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Facilitation, complexity growth, mode coupling, and activated dynamics in supercooled liquids

Contributed by Peter G. Wolynes, September 4, 2008 (received for review August 5, 2008)
Abstract
In lowtemperaturesupercooled liquids, below the ideal modecoupling theory transition temperature, hopping and continuous diffusion are seen to coexist. Here, we present a theory that shows explicitly the interplay between the two processes and shows that activated hopping facilitates continuous diffusion in the otherwise frozen liquid. Several universal features arise from nonlinear interactions between the continuous diffusive dynamics described here by the mode coupling theory (MCT)] and the activated hopping (described here by the random firstorder transition theory). We apply the theory to a specific system, Salol, to show that the theory correctly predicts the temperature dependence of the nonexponential stretching parameter, β, and the primary α relaxation timescale, τ. The study explains why, even below the mean field ergodic to nonergodic transition, the dynamics is well described by MCT. The nonlinear coupling between the two dynamical processes modifies the relaxation behavior of the structural relaxation from what would be predicted by a theory with a complete static Gaussian barrier distribution in a manner that may be described as a facilitation effect. Furthermore, the theory correctly predicts the observed variation of the stretching exponent β with the fragility parameter, D. These two predictions also allow the complexity growth to be predicted, in good agreement with the results of Capaccioli et al. [Capaccioli S, Ruocco G, Zamponi F (2008) J Phys Chem B 112:1065210658].
The glass transition is characterized by a number of interesting kinetic phenomena. Very slow and simultaneously nonexponential relaxation of time correlation functions over large time windows is one such important phenomenon. This relaxation is often approximated by the stretched exponential, Kohlrausch–William–Watts (KWW) formula, ø(t) = exp(−(t/τ))^{β}, with both β and τ exhibiting nontrivial temperature dependence. The origin of the stretching is usually attributed to the presence of dynamic heterogeneity in the system (1, 2). The temperature dependence of the typical relaxation time can be described by the the Vogel–Fulcher–Tamman (VFT) expression, τ = τ_{VFT} exp(DT_{o}/(T−T_{o})), where τ_{VFT} is the hightemperature relaxation time, T_{o} is the VFT temperature, and D is the fragility index. The fragility index, D, determines the degree of deviation from the Arrhenius law that is appropriate for simple activated events. Experimental and theoretical model studies have shown that β and D are correlated (3–5). The temperature dependence of τ has also been described by phenomenological mode coupling theory (MCT) expression, τ ∼ [(T−T_{c}^{fit})/T_{c}^{fit}]^{−γ}, (γ > 0), but this ultimately breaks down at low temperature. T_{c}^{fit} is referred to as the MCT transition temperature. Above T_{c}^{fit}, MCT is found to explain many experimental results (6–9), and below T_{c}^{fit}, the MCT picture of continuous diffusion fails eventually. It is conjectured that this breakdown is due to the ergodic to nonergodic transition in the dynamics and below T_{c}^{fit} activated dynamics becomes a dominant mode of transport. However in an elegant work, Brumer and Reichman (10) (BR) have recently shown that the idealized MCT using microscopic input breaks down at a much higher temperature, T_{c}^{0}, which corresponds to the temperature, T_{L}, where the landscape properties show a sharp change. Kob et al. (11) have shown that the structural MCT predicts the proper dynamics till very close to T_{c}^{fit} but needs static inputs to be calculated at a higher effective temperature. Stevenson et al. (12, 13) have shown that below the ergodic to nonergodic transition but before the predicted crossover to the activated dynamics, string or fractal excitations drive the dynamics and that this qualitatively describes the crossover temperature as occurring at a specific value of the configurational entropy.
