Dynamics of supercooled liquids and glasses: comparison of experiments with theoretical predictions

Transcription

1 Z. Phys. B 103, (1997) ZEITSCHRIFT FÜR PHYSIK B c Springer-Verlag 1997 Dynamics of supercooled liquids and glasses: comparison of experiments with theoretical predictions H.Z. Cummins, Gen Li, Y.H. Hwang, G.Q. Shen, W.M. Du, J. Hernandez, N.J. Tao Department of Physics, City College of the City University of New York, NY 10031, USA Received: 29 October 1996 Dedication. Five years ago, preliminary light scattering studies in our laboratory of the mixed-salt glassforming material CKN (calcium potassium nitrate) revealed spectral structure and scaling properties strikingly similar to the predictions of the mode coupling theory of the liquid glass transition developed by Wolfgang Götze and his coworkers. Initial correspondence and discussions in 1991 began a collaboration which has grown steadily and has been immensely satisfying and productive. Wolfgang Götze and his coworkers have constructed the first coherent dynamical theory of supercooled liquids and the glass transition, and he has been exceptionally helpful in working with experimentalists to make critical tests of the theory through detailed data analysis. I have learned much physics from our discussions and have come to deeply prize the friendship and warmth that has characterized our collaboration. Wolfgang and his students have given generously of their time and effort, and this generosity is deeply appreciated. Much work remains to be done. Wolfgang and his group have worked steadily to complete the theory and extend its range of validity; new experiments are needed to test their predictions in detail. I hope to continue to participate in this effort and look forward enthusiastically to our continued collaboration. Preface. Anyone entering the field of glass transition research will be immediately confronted by a large and frequently contradictory literature spanning over 100 years. Many theoretical papers have appeared, each claiming to explain the liquid-glass transition. These theories often contain unstated approximations or imprecisely defined quantities (e.g., the free volume), and their assumptions are usually incompatible with those of other competing theories. Experimental papers (with some exceptions) usually present a limited set of data which is compared with a single favored theory, bypassing the effort required to carry out detailed comparisons with different theories. There is clearly Current address: Ernst and Young LLp, 125 Chubb Ave., Lyndhurst, NJ 07071, USA Current address: Beijing Dong Dan, Tai Ji Chang, Tao Tiao #2, Beijing, People s Republic of China Current address: Dept. of Physics, Florida International University, Miami, FL 33199, USA a serious need to begin intercomparing the many competing current theories. We have recently begun to carry out a critical review of existing theories of the liquid-glass transition and the structure and dynamics of the glassy state. Our plan is to use a variety of experimental data, primarily dynamical data obtained from spectroscopic studies of various glasses and supercooled liquids, to carry out comparative fits to various theories. The goal of this program is to determine if one (or more) of the currently available theories, or some combination of these theories, can provide satisfactory descriptions of a range of different experimental data; concurrently, we will try to see if some theories can be definitely shown to be inapplicable on this basis. This is a large program which is still in its early stages. What follows, therefore, is a preliminary report that should be viewed as work in progress to be extended as the analysis proceeds. PACS: P; L I. Introduction Glass has existed on earth since geological times both as small solidified droplets (tektites) formed from molten rock heated and compressed during meteor impact, and as obsidian, an amorphous form of solidified volcanic lava. Manufacture of silica glass from sand is thought to have originated around 7,000 BC, with silica glass eventually becoming one of the most widely used man-made materials. Glassy forms of other materials including polymers, metals, and silicon have also become increasingly important with applications in many technical areas. Glassy phases have also been found in a variety of other systems, e.g. spin glass, dipolar ferroelectric glass, orientational glass, colloids, emulsions, and recently the vortex lattice in high-t C superconductors. However, while glasses form an extremely important class of materials and the glassforming ability is shared by many

2 502 different substances, the phenomena underlying the liquidglass transition and the nature of the glassy state remain incompletely understood. While SiO 2 is the best known structural glass, many other materials are also capable of forming structural glasses if cooled rapidly enough to avoid crystallization. Following a scheme suggested by Angell, structural glassforming materials can be classified by a fragility index m = [ d log<τ> d(t G /T ) ] T =T g where <τ>is the average α-relaxation time (or, equivalently, the viscosity), with strong materials containing networks of covalent bonds such as SiO 2 (m = 20) falling in the low-m region, and fragile materials composed of weakly interacting molecules such as orthoterphenyl (OTP) (m = 81) falling in the high m region [1]. During the past decade many new experimental studies of supercooled liquids and glasses have appeared, providing a wealth of new data, and a variety of theoretical approaches have been suggested to explain them. The data we will consider in this preliminary review fall into two categories. The first consists of the temperature dependence of particular properties such as the shear viscosity η S, structuralrelaxation time τ α, translational and rotational diffusion constants D T and D R, or the sound velocity C. Such data have been reported recently with precision and dynamic range far exceeding previously available results, particularly for τ α (T ) obtained from dielectric spectroscopy. The second category of experimental data consists of spectra, such as dielectric susceptibility, inelastic neutron scattering, and light scattering spectra, which have recently been extended to cover the large frequency range from below the primary α-relaxation peak up to the high-frequency region of microscopic dynamics dominated by vibrational modes. In the following discussion we will not consider either nonlinear phenomena or the ultra-slow stress relaxation effects (creep) in glasses associated with aging. We will also restrict our comparisons to fragile or intermediate glassforming materials, bypassing the strong covalently bonded network glasses which, while obviously important, are even more complicated. Among the theoretical approaches that have been proposed to explain such data, one finds simple empirical fitting functions, parametrized functions derived from physically plausible conjectures, and mathematical models based on either the theory of liquids or the theory of disordered solids (see Table 1). In some cases theories of the glass transition have found multiple applications. The vibration-relaxation coupling model, for example, which was originally developed to explain the central peak phenomenon in structural phase transitions in crystals, has recently reappeared as a theory of neutron and