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Spatial dimension and the dynamics of supercooled liquids

Edited by H. Eugene Stanley, Boston University, Boston, MA, and approved July 7, 2009 (received for review March 17, 2009)
Abstract
Inspired by recent theories that apply ideas from critical phenomena to the glass transition, we have simulated an atomistic model of a supercooled liquid in three and four spatial dimensions. At the appropriate temperatures and density, dynamic density correlation functions in three and four spatial dimensions correspond nearly exactly. Dynamic heterogeneity, quantified through the breakdown of the Stokes–Einstein relationship, is weaker in four dimensions than in three. We discuss this in the context of recent theories for dynamical heterogeneity. Because dimensionality is a crucially important variable, our work adds a stringent test for emerging theories of glassy dynamics.
When a liquid is cooled rapidly below its melting point (T_{m}), it may form a glass instead of a crystal (1, 2). As the temperature decreases by small fractions of T_{m}, relaxation times increase by orders of magnitude before becoming so large that it becomes impossible to keep the liquid in metastable equilibrium as cooling proceeds. Unlike the case of standard thermodynamic transitions, there are no simple structural changes that accompany this dramatic dynamical arrest (3). Indeed, the atomic structure of a glass is strikingly similar to that of the dense liquid. The approach to vitrification and the nature of the glassy state are topics of immense intellectual challenge to theorists and experimentalists alike. Because glassy dynamics have been observed in diverse fields ranging from materials science to biology, a comprehensive understanding of the glass transition will have broad and perhaps important practical consequences.
Progress toward a complete theory of the glass transition is hindered by the fact that fundamentally different pictures can often explain the same phenomena. Experimental temperature and time dependent data are usually not sufficient to rule out qualitatively different theoretical approaches or scenarios. In the past decade, researchers have considered new observables that reveal the underlying physics more clearly. Dynamic heterogeneity, where the relaxation time scales of particles separated by only a few molecular diameters can differ by orders of magnitude, has now been established as a robust feature of the glass transition (4–8). The quantitative features of dynamic heterogeneity, such as scaling exponents that connect growing dynamical length scales to growing times scales, may eventually provide the key information that singles out a particular picture or theory of the glass transition as correct and complete (7, 9, 10).
Computer simulations can test competing predictions of rival theories. In disordered magnetic systems, termed “spin glasses,” different pictures make different predictions for how various quantities depend on dimensionality (11–14). Clearly, dimensionality is a variable of limited range in experiments but conceptually unlimited in computer simulations. The only limitation on the study of high dimensional systems on the computer is one of computational expense. Perhaps due to this costliness, only very recently have simulations of higher dimensional liquids and glassy systems been reported. For example, Bruning et al. (15) have looked at thermodynamic phenomena in glassy systems of dimensionality ranging from d = 2 to d = 4. Van Meel et al. (16) have compared nucleation phenomena in 3 and 4 dimensions. To the best of our knowledge, no studies of the dynamics of structural glass forming systems have been studied in dimension greater than d = 3. Due to the recent focus on dynamical heterogeneity, the potential role of dimensionality in structural glass formers has come to the fore. Not surprisingly, because aspects of dynamic heterogeneity have analogs in the standard theory of critical phenomena, various pictures of dynamic heterogeneity in supercooled liquids have either an explicit or implicit dependence on spatial dimension (7, 17–24). Here, we take the crucial first steps in developing the study of dimensionality in a supercooled liquid. Current computational restrictions limit the scope of our current work to d = 4, but the system and simulation methodology should extend naturally to higher dimensions, thereby providing an approach to test predictions of emerging theories of the glass transition.
One important hallmark of dynamical heterogeneity in a supercooled liquid is the breakdown of the Stokes–Einstein relationship (19–21, 25–31). In a normal (meanfield) liquid, the product of a liquid's diffusion constant and its viscosity is proportional to temperature. It is often more convenient to study the relationship between the alpha relaxation time, τ, and the diffusion constant, D, than it is to examine the relationship between D and the shear viscosity. In supercooled liquids, D and τ are related by The scaling exponent, ω, is greater than zero and less than one in a dynamically heterogeneous environment. The magnitude of ω determines the strength of the violation. Recent theoretical approaches have made predictions of how different exponents, including ω, associated with dynamical heterogeneity depend on dimension (19–22). Although our study is not comprehensive in scope, we take the first steps to investigate such predictions.
