponent g in of the in-degree distribution as a function of g out [13]. We find that g in increases approximately linearly with increasing values of the exponent g out until it reaches the limiting value g in 2. Forg out 1.7, which is the empirically observed value for the WWW, we find g in 1.8, which does not agree with the empirical value of 1.25, so the power law decaying out-degree distribution alone cannot explain the results obtained for the WWW. (ii) Effect of fitness on g in. The preferential attachment mechanism is modified by a fitness factor [11]: Nodes have different fitness, and fitter nodes are more likely to receive incoming links than less fit nodes with the same value of k. Uniformly distributed fitness is known to lead to a smaller exponent g in [11], which is quite close to the value measured for the WWW. Hence, we assign to each node a fitness h i [11], reflecting the fact that for equal values of k some nodes are more attractive than others [14]. The probability that a new node will link to node i is γ in without fitness with fitness WWW γ out FIG. 3. Dependence of the in-degree distribution exponent g in on the out-degree distribution exponent g out. We show results for models (i) without fitness [h i const] and (ii) with fitness [h i uniformly distributed]. For the former case, g in increases initially approximately linearly with g out, and then saturates at g in 2 for g out. 2. This saturation of g in is to be expected as g in 2 for the case of a peaked distribution of n o. For the latter case, g in increases approximately linearly with g out initially, and then saturates at g in 1.25 for g out This saturation is to be expected as g in for the case of a peaked distribution of n o [11]

47 VOLUME 88, NUMBER 13 P H Y S I C A L R E V I E W L E T T E R S 1 APRIL 2002 h i k i p i, t P t1no j 1 h j k j. (5) We consider here the case in which h i is a uniformly distributed random variable [15]. Figure 3 shows that the in-degree distribution decays as a power law with values of g in, Forg out. 1.9, the exponent approaches the limiting value g in Interestingly, for g out 1.7, the empirical value for the WWW, we find g in 1.2, in agreement with the empirical value g in Our results for the model with fitness show that information filtering and node fitness are both necessary in order to approximate the empirical results. An open question is which type of filtering is more appropriate for the WWW, constant f or constant n? To answer this question one would need WWW data for a different sample size, which are not presently available to us. However, due to the sheer size of the WWW, it seems plausible that constant n would be the more appropriate case. Our key finding is that limited information-processing capabilities have a significant and quantifiable effect on the large-scale structure of growing networks. We find that information filtering leads to an exponential truncation of the in-degree distribution for networks growing under conditions of preferential attachment. Surprisingly, we find simple scaling relations that predict the in-degree distribution in terms of (i) the information-processing capabilities available to the nodes, and (ii) the size of the network. We also quantify the effect of a heterogeneous outdegree distribution on the in-degree distribution of networks growing under conditions of preferential attachment. We find that, for a power law decaying out-degree distribution with exponents g out, 2, the exponent g in characterizing the tail of the in-degree distribution will take values smaller than those predicted by theoretical calculations [2,3]. The exponential truncation we find may have dramatic effects on the dynamics of the system, especially for processes where the nodes with the largest degree have important roles. This is the case, for example, for virus spreading [7], where for networks with exponentially truncated indegree distributions there is a nonzero threshold for the appearance of an epidemic. In contrast, scale-free networks are prone to the spreading and the persistence of infections no matter how small the spreading rate. Our finding of a mechanism leading to an exponential truncation even for systems where before none was expected [16] indicates that the most connected nodes will have a smaller degree than predicted for scale-free networks leading, possibly, to different dynamics, e.g., for the initiation and spread of epidemics. In the context of network growth, the impossibility of knowing the degrees of all the nodes comprising the network due to the filtering process and, hence, the inability to make the optimal, rational, choice is not altogether unlike the bounded rationality concept of Simon [17]. Remarkably, it appears that, for the description of WWW growth, the preferential attachment mechanism, originally proposed by Simon [10], must be modified along the lines of another concept also introduced by him bounded rationality [17]. We thank R. Albert, P. Ball, A.-L. Barabási, M. Buchanan, J. Camacho, and R. Guimerà for stimulating discussions and helpful suggestions. We are especially grateful to R. Kumar for sharing his data. We thank NIH/NCRR (P41 RR13622) and NSF for support. [1] S. H. Strogatz, Nature (London) 410, 268 (2001). [2] R. Albert and A.-L. Barabási, Rev. Mod. Phys. 74, 47 (2002). [3] S. N. Dorogovtsev and J. F. F. Mendes, Adv. Phys. (to be published). [4] R. Albert, H. Jeong, and A.-L. Barabási, Nature (London) 401, 130 (1999). [5] B. A. Huberman and L. A. Adamic, Nature (London) 401, 131 (1999); R. Kumar et al., in Proceedings of the 25th International Conference on Very Large Databases (Morgan Kaufmann Publishers, San Francisco, 1999), p. 639; A. Broder et al., Comput. Netw. 33, 309 (2000); P. L. Krapivsky, S. Redner, and F. Leyvraz, Phys. Rev. Lett. 85, 4629 (2000); S. N. Dorogovtsev, J. F. F. Mendes, and A. N. Samukhin, ibid. 85, 4633 (2000); A. Vazquez, Europhys. Lett. 54, 430 (2001). [6] M. Faloutsos, P. Faloutsos, and C. Faloutsos, Comput. Commun. Rev. 29, 251 (1999); G. Caldarelli, R. Marchetti, and L. Pietronero, Europhys. Lett. 52, 386 (2000); A. Medina, I. Matta, and J. Byers, Comput. Commun. Rev. 30, 18 (2000); R. Pastor-Satorras, A. Vazquez, and A. Vespignani, arxiv:cond-mat/ ; L. A. Adamic et al., Phys. Rev. E 64, (2001). [7] F. B. Cohen, A Short Course on Computer Viruses (Wiley, New York, 1994); R. Pastor-Satorras and A. Vespignagni, Phys. Rev. Lett. 86, 3200 (2001); Phys. Rev. E 63, (2001). [8] F. Liljeros, C. R. Edling, L. A. Nunes Amaral, H. E. Stanley, and Y. Åberg, Nature (London) 411, 907 (2001). [9] A.-L. Barabási and R. Albert, Science 286, 509 (1999). [10] Y. Ijiri and H. A. Simon, Skew Distributions and the Sizes of Business Firms (North-Holland, Amsterdam, 1977). [11] G. Bianconi and A.-L. Barabasi, Europhys. Lett. 54, 436 (2001). [12] A. F. J. Van Raan, Scientometrics 47, 347 (2000). [13] We consider a modification to the network growth rule described earlier in the paper: at each time step t, the new node establishes m new links, where m is drawn from a power law distribution with exponent g out. [14] For h i const, one recovers the scale-free model of Ref. [9]. [15] It is known [11] that, for an exponential or fat-tailed distribution of fitness, the structure of the network becomes much more complex; in particular, the in-degree distribution is no longer a power law. Hence, we do not consider in this manuscript other shapes of the fitness distribution. [16] L. A. N. Amaral, A. Scala, M. Barthélémy, and H. E. Stanley, Proc. Natl. Acad. Sci. U.S.A. 97, (2000). [17] H. A. Simon, Models of Bounded Rationality: Empirically Grounded Economic Reason (MIT Press, Cambridge, 1997)

48 PHYSICAL REVIEW E, VOLUME 65, Dynamics and configurational entropy in the Lewis-Wahnström model for supercooled orthoterphenyl S. Mossa, 1,2 E. La Nave, 1 H. E. Stanley, 1 C. Donati, 2 F. Sciortino, 2 and P. Tartaglia 2 1 Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts Dipartimento di Fisica, INFM and INFM Center for Statistical Mechanics and Complexity, Università di Roma La Sapienza, Piazzale Aldo Moro 2, I-00185, Roma, Italy Received 27 November 2001; published 10 April 2002 We study thermodynamic and dynamic properties of a rigid model of the fragile glass-forming liquid orthoterphenyl. This model, introduced by Lewis and Wahnström in 1993, collapses each phenyl ring to a single interaction site; the intermolecular site-site interactions are described by the Lennard-Jones potential whose parameters have been selected to reproduce some bulk properties of the orthoterphenyl molecule. A system of N 343 molecules is considered in a wide range of densities and temperatures, reaching simulation times up to 1 s. Such long trajectories allow us to equilibrate the system at temperatures below the mode coupling temperature T c at which the diffusion constant reaches values of order cm 2 /s and thereby to sample in a significant way the potential energy landscape in the entire temperature range. Working within the inherent structures thermodynamic formalism, we present results for the temperature and density dependence of the number, depth and shape of the basins of the potential energy surface. We evaluate the total entropy of the system by thermodynamic integration from the ideal noninteracting gas state and the vibrational entropy approximating the basin free energy with the free energy of 6N 3 harmonic oscillators. We evaluate the configurational part of the entropy as a difference between these two contributions. We study the connection between thermodynamical and dynamical properties of the system. We confirm that the temperature dependence of the configurational entropy and of the diffusion constant, as well as the inverse of the characteristic structural relaxation time, are strongly connected in supercooled states; we demonstrate that this connection is well represented by the Adam-Gibbs relation, stating a linear relation between logd and the quantity 1/TS c. This relation is found to hold both above and below the critical temperature T c as previously found in the case of silica supporting the hypothesis that a connection exists between the number of basins and the connectivity properties of the potential energy surface. DOI: /PhysRevE PACS number s : Cb, Fs, Pf I. INTRODUCTION Understanding the dynamic and thermodynamic properties of supercooled liquids is one of the more challenging tasks of condensed matter physics for recent reviews see Refs. 1 4 and references therein. A significant amount of experimental 5 9, numerical 10 and theoretical work is being currently devoted to the understanding of the physics of the glass transition and to the associated slowing down of the dynamics. Among the theoretical approaches, an important role has been played by the mode coupling theory MCT 11,12, which, interpreting the glass transition as a purely dynamical phenomenon, has constituted a significant tool for the interpretation of both experimental 5,9,16 19 and numerical simulation results in weakly supercooled states. In recent years the study of the topological structure of the potential energy hyper- surface PES 23,24 and the connection between the properties of the PES and the dynamical behavior of glass-forming liquids has become an active field of research. Building on the inherent structure IS thermodynamic formalism proposed a long time ago by Stillinger and Weber 23, the PES can be uniquely partitioned in local basins and properties of the basins explored in supercooled states average basin depth and basin volume have been quantified. Studies have mainly focused on two fundamental questions: i which are the basins relevant for the thermodynamics of the system, i.e., which are the basins populated with largest probability? and ii which are the topological properties of the regions of the PES actually explored by the system during its dynamics? From this point of view, the PES approach has somehow unified, at least on a phenomenological level, the thermodynamic and dynamic approaches to the glass transition. Numerical analysis of the PES has shown that trajectories in configuration space can be separated into intrabasin and interbasin components 25,26. The time scales of the two components become increasingly separated on cooling. The intrabasin motion has been associated with the highfrequency vibrational dynamics, while the structural relaxation ( relaxation has been related to the process of exploration of different basins. It has also been shown that on lowering T, the system populates basins of lower and lower energy 27,28. The T dependence of the depth of the typical sampled basins follows a 1/T law for fragile liquids, and, for strong liquids, it appears to approach a constant value on cooling 32. The number of basins as a function of the basin depth e IS has also been recently evaluated for a few models 29,30,32 35, opening the possibility of calculating the so-called configurational entropy S c and its T dependence. S c, defined as the logarithm of the number of accessible basins S c k B log, has been successfully com X/2002/65 4 / /$ The American Physical Society

49 S. MOSSA et al. PHYSICAL REVIEW E pared with theoretical predictions 13,36. At the same time, the approaches and the techniques developed for the analysis of the PES of structural glasses have spread to the field of disordered spin systems, where similar calculations have been performed 37 and similar conclusions have been reached. The evaluation of S c for models of glass-forming liquids allows us to numerically check, in a very consistent way, the relation between S c and the systems characteristic time, proposed by Adam and Gibbs 38, and recently derived in a novel way 14. Numerical support for a relation between the T dependence of S c and the T dependence of, although limited to very few models, is providing new physical insight on the connection between thermodynamics and long time dynamical properties. The ideas developed within the inherent structure formalism have also been generalized to out-of-equilibrium conditions where the slow aging dynamics has been interpreted as the process of searching for basins of increasingly deep energy In this paper we study the properties of the PES for a rigid model Lewis and Wahnström LW of the fragile glass former orthoterphenyl OTP, first introduced by Lewis and Wahnström 43 and recently revisited by Rinaldi et al. 44. We have studied the properties of the PES in a temperature range in which the diffusion coefficient varies by more than four orders of magnitudes for five different density values. This work attempts to build a bridge between models of more direct theoretical interest, like Lennard-Jones LJ and soft spheres, and models which appear to reproduce, even if in a crude way, properties of complex materials. In this respect, orthoterphenyl is the best candidate, being one of the most studied glass-forming liquids 17. The LW model is a three-sites model, with intermolecular site-site interactions described by the LJ potential. This model is among the simplest models for a nonlinear molecule. The limitation constituted by the fact that it does not take into account the internal molecular degrees of freedom see 45 for a more realistic model, is overruled by the observation that its simplicity it can be considered as an atomic LJ with constraints allows one to reach simulation times of the order of s. Hence a significant sampling of the PES in a large temperature and density range is possible. Moreover, this model constitutes an ideal bridge between simple atomic models and molecular models, being possible to treat it under several approximations 44. The paper is structured as follows: In Sec. II we briefly recall the main results of the IS formalism. In Sec. III we show the calculation of the configurational entropy as a difference between the total entropy and the vibrational entropy. In Sec. IV we give some numerical details. We present our results in Sec. V, which is divided into subsections detailing the calculation of the total entropy by thermodynamical integration from the ideal gas state, the study of the vibrational properties of the PES, and the calculation of the configurational entropy. In the end we study the link between configurational entropy and the diffusion constant, investigating the validity of the Adam-Gibbs equation. In Sec. VI we finally discuss our results and we draw some conclusions. In Appendix A we report the analytical calculation of the total entropy of a system of LW molecules in the noninteracting ideal gas limit. II. INHERENT STRUCTURE THERMODYNAMICS FORMALISM In this section we briefly review the IS formalism in the NVT ensemble 23,46, the extension to the NPT ensemble poses no particular problems 23. This formalism has become an important tool in the numerical analysis of classical models since it is numerically possible to calculate in a very precise way the inherent structures defined as the local minima of the PES and hence compare the theoretical predictions with the numerical results. Given an instantaneous configuration of the system, a steepest descent path along the potential energy hypersurface defines the closest IS. In the IS formalism, the partition function of a system is written as a sum over all the PES basins. Basins of given IS energy contribute non-negligibly to the total sum if their IS energy is very low, if their volume is very large, and/or if they are highly degenerate, i.e., several basins are characterized by this IS energy. This corresponds to partitioning the phase space in the local energy minima of the PES and their basins of attraction. Such a partition is motivated by the fact that in supercooled states, the typical time scales of the intrabasin and interbasin dynamics differ by several orders of magnitude and hence the separation of intrabasin and interbasin variables becomes meaningful. In the 6N-dimensional configuration space, the partition function Z for a system of N rigid molecules can be written as Z x y z N 3N dq N exp V q N /k B T, 1 where q N denotes the positions and orientations of the molecules, V(q N ) is the potential energy, I, where x,y,z are the principal moments of inertia of the molecule, (2 I k B T) 1/2 /h, and h(2 mk B T) 1/2 is the de Broglie wavelength. Let (E IS ) denote the number of minima with energy E IS, and f (T,E IS ) the average free energy of a basin with basin depth E IS. f (T,E IS ), which takes into account both the kinetic energy of the system and the local structure of the basin with energy E IS, is defined by f T,E IS k B Tln x y z N 3N 1 E IS basins Rbasin dq N exp V E IS /k B T, where R basin is the configuration volume associated with the specific basin. The partition function can then be rewritten as a sum over all basins in configurational space, i.e.,

50 DYNAMICS AND CONFIGURATIONAL ENTROPY IN THE... PHYSICAL REVIEW E Z EIS EIS E IS exp E IS f T,E IS k B T exp TS c E IS E IS f T,E IS k B T, 3 E T 6 3 N k BT 2 e IS T U anh T, 9 S T S v T S c T S harm T S anh T S c T, 10 where the configurational entropy S c (E IS ) has been defined as S c E IS k B ln E IS. 4 and S harm 6 N 3 1 N 6N 3 ln n T n 1 k B T, 11 In the thermodynamic limit, the free energy of the liquid can be calculated using F e IS T e IS T f T,e IS T TS c e IS T, where e IS (T), the average value of the IS energy at temperature T, is the solution of the saddle point equation 1 f E IS T S c E IS 0. The liquid free energy expression Eq. 5 has a clear interpretation. The first term in Eq. 5 takes into account the average energy of the PES minimum visited, the second term describes the volume of the corresponding basin of attraction and the kinetic energy, and the third term is a measure of the multiplicity of the basin. It can be rigorously shown 31,46,29 that, if the density of state (E IS ) is Gaussian, and if the basins have approximately the same shape or are, to a good degree, harmonic, the important relation holds, e IS T 1 T. On lowering T, basins with lower E IS energies and lower degeneracy are populated, i.e., both e IS and S c decrease with T where the frequencies n are the square root of the eigenvalues of the Hessian matrix calculated in the inherent structures. Thus, the total entropy is the sum of two contributions: S c (T) which accounts for the multiplicity of basins of depth e IS (T), and S v (T) which accounts for the volume of the basins. The last equations give us, in a very transparent way, the physical meaning of the partition of the PES; moreover, they provide us with a very efficient way to calculate the configurational entropy as a difference between the total energy of the system and the vibrational entropy. The total entropy S can be evaluated via thermodynamic integration, starting from a known reference point. Every variation of total entropy can be generally written as the sum of variation along isochores and isotherms in the form S S V S T. 12 Then the change of entropy along an isochore between two temperatures T and T is TdT S V S V,T S V,T T T c v T 13 and the change along an isotherm between two volumes V and V is S T S V,T S V,T III. EVALUATION OF THE CONFIGURATIONAL ENTROPY 1 T E V,T E V,T V V dv P V,T. 14 The Eq. 5 provides a natural starting point for a numerical evaluation of the configurational entropy. Indeed, the free energy F(T,V) per molecule can be split in the usual way as a sum of an energy and an entropic contribution. Considering Eq. 5 we write F T E T TS T TS c T e IS T E v T TS v T, where the index v indicates the vibrational quantities intra basin components. In order to evaluate these quantities we calculate the basin free energy as the free energy of 6N 3 independent harmonic oscillators 34 plus a contribution that takes into account the basin anharmonicities. Then we can write 8 In the present case, to evaluate the total entropy of the liquid we start from the known expression of the ideal gas of LW molecules, reviewed in the Appendix. To evaluate the basin free energy f T,e IS (T), we select as a reference point the free energy of (6N 3) independent harmonic oscillators whose distribution of frequencies can be calculated evaluating the eigenvalues of the Hessian matrix evaluated in the IS structure and add corrections to take into account the basin anharmonicities.the harmonic contribution to the entropy is given by Eq. 11. Assuming that the anharmonic contribution is independent from the basin depth, the anharmonic corrections to the entropy at T can be calculated integrating the quantity du anh /T, where U anh is implicitly defined in Eq. 9, from T 0 tot see Eq

51 S. MOSSA et al. PHYSICAL REVIEW E TABLE I. Densities, volumes, and simulation box lengths calculated. k k (g/cm 3 ) V k (nm 3 ) L k nm IV. NUMERICAL DETAILS The LW OTP molecule 43 is a rigid three-site planar isosceles triangle; the length of the two short sides of the triangle is nm and the angle between them is 5 /12 (75 ). Each site represents an entire phenyl ring of mass m 6m C 78 amu, where m C is the mass of the carbon atom. For each pair of interacting molecules, nine sitesite interactions are evaluated according to the site-site interaction potential V r 4 12 r r r, 15 where r is the site-site distance, kj/mol, nm kj/mol and kj/ mol nm. The parameters of the potential are selected to reproduce some bulk properties of the OTP molecule 43 such as the temperature dependence of the diffusion coefficient and the structure. The values of 1 and 2 are selected in such a way that the potential and its first derivative are zero at r c nm. Such a potential is characterized by a minimum at r nm of depth kj/mol. The integration time step is 0.01 ps. The shake algorithm is implemented to account for the molecular constraints. We study a (N,V,E) system composed by N 343 molecules 1029 LJ interaction sites at five different densities see Table I for several temperatures at each density Table II. The overall total simulation time, comprising thermalization and production runs at all the thermodynamic points investigated, exceeds 10 s. We carefully check the thermalization of the system at the lowest temperatures. The lengths of the thermalization runs cover a time interval during which each molecule has moved on average a few times. This time is calculated by monitoring the mean square displacement. We study also the time dependence of the intermediate scattering function F(Q M,t) QM (t) * QM (0) ; here Q M is the value of momentum Q locating the first maximum of the static structure factor S(Q). We confirm that this correlation function has decayed to zero during the equilibration time. Moreover, we ensure that no drift in the one-time quantities is observed during the production run. The lengths of the equilibration runs range from a few nanoseconds at the highest temperatures to several hundred nanoseconds at the lowest temperatures. We have been able to equilibrate the system in a T range over TABLE II. Temperatures in K for which calculations are performed which the diffusion constant changes from 10 6 to cm 2 /s, i.e., over four orders of magnitude. After the thermalization run, the production run takes place. The length of each run is always several times longer than the estimated relaxation time. This allows us to collect, for each thermodynamic point, a set of configurations which are to a good extent uncorrelated from one an other. Two additional simulations are performed to connect the range of densities and temperature studied with the ideal gas reference point. The system at density 4 is simulated for temperatures ranging from 280 to 5000 K to evaluate the T dependence of the potential energy. A second set of simulations at constant T (T 5000 K) in the volume range nm 3 is performed to calculate the excess pressure i.e., the pressure beyond the ideal gas contribution. To calculate the inherent structures visited in equilibrium we perform conjugate gradient minimizations to locate the closest local minima on the PES. We use a tolerance of kj/mol in the total energy for the minimization. For each thermodynamical point we minimize at least 100 configurations and we diagonalize the Hessian matrix of at least 50 configurations to calculate the density of states. The Hessian is calculated choosing for each molecule the center of mass and the angles associated with rotations around the three principal inertia axis as coordinates. Error bars have been calculated for all the simulation results points presented in the paper 47. Due to the length of the production runs, several times longer than the relaxation times, only configurations sufficiently uncorrelated have been used to calculate the different observables. The error bars have then been calculated using the standard relation for calculating errors. We show the error bars only when the amplitude of the error is larger than the size of the symbol used for the data point. V. RESULTS A. Dependence of the total entropy on T and To estimate the total entropy for the model we proceed in three steps as shown in Fig. 1. The thermodynamic path has been chosen to avoid the liquid-gas first order line

52 DYNAMICS AND CONFIGURATIONAL ENTROPY IN THE... PHYSICAL REVIEW E FIG. 1. Thermodynamic integration paths used to calculate the total entropy at the thermodynamical points of interest starting from the ideal noninteracting gas state. Details are given in the text. 1 Integration along the isotherm T K from (T 0,V ) perfect gas to (T 0,V nm 3 ), corresponding to point C 0 in Fig. 1. The ideal gas contribution to the total entropy is discussed in the Appendix. The entropy at C 0 can be calculated as S T 0,V 4 S id T 0,V 4 U T 0,V 4 V T 0 4dV P T ex V,T 0, 0 16 where P ex is the pressure that exceeds the pressure of the ideal gas, i.e., the contribution to the pressure due to the interaction potential and U is the system potential energy. The values of the pressure P ex (T T 0,V,N 343) as a function of V are reported in Fig. 2 a. P ex (T T 0,V,N 343) has been fit using the virial expansion P ex T T 0,V,N 343 a k V (k 1). k The a k values are reported in Table III, from which we estimate the first virial coefficient at T 0 FIG. 2. a Excess pressure at T 5000 K as a function of volume. The open circles are the MD results. The dashed line is the the first term of the virial expansion to the excess pressure; the solid line is a third order polynomial fit to the entire set of data. b Potential energy at T 5000 K as a function of volume. U C 0 T J/ mol K Integration along the isochore V V 4 from T 0 to T* 380 K, corresponding to point C 1 in Fig. 1. To evaluate the entropy along this isochore we use B 2 T 0 a 1 / k B T 0 N nm In Fig. 2 b we plot the potential energy as a function of volume along the T T 0 isotherm. The total entropy at the reference point C 0 is S(C 0 ) J/(mol K), resulting from the sum of three contributions S T*,V 4 S T 0,V 4 3Rlog T*/T 0 T*dT T0 T U V 4,T. 22 T TABLE III. Fitting coefficients for the excess pressure as a function of 1/V at T 5000 K and at T 380 K. S id C J/ mol K, 19 i a i MPa nm 3(i 1) ) p i * MPa nm 3(i 1) ) and V 4dV T 0 P ex V,T J/ mol K,

53 S. MOSSA et al. PHYSICAL REVIEW E TABLE IV. Total entropy at five densities for the reference temperature T*. k S(T*) J/ mol K S T*,V S T*,V 4 S id T*,V S id T*,V 4 1 T* U T*,V U T*,V 4 V dv V4 T* P ex T*,V. 27 Figures 3 b and 3 c show, respectively, the potential energy and the excess pressure as a function of volume at T T*. For convenience we fit P ex with a third order polynomial FIG. 3. a Integration step 2. Potential energy open circles at the density 4 in the entire temperature range considered; the solid line is the fit of the data to Eq. 23. The inset shows the lowest temperature region in order to stress the accuracy of the fit. b and c Integration step 3. Potential energy b and pressure c. Figure 3 a shows the potential energy for the V V 4 isochore. To calculate the integral in Eq. 22, we fit the potential energy using the functional form which best interpolates the calculated points P ex T*,V p k *V k 1, k where the values of the coefficients p k * are given in Table III. The resulting total entropy at T* for all studied densities is reported in Table IV. These values are used as reference entropies for the T dependence of S. For each of the studied isochores, we calculate the T dependence of the total entropy according to Eq. 22. In this low T range, the potential energy is very well represented by the Rosenfeld-Tarazona law 48 U V 4,T u 0 u 1 T 3/5 u 2 T, 23 U V,T U 0 V V T 3/5 29 obtaining the values u ,u ,u energy in kj/mol. The total entropy at the reference point C 1 is S(C 1 ) J/(mol K), resulting from the sum of three contributions: and S C J/ mol K, 3R log T/ J/ mol K, T*dT T0 T U V 4,T 52.5 J/ mol K. T Integration along the isotherm T* from V 4 to a generic V. To determine the total entropy difference for all studied densities we calculate consistent with what was found for LJ systems. In Fig. 4 we show the temperature dependence of the potential energy at all densities. The best-fit U 0 (V) and (V) values are reported in Table V. The calculated total entropies at each considered density are plotted in Fig. 5. TABLE V. The first two columns are the coefficients for the potential energy U(T,V) U 0 (V) (V)T 3/5 ; the second two columns are the coefficients for the inherent structures e IS (V,T) A(V) B(V)/T. k U 0 kj/mol (kjk 3/5 /mol) A kj/mol B kj T/mol

54 DYNAMICS AND CONFIGURATIONAL ENTROPY IN THE... PHYSICAL REVIEW E FIG. 4. Potential energies at the different densities as a function of T 3/5. The straight solid lines show the validity of the Rosenfeld- Tarazona law, Eq. 29. FIG. 6. Energies of the inherent structures at the different densities as a function of 1/T. The straight lines confirm the validity of Eq. 30 in the entire temperature range considered. B. Dependence of the inherent structure energies on T and In Fig. 6 we show the temperature dependence of the energy of the calculated inherent structures together with a fit according to Eq. 7 in the form e IS V,T A V B V T 30 The values of the fitting coefficients A(V) and B(V) are reported in Table V. On lowering temperature the system populates minima of lower and lower energy. It is worth noting that, in contrast to the case of the actual potential energy, the slope of these curves varies strongly with densities. From the T and V dependence of e IS the anharmonic potential energy can be calculated according to Eq. 3. Figure 7 shows U anh (T) for two densities symbols. We also show a cubic extrapolation solid lines in the form of U anh T c 2 T 2 c 3 T As shown in Fig. 7, the anharmonic contribution is rather small, in agreement with previous findings for the LJ model. For this reason, the low signal to noise level does not allow a well-defined characterization of the c 2 and c 3 values. To decrease the number of free parameters, we consider c 2 to be volume independent, and we fit simultaneously, according to Eq. 31, c 2 and the V dependence of c 3. As we will show in the following, the anharmonic contribution to the entropy is much smaller than the harmonic one and hence the choice of c 2 and c 3 does not affect significantly the resulting configurational entropy estimate. C. Density of states and vibrational harmonic entropy In this section we study the shape of the basins by investigating the properties of the density of states and we calculate the vibrational harmonic entropy. In Figs. 8 a and 8 b we show the temperature and density dependence of the density of state, namely the histogram of the square root of the eigenvalues of the Hessian calculated for the inherent structures. The distribution is characterized by only one peak, not showing any clear separation between translational and rota- FIG. 5. Temperature dependence of the total entropy as calculated by thermodynamic integration from the ideal gas reference state. Only points in the temperature range where MD measurements have been performed are shown. The reference temperature T* 380 K is also shown dashed line. FIG. 7. Anharmonic contributions to the energies, at the two indicated densities, together with the appropriate cubic fit, Eq. 31. This contribution is integrated to directly calculate the anharmonic contribution to the vibrational entropy

55 S. MOSSA et al. PHYSICAL REVIEW E FIG. 8. a Density of states at fixed density 4 at the three indicated temperatures. This quantity is the histogram of the square root of the eigenvalues of the Hessian calculated for the inherent structures. b Density dependence of the density of state at fixed temperature T 320 K. The dashed line indicates the isosbestic frequency * 44 cm 1 at which all the curves intersect. The relevance of this feature is discussed in the text. FIG. 9. a Temperature dependence of the average basin curvatures N 1 6N 3 k 1 log( k / 0 ); this quantity, being a sum of logarithms, is very sensitive to the spectrum tails. 0 1 cm 1 sets the frequency scale. b Relation between the energy of the inherent structures and the average basin curvatures. The straight lines confirm the correlation between shape and depth of the inherent structures accessed by the system. tional dynamics; the width of the distribution increases on increasing temperature. The position of the maximum is found to be to a good extent independent of temperature; at variance it increases with density as the width does. These features show that the LW PES basins have shapes that are function of the energy depth and of the density. It is worth noticing one particular feature of Fig. 8 b ; all the curves cross at a value of the frequency * 44 cm 1. The presence of this isosbestic frequency in analogy with the well-know isosbestic frequency observed in the Raman spectrum of water 49 supports the possibility that a twostate model 50 may provide a reasonable description of the change of the density of states with temperature and, correspondingly, of the change of the density of states with the basin depth. In Figs. 9 a and 9 b we plot the quantity 6N 3 log( k / 0 ) as a function of T and of the e IS, re- N 1 k 1 spectively. The scale frequency 0 is chosen as 1 cm 1. This quantity is an indicator of the average curvature of the basins and, being a sum of logarithms, is very sensitive to the spectrum tails. As shown in Fig. 9 a N 1 6N 3 k 1 log( k / 0 ) increases with temperature along isochores and increases with density along isotherms. As noted previously for the LJ 51,29 and for the simplepoint charge extended SPC/E model for water 30, the dependence of N 1 6N 3 k 1 log( k / 0 ) from e IS can be well approximated by a linear dependence, i.e., 1 N 6N 3 k 1 ln n T k B T 0 a V b V e IS T, 32 TABLE VI. Coefficients of the fit to the form N 1 6N 3 k 1 log( k / 0 ) a(v) b(v)e IS (T). k a(v) b(v) mol/kj

56 DYNAMICS AND CONFIGURATIONAL ENTROPY IN THE... PHYSICAL REVIEW E FIG. 10. Main panel: Harmonic contribution to the vibrational entropy as calculated from the eigenvalues of the Hessian for the inherent structures. Inset: Anharmonic contribution to the vibrational entropy as calculated by integration of the anharmonic contribution to the potential energy, as discussed in the text. FIG. 11. Volume and temperature dependence of the configurational entropy S c calculated as the difference between the total and the vibrational entropy. Solid lines are interpolations of the calculated points to Eq. 34. where T 0 defines the T scale (T 0 1 K). The values of the coefficients a(v) and b(v) are reported in Table VI. This dependence indicates that deeper and deeper basins have larger and larger volumes their average frequency being smaller. The fact that basins of different depths have different volumes introduces an important contribution to Eq. 6 since the term f / e IS is different from zero. The implication of this nonzero contribution has been discussed recently in Refs. 29,51,52. In Fig. 10 we show the harmonic contribution to the entropy as calculated from Eq. 11. This contribution is obviously increasing with temperature and along isotherms increases decreasing density. The lines are interpolations of the data using the fits of Fig. 9. D. Vibrational anharmonic entropy Integration of the anharmonic energy U anh, obtained from Eq. 9 according to Eq. 13, gives directly the anharmonic contribution to the entropy. For the LW case, U anh is described by the polynomial in T of Eq. 31, and we obtain lowering density, in agreement with the evidence that a glass transition may be induced along an isothermal path by progressively increasing the pressure. Considering Eqs. 10, 11, 30, 32, and 33, the configurational entropy can be described in the entire density and temperature range considered by means of the functional form S c T S T 6 3 N a V b V A V B V T 2c 2 T 3 2 c 3T These curves are plotted in Fig. 11 as solid lines. In the range of temperatures and density studied, S c /R per molecule varies from about 4 to 3, a figure not very different from the estimated configurational entropy of orthoterphenyl, based on an analysis of the T dependence of the measured specific heat 53,54. We recall that the LW model represents each S anh T 2c 2 T 3 2 c 3T The inset of Fig. 10 shows the anharmonic contribution to the vibrational entropy as calculated by integrating the anharmonic contribution to the potential energy. This contribution is negative showing that, in the range of densities and temperatures studied, the leading anharmonic contribution acts in the direction to decrease the volume of the basin. E. The configurational entropy In Fig. 11 we plot the configurational entropy calculated subtracting the vibrational sum of the harmonic and anharmonic terms from the total entropy for the five studied isochores. As expected the degeneracy of basins increases on FIG. 12. Temperature dependence of the different contributions to the total entropy closed triangles at the fixed selected density 4 : harmonic open squares, configurational entropy closed circles, and anharmonic open diamonds

57 S. MOSSA et al. PHYSICAL REVIEW E FIG. 14. MCT parameters as calculated from the diffusion constants. Main panel: Critical temperature T c (V) open circles together with the value calculated in Ref. 44 closed circle. The dashed line is only a guide for the eye. Inset: Power law exponent (V). The D values calculated are shown in Fig. 13. Figure 13 a shows the dependence on T, while Fig. 13 b shows the dependence on 1/T. Figure 13 a also shows the best fits to the power law D T T T c 36 FIG. 13. Diffusion constants together with the corresponding power law fits solid lines predicted by the MCT. The breakdown of this prediction and the crossover to an activated dynamics is evident. See text for a discussion of this point. a As a function of temperature. b As a function of the inverse temperature in order to stress the exponential dependence at the lowest temperatures. phenyl group as one single interaction site and it does not account for the the molecule flexibility. The similar estimate of S c seem to suggest that steric effects are dominant in controlling the configurational entropy. Finally, in Fig. 12 we plot the temperature dependence of all the contributions to the entropy at 4. F. Diffusion and the Adam-Gibbs relation In order to investigate the connection between the long time dynamics of the system and the underlying PES, we calculate the center-of-mass diffusion coefficient D(T) from the mean-square displacement r 2 (t,t) via the Einstein relation 1 D T lim 6t r2 t,t 35 t To guarantee a proper diffusive regime, at all densities simulations are performed until the average mean square displacement is greater than 0.1 nm 2 at the lowest temperatures and 10 nm 2 at the highest. The inverse of the diffusion coefficient provides an estimate of the characteristic structural relaxation time of the LW model. predicted by the ideal MCT in weakly supercooled states. The consistency of the MCT prediction for a wide range of D values confirms the analysis of Rinaldi et al. 44 where explicit ideal MCT calculations were presented and successfully compared with the numerical results along one isobar. Figure 13 shows also that clear deviations from the ideal MCT take place when the diffusion value becomes smaller than 10 8 cm 2 /s. The representation of D as a function of 1/T shown in Fig. 13 b shows that the ideal MCT region is followed by a T region where new types of processes become effective in controlling the molecular dynamics. These processes, termed hopping processes, transform the ideal MCT divergence of characteristic times into a crossover. In the region of D values between 10 8 cm 2 /s and cm 2 /s, limited from below by the present numerical resources, data are consistent with an apparent Arrhenius dependence with parameters which could well become T dependent if studied in a larger range of D values 3. The ideal MCT critical temperatures and values, determined by the fit of the D values to Eq. 36, as a function of density are shown in Fig. 14. The density dependence of T c is almost linear. The exponent seems to increase on increasing density, but the noise does not allow us to rule out the possibility of a constant value. The filled circle indicates the value of the critical temperature T c 265 K determined from an isobaric run in Ref. 44. We finally study the link between configurational entropy and diffusion coefficient, investigating the validity of the Adam-Gibbs equation. Figure 15 shows log D as a function of 1/(TS c ); for all studied isochores, log D vs 1/(TS c ) is well described by a linear relation, with coefficients which are

58 DYNAMICS AND CONFIGURATIONAL ENTROPY IN THE... PHYSICAL REVIEW E FIG. 15. Test of the Adam-Gibbs relation log D(T) (1/TS c ) for five different densities. Note that this linear relation holds both above and below the estimated critical temperatures T c. volume dependent, as previously found for the LJ liquid 29, for the SPC/E model for water 34 and for the BKS model for silica 32. We note in passing that deviations from linear behavior are observed at large values of log D, where intrabasin and interbasin dynamics time scales are no longer separated. At high T, it has been proposed 55 that entropy as opposed to configurational entropy is the relevant thermodynamic quantity controlling dynamics. VI. DISCUSSION AND CONCLUSIONS In this article we have studied systematically the properties of the potential energy surface for a simple three-site rigid model designed to mimic the properties of the fragile glass-forming liquid ortho-terphenyl. The choice of this simple model, which collapses the entire phenyl ring into one interaction site, allows us to run very long trajectories and to study in supercooled states the molecular dynamics up to 1 s, allowing the determination of diffusion coefficients down to cm 2 /s. We have found that, as in the atomic LJ case, by cooling along an isochore, basins of the PES of deeper and deeper energy are explored. The basin volumes are functions of the depth in agreement with previous studies. Using the inherent structure thermodynamic formalism, we have calculated the number of basins of the PES and their depth, in the region of depth values probed by our simulations. As a result, we presented a full characterization of the the temperature and density dependence of the basin depth, degeneracy, and volumes. These results are used to provide a consistent model for the intrabasin vibrational entropy. This, together with the numerical calculation of the total entropy via thermodynamic integration starting from the ideal gas state, allow us to calculate the configurational entropy the difference between the total entropy and the vibrational one. This quantity is of primary interest both for comparing with the recent theoretical calculations 13,36 and to examine some of the proposed relation between dynamics and thermodynamics 38,14,56 connecting a purely dynamical quantity like the diffusion coefficient to a purely thermodynamical quantity (S c ). To examine such a possibility we compare for five different isochores the T dependence of D with the Adam- Gibbs relation. In the entire range of T and densities studied the Adam-Gibbs relation appears to provide a consistent representation of the dynamics for the LW model. It is important to observe that a linear relation between log D and 1/(TS c ) holds both above and below the ideal MCT critical temperature T c, in agreement with a similar finding for the silica case 32. Recent works based on the instantaneous normal mode technique 57 for several representative models provide evidence that above T c the system is always located in a region of the PES close to the border between different basins. The number of diffusive directions significantly decreases above T c and, if only data above T c are considered, the number of diffusive directions would appear to vanish at T c. Hence dynamics above T c is a dynamics of borders between basins and there is no clear reason why such dynamics should be well described by the Adam-Gibbs relation, which focuses on the number of basins explored. The observed validity of the AG relation both above and below T c reported in this manuscript supports the hypothesis that a direct relation exists between the number of basins and their connectivity 60,62. It is a challenge for future studies to confirm or disprove this hypothesis. ACKNOWLEDGMENTS We thank W. Kob for very useful discussions. We also thank INFM-PRA-HOP, INFM-Iniziativa Calcolo Parallelo, MIURST-COFIN-2000, and NSF Chemistry Program. APPENDIX: IDEAL GAS ENTROPY FOR THE LW MODEL In this appendix we calculate the partition function of a system of N LW molecules in the noninteracting ideal gas case. The three moments of inertia for the single molecule are and I x 2 3 m 2 cos kg m 2, I y 2m 2 sin kg m 2, A1 I z m cos2 2 2 sin kg m 2. We define the following quantities: A 6 mk B h 2, R 8 2 k B I h 2, A2 where denotes x, y, orz. The translational and rotational partition functions for the single molecule are, respectively 63,