light scattering spectra of glasses. Energy landscape approaches, based on the topology of the potential energy hypersurface in configuration space, were primarily developed for spin glasses and were adopted as a possible model for structural glasses. The avoided critical point approach recently proposed for structural glasses has also been applied to ferromagnetic and superconducting materials. Mode coupling theory (MCT), which was originally developed for critical phenomena and subsequently introduced into the kinetic theory of ordinary dense fluids, was extended Table 1. Theories of the Liquid-Glass Transition (Partial) Free Volume Theory [Turnbull, Cohen, Grest] Entropy Theory [Adam, Gibbs, DiMarzio] Avoided Critical Point Model [Kivelson et al.] Energy Landscape Theories Coupling Model [Ngai et al.] Vibration-Relaxation Coupling Model [Gochiyaev et al.] Vibration Relaxation Model [Buchenau et al.] Trapping Diffusion Model [Odagaki] Generalized Hydrodynamics Mode Coupling Theory [Götze, Sjogren, et al.] Two-Level Systems and the Soft Potential Model Phenomenological Functions Stretched Exponential [Kohlrausch, KWW]: δ(t) =δ(0)exp[( t/τ) β ] Vogel-Fulcher: η(t)=aexp[-b/(t-t 0 )] WLF Equation Superposition of Lorentzians (etc.) to include nonlinear coupling strong enough to bring about structural arrest resembling the ergodic-nonergodic transition of the liquid-glass transition. Our goal will be to compare these and other theories and to see how well they describe experimental data. In the process, we will look for those aspects of experimental data providing signatures of the different theoretical predictions. In discussing any theory, three questions should be kept in mind: (1) how well do its predictions fit experimental data?; (2) are the physical concepts it is based on reasonable?; and (3) is it mathematically self-consistent and logical? We should not, however, be surprised to discover that reality is too complicated to be described by any one model. II. Theories of η(t ) and τ α (T ) The most striking and best known signature of the approach to the liquid-glass transition is the huge increase in the viscosity η(t ) and structural relaxation time τ α (T ) which both increase by 14 orders of magnitude between the melting temperatures T m and the glass-transition temperature T g. These two quantities are usually assumed to be proportional to each other as suggested by the Maxwell theory of viscoelasticity in which τ = η/g where G is the highfrequency shear modulus. Figure 1 shows the viscosity of OTP which increases from 0.1 poise at T m =329Kto poise at T g = 243 K [2, 3]. Figure 2 shows τ α (T ) for salol determined in a recent high-precision dielectric experiment by Stickel et al. [4]. Various approaches for fitting such data have been discussed in the literature. For an overview see Sect. 2.3 of [5]. A. Vogel-Fulcher-Tammann equation Experimental η(t ) and τ α (T ) data have frequently been fit using the empirical three-parameter Vogel-Fulcher-Tammann (VF) equation η(t )=Ae B/(T T0). (2.1)

3 503 with increasing T has also been observed for other materials [7]. Further detailed analysis of this question appears in [4]. Another widely-used empirical three-parameter equation for η(t ), closely related to the VF (2.1), is the Williams- Landel-Ferry (WLF) equation [8] log[τ α (T )/τ α (T S )] = C 1 (T T S )/(C 2 + T T S ) (2.2) where T S is a material-dependent reference temperature. Williams et al. found that (2.2) gave good fits to viscosity data for a number of different materials with C 1 =8.86 and C 2 = 101.6, with the reference temperature T S approximately 50 above T g. Note that the VF equation (2.1) and WLF equation (2.2) are formally equivalent. If T S in (2.2) is fixed, then C 2 = T S T 0 and C 1 = B/C 2 where T 0 and B are the VF parameters. Fig. 1. Viscosity η of orthoterphenyl (OTP) vs T [from W.T. Laughlin and D.R. Uhlmann, J. Phys. Chem. 76, 2317 (1972), and M. Cukierman, J.W. Lane, and D.R. Uhlmann, J. Chem. Phys. 59, 3639 (1973)]. (T m = 329 K, T g 244 K). Additional OTP viscosity data digitized from the figures in R.J. Greet and J.H. Magill [J. Phys. Chem. 71, 1746 (1967)], together with data from several unpublished sources, is shown in the inset (data provided by W. Steffen) Fig. 2. Structural relaxation time τ α(t ) of salol from dielectric relaxation data. Inset: ɛ /ɛ max vs f at six temperatures [τα =(2πfmax) 1 ]. [From F. Stickel, E.W. Fischer, and R. Richert, J. Chem. Phys. 102, 6251 (1995).] (T m = 315 K, T g 218 K). (Data provided by E.W. Fischer) The VF equation with T 0 <T g provides good fits to data over limited temperature ranges, but is generally less successful for large temperature ranges. Since (2.1) becomes an Arrhenius law in the limit T 0 0, the VF equation provides a useful estimate of the extent to which η(t ) departs from Arrhenius behavior. Dixon [6] developed a procedure in which fits to (2.1) are carried out in a temperature window whose center is successively displaced. He found that for salol dielectric τ α data, T 0 tends to increase with increasing T, in contrast to the frequent assumption that Arrhenius behavior is a correct description at high temperatures. A similar increase of T 0 B. Entropy theory Adam and Gibbs [9] analyzed the relaxation behavior in glass-forming liquids based on the vanishing of configurational entropy predicted by Gibbs and DiMarzio [10]. They suggested that, with decreasing temperature, relaxation would involve cooperatively rearranging regions of increasingly larger size. Adam and Gibbs computed the temperature-dependent τ α (T ) for the allowed size distribution of cooperatively rearranging regions and obtained a result nearly indistinguishable from the WLF Eq. (2.2). Comparison of τ α (T ) for 20 liquids with (2.2) with C 1 =8.86 and C 2 = K were presented and were reported to agree within the experimental error over a temperature range of 100. Thus, the Adam-Gibbs entropy theory of cooperatively rearranging regions provides the theoretical basis for the WLF equation. The presence of spatial inhomogeneity implied by the existence of cooperatively rearranging regions has found support in recent experiments (e.g. Ref. 11). Such inhomogeneity is also predicted by the free volume theory described in the following section. In the Adam-Gibbs approach, the configurational entropy S config (the entropy in excess of the vibrational entropy) is assumed to vanish at a temperature T 2 identified by Gibbs and DiMarzio as a second-order phase transition temperature. (T 2 is closely related to the Kauzmann temperature T K.) T 2, in turn, is identified by setting S config (T )=S glass S crystal i.e., assuming that the vibrational entropy of the crystal is the same as that of the glass. However, since the vibrational spectra of glasses are now known to be significantly different from those of the corresponding crystals, this assumption is questionable. The configurational entropy and T 2 are therefore not well-defined quantities. An interesting test of the Adam-Gibbs approach was reported by J.H. Magill [12]. The specific heat of tri-anapthylbenzene was measured in the glassy, liquid, and crystalline phases, and was used to determine the configurational entropy. The viscosity of this material was then compared with the prediction of the Adam-Gibbs model directly (rather than via the WLF equation) and was found to agree better than with the VF or WLF equations.

