Energy landscapes of model glasses. II. Results for constant pressure

Transcription

1 JOURNAL OF CHEMICAL PHYSICS VOLUME 118, NUMBER 10 8 MARCH 2003 Energy landscapes of model glasses. II. Results for constant pressure Thomas F. Middleton and David J. Wales a) University Chemical Laboratories, Lensfield Road, Cambridge CB2 1EW, United Kingdom Received 12 July 2002; accepted 18 December 2002 New geometry optimization techniques are introduced for characterizing local minima, transition states, and pathways corresponding to enthalpy surfaces at constant pressure. Results are obtained for comparison with the potential energy surfaces of model glass formers studied in previous work. The constant pressure condition, where the the box lengths of the simulation cell vary, makes the enthalpy surface less rugged than the potential energy surface corresponding to the same mean density. Analysis of barrier heights as a function of pressure provides insight into transport and relaxation processes. Elementary rearrangements can be separated into diffusive and nondiffusive processes, where the former involve changes in the nearest-neighbor coordination of at least one atom, and the latter do not. With increasing pressure the barrier heights for cage-breaking rearrangements rise, while those for cage-preserving rearrangements appear relatively unchanged. The strong or fragile character of the system can therefore change with pressure because the barriers encountered vary in a systematic fashion. The geometric mean normal mode frequencies of a binary Lennard-Jones system decrease with increasing potential energy for constant pressure, rather than increase as they do at constant volume, in agreement with a simple model American Institute of Physics. DOI: / I. INTRODUCTION The complex phenomenology of supercooled liquids and glasses is determined at a fundamental level by the interatomic potential. 1 Explaining the dynamics and thermodynamics of such materials in terms of the underlying potential energy surface PES at an atomic level of detail is now an active field of research, 2 32 and many finite systems have already been characterized in this way. 33 The success of mode-coupling theories MCT 34,35 at relatively high temperatures is consistent with relaxation occurring through collective mechanisms involving small potential energy barriers, where many atoms move, but only through small distances. At lower temperature relaxation probably involves some mechanisms where a smaller number of atoms move through larger distances, and higher potential energy barriers must be overcome. The ideal MCT does not include such processes because they cannot easily be described in terms of density correlation functions. In a previous paper 28 we presented detailed analyses of the PES s of several model glasses at constant volume, paying particular attention to the distribution of barrier heights. We showed that the elementary rearrangements, which must combine to produce relaxation and diffusion, do indeed include mechanisms with relatively large barriers, which are more localized, and mechanisms with low barriers, where many atoms move through small distances, with a continuous spectrum in between these ideal limits. We classified these rearrangements as diffusive and nondiffusive, respectively, since the localized rearrangements generally lead to a change in the nearest-neighbor cage surrounding one or a Electronic mail: dw34@cam.ac.uk more atoms, while the low barrier processes do not. We might alternatively think of them as cage-breaking and cage-preserving mechanisms. The existence of many nondiffusive rearrangements, even for minima sampled by the system at low temperature, means that the configurational entropy does not vanish when the high barrier mechanisms first become frozen out on any given observation time scale. These processes also govern the distribution of free volume in the system. Furthermore, minima separated by nondiffusive rearrangements may have surprisingly different local barrier distributions. Keyes and Chowdhary 36 reached a similar conclusion using minima obtained from quenched molecular dynamics simulations. In reality, most experiments are conducted at constant pressure, rather than constant volume, and there is a growing body of data suggesting that changing the pressure can induce qualitative changes in transport and relaxation properties. To make contact with such experiments it is essential for theoreticians to investigate the appropriate enthalpy surface as a function of pressure. In the present contribution we describe new geometry optimization techniques that enable us to characterize the local minima, transition states, and rearrangement pathways of an enthalpy surface. We investigate these features for a number of popular model glass formers as a function of pressure, and compare the results with those obtained previously at constant volume. 28 The rearrangement mechanisms at constant pressure can once again be classified between two extremes, corresponding at one end to lowbarrier processes, where nearest-neighbor cages do not change, and at the other end to high-barrier cage-breaking processes. The barriers involved in the cage-breaking rearrangements generally increase with increasing pressure, /2003/118(10)/4583/11/$ American Institute of Physics