59 S. MOSSA et al. PHYSICAL REVIEW E Z T T,V V AT 3, Z R T,V 1 2 R xr y R z T 3, A3 A4 F id T,V,N k B T ln Z id T,V,N N 1 2 ln 2 ln V A 3 R x R y R z 3 lnt ln N 1 so the total partition function for an ideal gas of LW OTP molecules can be expressed as S id T,V,N 1 k B T F id T,V,N A6 Z id T,V,N Z TZ R N. A5 N! We approximate N! N N e N. The free energy F id and the entropy S id of the non-interacting system then become Nk B 4 1 ln ln 2 2 ln 3 V A 3 R x R y R z T A7 N where the term ln 2 is due to the two possible degenerate angular orientations of the molecule P.G. Debenedetti and F.H. Stillinger, Nature London 410, M. Mézard, in More is Different, edited by M. P. Ong and R. N. Bhatt Princeton University Press, Princeton, NJ, 2001 ; cond-mat/ G. Tarjus and D. Kivelson, e-print cond-mat/ P. G. Debenedetti, Metastable Liquids Princeton University Press, Princeton, NJ, G.Q. Shen, J. Toulouse, S. Beaufils, B. Bonello, Y.H. Hwang, P. Finkel, J. Hernandez, M. Bertault, M. Maglione, C. Ecolivet, and H.Z. Cummins, Phys. Rev. E 62, H.Z. Cummins, J. Phys.: Condens. Matter 11, A W. Götze, J. Phys.: Condens. Matter 11, A C.A. Angell, Science 267, R. Torre, P. Bartolini, and R.M. Pick, Phys. Rev. E 57, ; A. Taschin, R. Torre, M. A. Ricci, M. Sampoli, C. Dreyfus, and R.M. Pick, Europhys. Lett. 56, K. Binder et al., incomplex Behaviour of Glassy Systems, edited by M. Rubi and C. Perez-Vicente Springer-Verlag, Berlin, W. Götze, in Liquids, Freezing and the Glass Transition, edited by J. P. Hansen, D. Levesque, and J. Zinn-Justin North- Holland, Amsterdam, 1991 ; W.Götze and L. Sjörgen, Rep. Prog. Phys. 55, ; W.Götze, J. Phys.: Condens. Matter 11, A R. Schilling, in Disorder Effects on Relaxational Processes edited by A. Richert and A. Blumen Springer-Verlag, Berlin, 1994 ; W. Kob, in Experimental and Theoretical Approaches to Supercooled Liquids: Advances and Novel Applications, edited by J. Fourkas et al. ACS Books, Washington, D.C., M. Mézard and G. Parisi, Phys. Rev. Lett. 82, ; J. Phys.: Condens. Matter 12, X. Xia and P.G. Wolynes, Phys. Rev. Lett. 86, R. Speedy, J. Phys.: Condens. Matter 10, ; 9, ; 8, G. Hinze, David D. Brace, S.D. Gottke, and M.D. Fayer, Phys. Rev. Lett. 84, M. Kiebel, E. Bartsch, O. Debus, F. Fujara, W. Petry, and H. Sillescu, Phys. Rev. B 45, ;A.Tölle, H. Schober, J. Wuttke, and F. Fujara, Phys. Rev. E 56, ;A.Tölle, H. Schober, J. Wuttke, O.G. Randl, and F. Fujara, Phys. Rev. Lett. 80, ; G. Monaco, D. Fioretto, L. Comez, and G. Ruocco, Phys. Rev. E 63, J. Gapinski, W. Steffen, and A. Patkowski, J. Chem. Phys. 110, ; A. Aouadi, C. Dreyfus, and M. Massot, ibid. 112, K.L. Ngai, J. Chem. Phys. 110, ; R. Casalini, K.L. Ngai, and C.M. Roland, ibid. 112, ; J. Wuttke, M. Ohl, M. Goldammer, S. Roth, U. Schneider, P. Lunkenheimer, R. Kahn, B. Rufflé, R. Lechner, and M.A. Berg, Phys. Rev. E 61, ; M. Goldammer, C. Losert, J. Wuttke, W. Petry, F. Terki, H. Schober, and P. Lunkenheimer, ibid. 64, T. Gleim, W. Kob, and K. Binder, Phys. Rev. Lett. 81, L. Fabbian, A. Latz, R. Schilling, F. Sciortino, P. Tartaglia, and C. Theis, Phys. Rev. E 60, ; 62, ; C. Theis, F. Sciortino, A. Latz, R. Schilling, and P. Tartaglia, ibid. 62, F. Sciortino and W. Kob, Phys. Rev. Lett. 86, F.H. Stillinger and T.A. Weber, Phys. Rev. A 25, ; Science 225, ; F.H. Stillinger, ibid. 267, David J. Wales, Science 293, T.B. Schrøder, S. Sastry, J.C. Dyre, and S.C. Glotzer, J. Chem. Phys. 112, H. Fynewever, D. Perera, and P. Harrowell, J. Phys.: Condens. Matter 12, A S. Sastry, J. Phys.: Condens. Matter 12, P.S. Shah and C. Chakravarty, J. Chem. Phys. 115, S. Sastry, Nature London 409,

60 DYNAMICS AND CONFIGURATIONAL ENTROPY IN THE... PHYSICAL REVIEW E F.W. Starr, S. Sastry, E. La Nave, A. Scala, H.E. Stanley, and F. Sciortino, Phys. Rev. E 63, A. Heuer, Phys. Rev. Lett. 78, ; S.Büchner and A. Heuer, Phys. Rev. E 60, I. Saika-Voivod, P.H. Poole, and F. Sciortino, Nature London 412, F. Sciortino, W. Kob, and P. Tartaglia, Phys. Rev. Lett. 83, A. Scala, F.W. Starr, E. La Nave, F. Sciortino, and H.E. Stanley, Nature London 406, R.J. Speedy, J. Chem. Phys. 114, B. Coluzzi, G. Parisi, and P. Verrocchio, Phys. Rev. Lett. 84, ; B. Coluzzi and P. Verrocchio, J. Chem. Phys. 116, ; B. Coluzzi, M. Mézard, G. Parisi, and P. Verrocchio, ibid. 111, ; B. Coluzzi, G. Parisi, and P. Verrocchio, ibid. 112, A. Crisanti and F. Ritort, e-print cond-mat/ G. Adam and J.H. Gibbs, J. Chem. Phys. 43, W. Kob, F. Sciortino, and P. Tartaglia, Europhys. Lett. 49, S. Mossa, G. Ruocco, F. Sciortino, and P. Tartaglia, Philos. Mag. B to be published. 41 A. Scala and F. Sciortino, e-print cond-mat/ F. Sciortino and P. Tartaglia, J. Phys.: Condens. Matter 13, G. Wahnström and L.J. Lewis, Physica A 201, ; L.J. Lewis and G. Wahnström, Solid State Commun. 86, ; J. Non-Cryst. Solids , ; Phys. Rev. E 50, ; G. Wahnström and L.J. Lewis, Suppl. Prog. Theor. Phys. 126, A. Rinaldi, F. Sciortino, and P. Tartaglia, Phys. Rev. E 63, S. Mossa, R. Di Leonardo, G. Ruocco, and M. Sampoli, Phys. Rev. E 62, ; S. Mossa, G. Ruocco, and M. Sampoli, ibid. 64, ; S. Mossa, G. Monaco, and G. Ruocco, e-print cond-mat/ ; S. Mossa, G. Monaco, G. Ruocco, M. Sampoli, and F. Sette, J. Chem. Phys. 116, ; S. Mossa, G. Ruocco, and M. Sampoli, e-print cond-mat/ F. Sciortino, W. Kob, and P. Tartaglia, J. Phys.: Condens. Matter 12, In this article, all the solid lines found in the figures should be considered as interpolations of the calculated data points as in the case of the thermodynamic integration section or as a support to the interpretation of our results in the framework set by theoretical approaches as in the case of the fit of the diffusion coefficients according to the MCT. 48 Y. Rosenfeld and P. Tarazona, Mol. Phys. 95, G.E. Walrafen, M.S. Hokmabadi, and W.H. Yang, J. Chem. Phys. 85, ; see also P. Benassi, V. Mazzacurati, M. Nardone, M. A. Ricci, G. Ruocco, and G. Signorelli, ibid. 88, C.A. Angell, B.E. Richards, and V. Velikov, J. Phys.: Condens. Matter 11, A F. Sciortino and P. Tartaglia, Phys. Rev. Lett. 86, L. M. Martinez and C.A. Angell, Nature London 410, F.H. Stillinger, J. Phys. Chem. B 102, R. Richert and C.A. Angell, J. Chem. Phys. 108, M. Dzugutov, Nature London 381, ; J. Phys.: Condens. Matter 11, A M. Schulz, Phys. Rev. B 57, T. Keyes, J. Phys. Chem. 101, F. Sciortino and P. Tartaglia, Phys. Rev. Lett. 78, C. Donati, F. Sciortino, and P. Tartaglia, Phys. Rev. Lett. 85, E. La Nave, A. Scala, F.W. Starr, F. Sciortino, and H.E. Stanley, Phys. Rev. Lett. 84, E. La Nave, A. Scala, F.W. Starr, H.E. Stanley, and F. Sciortino, Phys. Rev. E 64, E. La Nave, H.E. Stanley, and F. Sciortino, Phys. Rev. Lett. 88, J. E. Mayer and M. G. Mayer, Statistical Mechanics John Wiley & Sons, New York,

61 PHILOSOPHICAL MAGAZINE B, 2002, VOL. 82, NO. 6, 695±705 Quenches and crunches: does the system explore in ageing the same part of the con guration space explored in equilibrium? Stefano Mossayz }, Giancarlo Ruoccoz, Francesco Sciortinoz and Piero Tartagli az y Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts 02215, USA z Dipartimento di Fisica and Istituto Nazionale per la Fisica della Materia, Universita di Roma La Sapienza, Piazzale Aldo Moro 2, I Roma, Italy Abstract Numerical studies are providing novel information on the physical processes associated with physical ageing. The process of ageing has been shown to consist of a slow process of explorations of deeper and deeper minima of the system s potential energy surface. In this article we compare the properties of the basins explored in equilibrium with those explored during the ageing process both for sudden temperature changes and for sudden density changes. We nd that the hypothesis that during the ageing process the system explores the part of the con guration space explored in equilibrium holds only for shallow quenches or for the early ageing dynamics. At longer times, systematic deviations are observed. In the case of crunches, such deviations are much more apparent. } 1. Introduction At the glass transition temperature T g (Debenedetti 1997), the characteristic relaxation time of a liquid becomes of the same order of the experimental time, thus preventing equilibrium studies at lower temperatures. Material properties below T g depend on the previous history (i.e. on the preparation technique, on the cooling and compression rates and so on) as well as on the time spent in the glass state. This time dependence, generically known as physical ageing, highlights the out-of-equilibrium condition of glasses and their extremely slow equilibration processes. A substantial amount of work has been devoted to the understanding and to the formal description of the supercooled liquid dynamics (Cummins 1999, GoÈ tze 1999), of the physics beyond the glass transition (Angell 1995) and of physical ageing (Cugliandolo et al. 1996, Bouchaud et al. 1998, Kurchan 2001, Latz 2000, 2001). Numerical simulations of supercooled states, both in equilibrium and in controlled out-of-equilibrium conditions, have played an important role in the present developments. For a recent review on the contribution of numerical simulation to the glass transition phenomenon see for example Kob (1999). For out-of-equilibrium studies see also Andrejew and Baschnagel (1996) and Wahlen and Rieger (2000). Although the time scales probed by numerical simulations are very diœerent from } mossa@argento.bu.edu Philosophical Magazine B ISSN 1364±2812 print/issn 1463±6417 online # 2002 Taylor & Francis Ltd DOI: /

62 696 S. Mossa et al. experimental scales (100 ns compared with seconds), the numerical `experiments appear to be able to reproduce features of real materials (Utz et al. 2000, Barrat and Berthier 2001). The description of the ageing dynamics as motion in con guration space has been very fruitful (Kob et al. 2000). It has been shown that, during ageing, the system explores deeper and deeper basins of the potential energy surface (PES). The search for deeper basins during ageing resembles the exploration of deeper and deeper basins which takes place in equilibrium on cooling. This similarity has been interpreted in terms of a decrease in the internal con gurational temperature T of the system under ageing (Kob et al. 2000). An analysis of the curvatures of the PES basins supported the possibility that during the ageing process the system visits the same type of minima as that visited in equilibrium. This analysis supported also the possibility of a thermodynami c description of the ageing system and a prediction of the internal con gurational temperature in full agreement with the numerical estimates (Sciortino and Tartaglia 2001a). The outcome of these studies, which are still far from being settled, suggests that the out-of-equilibrium glassy state, notwithstanding its out-of-equilibrium condition, can be uniquely determined by its kinetic temperature, its volume and the properties of the basin in which the system is trapped. These quantities allow us to develop a thermodynamic description of the glass state (Davies and Jones 1953, Nieuwenhuizen 1997, 1998, Speedy 1998, 1999, Cardenas et al. 1999, Me zard and Parisi 1999, Sciortino et al. 1999) which is currently being tested in experiments (Grigera and IsraeloŒ 1999, Knaebel et al. 2000) and further simulations (Sciortino and Tartaglia 2001b). From an experimental point of view, ageing experiments are performed at constant pressure. Moreover, often a change in pressure (and hence in density) is used to bring the system from an equilibrium to an out-of-equilibrium state. The change in density produces a much more dramatic change in the PES than a change in temperature does. This opens the possibility that the hypothesis of out-of-equilibrium dynamics as a succession of quasi-equilibrium states may not apply to the crunch experiments and hence that a more re ned thermodynami c approach is required for glass materials produced with a crunch route. In this paper we revisit the hypothesis that during the ageing process the system explores the same part of the con guration space by performing a more accurate analysis of the basin depth as a function of the basin curvature, made nowadays possible by the increased computational facilities. We also present a comparison of this relation for diœerent quenching temperatures and for a crunch (Di Leonardo et al. 2000, Angelani et al. 2001). } 2. The models We consider two models: an atomic model and a molecular model. The rst microscopic model that we consider is a binary (80 : 20) mixture of Lennard-Jones particles (BMLJ), which in the following we shall call type A and type B particles. The interaction between two particles of type a and b, with a; b 2 fa; Bg, is given by V ab ˆ 4 ab ¼ ab =r 12 ¼ ab =r 6 Š. The parameters ab and ¼ ab are given by AA ˆ 1:0, ¼ AA ˆ 1:0, AB ˆ 1:5, ¼ AB ˆ 0:8, BB ˆ 0:5 and ¼ BB ˆ 0:88. The potential is truncated and shifted at r cut ˆ 2:5¼ ab. ¼ AA and AA are chosen as the unit of length and energy respectively (setting the Boltzmann constant k B ˆ 1:0). Time is measured in units of m¼ 2 AA=48 AA 1=2, where m is the

63 Quenches and crunches 697 mass of the particles particles were placed in a box of side 9.4. The studied isochore has a density of 1.2. To study the quenches, we have equilibrated several independent con gurations at T ˆ T i in the NVT ensemble (Nose ±Hoover thermostat). Each of the con gurations has been quenched to T f by changing at t ˆ 0 the thermostat temperature to T f. The thermostat constant is chosen in such a way that, within 1000 molecular dynamics (MD) steps, the average kinetic energy thermalizes to T f. To study the crunches, we have equilibrated 280 independent con gurations at T ˆ 0:35 in the NVT ensemble at a density of The coordinates of each atom have been rescaled by at t ˆ 0 to simulate a 10% density increase. In this way, the nal density of the crunches coincides with the density of the quenches. Inherent structures of a system are calculated by conjugate gradient algorithms. The minimization procedure is iterated until the energy change is less than The density of states is then calculated by diagonalizing the corresponding Hessian matrix. The second model that we consider is a simple three-site molecular model, introduced by Lewis and WahnstroÈ m (LW) (1993, 1994a,b) and WahnstroÈ m and Lewis (1993, 1997). The model is constructed by glueing in a rigid molecule three identical Lennard-Jones (LJ) atoms. The shape of the molecule (an isosceles triangle) and the LJ parameters were chosen to mimic as closely as possible one of the most studied glass-forming liquids, o-terphenyl. The slow dynamics of this model have been recently revisited (Rinaldi et al. 2001) in great detail. For this model we present preliminary results for a quench, starting from T ˆ 480 K down to T ˆ 280 K at a constant density of kg m 3. The simulated system is composed of 343 molecules. The integration time step is 0.01 ps and the ageing dynamics are followed up to 1 ns. Averages over more than 50 independent realizations are presented. } 3. Binary (80 : 20) MIXTURE OF LENNARD-JONES PARTICLES 3.1. Energies Following a quench or a crunch, the system nds itself in a region of con guration space which is not explored in equilibrium under the externally imposed temperature and volume. The motion of the system in con guration space evolves in the attempt to reach the typical equilibrium con guration. A clear indicator of this evolution is the time evolution of the potential energy. In the PES paradigm, the potential energy of the system can be expressed as E ˆ e IS E vib, where e IS is the inherent structure (IS) energy and E vib describes the thermal excitations about the IS (harmonic plus anharmoni c vibrations). While the vibrational energy does not show a signi cant ageing dependence, the time evolution of e IS provides a clear indication of the search for deeper and deeper potential energy minima, as shown in gure l Curvatures of the basins In equilibrium, below T ˆ 1, the system start to explore deeper and deeper basins, whose shapes are depth dependent. A simple way to characterize the shape of the PES basins is to calculate the density of states in a harmonic approximation, expanding the potential energy around the local minimum con guration. The resulting distribution of frequencies characterizes, in a harmonic approximation, the volume in con guration space associated with the basin. The density of states allows one to estimate the vibrational free energy f basin, in a harmonic approximation, since

64 698 S. Mossa et al. (a) (b) (c) Figure 1. IS energy as a function of T (a) in equilibrium and (b), (c) as a function of waiting time t w following (b) a quench or (c) a crunch. In (b), T f ˆ 0:25 and T i ˆ 5, 0.8, 0.6, 0.55 and 0.5 from top to bottom. f basin T ˆ kbt 3N 3 log! N i : 1 iˆ1 A T-independent indicator of the basin shape can be de ned as S 3N 3 iˆ1 log! i = AA. Studies on several models of liquids shows that along isochoric paths, the depth dependence of the average curvature of the basins is model and density dependent. In the case of the BMLJ at the studied density, the average curvature of the basins becomes wider and wider on moving to deeper and deeper basins, as shown in gure 2. Figure 2 also shows S e IS ) during the ageing dynamics following quenches and crunches, from diœerent initial conditions. The important observation is that during temperature changes, there is an initial part of the ageing process where the system explores in ageing the same set of basins explored in equilibrium. For longer times, deviations start to take place and the system nds itself located in the region of con guration space which is not commonly explored in equilibrium conditions. The eœect is negligible (and indeed it was not noted in previous studies) at su ciently high T f values (shallow quenches) but becomes detectable in deeper quenches. In future studies, it will become important to correlate the time at which the ageing dynamics signi cantly separates from the equilibrium dynamics with the diœerent mechanism of exploration of con guration spaces which characterize also the dynamics of the system in equilibrium (i.e. saddle-dominated dynamics compared with activated dynamics (Sciortino and Tartaglia 1997, Angelani et al. 2000, Bhattacharya et al. 2000, La Nave et al. 2000, 2001)). A related interesting question is the validity of the description of the ageing system as composed of two systems in quasi-equilibrium at diœerent temperatures (Sciortino and Tartaglia 2001a,b) in

65 Quenches and crunches 699 Figure 2. (a) (b) 3N 3 Relation between S iˆ1 log! i = AA and e IS. (a) Data from Kob et al. (2000) and (b) new data from crunch and quench ageing. conditions where the ageing system explores basins which are not populated in equilibrium. In the crunch case, in the time window accessible to numerical simulations the system never explores regions of con guration space which are visited in equilibrium. The starting con guration is diœerent from any con guration visited in equilibrium. In this respect, the ensemble of con guration of the system after an initial crunch cannot be identi ed with an equilibrium ensemble at a diœerent temperature, not even at in nite temperature. Having said this, we call the reader s attention to the fact that the diœerences observed in S are smaller than the entire variation in S with ageing. We also recall that the time window accessible to numerical experiment is such that the system, even after the longest waiting time simulated, has not completely forgotten the initial con guration (or, in other words, the correlation functions never decays to zero in the simulated time period). In this respect, it is not that surprising that the basins explored in crunches are diœerent from the basins explored in ageing and in equilibrium. Finally we note that S, being the sum of logarithms, is driven by the lowfrequency values. The small frequencies are the most aœected by size eœects and by arti cial localization of the eigenmodes. Future studies should focus on the size eœect in the data reported in gure 2. We note in passing that some model potentials, for example the Van Beast, Kramer and van Santen (BKS) model for silica, do not show along an isochoric path any dependence of the basin shape on the potential depth (Saika-Voivod et al. 2001). This class of models (and their corresponding materials) may provide simpler ageing and crunching dynamics compared with the BMLJ model here studied.

66 700 S. Mossa et al Structure of the liquid in the inherent structures The IS energy is, by de nition, the integral of the pair potential over r, weighted by the radial distribution function g r. The changes in IS energy on cooling re ect changes in the local structure. In this section we look at these diœerences, with the aim of characterizing in a microscopic way the diœerences in states with the same IS but diœerent densities of states. Figure 3 shows the AB radial distribution function in equilibrium as evaluated in the IS con gurations, that is once the thermal distortion has been subtracted. The (a) (b) (c) Figure 3. Radial distribution function for the AB atoms for eight diœerent temperatures ranging from 2 to 0.446, evaluated in the IS con gurations. (b), (c) Enlargements of two diœerent r regions. A signi cant enlargement is required to highlight the very subtle changes accompanying the population of basins of deeper and deeper IS energy. Figure 4. Dependence on e IS of g AB (r ˆ 0:8675) in equilibrium and during ageing (crunch and quench). Note that states with same e IS value are realized with diœerent pair distribution functions.

67 Quenches and crunches 701 structure of the liquid changes in a very minor way and can be visualized only with a very ne resolution. The net eœect of cooling on g AB r appears as a shift of less than in the average interatomic distance. To emphasize the temperature changes we show in gure 4 the relation between the basin potential energy and g r for a xed r value. The data con rm that, under crunch, the local liquid order is diœerent from the equilibrium order. Figure 5 shows the enlargements of g r in equilibrium and contrasts them with those during ageing at the same e IS value. Again, we note that the quench con guration is closer to the equilibrium con guration than to the crunch con guration. } 4. The Lewis± Wahnström model In the case of the LW model for o-terphenyl (Lewis and WahnstroÈ m 1993, 1994a,b, WahnstroÈ m and Lewis 1993, 1997), all results refer to a quench case Energies The temperature dependence of the IS energy for the LW model is shown in gure 6, together with the time evolution under ageing. As for the BMLJ model, the ageing dynamics are characterized by a slow progressive reduction in the IS energy Curvatures of the basins As in the BMLJ case, the basin curvature is correlated with the basin depth. The temperature dependence of the average frequency in equilibrium and the t w, dependence in ageing are shown in gure 7. The frequency decreases on cooling or on ageing as in the BMLJ case. The e IS dependences of the local curvatures in equilibrium and in ageing are shown in gure 8. We note that for the case of the LW potential, in the time window (a) (b) Figure 5. Equilibrium, crunch and quench g AB at the same e IS values. To maximize the diœerences, the equilibrium g AB r at T ˆ 2:0 has been subtracted for all curves.

68 702 S. Mossa et al. (a) (b) Figure 6. (a) Equilibrium temperature dependence of the IS energy for the LW model and (b) the IS energy as a function of time during a quench from T ˆ 480 K to T ˆ 280 K. (a) Figure 7. (a) Equilibrium temperature dependence of the average frequency for the LW model and (b) the average frequency as a function of time during a quench from T ˆ 480 K to T ˆ 280 K. (b)

69 Quenches and crunches 703 Figure 8. Relation between e IS and the average basin curvature for the LW potential, in equilibrium and in ageing (! 0 ˆ 1 cm 1 ). explored by the numerical simulation and for the chosen T f, the basins explored during ageing coincide with the basins explored in equilibrium. } 5. Conclusions In this paper we have compared the properties of con guration space explored in equilibrium and in out-of-equilibrium conditions, with the aim of deepening our understanding of the physical mechanisms behind the ageing process in disordered materials. A careful analysis of the relations between curvature and depth of the potential energy basins reveals that basins which are not statistically explored in equilibrium are visited during the ageing dynamics, especially during the dynamics following a crunch. In quenches, such conditions appear to hold also in the case of a deep quench depth. At long times, the hypothesis that during the ageing process the system explores the same part of the con guration space does not seem to hold any longer, at least on the scale of present-day numerical calculations. The diœerences in the basins appear to be located in the region of very small frequencies and may not be clearly seen if the average frequency is used as indicator of the basin curvature. We have shown here that the average of the logarithm of the frequency (a quantity which weighs more the very-low-frequency spectrum) is indeed a better indicator. This quantity is important since it quanti es the vibrational free energy of the basin. On the other hand, the very-low-frequency part of the spectrum is very sensitive to size eœects. It could be that spurious localized modes are stabilized by the limited size of the simulated system. This calls for a size-dependence detailed analysis of the relation between curvature and depth in equilibrium and in ageing.

70 704 S. Mossa et al. Finally, we recall that the hypothesis that the parts of con guration space explored in ageing and in equilibrium are similar is an important element to be able to predict the value of the internal temperature of the system and the associated response of the ageing system to an external perturbation. Indeed, in the case where such detailed comparison was performed, the basin shape and curvature in ageing and equilibrium do coincide. Unfortunately, no evaluation of the internal temperature has been performed yet for t w values where the possibility of predicting the value of T int should fail. It would be very interesting to perform such an accurate study (i.e. the comparison between predictions and numerical calculations) in the near future, in particular for crunches, where the breaking of the assumption is apparent even at short waiting times. ACKNOWLEDGEMENTS We are supported by Istituto Nazionale per la Fisica della Materia Progetto Ricerca Avanzato±Hopping, Istituto Nazionale per la Fisica della Materia Parallel Computing Initiative and Ministero dell UniversitaÁ e della Ricerca Scienti ca e Tecnologica (COFIN2000). References Andrejew, E., and Baschnagel, J., 1996, Physica A, 233, 117. Angelani, L., Di Leonardo, R., Parisi, G., and Ruocco, G., 2001, Phys. Rev. Lett., 87, Angelani, L., Ruocco, G., Scala, A., and Sciortino, F., 2000, Phys. Rev. Lett., 85, Angell, C. A., 1995, Science, 267, Barrat, J. L., and Berthier, L., 2001, Phys. Rev., E, 63, Bouchaud, J. P., Cugliandolo, L. F., Kurchan, J., and Me zard, M., 1998, Spin Glasses and Random Fields, edited by A. P. Young (Singapore: World Scienti c), p Bhattacharya, K., Cavagna, A., Zippelius, A., and Giardina, I., 2000, Phys. Rev. Lett., 85, Cardenas, M., Franz, S., and Parisi, G., 1999, J. chem. Phys., 110, Cugliandolo, L. F., Kurchan, J., and Le Doussal, P., 1996, Phys. Rev. Lett., 76, Cummins, H., 1999, J. Phys.: condens. Matter, 11, A95. Davies, R. O., and Jones, G. O., 1953, Adv. Phys., 2, 370. Debenedetti, P. G., 1997, Metastable Liquids (Princeton University Press). Di Leonardo, R., Angelani, L., Parisi, G., and Ruocco, G., 2000, Phys. Rev. Lett., 84, Götze, W., 1999, J. Phys.: condens. Matter, 11, A1. Grigera, T. S., and Israeloff, N. E., 1999, Phys. Rev. Lett., 83, Knaebel, A., Bellour, M., Munch, J. P., Viasnoff, V., Lequeux, F., and Harden, J. L., 2000, Europhys. Lett., 52, 73. Kob, W., 1999, J. Phys.: condens. Matter, 11, R85. Kob, W., Sciortino, F., and Tartaglia, P., 2000, Europhys. Lett., 49, 590. Kurchan, J., 2001, C.r., hebd. SeÂanc. Acad. Sci., Paris (to be published). La Nave, E., Scala, A., Starr, F. W., Sciortino, F., and Stanley, H. E., 2000, Phys. Rev. Lett., 84, La Nave, E., Scala, A., Starr, F. W., Stanley, H. E., and Sciartino, F., 2001, Phys. Rev. E, 64, Latz, A., 2000, J. Phys.: condens. Matter, 12, 6353; 2001, condmat/ Lewis, L. J., and Wahnström, G., 1993, Solid St. Commun., 86, 295; 1994a, J. non-crystalline Solids, 172±174, 69; 1994b, Phys. Rev. E, 50, Me zard, M., and Parisi, G., 1999, J. Phys.: condens. Matter, 11, A157. Nieuwenhuizen, Th. M., 1997, Phys. Rev. Lett., 79, 1317; 1998, ibid., 80, Rinaldi, A., Sciortino, F., and Tartaglia, P., 2001, Phys. Rev. E, 63, Saika-Voivod, I., Poole, P., and Sciortino, F., 2001, Nature, 412, 514.

71 Quenches and crunches 705 Sciortino, F., Kob, W., and Tartaglia, P., 1999, Phys. Rev. Lett., 83, Sciortino, F., and Tartaglia, P., 1997, Phys. Rev. Lett., 78, 2385; 2001a, ibid., 86, 107; 2001b, J. Phys.: condens. Matter, (to be published). Speedy, R., 1998, J. Phys.: condens. Matter, 10, 4185; 1999, J. phys. Chem. B, 103, Utz, M., Debenedetti, P. G., and Stillinger, F. H., 2000, Phys. Rev. Lett., 84, Wahlen, H., and Rieger, H., 2000, J. phys. Soc. Japan, Suppl. A, 69, 242. Wahnström, G., and Lewis, L. J., 1993, Physica A, 201, 150; 1997, Prog. theor. Phys. Osaka, Suppl., 126, 261.

72 VOLUME 88, NUMBER 19 P H Y S I C A L R E V I E W L E T T E R S 13 MAY 2002 Interplay between Time-Temperature Transformation and the Liquid-Liquid Phase Transition in Water Masako Yamada, 1 Stefano Mossa, 1,2 H. Eugene Stanley, 1 and Francesco Sciortino 2 1 Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts Dipartimento di Fisica, INFM UdR and INFM Center for Statistical Mechanics and Complexity, Università di Roma La Sapienza, Piazzale Aldo Moro 2, I-00185, Roma, Italy (Received 6 February 2002; published 26 April 2002) We study the new water model proposed by Mahoney and Jorgensen [J. Chem. Phys. 112, 8910 (2000)], which is closer to real water than previously proposed classical pairwise additive potentials. We simulate the model in a wide range of deeply supercooled states and find (i) the existence of a nonmonotonic nose-shaped temperature of maximum density line and a nonreentrant spinodal, (ii) the presence of a low-temperature phase transition, (iii) the free evolution of bulk water to ice, and (iv) the time-temperature-transformation curves at different densities. DOI: /PhysRevLett Much effort has been invested in exploring the overall phase diagram of water and the connection among its liquid, supercooled, and glassy states [1 3], with particular interest in understanding the origin of the striking anomalies at low temperatures, such as the T dependence of the isothermal compressibility K T, the constant pressure specific heat C P, and the thermal expansivity a P. The stability limit conjecture attributes the increase of the response functions upon supercooling to a continuous retracing spinodal line bounding the superheated, supercooled, and stretched (negative pressure) metastable states [4]. This line at its minimum intersects the temperature of maximum density (TMD) curve tangentially. More recently, a different hypothesis has been developed, for which the spinodal does not reenter into the positive pressure region, but rather the anomalies are attributed to a critical point below the homogeneous nucleation line [5]. The TMD line, which is negatively sloped at positive pressures, becomes positively sloped at sufficiently negative pressures and does not intersect the spinodal. A line of first order phase transitions interpreted as the liquid state analog of the line separating low and high density amorphous glassy phases [3,5] develops from this critical point. Simulations of supercooled metastable states are possible because the structural relaxation time at the temperatures of interest is several orders of magnitude shorter than the crystallization time. It is difficult, but not impossible [6], to observe crystallization in simulations of molecular models [7] because homogeneous nucleation rarely occurs on the time scales reachable by present-day computers. Bulk water simulations have been crystallized by applying a homogeneous electric field [8] or placing liquid water in contact with preexisting ice [9,10], but spontaneous crystallization of deeply supercooled model water has not been observed in simulations. In contrast, experimental measurements of metastable liquid states are strongly affected by homogeneous nucleation. The nucleation and growth of ice particles from PACS numbers: Ja, Ns aqueous solution has been extensively studied, and the nose-shaped time-temperature-transformation (TTT) curves have been measured [1,11,12]. The nonmonotonic relation between crystallization rate and supercooling depth results from the competition between the thermodynamic driving force for nucleation and the viscous slowing down [1]. Crystallization hinders direct experimental investigation of pure metastable liquid water below the homogeneous nucleation line; only indirect measurements can be made by studying the metastable melting lines of ices [3]. This work attempts to unify the phenomena connected with the existence of a liquid-liquid phase transition and homogeneous nucleation in a single molecular dynamics simulation study. We simulate a system of N 343 molecules interacting with the transferable intermolecular potential with five points (TIP5P) [13]. TIP5P is a five-point, rigid, nonpolarizable water model, not unlike the five-point ST2 model [14]. The TIP5P potential accurately reproduces the density anomaly at 1 atm and exhibits excellent structural properties when compared with experimental data [13,15]. The TMD shows the correct pressure dependence, shifting to lower temperatures as pressure is increased. Under ambient conditions, the diffusion constant is close to the experimental value, with reasonable temperature and pressure dependence away from ambient conditions [13]. We perform equilibration runs at constant T (Berendsen s thermostat), while we perform production runs in the microcanonical (NVE where N is the number of particles, V is volume, and E is energy) ensemble. After thermalization at T 320 K we set the thermostat temperature to the temperature of interest. We let the system evolve for a time longer than the oxygen structural relaxation time t a, defined as the time at which F s Q 0, t a 1 e, where F s Q 0, t is the self-intermediate scattering function evaluated at Q 0 18 nm 21, the location of the first peak of the static structure factor. In the time t a, each molecule diffuses on average a distance of the order of the nearest (19) (4)$ The American Physical Society

73 VOLUME 88, NUMBER 19 P H Y S I C A L R E V I E W L E T T E R S 13 MAY 2002 neighbor distance. We use the final configuration of the equilibration run to start a production run of length greater than several t a and then analyze the calculated trajectory. We check that no drift in any of the studied quantities and no crystallization occurs during the production run. In Fig. 1 we show results for pressure along isotherms. At lower temperatures an inflection develops, which becomes a flat isotherm at the lowest temperature, T 215 K. The presence of a flat region indicates that a phase separation takes place, and we estimate the critical temperature T C K, the critical pressure P C MPa, and the critical density r C g cm 3. In Fig. 2(a) we plot the pressure along isochores. The curves show minima as a function of temperature; the locus of the minima is the TMD line, since P T V a P K T. It can be seen that the pressure exhibits a minimum if the density passes through a maximum a P 0. It is clear that, as in the case of ST2 water, TIP5P water has a TMD that changes slope from negative to positive as P decreases. Notably, the point of crossover between the two behaviors is located at ambient pressure, T 4 ± C, and r 1 g cm 3. We also plot the spinodal line. We calculate the points on the spinodal line fitting the isotherms (for T $ 300 K) of Fig. 1 to the form P T, r P s T 1 A r 2r s T 2, where P s T and r s T denote the pressure and density of Pressure P [ MPa ] T = 215 K ρ n = 10 n = P T = 320 K T = 215 K n = Density ρ [ g / cm 3 ] FIG. 1. Dependence on density of the pressure at all temperatures investigated (T 215, 220, 230, 240, 250, 260, 270, 280, 290, 300, and 320 K, from bottom to top). Each curve has been shifted by n MPa to avoid overlaps. An inflection appears as T is decreased, transforming into a flat coexistence region at T 215 K, indicating the presence of a liquid-liquid transition. Inset: A detailed view of the T 215 K isotherm. the spinodal line. This functional form is the mean field prediction for P r close to a spinodal line. For T # 250 K, we calculate P s T by estimating the location of the minimum of P r. The results in Fig. 2 show that the liquid spinodal line is not reentrant and does not intersect the TMD line [16]. A supercooled liquid is metastable with respect to the crystal, so it is driven to crystallize [1]. However, crystallization of model water has not been found in simulations because the homogeneous nucleation time far exceeds the CPU time. Fortunately, for TIP5P, crystallization times lie within a time window accessible to present-day simulations, and we observe crystallization at densities r 1.15 and 1.20 g cm 3 for a wide range of temperatures (Fig. 3). To quantify the crystallization process, we analyze four independent configurations thermalized at temperature T 320 K and instantaneously quenched to the temperature of interest. We monitor the potential energy as well as the time evolution of the structure factor S Q t r Q t r Q t N at all wave vector Q values, ranging Pressure [ MPa ] Pressure [ MPa ] TMD line Spinodal ( a ) Temperature [ K ] C TMD line liquid spinodal liquid gas critical point liquid liquid critical point 200 ( b ) Temperature [ K ] FIG. 2. (a) Pressure along seven isochores; the minima correspond to the temperature of maximum density line (dashed line). Note the nose of the TMD line at T 4 ± C. Stars denote the liquid spinodal line, which is not reentrant and terminates at the liquid-gas critical point. (b) The full phase diagram of TIP5P water. The liquid-gas critical point C is indicated by the filled square [16] and the liquid-liquid critical point C 0 by the filled circle. C

74 VOLUME 88, NUMBER 19 P H Y S I C A L R E V I E W L E T T E R S 13 MAY 2002 Temperature [ K ] τ cryst τ α ( a ) ρ = 1.15 g / cm 3 ( τ n, T n ) ( b ) ρ = 1.20 g / cm Time [ ps ] FIG. 3. Average crystallization time (open circles) as a function of temperature at the two densities (a) 1.15 g cm 3 and (b) 1.20 g cm 3. A well-defined nose shape is visible, as measured for water solutions [11]. We also show the structural relaxation times t a as calculated from the self-intermediate scattering function F s Q, t (closed circles). Shown as solid lines are the mode-coupling theory power law fits with T c 211 K, g 2.9 for r 1.15 g cm 3, and T c 213 K, g 2.13 for r 1.20 g cm 3. The interplay between these two curves is discussed in the text. of T. The resulting TTT curve shows a characteristic nose shape, arising from the competition between two effects, the thermodynamic driving force for nucleation and the viscous slowing down [1,6]. As temperature is lowered, both the thermodynamic driving force and the relaxation time increase, and it becomes more difficult for particles to diffuse to the energetically preferred crystalline configuration. For both densities, r 1.15, and 1.20 g cm 3, the T at which nucleation is fastest is around 240 K. At this T, the onset of crystallization requires about 3 ns. At the lowest studied T, the crystallization time has grown to 30 ns. Figure 3 also shows the relaxation times t a. The T dependence of t a can be described by a power law t a ~ T 2 T c g, in agreement with the prediction of mode coupling theory [17]. Since the relation t a ø t cryst holds at each temperature, including in the deeply supercooled region, equilibrium studies of metastable water can be achieved before nucleation takes place. The liquid can be connected to the deeply supercooled state via equilibrium metastable states if we choose a quench rate larger than the critical cooling rate R c T m 2 T n t n, where T m is the melting T and T n and t n locate the nose in the TTT curve [1]. For TIP5P, R c K s at the two studied densities. For r 1.10 g cm 3 and T 240 K, we observe only one (out of four) crystallization event within a time of 70 ns. For densities smaller than r 1.10 g cm 3,we observe no crystallization events within a time of 60 ns and hence we can only estimate that R c is smaller than 10 9 K s (the experimental value for water at ambient pressure is R c 10 7 K s [18]). from the smallest value 2p L allowed by the side L of the simulation box up to 50 nm 21. The oxygen density fluctuation r Q t is defined as P N i 1 exp iq r i, where r i is the oxygen coordinate of molecule i. The onset of crystallization coincides with the occurrence of (i) a sudden drop in potential energy and (ii) a sharp increase in the density fluctuations at one or more wave vector values. When crystallization occurs, the value of S Q t jumps from O 1 in the liquid to O N. Defining the crystallization time is somewhat arbitrary because of the stochasticity which accompanies the onset of crystallization and the definition of the critical nucleus. We define t cryst as the time at which any density fluctuation S Q t grows above a threshold value S 15 and remains continuously above the threshold for a time exceeding t 40 ps. This threshold prevents transient density fluctuations from being attributed to crystallization. We also perform calculations for other definitions of S and t, but the above values are sufficient to unambiguously identify the onset of crystallization without requiring excessive simulation time. Figure 3 shows the crystallization times t cryst, averaged over the four independent runs, for two different densities and for a broad range FIG. 4. For r 1.20 g cm 3, the energy minimum configuration, viewed along the c axis, of a crystal formed at T 270 K