4 504 C. Free volume theory Free volume theories have been proposed in various forms by a number of authors. In its simplest form this theory assumes that the free volume v f available to a molecule is the volume v within the transient cage formed by its neighbors minus the minimum volume v c (or v ) required by the molecule. Statistical redistribution of the free volume occasionally creates voids large enough for diffusive displacement which leads to an equation (the Doolittle equation) for the diffusion constant: D = Ae γv /v f. If the temperature dependence of v f is taken as v f (T T 0 ), then one recovers for D 1 (which is assumed to be proportional to η) the VF equation (2.1) [13]. In the extended form of the free volume theory, Cohen and Grest [14 17] analyzed the statistical distribution of clusters of liquidlike cells (those with v>v c ) and solidlike cells. Above the percolation threshold, there is an infinite cluster of liquidlike cells and the material can be considered as a liquid. Below this threshold only finite liquidlike clusters exist. Cohen and Grest s analysis of the extended free volume model led to an equation for the viscosity B log 10 [η(t )] = A + {T T 0 +[(T T 0 ) 2 +CT] 1/2 }. (2.3) This four-parameter function was found to give fits to viscosity data for six glassforming materials that were far superior to the VF equation (2.1). However, the free volume v f at the heart of this theory, while physically plausible, is mathematically ill-defined. D. Avoided critical point model D. Kivelson and his coworkers have recently suggested that high-temperature liquids have a locally preferred structure (e.g. icosahedral for Lennard-Jones systems) into which the liquid would crystallize except for geometrical constraints [18]. Consequently, there is conjectured to be a narrowly avoided critical point at a temperature T >T m. A dynamical scaling assumption led to the prediction for either η(t ) or τ α (T ): log[η(t )] = log[η ]+E(T)/T (2.4a) where E(T )=E (T>T ) (2.4b) E(T )=E +BT [(T T )/T ] 8/3 (T<T ) (2.4c) Comparison of (2.4) with viscosity and relaxation time data for a variety of materials was reported to give good fits [19]. This model has also been used to study antiferromagnetic effects related to high-t C superconductivity [20]. E. Comparison with experiment The four theoretical approaches described above each provide an equation for η(t )orτ α (T), but their underlying physical assumptions are very different. One distinction between these theories is the predicted high-temperature behavior of η(t ). While the avoided critical point model prediction (2.4) is Arrhenius for T>T, the free volume theory prediction (2.3) become Vogel-Fulcher (2.1) for (T T 0 ) 2 CT. Another is the temperature underlying the assumed singular behavior. While the entropy and free volume theories include a singularity at T < T g, the avoided critical point model includes a singularity at T > T m. No generally accepted experimental evidence exists, however, for a singularity either above T m or below T g. From an experimental viewpoint, the conventional method of evaluating competing theories is to test their relative abilities to fit experimental data, comparing both χ 2 values and systematic errors. We have carried out a limited comparison using these four theories for the set of salol τ α data shown in Fig. 2. In Fig. 3 we show fits for these four models for the salol τ α data of Fig. 2. From these fits, we see that the Vogel Fulcher (2.1) and WLF (2.2) give comparable fits (as expected), while the two four-parameter equations derived from the free volume theory (2.3) and the avoided critical point model of (2.4) give fits of comparable quality, both clearly superior to the first two fits with three parameters. Thus, on the basis of these fits, the free volume theory and avoided critical point model appear to provide much better descriptions of the salol τ α data than the entropy theory (at least as approximated by the WLF equation). However, since the free volume theory has an implicit low-temperature singularity at T 0 <T g while the avoided critical point model includes a high-temperature singularity at T >T m, the two theories are clearly incompatible. What conclusions can be drawn from these comparisons? In 1969, Goldstein [21] wrote, Recent attempts have been made to assess the relative merits of the free volume and entropy theories of viscous flow in glass-forming liquids by accurate measurement of viscosity over wide temperature ranges, and subsequent comparison with the equations derived from these theories. In the author s view, this effort is misguided...it is better to make qualitative or semiquantitative comparison of a wide variety of physical phenomena. Following Goldstein s suggestion, we will next turn to the dynamical information provided by various spectroscopies which is capable of providing a much more detailed testing ground for theoretical models. However, while the free volume and entropy theories do not provide direct predictions for the dynamics of structural relaxation, they do provide conceptual frameworks that will be useful in the subsequent discussion. The physical picture underlying the free volume theory is that of a molecule moving through a fluid composed of other identical molecules. At high temperatures, motion occurs relatively freely with frequent binary collisions. With decreasing temperature and increasing density, each molecule becomes progressively more trapped in the transient cage formed by its neighbors, and makes many collisions within its cage before finding a way out via cage diffusion. Successive cage diffusion events provide the long-range transport motion which slows down drastically as the cage effect strengthens. This picture suggests three dynamical regimes: first, a short-time microscopic (vibrational) regime in which the molecule vibrates inside of its transient cage; second, a longtime relaxational regime controlled by diffusion; and third,