Computer simulation studies seem to show the coexistence of continuous diffusion and hopping as mechanism of mass transport at temperatures much above T_{c}^{fit} (14–17). These studies show that individual hopping events are often followed by enhanced continuous diffusion or more hops of the surrounding atoms or molecules (14, 15, 18). Simulations have also shown that a single hopping event relaxes the local stress (16), hence, it is expected that hopping events are followed by continuous diffusive dynamics. From BR study we know the temperature at which the ergodic to nonergodic transition takes place (T_{c}^{0} = T_{L}) (10). However, there is no clear theoretical understanding of the dynamics in the range, T_{c}^{fit} < T < T_{c}^{0}. Furthermore, we need to understand, why, below T_{c}^{0}, the MCT still seems to explain the form of the dynamics rather well and, finally, what happens at T_{c}^{fit} that leads to the breakdown of the MCT as far as the temperature dependence is concerned. In an earlier work it was shown that the full dynamics is a synergetic effect of continuous diffusive motion and activated dynamics (19). Here, we show how the coupling between these two dynamical processes influences each other. The coupling leads to hoppinginduced diffusive motion below T_{c}^{0} and thus explains the validity of MCT below T_{c}^{0}. The study also explains the origin of the apparent breakdown of MCT at T_{0}^{fit}. The crucial result of the present study is that the coupling renormalizes the distribution of hopping barriers which participates in the dynamics. This provides a formal treatment of the facilitation effect discussed by Xia and Wolynes (4, 5) in their theory of the stretching exponent.
The present work uses a scheme of calculation similar to the one presented earlier (19) with a few modifications. For describing the diffusive motion we use the schematic F_{12} model of the MCT (20–22) but the activated hopping dynamics now have a static barrier height distribution (4, 5). The equation of motion for the total intermediate scattering function is written as,
In describing the activated dynamics we consider that there is a distribution of the hopping barriers in the system arising from the entropy fluctuation (4, 5). Thus, the total contribution from the multiplebarrier hopping events is written as,
where ℘^{static}(ΔF) is taken to be Gaussian,
We call this distribution, the static barrier height distribution. Here, ø_{hop}^{s} = e^{–tKhop(ΔF)} describes the activated hopping dynamics for a single hopping barrier. K_{hop}(ΔF) = ℱ(q)P_{hop}(ΔF). The expression for ℱ(q) is given by
The equation of motion for the MCT part of the intermediate scattering function, ø_{MCT}(t), is written as, Now, the MCT part of the intermediate scattering function, ø_{MCT}(t) is selfconsistently calculated with the full scattering function, ø(t). In describing the dynamics with schematic MCT the coupling between the different wave vectors is neglected and the contribution from the static and dynamic quantities is calculated at a single wavenumber q = q_{m} (q_{m} is the wave number where the peak of the structure factor appears) which is known to provide the dominant contribution. In the present formalism it is assumed that hopping opens up multiple relaxation channels for the otherwise frozen MCT dynamics.
Eqs. 1 and 4 together describe the full dynamics, which is similar in spirit to the extended MCT (21) and other recent approaches (24). The similarity of the present scheme to the extended MCT of Gotze and Sjogren was elaborately analyzed in our earlier work (19).
As in our earlier work (19), the values of Ω_{o} and γ are kept fixed at unity, neglecting their temperature dependence, and the scaling time is taken as 1 ps.
Next, we discuss the hopping dynamics for the Salol system as predicted from random firstorder transition (RFOT) theory. According to the RFOT theory, the freeenergy cost, F(r), which is used to calculate the mean barrier height, can be written as
Combining the static RFOT and MCT, we solve Eqs. 1 and 4 numerically.^{‖} We use a numerical method presented earlier (28) with a minor modification (29). The total relaxation time, τ_{total}, and the stretching parameter, β_{total}, are obtained from the coupled dynamics by fitting the long time part of ø(t) (obtained from Eq. 1) to a KWW stretched exponential function, ø(t) = A exp(−(t/τ_{total}))^{βtotal}. Similarly from ø_{MCT}(t) we obtain the MCT relaxation time, τ_{MCT}, and the corresponding stretching parameter, β_{MCT}.
