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JOURNAL OF CHEMICAL PHYSICS

VOLUME 112, NUMBER 23

15 JUNE 2000

A heterogeneous picture of ␣ relaxation for fragile supercooled liquids Pascal Viot and Gilles Tarjus Laboratoire de Physique Theorique des Liquides, Universite Pierre et Marie Curie, 4 Place Jussieu, Paris 75005, France

Daniel Kivelson Department of Chemistry and Biochemistry, University of California, Los Angeles, California 90095

共Received 4 January 2000; accepted 22 March 2000兲 We examine some of the consequences, and their connection to experiments on supercooled liquids, of a scaling model of heterogeneous relaxation that is based on the theory of frustration-limited domains. In particular, we focus on what appears to be the two slowest components of structural relaxation, the one usually described by a stretched exponential or a Cole–Davidson function and the somewhat faster, apparently power-law decay known as von-Schweidler relaxation. Based on our model we study the ␣-relaxation activation free energy, the imaginary part of the dielectric frequency-dependent susceptibility, the susceptibility-mastercurve of Dixon et al. 关Phys. Rev. Lett. 65, 1108 共1990兲兴, and the breakdown of the Stokes–Einstein relation for translational diffusion at low temperatures. We also obtain estimates for the characteristic domain sizes as a function of temperature. As with all mesoscopic approaches, a number of assumptions must be introduced, but they all fit the overall scaling picture that motivates this approach. The good agreement with experimental dielectric relaxation data on two representative supercooled liquids, salol and glycerol, though necessarily dependent upon adjustable parameters, gives support to the theory. © 2000 American Institute of Physics. 关S0021-9606共00兲50823-X兴

I. INTRODUCTION

Despite the lack of direct structural evidence 共such as that obtained by means of scattering experiments兲 for the existence of significant supermolecular domains in supercooled liquids,1 there are numerous reasons to suspect their existence. First of all, the dramatic slowing2 of structural or ␣-relaxation times with decreasing temperature suggests the presence of an increasing correlation length. In particular, the super-Arrhenius activated behavior in ‘‘fragile’’ glassforming liquids of the ␣-relaxation time, ␶ ␣ , i.e., increasing activation free energy with decreasing temperature, can most readily be understood in terms of domainlike structures whose growing size determines the change in ␣-relaxation time with decreasing temperature. 共Fragility is a measure of the super-Arrhenius behavior, the more fragile the liquid, the stronger the departure from Arrhenius behavior.2兲 Furthermore, it is found that the slow relaxation processes often become increasingly nonexponential with time as temperature is lowered;2 this nonexponentiality is simply accounted for by envisaging a polydisperse system of domains, each relaxing exponentially with its characteristic relaxation time determined by its size.3 The fact that the translational diffusion constant has a much weaker T-dependence than the rotational relaxation frequency 共‘‘breakdown of the Stokes– Einstein relation’’兲4,5 when approaching the glass transition can also be most readily understood in terms of different weighted averages over the polydisperse relaxing domains.6,7 Recently there have been a number of experiments that support more directly this picture of dynamic domains. These experiments include dielectric hole-burning,8 fourdimensional NMR studies,9,10 and photobleaching probe ro0021-9606/2000/112(23)/10368/11/$17.00

tation measurements;11 computer simulations also suggest domain-formation, but here, too, the evidence is dynamical and not structural, i.e., indicative of correlations in velocity and not structure.12–14 The size of the domains or heterogeneities can be estimated from optical studies of the rotational relaxation of different-sized probes,15 NMR measurements,16 the study of excess light scattering,17 and the influence of a well-defined three-dimensional confinement,18 all of which lead to a typical length of several nanometers 共i.e., of the order of five or so molecular diameters兲 near T g . Fluorescence-bleaching11 and aging19 experiments at temperatures around T g suggest that there may be an additional, even slower relaxation process, one possibly associated with domain lifetimes or interdomain exchange times; it has been proposed that at T’s 10–20 K above T g this time is of the order of ␶ ␣ , but that near T g it can be two orders of magnitude longer, which suggests that at least at low T’s, the ␣ relaxation occurs within long-lived intact domains.11 It should be mentioned that a peak at tens of cm⫺1 in the Raman and inelastic neutron scattering spectra, a feature denoted ‘‘the boson peak,’’ has been interpreted as a structural signature of domains.20 The experimental evidence for domains in supercooled liquids is persuasive enough, we believe, to justify modeling of the system in terms of domains. Despite the fact that no local structural order-parameter has yet been found1 共a local order-parameter being the structural quantity that allows one to distinguish one domain from another兲, it seems reasonable to postulate the existence of structurally distinguishable domains. Therefore, we examine a phenomenological extension of a theory, the theory of frustration-limited domains 共FLD兲,21 that envisages, below some crossover temperature

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A heterogeneous picture of ␣ relaxation

J. Chem. Phys., Vol. 112, No. 23, 15 June 2000

T * , the formation of structural domains that are polydisperse in size, whose average size increases as T is lowered toward T g , and whose individual relaxation times are associated with relaxation of the local order-parameter which is dependent upon domain size. In this article we consider the extension of the FLD theory based on static and dynamic scaling analysis in which there is an equilibrium distribution, ␳ (L,T), of domain sizes below a crossover temperature T * , and in which the local order parameter of each domain of size L relaxes with a typical time ␶ (L,T). Dynamic scaling analysis allows us to re-express ␶ (L,T) as ␶ ⬁ exp关E(L,T)/T兴, where ␶ ⬁ is a species-specific, T-independent parameter; this activation expression is appropriate for describing the crossover at T * from Arrhenius to super-Arrhenius T-dependence observed for the ␣ relaxation of ‘‘fragile’’ supercooled glassformers.2,22 For simplicity we assume that the relaxation of the local order-parameter in each domain is exponential, i.e., of the form exp关⫺t/␶(L,T)兴,23 and that the domains relax independently. 关It is certainly easier to satisfy this last requirement if the domain lifetime is longer than, but not necessarily much longer than, ␶ (L,T).兴11,24 The picture outlined above leads to the following expression for the normalized ␣-relaxation function associated with a local variable that completes its relaxation within a single domain, f ␣共 t 兲 ⫽

冕

⬁

0

冋

dL ␳ 共 L,T 兲 exp ⫺

冋

册册

t E 共 L,T 兲 exp ⫺ , ␶⬁ T

共1兲

where f ␣ (0)⫽1, and to the expression

␶ ␣⫽

冕

⬁

0

共2兲

dt f ␣ 共 t 兲

for the ␣-relaxation time. Actually, we focus on the imaginary part of the dielectric susceptibility,