5 505 Fig. 3a d. Fits of the salol τ α(t ) data of Fig. 2 to the four theoretical functions discussed in the text. The insets show the difference plots log 10 (τ th ) log 10 (τ exp). The reduced χ 2 values are computed as χ 2 = i [y i(exp) y i (th)] 2 /(N n) where y =log 10 (τ), N is the number of data points, and n is the number of parameters. a Vogel-Fulcher-Tamann Equation (2.1). A = 12.39, B = K,T 0 = K;[χ 2 =0.103]. b WLF Equation (2.2) (entropy theory). C 1 =6.02, C 2 =58.12, T S = ; [χ 2 =0.103]. c Free Volume Theory - Eq. (2.3). A = , B = , C =5.15, T 0 = K; [χ 2 = ]. d Avoided Critical Point Model - Eq. (2.4). log 10 τ = 13.58, E = 1268, B = 175.5, T = K; [χ 2 = ] an intermediate regime during which the cage (which is composed of molecules that are themselves surrounded by cages of their own neighbors) undergoes collective distortions that may open the way for the cage diffusion event. As the density continues to increase, the frequency of opening of escape paths in a cage may become vanishingly small, leading to a complete disappearance of ordinary diffusive motion. However, as noted by Goldstein [21], a molecule can still escape from its cage by activated hopping that carries it over a local potential minimum. Thus, as glass is warmed from the low-temperature amorphous solid state, thermally activated viscous flow will begin at temperatures where the average thermal energy of a molecule is still well below the energy minima of the surrounding potential barriers. At higher temperatures, the activated viscous flow will cross over to flow mediated by ordinary cage diffusion events. We will return to this crossover scenario in Sect. IV. F. Mode coupling theory of η and τ α MCT (to be discussed in Sect. IV) is primarily concerned with high-frequency dynamics; however, it also provides predictions for the temperature dependence of η(t )orτ α (T). In the original idealized version of the theory, one finds that for T>T C [5] τ α (T ) T T C γ (2.5) where T C is the critical (or crossover) temperature. Fits of viscosity data for various materials to (2.5) revealed the predicted power law behavior at high temperatures, with T C > T g [22]. For temperatures close to T C, however, the data no longer follow the power law prediction. [Discrepancies also occur at very high temperatures where the asymptotic expression (2.5) is not expected to apply.] In the extended version of MCT, an additional relaxation mechanism is included, controlled by a temperaturedependent hopping parameter δ(t ). This mechanism provides thermally-activated viscous flow below T C, as predicted by Goldstein, avoiding the unphysical divergence predicted by (2.5) [23]. In Fig. 4 we have plotted the temperature-dependent peak frequency of the dielectric data shown in Fig. 2 as (f max ) 1/γ vs T with γ determined by the fit to (2.5) indicated by the squares. Figure 4a shows the comparison with the I-MCT prediction. Close to and below T C = 262 K, the deviations mentioned above are apparent.

6 506 Fig. 4a,b. Frequency of the maximum in ɛ (f) of salol found by Stickel et al. [4] (from Fig. 2) plotted as (f max) 1/γ vs T with γ =1.962 (points). a Fit to the idealized MCT [(2.5)] with T C = 262 K (solid line). b Prediction for f max(t ) of the extended schematic F 12 model of MCT with δ(t )=δ 0 e A/T (T <T C ) and δ(t )=δ 0 e A/T C (T >T C ) (squares). Inset: enlarged view of the region near T C We have also attempted an approximate fit to the data using the schematic extended F 12 model of MCT [24]. The parameters were first adjusted using the results of the fit with δ = 0. We then included an Arrhenius T -dependence for δ(t ) for T < T C and a constant δ for T > T C.As shown in Fig. 4b, the inclusion of δ/= 0 greatly improves the agreement, eliminating the divergence at T C. Since this is a preliminary fit included to demonstrate the qualitative difference between the predictions of the two versions of MCT, and we have not attempted to optimize the parameters systematically as with the theories described in the preceding sections, we have not included a difference plot or χ 2 value for this fit. III. Dynamics of supercooled liquids A. Qualitative description When a viscoelastic liquid is subjected to a sudden shear strain ɛ S, a shear stress σ S appears. At high temperatures stress appears only when ɛ S is changing and the Newtonian relation σ S = η S ɛ S applies, where η S is the shear viscosity and ɛ S is the strain rate. At low temperatures, the (solid) glass obeys Hooke s law and σ S = G S ɛ S where G S is the shear modulus. In supercooled liquids at intermediate temperatures, if the strain is suddenly increased from zero to ɛ S, a stress σ S = G ɛ S appears and then relaxes to zero so that σ S (t) =G ψ(t)ɛ S where ψ(t) is a relaxation function. In the simplest theory of such viscoelastic behavior, due to Maxwell, the relaxation function ψ(t) is assumed to be a single exponential (Debye) function: ψ(t) =e t/τ. With this relaxation function, Fourier transformation gives a frequency-dependent complex shear modulus G(ω) or viscosity η(ω). G, the ω limit of G(ω), and η 0, the ω 0 limit of η(ω) are connected by the Maxwell relation η 0 = τg. In practice, the relaxation function ψ(t) is usually better described by the empirical stretched-exponential Kohlrausch- Williams-Watts (KWW) function ψ(t) =e (t/τ)β K (3.1) where the stretching parameter β K lies in the range 0 < β K 1. Similar relaxation functions occur for compression or uniaxial distortions. We will be particularly interested in spatially modulated longitudinal strains (density fluctuations) ρ q (t) = e iq r ρ(r, t)d 3 r (where ρ is the density) and their normalized time correlation functions φ q (t) =<ρ q (0)ρ q (t) >/< ρ q 2 > (3.2) In many experimental studies, the relaxation function ψ(t) [or the autocorrelation function φ(t)] can be fit to the KWW function (3.1) with an initial value f<1: φ(t) =fe (t/τ)β K (3.3) where the structural relaxation time τ and stretching coefficient β K are considered as temperature-dependent fitting parameters. Stretching (β K < 1) can be viewed either as the result of a superposition of exponential relaxations with different relaxation times (parallel relaxation) or, alternatively, as the form of a single nonexponential relaxation process (serial relaxation). The constant f<1presumably reflects the presence of a fast transient process that precedes the primary structural relaxation (α process). Fits of experimental relaxation data to (3.3) can be found in many publications and represents the conventional empirical approach to supercooled liquid dynamics.