HoppingInduced Continuous Diffusion
The plot for the relaxation time τ_{total} is given in Fig. 1, where we have also shown the experimental results (for Salol) and the fit to the MCT phenomenological expression. Already the temperature dependence of τ_{total} reveals several interesting physics. The τ_{total} compares well with the experimental results (8) and predicts the correct glass transition temperature, T_{g} = 220 K. Recall that in our calculation the ergodic to nonergodic transition takes place at T_{c}^{0} = 278 K where the activated dynamics also becomes significant. Thus, we know that the idealized MCT prediction breaks down at T = 278 K and below this temperature, its validity as such is unclear. However, when we fit the relaxation time to MCT expression (τ ∼ [(T−T_{c}^{fit})/T_{c}^{fit}]^{−γ}) we obtain a value, T_{c}^{fit} = 258 K, which is in excellent agreement with the experimental fits (6, 8, 9).
The advantage of the present scheme of calculation is that, in addition to addressing the coupling between the continuous diffusion and the hopping motion, we can separately analyze their relative contributions to the total dynamics and explore the origin of the apparent validity of the MCT below T_{c}^{0}. Due to the nonlinear coupling in Eqs. 1 and 4, the activated dynamics plays both a “direct” and a “hidden” role in the total relaxation. The direct role is the direct relaxation of ø via ø_{hop}^{static}. However, if we analyze the structure of Eq. 4, we find that the activated dynamics also acts to soften the growth of the longitudinal viscosity, which finally leads to the relaxation of the otherwise frozen ø_{MCT}. Thus ø_{hop}^{static} plays a hidden role in the relaxation by helping ø_{MCT} to relax. Earlier simulation studies of Bhattacharyya and Bagchi have already observed this hidden role of hopping in relaxing the local stress (16). We now analyze these two different roles of the hopping dynamics and their effect on the total relaxation.
The present study shows that both continuous and activated dynamics change continuously across T_{c}^{fit}. However, when we plot (τ_{MCT} − τ_{total})/τ_{total} against T (in Fig. 1 Inset) this quantity is seen to undergo a rapid increase in the temperature range of the phenomenological MCT transition temperature T_{c}^{fit}. This implies that although below T_{c}^{fit} the continuous diffusion still remains the dominant mode of relaxation, there is an increased contribution from the activated dynamics. This plot suggests that in the range, T_{c}^{fit} < T < T_{c}^{0}, where direct hopping contribution is small, the activated dynamics plays only an important hidden role, whereas below T_{c}^{fit}, it also plays a direct role in the relaxation process.
One important observation is that although idealized MCT breaks down at T_{c}^{0} = 278 K, the functional form of MCT continues to describe the dynamical characteristics until 258 K. The latter is a widely known result, usually obtained from fitting the MCT functional forms to the experimental data (6, 8, 9). Thus, our study provides an explanation of this intriguing result in terms of the interaction between the MCT and activated dynamics, which leads to hoppinginduced continuous diffusive motion. This also quantitatively explains the observations reported by Kob et al. (11). The authors find that until close to T_{c}^{fit} the dynamics can be described via MCT, although the static inputs need to be calculated at a higher effective temperature. The present study further shows that as we progressively lower the temperature below T_{c}^{fit}, although the continuous diffusive dynamics continues to play an important role in the relaxation, the increased contribution from the activated dynamics finally leads to the breakdown of the MCT predictions.
Renormalized Barrier Height Distribution
In Fig. 2 we plot the calculated temperature dependence of the stretching parameter β_{total}. In the same figure we also plot β_{MCT} and β_{hop}^{static} as predicted by the MCT and the static RFOT theory^{††} (by using static barrier height distribution), respectively. The experimental results on Salol are also plotted in the same figure (6, 7). It is clear that neither the MCT nor the static RFOT distribution can alone describe the proper temperature dependence of the stretching parameter. However, the coupled theory predicts a temperature dependence of β_{total} which is qualitatively similar to that found in experiments. The stretching parameter is known to provide a measure of the heterogeneity in the system. In our study the heterogeneity arises from the distribution of hopping barriers. However, below T_{c}^{0}, β_{total} is larger than β_{hop}^{static}, which means that the barrier height distribution that participates in the dynamics (dynamic barrier height distribution) is narrower than the initial assumption of distribution (static barrier height distribution). This modification of the barrier height distribution (or heterogeneity) takes place because of the coupling between the diffusive and the activated dynamics. The timescale analysis reveals that the smaller barriers contribute to the dynamics, which implies that, in the presence of the quasistatic dynamic heterogeneity, the coupling leads to a facilitation effect. Regions that would be slow are relaxed by their neighbors.