␹ ⬙ 共 ␻ 兲 ⫽⫺⌬ ⑀ 共 T 兲 Im

再冕

⬁

0

dt exp共 i ␻ t 兲

冎

d f ␣共 t 兲 , dt

共3兲

where ⌬ ⑀ (T) is the difference between the static dielectric constant and the high-frequency 共optical兲 dielectric constant. The dielectric relaxation experiments are carried out at very small wave-vector, and so the dependence on the wavevector has been neglected, a procedure not possible in the analysis of neutron scattering data. Equation 共1兲 has been widely applied, but what is supplied in this work are theoretically generated expressions for ␳ (L,T) and E(L,T). It should be noted that the basic FLD theory does not require that the domains relax either independently or exponentially, but the basic equations have yet to be totally solved so that the phenomenological extensions are currently needed for comparison with experiment. The phenomenologically extended FLD theory, as developed below, yields such expressions for ␳ (L,T) and E(L,T), i.e., for the relaxation of the local order parameter. The relaxation properties of the quantities actually measured are obtained via dynamic coupling to the local order parameter; as a result, this could lead to somewhat different relaxation properties for different kinds of experiments, e.g., NMR, dielectric

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relaxation, dynamic light-scattering, neutron scattering.25 In the interest of simplicity we assume that these differences, though possibly discernible, are not great when the measured quantity is local enough to complete its relaxation within a single domain since it is the underlying domain relaxation that is thought to control the slow ␣ relaxations in supercooled fragile glass-formers. If the relaxation is not completed within a single domain, as for reorientational motions of large solute molecules,15 for long-wavelength concentration fluctuations,4,5 or for diffusional motion over large distances, then different averaging over domain characteristics must be considered, averaging which leads to a motionally narrowed, faster relaxation process.6 The FLD theory used to describe the system should presumably be valid at temperatures sufficiently low that the system is composed of reasonably supermolecular domains, i.e., at a temperature somewhat below the crossover T * above which the domains are reduced to molecular size. We thus must consider with some care the low-temperature range over which our results might be valid. A related consideration is that of the time or frequency scale over which the results might be expected to hold. The theory is one relevant to the large, well-specified supermolecular domains in what is taken to be a system of polydisperse random domains; those domains with sizes placing them among the most probable are those that are thought to contribute dominantly to the ␣ relaxation and to be sufficiently large to be well described by the theory. This leads us to focus on the slow ␣ relaxations and to leave discussion of the faster ␤ relaxations for another day. By ␣ relaxation we mean the relaxations covered by the susceptibility master-curve of Dixon et al.;26 the fact that such a master-curve can be formed, even if not perfectly, suggests that the data included might possibly be described in terms of a single 共␣-兲 relaxation mechanism. The ␣ relaxation, as specified here, thus includes the slow, so-called stretched relaxation region where the relaxation function is often approximated by a KWW stretched exponential, exp关⫺(t/␶)␤兴, or by a Cole– Davidson function in frequency space, (1⫺i ␻ ␶ ⬘ ) ⫺ ␤ ⬘ , as well as the faster von-Schweidler or slow ␤-relaxation which is often represented by a power-law.27 Chamberlin3 was the first to show that with reasonable choices of ␳ (L,T) and E(L,T), one could make use of Eq. 共1兲 to describe this ␣-relaxation regime. Our aim here is to investigate the ␹ ⬙ ( ␻ ) line shape in terms of the phenomenologically extended FLD theory. The actual fitting formulas used for ␳ (L,T) and E(L,T) are consequences of the theory and differ markedly from those of Chamberlin. II. FRUSTRATION-LIMITED DOMAIN THEORY A. Basic FLD theory

In this theory a coarse-grained local order parameter representing the locally preferred structure of the liquid is mapped onto a spin variable; the spins are characteristic of those in a multiorientation clocklike model interacting via a short-range ferromagnetic ordering interaction 共strength J兲, but frustrated by a weak, long-range 共Coulombic兲 antiferro-

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magnetic interaction 共strength QⰆJ兲.21 The latter describes the strain-induced interaction that opposes the spatial extension of the locally preferred structure. The associated dynamics can be studied by introducing arbitrary short-time processes 共such as Glauber dynamics兲 and examining the longtime behavior. This problem has not yet been solved, but insights have been gleaned.21,28–31 The model in the absence of frustration gives rise to a critical point at a temperature T * of the order of J with a correlation length ␰ 0 ⬀a 0 (1⫺T/T * ) ⫺ ␯ which is characteristic of standard critical theories with a 0 being a molecular length. It appears that the long-range frustration causes the critical point to be avoided and isolated in the sense that in the limit of Q→0 ⫹ , no phase transition occurring at Q⫽0 goes continuously to that at T * and Q⫽0兲.32 For weak frustration, the critical point is narrowly avoided, which gives rise to a second supermolecular length, L * (T), that increases as T decreases below T * . Thus for T⬍T 1 ⬍T * , where T 1 is the temperature at which L * becomes greater than ␰ 0 , one has21 L * ⬎ ␰ 0 ⬎a 0 , and L * ⬀a 20

冉冊 Q J

⫺1/2

␰ ⫺1 0

⬀ a0

冉冊 冉 Q J

⫺1/2

T 1⫺ T*

冊

⫹␯

,

共 T⬍T 1 ⬍T * 兲 ,

共4兲

where as in all standard three-dimensional problems in the absence of quenched disorder one takes ␯ ⬇2/3. L * is the characteristic scale over which the physics is essentially that of the unfrustrated system 共which is ordered since T⬍T * 兲 and that can be identified with the typical domain size. The general picture is the following. The ordering induced by the short-range interaction J would give rise to a transition to a fully ordered 共ferromagnetic兲 phase at T * , i.e., to the reference ordered phase, were it not for the physically ubiquitous frustration; however, because of the frustration the transition is aborted, and the system breaks up into polydisperse supermolecular domains of characteristic size L * (T). At sufficiently low T the system undergoes a firstorder transition to a ‘‘defect-ordered crystal,’’ one with supermolecular unit cells which could well be related to the random domains in the supercooled liquid above this transition; whether or not this transition is observed depends in large part on whether the transition temperature to this phase T DO lies above or below the glass temperature T g . 33 It is thought that the dynamics, which are known experimentally to be activated,2,22,34 are associated with the size of the domains, and that the relaxations slow down with superArrhenius behavior as the domains grow. For this to be true, it is believed that one requires a discrete, multiorientation spin variable.35 Note the distinction between the avoided transition to the reference ordered phase at T * and the transition to the defect-ordered phase at much lower temperature T DO ; the avoided critical property discussed above means that limQ→0 ⫹ T DO(Q) is significantly less than T * . B. Phenomenological approach to the FLD theory: Static scaling analysis