2 4584 J. Chem. Phys., Vol. 118, No. 10, 8 March 2003 T. F. Middleton and D. J. Wales whereas those corresponding to the cage-preserving mechanisms are less sensitive. The observed dynamics of a supercooled liquid depend not only upon the barrier distribution but also on how these barriers are organized on the PES. 29 For super-arrhenius fragile behavior to be observed in canonical relaxation or transport properties the effective free energy barrier must increase as the temperature decreases. Increasing the pressure is expected to produce larger activation energies, but the change would have to be different for high and low potential energy regions of the PES for Arrhenius behavior to change to super-arrhenius, or vice versa. Increasing the pressure seems most likely to amplify the difference between cagebreaking and cage-preserving barriers, and the effect on observable dynamical properties will depend upon the relative importance of such processes in different regions of the potential energy or enthalpy surface. II. PRESSURE DEPENDENCE OF STRONG AND FRAGILE BEHAVIOR In Angell s scheme glass-forming liquids are described as either strong or fragile, depending on their behavior in the regime between the melting temperature of the crystal, T m, and the glass transition temperature, T g. 37 The principal criterion for this classification is the departure of the temperature behavior of transport properties, especially the viscosity, from the Arrhenius law, exp(a/rt). For many glassformers, these properties can be better described by the Vogel Tammann Fulcher equation: expa/tt 0, 1 where A is a constant and T 0 is a nonzero temperature at which the viscosity would diverge. The fragility in this context is the degree to which the viscosity departs from Arrhenius behavior. A strength parameter, D T, is sometimes defined as D T A/T 0. D T tends to infinity for strong liquids, because T 0 0 for materials that obey the Arrhenius law exactly. An analogous equation can be used to describe the pressure dependence, with a similarly defined pressure dependent fragility parameter, D P. 41,42 Strong liquids tend to be network formers, such as SiO 2 and GeO 2, while the bonding in fragile liquids, such as ortho-terphenyl, tends to be less directional. An alternative definition of fragility is the steepness index, 43 m T, which is defined as log m T, 2 T g /TTT g where is the characteristic relaxation time of the liquid, and T g is the glass transition temperature. Measurements of D T and m T are not always consistent: Paluch et al. found that D T was pressure independent in dynamic light scattering studies of the fragile glass former EPON 828, while m T decreased the liquid became less fragile as the pressure increased. 44 The sensitivity of the relaxation time and glass transition temperature to the pressure is given by the pressure coefficients (d ln /dp) P 0 and (dt g /dp) P 0, respectively. In general, both of these increase with increasing fragility. 45 The thermodynamic properties of strong liquids tend to change smoothly at the glass transition, while those of fragile liquids tend to exhibit a large jump in the heat capacity at a well-defined point. 37 Angell has associated the large jump in fragile liquids with a higher density of minima on the energy landscape per unit energy increase. 46 With a few exceptions, the thermodynamic and kinetic fragilities correlate extremely well. This correlation could either be coincidental or provide evidence that kinetic and thermodynamic fragilities have a common cause at a molecular level. Experimental studies of transport processes in glasses at high pressure reveal that the free volume model of Cohen and Turnbull 47 fits the experimental results reasonably well. In this model, the rate of diffusion in a hard sphere mixture depends on the probability of voids opening up through the redistribution of free volume. We consider an equation analogous to the Arrhenius law, DA expv*/v f, in which A and are constants, and v* is the activation volume. v f v v 0, the difference between the average volume per molecule in the liquid, v, and the van der Waals volume of the molecule, v 0. Experimentally, the fragility of some glass-formers is affected by pressure, but the origin of this dependence is unclear. Liquids that are classed as fragile at atmospheric pressure either have fragilities that are not affected by pressure within experimental error, such as ortho-terphenyl, 48 chlorobenzene-decalin, 49 polyvinyl acetate, and polyethyl acrylate; 50 increase with pressure, like glycerol and dibutyl phthalate; 51 or decrease with pressure, like epoxy resin, 42 polymethyl acrylate, polyvinyl chloride, and polystyrene. 50 Salol exhibits a maximum in its fragility as the pressure increases, which Schug et al. attributed to increased hydrogen bonding at high densities. 48 The fragility can even be path dependent: Huang et al. found that for some polymers the steepness index, m T, at the same (T g,p g,v g ) state point depended on whether it had been reached under isobaric or isochoric conditions. 50 Some liquids, such as silicon, water, SiO 2, and GeO 2 exhibit polyamorphism, where distinct liquid phases occur with different densities and fragilities. The corresponding liquid liquid transition is often obscured by crystallization or a glass transition. 52 Silica is believed to undergo a fragileto-strong transition on cooling from very high temperatures, 53 and forms low- and high-density polyamorphs. 54,55 Water exhibits similar polymorphism as well as its better known anomalous properties. 56 Analysis of stationary points in monoatomic glasses and silica has shown that increased pressure generally lowers the barriers between minima and can eventually destabilize them. 57,58 III. MODELS The first model glass former we considered was a binary Lennard-Jones BLJ mixture of 205 A atoms and 51 B atoms. 28 Recent simulations by Sastry 25 of the same system suggest that it becomes more fragile as the number density is increased, certainly if the volume is held constant. Sastry found that the vibrational contribution to the entropy de-