75 VOLUME 88, NUMBER 19 P H Y S I C A L R E V I E W L E T T E R S 13 MAY 2002 In Fig. 4 we show a typical crystal configuration. The crystal structure, after energy minimization at constant volume, is a proton-ordered structure similar to ice B, first observed by Baez and Clancy [10]. Ice B is a variant of the ice IX structure, which is the proton-ordered form of ice III. The density of ice IX and ice III is in fact 1.16 g cm 3, close to our value. We have shown that the liquid-liquid phase separation can be observed in metastable equilibrium (i) if the cooling rate is faster than R c and (ii) if the observation time is shorter than the crystallization time at the critical point. While both such conditions can be realized in numerical simulations as shown here they cannot be met in experiments. Our simulations also show that a continuity of states between liquid and glassy phases of water exists [19]. Liquid states below the homogeneous crystallization temperature can be accessed provided the cooling rate exceeds R c. We acknowledge useful discussions with D. R. Baker, G. Franzese, W. Kob, E. La Nave, M. Marquez, and C. Rebbi. We acknowledge support from the NSF Grant No. CHE M. Y. acknowledges support from NSF Grant No. GER as a Graduate Research Trainee at the Boston University Center for Computational Science. F. S. acknowledges support from MURST COFIN 2000 and INFM Iniziativa Calcolo Parallelo. Note added. Recently, Matsumoto et al. have reported molecular dynamics simulations of ice nucleation and growth processes for a system of 512 water molecules interacting with the TIP4P potential [20,21]. [1] P. G. Debenedetti, Metastable Liquids (Princeton University Press, Princeton, 1996). [2] R. S. Smith and B. D. Kay, Nature (London) 398, 788 (1999). [3] O. Mishima, J. Chem. Phys. 100, 5910 (1994); Phys. Rev. Lett. 85, 334 (2000); O. Mishima and H. E. Stanley, Nature (London) 392, 164 (1998); 396, 329 (1998). [4] R. J. Speedy, J. Chem. Phys. 86, 982 (1982). [5] P. H. Poole, F. Sciortino, U. Essmann, and H. E. Stanley, Nature (London) 360, 324 (1992); Phys. Rev. E 48, 3799 (1993); P. H. Poole, U. Essmann, F. Sciortino, and H. E. Stanley, Phys. Rev. E 48, 4605 (1993); F. Sciortino, P. H. Poole, U. Essmann, and H. E. Stanley, Phys. Rev. E 55, 727 (1997). [6] I. M. Svishchev and P. G. Kusalik, Phys. Rev. Lett. 75, 3289 (1995). [7] For the case of atomic systems, see, e.g., H. E. A. Huitema, J. P. van der Eerden, J. J. M. Janssen, and H. Human, Phys. Rev. B 62, (2000). [8] I. M. Svishchev and P. G. Kusalik, Phys. Rev. Lett. 73, 975 (1994). [9] I. Borzsák and P. T. Cummings, Chem. Phys. Lett. 300, 359 (1999). [10] L. A. Baez and P. Clancy, J. Chem. Phys. 103, 9744 (1995). [11] D. R. MacFarlane, R. K. Kadiyala, and C. A. Angell, J. Chem. Phys. 79, 3921 (1983). [12] M. Kresin and Ch. Körber, J. Chem. Phys. 95, 5249 (1991). [13] M. W. Mahoney and W. L. Jorgensen, J. Chem. Phys. 112, 8910 (2000); 114, 363 (2001). [14] F. H. Stillinger and A. Rahman, J. Chem. Phys. 60, 1545 (1974). [15] J. M. Sorenson, G. Hura, R. M. Glaeser, and T. Head- Gordon, J. Chem. Phys. 113, 9149 (2000). [16] We also calculated the TIP5P isotherms at high T to provide an estimate of the location of the liquid-gas critical point, the terminus of the liquid spinodal line. We find T C K, r C g cm 3, P C MPa, in rough agreement with the experimental values. [17] W. Götze, in Liquids, Freezing, and Glass Transition, Proceedings of the Les Houches Summer School, Session LI, edited by J. P. Hansen, D. Levesque, and J. Zinn-Justin (North-Holland, Amsterdam, 1991). [18] D. R. Uhlmann, J. Non-Cryst. Solids 7, 337 (1972). [19] A. Hallbrucker, E. Mayer, and G. P. Johari, J. Phys. Chem. 93, 4986 (1989). [20] M. Matsumoto, S. Saita, and I. Ohmine, Nature (London) 416, 409 (2002). [21] S. Sastry, Nature (London) 416, 377 (2002)

76 VOLUME 88, NUMBER 22 P H Y S I C A L R E V I E W L E T T E R S 3 JUNE 2002 Potential Energy Landscape Equation of State Emilia La Nave, 1,2 Stefano Mossa, 2,1 and Francesco Sciortino 1 1 Dipartimento di Fisica, INFM UdR and INFM, Center for Statistical Mechanics and Complexity, Università di Roma La Sapienza, Piazzale Aldo Moro 2, I-00185, Roma, Italy 2 Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts (Received 4 February 2002; published 16 May 2002) Depth, number, and shape of the basins of the potential energy landscape are the key ingredients of the inherent structure thermodynamic formalism introduced by Stillinger and Weber [F. H. Stillinger and T. A. Weber, Phys. Rev. A 25, 978 (1982)]. Within this formalism, an equation of state based only on the volume dependence of these landscape properties is derived. Vibrational and configurational contributions to pressure are sorted out in a transparent way. Predictions are successfully compared with data from extensive molecular dynamics simulations of a simple model for the fragile liquid orthoterphenyl. DOI: /PhysRevLett PACS numbers: Pf, Ja, Lc Recent years have seen a resurgence in studies devoted to modeling the thermodynamics of supercooled liquids [1 5]. Such studies aim to elucidate the physics of the liquid-glass transition, to develop a thermodynamic description of out-of-equilibrium systems and to provide keys for a deeper understanding of the dynamics of supercooled states [6]. Numerical studies are nowadays providing quantitative estimates for the free energy of simple model systems [7 10]. The availability of such data provides stringent tests of the theoretical predictions [9 11] and helps in the understanding of basic mechanisms associated with the behavior of thermodynamic and dynamic quantities close to the glass transition. Among the thermodynamic formalisms amenable to numerical investigation, a central role is played by the inherent structure (IS) formalism introduced by Stillinger and Weber [12]. Properties of the potential energy landscape (PEL), such as depth, number, and shape of the basins of the potential energy surface, are calculated and used in the evaluation of the liquid free energy [9 11,13]. In the IS formalism, the system free energy is expressed as a sum of an entropic contribution, accounting for the number of the available basins, and a vibrational contribution, expressing the free energy of the system when constrained in one of the basins [12]. Important progress has been made after the discovery that, for models of fragile liquids, the number V e IS of distinct basins of depth e IS in a system of N atoms or molecules is well described by a Gaussian distribution [10,14] V e IS e an e2 e IS2E o 2 2s 2 2ps (1) Here the amplitude e an accounts for the total number of basins. Numerical studies of models for fragile liquids have also shown that the basin free energy can be written as the depth e IS plus a vibrational contribution which, in the harmonic approximation, has the well-known form F vib e IS, T k B T MX ln b hv i e IS, (2) where v i e IS is the ith normal mode frequency i 1,...,M and b 1 k B T. The M normal mode frequencies define the shape of the basin. If relevant, anharmonic corrections can also be accounted for [11,13]. The quantity P M i 1 ln v i e IS v o (where v o is the frequency unit) is found to depend linearly on the basin depth [10], i.e., can be written, in terms of two parameters a and b, as MX ln v i e IS v o a 1 be IS. (3) i 1 i 1 Hence, the vibrational free energy can be written as F vib e IS, T F vib E o, T 1 k B Tb e IS 2 E o. (4) Within the two assumptions of Eq. (1) Gaussian distribution of basin depths and Eq. (4) linear dependence of the basin free energy on e IS an exact evaluation of the partition function can be carried out. The corresponding Helmholtz free energy is given by [10] F T 2TS conf T 1 e IS T 1 F vib E o, T 1 k B Tb e IS T 2 E o, (5) where e IS T E o 2 bs 2 2bs 2 e` 2bs 2, (6) and S conf T k B an 2 e IS T 2 E o 2 2s 2 S` k B 2 bbs 2 2b 2 s 2 2. (7) In the above equations, e` and S` are defined as the value of e IS and S conf at infinite T. Equations (5) (7) show that, along constant volume V paths, the behavior of the thermodynamic quantities is controlled by the values of the PEL properties, as given by a, s, E 0 [from Eq. (1)] and by a and b [from Eq. (3)]. In this Letter we study the volume dependence of Eq. (5) to provide an expression for the equation of (22) (4)$ The American Physical Society

77 VOLUME 88, NUMBER 22 P H Y S I C A L R E V I E W L E T T E R S 3 JUNE 2002 state (EOS) based completely on landscape properties (PEL-EOS) [15]. This paper provides a significant insight into the understanding of the pressure P, and favors a detailed comparison between experimental measurements (usually performed at constant P) and theoretical approaches based on the IS formalism. It may also help in developing an IS-based thermodynamic description of out-of-equilibrium (glass) states and a theoretical definition of the concepts of fictive P and T [16]. In thermodynamics, P is defined as the (negative) V derivative of the Helmholtz free energy. Hence P is fully determined by the V dependence of the landscape properties a, s, E 0, a, and b. Equation (5) shows that P can be split into three main contributions: a configurational one, P conf related to the change in the number of available basins with V; ane IS one, P eis related to the change in basin depth with V; a vibrational one, P vib related to the change in the shape of the explored basin with V. The T dependence of each contribution can then be studied independently. The explicit expressions for these contributions are S conf / R > [ kj / mol ] < e IS Σ i ln ( ω i / ω o ) / N V 1 V 2 V 3 V 4 V 5 ( a ) ( b ) P = 0 MPa P=200 MPa ( c ) T [ K ] FIG. 1. T dependence of (a) S conf,(b) P i ln v i v o N, and (c) e IS (per molecule) at the five studied volumes V k (symbols). Data are from Ref. [13]. The solid curves are simultaneous fits of the three sets of data according to Eqs. (7), (3), and (6). The dashed curves are the constant P paths (at P 0 and 200 MPa) calculated according to the IS-EOS, as discussed in the text. The frequency unit is v 0 1 cm 21. P conf T, V T V S` T P eis T, V 2 V e` 1 1 T P vib T, V 2k B T V bs2 V s2 2k B, (8) V a 1 be` 1 V s2 k B, (9) V bs2. (10) By grouping together all the contributions with the same T dependence, P can be expressed in terms of V derivatives of only three combinations of the five PEL parameters [18] P T, V P const 1 TP T 1 T 21 P 1 T, (11) where P const 2 e` V, P T S` V 2 k B V a 1 be`, and P 1 T V s2 2k B. The present formalism also provides a possible expression for the so-called inherent structure equation of state (IS-EOS), P IS T IS, V [15,19 21], i.e., the relation between the pressure and volume of the typical IS and the temperature T IS of the equilibrium ensemble from which configurations were extracted. Indeed, one can assume that the constant V steepest descent minimization procedure, which realizes the search for the closest local minimum starting from an equilibrium configuration, suppresses all the vibrational components (hence P vib 0) but keeps P conf and P eis frozen at their equilibrium initial value. As a result P IS, a purely mechanical quantity, can be expressed as e, 600[S k B (a+be )] V 5 V 4 V 3 V 2 σ 2 / 2 k B V [S k B (a+be )] 0.2 e Molar Volume [ cm 3 / mol ] FIG. 2. V dependence of the three combinations of PEL parameters contributing to the linear (squares), constant (circles), and T 21 (triangles) components of P, according to Eq. (11). The solid lines are the polynomial fit used to evaluate P const, P T, and P 1 T. Tables of the coefficients of the polynomial fit will be reported in Ref. [17]. Here e` is expressed in 10 6 J mol 21, S` 2 k B a 1 be` in 10 6 J mol 21 K 21, and s 2 2k B in 10 6 J K mol 21. σ 2 / 2 k B

78 VOLUME 88, NUMBER 22 P H Y S I C A L R E V I E W L E T T E R S 3 JUNE 2002 P IS T IS, V P conf 1 P eis 2 V E o 1 T IS V S` 1 1 k B T IS V s2 2. (12) Note that in this expression T IS is assumed to be the T controlling the configurational contribution. Different formulations for the out-of-equilibrium free energy [16] would lead to different expressions [17]. The present approach also predicts the behavior of P when an IS configuration is heated from T 0 at constant V. While the system remains in the same basin, P is given by P IS T IS, V 1 P vib T, T IS, V, where the only (linear) T-dependent part arises from P vib T, T IS, V 2k B T V a 1 be IS T IS. (13) We now apply the present derivation to the simple Lewis and Wanström (LW) model for the fragile molecular liquid orthoterphenyl (otp) [13,22]. The LW model is a rigid P [ MPa ] P IS P = P conf + P e IS + P vib P IS + P vib three-site model, with intermolecular site-site interactions described by the Lennard Jones (LJ) potential [22]. Its simplicity allows one to reach simulation times of the order of ms [13]. In this model, as in the LJ case, the anharmonic contributions are negligible, e IS T goes as 1 T, and P M i 1 ln v i e IS v o is linear in e IS [13]. Hence this model is a perfect candidate for testing the validity of the PEL-EOS proposed here. We use the excellent data base of state points presented in Ref. [13] (i) to calculate the V dependence of the PEL parameters; (ii) to derive the EOS for the otp model, and (iii) to compare it with the exact EOS calculated using the virial expression, as commonly implemented in molecular dynamics (MD) codes. Figure 1 shows, for five constant V paths, the simultaneous fit of the T dependence of S conf, the basin curvatures, and of the e IS T, according to Eqs. (7), (3), and (6). The possibility of fitting simultaneously, with the same P values of a, s, E 0, a, and b, the quantities e IS T, M i 1 ln v i e IS v o, and S conf T, supports the validity of the two main assumptions, i.e., Eqs. (1) and (4). The V dependence of the three combinations of fit parameters, 2e`, S` 2 k B a 1 be`, and s 2 2k B is shown in Fig. 2. P V, T can be calculated from the V derivative of these quantities according to Eq. (11) and compared with the MD data. Such a comparison is reported in Fig P = 50 MPa ( PEL EOS ) P V = V 2 ( PEL EOS ) P V = V 2 ( MD ) P IS V = V 2 ( MD ) P vib P [ MPa ] Equilibrium Theory Out of Equilibrium Theory T [ K ] FIG. 3. Comparison between P evaluated according to the V derivative of PEL properties (lines) and P evaluated in the MD simulation using the virial expression (symbols). The solid symbols are equilibrium values. For each V k, the symbol indicates P IS for the coldest equilibrated state point. The open symbols are MD data calculated during a constant heating procedure starting from the IS configuration marked with, asexplained in the text. Lines are PEL-EOS for equilibrium [solid lines, Eq. (11)] and for the heating procedure [dashed lines, Eq. (13)] P conf + P e IS T [ K ] FIG. 4. Total (P), vibrational (P vib ) and e IS plus configurational (P eis 1 P conf ) contributions to P along constant V (dashed lines) and constant P (solid lines) paths. At constant V, P vib is linear in T [Eq. (10)]. P

79 VOLUME 88, NUMBER 22 P H Y S I C A L R E V I E W L E T T E R S 3 JUNE 2002 P IS [ MPa ] The present EOS, based exclusively on PEL properties, can be used to address important issues in the thermodynamics of supercooled liquids [24], such as the study of the intrinsic limit of stability of the liquid state [15,20,21] and the IS-based thermodynamic description of aging [16]. We acknowledge support from the NFS Grant No. CHE , INFM Initiative Parallel Computing, and INFM PRA HOP / T IS [ K 1 ] FIG. 5. Inherent structure equation of state. Symbols are MD calculations, lines are PEL predictions. The T dependence of the individual contributions can be evaluated according to Eqs. (8) (10). The volume dependence of bs 2 and a 1 be` has been evaluated from the best fitofp 2 P IS which, according to Eq. (12), coincides with P vib. Figure 4 shows along a constant V and a constant P path P vib and P conf 1 P eis. We note that, at constant V, both P components are increasing with T while, at constant P, P conf 1 P eis decreases on heating to compensate for the increase of P vib. The PEL-EOS allows us to also contrast the isobaric Pand isochoric T dependence of S conf, e IS T and M i 1 ln v i v o. Such comparison is reported in Fig. 1. Here we note the faster decrease of S conf T and e IS T along constant P paths as well as the different trend in the change of the basin shape [23]. As a further check of the quality of the calculated EOS for the otp model, Fig. 5 compares the MD (virial) and PEL [Eq. (12)] IS-EOS. Since for the otp model the term linear in T in Eq. (12) is negligible [17], P IS follows, to a good extent, a 1 T IS law. The quality of the comparison supports the interpretation of P IS as the V derivative of the depth and number of basins sampled in the corresponding thermodynamic equilibrium state. Finally, Fig. 3 compares MD data (open symbols) and PEL (dashed lines) predictions in a run where T is increased starting from the T 0 IS configuration, as previously discussed. The entire simulation is much shorter than the time needed to change the basin, so that only the vibrational degrees of freedom are thermalized. Also in this case, the PEL expression accounts for the observed linear increase of P. [1] F. H. Stillinger, J. Phys. Chem. B 102, 2807 (1998). [2] R. J. Speedy, J. Chem. Phys. 110, 4559 (1999); J. Phys. Chem. B 103, 4060 (1999); R. J. Speedy and P. G. Debenedetti, Mol. Phys. 88, 1293 (1996). [3] L. M. Martinez and C. A. Angell, Nature (London) 410, 667 (2001). [4] B. Coluzzi et al., Phys. Rev. Lett. 84, 306 (2000); M. Mézard and G. Parisi, Phys. Rev. Lett. 82, 747 (1999). [5] D. J. Wales, Science 293, 2067 (2001). [6] P. G. Debenedetti and F. H. Stillinger, Nature (London) 410, 259 (2001). [7] F. Sciortino et al., Phys. Rev. Lett. 83, 3214 (1999). [8] S. Büchner and A. Heuer, Phys. Rev. E 60, 6507 (1999). [9] A. Scala et al., Nature (London) 406, 166 (2000). [10] S. Sastry, Nature (London) 409, 164 (2001). [11] I. Saika-Voivod et al., Nature (London) 412, 514 (2001). [12] F. H. Stillinger and T. A. Weber, Phys. Rev. A 25, 978 (1982); Science 225, 983 (1984). [13] S. Mossa et al., Phys. Rev. E 65, (2002). [14] A. Heuer and S. Buchner, J. Phys. Condens. Matter 12, 6535 (2000). [15] P. G. Debenedetti et al., J. Phys. Chem. B 103, 7390 (1999); Adv. Chem. Eng. 28, 21 (2001). [16] F. Sciortino and P. Tartaglia, Phys. Rev. Lett. 86, 107 (2001). [17] E. La Nave et al. (to be published). [18] Note that liquids with nonmonotonic behavior of density (as water) must be characterized by a positive P 1 T coefficient. [19] R. A. LaViolette, Phys. Rev. B 40, 9952 (1989). [20] C. J. Roberts et al., J. Phys. Chem. B 103, (1999). [21] S. Sastry et al., Phys. Rev. E 56, 5533 (1997); Phys. Rev. Lett. 85, 590 (2000). [22] G. Wahnström and L. J. Lewis, Phys. Rev. E 50, 3865 (1994). [23] Work is in progress [17] to elucidate the possibility that the 1 T dependence of the excess specific heat measured in constant P experiments [C. Alba et al., J. Chem. Phys. 92, 617 (1990)] can be understood using the PEL-EOS developed here. [24] Another interesting possibility is offered by a generalization to V of Eq. (1). The otp data allow us to eliminate the simple possibility that V e IS, V is a bivariate Gaussian [17]

80 JOURNAL OF CHEMICAL PHYSICS VOLUME 117, NUMBER 7 15 AUGUST 2002 Orientational and induced contributions to the depolarized Rayleigh spectra of liquid and supercooled ortho-terphenyl S. Mossa a) Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts and Dipartimento di Fisica, INFM UdR and INFM Center for Statistical Mechanics and Complexity, Università di Roma La Sapienza, Piazzale Aldo Moro 2, I-00185, Roma, Italy G. Ruocco Dipartimento di Fisica, INFM UdR and INFM Center for Statistical Mechanics and Complexity, Università di Roma La Sapienza, Piazzale Aldo Moro 2, I-00185, Roma, Italy M. Sampoli Dipartimento di Energetica and INFM, Università di Firenze, Via Santa Marta 3, Firenze, I-50139, Italy Received 12 February 2002; accepted 22 May 2002 The depolarized light scattering spectra of the glassforming liquid ortho-terphenyl have been calculated in the low frequency region using molecular dynamics simulation. Realistic system configurations are produced by using a recent flexible molecular model and combined with two limiting polarizability schemes, both of them using the dipole-induced-dipole contributions at first and second order. The calculated Rayleigh spectral shape are in good agreement with the experimental results in a large temperature range. The analysis of the different contributions to the Rayleigh spectra emphasizes that the orientational and the collision-induced translational terms lie on the same time scale and are of comparable intensity. Moreover, the cross correlation terms are always found to be an important contribution to the scattering intensity American Institute of Physics. DOI: / I. INTRODUCTION Depolarized light scattering DLS spectroscopy has been proven to be a valuable tool to explore the dynamical properties of molecular fluids. 1 In the case of supercooled and glassforming liquids, DLS has been used by several researchers with the aim to elucidate the dynamical mechanisms underlying the liquid-glass transition. 2 8 However, the uncertainties in the scattering mechanisms giving rise to the Rayleigh spectra, and the consequent hypotheses that have been introduced to interpret the experiments, leave open the problem of a reliable analysis. In some glassforming liquids, DLS spectra have been interpreted as arising primarily from interaction-induced mechanisms, namely dipole-induced-dipole DID, and so they have been related directly to the dynamics of the density fluctuations and compared with the outcomings of modecoupling theories MCT. 9 Following this line, Cummins et al. 3 have interpreted the spectra of salol as arising entirely from the DID mechanism; in a more recent work, 10 however, the authors assume that orientational contributions dominate the DLS spectra of salol at least up to 4000 GHz, and the connection with the density fluctuations dynamics has to rely on a strong coupling between rotational and translational degrees of freedom. The situation is similar also for other systems. In the case of an other prototype of fragile glassforming liquid, the ortho-terphenyl OTP, Patkoswski et al. 11 state that the low frequency part of the spectrum, up to about a Present address: Laboratoire de Physique Theorique des Liquides, Univirsite Pierre et Marie Curie, 4 Place Jussieu, Paris 75005, France. 10 GHz is due to collective reorientation, while at higher frequency the DID contribution is dominating. We think that a careful investigation of the scattering mechanisms which contribute to DLS spectra is needed in order to clarify what type of information is possible to extract from the experimental data, and to ascertain the connections with the density fluctuation correlators treated by MCT. In the present work we investigate the low frequency DLS spectra of liquid OTP for temperatures above and close to the MCT critical temperature T c. We use the new realistic intra- and intermolecular potential model set up by some of us very recently Various single particle correlation functions have already been studied through classical molecular dynamics MD simulations, yielding noticeable results. It is now possible to combine realistic dynamical configurations with plausible polarizability models, and try to discriminate among the different hypotheses on the origin of the different contributions to DLS experimental spectra. The paper is organized as follows: Sec. II is a brief recall of our potential model and of the methods used to produce the dynamical configurations. In Sec. III we recall the main polarizability models that have been used so far to calculate the DLS spectra of simple molecular liquids; moreover, we show how they can be used to estimate the DLS spectra for our realistic flexible model. The computational details are treated in Sec. IV, while the results and a discussion are presented in Sec. V. In Sec. VI we address some conclusions. II. THE MOLECULAR MODEL In our flexible model 12 the OTP molecule is constituted by three rigid hexagons phenyl rings of side L a 1.39 Å /2002/117(7)/3289/7/$ American Institute of Physics Downloaded 17 Oct 2002 to Redistribution subject to AIP license or copyright, see

81 3290 J. Chem. Phys., Vol. 117, No. 7, 15 August 2002 Mossa, Ruocco, and Sampoli Two adjacent vertices of the parent central ring are bonded to one vertex of the two side rings by bonds whose length at equilibrium is L b 1.5 Å. Each vertex of the hexagon is thought to be occupied by a fictious atom of mass M CH 13 amu representing a carbon hydrogen C H pair. In the isolated molecule equilibrium configuration, the two lateral rings lie in planes that form an angle of about 54 with respect to the central ring s plane. The three rings of a given molecule interact among themselves by an intramolecular potential V intra. This potential is chosen in such a way to preserve the molecule from dissociation, to give the correct relative equilibrium positions for the three phenyl rings, and to represent at best the isolated molecule vibrational spectrum. In particular, it is written in the form V intra k c k V k (r), where each term V k (r) controls a particular degree of freedom; the actual values of the coupling constants c k have been determined in order to have a realistic isolated molecule vibrational spectrum. Several internal motions have been taken into account, like the stretching along the ring ring bonds and between the side rings, and the rotation of the lateral rings along the ring ring bonds. The intermolecular part of the potential, concerning the interactions among the rings pertaining to different molecules, is described by a site site pairwise additive Lennard- Jones LJ 6-12 potential, each site corresponding to one of the six hexagons vertices. The details of the intramolecular and intermolecular interaction potentials, together with the values of the potential parameters, are reported in Ref. 12. It is worth noting that previous studies of the temperature dependence of the self-diffusion constant 12 and of the structural relaxation times, 12,13 indicate that this model is capable of quantitatively reproducing the dynamical behavior of the real system, but the actual simulated thermodynamical point has to be shifted by 20 K upward. In the following, when we compare the simulation results with the experiments, the reported MD temperatures are rescaled by such an amount. III. POLARIZABILITY MODELS In a liquid composed of N optical anisotropic units a unit can be the entire molecule, a group of atoms in the molecule, or a single atom inside the molecule, characterized by a permanent polarizability tensor, the classical low frequency DLS spectrum in the dipole approximation is proportional apart from a trivial frequency factor to the Fourier transform of the time correlation function of the traceless tensorial part 2 of the collective polarizability. can be approximated by the sum of two terms: the first, M, is the sum of all the permanent polarizabilities dependent on the orientational variables Q i of the units; the second, I, is due to all the interaction induced contributions, and its leading part can be written in term of the dipole propagator tensor T (2) (i, j) 2 R 1 ij where R ij is the distance between the ith and jth units, M I, 1 M i i Q i i Q i, 2 I i j i T 2 i, j j i j k i T 2 i, j j T 2 j,k k. Here M is written for units with symmetric top symmetry, and are the gas phase isotropic and anisotropic polarizabilities of the scattering units, and Q (3/2)û û (1/2), where û is a unit vector along the symmetry axis. Usually only the first order DID is considered in Eq. 3, the second order DID being sometimes completely disregarded. The relative importance of the higher order DID contributions with respect to the first one is dependent on the strength of the permanent polarizability tensor, and on the minimum approach distances of the units. 16 However, it is well known from the study of noble gas systems, that at short distances, when high order contributions become important, also other multipole and quantum electron correlation effects have to be considered. These terms are usually found to be of opposite sign with respect to the DID contribution. 17,18 Empirical diatomic induced polarizabilities have negative corrections at short distances for all noble gases, 19 and only for gaseous mercury a small positive correction to the second order DID has been used to fit the experimental spectra. 20 From these considerations, in the present work we discuss only the contributions from the first and second order DID. We want to underline that, in the case of OTP, the choice of the unit is not so obvious as for monatomic fluids or simple molecular fluids such as N 2. Thus, the subsequent choices are based on the following considerations. Even for simple molecular systems, like CO 2 or CS 2 fluids, we can consider as unit the entire molecule, each atom, or the peripheral most polarizable atoms. In the case more units are pertaining to the same molecule, these units are at short distance each other, so their induced contributions are tentatively treated in a more refined scheme. The major attempt in this way has been made by Applequist et al. 21 Their formalism, introduced to reproduce the molecular polarizabilities with transferable atomic polarizabilities, is equivalent to take into account the DID at all orders. Indeed the distances are so short that the major contributions come from higher order DID interactions. Irrespective to the relative success of the method, various authors have modified the scheme adding monopole polarizability, and/or using nonisotropic atomic polarizability, and/or separating atoms of the same species on the base of chemical bonds. Only a few attempts have been made so far to use the Applequist formalism to calculate the experimental spectra, 16 and with doubtful improvements. Thus, at present we are not confident in using classical dipole interactions at short distances. So far we have not tried to use a complex polarizability scheme for our molecular model, and we have chosen to adopt two models that can give a rough estimate of a lower and upper limit of the induced interactions. In our potential model, three six sites rigid rings form the molecule and this suggests to consider as units the rings themselves R scheme, or the six sites of each ring S scheme. We have always neglected the effects on the polarizability due to the 3 Downloaded 17 Oct 2002 to Redistribution subject to AIP license or copyright, see

82 J. Chem. Phys., Vol. 117, No. 7, 15 August 2002 Depolarized Rayleigh spectra of ortho-terphenyl 3291 change of one or two H C with a C C bond. In the R scheme, the polarizability tensor is taken equal to that of benzene (C 6 H 6 ). In the S scheme, we have simply divided by six the polarizability tensor of benzene, and we have neglected the DID contributions coming from atoms pertaining to the same ring. The induced contributions grow up at least with the sixth power of the inverse of the distance, so we expect the R scheme to be a lower limit to the induced contributions, while the S scheme to be an upper one. Indeed, in the S scheme the mean distances are shorter than in the R scheme. We want to underline that, if in the S scheme the DID interactions inside the ring were taken into account, this would have reduced the polarizabilities of the six sites. Using a notation similar to that of Stassen et al., 25 we can work out the collective polarizability correlation function for the depolarized scattering, C t 1 10V 2 t : 2 0, 4 where V is the volume occupied by the N units. C(t) can be divided in three terms, namely, the orientational term C or (t), the interaction-induced or DID term C ii (t), and the cross contribution term C cr (t), C t C or t C ii t C cr t, 5 with C or t 2 45V 2 Q 0 :Q t, C ii t 1 10V 2 I 0 : 2 I 0, C cr t 1 15V Q 0 : 2 I t I 2 0 :Q t. Here Q i Q i and all the sums are extended over the N units. Although the importance of the three terms in Eq. 6 has been found similar in various systems, 16 the cross term C cr (t) has been neglected, without sound arguments, in various works on glassforming liquids. The integrated intensities of the orientational contribution, i.e., the value at t 0 of the correspondent correlation function, can be written as C or t 0 N 15V 2 1 f 2, 7 where g 2 1 f 2 is the static angular pair correlation factor between our units, defined as 26 6 g 2 2 3N i j i Q i 0 :Q j 0. 8 It is well known that, if the units are spherically symmetric ( 0), the orientational and cross contributions do vanish, as in the case of DLS from monatomic fluids. In both R and S schemes, we expect a value of g 2 rather different from 1. Indeed, also in the R scheme we have a strong orientational correlation between different units rings pertaining to the same molecule. This is quite different from the case of simple molecular fluids, such as N 2, where only very small orientational correlations exist between different molecules, even in the liquid phase near the melting point. If all the rings of the OTP molecules were in the equilibrium positions, while there was no correlation between the orientation of different molecules, we can calculate the g 2 factor for both R and S models. Using the equilibrium values of the cosines between the normals to the ring planes n 1 n 2 n 1 n , and n 2 n , with the index 1 standing for the central ring 12, we have g R,eq , g S,eq 2 6 g R,eq Obviously, the obtained g 2 factors make the orientational contribution integrated intensity in this idealized condition independent of the adopted scheme. Since the relation g R,eq 2 g S,eq 2 /6 holds always, in the following we refer simply to g 2 in the R scheme. Further, we can evaluate the average value of the analogous of g 2 for the rings pertaining to the same molecules, i.e., K 2 3 2N i j i cos 2 ij 1/3, 10 where the sum is extended over the rings of a molecule, ij is the angle between the normals of the ith and jth ring, and means a configuration or time average. The value of K 2 for the isolated molecule at equilibrium is always K 2 eq IV. COMPUTATIONAL DETAILS We study a system composed by 108 OTP molecules 324 rings, 1944 LJ interaction sites enclosed in a cubic box with periodic boundary condition. The MD runs are performed in the microcanonical NVE ensemble. To integrate the equations of motion we treats each ring as a separate rigid body, identified by the position of its center-of-mass R i and by its orientation, expressed in terms of quaternions q i. 27 The standard Verlet leapfrog algorithm 27 has been implemented to integrate the translational motion while, for the most difficult orientational part, a refined algorithm has been used. 28 The integration time-step is t 2 fs, which gives rise to an overall energy conservation better than 0.01% of the kinetic energy. We consider two series of runs. For temperature T 420 K, to be compared with the experimental results, we perform one run 24 ns long with a saving time of 1 ps; using windows 800 ps wide we achieve a resolution of 0.04 cm 1. For the temperatures T 280, 290, 300, 310, 320, 330, 350, 370, 390, 410, and 430 K we run for 640 ps with a saving time of 0.04 ps; using windows of 320 ps we reach a resolution of 0.1 cm 1. The spectra have been numerically evaluated via the Wiener Kinchin theorem, i.e., through the square modulus of the time-fourier transform of the signals calculated along the MD trajectories. In the R scheme, the components of the polarizability tensor in the molecular fixed frame are Å 3, 6.35 Å 3, 1/3( 2 ) Å 3, 5.96 Å 3, where stands for orthogonal to the phenyl ring s plane, i.e., parallel to the symmetry axis. These values are taken from the polarizability tensor of the benzene. 29 For Downloaded 17 Oct 2002 to Redistribution subject to AIP license or copyright, see

83 3292 J. Chem. Phys., Vol. 117, No. 7, 15 August 2002 Mossa, Ruocco, and Sampoli FIG. 1. a First order DID spectrum calculated considering the carbon atoms model S at T 420 K together with the resolution curve see text for details. The insets show the data, renormalized to the peak intensity, in semilogarithmic left panel and logarithmic scale right panel, in order to stress the power law behavior of the tail. b DID spectrum calculated considering the phenyl rings centers-of-mass model R. the S scheme, to each site have been attributed the previous values divided by 6 and the same orientation. V. RESULTS AND DISCUSSION In Fig. 1 the DID spectrum at first order is reported in the liquid phase at 420 K for both S Fig. 1 a and R Fig. 1 b schemes. The dashed lines refer to the obtained resolution, which depends on the time length of the MD run. In the insets the same data, rescaled to the peak intensity, are shown in semilogarithmic left and logarithmic right scale in order to emphasize the power law behavior of the tails. As expected, the overall intensity in the S scheme is much higher than the one in the R scheme. On the contrary, the shape is very similar, although the relative intensity of the spectral wings at 10 cm 1 with respect to the peak intensity is about three times higher in the R scheme. As a consequence the absolute wing intensities are comparable. On the overall, the shape is similar to the induced contribution from heavy noble gases, i.e., an exponential decay of the form exp( / 0 ) with 0 of the order of 2 cm 1. Only for the S scheme the second order DID is of some importance, so in Fig. 2 the corresponding spectrum Fig. 2 b is compared with the first order DID Fig. 2 a. At all frequencies the second order contribution is a relatively small fraction of the first one; above 0.25 cm 1 it is less than about 10%, so the DID s at higher order can be safely neglected. FIG. 2. a As in Fig. 1 a. b Second order DID calculated for model S. The sum of the first and second order DID, and the orientational spectra in the S scheme are compared in Figs. 3 a and 3 b, respectively. As we can see, the relative shapes and intensities in this case are quite similar apart from the shape of the peak at frequency comparable to the spectral resolution, where the simulation statistical errors are large and prevent any reliable comparison. It is worth pointing out that, while the orientational spectra depends only on angular variables, the DID spectra depends on both translational and angular degrees of freedom. Therefore, the coincidence of the shapes of the different spectral contributions implies the absence of a clear time scale separation between translational and orientational dynamics. This is reflected also in the shape and high intensity of the cross contribution; at this temperature, indeed, the spectra of orientational, DID and cross contributions practically superimpose each other, as shown in Fig. 4. It is worth noting that the resulting cross-correlation term is found to be positive, in contrast to the case of simple liquids. 25 The reason for such behavior can be ascribed to the strong correlations present among the sites pertaining to the same molecule. In the case of the R scheme, the situation is quite different. Beyond the smaller relative importance of the DID and cross terms, at high frequency above 5 cm 1 the spectrum is dominated by the orientational contributions, even if at 10 cm 1 the ratio of DID, cross, and orientational contributions is not so high, being 1:2:4 see Fig. 5. Again, even in the R scheme our simulations are in poor agreement with the assumptions made by Patkowski et al. 11 about the spectral separation between orientational and DID Downloaded 17 Oct 2002 to Redistribution subject to AIP license or copyright, see

84 J. Chem. Phys., Vol. 117, No. 7, 15 August 2002 Depolarized Rayleigh spectra of ortho-terphenyl 3293 FIG. 5. Contributions to the total spectrum for the R model: first order DID circles, orientational contribution squares, and cross-correlation triangles calculated at T 420 K; the inset shows the same data, renormalized to the respective peak intensities, in a double-logarithmic scale. then in shape, and some help would come from an absolute intensity comparison; unfortunately, the OTP experimental absolute intensity has not been calculated so far. Anyway, the large uncertainties usually present in both experimental and FIG. 3. a Sum of first and second order DID terms for the S model. b Orientational contribution to the spectrum. contributions. Indeed, it is evident from Fig. 5 that no time scale separation can be effective and the cross term cannot be neglected. The resulting total simulated DLS spectra for the two schemes are reported in Fig. 6 a. In Fig. 6 b we show, in double logarithmic scale, a comparison of the two line shapes with the experimental spectrum at 444 K. 30 The agreement among the sets of data are quite good, considering the approximations we have made in the dynamical and polarizability model. This shape comparison gives some preference to the S scheme, but it is hard to really discriminate between these two limiting polarizability models on this basis. R and S DLS spectra differ essentially in intensity rather FIG. 4. Contributions to the total spectrum for the S model: first order DID circles, orientational contribution triangles, and cross-correlation squares calculated at T 420 K; the inset shows the same data, renormalized to the respective peak intensities, in a double-logarithmic scale. FIG. 6. a Total spectra for the S circles and R triangles models. b Total spectra for the S circles and R triangles models together with the experimental results V H spectra from Ref. 30 at T 444 K diamonds in a double logarithmic scale. A power law of exponent 1.5 dashed line is also plotted as a guide to the eye. Downloaded 17 Oct 2002 to Redistribution subject to AIP license or copyright, see

85 3294 J. Chem. Phys., Vol. 117, No. 7, 15 August 2002 Mossa, Ruocco, and Sampoli TABLE I. Temperature dependence of the integrated intensities of the different spectral contributions. T K DID S (1) Å 3 DID S (2) Å 3 DID R Å 3 OR Å simulated determinations of DLS absolute spectral intensities, could not allow this kind of comparison. In Table I we report the integrated intensities of the different spectral contributions, calculated by numerical integration of the spectral densities. Figure 7 shows the temperature dependence of the various relative integrated intensities. We see that the relative intensities do not show any significant trend with temperature, apart from a tendency of the orientational part to increase at the lowest investigated temperatures. The increase has to be attributed to a change in the relative orientation of the rings, i.e., to an increasing of g 2. FIG. 8. Static angular pair correlation factor g 2 (T) Eq. 8 as a function of temperature. In the inset, K 2 (T) Eq. 10 is shown as a function of temperature. This is clearly seen in Fig. 8, where g 2 main panel and K 2 inset are plotted as a function of temperature. It is interesting to note that also K 2 increases on decreasing temperature, being less than the reference value K 2 eq 1.14 at high temperatures and more than that value at low temperatures. On the basis of the value of the angles between the symmetric top axis of the rings, K 2 increases fast if there is a spread of n 1 n 2 and n 1 n 3, but decreases with the spread of n 2 n 3. The distribution of angle cosines is reported in Fig. 6 of Ref. 12. At high temperatures, the large spread of n 2 n 3 is, anyway, able to decrease K 2 under K 2 eq, but at low temperature the situation is reversed by the spread of n 1 n 2 and n 1 n 3. We expect a similar situation to hold for the DLS of a large class of molecular glassforming liquids; the DLS would be a mixture of orientational, DID, and cross contributions in the entire low frequency range, with no significant time scale separation. Further, the better agreement with the experimental results of the S scheme in the case of OTP, underlines the importance to take into account the internal degrees of freedom to obtain a realistic description of the DLS of glassforming liquids consisting of large flexible molecules. VI. CONCLUSIONS FIG. 7. a Ratios between the first order DID contribution to the S model and the other terms as a function of temperature: Orientational circles, DID calculated for model R full squares, and second order DID calculated for model S triangles. The arrows indicate the mean values that are, respectively, 1.22, 0.25, and b Relative intensity of the different contributions with respect to the total intensity calculated with both models. The arrows indicate the mean values of, respectively, 0.66, 0.32, 0.27, and In this paper we have studied, by means of molecular dynamics simulations, the orientational and induced contributions to the low frequency depolarized Rayleigh spectra of supercooled ortho-terphenyl. We have used a realistic flexible intramolecular model recently introduced by some of us, taking into account the most important internal degrees of freedom of the OTP molecule. Two polarizability models have been introduced, each of them considering a different scattering unit. In the ring s model R, we have assigned to each phenyl ring the polarizability of the benzene. In the site s model S, we have assigned to the sites of each ring the same polarizability divided by six. This procedure has allowed us to study the single contributions to the DLS spectra in both cases. Our main findings are the following: i Although in the two schemes the intensities of the DID contributions are very different, their overall shape is very similar; ii The second order DID is at all frequencies already a Downloaded 17 Oct 2002 to Redistribution subject to AIP license or copyright, see