The model used for this discussion is the phenomenological 共scaling兲 analysis of the theory of FLD.21,33 In the

FLD theory 共as described above兲 domains begin to form below a crossover temperature T * , and below a temperature T 1 , which is less than T * , they are large enough to be treated as polydisperse supermolecular entities. Each domain is a locally ordered equilibrated entity although the order is far from perfect because growth of a domain is associated with an increase of structural strain 共frustration兲. Of course, the domains in the supercooled liquid phase are equilibrated only in the sense that the stable crystalline phase is effectively blocked by a high nucleation barrier. The phenomenological treatment of the FLD theory for temperatures below T 1 is based on the following domain free-energy density,33 ⌽ 共 L,T 兲 ⫽

␴共 T 兲 ⫺ ␾ 共 T 兲 ⫹sL 2 , L

共 T⬍T 1 ⬍T * 兲 ,

共5兲

where L is a domain 共linear兲 size, ␴ (T)L 2 is the surface free energy required to form a domain, ␾ (T)L 3 is the free energy decrease associated with extension of the locally preferred structure found in the liquid, and s(T)L 5 is the superextensive strain term that accounts for the fact that growth of the domain with the locally preferred structure found in the liquid is accompanied by increasing strain due to the constraint that space cannot be tiled with this structure. 共The power 5 for the strain free energy is compatible with the Coulombic frustration.兲 This free energy is operative only at T’s sufficiently less than the crossover T * , i.e., where Eqs. 共4兲 and 共5兲 hold. T * being the narrowly avoided critical point indiand ␾ (T)⬀T * ␰ ⫺3 cated above, we expect ␴ (T)⬀T * ␰ ⫺2 0 0 , where the proportionality constants are dimensionless numbers of order one. The free-energy density given above is that for a single domain in a molecular liquid. If the system is entirely formed of such 共randomly oriented兲 domains, and if the interdomain interaction is weak enough so that it can be treated in a mean-field approximation, then the equilibrium state of the system is obtained by minimizing the free-energy density function, ⌽(L,T), where ⌽(L,T) is given by Eq. 共5兲 with a possible renormalization of the various proportionality constants. The minimum ⌽ min in ⌽(L,T) is attained for a domain of size L min⫽(␴/2s) 1/3. We now associate L min(T) with the characteristic domain size, L * (T), obtained above 关see Eq. 共4兲兴, with the result that s 共 T 兲 ⬀ a ⫺6 0 T*

冉冊 Q J

3/2

␰0 .

共6兲

From this we can also obtain the distribution of domain sizes, ␳ (L,T)⬀L 2 exp关⫺(⌽(L,T)⫺⌽min(T))L3 /T兴, in a scaled form,

␳ 共 L,T 兲 ⬀ x 2 exp关 ⫺ ␥ 共 T 兲共 ␬ x 2 ⫺ 23 x 3 ⫹ 12 x 5 兲兴 , where x⫽L/L * and

␥ 共 T 兲 ⫽C

冉

T* T 1⫺ T T*

冊

共7兲

8/3

共8兲

with C⬀(Q/J) ⫺1 . 关The 8/3 exponent arises because ␥ ⬀(L * / ␰ 0 ) 2 , L * ⬀ ␰ ⫺1 0 , and ␰ 0 is a correlation length with a critical exponent ␯ ⬇2/3.兴 The parameter ␬ is included to allow for small domain shape effects in a mean-field sense,

J. Chem. Phys., Vol. 112, No. 23, 15 June 2000

but we expect it to be close to one. Note that we have resorted to a mean-field treatment for the 共assumed兲 weak interdomain effects, but the basic physics associated with avoided critical behavior as well as the intradomain scaling laws are not mean-field features. C. Phenomenological approach to the FLD theory: Dynamic scaling analysis

Describing the dynamical behavior of a system is much more complicated than describing its thermodynamics because there are innumerable possible complicated, manyparticle reaction paths. Furthermore, describing the relaxation function f ␣ (t) is much more involved than merely describing the effective activation free energy because of its sensitivity to path. We do know, however, that the most efficient way for an ordered finite system of size L to change the orientation of its order parameter 共for a discrete spin variable as we envisage here兲 is to build a wall of defects through the system. The cost of building such a wall is of order ␴ L 2 , but in a frustration-limited domain some of this cost is regained by the reduction in strain 共frustration兲 achieved by construction of a wall; the free energy reduction due to reduction of strain scales as sL 5 . This leads to a Tand L-dependent domain activation free-energy, E(L,T), in the following scaled form: E 共 L,T 兲 ⫽E ⬁ ⫹bT ␥ 共 T 兲共 x 2 ⫺mx 5 兲 ,

共 T⬍T 1 ⬍T * 兲 , 共9兲

where E ⬁ describes the molecular contribution to the activation energy 共already discussed兲, b and m are dimensionless constants, ␥ (T) is given by Eq. 共8兲, and once again x ⫽L/L * . We expect both b and m to be of order one, although physically we expect m to be somewhat smaller than one. The above expression becomes meaningless for domains that are too large, i.e., for x⬍(5m/2) ⫺1/3, but this is not a problem because the distribution ␳ (L,T) cuts off contributions from such large domains. Above T * , one finds E(L,T)⫽E ⬁ . The E(L,T) in Eq. 共9兲 is the activation free energy for the ␣-relaxation of a domain, this latter being associated with the slow relaxation of the local order variable within the domain. The relaxation function, f ␣ (t), can be described in terms of Eqs. 共1兲, 共3兲, 共7兲–共9兲. With the reduction in the number of necessary independent adjustable parameters, one can also obtain the ␣-relaxation time, ␶ ␣ (T), or the viscosity, ␩ (T). From f ␣ (t) one can deduce the frequency-dependent susceptibility, ␹ ⬙ ( ␻ ), a quantity with identical information content to that of f ␣ (t). Finally, we can use this information to evaluate the translational diffusion constant, D(T), which involves a different average over domain sizes than does ␶ ␣ (T), and therefore provides an independent check of the theory. III. COMPARISON WITH EXPERIMENT A. Activation free energy