To quantify these points, we carry out the following analysis. Nonexponential relaxation in the system can be considered as arising from superposition of exponentials, It is now possible to find an effective distribution function of the relaxation times ℘^{dynamic}(τ) by Laplace inverting the stretched exponential. Once we obtain the distribution of relaxation times we can convert it into the distribution of barrier heights by using, ℘^{dynamic}(ΔF)dΔF = ℘^{dynamic}(τ)dτ, provided τ can be expressed in terms of ΔF. The details are presented in Materials and Methods. Note that it is possible to find an analytical relation between τ and ΔF only when the individual dynamics are exponential. Hence, we do this analysis considering only exponential MCT relaxation (i.e., for this analysis we choose λ such that β_{MCT} = 1.0). We now analyze the dependence of ℘^{dynamic}(ΔF) on the MCT relaxation time and show that ℘^{dynamic}(ΔF) not only depends on ℘^{static}(ΔF), but also depends on the MCT relaxation time τ_{MCT} = K_{MCT}^{−1}. We fix the ℘^{static}(ΔF) and change the τ_{MCT} by calculating the MCT dynamics [which is coupled to ℘^{static}(ΔF)] at different temperatures. The results are presented in Fig. 3.
At high temperatures, where MCT dynamics is fast, ℘^{dynamic}(ΔF) is narrow (β_{total} is large), and the position of the maximum, ΔF_{max}, is shifted to smaller ΔF (Fig. 3). We also find that, as the MCT dynamics slows down, the distribution becomes broader and more nonGaussian (β_{total} also becomes smaller), larger barriers contribute to the dynamics, and ΔF_{max} shifts to higher values (though it always remains smaller than ΔF_{o}). Thus, we find, in accord with the literature, that as the temperature is lowered (or the dynamics becomes slower) there is a growth in dynamic heterogeneity. We may say that ℘^{dynamic}(ΔF) provides a mean field estimation of the dynamic heterogeneity. Note that ℘^{dynamic}(ΔF) is always narrower than ℘^{static}(ΔF) and overlaps with it only in the low barrier side. Xia and Wolynes (4) have given a physical interpretation of this exclusion of the higher barriers from the dynamics by using the picture of dynamic mosaic structure. Because the exclusion of higher barriers hastens the hopping dynamics this can be described as a “facilitation effect.” The present theoretical model does not explicitly treat the spatial structure of the dynamical mosaic. However, we find that the presence of the continuous dynamics and its coupling to the activated dynamics does lead to such a facilitation effect. Note that the facilitation is strongest when the MCT dynamics is exponential and should be weaker when we have stretching in the MCT dynamics.
The MCT modification of the static barrier height distribution, as predicted by the present theory, plays a key role in describing the proper relationship between the fragility index, D, and the stretching parameter, β. The study of Bohmer et al. (3) finds a relationship between the fragility index, D, and the stretching parameter, β, at T = T_{g}. As discussed in ref. 4, β_{hop}^{static} is related to the width of the static Gaussian distribution of the barrier heights δΔF. δΔF can be related to the fragility, D,
The details of the calculation are presented in Materials and Methods. In Fig. 4 we plot the relationship between β_{total} and D as predicted by the present calculation. In the same plot we also show the experimental results (3). We find that the present theory captures the correct trend. This is to be contrasted with the predictions obtained from the static distribution of barrier heights. Note that, in all the cases, β_{total} > β_{hop}^{static}, which implies that ℘^{dynamic}(ΔF) is always narrower than ℘^{static}(ΔF). Earlier we showed that ℘^{dynamic}(ΔF) overlaps with ℘^{static}(ΔF) only on the low barrier side. Thus, the higher barriers do not participate in the dynamics leading to facilitation effect. Hence, for a wide range of systems the theory predicts a proper modification of the barrier height distribution. Note that Xia and Wolynes (4) showed that the static barrier height distribution should have a cutoff to describe the relationship between β and D. The authors gave a physical explanation of this “cutoff” by using the picture of dynamic mosaic structure.