86 J. Chem. Phys., Vol. 117, No. 7, 15 August 2002 Depolarized Rayleigh spectra of ortho-terphenyl 3295 relatively small fraction of the first order contributions, so that all the higher induced terms can be safely neglected; iii For the S model first order DID and orientational contributions are very similar in both shape and intensity; iv In both models the cross correlation between induced and orientational terms cannot be neglected. This fact is in striking contrast with the analysis of the experimental data of Ref. 11 and support the conclusion that v no time scale separation is present between DID and orientational contribution. In other words we expect that, similarly to the present case, for a broad class of molecular glassforming liquids the DLS spectra would be a superposition of DID, orientational, and mixed contributions, and none of them should be disregarded. Finally vi the better agreement of the S scheme calculated spectra with the experimental results underlines the importance of taking into account the molecular internal degrees of freedom in order to obtain a realistic description of complex molecular liquids. ACKNOWLEDGMENT This work was supported by MURST PRIN B. J. Berne and R. Pecora, Dynamic Light Scattering Wiley, New York, N. J. Tao, G. Li, X. Chen, W. M. Du, and H. Z. Cummins, Phys. Rev. A 44, G. Li, W. M. Du, X. K. Chen, H. Z. Cummins, and N. J. Tao, Phys. Rev. A 45, W. M. Du, G. Li, X. Chen, H. Z. Cummins, M. Fuchs, J. Toulouse, and L. A. Knauss, Phys. Rev. E 49, J. Wuttke, J. Hernandez, G. Li, G. Coddens, H. Z. Cummins, F. Fujara, W. Petry, and H. Sillescu, Phys. Rev. Lett. 72, A. D. Bykhovskii and R. M. Pick, J. Chem. Phys. 100, M. J. Lebon, C. Dreyfus, G. Li, A. Aouadi, H. Z. Cummins, and R. M. Pick, Phys. Rev. E 51, C. Alba-Simionesco and M. Krauzman, J. Chem. Phys. 102, W. Götze and L. Sjögren, Rep. Prog. Phys. 55, H. Z. Cummins, G. Li, W. Du, R. Pick, and C. Dreyfus, Phys. Rev. E 53, A. Patkowski, W. Steffen, H. Nilgens, E. W. Fisher, and R. Pecora, J. Chem. Phys. 106, S. Mossa, R. Di Leonardo, G. Ruocco, and M. Sampoli, Phys. Rev. E 62, S. Mossa, G. Ruocco, and M. Sampoli, Phys. Rev. E 64, S. Mossa, G. Monaco, G. Ruocco, M. Sampoli, and F. Sette, J. Chem. Phys. 116, S. Mossa, G. Monaco, and G. Ruocco, cond-mat/ preprint. 16 For a review of induced contribution, see Phenomena Induced by Intermolecular Interactions, edited by G. Birnbaum Plenum, New York, P. D. Dacre, Can. J. Phys. 59, ; 60, ; Mol. Phys. 36, ; 45, ; 45, ; 47, K. L. C. Hunt in Ref. 16, p F. Barocchi and M. Zoppi, in Ref. 16, p F. Barocchi, F. Hensel, and M. Sampoli, Chem. Phys. Lett. 232, J. Applequist, J. R. Carl, and K. K. Fung, J. Am. Chem. Soc. 94, M. L. Olson and K. R. Sundberg, J. Chem. Phys. 69, B. T. Thole, Chem. Phys. 59, J. Applequist, J. Phys. Chem. 97, H. Stassen, Th. Dorfmuller, and B. M. Ladanyi, J. Chem. Phys. 100, A. Ben-Reuven and N. D. Gershon, J. Chem. Phys. 51, M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids Clarendon, Oxford, G. Ruocco and M. Sampoli, Mol. Phys. 82, J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory of Gases and Liquids Wiley, New York, R. Angelini, thesis, Università di L Aquila, Downloaded 17 Oct 2002 to Redistribution subject to AIP license or copyright, see

87 EUROPHYSICS LETTERS 1 October 2002 Europhys. Lett., 60 (1), pp (2002) Vibrational origin of the fast relaxation processes in molecular glass formers S. Mossa 1,2,G.Monaco 3 and G. Ruocco 2 1 Center for Polymer Studies and Department of Physics Boston University, Boston, MA 02215, USA 2 Dipartimento di Fisica and INFM, Università di Roma La Sapienza Piazzale Aldo Moro 2, Roma, I-00185, Italy 3 European Synchrotron Radiation Facility BP 220 Grenoble Cedex, F-38043, France (received 19 December 2001; accepted in final form 12 July2002) PACS Pf Glass transitions. PACS Pd Molecular dynamics calculations (Car-Parrinello) and other numerical simulations. PACS Em Molecular liquids. Abstract. We study the interaction of the relaxation processes with the density fluctuations by molecular dynamics simulation of a flexible molecular model for o-terphenyl in the liquid and supercooled phases. We find evidence, besides the structural relaxation, of a secondary vibrational relaxation whose characteristic time, few ps, is slightly temperature dependent. This i) confirms the result by Monaco et al. (Phys. Rev. E, 62 (2000) 7595) of the vibrational nature of the fast relaxation observed in Brillouin Light Scattering experiments in o-terphenyl; and ii) poses a caveat on the interpretation of the BLS spectra of molecular systems in terms of a purely center-of-mass dynamics. After the development of the Mode Coupling Theory(MCT) [1], which aimed to give a microscopic interpretation of the slowing-down of the dynamics which takes place at the liquidto-glass transformation, different experimental and numerical studies have been devoted to test its predictions, especiallythrough the studyof the densityfluctuations correlation function, or of its space Fourier transform F (Q, t). These studies have covered a broad wave vector Q and temperature T range. In particular, the high-q region has mainlybeen probed by numerical techniques [2] and inelastic neutron scattering spectroscopies [3], while the low-q region has been investigated bybrillouin Light Scattering (BLS), depolarized light scattering, and dielectric spectroscopy[4, 5]. Despite this huge effort, the interpretation of the experimental tests of the main predictions of the MCT is still at the center of a strong debate, and a general consensus has not been reached yet. Indeed, a direct comparison of theoretical predictions with experimental results can be difficult, as it occurs in some light scattering experiments [4], in particular as far as the so-called β-region of the MCT is concerned. This is the time region where the first decayof F (Q, t), that is ascribed to the dephasing of the microscopic vibrational dynamics, merges with the earliest part of the decayassociated to the structural (α) relaxation process. The MCT makes specific predictions on the shape of the F (Q, t) in this region, but additional c EDP Sciences

88 S. Mossa et al.: Vibrational origin of the fast relaxation processes etc. 93 contributions not yet included in the theory have to be taken into account in order to satisfactorilydescribe the experimental spectra [4]. Another debated issue is the recent observation of a flat background (constant loss) in the dielectric absorption spectra at few GHz [5], which seems at odds with some of the MCT predictions. In this case, the compatibilitybetween the experimental results and the theorycan be recovered but at the expense of admitting that not all variables reach the asymptotics which the MCT predicts to be universal [5]. Recently, studying the BLS spectra of o-terphenyl (otp) [6], one of the prototypical fragile glass formers, some of us found evidences besides the usual structural or α relaxation process of a secondaryrelaxation, with a characteristic time τ f lying in the ten picosecond time scale and weaklytemperature dependent. This process lies in the frequencyregion where the features associated to the β-region are expected to show up. Bycomparing the BLS spectra in the glass and in the ordered crystalline phase, and by comparing the effect of this relaxation process on the longitudinal and transverse sound waves, the vibrational nature of the observed relaxation [7] has been suggested. If confirmed, this suggestion would indicate an important reason whythe MCT predictions are not always retrieved in the light scattering experiments. Indeed the MCT is, up to now, formulated for monatomic or rigid molecules [8, 9], the complexityof real non-rigid molecules with all the processes associated to the internal vibrational dynamics not yet being taken into account. The presence of a vibrational relaxation would indicate that the molecular glass formers are not suitable benchmarks to test the MCT on the short time scale. Indeed the presence of a vibrational relaxation dynamics lying in the ps time scale would mask the spectral features predicted for the β-region. This considerations could partiallyexplain the unsatisfactoryconsistencysometimes found between the MCT predictions and the BLS experiments [4]. Here we present a molecular dynamics simulation investigation of the relaxation processes active in otp in a wide temperature range, spanning the normal and supercooled liquid region. Using the recent flexible molecular model for otp [10,11], we investigate the coupling between the densityfluctuations and the intra-molecular (vibrational) degrees of freedom. We find that i) there is a strong coupling between the densityfluctuations and the vibrational excitations, and ii) that this coupling gives rise to a vibrational relaxation in the s time scale. These findings stronglysupport the hypothesis suggested in ref. [7], and confirm the existence of a vibrational relaxation in otp in the same frequencyregion where the signatures of the MCT β-region are expected. Moreover, byanalyzing the effects of an external perturbation on the individual contributions to the intra-molecular interaction potential, we identifythe phenyl-phenyl stretching as the internal vibration mainly responsible for such a relaxation. The absolute value of the relaxation time τ f turns out to be a factor 3 shorter than the experimental value, a discrepancythat is tentativelyexplained on the basis of the classical nature of the MD simulations. Finally, we investigate the temperature dependence of τ f, finding a behavior verysimilar to that observed in the BLS experiment [12]. We have studied a microcanonical system composed by 32 molecules (96 rings, 576 Lennard Jones interaction sites) enclosed in a cubic box with periodic boundaryconditions. In the utilized model [10] the otp molecule is constituted bythree rigid hexagons (phenyl rings) of side L a =0.139 nm. Two adjacent vertices of the parent ring are bonded to one vertex of the two lateral rings bybonds of equilibrium length L b =0.15 nm. Each vertex of the hexagons is occupied bya fictitious atom of mass M CH = 13 a.m.u. representing a carbon-hydrogen pair (C-H). In the isolated molecule equilibrium position the lateral rings lie in planes that form an angle of about 54 with respect to the parent ring s plane. The interaction among the rings pertaining to all the molecules is described bya site-site pairwise additive Lennard-Jones 6-12 potential, each site corresponding to the six hexagons vertices. Moreover, the three rings of a given molecule interact among one another byan intra-

89 94 EUROPHYSICS LETTERS molecular potential, such potential being chosen in such a wayas to preserve the molecule from dissociation, to give the correct relative equilibrium positions for the three phenyl rings, and to represent at best the isolated molecule vibrational spectrum. In particular it is written in the form V intra = k c kv k, where each term V k controls a particular degree of freedom. The contributions to the intra-molecular potential relevant to the present studyare: i) the stretching along the central ring - side ring bonds (S), ii) the bending of the central ring - side rings bonds (B), and iii) the in-phase (R 1 ) and out-of-phase (R 2 ) rotation of the lateral rings along the ring-ring bond ( 1 ). The first two terms are modeled bysprings, and the interaction with the other degrees of freedom (anharmonicity) is introduced by the site-site LJ potential of different molecules. The third term, on the contrary, has been modeled in a more realistic way. In this case the relevant variables are the two angles {Φ 1, Φ 2 } between the normals to the lateral rings and the parent ring plane. An ab initio calculation of the single-molecule potential energysurface as a function of these two angles has shown that two iso-energetic configurations exist separated byan energybarrier of height V S /k B = 580 K. The nature of the rotational motion at the temperatures of interest can be summarized as follows: the two side rings can pivot in phase around the bonds crossing from one minimum to the other degenerate one; at the same time theycan perform librational out-of-phase motions. In order to represent this potential surface we express the in-phase rotation of the two side rings with a high-order (6th) polynomial and the out-of-phase rotation by a quadratic (harmonic) potential energy. Other details of the intra-molecular and inter-molecular interaction potentials, together with the values of the involved constants, are reported in ref. [10]. Previous studies on the temperature dependence of the self-diffusion coefficient [10] and of the structural (α) relaxation times [10, 11] indicate that this molecular model is capable to quantitativelyreproduce the dynamical behavior of the real system, but the actual simulated temperatures must be shifted by 20 K upward. In what follows, as our aim is to compare the simulation results with the experiments, the reported MD temperatures are always shifted by such an amount. To investigate the effect of a long-wavelength densityfluctuation as those probed in a BLS experiment on the intra-molecular vibrational dynamics we make use of a technique based on the following considerations. The characteristic time which a densityfluctuation takes to relax towards equilibrium after interacting via temperature with one or more intra-molecular degrees of freedom can be shown to be proportional to the characteristic time which the energyof the involved internal degrees of freedom takes to relax towards equilibrium after a sudden densityincrease [13]. Moreover, if onlyfew internal degrees of freedom are involved, the two characteristic times are essentiallythe same. While the former relaxation time is the one which is directlyobtained in a BLS experiment, in the following we will determine the latter one, which is easier to work out bynumerical simulation. We then proceed as follows. After an equilibration run at a given temperature, we make a sudden densityvariation of the system ( crunch ), then we follow the time evolution of the intra-molecular potential energyduring the subsequent evolution. As an example, in fig. 1 we report such a time evolution for T = 325 K and averaged over 2000 statistically independent crunches. The value of the internal potential energyis close to that pertaining to 12 harmonic oscillators (the number of internal degrees of freedom), 6RT = 16.2 kj/mol, the slight deviation being associated to the anharmonicitypresent in the parameterizations of some of the internal degrees of freedoms. It is evident that, after the densitychange, the vibrational energyrelaxes toward its new equilibrium value on a time scale of 6 ps, a value not far from ( 1 )Another contribution to the intra-molecular potential is taken into account in the model, i.e., the tilt of the ring-ring bonds. This contribution is constituted by two terms which are not relevant in this context and are not shown. It is worth noting, however, that their contribution to the total intra-molecular energy turns out to be negative.

90 S. Mossa et al.: Vibrational origin of the fast relaxation processes etc T = 325 K V intra [ kj / mol ] τ = 5.8 ps t [ ps ] Fig. 1 Time dependence of the total intra-molecular potential energy after the density jump starting from a well-equilibrated configuration at T = 325 K. The data have been obtained averaging over 2000 realizations and the error bars represent the ±σ standard deviation of the different runs data. the experimental one: τ f 20 ps. We do not expect a much better agreement between these two values as i) the intra-molecular potential model has been parameterized to represent the vibrational spectrum of the isolated molecules and not on the dynamics in the condensed phase, and ii) the intra-molecular vibrational frequencies have values up to 500 cm 1 ( 720 K); therefore, at the investigated temperature, the quantum effects (population of the vibrational levels) certainlyplaya relevant role, and theyare not considered in the present classical simulation. It is worth noting that, at this T, theα-relaxation time of density fluctuations determined bybls is τα BLS 100 ps, much longer than the time scale identified here. Also the rotational relaxational times turns out to be comparable with τα BLS [14]. Overall, we conclude from fig. 1 that there is definitivelya coupling between the density fluctuations and the vibrational degrees of freedom i.e., a vibrational relaxation is active in the system and that the relaxation time favorably compares with the experimental findings. In order to identifywhich one of the internal degree(s) of freedom is more efficiently coupled to the densityfluctuation we cannot use the crunch technique, as a densityjump simultaneouslyaffects all the intra-molecular degrees of freedom. To selectivelyperturb a specific vibration, we proceed as follows. For each temperature, we equilibrate the system with a new set of elastic constants, where the constant of interest c k is scaled to a new value c k = λc k ( 2 ), and store the final configuration. Then, starting from this configuration (t = 0), we perform two runs; we a) continue to evolve the system for 100 ps with the constants {c n}, and b) evolve the system with the original set of elastic constants {c n }. In this wayrun a) is an equilibrium run, while run b) is a run where, at t = 0, the system has been suddendly perturbed bya change in the Hamiltonian (c k c k)( 3 ). After this perturbation is applied, we follow the time evolution of the specific term of the intra-molecular interaction potential energy, and measure the time needed to the energyto relax towards its equilibrium value. As an example, in fig. 2 (A-D), we report the time evolution of some of the terms V k (k = R 1,R 2,B,S)for ( 2 )Wehaveusedavalueofλ =1.2; this value has been chosen by some preliminary runs in such a way as to clearly observe a response to the introduced perturbation but to stay in the linear response regime. ( 3 )We have performed the runs applying the modification c k c k (instead of c k c k ) after the equilibration of the systems with c k to always follow the relaxation dynamics of the correct (c k)system. Itisworth noting that, by modifying the elastic constants, we are not perturbing a specific vibration, as the latter are combinations of the different internal variables (bonds lengths and angles). However, as each molecular eigenmode mainly projects on one of the internal variables, we can anyway identify the vibrational modes responsible for the relaxation process.

91 96 EUROPHYSICS LETTERS ( A ) T = 370 K V * R ( t ) 1 V R ( t ) 1 R R 1 ( t ) t 1.0 V k, V k * [ kj / mol ] ( B ) V * R 2 ( t ) V R 2 ( t ) R R 2 ( t ) t ( C ) V * B ( t ) V B ( t ) R B ( t ) t t [ ps ] [ kj / mol ] ( D ) V S, V* S 7.42 V * S ( t ) V S ( t ) t [ ps ] Fig. 2 (A) Signals (main panel) and relaxation function (inset) of the potential controlling the on-phase pivoting of the lateral rings at T = 370 K and λ =1.2. (B) As above for the out-phase librations. (C) As above for the bending of the central ring - side rings bond. (D) As above for the stretching between the sides and central ring along the molecular bonds; it is worth noting that V S decays on a time scale much longer than that of the other terms shown above. the selected temperature T = 370 K (squares). The value of the energies measured in runs a) (open triangles) are also shown for comparison. Finally, in the insets, the difference R k (t) (normalized to unityat t = 0) between runs b) and a) are shown. As can be seen, all the potential term but V S decayon a veryfast time scale ( 100 fs or less followed byan oscillating tail that vanishes in 0.5 ps). ThetermV S on the contrarydecays on a much longer (> 1ps) time scale. This term, controlling the stretching along the two bonds connecting the parent ring with the two side rings, is therefore the main candidate for the vibrational relaxation observed at few ten ps. It is worth noting that, as far as the ring-ring stretching vibration is

92 S. Mossa et al.: Vibrational origin of the fast relaxation processes etc R S ( t ) T = 325 [ K ] T = τ f [ ps ] 10 1 EXPT. ( Ref. [ 12 ] ) MD ( This work ) T = 440 T = t [ ps ] t [ ps ] Fig / T [ K -1 ] Fig. 4 Fig. 3 Relaxation functions R S(t) of the bond stretching potential at the four indicated temperatures. Fig. 4 Experimental (open circles, from ref. [12]) and MD (full diamonds) results for τ f (T ) reported in an Arrhenius plot. The MD data have been multiplied by a common factor F = 6 in order to show a T behaviour similar to that of the experimental data. concerned, the stretching potential energyat long time in run b) (i.e., with the set {c n } in the Hamiltonian) does not coincide with that of run a) (i.e., with the set {c n}). This indicates the presence of a relevant anharmonicityaffecting this specific degree of freedom (in the absence of anharmonicitythe two values are expected to be the same due to the energyequipartition). The functions R S (t) are plotted in fig. 3 at selected temperatures, together with their best fit to an exponential decay. A slight but evident temperature dependence of the relaxation time is present. The T -dependence of the relaxation time derived from these fits is reported in fig. 4, together with the relaxation times for the fast process experimentallydetermined in ref. [12]. In this figure the MD data have been multiplied bya common factor F = 6 in order to emphasize that their T behavior is verysimilar to that of the experimental data. Both sets of data show an Arrhenius behavior at low T (with activation energy E = 0.28 ± 0.01 kj/mol) and a different (steepest) trend at high T, where the fast relaxation process merges with the structural one. We notice the T -dependence agreement between the experimental data and the simulation after the appropriate rescaling. On the contrary, there is a discrepancy of a factor of about 6 between the MD and the experimental time scales. This difference could be explained, as noticed above, considering a non-perfect parameterizations of the intramolecular interaction potential and the fact that we expect strong quantum effects on the studied process. Moreover, in the present case further differences certainlyarise from the different perturbation utilized in the MD simulation (changes of force constants) and in the experiment (densityfluctuations). In conclusion, bystudying the coupling of external perturbations (a densityfluctuation and a fictitious coupling constant modification) with the intra-molecular vibrations in a flexible model of otp molecule we demonstrate i) the existence of a vibrational relaxation process with a non-negligible strength and a relaxation time in the few ps time scale; ii) that this relaxation process is mainly associated to the phenyl-phenyl stretching; and iii) that the relaxation time shows a temperature behavior verysimilar to that of the experimentallydetermined τ f [6,7,12]. The quantitative discrepancies between the simulate and the experimental relaxation times

93 98 EUROPHYSICS LETTERS can be tentativelyassigned to the quantum nature of the real vibration, for which, at room temperature, hω v /k B T 2. The present findings give strong support to the vibrational origin of the fast relaxation process observed in otp. It is tempting to generalize this conclusion to other systems where such a fast process has been observed (PC [15] and PB [16]). Given the depicted scenario, one should be extremelycareful in drawing conclusion on either the validity, or the failure, of the MCT bythe analysis of the isotropic light scattering spectra. This is especiallytrue when analyzing the MCT β-region, as it lies at frequencies that coincide with the typical fast process ones and as its main features (susceptibilityminima, a and b exponents, knees, etc.) can be masked bythe fast process itself ( 4 ). We thank R. Di Leonardo, G. Lacorata and F. Sciortino for veryuseful discussions and comments. REFERENCES [1] Götze W., Liquids, Freezing and the Glass Transition, editedbyhansen J. P., Levesque D. and Zinn-Justin J. (North-Holland, Amsterdam) 1991; Götze W. and Sjörgen L., Rep. Prog. Phys., 55 (1992) 241; Götze W., J. Phys. Condens. Matter, 11 (1999) A1. [2] Gleim T., Kob W. and Binder K., Phys. Rev. Lett., 81 (1998) 4404; Sciortino F. and Tartaglia P., J. Phys. Condens. Matter, 11 (1999) A261. [3] Petry W. et al., Z. Phys. B, 83 (1991) 175; Kiebel K. et al., Phys. Rev. B, 45 (1992) 10301; Tölle A. et al., Phys. Rev. E, 56 (1997) 809; Tölle A. et al., Phys. Rev. Lett., 80 (1998) [4] Gapinski J. et al., J. Chem. Phys., 110 (1999) 2312; Wiedersich J. et al., J. Phys. Condens. Matter, 11 (1999) A147; Aouadi A. et al., J. Chem. Phys., 112 (2000) 9860; Novikov V. N., Surovtsev N. V., Wiedersich J., Adichtchev S., Kojima S. and Rössler E., Europhys. Lett., 57 (2002) 838. [5] Ngai K. L., J. Chem. Phys., 110 (1999) 10576; Casalini R., Ngai K. L. and Roland C. M., J. Chem. Phys., 112 (2000) 5181; Wuttke J. et al., Phys. Rev. E, 61 (2000) 2730; Brand R. et al., Phys. Rev. B, 62 (2000) 8878; Goldammer M. et al., Phys. Rev. E, 64 (2001) [6] Monaco G. et al., Phys. Rev. Lett., 82 (1999) [7] Monaco G. et al., Phys. Rev. E, 62 (2000) [8] Schilling R. et al., Phys. Rev. E, 56 (1997) 2932; Franosh T. et al., Phys. Rev. E, 56 (1997) 5759; Fabbian L. et al., Phys. Rev. E, 60 (1999) [9] Chong S. H. et al., Phys. Rev. E, 58 (1998) 6188; Chong S. H. et al., Phys. Rev. E, 63 (2001) [10] Mossa S. et al., Phys. Rev. E, 62 (2000) 612. [11] Mossa S., Ruocco G. and Sampoli M., Phys. Rev. E, 64 (2001) ; Mossa S. et al., J. Chem. Phys., 116 (2002) 1077; Mossa S., Ruocco G. and Sampoli M., J. Chem. Phys., 117 (2002) [12] Monaco G. et al., Phys. Rev. E, 63 (2001) [13] Herzfeld K. F. and Litovitz T. A., Absorption and Dispersion of Ultrasonic Waves (Academic Press, London) 1965; Bhatia A. B., Ultrasonic Absorption (Clarendon Press, London) [14] Mossa S. et al., unpublished. [15] Di Leonardo R., Thesis, Università di L Aquila (1998), unpublished. [16] Fioretto D. et al., Phys. Rev. B, 65 (2002) ( 4 )We note that an extension of MCT to non-rigid molecules would allow to compare the present results and the theory. The interference of the fast relaxation (due to vibrations) with the critical law predicted by MCT for rigid molecules is not an evidence that present MCT is wrong, but only that it is not general enough.

94 arxiv:cond-mat/ v1 27 Jun 2002 WATER AT POSITIVE AND NEGATIVE PRESSURES H. E. STANLEY, 1 M. C. BARBOSA, 1,2 S. MOSSA, 1 P. A. NETZ, 3 F. SCIORTINO, 4 F. W. STARR, 5 and M. YAMADA 1 1 Center for Polymer Studies and Department of Physics, Boston University Boston, MA USA 2 Instituto de Física, Universidade Federal do Rio Grande do Sul, Caixa Postal , Porto Alegre, RS, Brazil 3 Departamento de Química, Universidade Luterana do Brasil , Canoas, RS, Brazil 4 Dipartimento di Fisica Università di Roma La Sapienza, Istituto Nazionale di Fisica della Materia, and INFM Center for Statistical Mechanics and Complexity, Piazzale Aldo Moro 2, Roma, Italy 5 Polymers Division and Center for Theoretical and Computational Materials Science National Institute of Standards and Technology, Gaithersburg, MD USA Abstract We review recent results of molecular dynamics simulations of two models of liquid water, the extended simple point charge (SPC/E) and the Mahoney-Jorgensen transferable intermolecular potential with five points (TIP5P), which is closer to real water than previously-proposed classical pairwise additive potentials. Simulations of the TIP5P model for a wide range of deeply supercooled states, including both positive and negative pressures, reveal (i) the existence of a non-monotonic nose-shaped temperature of maximum density (TMD) line and a nonreentrant spinodal, (ii) the presence of a low temperature phase transition. The TMD that changes slope from negative to positive as P decreases and, notably, the point of crossover between the two behaviors is located at ambient pressure (temperature 4 C, and density 1 g/cm 3 ). Simulations on the dynamics of the SPC/E model reveal (iii) the dynamics at negative pressure shows a minimum in the diffusion constant D when the density is decreased at constant temperature, complementary to the known maximum of D at higher pressures, and (iv) the loci of minima of D relative to the spinodal shows that they are inside the thermodynamically metastable regions of the phase-diagram. These dynamical results reflect the initial enhancement and subsequent breakdown of the tetrahedral structure and of the hydrogen bond network as the density decreases. 1

95 2 1. Introduction Water is an important liquid in nature, and is also fundamental in chemical and technological applications. Although the individual water molecule has a simple chemical structure, water is considered a complex fluid because of its anomalous behavior [1, 2, 3, 4, 5, 6]. It expands on freezing and, at a pressure of 1 atm, the density has a maximum at 4 C. Additionally, there is a minimum of the isothermal compressibility at 46 C and a minimum of the isobaric heat capacity at 35 C [7]. These anomalies are linked with the microscopic structure of liquid water, which can be regarded as a transient gel a highly associated liquid with strongly directional hydrogen bonds [8, 9]. Each water molecule acts as both a donor and an acceptor of bonds, generating a structure that is locally ordered, similar to that of ice, but maintaining the longrange disorder typical of liquids. Despite the extensive work that has been done on water, many aspects of its behavior remain unexplained. Several scenarios have been proposed to account for the the anomalous behavior of the thermodynamic response functions on cooling, each predicting a different behavior for the liquid spinodal, the line of the limit of stability separating the region where liquid water is metastable from the region where the liquid is unstable. (i) According the stabilitylimit conjecture [10, 11], the pressure of the spinodal line should decrease on cooling, become negative, and increase again after passing through a minimum. It reenters the positive pressure region of the phase diagram at a very low temperature, thereby giving rise to a line of singularities in the positive pressure region, and consequently the increase in the thermodynamic response functions on cooling in the anomalous region is due to the proximity of this reentrant spinodal. (ii) The critical point hypothesis [12, 13, 14, 15, 16, 17], proposes a new critical point at the terminus of a first-order phase transition line separating two liquid phases of different density. The anomalous increases of the response functions, compressibility, specific heat, and volume expansivity, is interpreted in terms of this critical point. (iii) The singularity-free hypothesis [9, 18, 19] proposes that actually there is no divergence close to the anomalous region; the response functions grow on lowering temperature but remain finite, attaining maximum values. Water properties and anomalies can be strongly influenced by the physical or chemical properties of the medium [1, 3, 4, 5, 20, 21]. The effect not only of applied pressure, but also of negative pressure ( stretching ) is remarkable. The study of the behavior of this fluid under nega-

96 Water at Positive and NegativePressures 3 tive pressures is relevant not only from the academic point of view, but also for realistic systems. For example, negative pressures are observed [22], and seem to play an important role in the mechanism of water transport in plants. Therefore, properties that modify the structure of water, especially if this modification is similar to the effect of stretching (as is the case in some hydrogels [20]), also influence its dynamical behavior. Dynamic properties, such as the diffusion constant, have been studied in detail for water systems at atmospheric and at high positive pressures, both experimentally [23, 24] as well as by computer simulations [25, 26, 27, 28, 29, 30, 31]. The increase of pressure increases the presence of defects and of interstitial water molecules in the network [26]. They disrupt the tetrahedral local structure, weakening the hydrogen bonds, and thus increasing the diffusion constant [30, 31]. However, a further increase in the pressure leads to steric effects which works in the direction of lowering the mobility. The interplay of these factors leads to a maximum in the diffusion constant [30, 31] at some high density ρ max. Above this density (or corresponding pressure), the diffusion of water is in some sense like that of a normal liquid, controlled by hindrance, with the hydrogen bonds playing a secondary role. However, the behavior at very low ρ is less well understood. 2. Location of the Spinodal at Positive and Negative Pressures Relatively few experimental works [32, 33] and simulations [12, 16, 29, 30, 35, 34] have been performed on stretched water. In this negative pressure region of the phase diagram the system is metastable, and becomes unstable beyond the spinodal line, so locating the spinodal we can ensure that the simulated state points lie in the metastable and not in the unstable region. Moreover, the shape of the spinodal can test the stability-limit conjecture against the critical point hypothesis and the singularity-free interpretation, so we first discuss the density and pressure of the spinodal, which we denote ρ sp (T ) and P sp (T ), respectively. Yamada and her coworkers [36] simulated a system of N = 343 molecules interacting with the TIP5P potential [37]. TIP5P is a fivesite, rigid, non-polarizable water model, not unlike the ST2 model [38]. The TIP5P potential accurately reproduces the density anomaly at 1 atm and exhibits excellent structural properties when compared with experimental data [37, 39]. The TMD shows the correct pressure dependence, shifting to lower temperatures as pressure is increased. Under ambient conditions, the diffusion constant is close to the experimental

97 4 value, with reasonable temperature and pressure dependence away from ambient conditions [37]. Equilibration runs were performed at constant T = 215 K P T = 320 K Pressure P [ MPa ] ρ n = 10 n = 5 0 T = 215 K n = Density ρ [ g / cm 3 ] Figure 1. Dependence on density of the pressure at all temperatures investigated (T = 215, 220, 230, 240, 250, 260, 270, 280, 290, 300, 320 K, from bottom to top). Each curve has been shifted by n 150 MPa to avoid overlaps. An inflection appears as T is decreased, transforming into a flat coexistence region at T = 215 K, indicating the presence of a liquid-liquid transition. Inset: A detailed view of the T = 215 K isotherm. Adapted from [36]. T. After thermalization at T = 320 K the thermostat temperature was set to the temperature of interest. The system evolved for a time

98 Water at Positive and NegativePressures 5 longer than the structural relaxation time τ α, defined as the time at which F s (Q 0, τ α ) = 1/e, where F s (Q 0, t) is the self-intermediate scattering function evaluated at Q 0 = 18 nm 1, the location of the first peak of the static structure factor. In the time τ α, each molecule diffuses on average a distance of the order of the nearest neighbor distance. We use the final configuration of the equilibration run to start a production run of length greater than several τ α and then analyze the calculated trajectory. Pressure [ MPa ] TMD line Spinodal ( a ) Temperature [ K ] Figure 2. Pressure along seven isochores; the minima correspond to the temperature of maximum density line (dashed line). Note the nose of the TMD line at T = 4 C. Stars denote the liquid spinodal line, which is not reentrant, and terminates at the liquid-gas critical point. Adapted from [36]. Figure 1 shows results for pressure along isotherms. At lower temperatures an inflection develops, which becomes a flat isotherm at the lowest temperature, T = 215 K. The presence of a flat region indicates that a phase separation takes place; the critical temperature is T C = 217 ± 3 K, the critical pressure is P C = 340± 20 MPa, and the critical density ρ C = 1.13 ± 0.04 g/cm 3. Figure 2(a) plots the pressure along isochores. The curves show minima as a function of temperature; the locus of the minima is the TMD line, since ( P/ T ) V = α P /K T. Note that the pressure exhibits a mini-

99 6 mum if the density passes through a maximum (α p = 0). It is clear that, as in the case of ST2 water, TIP5P water has a TMD that changes slope from negative to positive as P decreases. Notably, the point of crossover between the two behaviors is located at ambient pressure, T 4 C, and ρ 1 g/cm 3. Also plotted the spinodal line, obtained by fitting the isotherms (for T 300K) of Fig. 1 to the form P (T, ρ) = P s (T ) + A [ρ ρ s (T )] 2, where P s (T ) and ρ s (T ) denote the pressure and density of the spinodal line. This functional form is the mean field prediction for P (ρ) close to a spinodal line. For T 250K, P s (T ) is calculated by estimating the location of the minimum of P (ρ). The results in Fig.2 show that the liquid spinodal line is not reentrant and does not intersect the TMD line. 3. Dynamic Properties We next discuss results on the dynamics of stretched water recently obtained by Netz and his collaborators [40]. While there are a large number of intermolecular potential functions used to simulate water, each of which gives slightly different results, the overall thermodynamics picture obtained from these models is generally very similar. Since dynamic properties are particularly sensitive to the potential choice, the extended simple point charge (SPC/E) potential is used since it reproduces both the maximum in diffusivity under pressure as well as the power-law behavior of dynamics properties on cooling. For understanding the properties of water at negative pressure, simulations are particularly important since experiments are very difficult to perform in this region. The effect of extreme conditions on the flow of the liquid is assessed by calculating the diffusion constant D, defined by the asymptotic value of the slope of the mean square displacement versus time. We show D along isotherms in Fig. 3. For T 260 K, D has a minimum value at ρ 0.9 g/cm 3, which becomes more pronounced at lower T. This behavior can be understood considering the structural changes that occur with decreasing density. At low T, the decreased density enhances the local tetrahedral ordering, which leads to a decrease in D. Further decreases in density reduces the stability of the tetrahedral structure and causes an increase of D. The location of the minimum is near the ice Ih density g/cm 3, which is the density where the perfect tetrahedral order occurs. The behavior of the minimum of D, D min (T ), complements the known behavior of D max (T ) for the same model [30, 31, 35], where a maximum

100 Water at Positive and NegativePressures T = 260K 250K 240K D (10 5 cm 2 /s) (b) 230K 210K ρ (g/cm 3 ) Figure 3. Dependence of the diffusion constant D on ρ along isotherms (for ρ 1.0 g/cm 3 ). Open symbols are the new simulations we report, and filled symbols are from Ref. [30]. Adapted from [40]. occurs due to breaking hydrogen bonds at high pressure; the density of the D min (T ) increases slightly with increasing T, while the density of D max (T ) decreases with increasing T [31]. This is expected, since the range of densities where anomalous behavior occurs expands with decreasing T. We show the loci of D min (T ) and D max (T ), along with the spinodal and locus of density maxima in Fig. 4. Below the spinodal, D also increases, since the mobility of the gas is larger than that of the liquid. However, the simulations clearly show that D min for the liquid occurs prior to the onset of cavitation, and so the location of D min we estimate is not affected by phase separation. Recently, Ref. [35] estimated the location of D min for the same model along several isotherms, and associated D min with a maximum in orientational order.

101 8 400 D max 200 P (MPa) 0 T MD 200 D min Spinodal T (K) Figure 4. Relation of the loci of maxima and minima of D with T MD and the spinodal. Open symbols are from the present work, and filled symbols are from Ref. [30]. Adapted from [40]. 4. Conclusions Water exhibits a very complex structure and its properties and anomalies are strongly influenced by variations of pressure. For high densities (ρ > ρ max ), water behaves as a normal liquid and the decrease of D with increasing pressure is governed by steric effects. For ρ min < ρ < ρ max, as the pressure is decreased, the presence of defects and interstitial water decrease, the tetrahedral structure dominates, with stronger hydrogen bonds. This process reaches its maximum at ρ = ρ min ρ ice. Further stretching destabilizes the hydrogen bond network, leading to an increase in mobility. The locus of D min roughly tracks the spinodal, not surprising since the same breakdown of tetrahedral order that gives rise to D min also facilitates cavitation. Acknowledgments We thank D. R. Baker, S. V. Buldyrev, P. Debenedetti, G. Franzese, W. Kob, E. La Nave, M. Marquez and C. Rebbi for useful discussions,

102 Water at Positive and NegativePressures 9 and NSF Grant CHE , the Conselho Nacional de Desenvolvimento Cientifico e Technologico (CNPq), the Fundacao de Amparo a Pesquisa do Rio Grande do Sul (Fapergs) for support. MY thanks NSF Grant GER for support as a Graduate Research Trainee at the Boston University Center for Computational Science, FS thanks MURST COFIN 2000 and INFM Iniziativa Calcolo Parallelo, and FWS thanks the National Research Council. References [1] For elementary introductions to recent work on liquid water, the reader may wish to consult P. Ball, Life s Matrix: A Biography of Water (Farrar Straus and Giroux, New York, 2000) or P. G. Debenedetti and H. E. Stanley, The Novel Physics of Water at Low Temperatures, Physics Today (submitted). [2] V. Brazhkin. S. V. Buldyrev, V. N. Ryzhov, and H. E. Stanley [eds], New Kinds of Phase Transitions: Transformations in Disordered Substances Proc. NATO Advanced Research Workshop, Volga River (Kluwer, Dordrecht, 2002). [3] O. Mishima and H. E. Stanley, Nature 396, 329 (1998). [4] M.-C. Bellissent-Funel, ed., Hydration Processes in Biology: Theoretical and Experimental Approaches (IOS Press, Amsterdam, 1999). [5] H. E. Stanley, S. V. Buldyrev, N. Giovambattista, E. La Nave, A Scala, F. Sciortino, and F. W. Starr, [Proc. IUPAP Statphys21, Cancun] Physica A 306, (2002). [6] S. V. Buldyrev, G. Franzese, N. Giovambattista, G. Malescio, M. R. Sadr- Lahijany, A. Scala, A. Skibinsky, and H. E. Stanley [Proc. International Conf. on Scattering Studies of Mesoscopic Scale Structure and Dynamics in Soft Matter] Physica A 304, (2002). [7] R. C. Dougherty and L. N. Howard, J. Chem. Phys. 109, 7379 (1998). [8] A. Geiger, F. H. Stillinger, and A. Rahman, J. Chem. Phys. 70, 4185 (1979). [9] H. E. Stanley and J. Teixeira, J. Chem. Phys. 73, 3404 (1980). [10] R. J. Speedy, J. Chem. Phys. 86, 982 (1982); Ibid 86, 3002 (1992). [11] R. J. Speedy, J. Chem. Phys. 91, 3354 (1987). [12] P. H. Poole, F. Sciortino, U. Essmann, and H. E. Stanley, Nature 360, 324 (1992); Phys. Rev. E 48, 3799 (1993); F. Sciortino, P. H. Poole, U. Essmann, and H. E. Stanley, Ibid. 55, 727 (1997); S. Harrington, R. Zhang, P. H. Poole, F. Sciortino, and H. E. Stanley, Phys. Rev. Lett. 78, 2409 (1997). [13] O. Mishima, J. Chem. Phys. 100, 5910 (1994). [14] P. H. Poole, F. Sciortino, T. Grande, H. E. Stanley and C. A. Angell, Phys. Rev. Lett. 73, 1632 (1994); C. F. Tejero and M. Baus, Phys. Rev. E 57, 4821 (1998); T. M. Truskett, P. G. Debenedetti, S. Sastry, and S. Torquato, J. Chem. Phys (1999). [15] M.-C. Bellissent-Funel, Europhys. Lett. 42, 161 (1998); O. Mishima and H. E. Stanley, Nature 392, 192 (1998). [16] H. Tanaka, J. Chem. Phys. 105, 5099 (1996).