This article is organized as follows. We first outline our atomistic model and choice of thermodynamic parameters used in our simulations. We next discuss how structural and single particle dynamical properties change when going from three to four spatial dimensions. We then describe results for the violation of the Stokes–Einstein relationship as a function of dimensionality. Finally, we present our conclusions. Technical details are included in the the SI Appendix and Fig. S1.
Results
System and Simple Structural Properties.
A system that has been the focus of numerous simulations of supercooled liquid behavior is the binary LennardJones system of Kob and Andersen (KA) (32, 33). Here, for the sake of comparison, we investigate the KA system in 4d, and compare the results to those of the 3d system. Thus, the parameters of the potential are the same in 3d and 4d. We do not consider the 2d KA system because crystallization is too facile.
There are two technical issues that make the comparison of dynamical behavior in dimensions higher than three difficult. The first is the computational expense of adding another dimension. In 4d, the total number of particles increases by almost an order of magnitude because one wants to keep the length of the simulation box the same as in 3d. The number of particles that need to be considered in the calculation of the shortranged “force loop” also increases dramatically in 4d. We developed parallel computing techniques, discussed in the SI Appendix, to surmount these challenges. The second problem arises because one wants to compare 3d and 4d atomistic systems. In lattice models, such as kinetic facilitated models of glassy dynamics, it is straightforward to investigate the role of dimensionality because the thermodynamic properties depend on dimensionality in a simple way (17–19, 20, 34). In contrast, atomistic systems are “offlattice” and have nontrivial phase diagrams as a function of the thermodynamic control parameters, including complications from intervening phases (16). It is not clear a priori whether there is a mapping of thermodynamic control parameters (e.g., the density and temperature) such that dynamical behavior in one spatial dimension corresponds to that in another (35).
To address this problem we scaled the density and found the temperature range over which supercooling occurs. We inferred the dimensional dependence of the supercooling density in 4d from the random close packing (RCP) densities of a one component hard sphere system in different dimensions. Using the RCP densities given by Skoge et al. (36) in 3d and 4d, the standard density of ρ = 1.206 used in the KA model in 3d suggests the use of ρ = 1.474 in 4d. Both densities are 1.16% less than the RCP density in the respective dimension. For this density, we scan temperatures by equilibrating and monitoring the dynamics at reduced temperatures T = 0.55, 0.56, 0.58, 0.6, 0.62, 0.65, 0.7, 0.8, 0.9, 1.0, 2.0, 3.0, 4.0, and 5.0. The dynamics we simulate under these conditions are free of crystallization, aging, and phase separation. We note that a system at ρ = 1.206 (the density used in 3d) undergoes a liquidgas phase separation in 4d at low temperatures. In 3d, the system contains n = 1000 particles at a density ρ = 1.206 with temperatures ranging from T = 0.45 to 5.0. In 4d, there are n = 4096 particles at a density of ρ = 1.474 and temperatures between T = 0.55 and 5.0. As we will demonstrate, at these state points, the dynamics in 4d are remarkably similar to those in 3d.
We begin by discussing structural quantities. All quantities are displayed only for the majority “A” species. Fig. 1 shows g_{AA}(r) in 4d in the high temperature regime where τ ≈ 1, and in the supercooled regime, where τ ≈ 10^{4}. For comparison, Fig. 1 also shows g_{AA}(r) for the 3d KA system in the low temperature regime where τ ≈ 10^{4}. A characteristic feature of structural glass formers is that although dynamics slow down dramatically as supercooling progresses, structural features appear to change only modestly. The difference between the high and low temperature structure in 4d is even less pronounced than it is in 3d. In particular, the characteristic split second peak is barely discernible in 4d, reflecting different packing motifs in 4d compared with 3d (37).
Single Particle Dynamics.