The ␣-relaxation time ␶ ␣ is defined by Eq. 共2兲, and the effective activation free energy E ␣ can be defined by the relation,

A heterogeneous picture of ␣ relaxation

E ␣ 共 T 兲 ⫽T ln

冉 冊

␶ ␣共 T 兲 . ␶⬁

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共10兲

For T⬍T 1 , one expects that the parameter ␥ (T) is large enough so that Eq. 共2兲, with the use of the appropriate forms of ␳ (L,T) and E(L,T) in Eq. 共1兲, can be approximated by a steepest descent procedure, which leads to the following 8/3power-law T-dependence,

冉

E ␣ 共 T 兲 ⫽E ⬁ ⫹BT * 1⫺

T T*

冊

8/3

,

共 T⬍T 1 ⬍T * 兲 ,

共11兲

where the dimensionless parameter B goes as (J/Q). For temperatures above T * ,E ␣ is simply given by E ⬁ , and the theory then describes the crossover from Arrhenius to superArrhenius T-dependence observed in fragile glassformers. Note that in practice we also use Eq. 共11兲 to describe the activation free energy for temperatures between T 1 and T * . This effective activation free energy for ␣ relaxation and viscosity has been extensively studied22,34,36 in the context of Eq. 共11兲, including the cases of salol and glycerol,22,36 and here we merely summarize some of the results and consequences. Activation free-energy studies such as these require four species-dependent, but temperature-independent parameters 兵 ␶ ⬁ ,E ⬁ ,T * ,B 其 . We know of few fitting schemes, let alone theoretically generated ones, that can describe all supercooled liquids over the full range of measured temperatures 共including the noncritical, nonsupercooled regime兲 with so few parameters, and we know of none that can do so with fewer parameters.22 The two parameters 兵 ␶ ⬁ ,E ⬁ 其 are ‘‘molecular’’ parameters, whereas the two parameters 兵 T * ,B 其 are ‘‘collective’’ parameters associated with domain formation. When sufficient high-T data are available, 兵 ␶ ⬁ ,E ⬁ 其 may be obtained in an independent high-T experiment, while 兵 T * ,B 其 can be obtained in a second independent low-T experiment; in this case, we have two parameters in each of two independent experiments. 共To effect these measurements, E ⬁ and ␶ ⬁ must be determined from high-T data well above T * , and the E ⬁ and ␶ ⬁ so determined can then be used for a global datafit with only B and T * as parameters.兲 We draw attention to the nontrivial exponent 共8/3兲 characterizing the power law in Eq. 共11兲, a critical exponent that arises in the particular scaling form of the theory used here. The exponent was checked by simultaneously fitting all the data from all the liquids studied to Eq. 共11兲 with the exponent 8/3 replaced by an adjustable parameter; the best overall simultaneous fits for all the liquids were obtained with the 8/3 exponent.22 关Recall, however, that the value of the correlation length exponent ␯ used in the scaling analysis, e.g., in Eq. 共4兲, is only approximately equal to 2/3.兴 We interpret this as supportive of the FLD theory. The parameter B can be taken as a measure of ‘‘fragility,’’ large values of B corresponding to high fragility. Since the measure of ‘‘frustration’’ is (Q/J) and B is inversely proportional to (Q/J), fragility is inversely related to frustration. Note that the activation free energy as given by Eq. 共11兲 depends on a combination of the four parameters C 共or Q/J兲, ␬, b, and m that are compressed into the single param-

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eter B. This compression, as seen below, is not true for the frequency-dependent susceptibility which depends more sensitively on the details of the model. We comment briefly on the molecular parameters ␶ ⬁ and E ⬁ . Our focus here is on low-T collective behavior below T * , and we are interested in the molecular relaxation above T * only to the extent that we can extrapolate this behavior to temperatures below T * and subtract it from the observed low-T behavior, thereby leading to ‘‘experimental collective data’’ that can be compared with the results of the FLD theory. For fragile liquids, the two-parameter 兵 ␶ ⬁ ,E ⬁ 其 description of the high-T molecular behavior seems quite adequate. B. Domain size and polydispersity

Equation 共4兲 gives the FLD theory prediction of the characteristic domain size, L * (T). As a rough estimate, we set all parameters of order one appearing in the scaling analysis equal to 1, and we can rewrite Eq. 共4兲 as

冉

T L *共 T 兲 ⬇B 1/2 1⫺ a0 T*

冊

2/3

,

共 T⬍T 1 ⬍T * 兲 .

共12兲

Typically, T g ⬇2T * /3, and so at T g the domain length is about (1/2)B 1/2a 0 . For a very fragile liquid such as orthoterphenyl and salol with B⬇400,22,36 one therefore expects domains of order 10 molecular diameters, i.e., domains with about 103 molecules at T g . Furthermore, we see that as T is increased above T g , the domains shrink. The more fragile the liquid 共i.e., the less the frustration兲, the larger the domains. For instance, we expect a larger typical domain size for salol and orthoterphenyl than for glycerol 关L * ⬇5a 0 at T g , with B⬇90 共Ref. 22兲兴. The formula in Eq. 共12兲 is valid only if L * is significantly larger than a 0 , a situation that holds only if T⬍T 1 ⬍T * ; for T⬎T * we take L * ⬇a 0 . In addition, from Eqs. 共7兲, 共8兲, and 共11兲, we can estimate that the spread ⌬L(T) in domain size, i.e., the polydispersity, is given by

冉 冊 冉

T* ⌬L 共 T 兲 ⬇B ⫺1/2 L *共 T 兲 T

⫺1/2

1⫺

T T*

冊

⫺4/3

,

共13兲

where the same rough approximation introduced above has been utilized. We conclude that the more fragile the liquid, the less the polydispersity, and what is particularly intriguing, the lower the temperature, the less the polydispersity. The first question to pose is whether these numbers reflect physical reality. Since Eq. 共12兲 results from a scaling approach, it is valid only within a factor of order 1, and the predicted domain sizes are therefore uncertain to this degree. To within such uncertainty the above results are compatible with the measured sizes of the dynamic domains 共which, too, are uncertain兲 found in supercooled liquids.15–18 The other features mentioned in the preceding paragraph have not yet been checked experimentally, but they seem reasonable and are integral to the theory that has been checked in other ways. The next question is whether such rather small domains 共and they are smaller still for less fragile liquids and for liquids much above T g 兲 can be treated by a mesoscopic

