New Research In
Physical Sciences
Social Sciences
Featured Portals
Articles by Topic
Biological Sciences
Featured Portals
Articles by Topic
 Agricultural Sciences
 Anthropology
 Applied Biological Sciences
 Biochemistry
 Biophysics and Computational Biology
 Cell Biology
 Developmental Biology
 Ecology
 Environmental Sciences
 Evolution
 Genetics
 Immunology and Inflammation
 Medical Sciences
 Microbiology
 Neuroscience
 Pharmacology
 Physiology
 Plant Biology
 Population Biology
 Psychological and Cognitive Sciences
 Sustainability Science
 Systems Biology
Growing length and time scales in glassforming liquids

Edited by H. Eugene Stanley, Boston University, Boston, MA, and approved January 9, 2009 (received for review November 6, 2008)
Abstract
The glass transition, whereby liquids transform into amorphous solids at low temperatures, is a subject of intense research despite decades of investigation. Explaining the enormous increase in relaxation times of a liquid upon supercooling is essential for understanding the glass transition. Although many theories, such as the Adam–Gibbs theory, have sought to relate growing relaxation times to length scales associated with spatial correlations in liquid structure or motion of molecules, the role of length scales in glassy dynamics is not well established. Recent studies of spatially correlated rearrangements of molecules leading to structural relaxation, termed “spatially heterogeneous dynamics,” provide fresh impetus in this direction. A powerful approach to extract length scales in critical phenomena is finitesize scaling, wherein a system is studied for sizes traversing the length scales of interest. We perform finitesize scaling for a realistic glassformer, using computer simulations, to evaluate the length scale associated with spatially heterogeneous dynamics, which grows as temperature decreases. However, relaxation times that also grow with decreasing temperature do not exhibit standard finitesize scaling with this length. We show that relaxation times are instead determined, for all studied system sizes and temperatures, by configurational entropy, in accordance with the Adam–Gibbs relation, but in disagreement with theoretical expectations based on spinglass models that configurational entropy is not relevant at temperatures substantially above the critical temperature of modecoupling theory. Our results provide new insights into the dynamics of glassforming liquids and pose serious challenges to existing theoretical descriptions.
Most approaches to understanding the glass transition and slow dynamics in glass formers (1–10) are based on the intuitive picture that the movement of their constituent particles (atoms, molecules, polymers) requires progressively more cooperative rearrangement of groups of particles as temperature decreases (or density increases). Structural relaxation becomes slow because the concerted motion of many particles is infrequent. Intuitively, the size of such “cooperatively rearranging regions” (CRR) is expected to increase with decreasing temperature. Thus, the above picture naturally involves the notion of a growing length scale, albeit implicitly in most descriptions. The notion of such a length scale, related to the configurational entropy S_{c} (see Methods), forms the basis of rationalizing (1, 6, 7) the celebrated Adam–Gibbs (AG) relation (1) between the relaxation time and S_{c}.
More recently, a number of theoretical approaches have explored the relevance of a growing length scale to dynamical slow down (5, 7, 9). A specific motivation for some of these approaches arises from the study of heterogeneous dynamics in glass formers (11–14). In particular, computer simulation studies (12–14) have focused attention on spatially correlated groups of particles that exhibit enhanced mobility, and whose spatial extent grows upon decreasing temperature. The spatial correlations of local relaxation permits identification of a dynamical (time dependent) length scale, ξ, through analysis of a 4point correlation function first introduced by Dasgupta, et al. (15) (see Methods), and the associated dynamical susceptibility χ_{4} (16, 17). These quantities have been studied recently via inhomogeneous modecoupling theory (IMCT) (5) and estimated from simulation and experimental data (5, 10, 18–21).
The method of finitesize scaling, used extensively in numerical studies of critical phenomena (22), is uniquely suited for investigations of the presence of a dominant length scale. This method involves a study of the dependence of the properties of a finite system on its size. We study a binary mixture of particles interacting via the Lennard–Jones potential (23), originally proposed as a model for Ni_{80}P_{20}, and widely studied as a model glass former. We perform constant temperature molecular dynamics simulations at a constant volume [see Methods and (24) for details], for 7 temperatures, and up to a dozen different system sizes for each temperature. For each case, we calculate the dynamic susceptibility χ_{4}(t) as the second moment of the distribution of a correlation function Q(t), which measures the overlap of the configuration of particles at a given time with the configuration after a time t (see Methods).
Results
From previous work, it is now wellestablished that χ_{4}(t) has nonmonotonic time dependence, and peaks at a time τ_{4} that is proportional to the structural relaxation time τ. Such behavior is shown in Fig. 1A Inset. In Fig. 1A, we show the peak values χ_{4}^{p}≡χ_{4}(τ_{4}) vs. system size (number of particles) N for a range of temperatures. At each temperature, χ_{4}^{p} is an increasing function of N, saturating at large N. The saturation occurs at a larger value of N at lower temperatures. This is the expected finitesize scaling behavior of a quantity whose growth with decreasing temperature is governed by a dominant correlation length that increases with decreasing temperature.