The simplest dynamical quantities of interest in supercooled liquids are twopoint single particle time correlations. We focus on two examples; the mean squared displacement (MSD) 〈r^{2}(t)〉, and the selfintermediate scattering function, F_{S}(k*, t), of the majority particles (38). From these two quantities, we extract the selfdiffusion constant, D and the alpha relaxation time, τ. We follow the standard definition for the alpha relaxation time, extracting it from the 1/e time of the decay of F_{S}(k*, t), where k* is the magnitude of the wavevector for the first peak in S(k) (k* = 7.25 in 3d and 7.98 in 4d). The diffusion coefficient is extracted directly from the slope of the MSD, D = 〈r^{2}(t)〉/2dt, for times that correspond to an MSD > 1. We focus on these two quantities to establish the quantitative correspondence of the dynamics in 3d and 4d for the state points under investigation, and to characterize the breakdown of the Stokes–Einstein violation.
Fig. 2A shows a plot of the MSD. The results are qualitatively similar to 3d, where the MSD is subdiffusive at short times, plateaus at intermediate times, and is linear at long times where the dynamics are diffusive (32, 33). This twostep relaxation is typical in supercooled liquids. Fig. 2B is a semilog plot of the diffusion constant as a function of temperature, along with fits to a power law, log(D) = log(D_{0}) + alogT − Tc and Vogel–Tammann–Fulcher (VTF) function, ln(D) = ln(D_{v}) − Δ/T − T_{v} (1, 2). Our fits are only shown in the regime where they are of similar quality; we find that the VTF fit holds over a broader temperature range than the powerlaw fit. We emphasize that we are not advocating the physics commonly used to motivate such fits, and clearly there is no direct evidence here that there are any divergences in relaxation times (39). Our goal is simply to characterize the temperature dependence of relaxation times by some fiducial reference temperatures, and to then demonstrate that the scaled distances between these reference points are essentially identical in our 3d and 4d systems.
The characteristic temperatures for a supercooled liquid that we focus on are the onset temperature, T_{0}, the critical VTF temperature, T_{v}, and the modecoupling temperature, T_{c}. The onset temperature in 4d, as inferred both by examining the energies of quenched inherent structures and the onset of a “shoulder” in F_{s}(k*,t), is between 0.8 and 1.0. This value is similar to the onset temperature in 3d, where T_{0} ≈ 0.8 − 0.9. Indeed, the relaxation dynamics at temperatures above T = 1.0 in 4d are similar to those in 3d. The “critical” temperatures are also only slightly different in 3d and 4d. In 3d, our fitting procedure yields T_{v} = 0.32 (4) and T_{c} = 0.42 (2)^{†}. In 4d, the values are T_{v} = 0.37 (2) and T_{c} = 0.49 (2). The relative difference, given by the ratio (T_{c} − T_{v})/T_{v} is nearly the same in 3d and 4d. This, along with direct examination of the slope of the VTF curve at low temperatures, indicates that characteristics of the dynamics such as fragility are not altered for the state points we have chosen to compare the systems in 3d and 4d.
We now compare dynamics in 3d and 4d further by illustrating a remarkable similarity between the rescaled density fluctuations in different dimensions. Fig. 2C shows F_{s}(k*,t) in 4d. As in 3d, the relaxation dynamics are twostep, and slow dramatically below the onset temperature. In 3d, the KA system is known to obey timetemperature superposition, where the temperature dependence of F_{s}(k*,t) follows the temperature dependence of τ(T), even in the β regime. Fig. 2D shows F_{s}(k*, t/τ(T)) as a function of t/τ for both 3d and 4d. These data collapse to a master curve that is nearly identical in the two different spatial dimensions. Although the temperature values in 4d differ from those in 3d, the dynamical density fluctuations are thus directly comparable. The only control variable that is different between them is the density and, accordingly, the value of k*. Our 4d model indeed resembles our 3d model, but with supercooled temperature range that is slightly shifted toward higher values in 4d compared with 3d. Given this remarkable mapping of the single particle dynamics, we are in position to determine what role dimensionality has on Stokes–Einstein breakdown in the comparison of 3d and 4d.
Dynamical Heterogeneity and Stokes–Einstein Decoupling.