FIG. 1. Log10 of the imaginary part of the frequency-dependent dielectric susceptibility ␹ ⬙ ( ␻ ) of supercooled liquid salol vs log10( ␻ ) at several temperatures below the crossover T * between Arrhenius and super-Arrhenius behavior. The data 共shown as symbols兲 are taken from Ref. 26 and from private communication by S. Nagel and N. Menon. The solid lines represent the theoretically predicted curves fitted with ␬ ⫽0.84, b⫽1.46, m ⫽0.20, C⫽333. The other parameters, T * ⫽304 K and E ⬁ ⫽3220 K, are taken from Ref. 22 and ␻ ⬁ is obtained from data in the high-T region 共not represented in the figure兲. The intrinsic cutoff, as given by Eq. 共16兲, is marked by a change from solid to dotted line.

theory. A definitive answer cannot be given, but one should keep in mind that the frustration-limited domains are not isolated clusters but domains equilibrated in a bath of domains. The last question concerns the precise nature of the local order parameter that characterizes the frustration-limited domains. We have stressed in the Introduction that whereas there is now significant evidence that domains or heterogeneities play a major role at the dynamic level, no definitive structural signature for the existence of an increasing supermolecular correlation length with decreasing temperature has been found. Guided by the only well-documented example of topological frustration, i.e., the case of one-component systems of spherical particles,37 we speculate that the local order parameter associated with the locally preferred structure of the liquid is a multiparticle quantity specifying the symmetry of the local arrangement of the molecules. 共For spheres, it has been proposed that an appropriate local parameter can be constructed from bond-orientational variables that depend on the angles of the vector joining two neighboring particles.37兲 For symmetry reasons, such a quantity is likely to have only a weak structural coupling to the local density, and any associated correlation on a mesoscopic range would not show up in x-ray and neutron scattering structural data. However, the dynamic coupling of such measures of order to the density may still be strong and dominate relaxation at long times via what we have called ‘‘late-time-coupling through derivatives.’’ 33 In any case, elucidating the nature of the

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FIG. 2. Same as Fig. 1 for supercooled liquid glycerol. The data 共shown as symbols兲 are taken from Ref. 26 and from private communication by S. Nagel and N. Menon. The solid lines represent the theoretically predicted curves fitted with ␬ ⫽0.86, b⫽1.18, m⫽0.26, C⫽100. The other parameters, T * ⫽322 K and E ⬁ ⫽5180 K, are taken from Ref. 22 and ␻ ⬁ is obtained from data in the high-T region 共not represented in the figure兲. The intrinsic cutoff, as given by Eq. 共16兲, is marked by a change from solid to dotted line.

configurational variable describing the locally preferred structure of real, fragile glassforming liquids remains an open question and is currently under investigation. C. Frequency-dependent dielectric susceptibility

The imaginary part, ␹⬙ 共␻兲, of the dielectric susceptibility is given by Eq. 共3兲 in conjunction with Eqs. 共1兲, 共8兲, 共9兲, and 共10兲, i.e.,

␹ ⬙ 共 ␻ 兲 ⫽⌬ ⑀ 共 T 兲

冕

⬁

0

dx ␳ 共 x 兲

␻␶共 x 兲 , 1⫹ ␻ 2 ␶ 共 x 兲 2

共14兲

with ␳ (x) given by Eq. 共7兲 and ␶ (x) derived from Eq. 共10兲 as

␶ 共 x 兲 ⫽ ␶ ⬁ exp

冉

冊

E⬁ ⫹b ␥ 共 T 兲共 x 2 ⫺mx 5 兲 , T

共15兲

where ␥ (T) is specified in Eq. 共8兲. We now have the speciesdependent but temperature-independent parameters 兵 ␶ ⬁ ,E ⬁ ,C,T * ,b,m, ␬ 其 and the species- and temperaturedependent parameter 兵 ⌬ ⑀ (T) 其 . Recall that C, like B in the preceding sections, is proportional to the inverse frustration (J/Q) introduced in the basic theoretical FLD formulation. Dielectric susceptibility curves are given in Figs. 1 and 2. All such curves are characterized by a peak frequency, ␻ P (T)⬇ ␻ ⬁ exp(⫺E␣(T)/T), where ␻ ⬁ ⬀ ␶ ⬁⫺1 is a molecular high-T frequency and E ␣ (T) is given by Eq. 共11兲. As discussed above, ␻ P (T), ␶ ␣ (T), and the viscosity, ␩ (T), as well, can be treated interchangeably as far as the dominant temperature dependence is concerned. In Fig. 3 we plot log10关 ␻ P (T)/( ␻ ⬁ exp(⫺E⬁ /T))兴 vs the scaling variable ␥ (T) for all data included in Figs. 1 and 2. Up to a trivial multiplying factor, this is equivalent to plot-

FIG. 3. Log10 of the peak frequency ␻ P divided by the molecular contribution ␻ ⬁ exp(⫺E⬁ /T) versus the scaling parameter ␥ (T)⫽C(T * /T) 关 1 ⫺(T/T * ) 兴 8/3 for salol 共a兲 and glycerol 共b兲. The symbols represent the experimental data and the solid line is obtained from the predicted curves for the susceptibilities 共same as in Figs. 1 and 2兲.

ting 关 (E ␣ (T)⫺E ⬁ )/T 兴 vs ␥ as suggested by Eqs. 共11兲 and 共8兲, and the linearity of the plots confirms the significance of Eq. 共11兲. The parameter B that characterizes the superArrhenius T-dependence of the activation free energy for ␣ relaxation is not an independent parameter; it is a function of C,b,m, and ␬. 关Actually, to a good approximation B is independent of ␬, with Bb(1⫺m)⬇C.兴 As indicated in the Introduction, the FLD theory is one designed to be applicable only to the slow ␣ relaxations, and not to the faster ␤ relaxations. Thus, the fits depicted in Figs. 1 and 2 require a high-frequency cutoff. We do not wish to introduce an adjustable cutoff parameter determined by the point of breakdown of fits; instead we introduce a built-in cutoff procedure that takes into account the fact that the theory of ␣-relaxation should be applicable only to those domains that are sufficiently large, domains in which relaxation is slow compared to bare molecular behavior. Thus, we introduce a high-frequency cutoff, ␻ cut to our fits, where

␻ cut⫽

冋 册

1 E⬁ exp ⫺ . 3␶⬁ T

共16兲

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FIG. 4. Imaginary part of the frequency-dependent dielectric susceptibility ␹ ⬙ ( ␻ ) of salol vs log10( ␻ ) at several temperatures 共see Fig. 1兲. The dashed lines result from replacing the domain size distribution by a Gaussian function in the FLD theoretical expressions.