3 J. Chem. Phys., Vol. 118, No. 10, 8 March 2003 Energy landscapes in model glasses. II 4585 creased as the energy landscape was ascended, in contrast to what appears to be found in experiments on glass formers at constant pressure. 25 Other molecular dynamics MD simulations of the BLJ system at different temperatures and pressures found that the viscosity and diffusion varied exponentially with pressure as: 59 expp*. There appeared to be a discontinuity in, a fit parameter, as the pressure was increased at constant temperature. The pressure at which this discontinuity occurred increased with temperature, and the discontinuity itself was attributed to a crossover from smooth, viscous flow to a landscape dominated regime. The pressure dependence of the fragility was not clear from these results. Schober has recently studied the dependence of the activation volume for diffusion on pressure in this system. 60 The activation volume was 0.3 atomic volumes in both the supercooled liquid and the glass, which was interpreted in terms of a collective mechanism. We also employed the Stillinger Weber SW silicon potential 61 in the present work, increasing the three-body term by 50%, as suggested by Barkema and Mousseau. 2,5 This modification is believed to give a more appropriate structure for a-si and amorphous germanium. 62 Both the databases in the present work were generated using this modification. IV. CHARACTERIZATION OF POTENTIAL ENERGY SURFACES A. Stationary point searches By definition, the gradient vector vanishes at a stationary point on a PES. A minimum is a point with no negative Hessian eigenvalues no imaginary normal-mode frequencies, and we follow Murrell and Laidler and define a true transition state as a stationary point with precisely one negative Hessian eigenvalue. 63 For each transition state, two barrier heights are then determined by the energy differences between the transition state and the two minima that are connected to it by steepest-descent paths. As usual, we will refer to the larger barrier for a given transition state as the uphill barrier, and the smaller as the downhill barrier. The constant volume PES of a system with N atoms and periodic boundary conditions is a 3N1 dimensional hypersurface. Under constant pressure conditions, the dimensions of the simulation supercell must be allowed to vary independently, and so there are three more degrees of freedom. Hence we now use the enthalpy surface, defined by HE PV. In our calculations the pressure, P, is an input parameter, and V is the volume of the simulation supercell. The resulting pressure-density dependence is not exactly the same as for isobaric MD studies, 59,64,65 nor did we expect it to be. Obviously, the minima and transition states are more compact than vibrationally excited instantaneous MD configurations. Nevertheless, we expect our results to reveal the underlying structural rearrangement mechanisms, and the effect of increasing pressure and density. We divided the degrees of freedom into 3N-dimensional atomic coordinate space, and three-dimensional box length space. We then successively optimized the atomic coordinates and the box lengths. We minimized the enthalpy with respect to the box lengths using an eigenvectorfollowing technique, employing numerical derivatives of the energy. The enthalpy gradients with respect to the box lengths were often very large, and we found that the searches converged more quickly if several steps were taken in box length space between each step in atomic coordinate space, to converge the gradient with respect to each box length to less than was also the convergence criterion for the gradient with respect to the box lengths at a stationary point. Analytic derivatives of the enthalpy were also programmed see Appendix A, but did not speed up the calculations significantly. All the other parameters and convergence criteria were exactly the same as in Ref. 28. The above-mentioned approach differs from that of Kopsias and Theodorou, 70 who found stationary points keeping the box lengths constant, and then optimized the latter quantities without changing the structure. It is conceivable that minima and transition states found using such a technique would not be linked by steepest-descent paths for which the box dimensions are fully optimized, as the convergence of the two above-defined spaces takes place in two distinct stages. This is why the box lengths are optimized between each step in configuration space in our calculations. The stationary points that we characterize on an enthalpy surface at constant pressure correspond to configurations where the derivatives of the enthalpy with respect to both atomic coordinates and box lengths vanish. These points are independent of the choice between scaled and unscaled atomic coordinates Appendix A. The enthalpy surface considered is the zero temperature surface that would underlie a simulation performed in an isobaric ensemble, just as the PES provides a reference for isochoric simulations. The mapping between instantaneous configurations and stationary points of a PES is unambiguous if the true steepestdescent equations are solved, because the uniqueness theorem applies to these first order differential equations. 