We finally turn to perhaps the most interesting and important aspect of glassy behavior, namely those phenomena associated with dynamically heterogeneous relaxation. We first demonstrate that in our 4d system the qualitative hallmarks of dynamical heterogeneity are present. In Fig. 3 we present the dynamical susceptibility, χ_{4}(t), calculated as the mean squared fluctuations in F_{s}(k*,t) (7, 9, 10, 17, 31). Similar to the behavior known in 3d, the peak of χ_{4}(t) grows as T lowers and goes through a maximum near t = τ. We also show the van Hove functions plotted on a logarithmic scale that make clear the palpable exponential tails in the distribution. These tails are usually associated with heterogeneous particle hopping dynamics (40–42). These features, clearly present in our 4d system, signify dynamical heterogeneity in lower dimensional systems.
Having established some qualitative features associated with dynamical heterogeneity, we turn to the quantitative role that dimensionality itself plays. Indeed, because dynamical heterogeneity may be viewed as a fluctuationdriven process akin to a type of critical phenomena, it is natural to investigate how the exponents associated with scaling of heterogeneous quantities depend on the spatial dimension (18, 19, 20, 22). We focus here the breakdown exponent, ω of the Stokes–Einstein relationship in Eq. 1. In meanfield systems, ω is zero (19, 20, 22, 31). Fig. 4 shows the diffusion constant as a function of τ. Although the values of the parameters in the power law fits to τ and D alone can depend strongly on the temperature range used in the fit, the plot of D vs. τ should not. First, power law fits of D vs. τ do not deviate significantly from a powerlaw form at low temperatures for our data. Furthermore, in a recent study on the KA system in 3d, Flenner and Szamel have demonstrated that standard powerlaw fits of the form τ = τ_{0}T − T_{c}^{−γ(T)} and D = D_{0}T − T_{c}^{α(T)} yield exponents α and γ that depend sensitively on the temperature range used (43). In this form, however, the dependence of D on τ is D ∼ τ^{−α(T)/γ(T)}, and although α and γ can vary with temperature range by as much as 60%, the ratio of α/γ changes by <10%; even at the lowest temperatures studied. We find that the breakdown exponent is 0.160 (8) in 3d and 0.14 (1) in 4d. Assuming that τT is a surrogate for η,^{‡} the Stokes–Einstein relationship in Eq. 1 can be written in a more standard form, D/T ∼ η^{−1+ν}. In this form, we find an exponent ν with the value 0.18 (1) in 3d and 0.15 (1) in 4d. In either case, the breakdown exponent is smaller in 4d than it is in 3d by ≈15%. We used the bootstrap method, discussed in the SI Appendix, to assess the significance of our results. Fig. 4Inset shows the likelihood distributions of ω in 3d and 4d obtained from bootstrap Monte Carlo (44). Significance can be inferred from these distributions. In particular, the righttailed t test rejects the hypothesis that the mean of the bootstrapped data for ω in 3d is less than the mean of ω in 4d at a confidence level >99% (44). Thus, we can conclude that the decrease of the violation exponent in 4d is statistically significant.
Conclusions
In this article, we have simulated the supercooled KA system in four spatial dimensions and compared the evolution of dynamical quantities to the same system in three dimensions. We have carefully matched the systems by adjusting the density and temperature so that the dynamic behavior and temperature dependence of density fluctuations matches closely between 3d and 4d. We find that although dynamics of density fluctuations are matched (by construction), static structural properties are altered by the dimensional dependence of packing motifs.
With regard to Stokes–Einstein violations, several recent approaches do address the role of dimensionality in determining the magnitude of departures from meanfield behavior. Jung et al. (45) have investigated decoupling phenomena in kinetic facilitated models (19, 20). For the East model, a model of fragile glass behavior, they have demonstrated that dimensionality plays essentially no role in altering the violation exponent, ω. This fact arises from the quasione dimensional nature of excitation lines, a feature which is unchanged in higher spatial dimension. Our results are not entirely inconsistent with this, given that the change in violation exponent is not drastic. However, there is a clear, statistically significant change in ω, which is presumably larger than what would be predicted by the East model. It would be most interesting to see how Stokes–Einstein violations evolve with dimensionality in other kinetic facilitated models, such as the Kob–Andersen lattice model, which do not incorporate explicit directional motion into the kinetic rules (46).