By this procedure we avoid introducing the cutoff as an extra adjustable parameter, and we also set a limit on the frequency range over which comparison of theory and experiment should be made. Although, as indicated above, no cutoff in x is required, to simplify calculations we have cut the integration in Eq. 共14兲 at x⫽(5m/2) ⫺1/3. Changes in this cutoff did not produce any detectable changes in the results. Our fits to the frequency-dependent susceptibilities of salol and glycerol are given in Figs. 1, 2, 4, and 5. The log–log plots in Figs. 1 and 2 enable us to examine the line shapes over a wide range of frequencies, whereas linear-

Viot, Tarjus, and Kivelson

log10关 ␻ 兴 plots in Figs. 4 and 5 enable us to examine carefully the peak frequency ␻ P , the susceptibility amplitude ⌬⑀, and the half-width at half-height. In all cases, the agreement between theory and experiment is good, especially if one considers the range of frequencies over which the fit applies 共13 decades!兲, a range that includes Debye-type behavior at low frequencies below the maximum, stretchedexponential or Cole–Davidson behavior around the maximum, and von Schweidler power-law behavior at higher frequencies. Eight adjustable parameters 兵 ␶ ⬁ ,E ⬁ ,C,T * ,b,m, ␬ , ⌬ ⑀ (T) 其 for each substance may seem a great many, but it should be noted that the fits are nontrivial and that all the parameters except the structural 共nondynamic兲 amplitudes, ⌬ ⑀ (T), are temperature-independent. In considering these parameters recall that ␶ ⬁ and E ⬁ are ‘‘molecular’’ parameters, C and T * are the primary ‘‘collective’’ parameters, (b,m, ␬ ) are dimensionless parameters that are all of order one 关with m⬍1/(2b)兴 and do not seem to change much from substance to substance. The amplitude, ⌬ ⑀ (T), could, in principle, be measured in an independent experiment on dielectric constant. In any case, one might expect ⌬ ⑀ (T) to be given by the Debye Law, i.e., ⌬ ⑀ (T)⫽A/T, where A has only a slight T-dependence associated with the T-dependence of the Kirkwood g-factor, or perhaps a Curie–Weiss law;4 the extent to which these laws are obeyed within our fits is indicated in Fig. 6. Moreover, to constrain the fitting procedure, the parameters E ⬁ and T * have been taken from the fit to the ␣-relaxation activation free energy discussed earlier, and ␶ ⬁ is determined from high-T susceptibility data above T *. The number of adjustable parameters can be reduced if one restricts the fits to those frequencies that dominate the linear-log(␻) plots in Figs. 4 and 5, i.e., to relaxations described by stretched-exponential/Cole–Davidson functions. The behavior in this restricted range, which is that best described by the scaling model, is, according to the theory, due to domains whose sizes are close to the most probable size. In such a case, one can replace the full distribution of domain sizes, Eq. 共7兲, by a Gaussian distribution, thereby reducing the number of adjustable parameters by 1. The results are displayed as dashed curves in Figs. 4 and 5. In the literature we have not found comparable agreement between theory 共fits兲 and experiment. 共Recall that the description of the ␣ relaxation by the mode-coupling theory27 is limited to the moderately supercooled regime of glassforming liquids where ␣-relaxation times are in the nanosecond range or smaller.兲 To illustrate the significance of the discrepancy between our fits and the experimental data, we show in Figs. 7 and 8 the effect of changing the temperature by only 1 K in the theoretically predicted susceptibility curves. D. Relative narrowness of relaxation-time distribution

FIG. 5. Imaginary part of the frequency-dependent dielectric susceptibility ␹ ⬙ ( ␻ ) of glycerol vs log10( ␻ ) at several temperatures 共see Fig. 2兲. The dashed lines result from replacing the domain size distribution by a Gaussian function in the FLD theoretical expressions.

A point that has not received much attention is that the stretching of the ␣ relaxation 共or, alternatively, the broadening of the ␣ peak in frequency space兲 is relatively small in light of the extremely rapid variation with temperature of the

J. Chem. Phys., Vol. 112, No. 23, 15 June 2000

A heterogeneous picture of ␣ relaxation

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FIG. 7. Illustration of the effect of a 1 K change in temperature in the theoretically predicted curves on the same plot as in Fig. 1. T is changed from 233 K to 232 K 共dashed line兲. Recall that apart from the amplitude ⌬⑀ all the adjustable parameters entering the FLD theoretical expressions used here are T-independent.

of relaxation times, as predicted by Eq. 共15兲, is opposed by the tendency of the domain-size distribution to become less and less polydisperse 关see Eqs. 共7兲 and 共13兲兴. Since both these opposing effects are exponentially dependent upon ␥ (T), the net result is the observed stretched-exponential or Cole–Davidson behavior of the ␣-relaxation function and dielectric susceptibility. It is the relative narrowness of the relaxation-time distribution that accounts for the close similarity of the ␣-relaxation activation free energies obtained from different types of experiments, such as NMR, dielectric relaxation,

FIG. 6. Inverse of the amplitude factor ⌬⑀ of the dielectric susceptibility curves vs temperature for salol 共a兲 and glycerol 共b兲.

␣-relaxation time itself.38 Unusually strong slowing down of relaxation, with exponentially growing times described by a super-Arrhenius T-dependence, is found in some disordered systems, like the random field Ising model, near their critical points, and this phenomenon is known as ‘‘activated dynamic scaling.’’ 39 However, in this latter case the growth of the activation free energy when temperature is decreased towards the critical point comes with a striking stretching of the relaxation that occurs on a logarithmic scale; instead of the stretched-exponential behavior observed in supercooled liquids, the logarithm of the relaxation function goes as some power of 1/ln(t).40 What then limits the stretching of the ␣ relaxation for supercooled liquids? In the scaling model presented here, the stretching with decreasing T is limited by the fact that the increasing spread

FIG. 8. Same as Fig. 7 for glycerol. The 1 K change is between T ⫽201 K and T⫽200 K 共dashed line兲.