103 10 [17] A. Scala, F. W. Starr, E. La Nave, H. E. Stanley and F. Sciortino, Phys. Rev. E 62, 8016 (2000). [18] S. Sastry, P. G. Debenedetti, F. Sciortino, and H. E. Stanley, Phys. Rev. E 53, 6144 (1996). [19] L. P. N. Rebelo, P. G. Debenedetti, and S. Sastry, J. Chem. Phys. 109, 626 (1998). [20] P. A. Netz and Th. Dorfmüller, J. Phys. Chem. B 102, 4875 (1998). [21] K. Koga, X. C. Zeng, and H. Tanaka, Chem. Phys. Lett. 285, 278 (1998). [22] W. T. Pockman, J. S. Sperry, and J. W. O Leary, Nature 378, 715 (1995). [23] J. Jonas, T. DeFries, and D. J. Wilbur, J. Chem. Phys. 65, 582 (1976). [24] F. X. Prielmeier, E. W. Lang, R. J. Speedy, and H.-D. Lüdemann, Phys. Rev. Lett. 59, 1128 (1987); Ber. Bunsenges. Phys. Chem. 92, 1111 (1988). [25] M. Rami Reddy and M. Berkovitz, J. Chem. Phys. 87, 6682 (1987). [26] F. Sciortino, A. Geiger, and H. E. Stanley, Nature 354, 218 (1991); Ibid., J. Chem. Phys. 96, 3857 (1992). [27] N. Giovambattista, F. W. Starr, F. Sciortino, S. V. Buldyrev, and H. E. Stanley, Phys. Rev. E 65, (2002) cond-mat/ [28] E. La Nave, A. Scala, F. W. Starr, H. E. Stanley and F. Sciortino, Phys. Rev. E 64, (2001); E. La Nave, H. E. Stanley and F. Sciortino, Phys. Rev. Letters 88, to (2002) cond-mat/ [29] P. Gallo, F. Sciortino, P. Tartaglia, and S.-H. Chen, Phys. Rev. Lett. 76, 2730 (1996). [30] F. W. Starr, F. Sciortino, and H. E. Stanley, Phys. Rev. E 60, 6757 (1999); F. W. Starr, S. T. Harrington, F. Sciortino, and H. E. Stanley, Phys. Rev. Lett., 82, 3629, (1999). [31] A. Scala, F. W. Starr, E. La Nave, F. Sciortino and H. E. Stanley, Nature 406, 166 (2000). [32] S. J. Henderson and R. J. Speedy, J. Phys. E: Scientific Instrumentation 13, 778 (1980). [33] J. L. Green, D. J. Durben, G. H. Wolf, and C. A. Angell, Science 249, R649 (1990). [34] I. I. Vaisman, L. Perera, and M. L. Berkovitz, J. Chem. Phys. 98, 9859 (1993). [35] J. R. Errington and P. G. Debenedetti, Nature 409, 318 (2001). [36] M. Yamada, S. Mossa, H. E. Stanley, F. Sciortino, Phys. Rev. Letters 88, (2002); cond-mat/ [37] M. W. Mahoney and W. L. Jorgensen, J. Chem. Phys. 112, 8910 (2000); Ibid. 114, 363 (2001). [38] F. H. Stillinger and A. Rahman, J. Chem. Phys. 60, 1545 (1974). [39] J. M. Sorenson, G. Hura, R. M. Glaeser, and T. Head-Gordon, J. Chem. Phys. 113, 9149 (2000). [40] P. A. Netz, F. W. Starr, H. E. Stanley, and M. C. Barbosa, J. Chem. Phys. 115, (2001); cond-mat/ ; P. A. Netz, F. W. Starr, H. E. Stanley, and M. C. Barbosa, cond-mat/ ; P. A. Netz, F. Starr, M. C. Barbosa, H. E. Stanley, cond-mat/

104 Physica A 315 (2002) Statistical physics andliquidwater at negative pressures H. Eugene Stanley a;, M.C. Barbosa a;b, S. Mossa a, P.A. Netz c, F. Sciortino d;e, F.W. Starr f, M. Yamada a a Center for Polymer Studies, Department of Physics, Boston University, 590 Commonwealth Avenue, Boston, MA 02215, USA b Instituto de Fsica, Universidade Federal do Rio Grande do Sul, Caixa Postal 15051, Porto Alegre, RS, Brazil c Departamento de Qumica, Universidade Luterana do Brasil, Canoas, RS, Brazil d Dipartimento di Fisica, Universita di Roma La Sapienza, Istituto Nazionale di Fisica della Materia, Italy e INFM Center for Statistical Mechanics and Complexity, Piazzale Aldo Moro 2, Rome, Italy f Polymers Division and Center for Theoretical and Computational Materials Science, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA Abstract Angell and his collaborators have underscored the importance of studying water under all extremes of pressure squeezing to high pressures andstretching to negative pressures. Here we review recent results of molecular dynamics simulations of two models of liquid water, the extendedsimple point charge (SPC/E) andthe Mahoney-Jorgensen transferable intermolecular potential with ve points (TIP5P), which is closer to real water than previously proposedclassical pairwise additive potentials. In particular, we describe simulations of the TIP5P model for a wide range of deeply supercooled states, including both positive and negative pressures, which reveal (i) the existence of a non-monotonic nose-shaped temperature of maximum density (TMD) line anda non-reentrant spinodal, (ii) the presence of a low-temperature phase transition. The TMD that changes slope from negative to positive as P decreases and, notably, the point of crossover between the two behaviors is locatedat ambient pressure (temperature 4 C, and density 1g=cm 3 ). We also describe simulations of the dynamics of the SPC/E model, which reveal (iii) the dynamics at negative pressure shows a minimum in the diusion constant D when the density is decreased at constant temperature, complementary to the known maximum of D at higher pressures, and(iv) the loci of minima of D relative to the spinodal shows that they are inside the thermodynamically metastable regions of the phase diagram. These dynamical results Corresponding author. Tel.: ; fax: address: hes@bu.edu (H.E. Stanley) /02/$ - see front matter c 2002 Elsevier Science B.V. All rights reserved. PII: S (02)

105 282 H.E. Stanley et al. / Physica A 315 (2002) reect the initial enhancement and subsequent breakdown of the tetrahedral structure and of the hydrogen bond network as the density decreases. c 2002 Elsevier Science B.V. All rights reserved. Keywords: Water; Diusion 1. Introduction Water is an important liquidin nature, andis also fundamental in chemical and technological applications. Although the individual water molecule has a simple chemical structure, water is considereda complex uidbecause of its anomalous behavior [1 9]. It expands on freezing, and at a pressure of 1 atm, the density has a maximum at 4 C. Additionally, there is a minimum of the isothermal compressibility at 46 C anda minimum of the isobaric heat capacity at 35 C[10]. These anomalies are linked with the microscopic structure of liquidwater, which can be regardedas a transient gel a highly associated liquid with strongly directional hydrogen bonds [11 13]. Each water molecule acts as both a donor and an acceptor of bonds, generating a structure that is locally ordered, similar to that of ice, but maintaining the long-range disorder typical of liquids. Despite water s ubiquitous role in disciplines as disparate as biochemistry and manufacturing, the development of a coherent view of the behavior of aqueous systems remains one of chemistry s greatest research challenges. This is particularly true for non-ambient conditions, where far less research has occurredandfar less intuition exists. Nevertheless, water under extremes of pressure, temperature, and solute concentration is of interest to many disciplines. For example, water serves as an eective medium for biochemical processes even at hyperbaric and hyperthermic conditions. Life goes on every day at truly remarkable extremes of pressure and temperature near deep sea hydrothermal vents; in fact, biochemical reactions sucient to sustain microorganisms have been foundat 113 C and1100 atm [14]. Despite the extensive work that has been done on water, many aspects of its behavior remain unexplained. Several scenarios have been proposedto account for the anomalous behavior of the thermodynamic response functions on cooling, each predicting a dierent behavior for the liquidspinodal, the line of the limit of stability separating the region where liquid water is metastable from the region where the liquidis unstable. (i) According to the stability-limit conjecture [15 17], the pressure of the spinodal line should decrease on cooling, become negative, andincrease again after passing through a minimum. It reenters the positive pressure region of the phase diagram at a very low temperature, thereby giving rise to a line of singularities in the positive pressure region, andconsequently the increase in the thermodynamic response functions on cooling in the anomalous region is due to the proximity of this reentrant spinodal. (ii) The critical point hypothesis [18 29] proposes a new critical point at the terminus of a rst-order phase transition line separating two liquid phases of dierent density. The anomalous increases of the response functions, compressibility, specic heat, andvolume expansivity are interpretedin terms of this critical point. (iii) The singularity-free hypothesis [11,13,30,31]

106 H.E. Stanley et al. / Physica A 315 (2002) proposes that actually there is no divergence close to the anomalous region; when temperature is decreased, response functions grow but remain nite, attaining maximum values. Water properties andanomalies can be strongly inuencedby the physical or chemical properties of the medium [1 6,32,33]. The eect of not only appliedpressure, but also negative pressure ( stretching ) is remarkable. The study of the behavior of this uid under negative pressures is relevant not only from the academic point of view, but also for realistic systems. For negative, example pressures are observed[34,35] and seem to play an important role in the mechanism of water transport in plants. Therefore, properties that modify the structure of water, especially if this modication is similar to the eect of stretching (as is the case in some hydrogels [32]), also inuence its dynamical behavior. Dynamic properties, such as the diusion constant, have been studied in detail for water systems at atmospheric andat high positive pressures, both experimentally [36 38] as well as by computer simulations [39 50]. The increase of pressure increases the presence of defects and of interstitial water molecules in the network [40,41]. They disrupt the tetrahedral local structure, weakening the hydrogen bonds, and thus increasing the diusion constant [48 50]. However, a further increase in the pressure leads to steric eects which works in the direction of lowering the mobility. The interplay of these factors leads to a maximum in the diusion constant [48 50] at some high density max. Above this density (or corresponding pressure), the diusion of water is in some sense like that of a normal liquid, controlled by hindrance, with the hydrogen bonds playing a secondary role. However, the behavior at very low is less well understood. 2. Location of the spinodal at positive and negative pressures Relatively few experimental works [34,51] andsimulations [18 21,28,47 49,52,53] have been performedon stretched water. In this negative pressure region of the phase diagram the system is metastable, andbecomes unstable beyondthe spinodal line, so by locating the spinodal we can ensure that the simulated state points lie in the metastable andnot in the unstable region. Moreover, the shape of the spinodal can test the stability-limit conjecture against the critical point hypothesis andthe singularity-free interpretation, so we rst discuss the density and pressure of the spinodal, which we denote sp (T) and P sp (T ), respectively. Yamada and her coworkers [54] simulateda system of N = 343 molecules interacting with the transferable intermolecular potential with ve points (TIP5P) [55,56]. TIP5P is a ve-site, rigid, non-polarizable water model, not unlike the ST2 model [57]. The TIP5P potential accurately reproduces the density anomaly at 1 atm and exhibits excellent structural properties when comparedwith experimental data [55,56,58]. The temperature of maximum density (TMD) shows the correct pressure dependence, shifting to lower temperatures as pressure is increased. Under ambient conditions, the diusion constant is close to the experimental value, with reasonable temperature and pressure dependence away from ambient conditions [55,56]. Equilibration runs were

107 284 H.E. Stanley et al. / Physica A 315 (2002) T = 215 K P T = 320 K Pressure P [MPa] ρ n = n = 5 0 T = 215 K n = Density ρ [g / cm 3 ] Fig. 1. Dependence on density of the pressure at all temperatures investigated (T = 215; 220; 230; 240; 250; 260; 270; 280; 290; 300; 320 K, from bottom to top). Each curve has been shiftedby n 150 MPa to avoid overlaps. An inection appears as T is decreased, transforming into a at coexistence region at T =215 K, indicating the presence of a liquid liquid transition. Inset: A detailed view of the T = 215 K isotherm. Courtesy of M. Yamada. performedat constant T. After thermalization at T = 320 K the thermostat temperature was set to the temperature of interest. The system evolvedfor a time longer than the structural relaxation time, dened as the time at which the self-intermediate scattering function evaluatedat the location of the rst peak of the static structure factor, has decreased by a factor of 2.7. In the time, each molecule diuses on average a distance of the order of the nearest-neighbor distance. We use the nal conguration of the equilibration run to start a production run of length greater than several and then analyze the calculatedtrajectory. Fig. 1 shows results for pressure along isotherms. At lower temperatures an inection develops, which becomes a at isotherm at the lowest temperature, T = 215 K. The presence of a at region indicates that a phase separation takes place; the critical temperature is T C = 217 ± 3 K, the critical pressure is P C = 340 ± 20 MPa, andthe critical density C =1:13 ± 0:04 g=cm 3. Fig. 2(a) plots the pressure along isochores. The curves show minima as a function of temperature; the locus of the minima is the TMD line, since (9P=9T) V = P =K T. Note that the pressure exhibits a minimum if the density passes through a maximum ( P = 0). It is clear that, as in the case of ST2 water, TIP5P water has a TMD that

108 H.E. Stanley et al. / Physica A 315 (2002) Pressure [MPa] TMD line Spinodal Temperature [K] Fig. 2. Pressure along seven isochores; the minima correspondto the TMD line (dashedline). Note the nose of the TMD line at T =4 C. Stars denote the liquid spinodal line, which is not reentrant, and terminates at the liquid gas critical point. Courtesy of M. Yamada. (a) changes slope from negative to positive as P decreases. Notably, the point of crossover between the two behaviors is locatedat ambient pressure, T 4 C, and 1g=cm 3. We also plottedthe spinodal line, obtainedby tting the isotherms (for T 300 K) of Fig. 1 to the form P(T; )=P s (T )+A[ s (T)] 2, where P s (T) and s (T) denote the pressure and density of the spinodal line. This functional form is the mean eld prediction for P() close to a spinodal line. For T K, P s (T) is calculatedby estimating the location of the minimum of P(). The results in Fig. 2 show that the liquidspinodal line is not reentrant anddoes not intersect the TMD line. The most natural response to the concept of a secondcritical point in a liquidis baement such a thing just does not make sense. To make the concept more plausible, we oer the following remarks. Consider a typical member of the class of intermolecular potentials that go by the name of core-softenedpotentials [59 61]. Recently, such potentials have been revisited[24,62 76]; they are attractive to study because they can be solvedanalytically in one dimension andare tractable to study using approximation procedures (and simulations) in higher dimensions. They are also more realistic than one might imagine at rst sight, andindeedmay reect what matters in water water interactions, since the repulsive soft core mimics the eect of the small number (four) of nearest neighbors in liquids with a local tetrahedral structure. Although such a picture may seem to be oversimplied, it is consistent with neutron data [26,77 79]. Also, simulation results are in goodaccordwith neutron results (see, e.g., [80]), and Sasai relates these two distinct local structures to dynamic properties [81]. 3. Dynamic properties We next discuss results on the dynamics of stretched water recently obtained by Netz andhis collaborators [82 84]. While there are a large number of intermolecular

109 286 H.E. Stanley et al. / Physica A 315 (2002) T = 260K 250K 240K D (10 5 cm 2 /s) (b) 230K 210K ρ (g / cm 3 ) Fig. 3. Dependence of the diusion constant D on along isotherms (for 6 1:0 g=cm 3 ). Open symbols are the new simulations we report, andlledsymbols are from Refs. [48,49]. Courtesy of P.A. Netz. potential functions usedto simulate water, each of which gives slightly dierent results, the overall thermodynamics picture obtained from these models is generally very similar. Since dynamic properties are particularly sensitive to the potential choice, the extendedsimple point charge (SPC/E) potential is usedsince it reproduces both the maximum in diusivity under pressure and the power-law behavior of dynamics properties on cooling. For understanding the properties of water at negative pressure, simulations are particularly important since experiments are very dicult to perform in this region. The eect of extreme conditions on the ow of the liquidis assessedby calculating the diusion constant D, dened by the asymptotic value of the slope of the mean square displacement versus time. We show D along isotherms in Fig. 3. For T K, D has a minimum value at 0:9 g=cm 3, which becomes more pronouncedat lower T. This behavior can be understood considering the structural changes that occur with decreasing density. At low T, the decreased density enhances the local tetrahedral ordering, which leads to a decrease in D. Further decrease in density reduces the stability of the tetrahedral structure and causes an increase of D. The location of the minimum is near the ice Ih density 0:915 g=cm 3, which is the density where the perfect tetrahedral order occurs. The behavior of the minimum of D, D min (T), complements the known behavior of D max (T) for the same model [48 50,53], where a maximum occurs due to breaking hydrogen bonds at high pressure; the density of the D min (T ) increases slightly with increasing T, while the density of D max (T) decreases with increasing T [50]. This is expected, since the range of densities where anomalous behavior occurs expands with decreasing T. We show the loci of D min (T) and D max (T ), along with the spinodal and locus of density maxima in Fig. 4. Below the spinodal, D also increases, since the mobility of the gas is larger than that of the liquid. However, the simulations clearly show that D min for the liquid occurs prior to the onset of cavitation, andso our estimate of the location of D min is not aectedby phase separation. Recently, Ref. [53] estimatedthe location of D min

110 H.E. Stanley et al. / Physica A 315 (2002) D max P (MPa) 0 T MD 200 D min Spinodal T (K) Fig. 4. Relation of the loci of maxima andminima of D with TMD andthe spinodal. Open symbols are from the present work, andlledsymbols are from Refs. [48,49]. Courtesy of P.A. Netz. for the same model along several isotherms, and associated D min with a maximum in orientational order. 4. Conclusions Water exhibits a very complex structure andits properties andanomalies are strongly inuencedby variations of pressure. For high densities ( max ), water behaves as a normal liquidandthe decrease of D with increasing pressure is governedby steric eects. For min max, as the pressure is decreased, the presence of defects and interstitial water decrease, the tetrahedral structure dominates, with stronger hydrogen bonds. This process reaches its maximum at = min ice. Further stretching destabilizes the hydrogen bond network, leading to an increase in mobility. The locus of D min roughly tracks the spinodal, not surprising since the same breakdown of tetrahedral order that gives rise to D min also facilitates cavitation. Acknowledgements We thank D.R. Baker, S.V. Buldyrev, P. Debenedetti, G. Franzese, W. Kob, E. La Nave, M. Marquez andc. Rebbi for useful discussions, andnsf Grant CHE , the Conselho Nacional de Desenvolvimento Cientico e Technologico (CNPq), the Fundacao de Amparo a Pesquisa do Rio Grande do Sul (Fapergs) for support. M.Y. thanks NSF Grant GER for support as a Graduate Research Trainee at the Boston University Center for Computational Science, F.S. thanks MURST COFIN 2000 andinfm Iniziativa Calcolo Parallelo.

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113 Eur. Phys. J. B 30, (2002) DOI: /epjb/e THE EUROPEAN PHYSICAL JOURNAL B Aging and energy landscapes: application to liquids and glasses S. Mossa 1,2,a,E.LaNave 1,F.Sciortino 1, and P. Tartaglia 1 1 Dipartimento di Fisica and INFM Udr and Center for Statistical Mechanics and Complexity, Universitá diroma La Sapienza, Piazzale Aldo Moro 2, 00185, Roma, Italy 2 Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts 02215, USA Received 7 August 2002 / Received in final form 8 October 2002 Published online 19 December 2002 c EDP Sciences, Società Italiana di Fisica, Springer-Verlag 2002 Abstract. The equation of state for a liquid in equilibrium, written in the potential energy landscape formalism, is generalized to describe out-of-equilibrium conditions. The hypothesis that during aging the system explores basins associated to equilibrium configurations is the key ingredient in the derivation. Theoretical predictions are successfully compared with data from molecular dynamics simulations of different aging processes, such as temperature and pressure jumps. PACS Pf Glass transitions Ja. Computer simulation of liquid structure Lc. Time-dependent properties; relaxation Glasses are out-of-equilibrium (OOE) systems, characterized by slow dynamical evolution (aging) and history dependent properties. Their thermodynamic and dynamic properties depend on the preparation method (cooling or compression schedules) as well as the time spent in the glass phase [1]. The extremely slow aging dynamics has often been considered an indication that a thermodynamic description of the glass state can be achieved by adding one or more history-dependent parameters to the equation of state (EOS) [2 13]. Recent theoretical work mostly based on mean field models of structural glasses, where analytic solutions of the out-of-equilibrium dynamics can be explicitly worked out [14] supports such possibility. Here we introduce an equation of state for out-ofequilibrium conditions, based on a generalization of the potential energy landscape (PEL) thermodynamic approach [15 17]. The equation of state is derived under the hypothesis that during aging the system explores basins associated with equilibrium configurations, a condition which, as shown later, is simple to implement in the PEL context. The proposed equation of state for out-ofequilibrium conditions depends on one additional parameter, which for example can bechosentobetheaverage depth of the explored local minima of the PEL (the socalled inherent structures (IS) [15,16]). Our formulation allows for the first time a detailed comparison between a Present address: Laboratoire de Physique Théorique des Liquides, Université Pierre et Marie Curie, 4 place Jussieu, Paris 75005, France mossa@lptl.jussieu.fr predictions and exact results from out-of-equilibrium molecular dynamics simulations. We report such a comparison for a realistic model of the fragile liquid orthoterphenyl, one of the most studied glass forming liquids [18]. The potential energy landscape formalism [15,16] allows for a clear separation of vibrational and configurational contributions in thermodynamical quantities. Indeed, the equilibrium liquid free energy F (T,V) at temperature T and volume V is expressed as a sum of a vibrational contribution f vib, representing the intrabasin free energy [15], and a configurational contribution ( TS conf + e IS ). Here the configurational entropy S conf accounts for the number of explored basins of the PEL of average energy depth e IS. f vib,forasystemofn atoms, depends on the curvature of the potential energy surface, i.e., onthe3n eigenvalues ωi 2 of the Hessian matrix evaluated at the inherent structure. Recently, calculating the V derivative of F (T,V), an analytic equation of state has been derived, based only on the statistical properties of the landscape [19,20]. The separation in vibrational and configurational parts can be carried out for several properties. In our case, it is particularly useful to separate the instantaneous pressure as sum of two contributions: the pressure felt by the system in the inherent structure configuration (P IS ), and an additional contribution related to the finite temperature, which we refer to as the vibrational contribution (P vib ) [21] P (T,V,e IS )=P IS (e IS,V)+P vib (T,V,e IS ). (1)

114 352 The European Physical Journal B In equilibrium e IS is a well defined function of T and V and, as well known, P depends only on T and V.Note that P IS, being the pressure experienced by the liquid in the IS, depends only on the basin s depth and on V, independently from the equilibrium or out-of-equilibrium state of the system. Instead, P vib depends on the basin s depth as well as on T and V [22]. For models for which the statistical properties of the PEL, i.e., the distribution of shapes and numbers of the inherent structures [19,20] are known, P IS and P vib can be calculated theoretically. In other cases, P IS and P vib can be calculated numerically. In all cases, a landscape-based equation of state can be derived, according to equation (1). The separation of the pressure in two contributions is crucial for the possibility of extending the equilibrium equation of state to out-of-equilibrium conditions, since it provides a direct way to rebuild the pressure once T, V and e IS are known. In the PEL formalism, the hypothesis that the system explores basins associated with typical equilibrium configurations, i.e., configurations sampled in equilibrium conditions, is equivalent to assuming that the relations calculated in equilibrium linking P IS and P vib to e IS and V are valid also in outof-equilibrium conditions. If this is the case, the knowledge of e IS, T and V is sufficient to calculate both P IS, P vib and their sum P according to equation (1). Similarly, the values of P, T and e IS are sufficient to predict V, since both P IS (e IS,V)andP vib (e IS,T,V) can be estimated as a function of V. The predicted V is the value for which P IS (e IS,V)+P vib (e IS,T,V) matches the external pressure. Next we apply the ideas discussed above to a system of N = 343 molecules interacting via the Lewis and Wahnström (LW) model for the fragile molecular liquid orthoterphenyl (OTP) [23,24]. The LW model is a rigid three-site model, with intermolecular site-site interactions modeled by the Lennard-Jones (LJ) potential [24]. We refer to reference [23] for all numerical details. An equilibrium equation of state, based on the assumption of a Gaussian landscape, has been recently presented for this model [19], and successfully compared with the exact equation of state calculated from molecular dynamics simulations. Figure 1 shows the relation between P IS and e IS in equilibrium along different isochores, and P vib as a function of T for the LW model. Both quantities can be parameterized in a quite accurate way. Indeed, for this model, P vib is weakly dependent on V and e IS and it is well represented by a linear T -dependence. The parameterization of P IS and P vib offers the possibility of estimating the system pressure once T, V and e IS are known. We study four different out-of-equilibrium protocols via computer simulation. The imposed external conditions are illustrated in Figure 2. We study i) a T -jump at constant V ; ii) a P -jump at constant T ; iii) a P -jump at constant T in the glass phase and iv) an isobaric heating of a glass. For the cases i) and ii), conditions are chosen in such a way that within the simulation time window (50 ns) the system reaches the equilibrium state. In all studied cases, Fig. 1. (a) Equilibrium relation between basin pressure P IS and basin depth e IS. Symbols are simulation results at the five volumes per molecule nm 3 (down triangles), nm 3 (circles), nm 3 (diamonds), nm 3 (up triangles), and nm 3 (squares) from reference [23]. Dashed lines are calculated using the parameterization discussed in reference [19]. Results at four intermediate volumes are also shown. The local minima are calculated using standard minimization algorithms [23]. (b) P vib as a function of T for the same five volumes. Each curve has been shifted by n 20 MPa to avoid overlaps. Dashed lines show a suitable parameterization of the e IS, V and T dependence of P vib, as discussed in reference [19]. averages over 50 independent simulations have been performed. Case (i). Figure 3 shows the comparison between the theoretical predictions and the numerical results for the constant-v T-jump case. The numerical experiment is performed by changing at t = 0 the thermostat value; the time constant of the thermostat has been fixed to 20 ps. The e IS values, calculated from the simulation (Fig. 3a), are used together with the values of T and V as input to predict the evolution of P (Fig. 3b). Figures 3c and d also show the comparison for P IS and W. W(e IS ), defined as 3N i=1 ln(ω i(e IS )), provides an indication of the similarity in basin shape (at the level of the harmonic approximation) between basins explored in equilibrium and in outof-equilibrium conditions. The quality of the predictions indeed supports the validity of the equation of state and the hypothesis that the basins sampled during aging have the same relationship W(e IS ) as in equilibrium. Case (ii). Figure 4 shows the evolution of the system after a constant-t pressure jump. In this case, the e IS

115 S. Mossa et al.: Aging and energy landscapes: application to liquids and glasses 353 Fig. 2. Paths in the P T plane of the four out-of-equilibrium simulations discussed in the text. Arrows indicate: (i) a T - jump at constant V between liquid states; (ii) a P -jump at constant T between liquid states; (iii) a P -jump at constant T starting from a glass; (iv) a constant P heating of a glass. P and T -jumps have been simulated by changing the thermostat and barostat value at t = 0. The time constant of the thermostat and barostat has been fixed to 20 ps. The full lines indicate the equilibrium P (T ) at five V values, from reference [23]. The dashed lines connect the equilibrium state points and the glass states obtained by fast constant-v cooling. Fig. 4. Dynamics after a P -jump at T = 320 K from 13.4 MPa to 60.7 MPa.(a)e IS; (b)v ;(c)p IS; (d)w. Open symbols are simulation results, closed circles are theoretical predictions based on the out-of-equilibrium PEL-equation of state, using as input the data shown in panel (a). Horizontal dashed lines indicate the known equilibrium values at the initial and final state. The vertical dashed lines indicate the time at which the external pressure reaches the final value. Panel (e) shows the actual path of the aging process in the P IS-e IS plane. The open symbols are the P IS-e IS equilibrium results at the two densities characterizing the initial and final state of the P -jump. Fig. 3. Dynamics after a T -jump, at volume per molecule V =0.345 nm 3, from 480 K to 340 K. (a) e IS; (b)p ;(c)p IS; (d) W (ω o =1cm 1 ). Open symbols are simulation results, closed circles are theoretical predictions based on the out-ofequilibrium PEL-equation of state and using as input the data shown in panel (a). Dashed lines indicate the known equilibrium values at the initial and final state. The error in the prediction at short and long times, when equilibrium conditions are met, provides a measure of the quality of the equilibrium equation of state [19], which agrees with the simulation data within ±5MPa. values calculated from the simulation are used together with the values of T and P as input to successfully predict the evolution of V. For times shorter than 20 ps, the barostat time constant, the fast increasing external pressure forces the system to change the volume with a solid-like response. In this time window, the initial basins in configuration space are continuously deformed by the rapid volume change, in agreement with the findings of reference [25]. For time longer than 20 ps, when the external pressure has reached the equilibrium value,

116 354 The European Physical Journal B Fig. 5. Top: Simulation of a P -jump from MPa to MPa, starting from a glass configuration, obtained by quickly cooling to 70 K equilibrium configurations at volume per molecule V =0.369 nm 3 and T = 170 K. Bottom: Isobaric heating, starting from the IS at V =0.337 nm 3 per molecule and T = 320 K. The heating rate is 1 K / ps. Circles are the e IS values used as input to predict the evolution of V (lines). Squares are the V calculated directly from the simulations. In both cases and for all times, the out-of-equilibrium equation of state is able to predict quantitatively the V changes. In summary, we have derived an extension of the PEL equation of state to model the thermodynamic properties of liquids under out-of-equilibrium conditions. Such an extension based on the hypothesis that an aging system evolves through regions of configuration space which are typically sampled in equilibrium requires an additional thermodynamic parameter which, for convenience, we have associated with the IS energy of the explored basins. The results reported in this paper (Figs. 3 5) show that the proposed generalization of the PEL equation of state is successful in predicting out-of-equilibrium thermodynamics, at least under the conditions and time scales probed by state-of-the-art numerical simulations. Larger T or P jumps and/or longer aging time could require a more detailed thermodynamic description, with more than one additional parameter. Despite such a possibility, the energy landscape approach developed in this paper is a promising starting point for looking into more complex problems in the physics of out-of-equilibrium liquids and glasses. We thank A. Angell, P. Debenedetti, T. Keyes, L. Leuzzi, G. McKenna, G. Ruocco, S. Sastry, A. Scala, and M. Yamada for discussions and comments. We acknowledge support from MIUR-COFIN, INFM-PRA and INFM Initiative Parallel Computing. the system starts to explore basins different from the (deformed) original ones, and the time evolution is controlled by the aging kinetics. To support this interpretation we show in Figure 4e the actual path in the e IS -P IS plane, where the change around 20 ps is clearly evidenced. As shown in Figure 4e, P IS is almost constant at long times. This is due to the fact that the vibrational component to the pressure is mostly controlled by the temperature. Hence, no significant changes in P vib are observed when the system explores states of lower energy. As a result, when the total pressure has reached the value fixed by the barostat (in about 20 ps), P IS must also have reached its final value; the system will then evolve by changing volume and energy depth keeping P IS almost constant. Cases (iii) and (iv). The last two cases are respectively a constant TP-jump and a heating at constant P (with an heating rate of 1 K/ps), both starting from glass configurations. These initial configurations are generated by rapid constant-v quenches of equilibrium configurations, as shown schematically in Figure 2. Figure 5 shows the comparison between the theoretical predictions and the numerical calculations for V,usinge IS as input. In the case of a P -jump (Fig. 5-top), again two different dynamical behaviors are observed: a fast dynamics process describing the mechanical response of the glass to the external pressure change, followed by an extremely slow aging dynamics, during which basin changes are taking place. References 1. P.G. Debenedetti, Metastable liquids (Princeton Univ. Press, Princeton, 1996) 2. A.Q. Tool, J. Am. Ceram. Soc. 29, 240 (1946) 3. R.O. Davies, G.O. Jones, Adv. Phys. 2, 370 (1953) 4. A.J. Kovacs, Fortschr. Hochpolym.-Forsch. 3, 394 (1964) 5. G.B. McKenna, in Comprehensive Polymer Science, Vol.2, edited by C. Booth, C. Price (Pergamon, Oxford, 1989), p R.J. Speedy, J. Chem. Phys. 100, 6684 (1994) 7. L.F. Cugliandolo, J. Kurchan, L. Peliti, Phys. Rev. E 55, 3898 (1997) 8. Th.M. Nieuwenhuizen, Phys. Rev. Lett. 80, 5580 (1998) 9. A.C. Angell, K.L. Ngai, G.B. McKenna, P.F McMillan, S.W. Martin, J. Appl. Phys. 88, 3113 (2000) 10. T.S. Grigera, N.E. Israeloff, Phys. Rev. Lett. 83, 5038 (1999) 11. L. Bellon, S. Ciliberto, C. Laroche, Europhys. Lett. 53, 511 (2001) 12. S. Franz, M.A. Virasoro, J. Phys. A 33, 891 (2000) 13. L. Leuzzi, Th.M. Nieuwenhuizen, J. Phys.: Cond. Matt. 14, 1637 (2002) 14. L.F. Cugliandolo, J. Kurchan, P. Le Doussal, Phys. Rev. Lett. 76, 2390 (1996) 15. F.H. Stillinger, T.A. Weber, Phys. Rev. A 25, 978 (1982)

117 S. Mossa et al.: Aging and energy landscapes: application to liquids and glasses P.G. Debenedetti, F.H. Stillinger, Nature 410, 259 (2001) 17. S. Sastry, Nature 409, 164 (2001) 18. A. Tölle, Rep. Prog. Phys. 64, 1473 (2001) 19. E. La Nave, S. Mossa, F. Sciortino, Phys. Rev. Lett. 88, (2002) 20. P.G. Debenedetti, T.M. Truskett, C.P. Lewis, F. Stillinger, Adv. Chem. Eng. 28, 21 (2001) 21. In equilibrium, P can be unambiguously separated in a configurational V [TS conf (T,V ) e IS(T,V )] and a vibrational V f vib contribution. In the inherent structure configuration the vibrational part is strictly zero. Still, the configurational component may not coincide with P IS.Different proposals for P IS arise depending on the meaning of T in the TS conf term above. See references [19,26]. Such ambiguity affects the theoretical expression for P IS based on the statistical properties of the landscape. Luckily, this ambiguity does not hamper the possibility of developing an out-of-equilibrium equation of state when P IS is calculated directly from the simulation data 22. For example, in an harmonic solid, P vib is related to the V derivative of the eigenfrequencies as P vib (T,V,e IS) k BT V P 3N 3 i=1 ln(β~ω i(e IS)) 23. S. Mossa, E. La Nave, H.E. Stanley, C. Donati, F. Sciortino, P. Tartaglia, Phys. Rev. E 65, (2002) 24. L.J. Lewis, G. Wahnström, Phys. Rev. E 50, 3865 (1994) 25. D.L. Malandro, D.J. Lacks, Phys. Rev. Lett. 81, 5576 (1998) 26. F. Sciortino, P. Tartaglia, Phys. Rev. Lett. 86, 107 (2001)

118 INSTITUTE OF PHYSICSPUBLISHING JOURNAL OFPHYSICS: CONDENSED MATTER J. Phys.: Condens. Matter 15 (2003) S351 S357 PII: S (03) Equilibrium and out-of-equilibrium thermodynamics in supercooled liquids and glasses SMossa 1,2,ELaNave 2,PTartaglia 2 and F Sciortino 2 1 Laboratoire de Physique Théorique des Liquides, Université Pierre et Marie Curie, 4 Place Jussieu, Paris Cédex 05, France 2 Dipartimento di Fisica, INFM Udr and Centre for Statistical Mechanics and Complexity, Università di Roma La Sapienza, Piazzale Aldo Moro 2, I-00185, Roma, Italy Received 16 October 2002 Published 16 December 2002 Online at stacks.iop.org/jphyscm/15/s351 Abstract We review the inherent structure thermodynamical formalism and the formulation of an equation of state (EOS) for liquids in equilibrium based on the (volume) derivatives of the statistical properties of the potential energy surface. We also show that, under the hypothesis that during ageing the system explores states associated with equilibrium configurations, it is possible to generalize the proposed EOS to out-of-equilibrium (OOE) conditions. The proposed formulation is based on the introduction ofone additional parameter which, in thechosen thermodynamic formalism, can be chosen as the local minimum where the slowly relaxing OOE liquid is trapped. 1. Introduction The possibility of a consistent description of the thermodynamics of equilibrium and outof-equilibrium (OOE) (glass) supercooled liquids has been and is an important research direction [1 10]. In recent years, the inherent structure (IS) formalism of Stillinger and Weber [11] has significantly contributed to the understanding of the physics of supercooled liquids and appears to offer a powerful and simple approach for developing a thermodynamics of OOE states. Indeed, on one hand, the IS formalism provides a transparent way to write the partition function in terms of the basis of the local minima (IS) of the underlying potential energy landscape (PEL). On the other hand, state-of-the-art computer simulations provide the possibility of a statistically complete sampling of the PEL explored in equilibrium conditions over a wide temperature range. The numerical analysis of configurations extracted from the canonical ensemble allows us to calculate the energy depth of the ISs explored during the dynamical evolution, to characterize the volumes of their basins of attraction, and to give estimates of their degeneracy (configurational entropy). Eventually it allows us to directly estimate the free energy of the system in terms of landscape properties. Recently [9] it has been shown that it is possible to write down a model equation of state (EOS) [12] expressed only in terms of quantities describing the statistical properties of /03/ $ IOP Publishing Ltd Printed in the UK S351

119 S352 SMossa et al the PEL. The crucial step in this process is the evaluation/modelling of the volume dependence of the number of basins, of their energy distribution, and of their volume. The landscape-based PEL EOS is able to predict the thermodynamics of the system in equilibrium [9] and, even more interestingly, OOE [10] when the ageing system explores states related to equilibrium configurations. Although the calculations that we have explicitly performed deal with the simulation of the Lewis and Wahnström (LW) model for the fragile molecular liquid orthoterphenyl (OTP) [13, 14], our results are general. Here we review and discuss some general implications of our results for the understanding of the thermodynamics of supercooled liquids and glasses. 2. The constant-volume free energy In pioneering papers [11], Stillinger and Weber have shown that the partition function Z(T ) at constant volume V can be written as Z(T ) = de IS (e IS )e βe IS e β F vib(e IS,T ), (1) where β = 1/k B T, e IS is the depth of the local potential energy minimum (IS) of the PEL, (e IS ) de IS is the number of potential energy minima with energy between e IS and e IS +de IS, and F vib (e IS, T ) describes the free energy of the system constrained in one of the basins of depth e IS,averagedover all basins of depth e IS. Following Stillinger and Weber [11], abasinis defined as the set of points in configuration space which lead to the same local minimum under a steepest-descent path. The power of this formulation relies on the fact that the procedures used to associate with each system configuration the corresponding IS are operationally well defined through constant-volume minimization techniques. Numerical evaluation of the density of states in the local minimum allows us to calculate the harmonic contribution to the basin free energy. Starting from equation (1), the free energy of the system can be written as F(T ) = e IS (T ) TS con f ( e IS (T ) ) + F vib (T, e IS (T ) ); (2) here e IS (T ) is the solution to the saddle point approximation to equation (1), and S con f = k B ln( (e IS )) is the configurational entropy. F vib,theintrabasin vibrational free energy, is usually written in the harmonic approximation as F vib = k B T M i=1 ln(β hω i(e IS ), where ω i (e IS ) is the ith normal-mode frequency (i = 1,...,M)) evaluated at the IS, and h is the Planck constant. The sum of the logarithms of the normal-mode frequencies describes the volume (via the curvature) of the basin of attractionoftheisin the harmonicapproximation. Computer simulation results and theoretical insight [15, 16] provideus with valid models forthe two crucial quantities (e IS ) and F vib,namely: (e IS ) de IS = e αn e (e IS E 0 ) 2 /2σ 2 de IS (3) 2πσ 2 F vib (e IS, T ) = k B T [(a + be IS ) ln( hβ)]. (4) The hypothesis of a Gaussian landscape is supported by the consideration that, if no correlation length diverges, e IS can be thought of as sum of the IS energies of several independent subsystems. In this case the central limit theorem suggests that, since the variance of the energy distribution in each of these independent subsystem is finite, a Gaussian distribution will describe the distribution of e IS -values [15]. We note that this hypothesis will break down in the very low-energy tail, where differences between the

120 Equilibrium and out-of-equilibrium thermodynamics in supercooled liquids and glasses S353 Figure 1. The T -dependence of e IS (a) and S conf (b) in the approximation of a Gaussian distribution of basin depths and e IS -independence of the basin volume. Curves for α = 1.0 and three different values of σ 2 /E0 2 are shown. Gaussian distribution and the actual distribution become relevant. The second hypothesis ( M i=1 ln(ω i(e IS )) = a + be IS )isnot crucial, but it is supported by the results of numerical studies. Substituting in equation (2) and solving, one obtains [16] e IS (T ) =(E 0 bσ 2 ) σ 2 /k B T (5) ( b 2 σ 2 ) TS conf ( e IS (T ) ) = k B T αn + bσ 2 σ 2 (6) 2 2k B T F vib (T, e IS (T ) ) = F 0 (E 0, T ) k B Tbσ 2 (b + β). (7) Therefore, F(T, V ) is expressed only in terms of proper combinations of the parameters α, E 0, σ, a, andb which are related to the statistical properties of the PEL [9] and to a particular relation between volume and depth of the basins. We also note that from a plot of e IS versus 1/T it is possible to evaluate σ 2 and E 0.Acomparison between numerical data and equation (6) allows us to estimate α. In the case where all basins have the same volume (b = 0), equations (5) and (6) simplify considerably, and in terms of scaled quantities one obtains e IS (T ) /E 0 = 1 σ 2 /E 2 0 (k B T/E 0 ) (8) S con f ( e IS (T ) )/k B = α σ 2 /E0 2 2(k B T/E 0 ). (9) 2 Within the Gaussian approximation, the lowest e IS -value, e K characterized by S con f (e K ) = 0 is the Kauzman energy: e K (T K ) /E 0 = 1 2α σ 2 E0 2, (10)

121 S354 SMossa et al Figure 2. Comparison between the different contributions to the pressure calculated according to the theory (solid curves) and by MD simulations (symbols) for the LW model: (a) IS contribution; (b) vibrational contribution. The curves have been shifted by n 20 MPa to avoid overlaps. (c) Total pressure. Details on the calculations of these quantities can be found in [9, 10]. and it is reached at a Kauzman temperature T K given by (σ k B T k /E 0 = 2 /E0 2). (11) 2α The behaviours of e IS and S con f (T ) as functions of T,inreduced units, are shown in figure 1. We note in passing that recent works by Speedy [17] and Sastry [16] have attempted to correlate kinetic fragility with thermodynamic fragility [18], suggesting that σ and α are the statistical properties of the PEL which control the material fragility. 3. Equilibrium equation of state The generalization of equation (2) to the volume-dependent case requires the determination of the volume dependence of equation (3), i.e., the formulation of an ansatz for the joint probability P(e IS, V ) of finding a value of e IS at a given volume V.Wefollow the equivalent approach of fitting simultaneously the lhs of equations (4) (6) determined by MD simulations at different volumes [9]. This procedure allows us to calculate directly the volume dependence of the parameters α, E 0, σ, a,andb introduced above. Substituting in equation (2) we obtain F(T, V ) for the model considered, and the EOS can be finally calculated via P(T, V ) = V F(T, V ) at T constant. From equation (2) it is immediately clear that P can be split into two contributions: a configurational part, P conf, related to the change in the number and depth of available basins with V,andavibrational part, P vib,related to the change in the volume of the basin with V as P(T, V, e IS ) = P conf (V, e IS ) + P vib (T, V, e IS ). (12) Figure 2 shows the comparison among the MD estimates of the different contributions to the pressure (symbols) and the predictions of the above theory, for the case of the LW model. The excellent agreement between the two sets of data confirms the validity of the procedure introduced above which provides us with an effective EOS for the system under study based on the statistical properties of the landscape as expressed in equations (3) and (4).