Biroli and Bouchaud have put forth a different approach to calculating the breakdown of the Stokes–Einstein relationship between the diffusion constant and the viscosity (22). In their theory, the violation of the Stokes–Einstein relation in mildly supercooled liquids results from critical fluctuations around the (avoided) meanfield critical point of modecoupling theory. For a given value of the upper critical dimension d_{c}, the theory predicts that the ratio of the violation exponents in 3d and 4d is ν(3)/ν(4) = (d_{c} − 3)/(d_{c} − 4). Using the value d_{c} = 8 for MCT, as determined from their theory, the expected ratio of violation exponents is ν(3)/ν(4) = 1.25 From our data we find from direct simulation, ν(3)/ν(4) = 1.2(1) Although our results are entirely consistent with these predictions, we voice several words of caution regarding this agreement. First, it is not clear how well such a prediction should work far from the upper critical dimension, nor is it clear whether the modecoupling picture should hold without interference from local activated processes in the regimes simulated.^{§}
Several other theoretical approaches such as the random firstorder theory (RFOT) (23), the frustrationlimited domain theory (47), and the recent theory of Schweizer and coworkers (48) all make predictions concerning the influence of spatial dimensionality on supercooled liquid dynamics. It would be most interesting to test these predictions via the approach taken here. We hope that our work stimulates effort along these lines. Although quite computationally challenging, extension to still higher spatial dimensions would be quite valuable as a means of sensitively testing predictions of a variety of theories of relaxation in supercooled liquids.
In conclusion, we have simulated the canonical KA system in 4d. By carefully choosing the density and temperature in 4d, we can meaningfully compare the KA system in 4d to that in 3d. We find that there is a decrease in the violation exponent characterizing the breakdown of the Stokes–Einstein relationship in 4d relative to 3d. Although this exponent differs by only ≈15% between 3d and 4d, our results are statistically significant. Although seemingly modest, a 15% difference in the violation exponent would change the absolute size of decoupling at the glass transition by a factor of 2 to 3. Our results suggest a finite upper critical dimension, which is consistent with theories that relate the Stokes–Einstein violation to criticallike fluctuations. Quantitatively, the changes we observe are in the range predicted from those theories. We have shown that dimensionality is indeed an important and accessible variable in atomistic simulations of supercooled liquids, and suspect that the 4d KA system and simulation methodology presented here will be quite useful for testing different theoretical predictions of dynamic heterogeneity.
Acknowledgments
We thank Ludovic Berthier, Giulio Biroli, JeanPhilippe Bouchaud, Andreas Heuer, and Grzegorz Szamel for interesting discussions. This work was funded by National Science Foundation Grant CHE0719089
Footnotes
 ^{1}To whom correspondence should be addressed. Email: drr2103{at}columbia.edu

Author contributions: D.R.R. designed research; J.D.E. performed research; J.D.E. contributed new reagents/analytic tools; J.D.E. and D.R.R. analyzed data; and J.D.E. and D.R.R. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/cgi/content/full/0902888106/DCSupplemental.

↵† We used temperatures between 0.47 and 0.9 for the power law fit and T = 0.45 to T = 1.0 for the VFT fit in 3d. The parameters for the power law fit depend on the range of the fitting temperature used, but as we argue in the text, the SE violation is not sensitive to this range as long as the temperatures used are below the onset temperature. Error bars correspond to one standard deviation, or a 68% confidence interval.

↵‡ It is impractical to calculate η by nonequilibrium simulations in the 4d system. Instead, we can again use τ as a proxy for η. For soft spheres, the relationship η ∼ T τ was found to hold for over more than four decades (50).

↵§ A rather interesting additional quantitative fact emerges in our data with regard to the precise numerical value of the violation exponents in 3d and what MCT would predict. According to ref. 31 above, the violation exponent is expected to have a relationship to the exponent z extracted from 4point functions of the type discussed in refs. 7, 9, 10 via ω=β/z, and β a number numerically determined from direct simulation of diffusive behavior. For the system discussed here in 3d, β=1.6. As discussed in ref. 49, for the same system in 3d, MCT predicts z=9.2. Thus, one would expect a violation exponent ω=0.17, which is very close to what we have measured directly. We have not carried out this analysis in 4d (which would require recalculating β) or in other systems, and thus we have no way to determine whether this is merely coincidental.
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