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Viot, Tarjus, and Kivelson

viscosity, or light scattering. However, it is found that the effective activation free energy for translational diffusion is quite different.

E. Translational diffusion

The temperature dependence of the relaxation time ␶ ␣ (T) discussed above should be applicable to rotational relaxation processes because molecular reorientation is likely to take place while the molecule remains within a domain. On the other hand, translational diffusion is often measured over distances comparable to the wavelength of light, and the molecules are, therefore, likely to find themselves in many different domains during the experiment. The required averaging over domains is thus quite different for rotations and translations. For long-lived domains with sharp boundaries 共an assumption that tends to overestimate diffusivity兲, the three-dimensional translational diffusion constant, D, is6 D⬀

冕

⬁

0

冋

dL ␳ 共 L,T 兲 exp ⫺

册

E 共 L,T 兲 , T

共17a兲

whereas, as already stated, the rotational correlation time, ␶ rot⬀ ␻ ⫺1 P is

␶ rot ⬀

冕

⬁

0

冋

dL ␳ 共 L,T 兲 exp

册

E 共 L,T 兲 . T

共17b兲

At the same time, the Stokes–Einstein and Stokes–Einstein– Debye theories lead to the results

␶ rot ⬀

␩ T

T

,

共18a兲

,

共18b兲

D ␶ rot⫽constant.

共18c兲

D⬀

␩

Although there is no compelling reason why these relations should be obeyed at the molecular level, they are followed quite well at high temperatures even though D, ␶ rot , and ␩ may individually change by many orders of magnitude.2 However, these expressions break down seriously for deeply supercooled liquids, particularly for liquids near T g . An explanation of the breakdown is given above, and use of Eqs. 共17a兲 and 共17b兲 enables us to calculate the nonconstancy of D ␶ rot . The values of log(D␶rot) predicted by Eqs. 共17a兲 and 共17b兲 with the parameters determined by fits of the dielectric susceptibility are given for salol and glycerol in Fig. 9. The figure displays the same qualitative features as those observed experimentally for another fragile glassformer, orthoterphenyl. In particular, log(D␶rot) remains astonishingly close to zero down to temperatures well below T * and approaching T g , and at T g the quantity log关D␶rot兴 is between 3 and 5. Although the curves in Fig. 9 are those predicted by the FLD theory, it seems at first that log(D␶rot) should scale linearly with ␥ (T), i.e., with (1⫺T/T * ) 8/3; the fact that this is not the case can be understood by noting that the integral in Eq. 共18b兲 can be evaluated by steepest descent, and such

FIG. 9. Log10 of the product of the translational diffusion constant D by the rotational relaxation time ␶ rot vs temperature for T g ⭐T⭐T * ; 共a兲 salol and 共b兲 glycerol. D is obtained from Eq. 共17a兲 and ␶ rot from Eq. 共17b兲, with the parameters determined from fits to the dielectric susceptibility curves. Note that the decoupling of the T-dependencies of D and ␶ rot only becomes apparent some 30–40 K below T * .

an approach leads to a near linear ␥ (T)-dependence of ␶ rot(T), whereas the integral in Eq. 共17a兲 cannot be evaluated by this method. It is remarkable that the theoretical values of D ␶ rot are so reasonable because the translational diffusion process is dominated by the faster motions, i.e., by those associated with the smaller domains and by ␤-relaxations; the latter have been completely neglected here and the former should not be well described by the theory. Despite this, we believe that no other comparable predictions of the T-dependence of

J. Chem. Phys., Vol. 112, No. 23, 15 June 2000

FIG. 10. Masterplot of Dixon et al. 共Ref. 26兲 for all theoretically predicted frequency-dependent dielectric susceptibility curves of salol and glycerol shown in Figs. 1 and 2; ␻ P is the peak frequency, W is the width at halfheight of ␹ ⬙ (log10( ␻ )) divided by 1.14, and ⌬⑀ is the amplitude factor. Note that the frequency range extends over 13 decades.

D ␶ rot based exclusively on parameters determined independently by fits to the susceptibility have been made. F. Susceptibility mastercurve

Nagel and co-workers26 have shown that the frequencydependent susceptibilities of many different fragile glassforming liquids could be placed with good accuracy onto a single curve for which the data are scaled by means of three temperature- and species-dependent parameters; these parameters correspond closely to the ␣-peak position ␻ P (T), the width at half-height of ␹ ⬙ (log(␻)) normalized by 1.14 共the width of the Debye spectrum兲, W(T), and the amplitude factor ⌬ ⑀ (T). The data cover a 13-decade range in frequency that encompasses the low-frequency Debye, the Cole–Davidson, and the von Schweidler regimes, thereby suggesting that these regimes are part of the same ␣-relaxation mechanism. The scaling model developed in the previous section does not lead explicitly to a susceptibility mastercurve. However, the predicted curves obtained above by fitting to the experimental data on salol and glycerol can be analyzed according to the procedure of Dixon et al. For each temperature, we have determined ␻ P (T), W(T), and ⌬ ⑀ (T). The result is shown in Fig. 10, where all our susceptibility curves for salol and glycerol at all temperatures displayed in Figs. 1 and 2 are reasonably well collapsed onto a single curve.