122 Equilibrium and out-of-equilibrium thermodynamics in supercooled liquids and glasses S355 Figure 3. OOE simulation protocols: (a) pressure evolution after a T -jump at constant volume per molecule V = nm 3 from 480 to 340 K; (b) volume evolution after a P-jump at constant temperature T = 320 K from 13.4 to 60.7 MPa. Details can be found in [10]. Dashed lines are the equilibrium values at the initial and final state. 4. Out-of-equilibrium equation of state The possibility of a proper thermodynamical description of OOE systems has been widely debated [1 3, 19, 20]. In particular it has been recognized that this should be possible by adding one or more history-dependent parameters to the equilibrium EOS. The arguments discussed above allow us to go further in this direction [10]. Indeed the only hypothesis that we made is that equations (3) and (4) are valid. In all the OOE conditionswhere these two conditions are met, i.e., the (gently) system-driven OOE explores states which are typical at equilibrium, the theory is expected to hold at the expense of adding one parameter to the equilibrium EOS. Looking at equations (4) (6), the choice of the basin depth e IS as the additional parameter turns out to be very natural. To the extent of this extension, the validity of equation (12) in OOE conditions is crucial, allowing us to link P con f and P vib to e IS and V.Ifthisisthe case, the knowledge of e IS and V is sufficient for calculating both P conf, P vib and their sum P according to equation (12). Similarly, the values of P, T,ande IS are sufficient for predicting V,sinceboth P con f (e IS, V ) and P vib (e IS, T, V ) can be estimated as functions of V.Thepredicted V is the value for which P con f (e IS, V ) + P vib (e IS, T, V ) matches the external (fixed) pressure. In figure 3 we show the comparison among MD results and the predictions of the OOE EOS for two different OOE protocols via computer simulation. In particular, we consider in figure 3(a) the case of a T -jump at constant volume, and in figure 3(b) a P-jump at constant temperature. In the first case the dynamical evolution of e IS together with the (fixed) values of V and T allow us to predict the dynamical evolution of P; inthesecond one, the time dependence of e IS together with P and T allow us to predict the evolution of P. An interesting representation of the ageing processes discussed above is the parametric plot in the P IS e IS plane. In figure 4 we show the path followed by the ageing system for the two protocols discussed above (figures 4(a) and (b)). Panels (c) and (d) report the comparison between the basin volume described by the quantity M i=1 ln(ω i(e IS )) during the ageing process and the basin volume of the corresponding basin (same e IS and V )explored in equilibrium conditions. In all cases the agreement between the calculated quantities and the theoretical prediction is quite good, confirming the validity of our approach.

123 S356 SMossa et al Figure 4. Paths of the ageing process in the P IS e IS plane for the OOE protocols considered in figure 3: (a) a T -jump at constant pressure; (b) a P-jump at constant temperature. The arrows indicate the time evolution direction. Details can be found in [9, 10]. Panels (c) and (d) report the comparison between the basin volume during the ageing process and the basin volume of the corresponding basin (same e IS and same V )explored in equilibrium conditions. The basin volume is described by M i=1 ln(ω i (e IS )). Figure 4(b) is of particular interest,showing that in the OOE dynamics following a pressure jump two different regimes can be recognized. For times shorter than the barostat time constant (see [10] for details) the system responds to the external increase of pressure in a solid-like fashion, i.e., the PEL basins initially populated are only deformed by the volume change. Only for longer times, when the pressure has reached the equilibrium value, does the system start to age among basins different from the original ones. 5. Conclusions In this paper we have reviewed some recent results on a general approach to the thermodynamics of equilibrium supercooled liquids and glasses [9, 10]. We have discussed how it is possible to formulate an equilibrium EOS in terms of quantities describing the statistical properties of the PEL. These findings allow us to better understand the nature of the different terms contributing to thetotal pressure of the system, and fill the gap usually found among experiments (usually performed at constant P) andcomputer simulations (usually performed at constant V ). The generality of the hypothesis that we have introduced allows us to generalize our approach to OOE conditions. In all the cases where the hypotheses introduced are met, i.e., the system ages among states typical at thermodynamical equilibrium, it is possible to write down an OOE EOS at the expense of the addition of one more parameter. This quantity can be naturally chosen as the depth of the explored IS. The correctness of this generalization has been checked under several OOE conditions. Its limits of validity under more severe OOE conditions where more then one additional parameter is needed for a consistent description (like in the so-called Kovacs memory experiments [19 21]) are currently under investigation. We foresee the possibility that, under large variations of the temperature and/or pressure, different parts of the system will age with different speeds producing, as a net result, a material characterized by an e IS -distribution of the component subsystems different from the equilibrium one, and/or

124 Equilibrium and out-of-equilibrium thermodynamics in supercooled liquids and glasses S357 amaterial for which the relation between basin volume and depth is different from that at equilibrium. References [1] Davies R O and Jones G O 1953 Adv. Phys [2] Cugliandolo L F, Kurchan J and Peliti L 1997 Phys. Rev. E [3] Nieuwenhuizen Th M 1998 Phys. Rev. Lett [4] Sciortino F and Tartaglia P 1999 Phys. Rev. Lett Sciortino F and Tartaglia P 2001 Phys. Rev. Lett Sciortino F and Tartaglia P 1997 Phys. Rev. Lett [5] Kob W, Sciortino F and Tartaglia P 2000 Europhys. Lett [6] Scala A et al 2000 Nature Saika-Voivod et al 2001 Nature [7] Barrat J L and Kob W 1999 Europhys. Lett [8] Di Leonardo R et al 2000 Phys. Rev. Lett Di Leonardo R et al 2001 Phys. Rev. Lett [9] La Nave E, Mossa S and Sciortino F 2002 Phys. Rev. Lett [10] Mossa S et al 2002 Eur. J. Phys. Batpress [11] Stillinger F H and Weber T A 1982 Phys. Rev. A [12] Debenedetti P G et al 2001 Adv. Chem. Eng [13] Mossa S et al 2002 Phys. Rev. E [14] Lewis L J and Wahnström G 1994 Phys. Rev. E [15] Heuer A and Buchner S 2000 J. Phys.: Condens. Matter [16] Sastry S 2001 Nature [17] Speedy R J 1999 J. Phys. Chem. B [18] Martinez L M and Angell A C 2001 Nature [19] McKenna G B 1989 Comprehensive Polymer Science vol 2, ed C Booth and C Price (Oxford: Pergamon) p 311 [20] Angell A C et al 2000 J. Appl. Phys [21] Kovacs A J 1964 Fortschr. Hochpolym.-Forsch

125 Journal of Statistical Physics, Vol. 110, Nos. 3 6, March 2003 ( 2003) Application of Statistical Physics to Understand Static and Dynamic Anomalies in Liquid Water 1 H. E. Stanley, 2 S. V. Buldyrev, 2 N. Giovambattista, 2 E. La Nave, 2, 3 S. Mossa, 2 A. Scala, 2, 3 F. Sciortino, 3 F. W. Starr, 4 and M. Yamada 2 Received March 15, 2002; accepted September 10, 2002 We present an overview of recent research applying ideas of statistical mechanics to try to better understand the statics and especially the dynamic puzzles regarding liquid water. We discuss recent molecular dynamics simulations using the Mahoney Jorgensen transferable intermolecular potential with five points (TIP5P), which is closer to real water than previously-proposed classical pairwise additive potentials. Simulations of the TIP5P model for a wide range of deeply supercooled states, including both positive and negative pressures, reveal (i) the existence of a non-monotonic temperature of maximum density line and a non-reentrant spinodal, (ii) the presence of a low-temperature phase transition. The take-home message for the static aspects is that what seems to matter more than previously appreciated is local tetrahedral order, so that liquid water has features in common with SiO 2 and P, as well as perhaps Si and C. To better understand dynamic aspects of water, we focus on the role of the number of diffusive directions in the potential energy landscape. What seems to matter most is not values of thermodynamic parameters such as temperature T and pressure P, but only the value of a parameter characterizing the potential energy landscape just as near a critical point what matters is not the values of T and P but rather the values of the correlation length. KEY WORDS: Mode coupling theory; low-density liquid; high-density liquid; homogeneous nucleation; structural heterogeneities; instantaneous normal mode. 1 Dedicated to Michael E. Fisher on the occasion of his 70th birthday. 2 Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts 02215; hes@meta.bu.edu 3 Dipartimento di Fisica Università di Roma La Sapienza, Istituto Nazionale di Fisica della Materia and INFM Center for Statistical Mechanics and Complexity, Piazzale Aldo Moro 2, Roma, Italy. 4 Polymers Division and Center for Theoretical and Computational Materials Science, National Institute of Standards and Technology, Gaithersburg, Maryland /03/ / Plenum Publishing Corporation

126 1040 Stanley et al. 1. THE GOAL: UNDERSTANDING WHAT MATTERS Many physicists are attracted to physics because of the focus on understanding just enough of a subject to comprehend the key features that really matter. As soon as some physicists feel they understand what matters, insatiable appetites for novelty force attention to new puzzles. Among most exciting new developments in the 1960 s (when the first author met Michael Fisher) was the degree to which the principle of scale invariance provided increased understanding of what matters near a critical point. The key point was that what matters near a critical point is the correlation length for statics, and the correlation time for dynamics. An exciting question these days is what matters in understanding the statics and dynamics of liquid water, and important clues are emerging when one focuses on behavior in the deeply supercooled region, especially just above and just below the critical temperature T MCT predicted by mode coupling theory. 2. STATICS: WHAT MATTERS IS LOCAL TETRAHEDRAL GEOMETRY 2.1. Introduction Liquid water is not a typical liquid. However, some progress has occurred in understanding its highly anomalous equilibrium and dynamical properties. (1 6) Water is a space-filling hydrogen bond network, as expected from continuum models of water. However when we focus on the wellbonded molecules, we find that water can be regarded as having certain clustering features the clusters being not isolated icebergs in a sea of dissociated liquid (as postulated in mixture models dating back to Röntgen) but rather patches of well-bonded molecules embedded in a highly connected network or transient gel. (7 10) Similar physical reasoning applies if we generalize the concept of well-bonded molecules to molecules with a smaller than average energy (11) or to molecules with a more ordered than average local structure. (12) 2.2. Liquid Liquid Phase Transition Hypothesis Poole et al made computer simulations of the ST2 model of water, with the goal of exploring in detail what might happen in the low-temperature region, (13) and discovered in this artificial computer water the existence of a second critical point CŒ, below which the liquid phase separates into two distinct phases a low-density liquid (LDL) and a high-density

127 Static and Dynamic Anomalies in Liquid Water 1041 liquid (HDL). It is not required that the system is exactly at its critical point for the system to exhibit remarkable behavior, such as the phenomenon of critical opalescence discovered and correctly explained in 1869 by Andrews (14) in terms of increased fluctuations away from (but close to) the critical point. Thus, although in experiments one cannot get closer than (5 15), CŒ nonetheless exerts a strong effect in the experimentally-accessible region of the phase diagram. If we have a singularity in our phase diagram at a well-defined critical point, it s going to have an effect on an entire region around it a critical region Experimental Work When liquid water is supercooled below the homogeneous nucleation temperature T H ( 38 CatP=1 atm), crystal phases nucleate homogeneously, and the liquid freezes spontaneously to the crystalline phase. Mishima creates 1 cm 3 high-pressure ices in a piston-cylinder apparatus, decompresses the sample at a constant rate of 0.2 GPa/min, and because melting is endothermic observes melting transitions of the ice polymorphs using a thermocouple to detect a change in the sample temperature during the decompression. (15, 16) He then determines melting pressures at different temperatures. The melting curves he obtains agree with previously-reported data for stable melting lines, (17, 18) and extend our knowledge of the location of metastable melting lines to much lower temperatures. The Gibbs potential G of the ice polymorphs is known. Since G is identical in coexisting phases, locating the melting lines of the ice polymorphs is sufficient to learn G for water along these lines. By interpolating data for G obtained along these melting lines, one can find the approximate experimental G for a wide range of temperatures and pressures in the noman s land below T H. (15) After finding G as a function of pressure P and temperature T, one can find by differentiation the volume as a function of P and T. Volume as a function of T is just what we want this is the equation of state of liquid water. The P-V-T relation found is consistent with the existence of a line of first-order liquid liquid transitions which continues from the line of low-density amorphous high-density amorphous transitions and terminates at an apparent critical point CŒ. The P-V-T relation is also consistent with other known experimental data (19 25) and also (12, 13, 26 35) with a number of theoretical and simulation results Theoretical Work The most natural response to the concept of a second critical point in a liquid is bafflement such a thing just does not make sense. To make the

128 1042 Stanley et al. concept more plausible, we offer the following remarks. Consider a typical member of the class of intermolecular potentials that go by the name of core-softened potentials. (36 38) Recently such potentials have been revisited; (27, 39 52) they are attractive to study because they can be solved analytically in one-dimension and are tractable to study using approximation procedures (and simulations) in higher dimensions. They are also more realistic than one might imagine at first sight, and indeed may reflect what matters in water-water interactions, since the repulsive soft core mimics the effect of the small number (four) of nearest neighbors in liquids with a local tetrahedral structure. Although such a picture may seem to be oversimplified, it is consistent with neutron data. (21 24) Also, simulation results are in good accord with neutron results (see, e.g., ref. 53), and Sasai relates these two distinct local structures to dynamic properties. (54) 2.5. Simulations The shape of the spinodal in the negative-pressure region can be used to test the liquid liquid phase transition hypothesis, so we briefly discuss recent calculations of the density and pressure of the spinodal, which we denote r sp (T) and P sp (T), respectively. Relatively few experimental works (55, 56) and simulations (13, 57 61) have been performed on stretched water. Recently, Yamada and her coworkers (30) simulated a system of N=343 molecules interacting with the TIP5P potential. (62) TIP5P is a fivesite, rigid, non-polarizable water model, not unlike the ST2 model. (63) The TIP5P potential accurately reproduces the density anomaly at 1 atm and exhibits excellent structural properties when compared with experimental (62, 64) data. The TMD shows the correct pressure dependence, shifting to lower temperatures as pressure is increased. Under ambient conditions, the diffusion constant is close to the experimental value, with reasonable temperature and pressure dependence away from ambient conditions. (62) Figure 1 shows results for pressure along isotherms. At lower temperatures an inflection develops, which becomes a flat isotherm at the lowest temperature, T=215 K. The presence of a flat region indicates that a phase separation takes place; the critical temperature is T CŒ =(217 ± 3) K, the critical pressure is P CŒ =(340 ± 20) MPa, and the critical density r CŒ = (1.13 ± 0.04) g/cm 3. Figure 2 plots the pressure along isochores. The curves show minima as a function of temperature; the locus of the minima is the TMD line, since ( P/ T) V =a P /K T, the ratio of the thermal expansivity to isothermal compressibility. Note that the pressure exhibits a minimum if the density passes through a maximum (a P =0). It is clear that, as in the case of ST2

129 Static and Dynamic Anomalies in Liquid Water T = 215 K P T = 320 K Pressure P [ MPa ] ρ n = 10 n = 5 0 T = 215 K n = Density ρ [ g / cm 3 ] Fig. 1. Dependence on density of the pressure at all temperatures investigated (T= [215, 220, 230, 240, 250, 260, 270, 280, 290, 300, 320] K, from bottom to top). Each curve has been shifted by n 150 MPa to avoid overlaps. An inflection appears as T is decreased, transforming into a flat coexistence region at T=215 K, indicating the presence of a liquid liquid transition. Inset: A detailed view of the T=215 K isotherm. Courtesy of M. Yamada. water, TIP5P water has a TMD that changes slope from negative to positive as P decreases. Notably, the point of crossover between the two behaviors is located at ambient pressure, T % 4 C, and r % 1 g/cm 3. Also plotted is the spinodal line, obtained by fitting the isotherms (for T \ 300 K) of Fig. 1 to the form P(T, r)=p s (T)+A[r r s (T)] 2, where P s (T) and Pressure [ MPa ] TMD line Spinodal ( a ) Temperature [ K ] Fig. 2. Pressure along seven isochores of density (0.90, 0.95,..., 1.20) g/cm 3. The minima correspond to the temperature of maximum density line (dashed line). Note the nose of the TMD line at T=4 C. Stars denote the liquid spinodal line, which is not reentrant, and terminates at the liquid-gas critical point. Courtesy of M. Yamada.

130 1044 Stanley et al. r s (T) denote the pressure and density of the spinodal line. This functional form is the mean field prediction for P(r) close to a spinodal line. For T [ 250K, P s (T) is calculated by estimating the location of the minimum of P(r). The results in Fig. 2 show that the liquid spinodal line is not reentrant and does not intersect the TMD line Outlook Before concluding this brief discussion of statics, we ask What is the requirement for a liquid to display a liquid liquid phase transition? By the arguments presented above, some other liquids should display liquid liquid phase transitions, namely systems that at low temperature and low pressure have anticorrelated entropy and specific volume fluctuations. Thus a natural extension to our work is to consider other tetrahedrally-coordinated liquids. Since other tetrahedral liquids have that similar features, we might anticipate similar liquid liquid phase transitions occur on the liquid free energy surface of these liquids. Evidence in favor of this possibility has been reported for SiO 2, (66, 67) amorphous GaSb, (68, 69) C, (70, 71) and Si. (72) Recently, clear experimental evidence for a liquid liquid phase transition has been reported in phosphorus, where the low-density liquid phase is a molecular liquid of tetrahedral P 4 molecules. (73, 74) With a change in pressure, the low-pressure, low-density molecular liquid transforms to a high-pressure, high-density polymeric liquid. During the transformation, two forms of liquid coexist, showing that phosphorus has a first-order liquid liquid phase transition. (75) A careful analysis of tetrahedral liquids with and without liquid liquid phase transitions has recently been published, (76) who compares the density maxima of the four best researched examples of tetrahedral liquids (SiO 2, BeF 2,H 2 O, and liquid Si), compares their special liquid state heat capacity behavior, and places them in a semi-quantitative relationship in their anomalous transport properties. 3. DYNAMICS ON THE POTENTIAL ENERGY LANDSCAPE: WHAT MATTERS IS THE NUMBER OF DIFFUSIVE DIRECTIONS 3.1. Introduction The study of the dynamics in supercooled liquids is receiving great interest (77) due to novel experimental techniques, (78, 79) detailed theoretical predictions, (80) and by the opportunity to follow the microscopic dynamics via computer simulation. (81, 82) Mode coupling theory (80) quantitatively predicts the time evolution of correlation functions and the dependence on

131 Static and Dynamic Anomalies in Liquid Water 1045 temperature T of characteristic correlation times. Unfortunately, the temperature region in which mode coupling theory is able to make such predictions for the long time dynamics is limited to weakly supercooled states. Parallel with the development of mode coupling theory, theoretical work (83 87) has called attention to thermodynamic approaches to the glass transition, and to the role of configurational entropy in the slowing down of dynamics. (88 90) These theories, which build on ideas put forward some time ago, (91 93) stress the relevance of the topology of the potential energy landscape explored in supercooled states. Detailed studies of the potential energy landscape may provide insights into the slow dynamics of liquids, and new ideas for extending the present theories to the deep supercooling regime Instantaneous Normal Modes and the Topology of the Potential Energy Landscape One approach to understanding the role of the potential energy landscape is to study the connectivity between different local configurations using the instantaneous normal mode formalism. (94) Analogous to the standard normal mode theory for solids, an instantaneous normal mode is the eigenfunction of the Hessian, which is the matrix of the second derivatives of the potential energy with respect to all 6N atomic coordinates. In a liquid state, the eigenvalues of the Hessian matrix are not all generally positive; the negative eigenvalues indicate a downward curvature of the potential energy landscape, i.e., indicate unstable directions for the system. Previous studies using the instantaneous normal mode formalism indicate that the number of directions with negative curvature is reduced on cooling, motivating theories relating diffusion in liquids to the instantaneous normal mode density of states. (95, 96) Low temperature liquid dynamics involve the superposition of fast oscillations around quasi-equilibrium positions (intra-basin motion) and the rearrangement of the system between these positions (inter-basin motion). The typical oscillation period is much shorter than the typical time needed by the system to rearrange itself, i.e., the structural relaxation time. instantaneous normal mode theories for diffusion relate the diffusion of the system in configuration space to activated processes of inter-basin motion. In this respect, the unstable modes are considered representative of the barriers crossed when the system changes basins. One approach (97, 98) among many (99, 100) for separating the diffusive modes (basin changes in configuration space) from the non-diffusive modes (no basin changes) is classifying the modes according to their potential energy profile (Fig. 3), and partition those unstable modes into two groups:

132 1046 Stanley et al. Fig. 3. Schematic sketch of the possible shapes of the potential energy landscape associated with imaginary eigenvalues. Unstable modes are first separated into shoulder and double well modes. Furthermore, double well modes are split into diffusive and non diffusive ones. Courtesy of E. LaNave. (i) unstable normal modes due to the anharmonicities (shoulder modes) and (ii) modes along which the system is crossing a saddle (double-well modes). (99) In order to distinguish between shoulder and double-well modes, the potential energy profile is calculated along straight paths that follow the direction of the eigenvector. Furthermore, to distinguish the false and true double wells, we calculate the steepest descent trajectories starting from the opposite sides of the saddle. A mode represents true double well, and this is called a diffusive mode if these trajectories end up in two distinct local minima Results Next we discuss the numerical relationship between D and the number of diffusive modes f diff in the vicinity of the fragile-to-strong crossover temperature T. We review recent work on two different models of tetrahedral liquids, the SPC/E extended simple point charge model for water (98, 101) and the BKS model of silica. (102) For silica (Fig. 4), the fragileto-strong transition temperature T coincides numerically ( ) with the critical temperature T MCT identified by mode coupling theory. For both models, it appears that D depends on T and P only through f diff the

133 Static and Dynamic Anomalies in Liquid Water 1047 D (cm 2 s 1 ) (a) f diff D/T (cm 2 s 1 K 1 ) T x (c) T x (b) /T (K 1 ) α= Fig. 4. Arrhenius plot of (a) the diffusion constant D for Si atoms in SiO 2 and (b) f diff. The crossover to the straight line Arrhenius behavior below T represents the fragile-to-strong crossover for silica. Part (c) shows the parametric relation D/T vs f diff in a log-log scale. The data are smooth through the mode-coupling crossover temperature T. Courtesy of E. LaNave. analog of the magnetization M(H, T) of a ferromagnet depending on magnetic field H and temperature T only through the correlation length t. Specifically, for both models it appears that D follows a general power-law relation of the form f diff D/T (f diff ) a, (1) for roughly two decades in f diff and three decades in D/T. For the water model, a % 2 while for the silica model it appears that a % 1.3. In the case of silica, the identical functional form describes the relationship between D and f diff both above and below T, showing that while the T dependence of both D and f diff is sensitive to the microscopic mechanisms controlling the dynamics, the fragile-to-strong transition does not affect the relation between D and f diff. The exponent value a=2 found for water has recently been theoretically interpreted. (104) In summary, then, two different dynamical mechanisms affect the slowing down of the dynamics in supercooled states: (98) (i) In the weakly supercooled region, the slowing down of the dynamics arises from the progressive reduction in the number of directions where free exploration of configuration space is possible. The system is always located close to a multi-dimensional ridge between different basins,

134 1048 Stanley et al. and the time scale of the long-time dynamics is set by the time required to probe one of the free directions. In this range of T, the diffusion is not limited by the presence of energy barriers that must be overcome by thermally activated processes, but is controlled by the limited number of directions leading to different basins along almost constant potential energy paths. Furthermore, the number of free directions completely determines the value of D, independent of the thermodynamic parameters T and r. (ii) Close to T MCT, the system starts to sample regions of configuration space that have no free directions. The change in the dynamics above and below T MCT can be viewed as a change in the properties of the potential energy landscape sampled in equilibrium, from configurations always close to a ridge of progressively lower and lower dimension to configurations far from any ridge. (105, 106) Below T MCT, the system must go close to the ridge and then select the right direction. The search for the ridge below T MCT, i.e., the search for a rare event, can be probably described as an activated process, which corresponds to Arrhenius behavior of the diffusion constant. (iii) The relation between connectivity and number of local minima in the potential energy landscape which can be calculated in theoretical models as recently done for the random energy model (107) may help build on the existing ideas bridging thermodynamics and dynamics. (108) 4. DYNAMICS BELOW THE MODE COUPLING THEORY: WHAT MATTERS IS COOPERATIVE MOTION 4.1. Introduction As a supercooled liquid is cooled toward the glassy state, the system is increasingly found near local potential energy minima, called inherent structure configurations. (91) In this description, in the glassy state, the (103, ) system is localized in one of the potential energy basins. While such a picture of liquid dynamics is difficult to verify experimentally, computer simulation offers an excellent opportunity to explore these ideas. For a pre-defined liquid potential, a liquid trajectory can be generated via molecular dynamics simulation and the local potential energy minima can be evaluated by an energy minimization method. (91) With this procedure, the motion in phase space is converted into a minimum-to-minimum trajectory, or inherent structure trajectory. A general picture of the system moving among a set of basins surrounding the multitude of local minima has evolved. More specifically, simulations have shown that both the depth

135 Static and Dynamic Anomalies in Liquid Water 1049 of the minima sampled by the system, as well as the number of these (111, 112) minima, decrease on cooling. The description of the real motion of the system as an inherent structure trajectory becomes a powerful way of separating the vibrational contribution, responsible for the thermal broadening of instantaneous measurements from the slow structural component. (113) Such an approach becomes even more powerful below T MCT, since most of the instantaneous configurations are far from saddles, making correlation functions calculated from the inherent structure trajectory fully account for the a-relaxation dynamics. (109) 4.2. Results Recent results (114) are based on molecular dynamics simulations of the SPC/E model (115) of water for 216 molecules, at fixed density r=1 g/cm 3. The numerical procedure is described in ref. 59. The trajectories are analyzed at T=180 K, and the mode coupling temperature for this density is T MCT =193.6 K, (59) so the system is in the deep supercooled liquid state. At this temperature, the diffusion coefficient is four orders of magnitude smaller than its value at T=300 K and only a few molecules move significantly (with displacements larger than nm) at each simulation time step. To aid in understanding the distribution of the displacements during the IS changes, Fig. 5(a) shows the displacements u of all 216 individual molecules for a typical inherent structure transition. In fact, there is a relatively small set of molecules with a large displacement. A snapshot of the eight molecules with the largest displacement is shown in Fig. 6. Interestingly, this set of molecules forms a cluster of bonded molecules. Indeed, for all cases studied, the set of molecules which displace most forms a cluster of bonded molecules. The observed clustering phenomenon characterizes the inherent structure transitions in water and can be interpreted as the analog of the string-like motion observed in simple atomistic liquids, (109) connected to the presence of dynamical heterogeneities. ( ) Similar results were found by Ohmine et al. using the TIP4P and TIPS2 models for water. (120) To characterize the distribution of individual molecular displacements between different inherent structures more carefully, Fig. 5(b) shows the distribution of displacements u of the oxygen atoms P(u). Note that P(u) was previously studied by Schrøder et al. for a binary Lennard-Jones mixture. (109) Analysis of the changes in hydrogen bond connectivity associated with inherent structure changes reveals that these transitions are associated with the breaking and reformation of hydrogen bonds.

136 1050 Stanley et al. u [nm] P(u) (a) molecule number (b) ~ u u [nm] Fig. 5. (a) Displacement of each of the 216 molecules during the course of a transition from one inherent structure to another. (b) Distribution of displacements u of the oxygen atoms between inherent structure changes, P(u), sampled along a 30 ns trajectory in 20,000 inherent structure changes. The exponential tail of P(u), with a characteristic length of about 0.02 nm, is mostly due to the highly mobile molecules, while the power law with exponent 2.5 would correspond to an elastic response of the system to these highly mobile molecules. (123) Courtesy of N. Giovambattista. Fig. 6. Snapshot of the system in one inherent structure. Only the eight molecules with displacement larger than nm [Fig. 3(a)] are shown here. Hydrogen-bonded molecules are connected by tubes. Note that all 8 molecules are nearby and form a cluster, which unlike the Lennard Jones case, are bounded and less string-like. Courtesy of N. Giovambattista.

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141 INSTITUTE OF PHYSICSPUBLISHING JOURNAL OFPHYSICS: CONDENSED MATTER J. Phys.: Condens. Matter 15 (2003) S1085 S1094 PII: S (03) Numerical evaluation of the statistical properties of a potential energy landscape ELaNave 1,FSciortino 1,PTartaglia 1,CDeMichele 2 and SMossa 3 1 Dipartimento di Fisica, INFM and Centre for Statistical Mechanics and Complexity, Università di Roma La Sapienza, Piazzale Aldo Moro 2, I-00185, Roma, Italy 2 Dipartimento di Scienze Fisiche and INFM, Universitá dinapoli Federico II, ViaCinthia (Monte S Angelo) Ed G, I Napoli, Italy 3 Laboratoire de Physique Theorique des Liquides, Université Pierre et Marie Curie, 4place Jussieu, Paris 75005, France Received 17 October 2002 Published 10 March 2003 Onlineat stacks.iop.org/jphyscm/15/s1085 Abstract The techniques which allow the numerical evaluation of the statistical properties of the potential energy landscape for models of simple liquids are reviewed and critically discussed. Expressions for the liquid free energy and its vibrational and configurational components are reported. Finally, a possible model for the statistical properties of the landscape, which appears to describe correctly fragile liquids in the region where equilibrium simulations are feasible, is discussed. 1. Introduction Understanding the dynamic and thermodynamic properties of supercooled liquids is one of the more challenging tasks of condensed matter physics [1 6]. A significant amount of experimental [7 10], numerical [11], and theoretical work [12 16] is being currently devoted to the understanding of the physics of the glass transition and to the associated slowing down of the dynamics. In recent years the study of the topological structure of the potential energy landscape (PEL) [17] and the connection between the properties of the PEL and the dynamical behaviour of glass forming liquids has become an active field of research. Among the thermodynamic formalisms amenable to use in numerical investigation of the PEL properties, a central role is played by the inherent structure (IS) formalism introduced by Stillinger and Weber [17]. Properties of the PEL, such as depth, number, and shape of the basins of the potential energy surface are calculated and used in the evaluation of the liquid free energy in the supercooled state [18 21]. In the IS formalism, the system free energy is expressed as a sum of an entropic contribution, accounting for the number of explored basins, and a vibrational contribution, expressing the free energy of the system when constrained in one of the basins [17] /03/ $ IOP Publishing Ltd Printed in the UK S1085

142 S1086 ELaNaveet al In this work we review the numerical techniques which allow the evaluation of the statistical properties of the PEL for atomic and molecular systems [18, 19, 21 24]. The paper is organized as follows. Section 2 provides a brief introduction to the IS formalism, introduced by Stillinger and Weber [17]. Within this formalism, an exact expression for the liquid free energy, based on landscape properties, can be derived. Section 3 reviews the numerical techniques which allow a precise numerical evaluation of the liquid free energy. Section 4 describes the numerical techniques required for the evaluation of the IS energies. Section 5 discusses techniques for evaluating the vibrational component of the free energy. Section 6 shows how, from the previous information, is possible to evaluate the configurational entropy. Section 7 discusses a possible modelling of the statistical properties of the landscape, based on the hypothesis of a Gaussian distribution of the basin s depth [19, 21, 25], and compares the predictions of the model with numerical results for a molecular system. 2. The free energy in the IS formalism In the IS formalism [17], the free energy of a supercooled liquid is expressed in terms of the statistical properties of the PEL. The potential energy surface is partitioned into so-called basins, each defined as the set of points such that a steepest descent path ends in the same local minimum. The configuration at the minimum is called the IS and its energy and pressure are usually indicated as e IS and P IS.The partition function can be expressed as a sum of the Boltzmann weight over all the basins, i.e., as a sum over the basin partition functions. As a result, the Helmholtz liquid free energy F(T, V ) can be written as [17] where F(T, V ) = e IS (T, V ) TS con f (T, V ) + f vib (T, V ), (1) e IS is the average energy of the local minima explored at temperature T and volume V ; f vib is the vibrational free energy, i.e., the free energyofthe system constrained in one basin, a quantity depending on the shape of the basins explored; S conf is theconfigurational entropy, which counts the number of basins explored. The numerical evaluation of F(T, V ), e IS (T, V ), and f vib (T, V ) is sufficient for calculating S con f and, from it, the number of basins (e IS ) de IS with depth between e IS and e IS +de IS.Indeed, in the thermodynamic limit, ln (e IS ) can be derived from a plot of S con f versus e IS (parametric in T ). This quantity, together with the e IS dependence of f vib, provides a precise quantification of the statistical properties of the landscape. 3. Numerical evaluation of F (T,V ) This section describes the numerical techniques used to evaluate the liquid free energy, based on thermodynamic integration [18, 19, 21, 22, 26]. First, a path in the (T, V ) plane, connecting the ideal gas state to the desired state point, has to be selected. The selected path must avoid the liquid gas first-order line. A convenient choice is a constant-temperature path (with T = T 0 higher than the liquid gas critical temperature) from infinite volume to the desired volume, followed by a constant-volume path from T 0 down to the range of temperature of interest. In the general case of a system of N rigid molecules, the ideal gas free energy is { F ig (T, V, N) = Nk B T ln π ln ν +ln [ V A 3 R x R y R z N T 3 ]}, (2)

143 Numerical evaluation of the statistical properties of a potential energy landscape S1087 where A 2πmk B /h 2, R µ 8π 2 k B I µ /h 2 (with µ denoting x, y, orz), I µ is the inertia moment of the molecule with respect the axis µ,andlnν accounts for the molecular symmetry. In the case of C 2v molecules (such as water), ν = 2, due to there being two possible degenerate angular orientations of the molecule [27]. To perform the thermodynamics integration along the isotherm T 0, one needs to select about state points at different volumes (figure 1(a)). Of course, the smallest volume chosen must coincide with the final volume V 0. The largest V -value (V ) must be chosen in such a way that the vast majority of the molecular interactions are binary collisions, i.e., such that the volume dependence of the pressure is well described by the (first-order) virial expansion. At large volumes, although the dynamics is very fast, care has to be taken to run the simulation for long enough to sample a large number of binary collisions. The free energy at (T 0, V 0 ) can be calculated as V0 F(T 0, V 0 ) = F ig (T 0, V 0 ) dv P ex (T 0, V ) + U(T 0, V 0 ), (3) T 0 where U(T 0, V 0 ) is the potential energy and P ex (T 0, V ) is the excess pressure,i.e., the pressure in excess of the ideal gas pressure. The calculated P ex (T 0, V ) curve can be fitted according to the polynomial in powers of V 1 (figure 1(b)): n P ex (T 0, V ) = a k (T 0 )V (k+1), (4) giving k=1 F(T 0, V 0 ) = F ig (T 0, V 0 ) + n k=1 a k (T 0 )V k 0 k + U(T 0, V 0 ) T 0. (5) To perform the thermodynamic integration along a constant-v 0 path, it is necessary to evaluate the internal energy U(T, V 0 ) as a function of T, from T 0 down to the lowest state where equilibration of the system is feasible (figure 1(c)). The resulting free energy F(T, V 0 ) can be calculated as F(T, V 0 ) = F(T 0, V 0 ) +3R ln(t/t 0 ) + T T 0 dt T U(T, V 0 ). (6) T The 3R ln(t/t 0 ) term accounts for the ideal gas contribution to the free energy. Again, a fit of U(T, V 0 ) versus T is required to evaluate the integral in the above expression. One possibility, which has been often found to be very successful for dense systems (small V 0 )[22, 26], is to fit U(T, V 0 ) versus T according to the Tarazona law [28], i.e., U(T, V 0 ) = b 0 (V 0 ) + b 1 (V 0 )T 3/5. Of course, for the present purposes, any functional form which correctly represents U(T, V 0 ) can be selected. In summary, performing thermodynamic integration, an accurate numerical expression for F(V, T ) can be obtained. 4. The average IS energy e IS This section describes how to calculate the average IS energy e IS (T, V ). Recently, it has been shown that, for cooling at constant volume, on entering the supercooled region, the system starts to explore basins of lower and lower e IS [32]. The T -dependence of the average explored basin depth follows a T 1 -law [19, 21, 23, 33] for fragile liquids. Note that for silica, the prototype of a strong liquid, the T 1 -law is not observed and e IS (T, V ) appears to approach a constant value on cooling [20].