7 507 Fig. 5. Photon correlation data [c(t)] 1/2 (the field correlation function) for a dilute colloidal polystyrene suspension (upper figures) and for the glassforming liquid orthoterphenyl at T = 260 K (lower figures) with fits to (3.3). The stretched exponential fit for the colloidal suspension gives f 1, while for orthoterphenyl, f = 0.55 In Fig. 5 we illustrate these concepts with photon correlation data (a low-frequency light scattering technique) for two systems. The upper figures show the optical correlation function φ(t) for a dilute colloidal suspension with a linear time scale (left) and a logarithmic time scale (right). Since the colloidal particles undergo independent Brownian motion, φ(t) should be a simple exponential with intercept f = 1. Actually, slight stretching is observed (β K =0.95) due to some heterogeneity of the colloid, but the t 0 intercept is f = 1 within experimental error. The lower figures show φ(t) for the glass-forming liquid orthoterphenyl (OTP) at 260 K, approximately 16 K above T g. There is marked stretching (β K =0.46) and the intercept is f =0.55, far below the colloid value of f = 1. There must therefore be another short-time component to the OTP dynamics to account for the decay of φ(t) from φ(0) = 1 to 0.55, but it is not visible in the experimental window available. PCS (photon correlation spectroscopy) data for concentrated colloids, obtained by van Megen and Underwood [25] as illustrated in Fig. 6, exhibit both the short-time and long-time dynamical relaxation regimes. Similarly, φ(t) data for OTP and other materials obtained from neutron scattering experiments also exhibits additional short-time dynamics in the picosecond regime preceding the primary α relaxation (e.g. Ref. 26). Such observations have frequently been desribed as a twostep relaxation scenario. During the past decade there has been considerable interest in the fast dynamical regime preceding the final α-decay process. This aspect of supercooled liquid dynamics is addressed by several of the theories we will discuss in this review. To illustrate the qualitative aspects of the dynamics, we have plotted a schematic correlation function φ(t), its Fourier transform (which is the spectrum) S(ω), and the susceptibility spectrum χ (ω) =ωs(ω) in Fig. 7. In the upper figure (a) φ(t) is shown on a linear time scale with a fit to the KWW stretched exponential function [(3.3)] and, in the inset, on a log time scale. The inflection point in φ(t) vs log(t) corresponds approximately to the minimum in Fig. 6. Photon correlation data φ(t) for concentrated colloidal solutions near the colloidal glass transition showing both the long-time α decay and another short-time decay with strengths f and 1 f, respectively. The solid lines are MCT fits. [From W. van Megen and S.M. Underwood, Phys. Rev. E 49, 4206 (1994)] χ (ω) seen in the susceptibility spectrum (c). Susceptibility minima resembling Fig. 7c have been observed by several experimental techniques and will be discussed in later sections of this review. In Fig. 8 we show OTP susceptibility data obtained in neutron scattering experiments by Kiebel et al. [26] scaled so that their minima coincide. Recently, such susceptibility minima have also been observed in dielectric susceptibility experiments (e.g. [27]) as illustrated in Fig. 9. Another experimental approach has been employed recently to study the dynamics of supercooled liquids approaching the glass transition, based on laser light-scattering spectroscopy. Depolarized near-backscattering spectra are collected with a conventional Raman grating spectrometer and also with a tandem Fabry-Perot interferometer at several different settings, and the spectra for a given temperature are spliced together to form a single broad frequency range DLSS (depolarized light scattering spectrum). (This geometry is used because the longitudinal and transverse Brillouin components are nominally forbidden.) The I(ω) intensity spectra are then divided by the Bose factor (effectively multiplied by ω/t) to produce susceptibility spectra χ (ω). A set of χ (ω) spectra for the mixed-salt glassformer CKN [CaKNO 3 ] obtained with this procedure is shown in Fig. 10, from [28, 29]. Similar experimental light scattering data has been obtained for salol [30], PC (propylene carbonate) [31], ZnCl 2 [32], glycerol [33], OTP [34, 35], and metaluluidine [36]. In this preliminary review, we will primarily use the CKN data shown in Fig. 10 to compare with the predictions of several theories in the following sections. B. Superposition fits The relaxation function ψ(t) [or correlation function φ(t)] and corresponding susceptibility spectrum χ (ω) described in Sect. IIIA can be considered to consist of three regions. (1) At high frequencies (short times) a nearly temperatureindependent microscopic structure (sometimes called the bo-

8 508 Fig. 8. Scaled χ (ω) spectra of OTP from neutron scattering data. [From M. Kiebel et al., Phys. Rev. B 45, (1992)] Fig. 9. Dielectric χ (ω) data for glycerol exhibiting the susceptibility minimum. [From P. Lunkenheimer, A. Pimenov. M. Dressel, Yu.G. Goncharov, R. Bohmer, and A. Loidl, Phys. Rev. Lett. 77, 318 (1996)] Fig. 7. a Schematic correlation function φ(t), b intensity spectrum S(ω), and c susceptibility spectrum χ (ω) for a supercooled liquid. In a φ(t) (points) is shown with a fit (solid line) to (3.3) that gave φ(t) = 0.3 exp[ (t/ ) 0.57 ]. (Computed with the F 12 schematic model of MCT, f 0 =10 12 Hz) son peak) associated with vibrational dynamics; (2) at low frequencies (long times) a strongly temperature-dependent α-relaxation peak associated with structural relaxation and diffusive transport; (3) between (1) and (2) a minimum in χ (ω) [or plateau in φ(t)] whose amplitude and position is also temperature dependent. These three regions are visible in Figs. 6 and 10. Several authors have proposed that these spectra can be viewed as superpositions of