71 In practice, however, more efficient energy minimization schemes are commonly employed, which usually lead to the same minimum as the steepest-descent path. In the present work it was convenient to minimize the enthalpy and calculate pathways under conditions in which the box lengths remained practically in equilibrium as the atomic coordinates changed. Alternative schemes can certainly be envisaged, but we do not expect our principal conclusions regarding barrier height distributions and classification of the elementary rearrangements to be affected. We simulated the Stillinger Weber system at standard pressure, which corresponds to The pressure was set to 0.6 AA AA for the binary Lennard-Jones system, because we had found from previous work that this value gave a number density close to 1.2 AA, the most popular density for simulation of this system. A further database was collected at P1.2 AA AA for comparison. The energy derivatives with respect to box lengths are

4 4586 J. Chem. Phys., Vol. 118, No. 10, 8 March 2003 T. F. Middleton and D. J. Wales TABLE I. Details of the databases studied. H 1 is the enthalpy of the initial minimum, H low and H high are the enthalpies of the lowest and highest lying minima, N min is the number of minima in the database, and N ts is the number of transition states. All the enthalpies are in per atom for supercells of 256 and 216 atoms for the BLJ and SW systems, respectively. H min and H ts are the mean energies of the minima and transition states, respectively. Database H 1 H low H high N min min H min H ts Binary Lennard-Jones, partially crystalline starting minimum, P0.6 AA AA BLJLP(x) Binary Lennard-Jones, amorphous starting minimum, P0.6 AA AA BLJLP BLJLP BLJLP Binary Lennard-Jones, amorphous starting minimum, P1.2 AA AA BLJHP Adjusted Stillinger Weber Si potential, amorphous starting minimum, P410 5 SWP SWP given in Appendix A, and the parameters employed in the present calculations are summarized in Appendix B. A more detailed account of the algorithms is available in Ref. 28. B. Sampling schemes and starting configurations We have previously characterized rearrangements in the same model glasses at constant volume. 28 There we employed three different sampling schemes to generate databases of local minima, transition states, and pathways. These were denoted SS1, which provided an overview of configuration space, SS2, which probed a smaller region of configuration space more thoroughly, and an intermediate scheme, SS3. SS1 employed eight transition state searches per minimum, SS2 400, and SS3 employed up to 40 transition state searches per minimum, accepting downhill moves to new minima. We used SS3 to generate the databases presented here because out previous results suggest that it is the most efficient. The resulting databases of minima, transition states, and pathways will be labeled as follows: 1 The first part of the label denotes the potential. Thus, binary Lennard-Jones and Stillinger Weber databases are labeled BLJ and SW, respectively. 2 For the BLJ databases there then follows a two-letter code according to whether the pressure was low LP, or high HP. The low and high pressures were 0.6 AA AA and 1.2 AA AA, respectively. We simply add a P to the SW codes to differentiate them from the databases discussed in Ref. 28. Any additional labels provide information about the starting minimum for the sampling. The BLJ database that was started from a partially crystalline minimum is denoted BLJLPx. The other databases are numbered according to the energy of the starting minimum, in ascending order. The starting minimum for database BLJLPx was the lowest energy partially crystalline minimum for the 256- atom system found in Ref. 72. For databases BLJLP1-3, we used starting configurations generated from a MD cooling run obtained in Ref. 28. Databases BLJLP1-3 correspond to configurations obtained from MD trajectories at total energies of AA, AA, and AA per atom, respectively. We started sampling database BLJHP from the same minimum as BLJLP1, but set the pressure at 1.2 AA AA. The starting configuration for database SWP1 corresponds to an intermediate energy from a database generated with sampling scheme SS3, using an amorphous minimum obtained from a quenched MD configuration as the seed. A MD run was used to produce a liquid-like starting configuration for database SWP2. The time step was 0.05 m 2 /, and the mean temperature 0.08, corresponding to 2000 K. The number density was constant at 0.48, which is the optimum density for the liquid. The system was equilibrated for time steps, and the final configuration was used to initiate the sampling process. We considered the databases of stationary points sufficiently large when transition states had been obtained. V. RESULTS General properties of the databases of stationary points obtained at constant pressure are summarized in Tables I III. The lowest minima in databases BLJLPx and BLJLP1 are only 0.02 AA and 0.03 AA lower than the starting minima, respectively, implying that the starting minima are close to the bottom of a funnel. This result is not altogether surprising, as the BLJLPx database was initiated from a minimum with significant crystalline character obtained by global optimization in Ref. 72, and the starting minimum for BLJLP1 was found to be close to the bottom of a megabasin at constant volume as well. 28 The lowest energy minimum in database BLJLPx is very similar to the starting minimum, and retained the layer of B atoms coordinated by eight A atoms. 72 The other BLJ databases generated at a pressure of 0.6 AA AA relax significantly to lower potential energy.