A heterogeneous picture of ␣ relaxation

10377

of frequency corresponding to the ␣ relaxation that includes the stretched-exponential/Cole–Davidson and von Schweidler regimes, i.e., regimes that are incorporated in the mastercurve of Nagel and co-workers.26 Besides the amplitude factor ⌬ ⑀ (T), the fit requires the two ‘‘molecular’’ T-independent parameters, ␶ ⬁ and E ⬁ , and the five ‘‘collective’’ T-independent parameters, C,T * ,b,m, ␬ , the latter three varying little from species to species. 关Recall that C, just like B, is inversely proportional to the frustration parameter (Q/J) introduced in the basic FLD theory.兴 The quality of the resulting fits, considering that we use a theoretically generated fitting formula and that we apply it to nontrivial lineshape features, gives support to the theory. To our knowledge, no other theories or models are able to describe the same wide range of phenomena down to T g . 共This is true in particular of the mode-coupling theory whose application is limited to the moderately supercooled regime of glassforming liquids.兲 The FLD theory associates the crossover to superArrhenius behavior with activated domain-relaxation processes dependent upon domain size, it associates nonexponential relaxation with domain-size polydispersity, and it associates both the apparent power-law 共von Schweidler兲 regime and the decoupling of translational diffusion from viscosity and rotational relaxation to the role of the smaller domains. The phenomenological extension of the FLD theory developed here ties all these phenomena together quantitatively. One objective that researchers in this field have long had is to relate the fragility 共B in the present case兲 to the nonexponential stretching coefficient 共taken as ␤ in the stretched exponential f ␣ (t)⫽exp关⫺(t/␶)␤兴 or as the width W of the ␣ peak on the ␹ ⬙ (log ␻)-plot兲. Although the FLD theory does not incorporate ␤ or W as a primary parameter, the fact that it gives fits to the frequency-dependent suceptibility is equivalent to obtaining an implicit relation between B and W 共or ␤兲. One might expect W(T) to be dominantly determined by ␥ (T) and for W(T) to decrease as ␥ (T) increases; and indeed this is roughly true.22 However, W(T) also depends on the parameters ␬, b, and m. This paper represents a phenomenological analysis by means of the FLD theory of all aspects of the long-time dynamics 共␣ relaxations兲 of fragile supercooled liquids. The basics of the theory,21,33 its treatment of thermodynamic phenomena 共entropy兲,41 and its application to the study of the predicted defect-ordered phases42 are discussed elsewhere. Here we have not studied the local order parameter to be associated with the locally preferred structure and with the putative structural dynamics; as discussed previously, the issue of the nature of the local order parameter has not been resolved and is currently under study. ACKNOWLEDGMENTS

IV. CONCLUSION

We have made use of an expression derived by a phenomenological extension of the frustration-limited domain theory to fit the dielectric susceptibility data for fragile supercooled liquids. The fits are made over a 13 decade range

We would like to thank Jim Sethna for getting us to think about this problem and S. Nagel and N. Menon for providing us their experimental data. We benefited greatly from discussions with Mark Ediger and Christiane AlbaSimionesco, and we wish to acknowledge the many insight-

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ful comments made to us by Steven Kivelson. We are indebted to the NSF, the C.N.R.S. and NATO for their support. 1

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Viot, Tarjus, and Kivelson hand generally show intrinsic intradomain nonexponential relaxation. See Ref. 24. 24 H. Wendt and R. Richert 共preprint, 1999兲. 25 I. Chang and H. Sillescu, J. Phys. Chem. B 101, 8794 共1997兲. 26 P. K. Dixon, L. Wu, S. R. Nagel, B. D. Williams, and J. P. Carini, Phys. Rev. Lett. 65, 1108 共1990兲; L. Wu, P. K. Dixon, S. R. Nagel, B. D. Williams, and J. P. Carini, J. Non-Cryst. Solids 131-133, 32 共1991兲; S. R. Nagel, in Phase Transitions and Relaxation in Systems with Competing Energy Scales, edited by T. Riste and D. Sherrington 共Kluwer Academic, The Netherlands, 1993兲, p. 259. 27 W. Go¨tze, in Liquids, Freezing, and the Glass Transition, edited by J. P. Hansen, D. Levesque, and J. Zinn-Justin 共North–Holland, Amsterdam, 1991兲, p. 287. 28 L. Chayes, V. J. Emery, S. A. Kivelson, Z. Nussinov, and G. Tarjus, Physica A 225, 129 共1996兲. 29 Z. Nussinov, J. Rudnick, S. A. Kivelson, and L. Chayes, Phys. Rev. Lett. 83, 472 共1999兲. 30 P. Viot and G. Tarjus, Europhys. Lett. 44, 423 共1998兲. 31 M. Grousson 共private communication, 1999兲. 32 This property has been shown rigorously for the spherical version of the model 共Ref. 28兲 and for the O(n) version with n strictly larger than 2 共Ref. 29兲. For Ising spins (n⫽1), the transition line for Q⫽0 approaches the critical point at T * and Q⫽0 in a continuous, albeit nonanalytical way. However, the transition in the presence of frustration is not continuous: it is driven first-order by the fluctuations 共Ref. 30兲. 33 G. Tarjus, D. Kivelson, and S. A. Kivelson, in Supercooled Liquids: Advances and Novel Application, edited by J. Fourkas et al. 共American Chemical Society, Baltimore, 1997兲, p. 67; D. Kivelson, G. Tarjus, and S. A. Kivelson, Prog. Theor. Phys. Suppl. 126, 289 共1997兲; D. Kivelson and G. Tarjus, Philos. Mag. B 77, 245 共1998兲. 34 G. Tarjus, C. Alba-Simionesco, M. L. Ferrer, H. Sakai, and D. Kivelson, in Slow Dynamics in Complex Systems, edited by M. Tokuyama and I. Oppenheim 共AIS, New York, 1999兲, p. 406; M. L. Ferrer, H. Sakai, D. Kivelson, and C. Alba-Simionesco, J. Phys. Chem. 103, 4191 共1999兲. 35 Current computer simulations strongly suggest that a multistate clock model with long-range 共Coulombic兲 frustration does indeed lead to superArrhenius activated dynamics 共Ref. 31兲. 36 H. Z. Cummins, Phys. Rev. E 54, 5870 共1996兲; D. Kivelson, G. Tarjus, X-L Zhao, and S. A. Kivelson, ibid. 54, 5873 共1996兲. 37 F. C. Frank, Proc. R. Soc. London, Ser. A 215A, 43 共1952兲; M. Kleman and J. F. Sadoc, J. Phys. 共France兲 Lett. 40, L569 共1979兲; D. R. Nelson and F. Spaepen, Solid State Phys. 42, 1 共1989兲. 38 In this particular discussion, we do not consider the ‘‘stretching’’ that occurs at shorter times or higher frequencies corresponding to the von Schweidler power-law regime. 39 D. S. Fisher, G. M. Grinstein, and A. Khurana, Phys. Today 41, 56 共1988兲. 40 J. Villain, J. Phys. 共Paris兲 46, 1843 共1985兲; D. S. Fisher, Phys. Rev. Lett. 56, 416 共1986兲; A. T. Ogielski and D. A. Huse, ibid. 56, 1298 共1986兲. 41 D. Kivelson and G. Tarjus, J. Chem. Phys. 109, 5481 共1998兲. 42 D. Kivelson, J.-C. Pereda, K. Luu, M. Lee, H. Sakai, A. Ha, I. Cohen, and G. Tarjus, in Supercooled Liquids: Advances and Novel Applications, ACS Symposium Series 676, edited by J. Fourkas et al. 共American Chemical Society, Washington, D.C., 1997兲, p. 224.