elementary functions representing different independent scattering processes. The simplest such models consider the spectra as the additive superposition of two components: a high frequency microscopic component (vibration) and a low-frequency structural relaxation component (α relaxation). The susceptibility minimum at ω min is then identified as a crossover due to the superposition of the temperature-dependent α peak and the temperature-independent microscopic peak. (See, for example, [37].) This simple additive superposition model does qualitatively describe the susceptibility spectra found in neutron and light scattering measurements and in recent dielectric susceptibility data. However, its predicted minimum turns out to be much too deep. Figure 11 shows χ (ω) for CKN at 150 C from Fig. 10, illustrating the enhancement of the observed χ (ω) minimum relative to the simple superposition prediction [38]. In constructing this figure, we have estimated the microscopic contribution from the T =23 C spectrum, well below T g where only vibrational dynamics are relevant. The α peak of the 150 C spectrum was fit to the KWW function [Eq. (3.1)], extrapolated to high frequency, and added to the vibrational contribution which is assumed to be temperature independent. The minimum predicted by the superposition (dashed line) is too deep, indicating that the model is too simple. Lunkenheimer et al. (1996) [27] have also examined the ɛ (ω) minimum in their extended dielectric spectra of glycerol, again considered as a simple crossover from the low-frequency structural relaxation to the high-frequency microscopic response represented, respectively, by a Cole- Davidson fit for the α process (approximated by a power law well above the peak) and a high frequency spectrum ω as the beginning of the microscopic structure. Note that the spectrum S(q, ω) leading up to the microscopic (or boson) peak must have a positive slope 0 (by definition for a peak to exist) so that, since χ (ω) ωs(q, ω), the slope in χ (ω) below the microscopic peak must be 1. Fitting the α peak to a stretched exponential and representing the spectrum above the minimum by χ (ω) ω therefore produces a minimum which is as shallow as is theoretically possible.

9 509 Fig. 10. χ (ω) spectra of CKN obtained from depolarized light scattering spectroscopy (DLSS) experiments with θ = 173. [From G. Li et al., Phys. Rev. A 45, 3867 (1992) and H.Z. Cummins et al., Physica A 201, 207 (1993)] Fig. 11. Comparison of DLSS χ (ω) spectra of CKN at T = 150 C with the superposition of α relaxation and vibrational spectra. Upper curve: experimental 150 C χ (ω) data. Lower curve: microscopic/vibrational contributions given by the T =23 C spectra. The 150 α-peak at the left was extended to higher frequencies with a Kohlrausch fit (τ =0.14 ns, β K =0.53). The sum of the α peak and vibrational spectrum is indicated by the dashed line. [From H.Z. Cummins et al., Transport Theory and Statistical Physics 24, 981 (1995)] Fig. 12. Dielectric spectra ɛ (ω) of glycerol fit to a simple superposition of α-relaxation plus a microscopic process a, and with an additional temperature-dependent constant C 3 b. The tempeature dependence of C 3 is shown in the inset. [From P. Lunkenheimer, A. Pimenov. M. Dressel, Yu.G. Goncharov, R. Bohmer, and A. Loidl, Phys. Rev. Lett. 77, 318 (1996)] As shown in Fig. 12a by curves (1) and (2), this fit does not work; the predicted minimum is again too deep and the temperature dependence of the minimum is wrong. A good fit was obtained, in [27], as shown in Fig. 12b, by including an additional temperature dependent constant C 3. An alternative superposition ansatz was proposed by Zeng, Kivelson and Tarjus [39]. In their model, an additional component was added between the low-frequency stronglytemperature-dependent α process and the high-frequency temperature-independent molecular process (boson peak). This feature was assumed to have the form χ (ω) =Bω a for ω>τ 1 β and χ = Bτ (1 a) β ω for ω<τ 1 β. With this additional feature included, the predicted depth and temperature dependence of the susceptibility minimum can be made to agree quite well with experimental data, if the temperature dependence of τ 1 β is chosen correctly. The predictions of this ansatz, with the appropriate choice of τ β (T ), exactly reproduce the mode coupling results discussed in Sect. IVA. In [40], Steffen et al. analyzed depolarized light scattering spectra of OTP as the sum of two Lorentzians identified, in [41], as a broad Lorentzian with temperatureindependent width and temperature-dependent intensity arising from dipole-induced-dipole interactions and a narrow Lorentzian with strongly temperature-dependent width and temperature-independent intensity due to molecular rotations. (The α-peak Lorentzian of this superposition was subsequently modified to a KWW function in [34].) The physical basis of this superposition is questionable, however, because of the microscopic identification of the two components. Recent computer simulations for OTP by Wahnstrom and Lewis [42] show that the orientational and translational dynamics of supercooled OTP slow down similarly with decreasing temperature, while Watson and Madden [43] showed that the DID spectrum is dominated by terms which are quadratic in the density correlators. Thus, the assignment of the broad temperature-independent component to DID scattering and the narrow temperature-dependent com-