5 J. Chem. Phys., Vol. 118, No. 10, 8 March 2003 Energy landscapes in model glasses. II 4587 TABLE II. Potential energies of the first, lowest, and highest lying minima for the databases of Table I, E HPV, where P is the pressure, V the volume of the supercell, and H the enthalpy. The mean potential energies of the minima and transition states are also given. Database E 1 E low E high E min E ts Binary Lennard-Jones, partially crystalline starting minimum, P0.6 AA AA BLJLP(x) binary Lennard-Jones, amorphous starting minimum, P0.6 AA AA BLJLP BLJLP BLJLP Binary Lennard-Jones, amorphous starting minimum, P1.2 AA AA BLJHP Adjusted Stillinger-Weber Si potential, amorphous starting minimum, P410 5 SWP SWP However, the lowest energy minimum in database BLJHP is only 510 AA lower than the starting minimum. The lowest energy minimum in the SWP1 database has energy 2.000, and is a perfect crystal with the diamond structure. Thus, at constant pressure, the crystal was rapidly located using the present sampling scheme. The disconnectivity graph, 28,33,73 76 which is omitted for brevity, shows that the region of the PES around the crystal has very strong funnelling properties, as expected. The average densities of the minima in the BLJ databases show no surprising variations, except that the partially crystalline database BLJLPx has a 2% higher density. The mean density of the minima in SWP1 is 0.45, as expected for the crystal, while the corresponding result for SWP2 is surprisingly low, possibly signifying the onset of crystallinity in that database too. There is no significant difference in the mean number densities of transition states compared to minima within each database. A. Barrier distributions Every transition state is associated with an uphill larger and a downhill smaller barrier Sec. IV A, except for degenerate rearrangements 77 where these are the same. We present the barrier height distributions using a Gaussian for each data point: n f b 1 n i1 e (bb i )2 /2s 2 2s 2. 4 This convolution produces a smooth function when the Gaussian width, s, is chosen appropriately. An approximate representation of the probability distribution, f (b), for the barrier height, b, is thereby obtained from the n observed barrier heights, b i, in the database. Figures 1 5 show the resulting enthalpy barrier distributions, with some constant volume results from Ref. 28 for comparison. The positions of the largest maxima in the TABLE III. Pathway statistics for all the databases studied. We give the mean values of the integrated path lengths, S, in separations of connected minima, D, in; cooperativity indices, Ñ; the mean density of the transition states in the database, in ; and the volume difference between each pair of minima linked by a single transition state, v in. These quantities are defined in Sec. V C. Database S D Ñ ts v Binary Lennard-Jones, partially crystalline starting minimum, P0.6 AA AA BLJLP(x) Binary Lennard-Jones, amorphous starting minimum, P0.6 AA AA BLJLP BLJLP BLJLP Binary Lennard-Jones, amorphous starting minimum, P1.2 AA AA BLJHP Adjusted Stillinger Weber Si potential, amorphous starting minimum, P410 5 SWP SWP

6 4588 J. Chem. Phys., Vol. 118, No. 10, 8 March 2003 T. F. Middleton and D. J. Wales FIG. 1. Uphill top panel and downhill bottom panel enthalpy barrier distributions for the binary Lennard-Jones samples. The energy barrier distributions for two databases collected at constant volume in previous work Ref. 28, BLJ1 and BLJ14, are included for comparison. The Gaussian width, s0.05 AA and the barriers are in AA per supercell. FIG. 2. Uphill top panel and downhill bottom panel enthalpy barrier distributions for the two Stillinger Weber silicon databases. Included for comparison are the corresponding potential energy barrier distributions for a database collected at constant volume in previous work Ref. 28, SW1.5. The Gaussian width, s0.005 and the barriers are in per supercell. downhill and uphill enthalpy and energy barrier distributions are tabulated in Table IV. Because of the superposition of Gaussians used to generate these distributions the amplitude does not fall to zero precisely at zero barrier height, but this is simply an artifact of the representation employed. Comparison of the barrier distributions is informative. The bottom panel in Fig. 1 shows that the downhill barrier distributions of the amorphous BLJ databases, BLJLP1-3, peak at similar energies to databases obtained at constant volume in previous work, but with much larger amplitude. Thus, there appear to be many more low barrier rearrangements at constant pressure than at constant volume. It is perhaps surprising that the uphill and downhill barrier distributions for database BLJHP both peak at lower energy than for BLJLP1 Fig. 3. We will explain this result in Sec. V B. The uphill and downhill potential energy barrier distributions for a given database are shifted toward lower energy compared to the corresponding enthalpy barrier distributions, but their shapes are essentially identical. Thus, the effect of the PV term makes only a small difference to the barrier height. We conclude that allowing the box lengths to vary stabilizes transition states more than the minima to which they are connected, thus reducing the barrier heights somewhat. The same effect is visible if the downhill barrier distributions of samples SWP1 and SWP2 are compared with previous results obtained at constant volume. If barrier distributions obtained at constant volume are compared with constant pressure results for systems with the same average density then we find that the downhill enthalpy and potential energy barriers are very similar, while the uphill enthalpy barriers are smaller, except for database BLJHP. In