The combination of distributed activated hopping and mode coupling facilitation also explains some recent exciting experimental observations of Capaccioli et al. (31). These authors recognized that, in those theories of glassy dynamics based on an underlying entropy crisis, the complexity of a dynamically correlated region is a key quantity. They also used earlier insights (32, 33) to show how this quantity can be roughly inferred from experiment. This inference is independent of chemical details, such as bead count, which have previously been used for comparisons of RFOT theory with experiment.
The complexity, α_{CRR} (the logarithm of the number of accessible states in a cooperatively rearranging region) is obtained by combining the configurational entropy density, S_{c}, inferred from calorimetry, with the temperature dependence of dynamics as reflected in the alpha relaxation time, τ_{α}(T), and stretching exponent of dielectric relaxation or viscosity, β(T) (31).
In those purely kinetic models of the glass transition that are based on the facilitation concept but have trivial underlying thermodynamics (34), one would not expect this inferred complexity to be a universal function of the relaxation time. However, according to the Adam–Gibbs argument, the complexity should be a constant for all values of the relaxation time but could depend on substance. The static RFOT theory predicts that the logarithm of the relaxation timescales should be a universal function of the complexity of the dynamically reconfiguring regions for all glass formers. In contrast to the expectations from thermodynamically trivial kinetic models, Capaccioli et al. (31) showed a very strong data collapse of this complexity, σ_{CRR} versus log_{10}(τ_{α}) for many substances. The Adam–Gibbs prediction of a constant value is not borne out, however. In contrast also to static RFOT, which predicts an asymptotically linear relation, the experimental data show a distinct curvature.
In Fig. 5 we show the prediction of σ_{CRR} versus log_{10}(τ_{total}/τ_{0}) that is obtained from the present calculation^{‡‡} for Salol using the analysis of Capaccioli et al. along with their results (31). In harmony with the experimental observation we see that the combined MCT/activated RFOT mechanism does give a curvature. The universal curve obtained experimentally was fit by an assumed scaling form containing several adjustable exponents. The present calculations suggest that the true asymptotic region may not yet have been reached experimentally. At the same time, we must be clear that the present calculation is still a loworder one. It is conceivable that a renormalization group treatment, in which the static RFOT barriers themselves are modified by the MCT effects that occur on shorterlength scales, could ultimately give anomalous scaling even in the ultimate asymptotic regime.
The present study enriches our understanding of the dynamics below the ergodic to nonergodic transition temperature, T_{c}^{0}. It predicts the presence of hoppinginduced diffusive dynamics below T_{c}^{0}. It also predicts that this diffusive dynamics continues to exist even below T_{c}^{fit} and provides a dominant contribution to the total dynamics. However, in this temperature range there is an increased contribution from the activated dynamics that leads to the breakdown of the MCT. The diffusive dynamics is found to modify the activated dynamics by redefining the barrier height distribution of the hopping events in a way such that the higher barriers are excluded from the total dynamics. Thus, we find that in this unified theory the two dynamical processes do not just act as parallel channels of relaxation, but they interact with each other and modify each other's behavior, which leads to the facilitation of the total dynamics.
Materials and Methods
Relationship Between τ_{total} and ΔF.
According to the present theory,
Fragility Dependence of β_{total}.