10 510 ponent to orientational dynamics appears to be inconsistent with the underlying physics. Other authors have proposed superposition fits employing various combinations of Lorentzians and KWW functions to fit I(ω) orχ (ω) spectra, in some cases with excellent results. We note, however, that the relaxation function φ(t) beyond the short-time microscopic regime is generally a complete monotone function (a monotonically decreasing function of time with appropriate restrictions on the derivatives d n lnφ(t)/dt n ) and can therefore always be fit to arbitrary accuracy by a superposition of exponentials [44]: φ(t) =Σ n C n e t/τn (3.4) Fourier transformation of (3.4) shows that I(ω) can therefore always be represented by a superposition of Lorenzians centered at zero. Thus, such superpositions can always provide accurate descriptions of the experimental data if a sufficient number of Lorentzians is used, but the significance of such superpositions is dubious. Note that (3.4) is equivalent to a parallel relaxation process in which each entity or process relaxes as a single exponential with (Debye) relaxation time τ n even though φ(t) describes a single serial process. C. Separating vibration and relaxation The theory of vibrational dynamics of crystalline solids, based on the harmonic approximation, provides a description in terms of normal modes with well-defined wavevector q and eigenfrequencies Ω q. The vibrational modes of low-temperature amorphous solids, which lack the translationally periodic structure of crystals, are not classifiable by wavevector q, but still presumably possess well defined eigenfrequencies Ω. (The vibrational modes can be either propagating or localized.) Liquids present a less straightforward problem. Zwanzig (1967) [45] proposed that liquids can have dynamical variables with the character of longitudinal and transverse phonons. He noted, however, that the lifetimes of such excitations are expected to be very short, and that a vibrational description is only meaningful for modes whose vibrational periods are short compared to their lifetimes. Recent publications have analyzed liquid dynamics in terms of instantaneous normal mode approaches based on the topology of the potential energy surface [46, 47]. This description is also restricted to short times, however, since the potential energy surface of a liquid is itself highly transient. Several of the recent theoretical approaches to the dynamics of supercooled liquids assume that the short-time (high frequency) dynamics are due to vibrations and can be separated from the relaxational dynamics that dominate at longer times, i.e. that vibration and relaxation are parallel, rather than serial, processes. Kiebel et al. [26], in their study of OTP by incoherent neutron scattering, introduced a data analysis procedure to effect this separation. They assumed that the intermediate scattering function I(Q, t) is the product of a vibrational part I vib (Q, t) and a relaxational part I rel (Q, t). A spectrum measured at low T, where relaxation should be negligible, was scaled to the temperature being studied by the appropriate ratio of Bose-Einstein and Debye-Waller factors. I(Q, t) was then divided by the scaled I vib (Q, t) to obtain the relaxational part I rel (Q, t) which was fit to theory. This same procedure was followed in a later coherent neutron scattering study of OTP by the same group [48]. However, as subsequently noted by Wuttke et al. [49], this separation procedure relies on the assumption of harmonic behavior for the vibrational modes and cannot reliably be extended to higher temperatures where anharmonic effects generally become increasingly important. Nevertheless, similar separation procedures have been employed by other authors as discussed in the following section. D. The coupling model of Ngai In an extensive series of papers beginning in 1979, K.L. Ngai and his coworkers have developed an ansatz they call the coupling model (CM), e.g. [50 57], and applied it to a variety of problems. In contrast to the parallel relaxation ansatz of (3.4), the CM assumes that, apart from vibrations, there is a single relaxation function φ(t) whose Fourier transform determines the susceptibility χ (ω) [50]. Ngai assumes that at short times t<t C, the individual structural units relax independently and exponentially with a constant relaxation rate W 0. At longer times, the effects of mutual interactions between the units take hold and slow down the relaxation rate to W (t) (t/t C ) n which decreases with time, where the coupling parameter (0 <n<1) can be considered as an indicator of the degree of correlation or cooperativity. (For polymers, Hodge [58] defined cooperativity in terms of the number of chain segments involved in the relaxation process.) This result, in turn, led to the KWW function [Eq. (3.3)] for φ(t) at long times [52]. Thus, after an initial phase of local and exponential relaxation, coupling among the relaxing units leads to a relaxation time which is itself timedependent [59]. The essential working conclusion of the CM is that the relaxation function at short times is exponential φ(t) = exp( t/τ 0 ) (t<t C ) (3.5) while after some temperature-independent crossover time t C, it becomes a stretched exponential φ(t) = exp[ (t/τ ) 1 n ] (t>t C ) (3.6) where n is the coupling parameter which lies in the range 0 n<1. [Note that 1 n of (3.6) is identical to KWW β K.] Continuity of φ(t) at the crossover time t C leads to the continuity relation τ =[t n C τ 0] 1/(1 n) (3.7a) or, equivalently, (t C /τ 0 )=(t C /τ ) β (3.7b) Equations (3.5), (3.6), and (3.7) constitute the usual approximate version of the CM. Recent analysis of a system of globally coupled nonlinear oscillators has provided evidence for such a crossover from exponential to stretched exponential relaxation [57, 53]. Palmer et al. [60] discussed models