Stillinger s megabasin picture of strong and fragile liquids, 78 this result corresponds to both the BLJ and SW systems becoming stronger at lower densities. This conclusion is consistent with results obtained using MD. 25 B. Nondiffusive and diffusive rearrangements In Ref. 28 we identified two types of rearrangement mechanism: nondiffusive, which involve breaking the nearest-neighbor cage of at least one atom; and diffusive, which do not. We developed a criterion to differentiate between these cases, defining nondiffusive rearrangements as those in which no single atom moves further than a threshold distance. We repeated this analysis for the constant pressure pathways in databases BLJLP1 and BLJHP, applying the distance criterion with a threshold value of 0.8 AA. The results are presented in Table V and Fig. 4. The enthalpy barrier distributions in BLJLP1 and BLJHP are very similar for the nondiffusive processes, but the barriers to diffusive rearrangements increase significantly with pressure. The number of

7 J. Chem. Phys., Vol. 118, No. 10, 8 March 2003 Energy landscapes in model glasses. II 4589 FIG. 3. Enthalpy barriers distributions for database BLJLP1 dashed lines and BLJHP full lines. The Gaussian width, s0.005 AA and the barriers are in AA per supercell. diffusive rearrangements in the BLJHP database is approximately twice that in BLJLP1: 1147 and 630, respectively. We have no reason to believe that the pressure introduces any bias in the sampling process towards either nondiffusive or diffusive rearrangements. Hence our results suggest that increasing the pressure increases the number of diffusive rearrangements relative to nondiffusive and increases the barrier heights, especially for the downhill barriers. The increased separation between the enthalpy barriers of nondiffusive and diffusive rearrangements at higher pressure explains the smaller peak height in the overall BLJHP barrier distribution at low enthalpy. In the databases collected at lower pressure, BLJLPx-4, the low energy tail of the diffusive rearrangements and the high energy tail of the nondiffusive rearrangements overlap, shifting the peak to slightly higher enthalpy. In database BLJHP the two classes of rearrangement are more widely separated in enthalpy, and so this overlap is not present. In MD simulations Mukherjee et al. 59 found that, as expected, increased pressure decreases the diffusion constant and increases the viscosity. The barrier distributions we have presented here are consistent with that observation. The increase in the barrier heights corresponding to diffusive rearrangements, and the constancy of those corresponding to nondiffusive processes, support our suggestion in Ref. 28 that only the diffusive processes contribute significantly to transport processes. FIG. 4. Enthalpy and potential energy barrier distributions for diffusive rearrangements in the binary Lennard-Jones system, at a pressure of 1.2 AA A A full lines, and 0.6 AA A A dotted lines. As usual, the uphill distributions are in the top panel, and the downhill distributions in the bottom panel. The potential energy barrier distributions are shifted to lower energy relative to the enthalpy distributions. The barriers have units of AA and the Gaussian width, s0.25 AA. FIG. 5. Geometric mean normal mode frequencies plotted against potential energy for all the minima in the binary Lennard-Jones databases. Included for comparison are the constant volume databases BLJ11-14 from Ref. 28. The change from an upward trend with increasing potential energy in the constant volume case to the opposite trend at constant pressure is clearly visible. For clarity, we have omitted the results from BLJHP, which show a similar trend to the other databases collected at constant pressure, but are shifted upwards slightly in frequency.

8 4590 J. Chem. Phys., Vol. 118, No. 10, 8 March 2003 T. F. Middleton and D. J. Wales TABLE IV. Positions of the largest maxima in the downhill and uphill enthalpy barrier distributions, H max (up) and H max (down), respectively. The potential energy components of these barriers, E max (up) and E max (down), are also given. All energies are in per supercell. Database H max (up) H max (down) E max (up) E max (down) Binary Lennard-Jones, partially crystalline starting minimum, P0.6 AA AA BLJLP(x) Binary Lennard-Jones, amorphous starting minimum, P0.6 AA AA BLJLP BLJLP BLJLP Binary Lennard-Jones, amorphous starting minimum, P1.2 AA AA BLJHP Adjusted Stillinger Weber Si potential, amorphous starting minimum, P410 5 SWP SWP It is difficult to draw definitive conclusions about the dependence of fragility on pressure by comparing BLJLP1 and BLJHP. Jagla 79 suggested that single particle motion, i.e., our diffusive processes, are characteristic of strong liquids, while cooperative processes are characteristic of fragile liquids. The term cooperative appears to have been used in many different ways in the glasses literature: Jagla uses it to describe processes in which the motion of a number of rearranging atoms in a cluster is cooperative and collective. This motion may correspond to a sequence of nondiffusive rearrangements, in which none of the atoms in a rearranging cluster moves a large difference in a single step. Thus, it appears that increasing pressure will make our BLJ system more fragile, as the diffusive processes will be frozen out at higher temperature. However, the downhill barrier heights increase more than the uphill barrier heights, implying that the landscape becomes less rugged, which may be more characteristic of a strong liquid. Our results can also be discussed in terms of free volume. While that theory was derived for hard spheres, 47 it is obvious that increased density at the transition state will force the rearranging atoms closer together, toward the repulsive part of the potential, increasing barrier heights and hence slowing down dynamics. C. Pathway statistics The PES can be further analyzed using the integrated path length, S, and the distance in configuration space, D, between two connected minima. The integrated path length, S, is estimated as a sum over steps, from the approximate steepest-descent paths, while D is simply the modulus of the 3N-dimensional vector separating the two minima in configuration space. 