144 S1088 ELaNaveet al MD results polynomial fit Virial (c) 0 20 P ex [ MPa ] (T o,v o ) U [ kj / mole ] (b) T V (T,V o ) V [ nm 3 ] T [ K ] Figure 1. (a) Thermodynamic integration paths used to calculate the total free energy at the thermodynamical points of interest, starting from the ideal non-interacting gas state. (b) Excess pressure at T = T 0 as a function of volume. The open circles are the MD results. The dashed line is the first term of the virial expansion for the excess pressure; the solid curve is a third-order polynomial fit to the entire set of data. (c) The potential energy (open circles) at the volume V 0 over the entire temperature range considered; the solid curve is the fit of the data. The data are from our simulation [21] of a system of N = 343 molecules modelled by the Lewis and Wahnström model for orthoterphenyl [29], whose dynamics [30] and thermodynamics [21, 31] features have been studied in detail. (a) 80 In order to evaluate e IS (T, V ), one needs to perform steepest descent potential energy minimizations for a statistically representative ensemble of equilibrium configurations, to locate their corresponding IS, i.e., local minima. For efficiency reasons, the search for the closest local minima is performed using the conjugate gradient algorithm [34]. In this algorithm, the system evolves along a sequence of straight directions until the minimum is reached. In each step, the new search direction recalls the directions already explored, improving the algorithm efficiency. In rigid molecule systems, each step is composed via a sequence of minimizations of the centre of mass coordinates, followed by a minimization of theangular coordinates. Rotations around the principal axis of the molecule are often chosen. The minimization procedure is continued until the energy changes by less than a preselected precision. Since the change in e IS (T ) in supercooled states is often less than one per cent of its own value, a high precision is required in the minimization procedure. In figure 2 we show e IS (a) as a function of T and (b) as a function of 1/T for a rigid molecularmodel. 5. The vibrational free energy The vibrational free energy f vib (e IS, T, V ) = U vib (e IS, T, V ) TS vib (e IS, T, V ) is the free energy associated to the exploration of a basin of depth e IS at temperature T and volume V. f vib (e IS, T, V ) takesinto account both the kinetic energy of the system and the local structure

145 Numerical evaluation of the statistical properties of a potential energy landscape S1089 Figure 2. e IS as a function of T (a) and as a function of 1/T (b). The data are from simulations [21] of a system N = 343 molecules modelled by the Lewis and Wahnström model for orthoterphenyl [29]. of the basin with energy e IS.Fromaformal point of view, it is defined as ( x y z V f vib (e IS, T, V ) = k B T ln basin exp( β[v (r N ) e IS ]) dr N λ 3N (e IS ) de IS ), (7) where is the sum over all the basins with energy depth e IS.Theintegration of the Boltzmann factor is performed over all points in configuration space associated with the selected basin. Here µ (2π I µ k B T ) 1/2 /h, λ h(2πmk B T ) 1/2 is the de Broglie wavelength, and V (r N ) is the potential energy. The evaluation of the integral requires the exact knowledge of the shape of the PEL in the basin and, in general, it will give rise to a complex T -dependence of the vibrational energy. The best that can be done at the present time is to assume that the e IS -dependence in f vib is captured by the e IS -dependence of the density of states of the basin, evaluated at the IS configuration [21, 31]. In other words, the vibrational free energy is split into a harmonic contribution (which depends on the curvature of the potential energy at the minimum) and an anharmonic contribution,which is often assumed basin independent. In molecular systems, the Hessian, the matrix of the second derivatives of the potential energy, is calculated numerically, selecting as molecularcoordinates the centre of mass and the angles associated with the rotations around the three principal molecular inertia axes. Diagonalization is performed with standard numerical routines. In the harmonic approximation, the free energy associated with a single oscillator at frequency ω is k B T ln(β hω). Hence, the basin free energy can be written as 6N 3 f vib (e IS, T, V ) = k B T ln(β hω i (e IS )) + F anh (T, V ), (8) i=1

146 S1090 ELaNaveet al with U vib (T, V ) = (6N 3) k BT 2 + U anh(t, V ), (9) and 6N 3 ( ]) [ hωi (e IS ) S vib (e IS, T, V ) = 1 ln + S anh(t, V ); (10) k i=1 B T here the ω i (e IS ) are the frequencies of the 6N 3independent harmonicoscillators given by the square roots of the 6N 3 non-zero eigenvalues of the Hessian matrix evaluated in the IS. is the average over all the basins with the same energy e IS.Notethat the above equations are derived assuming 6N 3 ( 6N 3 ln(β hω i (e IS )) = ln exp ln(β hω i (e IS ))). (11) i=1 For the molecular systems studied so far, this approximation introduces an error smaller than 1%. The relevant approximation consists in dropping the e IS -dependence in the anharmonic contribution to the vibrational free energy F anh (and of course in U anh (T, V ) and S anh (T, V )). In other words, the anharmonicities are assumed to be identical in all basins. Under such an approximation, U anh (T, V ) can be calculated from the simulation data as U anh (T, V ) = U(T, V ) e IS (T, V ) (6N 3) k BT 2, (12) and it can be well fitted byanexpansion in powers of T,starting from T 2,as N c U anh (T, V ) = c k (V )T k. (13) k=2 Correspondingly, S anh (T, V ) can be estimated by thermodynamic integration along the isochore between temperatures 0 and T as S anh (T, V ) = T 0 i=1 dt U(T, V ) = T T N c k=2 kc k (V ) k 1 T k 1. (14) An alternative method for estimating the anharmonicities of the system is to assume that all the basins are quasi-harmonic and that the T -dependence of U anh (T, V ) arises from the e IS -dependence of the anharmonicity, i.e. U anh (T, V ) = D(e IS )T 2, (15) and S anh = 2D(e IS )T 2. (16) D(e IS ) can be calculated from a parametric plot of U anh (T, V )/T 2 versus e IS. By incorporating the anharmonic corrections, which in the models of simple fragile liquids studied so far are not particularly significant [21, 22], a good estimate of the basin free energy is obtained. We note on passing that for the cases of network forming liquids, anharmonic corrections are relevant [20, 23] and must be taken into account. In the assumption of U anh (T, V ) independent of e IS,allthe e IS -dependence in the basin free energy is carried by the term V(V, e IS ) 6N 3 i=1 ln ω i (e IS ). Aparametrization of such quantity as a function of e IS allows one to simply connect the basin free energy to the basin depth. Although in all models studied so far [19, 21, 31, 35] a linear relation between basin depth and basin shape V satisfactorily describes their relation, here we use the moregeneral expression V = a(v ) + b(v )e IS + c(v )e 2 IS, (17) which best describes the simulation data (figure 3(a)).

147 Numerical evaluation of the statistical properties of a potential energy landscape S1091 Figure 3. V (a) and S con f (b) as a function of e IS. 6. The statisticalproperties of the landscape In the previous section wehavediscussedhowtorelate the basin shape to the basin depth. In this section, we exploitthe formulationoftheliquidfreeenergyinthe ISformalismtoevaluate the number of PEL basins as a function of the basin depth. This quantity is of primary interest both for comparing with the recent theoretical calculations [15, 26] and for examining some of the proposed relations between dynamics and thermodynamics [16, 36, 37] connecting a purely dynamical quantity such as the diffusion coefficient to a purely thermodynamical quantity (S conf ). The number of basins as a function of the basin depth e IS has been recently evaluated for a few models [18 24, 38], and from the analysis of experimental data [39 41]. S con f (T, V ) the logarithm of the number of basins explored can be calculated as the difference of the entropic parts of equations (6) and (10), i.e., as S con f (T, V ) = S(T, V ) S vib (T, V ) S anh (T, V ). (18) In the thermodynamic limit, when fluctuations are negligible, a parametric plot in T of S con f (T, V )/k B versus e IS (T, V ) provides an accurate estimate of the number of basins of depth e IS. This information, together with the information on the e IS -dependence of the basin shape (or volume) (equation (17)), completely defines the statistical properties of the landscape, at least in the range of e IS -values sampled by the system in the T -region studied. The availability of S conf (e IS ) (figure 3(b)) and V(e IS ) (figure 3(a)) opens the possibility of a modelling of the thermodynamic of the system in terms of landscape properties, as discussed in the next section.

148 S1092 ELaNaveet al 7. The random energy model: the Gaussian landscape Amodelling of the statistical properties of the landscape is the next conceptual step in the development of a thermodynamic description of the liquid in the IS formalism. A possible modelling, which appears to be consistent with the numerical evidence for fragile liquids, is based on the hypothesis that the number (e IS ) de IS of distinct basins of depth between e IS and e IS +de IS in a system of N atoms or molecules is described by a Gaussian distribution [19, 23, 25, 31, 42], i.e., (e IS ) de IS = e αn e (e IS E 0 ) 2 /2σ 2 de (2πσ 2 ) 1/2 IS. (19) Here the amplitude e αn accounts for the total number of basins, E 0 has the role of an energy scale, and σ 2 measures the width of the distribution. One can understand the origin of such adistribution by invoking the central limit theorem. Indeed, in the absence of a diverging correlation length, in the thermodynamic limit, each IS can be decomposed into a sum of independent subsystems, each of them characterized by its own value of e IS. The system IS energy, in this case, will be distributed according to equation (19). We note that this hypothesis will break down in the very low-energy tail,where differences between the Gaussian distribution and the actual distribution become relevant. As discussed in [25], the system Gaussian behaviour reflects also some properties of the independent subsystems. Within the assumptions of equation (19) Gaussian distribution of basindepths and of the quadratic dependence of thebasin free energy on e IS, both of the harmonic term (equation (17)) and of the anharmonic contribution (i.e. D(e IS ) = d 0 + d 1 e IS + d 2 e 2 IS ), an exact evaluation of the partition function can be carried out. The corresponding Helmholtz free energy is given by F(T, V ) = TS conf (T, V ) + e IS (T, V ) + f vib (E 0, T ) ( + k B T b(v ) d ) 1(V )T ( e IS (T, V ) E 0 ) k B ( + k B T c(v ) d ) 2(V )T ( e IS (T, V ) 2 E0 2 ). (20) k B Moreover, using the notation B 1 = b(v ) d 1 (V )T /k B and B 2 = c(v ) d 2 (V )T /k B we have e IS (T ) = (E 0(V ) (B 1 (V ) + β)σ 2 (V )), (21) 1+2B 2 (V )σ 2 (V ) and S con f (T )/k B = α(v )N ( e IS (T, V ) E 0 (V )) 2 /2σ 2 (V ). (22) Note that, when we can neglect the anharmonic contributions to F(T, V ) and the quadratic term of equation (17), from a plot of e IS (T ) versus 1/T, one can immediately evaluate two of the parameters of the Gaussian distribution, σ 2 (from the slope) and E 0 (from the intercept). Similarly, from fitting S con f (T ) according to equation (22), one can evaluate the last parameter α (see figure 3). The fitting parameters α(v ), E 0 (V ),andσ 2 (V ) depend in general on the volume. A study of the volume dependence of these parameters, associated with the V -dependence of the shape indicators (a and b in equation (17)) provides a full characterization of the volume dependence of the landscape properties of a model, and offers the possibility of developing a full equation of state based on statistical properties of the landscape. When comparing numerical simulation data and theoretical predictions equations (21) and (22) the range of temperatures must be chosen with great care. Indeed, at high T,

149 Numerical evaluation of the statistical properties of a potential energy landscape S1093 the harmonic approximation will overestimate the volume in configuration space associated with an IS. While in harmonic approximation such a quantity is unbounded, the real basin volume is not. Indeed, the sum of all basin volumes is equal to the volume of the system in configuration space. Anharmonic corrections, ifproperly handled, shouldcompensate for such overestimation, but at the present time no model has been developed that correctly describes the high-t limit of the anharmonic component. Numerical studies have shown that the range of validity of the present estimates of the anharmonic correction does not extend beyond the temperatures at which the system already shows a clear two-step relaxation behaviour in the dynamics. Indeed, the presence of a two-step relaxation is a signature of the system spending a timelarger than the microscopic characteristic times around a well defined local minimum. 8. Conclusions In this paper we have discussed the numerical techniques employed to evaluate the statistical properties of the PEL for molecular systems. These numerical calculations are limited to the region of temperatures and volumes where equilibrium configurations can be numerically generated. Still, very simple arguments can be presented which allow one to generalize the results and formulate a full thermodynamic description of the supercooled liquid state, just in terms of the statistical properties of the PEL. The possibility of partitioning the free energy and its thermodynamic derivatives as a sum ofconfigurational and vibrational degrees of freedom has been recently exploited to derive a satisfactory description of the equation of state [31, 43] for supercooled liquids just in terms of PEL properties. A better understanding of the nature of each contribution (configurational and vibrational) to quantities such as the total pressure of the system is achieved. At the same time, the availability of detailed estimates for the landscape properties strongly suggests a generalization ofthis approach to out-of-equilibrium conditions. It has been recently shown [44] that if the system ages exploring the same basins as were visited in equilibrium, it is possible to give an out-of-equilibrium equation of state expressing P not only as a function of V and T but also as a function of the (time-dependent) depth of the basin explored. The availability of numerical estimates for the statistical properties of the PEL in models of simple liquids should encourage theoreticians to develop schemes for the analytic evaluation of these quantities. If this goal were reached, the understanding of the thermodynamics of supercooled liquids and glasses would be improved significantly. Acknowledgments We acknowledge support from MIUR-COFIN 2000 and FIRB. FS thanks Sharcnet for a visiting professor fellowship. References [1] Debenedetti P G 1997 Metastable Liquids (Princeton, NJ: Princeton University Press) [2] Debenedetti P G and Stillinger F H 2001 Nature [3] Mézard M 2001 More is Different ed M P Ong and R N Bhatt (Princeton, NJ: Princeton University Press) [4] Heuer A 2003 J. Phys.: Condens. Matter 15 [5] Tarjus G 2003 J. Phys.: Condens. Matter 15 [6] Sastry S 2003 J. Phys.: Condens. Matter 15 [7] Angell A 1995 Science [8] Torre R, Bartolini P and Pick R M 1998 Phys. Rev. E

150 S1094 ELaNaveet al Taschin A, Torre R, Ricci M A, Sampoli M, Dreyfus C and Pick R M 2001 Europhys. Lett [9] Götze W 1999 J. Phys.: Condens. Matter 11 A1 [10] Cummins H Z 1999 J. Phys.: Condens. Matter 11 A95 [11] Binder K et al 1997 Complex Behavior of Glassy Systems ed M Rubi and C Perez-Vicente (Berlin: Springer) [12] Schilling R 1994 Disorder Effects on Relaxational Processes ed A Richert and A Blumen (New York: Springer) Kob W 1997 Experimental and Theoretical Approaches to Supercooled Liquids: Advances and Novel Applications ed J Fourkas et al (Washington, DC: American Chemical Society) [13] Speedy R 1998 J. Phys.: Condens. Matter Speedy R 1997 J. Phys.: Condens. Matter Speedy R 1996 J. Phys.: Condens. Matter [14] Götze W 1991 Liquids, Freezing and the Glass Transition ed J-P Hansen, D Levesque and J Zinn-Justin (Amsterdam: North-Holland) Götze W and Sjörgen L 1992 Rep. Prog. Phys Götze W 1999 J. Phys.: Condens. Matter 11 A1 [15] Mézard M and Parisi G 1999 Phys. Rev. Lett Mézard M and Parisi G 2000 J. Phys.: Condens. Matter [16] Xia X and Wolynes P G 2001 Phys. Rev. Lett [17] Stillinger F H and Weber T A 1982 Phys. Rev. A Stillinger F H and Weber T A 1984 Science Stillinger F H 1995 Science [18] Scala A, Starr F W, La Nave E, Sciortino F and Stanley H E 2000 Nature [19] Sastry S 2001 Nature [20] Saika-Voivod I, Poole P H and Sciortino F 2001 Nature [21] Mossa S, La Nave E, Stanley H E, Donati C, Sciortino F and Tartaglia P 2002 Phys. Rev. E [22] Sciortino F, Kob W and Tartaglia P 1999 Phys. Rev.Lett [23] Starr F W, Sastry S, La Nave E, Scala A, Stanley H E and Sciortino F 2001 Phys. Rev. E [24] Speedy R J 2001 J. Chem. Phys [25] Heuer A and Büchner S 2000 J. Phys.: Condens. Matter [26] Coluzzi B, Parisi G and Verrocchio P 2000 Phys. Rev. Lett [27] Mayer J E and Mayer M G 1963 Statistical Mechanics (New York: Wiley) [28] Rosenfeld Y and Tarazona P 1998 Mol. Phys [29] Wahnström G and Lewis L J 1997 Prog. Theor. Phys. (Suppl.) [30] Rinaldi A, Sciortino F and Tartaglia P 2001 Phys. Rev. E [31] La Nave E, Mossa S and Sciortino F 2002 Phys. Rev. Lett [32] Sastry S 2000 J. Phys.: Condens. Matter [33] Heuer A 1997 Phys. Rev.Lett Büchner S and Heuer A 1999 Phys. Rev. E [34] Numerical Recipes, switcher.html [35] Sciortino F and Tartaglia P 2001 Phys. Rev. Lett [36] Adam G and Gibbs J H 1965 J. Chem. Phys [37] Schulz M 1998 Phys. Rev. B [38] Speedy R J 1998 Mol. Phys [39] Stillinger F H 1998 J. Phys. Chem. B [40] Richert R and Angell C A 1998 J. Chem. Phys [41] Speedy R J 2001 J. Phys. Chem. B [42] Deridda B 1981 Phys. Rev. B [43] Debenedetti P G, Truskett T M, Lewis C P and Stillinger F 2001 Adv. Chem. Eng [44] Mossa S, La Nave E, Sciortino F and Tartaglia P 2002 Preprint cond-mat/

151 INSTITUTE OF PHYSICSPUBLISHING JOURNAL OFPHYSICS: CONDENSED MATTER J. Phys.: Condens. Matter 15 (2003) S1051 S1068 PII: S (03) Potential energy, relaxation, vibrational dynamics and the boson peak, of hyperquenched glasses CAustenAngell 1,Yuanzheng Yue 2,Li-MinWang 1,JohnRDCopley 3, Steve Borick 4 and Stefano Mossa 5,6 1 Department of Chemistry and Biochemistry, Arizona State University, Tempe, AZ 85287, USA 2 Department of Chemistry, Aalborg University, 9220 Aalborg, Denmark 3 National Institute of Standards and Technology, Gaithersburg, MD , USA 4 Scottsdale Community College, Scottsdale, AZ , USA 5 Laboratoire de Physique Théorique des Liquides, Université Pierre et Marie Curie, 4Place Jussieu, 5252 Paris Cedex 05, France 6 Dipartimento di Fisica, INFM Udr and Centre for Statistical Mechanics and Complexity, Università di Roma La Sapienza, Piazzale Aldo Moro 2, I-00185, Roma, Italy Received 29 December 2002 Published 10 March 2003 Onlineat stacks.iop.org/jphyscm/15/s1051 Abstract We describe a combination of laboratory and simulation studies that give quantitative information on the energy landscape for glass-forming liquids. Both types of study focus on the idea of suddenly extracting the thermal energy, so that the system obtained for subsequent study has the structure, and hence potential energy, of a liquid at a much higher temperature than the normal glass temperature T g. One type of study gives information on the energy that can be trapped in experimental glasses by hyperquenching, relative to the normal glass, and on the magnitude of barriers separating basins of attraction on the landscape. Stepwise annealing studies also give information on the matter of energy heterogeneity and the question of nanogranularity in liquids near T g.theother type of study gives information on the vibrational properties of asystemconfined to a given basin, and particularly on how that vibrational structure changes with the state of configurational excitation of the liquid. A feature in the low frequency ( boson peak ) region of the density of vibrational states of the normal glass becomes much stronger in the hyperquenched glass. Qualitatively similar observations are made on heating fragile glass-formers into the supercooledand stableliquid states. Thevibrationaldynamicsfindings are supported and elucidated by constant pressure molecular dynamics/normal mode MD/NM simulations/analysis of the densities of states of different inherent structures of a model fragile liquid (orthoterphenyl (OTP) in the Lewis Wahnstrom approximation). These show that, when the temperature is raised at constant pressure, the total density of states changes in a manner that can be well represented by a two-gaussian excitation across thecentroid, leaving a thirdand major Gaussian component unchanging. The low frequency Gaussian component, which grows with increasing temperature, has a constant peak /03/ $ IOP Publishing Ltd Printed in the UK S1051

152 S1052 C A Angell et al frequency of 18 cm 1 and is identified with the Boson peak. It is suggested that the latter can serve as a signature for configurational excitations of the ideal glass structure, i.e. the topologically diverse defects of the glassy solid state. The excess vibrational heat capacity associated with this generation of low frequency modes with structural excitation is shown to be responsible for about 60% of the jump in heat capacity at T g,mostofthe remainder coming from configurational excitation. 1. Introduction The energy landscape [1 3] has provided an important conceptual route to dissecting the properties of liquids into separate quasi-independent contributions, namely those due to structure and those due to vibrational dynamics. Thus the system is thought of as being characterized by two temperatures, a fictive temperature T f [4] that relates to potential energy and a real temperature T that relates to kinetic energy. When the two temperatures are the same, the system is a liquid. Otherwise it is a glass. The possibility of such a dissection depends on the different timescales for vibration and relaxation that characterize viscous liquids. Because of these differences, it is possible to trap the liquid in different states of configurational excitation by cooling at different rates and then to study the effect of trapped structure (or level on the landscape) on the vibrational dynamics of the system. In this paper we use hyperquenching methods (melt spinning of mineral glasses [5 7] and electrospray quenching of molecular glasses [8]) to cool liquids ten million times faster than normal, and thus to trap them in states approaching that of the mode coupling theory T c,nowcalled the crossover temperature. One objective is to obtain glasses produced on timescales comparable to the most slowly cooled computer glasses. Another is to observe the changes in potential energy, and vibrational dynamics, as a trapped glass finds its way back towards the standard glassy state [8, 9] producedby steady coolingof the liquid at about 0.33 K s 1 (20 K min 1 ). We will need to interpret the finding that, in fact, it never reaches that state unless, first, all memory is removed by returning to the metastable liquid state. 2. Experimental section Samples for study were prepared using two distinct methods, appropriate to the samples that it was desired to study. Most of the measurements were performed on a room-temperature-stable, micrometre-diameter, fibre material produced from a mineral glass of complex composition (major components in mass% (SiO , Al 2 O , FeO 11.7, CaO 10.4, MgO 5.5, Na 2 O3.9)) which is produced commercially as an insulating material, Rockwool. This material is produced from the molten state at 1700 K by the cascade spinner, which consists of a series of rapidly rotating metal discs [7]. Onto the first of these is poured a thin stream of white hot molten material. The discs rotate at approximately 6000 rpm spinning off droplets of liquid, each drawing behind a fine fibre of glass. Melt that does not attach to the first spinning disc is passed to the next and so on until virtually all the melt has been fibred. This product is then sieved ( 63 µm) in order to separate the glass droplets from the fine fibres. The cooling rate for the fibres is found to be about 10 6 Ks 1 [6]. With respect to the volume distribution, the fibres of diameters d < 4.1 µm account for 16 vol%, those of d < 7.7 µm 50vol%and those of d < 12.6 µm 84vol%.

153 Potential energy, relaxation, vibrational dynamics and the boson peak, of hyperquenched glasses S1053 From viscosity measurements the precursor liquid state is intermediate in strength, with a F 1/2 fragility of 0.61, close to that of glycerol, 0.54 [10]. In a simple case like glycerol a F 1/2 fragility of 0.54 corresponds to an m fragility (or steepness index ) of 53 [10]. In the second method described in more detail elsewhere [8] a spray of micro-droplets is directed onto the interior surface of a large-sample differential scanning calorimetry (DSC) calorimeter pan, held at liquid nitrogen temperature, using a field of about 10 kv cm 1 to electrostatically destabilize the thin stream of liquid emerging from a 200 µm stainless steel capillary tube. The hyperquenched droplets are collected during several distinct bursts. After each burst the surface recovers its initial low temperature. Data for propylene glycol, analysed in [8], show that the cooling rate obtained is between 10 5 and 10 6 Ks 1. To observe the course of the energy evolution during the descent of the landscape at low temperature, we use DSC. We compare the apparent heat capacity during steady upscans of the hyperquenched glass, or some annealed variant of it, with scans of a standard glass obtained by cooling an initially equilibrated liquid sample of the same material into the glassy state at the standard rate, 0.33 K s 1.Thelatter rate is chosen so that the enthalpy relaxation time at the fictive temperature is 100 s [5, 8, 9]. Details of the protocols used [6] will be given below. To study the vibrational dynamics of hyperquenched and annealed glasses we use cold neutron scattering time-of-flight measurements carried out using the Disk Chopper Spectrometer at the NIST (National Institute of Standards and Technology) Center for Neutron Research. The measurements were confined to just one of the hyperquenched glasses of this study, namely the Rockwool mineral glass. Typically g of material (or 30 g when crystallized) were packed in an 18 mm diameter, 100 mm tall, aluminium can and measurements were performed at room temperature using 4.1 Å incident neutrons. Scattered neutrons were counted in 913 detectors placed 4 m from the sample and their energies were determined by time-of-flight spectroscopy, sorting events into µs time channels. Due to inherent technical limitations, and because of our primary concern with effects in the vicinity of the Boson peak, we concentrate on the low frequency part of the spectrum. 3. Results In figure 1 we show the upscans of hyperquenchedpropylene glycol glass compared with the standard scan for the same sample. The latter is run immediately after the initial scan. The area between the two scans corresponds to the energy difference between the sample in the hyperquenched state and the energy of the standard glass which can only be determined when both have been returned to the same state by heating to the upper end of the transformation range, about 10% above the onset T g. The onset T g corresponds, within 0.5 K, with the fictive temperature of the standard glass, obtained by the usual equalareaconstruction [11]. From these data we can derive the fictive temperature, T f,ofthe hyperquenched glass, at which the state of equilibrium of the original liquid was frozen in during the hyperquench. As reported elsewhere [8] this proves to be 15% higher than the standard T g.comparable results reported [5, 6] for the hyperquenchedsilicate glass show that, for this less-fragile glass-former, fictive temperatures as high as 1.25 T g can be obtained. The hyperquenched glass can be returned to the standard glass energy in a series of steps, making it possible to obtain a measure of that part of the total frozen-in energy that is released by annealing the hyperquenched glass at a specific temperature for a specific time. For instance, figures 2(a) and (b) show the manner in which the energy lost, during a specific anneal, is determined by scanning a series of samples (a h) up to the liquid state at 1.1 T g,after annealing for the temperatures indicated for a specific time. The annealing time is 90 min in figure 2(a) and eight days in figure 2(b). The difference between each of these post-annealing

154 S1054 C A Angell et al 4 Hyperquenched Apparent C p (J/K*g) H H hq Standard H s T g T Standard -/+ 20K/min T f s =169.4 K T f 0 Hyperquenched (Spray)/+ 20 K/min T f =195 K T(K) Figure 1. DSC upscans at the standard rate (20 K min 1 ) of hyperquenched glass of propylene glycol (lower curve) compared with the standard upscan of standard glass (i.e. glass formed by cooling at the standard rate). Integration over the area between the two curves gives the difference in energy between the two glasses. Inset: enthalpy temperature diagram for glass-formers cooled at different rates, with common representation of energy landscape section superposed. upscans and the original hyperquenched glass upscan yields, by integration, the amount of energy released during the anneal. The effect of annealing time at a given temperature can be seen by comparing scans such as (h) of figure 2(a) with (f) of figure 2(b). There are several points of great interest in these data that will be discussed in the next section. Important aspects of the way in which the vibrational dynamics of the glasses in these different trapped states changes with the potential energy of the glass can be revealed by neutron scattering studies [12 15]. The results of our cold neutron inelastic scattering (time-of-flight) measurements are presented in figures 3 and 4 as the function Z(ω) = A hω (1 e hω/kt ) (Q 4 max Q4 min ) θmax θ min ( d 2 σ d de f ) sin θ dθ, (1) where Q and hω are the wavevector transfer and energy transfer, respectively, T is the temperature, (d 2 σ/d de f ) is the double differential neutron scattering cross section per unit solid angle and final energy E f, Q min and Q max are ω-dependent extreme values of Q, θ is the scattering angle and A is an arbitrary constant; Z(ω) may be regarded as a crude representation of an effective vibrational density of states (VDOS), G(ω),ignoring corrections for effects such as multiphonon and multiple scattering. In our experiments Q min and Q max for elastic scattering (i.e. for scattering with ω = 0) were 0.3and 2.9Å 1,respectively. In figure 3 we present data for the hyperquenched glass, and for lower energy states of the same sample produced by annealing. In figure 3(a), the full circles are for the hyperquenched state, the open symbols are for an annealed version of the initial hyperquenched glass which should approximate the standard glass state and the full line is for a super-annealed glass

155 Potential energy, relaxation, vibrational dynamics and the boson peak, of hyperquenched glasses S1055 C p (Jg -1 K -1 ) T a (K) a: non-aged b: 573 c: 623 d: 673 e: 723 f: 773 g: 798 h: 823 t a =90 min T onset =710 K standard upscan (a) 1.0 a b c d e f g h T(K) Cp (Jg -1 K -1 ) T a (K) a: 573 b: 623 c: 673 d: 723 e: 773 f: 823 g: 873 h: 923 i: 973 non-aged t a =8 days a b c standard f d e g h i (b) T(K) Figure 2. (a) Standard DSC upscans of hyperquenched mineral glasses, after an initial annealing treatment for 90 min at the temperatures designated in the legend. Curve marked standard upscan is for glass cooled from above the transformation range at standard rate of 20 K min 1. It has onset glass temperature and fictive temperature of 944 K. (b) Standard DSC upscans (apparent heat capacities in J g 1 K 1 )ofhyperquenched mineral glasses after an initial annealing treatment for 8days at each of the temperatures noted in the legend. Note that in this case the selected annealing temperatures extend to temperatures above the standard T g of 944 K. Note that, for annealing temperatures greater than 823 K, the scans lose the crossover to exothermic responses characteristic of scans (a) (e) and develop the overshoot usually associated with annealed glasses. obtained by holding the sample for 21 h at a temperature of 894 K, which is 5.9% below the normal T g.finally, the full triangles are for the crystallized material obtained by holding the glass at 1156 K for 150 min. The excess Z(ω) at low frequencies of the hyperquenched glass over the value for the annealed glasses (and also the crystal) is the focus of our interest, for reasons that will become especially clear in the final paragraphs of our discussion. Figure 3(b) shows how this excess is emphasized by displaying Z(ω) for a restricted range of Q values, namely Q = Å 1. The physics appears to be most interesting on length scales of 2π/ Q = 9 16 Å.

156 S1056 C A Angell et al (a) (b) Figure 3. (a) Z(ω) up to 200 cm 1 for the mineral glass in hyperquenched, normal and wellannealed states, and the crystal state, as described in the text. For a monatomic glass corrected for multiple scattering and multiphonon scattering, Z(ω) approximates the VDOS. (b) Z(ω) up to 160 cm 1 for the same samples asinpart(a)except with Q restricted to the range Q = Å 1. Note the sharp maximum developed at 40 cm 1 in the case of the hyperquenched glass. In figure 4 we show Z(ω) at different temperatures for the fragile aqueous solution glassformer, Ca(NO 3 ) 2 8H 2 O, which has been well studied by other methods [16]. These plots represent the uncorrected, multicomponent, equivalent of the density of states G(ω) shown for the elemental glass-former Se by Phillips et al [12] in glassy and liquid states. A better analogue might be the DOS for the glassy and liquid states of the fragile molecular glassformer OTP by Wuttke et al [15]. The latter authors found that the DOS was independent of temperature for T < T g,butincreases at low frequencies for T > T g. The spectra seen in figure 4 are for the glass at 165 K (below T g of 183 K), at two temperatures 210 and 240 K in the supercooled state, and at ambient temperature, 298 K, where the solution is thermodynamically

157 Potential energy, relaxation, vibrational dynamics and the boson peak, of hyperquenched glasses S1057 (a) (b) Figure 4. (a) Z(ω) up to 100 cm 1 for glass, supercooled liquid and stable liquid states of the hydrated salt Ca(NO 3 ) 2 8H 2 O, as described in the text, showing that excess low frequency modes in high fictive temperature states are restricted to ω<70 cm 1.Standard T g is 183 K. (b) Z(ω) for extended frequency range for the same system and temperatures as in (a), showing the major temperature dependence of DOS in the range cm 1. stable. The sample at 180 K is below the standard glass temperature of 183 K but because of the time taken to set up the experiment and obtain the spectrum it is actually in the ergodic (supercooled liquid) state. It has a spectrum that is indistinguishable from that of the glassy sample at 165 K. The point to which we address attention in figure 4(a) is the excess Z(ω) for liquid samples over glassy states at low frequencies, like that of high fictive temperature glasses over low fictive temperature glasses infigure 3. We see the merging of the curves for frequencies

158 S1058 C A Angell et al above 70 cm 1 in figure 4(a) as the analogue of the crossover in the density of vibrational states of laboratory OTP at 3 4 mev (24 32 cm 1 ) reported bywuttke et al [15]. A corresponding crossover in the density of states, at 40 cm 1,will be seen in another set of DOS data, of much more precisely defined character, obtained from a simulation by one of us [17] to be introduced in the next section, based on the Lewis Wahnstrom model of OTP [18]. Figure 4(b) shows an extended frequency range for data in the Ca(NO 3 ) 2 8H 2 Osystem, which shows that something complicated, and distinct from the molecular glass-formers, happens to the DOS at intermediate frequencies. There is a major increase in Z(ω) with temperature in the frequency range cm 1.Thiseffect, which prevents the merging at 70 cm 1 from becoming an isosbestic point (crossover) as in the molecular OTP case, is probably associated with water hydrogen bonding and solvation structures. While this is a matter of considerable interest in connection with the thermodynamics and fragility of these hydrated melts (because of the extra increases in vibrational entropy with fictive temperature that it implies) it will not be considered further in this paper. 4. Discussion 4.1. Relative energy of trapped glassy state, and the trap depth dependence on fictive temperature First we discuss the relation of the energy trapped in the glass during hyperquenching relative to other quantities such as the energy of fusion and the energy of exciting the liquid to the top of the energy landscape appropriate to the density of the glass. We will compare this with the molar energy of escape from the energy minimum in which it was trapped during the quench. The energy trapped in the glass by hyperquenching is first assessed relative to the energy of the standard glass, by determining the integral over the difference between the two curves of figure 1. The result is 1.9 kjmol 1,which is small compared with the energy of fusion of about 12.5kJmol 1 (based on the value for the 1, 3-propylene glycol, which is crystallizable [19]). This enthalpy in excess of the standard glass is even smaller relative to the energy needed to excite the system from the normal glass temperaturetothetopofthesystem s energylandscape (at the density of the glass). While a precise value for this latter energy is not available (partly because the heat capacity at constant volume is not available over a wide range of temperature) it should be of the order of 20 kj mol 1 according to an argument given in [8]. Thus the range of glass potential energies that can be explored by the hyperquenching method is very limited, relative to the range of inherent structure energies that can be explored by computer simulation methods [18, 20]. Notwithstanding this limitation, the fact that hyperquench experiments explore the energetics near the low temperature end of the liquid range where the behaviour is most solid-like, and is also inaccessible to simulation studies at this time, offers special advantages, as we shall see below. Before considering the exploration of this range in more detail, we need to consider information about the height of the barriers trapping the hyperquenched glass that is available from figure 1. We can estimate this from the temperature, T esc,atwhichthesystemstarts to relax (i.e. escapes from the trap) during heating at 20 K min 1.FromotherDSCstudies we know this temperature corresponds to the temperature at which the system has an enthalpy relaxation time of 100 s. Assuming the escape is attempted on the inverse vibration frequency timescale, s, and that the probability of escape is a Boltzmann function of temperature, we obtain the relation τ = 100 s = exp(e trap /RT esc ) (2)

159 Potential energy, relaxation, vibrational dynamics and the boson peak, of hyperquenched glasses S1059 Figure 5. Assessment of the activation energy for relaxation of quenched glasses out of their trap sites, using the scaled fictive temperature, T f s/t f,shownby the open circle marked T f s/t f = Extrapolation of the straight line construction to log τ = 2predictsthetemperature at which a glass of this fictive temperature will start to relax during warm-up at the standard DSC heating rate of 20 K min 1 (see arrows). The second open circle, T f s/t f = 0.95, is for a sample in a sealed DSC pan, quenched (at much lower dt/dt) by dropping into liquid nitrogen (see [8]). which yields the value E trap = 39.9 kjmol 1,whenT esc = 125 Kissubstituted. Since this barrier height, expressed in molar units, is considerably greater than the height of the energy landscape given above, also in molar units, there is clearly some problem of interpretation. We therefore point out that our estimate of the trap depth is consistent with the estimate of the true activation energy for relaxation in viscous liquids suggested by Dyre [21]. It is also consistent with observations on the ionic conductivity of glasses that exhibit some degree of decoupling of conduction modes from viscous flow modes [22]. This activation energy is the one obtained from the slope of the Arrhenius straight line connecting the point of interest on the log(relaxation time) versus inverse temperature plot, to the attempt frequency, s(asnoted above). That the equation (2) activation energy is consistent with these considerations is seen by taking the relaxation time at the fictive temperature of the hyperquenched glass (obtained in the manner described in [6] and [8]), and connecting it to the s point at 1/T = 0. The activation energy obtained from the slope of this line is 40.0 kjmol 1,essentially the same as from the analysis of equation (2). The construction is shown in figure 5. Its support of the value obtained from equation (2) is seen most clearly from the fact that the extrapolation of the straight line to lower temperatures, to the value τ = 100 s, yields the temperature 125 K. This temperature is almost the same as that used in equation (2) at which relaxation is seen to begin, during the initial upscan at 20 K min 1 (figure 1). The problem of interpretation of these trap depth values will be dealt with elsewhere. We point out, however, that, according to the construction used above, the activation energy for relaxation will be larger when T f = T g (standard). At T g, E esc = 52 kj mol 1. We note also that, by this construction, the activation energy, i.e. trap depth, at T g will be universal, at 37RT g,solong as the pre-exponent remains s. It is interesting to compare these trap depth values with the activation energy for diffusion of H 2 Omolecules in ice, a single-particle process. The activation energy is 58.5kJmol 1 [23],

160 S1060 C A Angell et al which is somewhat in excess of the sublimation energy (51 kj mol 1 [24]). The activation energy for a number of other relaxation processes in ice has the same value [23]. Water, like PG, has two OH groups per molecule Stepwise descent of the landscape, and the possibility of nanogranularity in glass In figure 1 we have seen the way in which the energy difference between the hyperquenched glass and the normal glass may be manifested in one continuous scan. An opposite extreme can be imagined in which the energy difference is manifested and recorded during long observation at a single temperature. The temperature would be raised suddenly to a temperature of, say, 140 K (somewhat above the Kauzmann temperature of 116 K) and then the release of energy monitored isothermally. One might imagine that, in such an experiment, all the potential energy states usually observed during decrease oftemperature from the fictive temperature of the hyperquenched glass to the normal glass temperature could be observed as a function of time alone. The problem with such a putative experiment is not only that the time required to complete the observation would become excessively long (10 25 storelax (1/e)th of the way to the equilibrium state, according to the parameters of the well known Vogel Fulcher Tammann equation for the most probable relaxation time, noted in figure 6), but that both the initial jump and the initial recording of relaxation would have to be made extremely rapidly because (according to the activation energy determined in the previous section) the initial relaxation would occur on a timescale of 10 6 s. A compromise experimental protocol is that used in obtaining figure 2, in which the isothermal annealing is carried out in a series of stages. In each stage, annealing is allowed to occur for a fixed period of time at a succession of temperatures. Because the DSC is not well suited for detecting the small energy releases directly, such measurements are most accurately performed by carrying out an upscan, at the standard rate, at the end of each annealing period. The effect of theanneal is then obtained from the difference between the post-anneal scan and the scan on a sample which has not been annealed at all. Because this protocol requires a separate sample for each anneal, it is best carried out on asystem for which a large amount of hyperquenched material is available. Thus we use the hyperquenched mineral glass to record the annealing behaviour. Figure 2 includes the scan for a standard glass, formed by cooling the sample from the supercooled liquid to the glassy state at the standard 0.33 K s 1 (20 K min 1 ). It is in the comparison of the results of the hyperquenched glass series with the standard glass that this series reveals its most interesting aspect, discussed below. An alternative to the above protocol is available. This is to choose a fixed annealing temperature, and anneal for a series of different times. Then the fraction of total trapped energy that is released during each anneal can be obtained. Such a series, which also requires a number of identical samples, has been shown already in [6]. In that work, a plot of the fraction of the trapped energy released versus log (time of anneal) demonstrated the non-exponential nature of the relaxation process: (t) = 0 exp (t/τ) β (3) and allowed a stretching exponent β to be determined. The stretching exponent obtained is very small, 0.16, relative to those determined in experiments in the linear response regime (and to that expected, 0.7, from the approximate correlation of β with the fragility of the glass-former [10]). The difference is a manifestation of the non-linear nature of the relaxation in systems far from equilibrium. The non-linearity causes the relaxation of the hyperquenched glass to be much faster than that of an equilibrated glass at the same temperature (as if the glass

161 Potential energy, relaxation, vibrational dynamics and the boson peak, of hyperquenched glasses S1061 (a) (b) Figure 6. TheVDOS for the inherent structures of OTP, in the Lewis Wahnstrom model, obtained by steepest descent quenching of structures equilibrated at the designated temperatures (a) at 0.1 MPa (1 atm) and (b) at 200 MPa. Note the isosbestic point at 40 cm 1. The model has no internal degrees of freedom. is aware of the excess freevolume, or entropy, that it possesses). The faster relaxation at short times makes the relaxation look more non-exponential than the linear process actually is. Returning to the data of figure 2, we note the manner in which annealing at each temperature relaxes out a well defined portion of the total trapped energy, leaving the kinetics of relaxation of the remaining trapped energy completely unaffected. The manner in which the upscans of the samples that were subject to higher temperature anneals, abruptly crossover from a position which is endothermic with respect to the standard glass to one which is strongly exothermic, is quite striking. See, e.g., curve h in figure 2(a). Since Richert [25] has made a strong case for the heterogeneity of relaxation as the source of its non-exponential character, and since the heterogeneity is evidently spatial in character near T g [26 28], the figure 2 demonstration that the relaxation process is also energetically heterogeneous lends credence