11 511 of hierarchically constrained dynamics for glassy materials embodying the concept that smaller subunits must relax first in order that larger subunits can relax, and showed that at long times the KWW law of (3.3) follows. Their approach is closely related to the CM [59]. The coupling model, as applied to supercooled liquids, provides no predictions for the parameters τ 0, t C, and n. Therefore, tests of the model consist of comparing experimental data with (3.5), (3.6), and (3.7) treating t C, n, and τ 0 (T ), as adjustable fitting parameters. Furthermore, in order to make comparisons with experimental data, the vibrational contribution must be included either by incorporating a density of states model or by separation of vibration and relaxation using scaled low temperature spectra. (This procedure ignores the problem of temperature-dependent anharmonicity noted in the previous section with its implications for separating vibrational and relaxational contributions to the total spectrum.) Therefore, while the CM is a serial model for relaxation, the vibrational contribution is assumed to be parallel, with the two processes occurring independently. Four such comparisons have been reported. Colmenero, Arbe and Alegria [61] analyzed quasielastic incoherent neutron scattering data of the glass-forming polymer PVC. Assuming that the intermediate scattering function can be expressed as a product φ(t) =φ rel (t)φ vib (t) (3.8) (where φ rel is the relaxation process and φ vib is the vibrational process), they Fourier transformed their S(q, ω) data and divided by φ vib (t) found from the low-temperature vibrational spectra. The resulting φ rel (t) appeared to cross over from short-time exponential decay to long-time stretchedexponential decay with β K 0.23 at a temperature-independent crossover time of τ C 1.7 ps. These authors also reported similar results for two other polymers, PVME and PH [62]. Zorn et al. [63] applied the same procedure to neutron scattering data of the polymers polybutadiene and polyisoprene. For both materials a temperature- and wavevectorindependent crossover time t C 2 psec was found, separating short-time exponential relaxation for t < t C and stretched exponential relaxation for t>t C with β K =0.3 for PB and β K =0.2for PI. (In this paper, the data was also analyzed with a different model, the vibration-relaxation model of Buchenau et al..) Finally, Roland, Ngai, and Lewis [56] analyzed molecular dynamics (MD) simulation results for the molecular glassformer OTP using a Debye density of states model for φ vib (t) and (3.5) and (3.6) for φ rel (t). They compared the resulting φ(t) [(3.8)] and χ (ω) with the MD data, obtaining qualitatively reasonable agreement. Comparison of neutron scattering and molecular dynamics results with the predictions of the coupling model was further discussed by Ngai et al. in [64]. We have carried out a preliminary comparison of the coupling model approach to the CKN DLSS data of Fig. 10 using a related procedure. As shown in Fig. 11, the χ (ω) spectrum at 23 C shows no relaxational contribution and should therefore be a reasonable approximation to χ vib (ω). Rather than performing the Fourier transformation implied by (3.8), we have subtracted the 23 C χ (ω) spectrum from the 150 C χ (ω) spectrum considering the difference as χ rel (ω). (This procedure assumes that the vibrational and relaxational dynamics are independent processes rather than being statistically independent decays of the same variable, but the difference between the two procedures should be small.) Figure 13a shows the same χ (ω) spectra at 150 C and 23 C as Fig. 11, and the difference spectrum χ rel (ω) =χ 150 χ 23 [with the Brillouin component removed]. Figure 13b shows the χ rel (ω) spectrum together with a KWW fit carried out over the range GHz, and then extended to higher frequencies (dashed line). Figure 13c shows the φ KWW (t) correlation function corresponding to this fit (solid line), together with the coupling model φ CM (t) which is the same φ KWW (t) for T>t C joined onto a single exponential for t<t C =1.7 psec. The χ CM (ω) spectrum obtained by Fourier transforming χ CM (t) is also shown in Fig. 13b (solid line). The differences between χ KWW (ω) andχ CM (ω) are small, and occur at frequencies above 20 GHz where the vibrational contributions are important. (Some of the added intensity in the spectrum of Fig. 13b in excess of either prediction probably results from the failure of the separation of relaxational and vibrational contributions due to temperature-dependent anharmonicity.) Comparing Figs. 11 and 13, we see that the coupling model, like the simple superposition models, predicts a susceptibility minimum which is too deep. The challenge for any theory is not predicting that a χ (ω) minimum exists, but correctly explaining its depth and temperature dependence. While this preliminary comparison does not constitute a test of the coupling model, it does illustrate a major problem inherent in the CM approach. The effects of the proposed CM crossover appear in a frequency range where the vibrational dynamics are important, and, as noted in the preceeding section, it is not straightforward to separate vibrational dynamics from the full spectrum because of anharmonic effects. Therefore, establishing the existence of the CM crossover from experimental data requires considerable caution. Furthermore, the model provides no insights into the origin of the dramatic slowing down of structural relaxation with decreasing temperature. Such a connection might be included by combining the CM with the entropy theory to incorporate the size distribution of cooperatively rearranging regions. E. Vibration-relaxation coupling model of Gochiyaev et al. While the superposition models assume that the observed I(ω) spectra observed in light scattering or neutron scattering experiments represent the sum of scattering from vibrational and relaxational processes, some authors assume that the relaxing variable does not scatter light directly but can couple to the vibrational modes and modify their spectra. This approach was proposed by Gochiyaev et al. [65] who applied it to glycerol, dibutyl phthalate, n-butyl alcohol, and ethanol, with the structural relaxation process represented by a single exponential with an adjustable relaxation time τ 0 =1/γ. Each vibrational mode then has the spectrum S j (ω) =(k B T/~ω)Im[ωj 2 ω 2 iωγ (ω)] 1 (3.9) in which Γ (ω) is taken to be Γ (ω) =δ 2 /(γ iω). (3.10)