28 A measure of the localization of rearrangement i is given by the cooperativity index, Ñ i, 80,81 which varies between 1 and N. These two extremes correspond to rearrangements in which only one atom moves and those in which all atoms move an exactly equal distance, respectively. Statistics for the above-mentioned quantities are presented in Table III. There seems to be little variation of S, D, and Ñ within databases BLJLP1-3. However, the databases obtained in previous work at constant volume exhibit much larger values of all three quantities. By Jagla s hypothesis, 79 this result would also suggest that the databases BLJLP1-3 have stronger characteristics than their constant volume counterparts, as the cooperativity of the rearrangements has decreased significantly. This trend also holds for the SW silicon databases. The ratio of S to D for BLJHP is considerably higher than for BLJLP1, implying that higher pressure increases the curvature in configuration space of the transition paths. S, D, and Ñ are even lower for database BLJLPx than for BLJLP1-3. Ñ is slightly less than 2 from inspection it appears that a very large number of the rearrangements in this database are permutations of one A and one B atom. The variations in the volume difference between pairs of minima linked by a transition state are small: and the differences are significantly greater for the SW system than for BLJ. Other authors 64,65 have previously observed large pressure and volume variations in SW silicon. Given this result, and the increase in the funneling properties of the PES by allowing the box lengths to vary Sec. V, the high value of V is not surprising. D. Vibrational properties Figures 5 and 6 present the geometric mean vibrational normal mode frequency at each minimum. The geometric mean of the 3N normal modes,, is given by 3N i1 ( i ) 1/(3N). We confirmed the surprising result that the geometric mean frequency increases with increasing energy of the corresponding minimum for the BLJ system, at constant volume. 25,28 Sastry has attributed the increasing fragility of this system as the number density increases to the anomalous vibrational behavior. Here we compare the results for BLJLPx-4 with constant volume databases from Ref. 28. The constant volume trend is reversed at constant pres- TABLE V. Peaks in the barrier distributions for separated diffusive and nondiffusive rearrangements in samples BLJLP1 and BLJHP, which were obtained at pressures of 0.6 and 1.2 AA AA, respectively. Database Process type H max (up) H max (down) E max (up) E max (down) BLJLP1 Diffusive BLJHP Diffusive BLJLP1 Nondiffusive BLJHP Nondiffusive

9 J. Chem. Phys., Vol. 118, No. 10, 8 March 2003 Energy landscapes in model glasses. II 4591 FIG. 6. Geometric mean normal mode frequencies at each minimum,, plotted against potential energy for all the minima in the SW silicon constant pressure databases SWP1 and SWP2, with the constant volume database SW1.5 from Ref. 28 for comparison. sure, as expected. The crystalline database, BLJLPx, has considerably higher geometric mean frequencies, but appears to follow the same average trend as the amorphous databases, BLJLP1-3. Low frequency modes, which may contribute to the boson peak, affect the geometric mean frequency. A correlation between increasing disorder, increasing energy, and increasing amplitude of the boson peak has been reported by Angell. 82 Our results appear to be consistent with this correlation. For clarity, the results from BLJHP have been omitted from Fig. 5. They show a similar spread and energy dependence to the results from BLJLP1, with a shift upwards in frequency of 0.04 reduced units. The vibrational properties of the SW databases exhibit the expected behavior, with only a downwards shift of about 0.02 reduced units from constant volume to constant pressure. A simple model, described in Appendix B, predicts that the leading contribution to the density dependence of the frequency should scale as 7/3. For the two fixed volume Lennard-Jones systems considered in previous work this scaling suggests that the frequencies should differ by about 20%, 83 in reasonable agreement with the results in our previous report. 28 A particle-in-a-box model has also been used to estimate how the zero-point energy of a quantum solid should scale with density, and this analysis predicts a 2/3 dependence, which does not fit our data very well. VI. CONCLUSIONS Using new geometry optimization techniques we have reported the first calculations of local minima, transition states, and rearrangement pathways for constant pressure conditions. In previous experiments and simulations diffusion has been observed to slow down as the pressure increases. 41,42,59 This result is consistent with our finding that the barrier height for diffusive processes is greater in the binary Lennard-Jones BLJ database BLJHP, with P1.2 AA AA, than in database BLJLP, with P 0.6 AA AA. Taken together with the apparent insensitivity of barrier heights for nondiffusive processes to pressure, we conclude that the threshold distance criterion is successful in distinguishing the rearrangements that contribute significantly to diffusion. These results are also consistent with the BLJ studies of Mukherjee et al. and Sastry. 