162 S1062 C A Angell et al to the nanogranularity concept of liquids near T g. By this we mean the notion that there are independent micro-regions in the glass which are widely distributed in size (and local density) and stable in dimension. That they are stable in dimensions is suggested by the finding, detailed elsewhere [29], that once the apparent heat capacity becomes higher than that of the standard glass, the properties of the glass up to this temperature, become reproducible to cycling in temperature. The introduction of probesoflocal structure (e.g. cobalt ions [30]) into the parent oxide glass could make possible the demonstration that the structures of these micro-regions also becomes stable on annealing to this temperature. Figure 2 shows that the fast relaxing micro-regions can achieve states that are of low energy relative to the rest of the glass (the total energy of which remains above that of the standard glass) and then can re-absorb that energy during heating at what looks like a mini glass transition. This behaviour cannot be produced by annealing of the standard glass at the same temperature so it, and the annealing pre-peak (or shadow glass transition) that it produces, requires the structure of the unrelaxed portion of the hyperquenched glass for its manifestation (vault effect [31]?). Elsewhere [32], two of us have argued that this is the likely source of the weak endothermic rise in heat capacity observed in annealed forms of amorphous water formed by highly non-equilibrium processes. The endotherm has long been ascribed to astandard glass transition, which is now in question. The existence of independent micro-regions is also suggested by energy landscape considerations, as follows. A system at constant volume has a unique energy landscape that is fixed by the intermolecular potentials for the particles of the system. In configuration space the system is represented by a point that movesonthis surface. The point can only move in one direction at one time. However, figure 2 (and its counterpart in which time is varied at fixed temperature [7]) show that, for partially annealed glasses, the direction in which the energy changes with time during annealing, relative to a standard glass, depends on the timescale on which it is observed. Fast parts of the system increase in energy while slow parts decrease in energy, relatively. In the real space interpretation, one could argue that the independent nanograins can have different properties, fast parts (perhaps the smallest grains) behaving like homogeneous systems of high volume and enthalpy, while slow parts behave like homogeneous systems of low volume and enthalpy. While this may sound like an ensemble of landscapes, Stillinger (private communication) points out that it is better regarded as an ensemble of projections onto lower dimensions of the single higher-dimensional landscape that must encompass the full many-body behaviour of the system of interest. It is such complexity that is responsible for the fact (illustrated adequately by figures 2(a) and (b), and long known in glass science) that a glass formed by some arbitrary path cannot fully recover the state of the standard glass unless it is first brought into a state of complete internal equilibrium. This means it must first be heated above the glass transformation range, and then cooled to below T g at the standard rate. (A glass fully equilibrated by annealing long enough at a temperature below the standard T g will have a lower total energy than the standard.) The real space picture evoked above has much in common with the mosaic model used by Xia and Wolynes [33] who described the glass transition as a random first-order phase transition. It should be noted that the above amounts to an attempt to put into microscopic terms what is usually described by phenomenological models such as the Tool Narayanaswamy Moynihan [34], Kovacs Aklonis Hutchinson Ramos [35] and Scherer Hodge [36] models. All of these models contain parameters representing the non-exponentiality of relaxation and non-linearity of relaxation. Each model is capable of predicting the crossover phenomenon, and also the occurrence of the annealing pre-peak [37],but none provides a microscopic account

163 Potential energy, relaxation, vibrational dynamics and the boson peak, of hyperquenched glasses S1063 of the source of the non-exponentiality, and only the Scherer Hodge model accounts for the origin of non-linear behaviour Vibrational characteristics of glasses in different states of configurational excitation In this section we discuss a further manifestationofthe different structures of hyperquenched and annealed glasses. This is the redistribution of frequencies in the VDOS such as to favour, in high fictive temperature glasses, frequencies near the Boson peak frequency [38], Hz ( 50 cm 1 ) [38, 39] or 30 cm 1,asseen in figures 3 and 4. The Boson peak was originally identified from light scattering data [39] and the intensity of scattering is found to relate to the total density of states G(ω) tot as G(ω) tot /ω 2 [39]. Division of Z(ω) in figure 3(a) by ω 2 would certainly emphasize the excess of the Z(ω) of the hyperquenched glass over that of the standard glass, so one would say that the Boson peak has been greatly enhanced in the hyperquenched glass. In liquids the Boson peak is obscured by increased quasielastic scattering so investigation of glasses in which the high temperature structure is retained but the quasielastic scattering is suppressed as in hyperquenched glasses should be a good way to investigate the Boson peak. The low frequency vibrational spectra of glasses with different quench histories have been studied before. These studies, by Suck [12, 13], have been made on metallic glass-formers in which the fragilities are now known [40] to be significantly less than those of the present system. The findings of Suck and co-workers were qualitatively similar for the different glasses. Each of the hyperquenched states had an excess G(ω) at low frequencies, and diminished G(ω) around the Debye frequency, relative to the spectrum for the same glass after annealing. These workers utilized quenching rates of the same magnitude as ours (obtained by melt spinning onto cold rotating metal drums). However the DOS effects were quite small relative to those seen in figure 3, and more in line with those reported by Vollmayr and Kob [41] for simulated binary LJ glasses (the potentials for which were originally modelled on the Ni P metallic glass-former). The possibility of a distinct maximum emerging in the restricted Q density of states, seen in figure 3(b), has not arisen in previous studies. This enhanced feature is worthy of further study for the additional information onthenature of the Boson peak it may provide. It will be interesting to see how general such findings might be in silicate glasses. Since successful computersimulation studiesof silicate glass systems havebeenmade by a number of workers [42 45], it is reasonable to hope that studies of inherent structure densities of states, in systems of different compositions, might help clarify what sort of modes are involved. The suggestion of the present work is that there is a distinct configurational excitation that involves structures in which low frequency modes, presumably with transverse character, are generated. The simplest description of this source of Boson peak oscillation would be that of resonance modes that accompany defects with the character of interstitials in crystals [46]. This has been the suggestion of Granato [47] in advancing his interstitialcy theory of liquids. While Granato s description seems too simple to satisfy the requirements of liquid theory, some topologically diverse form ofinterstitial excitation, the spectral signature of which would be Gaussian in form as seen below, may indeed eventually prove adequate to describe the observed phenomena. In this respect the observations made on a model of the fragile glass-former orthoterphenyl (OTP) [17, 18] seem relevant. Before introducing these, however, we must correlate the observations of figure 4 with those of experimental OTP in order to suggest a certain generality for the simple picture to be presented. We earlier noted the similarity of the closing of the excess Z(ω) domain in Ca(NO 3 ) 2 8H 2 Odataoffigure 4(a) with the crossing point at 32 cm 1 observed by Wuttke et al [15] for the DOS of glassy and liquid OTP. Wuttke et al [15] paid little attention to their observation, on the basis that the DOS can depend on

164 S1064 C A Angell et al temperature only if there are deviations from harmonicity. However we see their crossing point reproduced below in the DOS of inherent structures of the model OTP, where certainly more than anharmonicity is involved. In [17], Mossa et al show the DOS for inherent structures of OTP explored by the system at three different temperatures, at a series of fixed densities. These were obtained from NM analysis of the inherent structures. They noted the presence of an isosbestic point at cm 1.Asfound by Kob et al [48] and Sastry [49] for the mixed Lennard-Jones model, and bymossa et al for OTP [17] high temperatures favour higher frequencies for systems held at constant volume. However, when the volume is adjusted to keep the pressure constant, as happens in most experiments, the reverse behaviour is found [50], as seen also in figures 6. The isosbestic point remains, unchanged in frequency, as shown in figure 6(b). In figure 7 we show that each of these total DOSs can be well represented by a sum of three Gaussian functions. An interesting observation is made. The centre Gaussian of each DOS, which has the largest area of the three components,isinvariantwith change in temperature. The differences responsible for the isosbestic point come from the compensating changes occurring in the high and low frequency components. This is the same phenomenon that has been reported recently for the O D overtone vibrations in water [51] and some time earlier [52, 53] for the overtone O H stretching mode in deuterated water. A related phenomenon seen in simulated water in the rigid molecule ST2 potential [54] was called exciting across the centroid by Rahman and Stillinger [54], and this seems a very appropriate description for the present observation. The system behavesas if molecules in somegaussian distribution of strained sites snap into some rearrangedgroup whose low frequency vibrational modes are also distributed in Gaussian form. (A simple two-gaussian strained site model, with predictive qualities, based on these observations will be described elsewhere [55].) The peak frequency of the low energy Gaussian is 18 cm 1,independent of temperature, while the high energy component has a peak frequency that decreases with increase in temperature. The peak frequency of the lower energy Gaussian is typical of the Boson peak frequency, and its increase in intensity with decreasing temperature is comparable to that seen in figure 3. Since the two cases have in commonthatthey are properties of systems trapped in high energy configurations and studied at low temperatures, there are good reasons to see them as the same phenomenon. In view of the interest in resolving the nature and origin of the Boson peak, it would seem that studies of hyperquenched inorganic glasses of simpler constitution than the present case should be rewarding. Likewise, DOS studies and restricted Q DOS studies of simulated glasses of different potentials, particularly including simple silicates, are obvious targets for future study. It is possible that these observations will prove consistent with, and provide spectroscopic signatures for, the Gaussian trap models for glassy systems that are currently being discussed in the literature [56, 57]. Before leaving the subject of the inherent structure densities of states seen in figure 6 it is important to take note of the manner in which they permit us to assess the vibrational contributions to the excess properties of the supercooled liquid. While details will be given elsewhere [55], we may summarize the essential findings here. The entropy-rich low frequency modes generated with increasing temperature (figure 7(b)) provide a large part of the excess entropy of the liquid over crystal (or glass) as the system temperature rises above T g.theyaretherefore responsible for much of the jump in heat capacity seen at the glass transition. In the present casethisjump in heat capacity, measured at 380 K, is found to be 61 J mol 1 K 1 (experimental value C p at 380 K = 68 J mol 1 K 1 ) and of this, afull 60% originates in the change of VDOS with temperature. This increase in vibrational entropy with increasing temperature becomes an important part of the thermodynamic drive for the system to reach the top of the landscape, as foreseen in 1976 by Goldstein [58], and

165 Potential energy, relaxation, vibrational dynamics and the boson peak, of hyperquenched glasses S1065 (a) (b) Figure 7. (a) Gaussian decomposition of the 320 K, 200 MPa DOS of figure 6, showing the temperature-independent central component by a broken curve and the match of the total DOS to the three Gaussian representations by a full curve. (b) The three DOS of figure 6(b) after subtraction of the common central component. Excitation of the structure, by increasing temperature at constant pressure, is seen to be associated with transfer of oscillators from the high temperature Gaussian component to the low temperature Gaussian component, which is called excitation across the centroid [53]. Note that the low frequency component has a peak frequency that is independent of temperature. This frequency, 18 cm 1,istypical of the Boson peak. as recently emphasized in [59]. It therefore is deeply involved in determining the fragility of the liquid state. The behaviour of OTP (fragile) makes an interesting contrast with that of mixed LJ [41] which, as argued elsewhere [60], is a relatively strong liquid. It also contrasts with that of the laboratory metallic glass PdSiP [14], which is known to be a rather strong liquid [40]. In these, the DOS changes only weakly with fictive temperature. The contrast is even stronger with the behaviour of the same system, OTP, under constant volume conditions. There [17], behaviour opposite to that seen in figure 6 is encountered: increase of temperature depresses

166 S1066 C A Angell et al thelow (boson peak) frequency component of the DOS and enhances higher frequencies, while preserving an isosbestic point at the same point as in figure 6 7.Itisonrecord that liquids that are highly fragile according to normal (constant pressure) measurements, appear much less fragile when studied at constant volume. A largediminution was documented recently for the ionic glassformer CKN [61]. Clearly, then, a proper understanding of the boson peak is the keytounderstanding important parts of the viscous liquid problem. 5. Concluding remarks How to relate the excitations inferred here (from the low frequency build-up in the DOS) to the evidence for energetic heterogeneity obtained from the anneal-and-scan studies is an unanswered problem at this time. Are the configurational changes involved in the excitations across the centroid, which are apparently involved in the boson peak, located in the interiors of thenanodomains or do they form part of their boundaries? Hopefully this and other questions arising from this work will find their answers in follow-up studies on hyperquenched glasses from other carefully chosen systems. It is clear that a much wider range of quenched-in structures can be investigated by computer simulation than by experiment, and since more simply constituted systems can be vitrified by simulation, it must be expected that it is from this quarter that the most rapid progress will be made. On the other hand, it is not clear that simulations can be conducted in a low enough temperature range for the non-exponentiality needed to show the phenomena of figure 2 to have developed. Acknowledgments We are indebted to several organizations for support of this research. CAA and L-MW acknowledge the NSF Solid State Chemistry program under grant no DMR Y-ZY acknowledges the support of Rockwool International A/S (Denmark). The measurements at NIST utilized facilities supported in part by the National Science Foundation under agreement no DMR The simulations on OTP were carried out as part of a wider study under NSF auspices, reported in [17]. We are grateful to Frank Stillinger, Francesco Sciortino, Andreas Heuer and Jeppe Dyre for very helpful discussions of the configuration space implications of our observations, and to Ranko Richert for educating one of us on spatial aspects of dynamic heterogeneity. References [1] Goldstein M 1969 J. Chem. Phys [2] Stillinger F H and Weber T A 1984 Science Stillinger F H 1995 Science [3] Sastry S, Debenedetti P G and Stillinger F H 1998 Nature [4] Tool A Q 1946 J. Am. Ceram. Soc [5] Yue Y-Z, Christiansen J dec and Jensen S L 2002 Chem. Phys. Lett [6] Yue Y-Z, Jensen S L and Christiansen J dec 2002 Appl. Phys. Lett In this respect there is a conflict with the behaviour of the soft sphere system studied by Parisi [62, 63] at constant volume. Parisi found that increase of temperature caused changes to the DOS which were superficially like those for OTP at constant volume but with a slight excess at low frequencies. Thus where the fragile liquid OTP at constant volume has a Boson peak that decreases in strength with increasing temperature, mixed soft spheres behave weakly in the opposite fashion. Indeed, the presence of large system size effects in the soft sphere system (studied at constant volume) [64], and pronounced boson dips [38] in the self part of the density autocorrelation function of the 2D mixed soft sphere system [65, 66] (which are amplified by system size effects as in [38] figure 26, for SiO 2 ), suggests that the mixed soft sphere system is not at all a fragile liquid, as unexpected as this might seem at first sight.

167 Potential energy, relaxation, vibrational dynamics and the boson peak, of hyperquenched glasses S1067 [7] Axten C W, Bauer J M, Boymel P M, Copham J D, Cunningham R N, Kamstrup O, Koenig A, Konzen J L, Ohberg I, Roe C, Sacks J, Singh T M and Wolf W Man-Made Vitreous Fibers: Nomenclature, Chemical and Physical Properties ed W Easters (Stamford, CT: Thermal Insulation Manufacturers Association (TIMA Inc.)) p 17 [8] Wang L-M, Borick S and Angell C A Phys. Rev. Bsubmitted [9] Velikov V, Borick S and Angell C A 2002 J. Phys. Chem Velikov V, Lu Q and Angell C A 2003 at press [10] Green J L, Ito K, Xu K and Angell C A 1999 J. Phys. Chem. B Bohmer R, Ngai K L, Angell C A and Plazek D J 1993 J. Chem. Phys [11] DeBolt M A, Easteal A J, Macedo P B and Moynihan C T 1976 J. Am. Ceram. Soc Moynihan C T 1976 J. Am. Ceram. Soc [12] Phillips W A, Buchenau U, Nucker N, Dianoux A-J and Petry W 1989 Phys. Rev. Lett [13] Suck J B 1989 Dynamics of Amorphous Materials (Springer Proc. Phys. vol 37) ed D Richter, A J Dianoux, WPetryand J Teixera (Heidelberg: Springer) p 182 Suck J B and Rudin H 1983 Glassy metals II (Springer Topics in Applied Physics vol 53) ed H Beck and H-J Güntherodt (New York: Springer) p 217 [14] Suck J-B 1993 J. Non-Cryst. Solids 153/ Suck J-B 2002 Adv. Solid State Phys [15] Wuttke J, Kriebel M, Bartsch E, Fujara F, Petry W and Sillescu H 1993 Z. Phys. B [16] Ambrus J H, Moynihan C T and Macedo P B 1972 J. Phys. Chem [17] Sciortino F, Kob W and Tartaglia P 1999 Phys. Rev.Lett La Nave E, Mossa S and Sciortino F 2002 Phys. Rev. Lett Mossa S, La Nave E, Stanley H E, Donati C, Sciortino F and Tartaglia P 2002 Phys. Rev. E Mossa S, La Nave E, Sciortino F and Tartaglia P 2003 Eur. Phys. J. B [18] Lewis L J and Wahnstrom G 1994 Phys. Rev. E [19] Takeda K, Yamamuro O, Tsukushi I, Matsuo T and Suga H 1999 J. Mol. Struct [20] Sastry S, Debenedetti P G and Stillinger F H 1998 Nature [21] Dyre J 1995 Phys. Rev. B [22] Videa M and Angell C A 1999 J. Phys. Chem. B [23] Franks F 1972 The properties of ice Water: A Comprehensive Treatise vol 1, ed F Franks (London: Plenum) pp [24] Davy J G and Somorjai G A 1971 J. Chem. Phys [25] Richert R 2002 J. Phys.: Condens. Matter 14 R Yang M and Richert R 2001 J. Chem. Phys [26] Reinsberg S A, Qiu X H, Wilhelm M, Spiess H W and Ediger M D 2001 J. Chem. Phys Tracht U, Wilhelm M, Heuer A, Feng H, Schmidt-Rohr K and Spiess H W 1998 Phys. Rev.Lett [27] Cicerone M T and Ediger M D 1996 J. Chem. Phys Ediger M D 2000 Annu. Rev. Phys. Chem [28] Bohmer R 1998 Curr. Opin. Solid State Mater. Sci [29] Yue Y-Z et al 2003 at press [30] Martinez L-M and Angell CA 2002 Physica A Lin TCand Angell C A 1984 Commun. Am. Ceram. Soc. 67 C33 [31] Donth E 2002 The Glass Transition (Heidelberg: Springer) p 104 [32] Yue J-Z and Angell C A Nature (under review) [33] Xia X and Wolynes P G 2000 Proc. Natl Acad. Sci. USA [34] Tool A Q 1946 J. Res. Natl Bur. Stand Narayanaswamy O S 1971 J. Am. Ceram. Soc Moynihan C T et al 1976 Ann. NY Acad. Sci [35] Kovacs A J, Aklonis J J, Hutchinson J M and Ramos A R 1979 J. Polym. Sci. Polym. Phys. Edn [36] Hodge I M 1994 J. Non-Cryst. Solids [37] Hodge I M and Berens A R 1982 Macromolecules [38] Angell C A, Ngai K L, McKenna G B, McMillan P F and Martin S W 2000 J. Appl. Phys See section D, and in particular figures 25 and 28 [39] Sokolov A P, Kisliuk A, Quitmann D, Kudlik A and Roessler E 1994 J. Non-Cryst. Solids Engberg D, Wischnewski A, Buchenau U, Borjesson L,Dianoux A J, Sokolov A P and Torell L M 1998 Phys. Rev. B [40] Schroter K, Wilde G, Willnecker R, Weiss M, Samwer K and Donth E 1998 Eur. Phys. J. B 5 1 [41] Vollmayr K, Kob W and Binder K 1996 Phys. Rev. B

168 S1068 C A Angell et al [42] Soules T F 1979 J. Chem. Phys [43] Angell C A, Cheeseman P A and Tamaddon S 1982 Science [44] Vessal B, Greaves G N, Marten P T, Chadwick A V, Mole R and Houdewalters S 1992 Nature [45] Horbach J and Kob W 2002 J. Phys.: Condens. Matter [46] Dederichs P H, Lehman C, Schober H R, Scholtz A and Zeller R 1978 J. Nucl. Mater [47] Granato A V 1992 Phys. Rev. Lett [48] Kob W, Sciortino F and Tartaglia P 2000 Europhys. Lett [49] Sastry S 2001 Nature [50] Sastry S 2002 New kinds of phase transitions: transformations in disordered substances Proc. NATOAdvanced Research Workshop (Volga River, 2002) ed V V Brazhkin, S V Buldyrev, V N Ryzhov and H E Stanley (Dordrecht: Kluwer) pp see figure 4 [51] Khoshtariya D E, Dolidze T D, Lindqvcist-Reis P, Neubrand A and van Eldik R 2002 J. Mol. Liq. 96/ [52] Luck W A P and Ditter W 1969 Z. Naturf. B [53] Angell C A and Rodgers V 1984 J. Chem. Phys [54] Rahman A and Stillinger F H 1972 J. Chem. Phys [55] Mossa S, Matyushov D, Wang L-M and Angell C A 2003 in preparation [56] Denny R A, Reichman D R and Bouchaud J-P 2003 Phys. Rev. Lett. at press Denny R A, Reichman D R and Bouchaud J-P 2002 Preprint cond-mat/ [57] Doliwa B and Heuer A 2003 Phys. Rev. Esubmitted (Doliwa B and Heuer A 2002 Preprint cond-mat/ ) [58] Goldstein M 1976 J. Chem. Phys [59] Martinez L-M and Angell C A 2001 Nature [60] Angell C A, Richards B E and Velikov V 1999 J. Phys.: Condens. Matter 11 A75 [61] Angell C A and Borick S 2002 J. Non-Cryst. Sol [62] Grigera T S, Martin-Mayor V, Parisi G and Verrocchio P 2003 Preprint cond-mat/ [63] Parisi G 2003 Preprint cond-mat/ [64] Reichmann D, private communication [65] Muranaka M and Hiwatari Y 1995 Phys. Rev. E 51 R2735 [66] Perrera D N and Harrowell P 1999 Phys. Rev. E

169 JOURNAL OF CHEMICAL PHYSICS VOLUME 119, NUMBER OCTOBER 2003 Locally preferred structure in simple atomic liquids S. Mossa and G. Tarjus Laboratoire de Physique Théorique des Liquides, Université Pierre et Marie Curie, 4 place Jussieu, Paris 75005, France Received 27 May 2003; accepted 7 July 2003 We propose a method to determine the locally preferred structure of model liquids. The latter is obtained numerically as the global minimum of the effective energy surface of clusters formed by small numbers of particles embedded in a liquidlike environment. The effective energy is the sum of the intracluster interaction potential and of an external field that describes the influence of the embedding bulk liquid at a mean-field level. Doing so we minimize the surface effects present in isolated clusters without introducing the full blown geometrical frustration present in bulk condensed phases. We find that the locally preferred structure of the Lennard-Jones liquid is an icosahedron, and that the liquidlike environment only slightly reduces the relative stability of the icosahedral cluster. The influence of the boundary conditions on the nature of the ground-state configuration of Lennard-Jones clusters is also discussed American Institute of Physics. DOI: / I. INTRODUCTION A liquid is said to be supercooled when it is possible to cool it below its melting temperature T m without crystallizing. The supercooled liquid phase is metastable with respect to the underlying crystal, and it is characterized by a dramatic increase of the viscosity and the relaxation times upon lowering the temperature, an increase that eventually leads to glass formation. These dramatic changes in the dynamic properties are not accompanied by strong signatures in the structural quantities, such as the static structure factor. Yet, it has been suggested that both supercooling and glass formation were deeply connected to the structure of the liquid, more precisely to a competition between extension of a local liquid order, different than that of the crystal, and global constraints associated with tiling of the entire space. 1 3 This competition has been termed geometric or topological frustration. 1,2 Some fifty years ago, Frank put forward the following argument to explain supercooling of liquids. 4 If one considers atomic liquids in which atoms interact via spherically symmetric potentials like the Lennard-Jones potential, the local arrangement of the atoms that is preferred is not the clusters associated with crystalline order face-centered cubic and hexagonal close-packed lattices, but a polytetrahedral packing, the icosahedron formed by 13 atoms. The energy of such an icosahedral arrangement interacting via the Lennard- Jones potential is indeed 8.4% lower than the close-packed crystalline clusters. Local icosahedral order should then be prevalent in liquids but, because of the fivefold rotational symmetry of the icosahedron, it cannot tile the entire space and form a crystal; this is a manifestation of geometrical frustration. Crystallization then requires a rearrangement of the local structure of the liquid, which leads to a strong firstorder freezing transition and allows supercooling of the liquid. Since then, there has been a large body of experimental and simulation work confirming the prevalence of local icosahedral, or more generally polytetrahedral, order in atomic liquids and metallic glasses. 2,5 16 The tendency to form icosahedral order has been shown to increase as the temperature is lowered. 2,8,10 16 Frank s argument has, however, one shortcoming: 17 the 13-atom cluster considered is isolated, so that most of the energetics is related to the surface, a situation that of course does not occur in bulk liquids. How can one remedy this problem? An a priori easy way would be to study directly the local arrangement of the atoms in a bulk liquid. In such a case, however, it is found that the proportion of icosahedra among all the 13-atom groups is very small, typically a few percent. 11 This is indeed to be expected since geometrical frustration is present, which opposes the growth of icosahedral order and distorts the local polytetrahedral arrangements. 2,18 How to disentangle then the determination of the locally preferred structure from frustration effects? The method we propose in this work is to consider the influence of the bulk liquid on a given 13-atom cluster at some mean-field level, so that surface effects can be reduced and made more realistic for describing a condensed phase, whereas geometrical frustration is strongly inhibited. The main advantage of this method is that it can be extended to study the locally preferred structure of molecular liquids, for which a priori topological arguments do not easily provide the symmetries of all possible local arrangements of the molecules, nor the nature of geometrical frustration. More specifically, in this paper we consider the ground state of a cluster of 13 atoms interacting via a Lennard-Jones potential smaller and larger clusters are also considered ; the atoms are placed in a cavity and are subject to an external field that mimics the interaction with the rest of the liquid. The structure of the outside liquid only enters the calculation via the bulk pair distribution function known from previous simulation studies. By means of an optimization algorithm 19,20 we find that the ground state of the cluster, i.e., the global minimum of the effective energy surface formed /2003/119(15)/8069/6/$ American Institute of Physics Downloaded 04 Oct 2003 to Redistribution subject to AIP license or copyright, see

170 8070 J. Chem. Phys., Vol. 119, No. 15, 15 October 2003 S. Mossa and G. Tarjus by the intracluster interactions and the external field, is of icosahedral symmetry, therefore generalizing Frank s result. For sake of comparison we consider in addition other boundary conditions for the cluster. These conditions describe different types of environments: free boundary conditions isolated cluster, periodic boundary conditions periodic tiling of space, and icosahedral-like boundary conditions hypothetical non-frustrated system. II. METHOD AND CHOICE OF THE BOUNDARY CONDITIONS We consider a system of atoms interacting via a pairwise additive spherically symmetric potential (r), where r is the center-to-center distance. A number N of atoms are placed in a spherical cavity C of radius R C, that we envisage as surrounded by bulk liquid made of the same atoms, and characterized by the temperature T and the density. As explained above, we do not want to fully account for the liquid structure because geometrical frustration would obscure the nature of the local order. We rather resort to a mean-field type of description in which the liquid outside the cavity is considered as a continuum, characterized by a known pair distribution function g(r;t, ) that is not affected by the fact that a cavity has been carved out. The potential energy acting on a given atom at position r inside the cavity due to the outside liquid is thus described as W r;r C 2 r C d 3 r g r r r r, where the integral is over all positions outside the cavity. Taking the center of the cavity as origin and transforming to spherical coordinates, one finds after standard manipulations R W r;r C C r dxx 2 g x x 1 u x;r,r C RC r, 2 2 dxx RC g x x 2 r where u x;r,r C R C 2 r 2 x rx Note that u(x R C r) u(x R C r) 1. The detail of the calculation is given in the Appendix. In view of the implementation of the optimization algorithm, 19,20 we need an explicit expression for the first derivatives of the external potential, both with respect to R C and r; we obtain W r;r R C C R R C r C r dxxg x x, 4 r RC r and R C r dxx 2 g x x r W r;r C R C RC r R C 2 r 2 x 2 2xr The second derivatives, needed for the calculation of the Hessian matrix and the study of the transition states, 19,20 can be obtained in a similar way. The total potential energy for the N atoms of the embedded cluster is the sum of the atom-atom interaction potentials inside the cavity and of the external potential, N N U r j 1,...,N ;R C i r i r j j 1 W r j ;R C, j 1 6 where W(r;R C ) is given by Eqs. 2 and 3. Finding the ground-state configuration for the N-atom cluster embedded in a liquidlike environment at a given T and amounts to determining the global minimum of U with respect to variations of the positions of the N atoms. Note that T and only enter through the external potential W(r;R C ), both explicitly, see Eq. 2, and through the state dependence of the pair distribution function g(r).] It is worth stressing that, contrary to the case of an isolated cluster for which the radius of the cavity R C is merely fixed to avoid evaporation of the atoms, R C becomes a relevant variable in a liquidlike environment: to preserve a realistic description of the liquid, R C should adjust to global contractions or expansions of the N-atom cluster, which should then be taken into account in the minimization procedure. This point will be further discussed below. To obtain the lowest energy minimum of U, we have used a slightly modified version of the basin-hopping algorithm introduced by Wales and co-workers. 19,20 The algorithm consists of a constant-temperature Monte Carlo simulation performed with an acceptance criterion based not upon the energy of the proposed new configuration, but upon the energy of the closest minimum of the potential-energy surface, obtained by a local minimization starting from that configuration. This algorithm turns out to be a very efficient method for exploring directly the minima of the potentialenergy surface, and it allows one to locate the ground state with relatively little effort. Finally we have also considered other boundary conditions for the N-atom cluster: (i) Free boundary conditions (isolated cluster). This is the standard case studied in the literature. It simply corresponds to the above situation in which the external potential is set to zero: W r;r C 0. 7 (ii) Periodic boundary conditions (periodic replication of the local cluster). Each atom of the cluster now interacts also with the images of the other atoms of the cluster, W r 1 r s 2 j, 8 j C where s j is the position of the image of atom j selected through the minimum image criterion, i.e., among all possible images of atom j, only the closest is selected. We have used a cubic elementary cell. (iii) Icosahedral-like boundary conditions (hypothetical nonfrustrated system). As mentioned in the Introduction, icosahedral order cannot be extended to the entire space. One Downloaded 04 Oct 2003 to Redistribution subject to AIP license or copyright, see

171 J. Chem. Phys., Vol. 119, No. 15, 15 October 2003 Locally preferred structure in liquids 8071 can however introduce icosahedral-like boundary conditions by embedding the 13-atom cluster in the center of a large 147-atom cluster with icosahedral (I h ) symmetry in which the central atom and its first layer have been removed. Recall that atomic clusters are characterized by magic numbers of atoms 13, 55, 147 for which the global minimum has icosahedral symmetry. 21,22 The external potential is now written as W r 1 2 r r k, 9 ico ico k S 2,S3 where S ico 2 and S ico 3 are the second and third shells of the 147-atom icosahedron. For completeness, we have also considered the 55-atom Mackay icosahedron, as well as the correction due to embedding the 55-atom or 147-atom icosahedron in bulk liquid with the resulting external potential treated at a mean-field level see above. III. GLOBAL MINIMUM OF THE LENNARD-JONES CLUSTERS We specialize our investigation to the case of the Lennard-Jones pair potential, LJ r 4 12 r r 6, 10 where and 2 1/6 are the well depth and the separation at the minimum of the potential, respectively. In what follows we set 1. In order to evaluate the liquidlike external potential acting on the N-atom cluster, Eqs. 2 and 3, and its derivatives, Eqs. 4 and 5, one needs a model for the pair distribution function g(r). We use the seven-parameter parametrization of Verlet s molecular-dynamics simulation data on the Lennard-Jones liquid 23 proposed by Matteoli and Mansoori: 24 g y 1 y m g d 1 y 1 y exp y 1 cos y 1 11 for m 1, y 1, and g y g d exp y for y 1. Here y r/d is the dimensionless intermolecular distance where d 2 1/6, and h, m,,,,, g(d) are adjustable parameters. The terms y m and exp (y 1) 2 describe the decay of the first peak, while the term exp (y 1) cos (y 1) provides the damped oscillations observed at larger distances. 24 We have taken the values h 1.065, m 13.42, g(d) 2.830, , 1.579, 6.886, that allow to reproduce the pair distribution function for the liquid at and T in usual reduced Lennard-Jones units. 23,24 The resulting external potential W(r;R C ) is shown in Fig. 1 for several values of the cavity radius R C. The shape of the r dependence changes with R C so that no rescaling of the curves is possible. For R C 1.2, the potential has a minimum at r 0 because the central atom sits at the FIG. 1. Mean-field external potential as a function of the distance r from the center of the cavity; different curves are for different values of the radius of the cavity R C. The dotted line is the value of the potential at the surface of the cavity. minimum or very close to it of the pair interactions due to liquid atoms at the boundary of the cavity; this is no longer true for larger cavity radii, and W decreases monotonically with r, the most favorable position inside the cavity being at its edge where the attractive interaction due to the nearby liquid particles is the strongest. By construction, when R C 0, W(r 0,R C ) becomes equal to the total potential energy of the Lennard-Jones liquid at the considered state point, E 5.7, whereas when R C, W (r R C,R C ) is equal to half this energy. As we discussed above, the radius of the cavity R C must be adjusted to global contractions or expansions of the cluster. A reasonable way to implement this is to take at each minimization step, i.e., for each configuration of the N atoms, R C r max, 13 where r max is the distance of the outermost atom from the center of the cavity, 25 and is a constant chosen to account for the fact that repulsive interactions between atoms make very unlikely the presence of bulk liquid atoms when their centers are too close to those of the cavity atoms; we have taken 0.5, but we have checked that the results are independent of the actual value, in runs with different values of between 0.1 and 1.0. A typical minimization run for a 13-atom cluster in the presence of a mean-field liquidlike environment is shown in Fig. 2, where we have plotted the evolution of the energy and its intracluster and external-field contributions bottom, together with the evolution of the cavity radius R C top. During the optimization run that starts from a random configuration of 13 atoms, the structure of the cluster becomes more and more compact, and its energy decreases. The final optimized configuration is found to have icosahedral symmetry. Both the final cluster radius, r max 1.08, and the final intracluster energy, U intra , are very close to those found for the ground state of the isolated 13-atom cluster. 21 The total energy of the icosahedral cluster in the presence of a liquidlike environment is, however, much lower by almost Downloaded 04 Oct 2003 to Redistribution subject to AIP license or copyright, see

172 8072 J. Chem. Phys., Vol. 119, No. 15, 15 October 2003 S. Mossa and G. Tarjus FIG. 3. Energy of a 13-atom cluster with mean-field liquidlike environment as a function of R C r max diamonds : a icosahedral (I h ) cluster, b fcc (O h ) cluster. We show separately the contributions of the intracluster interaction energy circles and that of the external field squares. FIG. 2. Evolution of the cavity radius top and of the cluster energy, together with its intracluster and external-field contributions bottom, during a minimization run starting from a random 13-atom Lennard-Jones cluster with mean-field liquidlike environment. a factor 2 because of the external field that compensates for the deficiency of nearest neighbors of the 12 surface atoms. We have repeated the procedure for different starting random configurations and we have always obtained the icosahedron with r max 1.08 as the global minimum. The same is true for a whole range of liquid density and temperature T ( , 0.6 T 3.6). In addition to carrying out a global optimization procedure, one may also make a calculation in the spirit of Frank s pioneering work. 4 We have considered two potential candidates for the ground-state configuration of the 13-atom cluster, namely the icosahedron and the cuboctahedral cluster with O h symmetry that is associated with the face-centeredcubic close-packed lattice and we have compared their energies in a liquidlike environment. The energies of the two, I h symmetric and O h symmetric, clusters are shown in Fig. 3 as a function of the cavity radius R C or, equivalently, as a function of r max, a unique distance being enough to fully determine the whole cluster once the symmetry, I h or O h,is chosen. The intracluster contribution to the energy has a minimum for r max 1.08 for the icosahedron, and r max 1.10 for the O h cluster, whereas the external-field contribution monotonously increases with R C in both cases. One then finds that the icosahedral cluster has a minimum total energy E Ih for r max 1.08 and that this energy is about 4.8% lower than the lowest energy (E Oh ) found for the O h cluster when r max When compared to Frank s result (E Ih , E Oh , i.e., a relative change of 8.4%, the relative energy difference between the two types of clusters is thus not drastically modified: a mean-field liquidlike environment only slightly reduces the relative stability of the icosahedral order. Finally, as a mere check of our global optimization procedure, we have verified that the icosahedral ground state found here is identical to that discussed above. The influence of the boundary conditions on the ground state of a 13-atom cluster can be investigated by using again the optimization algorithm always starting with a random initial configuration with the appropriate conditions, free, periodic, and icosahedral-like, described in the previous section. As already well known, the ground state of the isolated cluster is an icosahedron and, as anticipated, that of the cluster in the presence of icosahedral-like boundary conditions is also an icosahedron. The ground-state energies are shown in Table I, and can be compared to that of the icosahedral cluster in a mean-field liquidlike environment. Not surprisingly, this latter is much lower than that of the isolated cluster see above, but it is higher than that of icosahedra embedded in larger icosahedral structures. We note on passing that, in the case of an icosahedrallike environment, the change in the structure of the 13-atom and 55-atom clusters from random to icosahedral during the optimization run starts from the outside and propagates inward. This is reminiscent of what has been observed in the simulation of gold nanoclusters. 26 There, it has been found that, just after freezing, ordered nanosurfaces with fivefold rotational symmetry are formed, while interior atoms remain in a disordered state. On lowering the temperature, the crystallization of the interior atoms proceeds from the surface toward the core region, eventually producing an icosahedral TABLE I. Ground-state energy and symmetry of the 13-atom cluster with various boundary conditions. The icosahedral-like conditions correspond to the 55-atom one layer and 147-atom two layers icosahedral clusters, and the same structures embedded in a mean-field liquidlike environment. Boundary conditions Energy Point group Free I h Periodic O h Liquidlike I h Icos. 1 lay I h Icos. 2 lay.s I h Icos. 1 lay. & liquid I h Icos. 2 lay.s & liquid I h Downloaded 04 Oct 2003 to Redistribution subject to AIP license or copyright, see

173 J. Chem. Phys., Vol. 119, No. 15, 15 October 2003 Locally preferred structure in liquids 8073 ground state. For small N, the mean-field description of the liquid environment is probably too crude to give sensible results, but it is nonetheless significant that for a large range of N, the icosahedral cluster is properly selected as the locally preferred structure of the Lennard-Jones liquid. FIG. 4. Structure of the icosahedral a and cuboctahedral b 13-atom clusters. structure. 26 This is at variance with the classical picture of homogeneous nucleation and rather represents a surfaceinduced heterogeneous crystallization. Periodic boundary conditions lead to a quite different picture. Since icosahedra cannot tile space by periodic replication, such conditions should favor the symmetries that allow a complete filling of space with true long-range order. It is indeed what we have found: the global minimum is then a cuboctahedral cluster with O h symmetry, that leads to a facecentered-cubic close-packed lattice when periodically replicated see Fig. 4 ; the corresponding ground-state energy is given in Table I, and it is found lower than that of an isolated icosahedron because of the lack of neighbors already mentioned for this latter case but higher than icosahedra in either liquidlike or icosahedral-like environment. Finally, we have considered the effect of varying the number of atoms present in the cluster. The main motivation for this study is to check that without a priori knowledge of the preferred local structure in the presence of a mean-field liquidlike environment, hence of the number of atoms involved in this structure, the global minimization method will help select the proper preferred configuration. This will be important when considering molecular liquids. We have thus studied the global minima of N-atom clusters with N ranging from 2 to 23. As shown in Fig. 5, the energy per atom, the only relevant quantity for comparing local structures in a liquidlike environment, is lowest for the N 13 icosahedral IV. CONCLUSION Local icosahedral order has been found both in bulk condensed phases and in clusters formed by spherical particles. The connection between bulk and cluster studies is, however, obscured by two facts: first, geometrical frustration strongly hinders the spatial extension of local icosahedral order and distorts the local icosahedra in bulk conditions; second, the energetics of isolated clusters is partly determined by surface effects that are of course absent in the bulk. In this work we have tried to bypass those problems. We have proposed to determine the locally preferred structure of a liquid by finding the ground-state configuration of N-particle clusters embedded in a liquidlike continuum, characterized by the proper density and pair distribution function of the bulk liquid at the chosen thermodynamic state point. This meanfield-like procedure minimizes the surface effects without introducing full blown geometrical frustration. In terms of potential-energy surface, we have therefore introduced an effective energy surface that contains the usual intracluster potential-energy contribution plus an external field that accounts for the interaction with the outside liquid at a mean-field level. By a global optimization algorithm we have then located the lowest-energy minimum of the effective energy surface. For Lennard-Jones pair interactions, we have found that the locally preferred structure is indeed an icosahedron, the effect of the liquidlike environment being to only slightly reduce the relative stability of the icosahedral structure when compared to Frank s calculation for an isolated cluster. 4 We have also shown the importance of the boundary conditions used for the cluster: whereas icosahedral-like boundary conditions stabilize even more the local icosahedral cluster, periodic boundary conditions make the cuboctahedral cluster more stable, a consequence of geometrical frustration that prevents tiling of space by icosahedra, and therefore favors long-range order associated with face-centered cubic or hexagonal close-packed lattices. The present findings for the Lennard-Jones liquid suggest that the proposed method could be efficient as well for determining the locally preferred structure of molecular liquids, in cases where both translational and rotational degrees of freedom are involved, and a priori knowledge about the putative local order is scarce. ACKNOWLEDGMENTS The authors thank D. J. Wales and F. Sciortino for fruitful comments. FIG. 5. Energy per atom of the ground state of N-atom clusters with meanfield liquidlike environment. Inset: total energy as a function of N. APPENDIX: MEAN-FIELD LIQUID EXTERNAL POTENTIAL Here we give some details about the calculation of the mean-field liquid external potential W(r;R C ) discussed in Sec. II. Downloaded 04 Oct 2003 to Redistribution subject to AIP license or copyright, see