The mean barrier height depends on the configurational entropy and is almost independent of the system; thus, for all the systems (12), we calculate ΔF_{o} by using Salol parameters (s_{fit} = 2.65, T_{K} = 175 K, T_{c}^{0} = 278 K). δΔF is fixed according to the value of D, which is taken from Bohmer et al. (3). For the Salol system τ_{o} is used as a fitting parameter to describe the proper timescale of the dynamics. In this calculation we also vary τ_{o} for each system in such a way that at T = T_{g} the relaxation time is of the order of 100 s. We find that, for a more fragile system, we need a larger value of τ_{o}.
Above T_{c}^{fit}, the dynamics is expected to be described completely by the MCT. Thus, the β_{total} above T_{c}^{0} should be equal to the β_{MCT} value. For the systems like Salol (6), Glycerol (35), and Silica [Silica is known to be a strong liquid with β ≃ 1.] where the value of β_{MCT} at high temperatures is known, we use these values to define β_{MCT}. For the systems where β_{expt} at high temperatures are not known, we use a reasonable value [definitely larger than β_{expt}(T = T_{g})] and also vary it in small amounts along with τ_{o} to get the proper relaxation timescale at T_{g}. T_{c}^{0} is kept same as that for the Salol system. So, our only constraint is that the timescale should be of the order of 100 s at T_{g}.
Acknowledgments
S.M.B. thanks Prof. K. Miyazaki for discussions. This work was supported in part by Department of Science and Technology, J.C. Bose Fellowship, and the National Science Foundation.
Footnotes
 ^{§}To whom correspondence should be addressed. Email: pwolynes{at}ucsd.edu

Author contributions: S.B., B.B., and P.G.W. designed research; S.B., B.B., and P.G.W. performed research; S.B., B.B., and P.G.W. contributed new reagents/analytic tools; S.B., B.B., and P.G.W. analyzed data; and S.B., B.B., and P.G.W. wrote the paper.

The authors declare no conflict of interest.

↵¶ In the previous article (19) there was a mistake in the derivation of the hopping kernel. The correct expression is presented here.

↵‖ The results converge properly for the total time step N = 510 and the initial timestep for the integration is taken to be 10^{−9} ps.

↵†† When the full static barrier height distribution is considered we can write β_{hop}^{static} = [1 + (δΔF/k_{B}T)^{2}]^{−1/2} (4, 30).

↵‡‡ σ_{CRR} is calculated from Eq. 6. In the calculation, the temperature dependence of S_{c} and ΔC_{P} are calculated by using the empirical forms (for details, see text) (26). The temperature dependence of τ_{total} is obtained from the theory (Fig. 1). In our theory, although we predict a temperature dependence of β(T), to compare our results with Capaccioli et al. (31) we have considered, following those authors, β(T) to be independent of T. We fixed it to its value at T = T_{g}. For this plot we take τ_{o} = 10^{−3.9} ps (31).
 © 2008 by The National Academy of Sciences of the USA
References
 ↵
 ↵
 ↵
 ↵
 ↵
 Xia X,
 Wolynes PG
 ↵
 ↵
 ↵
 ↵
 ↵
 Brumer Y,
 Reichman DR
 ↵
 Kob W,
 Nauroth M,
 Sciortino F
 ↵
 ↵
 ↵
 ↵
 ↵
 Bhattacharyya S,
 Bagchi B
 ↵
 Denny RA,
 Reichman DR,
 Bouchaud JP
 ↵
 Chaudhuri P,
 Berthier L,
 Kob W
 ↵
 Bhattacharyya SM,
 Bagchi B,
 Wolynes PG
 ↵
 Gotze W
 Hansen JP,
 Levesque D,
 ZinnJustin J
 ↵
 ↵
 ↵
 ↵
 Greenall MJ,
 Cates ME
 ↵
 ↵
 ↵
 Landau LD,
 Lifshitz EM
 ↵
 ↵
 Bhattacharyya SM,
 Bagchi B,
 Wolynes PG
 ↵
 ↵
 Capaccioli S,
 Ruocco G,
 Zamponi F
 ↵
 Berthier L,
 et al.
 ↵
 DalleFerrier C,
 et al.
 ↵
 ↵
 Leon C,
 Ngai KL
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