25,59 Our study of vibrational properties confirms previous observations, 25 and is consistent with the BLJ system becoming stronger when simulated at constant low pressure, rather than at constant density. A simple model has been derived for the density dependence of the vibrational frequencies. The results from both the binary Lennard-Jones and Stillinger Weber SW silicon systems suggest that allowing the dimensions of the simulation supercell to relax tends to reduce rearrangement barriers compared to those obtained at constant volume. This reduction is particularly marked for the uphill barriers, and so we conclude that the PES is less rugged at low pressure than at constant volume or high pressure. Following Stillinger s superbasin description of the PES s of strong and fragile glasses, 78 this result suggests that the system is stronger when simulated at constant, low pressure than at constant volume. The mean cooperativity index, Ñ, also decreases significantly when the supercell dimensions are allowed to vary. If high pressure increases the barriers to diffusion in regions of the energy landscape that are sampled at lower temperature, relative to those that are sampled at higher temperature, then the system will become more fragile. However, it is also conceivable that increasing the pressure could increase barriers to diffusion more in the high temperature region, and make the system stronger. Hence the effect of pressure could be system dependent, which may explain the complicated trends observed experimentally. ACKNOWLEDGMENTS The authors thank Mark Miller and Paul Mortenson for continuing use of their programs, FAIR PHYLLIS, and her sister, FILTHY PHYLLIS. T.F.M. is grateful to the EPSRC for financial support. APPENDIX A: ANALYTIC DERIVATIVES OF THE ENTHALPY WITH RESPECT TO THE BOX LENGTH We follow Parinello and Rahman, 84 and allow the three box lengths of the supercell to vary independently, while keeping the supercell orthorhombic. Derivatives of the enthalpy with respect to the dimensions of the supercell can be derived using scaled or unscaled atomic coordinates. A. Scaled coordinates We define the scaled position vector of atom i in terms of its unscaled coordinates, (x i,y i,z i ), and the box lengths, (,L y,l z ), as follows: x i,ȳ i,z ix i /,y i /L y,z i /L z. A1 The distance between a pair of atoms,, is given by r 2 ij x 2 ij y 2 ij z 2 ij, A2

10 4592 J. Chem. Phys., Vol. 118, No. 10, 8 March 2003 T. F. Middleton and D. J. Wales where x ij x i x j NINT((x i x j )/ ). The function NINT(u)n, where n is an integer and n0.5un H L 0.5: i.e. NINT yields the nearest integer to u. The term x H E x L sc x ij unsc i j x ix j containing the NINT function is present to ensure that we follow the minimum image convention. If x ij is defined 1 E x 2 ij i ji x ix j. A10 similarly then r 2 ij x ij 2 L 2 x ȳ 2 ij L 2 y z ij 2 L 2 z. A3 In order to obtain the derivatives of the enthalpy with respect to the box lengths, we have to treat NINT(x i x j )/ as a constant. Thus / x ij 2 /. The enthalpy, H, is given by HEPV, where E is the potential energy, P the pressure, and V L y L z, the volume of the simulation supercell. Using the relation that E/ (E/ )( / ), we obtain H E PL L y L z L x x i j 2 E E 2 x ij PL r y L z A4 ij and the second derivatives are 2 H 2 i j 2 1 E x ij E 2 x ij, and 2 H L y PL z i j 2 E 2 1 B. Unscaled coordinates 2 2 ȳ ij A5 E L y x ij 2. A6 The derivation is similar if we use unscaled coordinates, (x i,y i,z i ). x ij x i x j NINT((x i x j )/ ), and so x ij / NINT((x i x j )/ ), and /x ij x ij /. Hence, H i j E x ij and the second derivatives are and 2 H 2 i j 2 H 2 E 2 1 NINT x ix j L y PL z i j 2 E NINT x ix j PL y L z, A7 E x ij E 1 2, A8 2 1 NINT x ix j E x ijy ij 2 NINT y iy j L y. A9 The derivatives obtained for scaled and unscaled coordinates are not, in general, the same. The difference between the first derivatives is given by We then use the relations (x ij /x i )1 and /x ij x ij / to obtain E E x ij x i. A11 r L ij ji Separating Eq. A10 into two sums, and substituting in Eq. A11, we obtain H H L sc x unsc 1 x i 2 i E x i L 1 x j 2 j E x j L A12 x E, A13 where x is the vector (x 1,0,0,x 2,0,...). Thus the overall difference between the box length gradients calculated using scaled and unscaled coordinates is given by the vector x E, y E, z E L y L z. A14 At a stationary point with respect to the atomic configuration, E0 by definition, and so the gradient of the energy with respect to the box dimensions is exactly equal for derivatives calculated with both scaled and unscaled coordinates. We used a test set of ten different starting configurations to search for transition states and subsequently generate paths, for numerical derivatives and analytic derivatives using both scaled and unscaled coordinates. The same stationary points were found in each case. APPENDIX B: SCALING OF VIBRATIONAL FREQUENCIES WITH DENSITY In view of the likely importance of the variation in the vibrational frequencies of the system with density we have developed a simple theory to predict this property. Consider a single atom that interacts with the walls of a spherical container, radius R, according to a potential V. For an infinitesimal element of the surface distance d from the atom the contribution to the potential energy is VdR 2 sin d d 4R 2, B1 where and are spherical polar coordinates. The distance from the surface element, d, and the displacement of the atom from the origin, r, are related by d 2 r 2 R 2 2rR cos. Hence the integral over the sphere can be performed analytically when V corresponds to a Lennard-Jones or a Morse potential. The Lennard-Jones result is

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