We have estimated the correlation length ξ from finitesize scaling of χ_{4}^{p}(T,N), which also involves estimating the value of χ_{4}^{p} as N → ∞. Because the latter estimation is a potential source of error in estimating ξ, we employ the Binder cumulant of the distribution of Q(τ_{4}) to estimate ξ. The Binder cumulant (25), defined (see Methods) in terms of the 4th and second moments of the distribution, vanishes for a Gaussian distribution, whereas it acquires negative values for bimodal distributions. The Binder cumulant has been used extensively in finitesize scaling analysis in the context of critical phenomena, owing to its very useful property that in systems with a dominant correlation length ξ, it is a scaling function only of L/ξ (or equivalently, of N/ξ^{3}), where L is the linear dimension of the system. The distributions themselves are shown in Fig. 1B Inset, for 2 different system sizes for temperature T = 0.47. We see that the distribution is unimodal for the large system size of N = 1600 whereas it is strongly bimodal for the small system size of N = 150. The same trend is observed as temperature is decreased for a fixed size of the system. The data collapse of the Binder cumulant, from which we extract the correlation length ξ(T), is shown in Fig. 1(b). The collapse observed is excellent, confirming that the growth of χ_{4}^{p} with decreasing T is governed by a growing dynamical correlation length. The values of ξ obtained from this scaling analysis are consistent with less accurate estimates obtained from a similar analysis of the N dependence of χ_{4}^{p}(T,N), and from the wavevector dependence of the 4point dynamic structure factor S_{4}(q,τ_{4}) (see, e.g., ref. 5). Because the data collapse of the Binder cumulant is not affected by a uniform rescaling of L/ξ for all temperatures, we can determine ξ(T) only up to an unknown multiplicative constant that is common to all of the temperatures. The unknown multiplicative constant has been fixed so that the ξ value from finitesize scaling matches the value obtained from analysis of S_{4}(q,τ_{4}) at one temperature. Estimated values of χ_{4}^{p} as N → ∞ compare very well with the q → 0 limit of S_{4}(q,τ_{4}), up to a proportionality constant (described elsewhere).
The value of ξ increases from 2.1 to 6.2 as T decreases from T = 0.70 to T = 0.45. We find that both ξ and the asymptotic, N → ∞ value of χ_{4}^{p} deviate from power law behavior as the critical temperature T_{MCT} of modecoupling theory (T_{MCT} ≃ 0.435 in our system) is approached (consistently with previous observations). However, the powerlaw relationship between χ_{4}^{p} and ξ predicted in IMCT is satisfied by our data. Because the range of the measured values of ξ is small, it is difficult to obtain accurate estimates of the exponents of these power laws.
Next we consider the dependence of the relaxation time τ on T and N. For each case, we calculate the relaxation time from the decay of 〈Q(t)〉. The results for τ are displayed in Fig. 2, which shows that τ increases as the temperature decreases, as expected. However, the observed increase in τ with decreasing N for small values of N at fixed T is not consistent with standard dynamical scaling for a system with a dominant correlation length (e.g., near a critical point): dynamical finitesize scaling would predict a decrease in τ as the linear dimension L of the system is decreased below the correlation length ξ*. Similar finitesize effects on relaxation times have been observed in previous simulations of realistic glass formers (e.g., ref. 26) but have not been analyzed in detail. Due to computational limitations, our simulations cover a (relatively) hightemperature regime, the lowest temperature considered being slightly above the modecoupling temperature T_{MCT} for this system. However, it is clear from Fig. 2 that the N dependence of τ becomes stronger and persists to larger values of N as the temperature is decreased. Therefore, the deviations of the N dependence of τ from standard dynamical finitesize scaling are expected to be more pronounced at temperatures near the actual glass transition. The N dependence of τ shown in Fig. 2 is opposite to that found in finitesize scaling studies of some spinglass models (27) but similar to that found in other studies (ref. 28 and Biroli G, personal communication).
Fig. 3Inset shows the largeN value of τ plotted as a function of (bottom curve) the correlation length ξ on a doublelog scale, and (top curve)
Motivated by the AG relation (1), τ∝ exp
Discussion
A central role for the configurational entropy, along with an analysis of a length scale relevant to structural relaxation, are the content of the random first order theory, developed by Wolynes and coworkers (7). According to RFOT, the length scale of dynamical heterogeneity is the “mosaic length” ξ_{m} that represents the critical size for entropy driven nucleation of a new structure in a liquid. Meanfield arguments based on known properties of infiniterange models suggest that the RFOT mechanism is operative for temperatures lower than T_{MCT}. In this regime, the dynamics of the system is activated, with the relaxation time expected to vary as τ = τ_{0} exp
RFOT focuses on behavior near the glass transition, and in the limiting case of the spin glass models where theoretical perditions are available, configurational entropy plays no role in the behavior of the system above the modecoupling temperature. However, there have indeed been attempts to extend the RFOT analysis to temperatures above the modecoupling temperature (30–32) and to estimate a mosaic length scale at such temperatures, and we thus compare our results with predictions arising from these analyses. Stevenson, et al. (30) have considered the change in morphology of rearranging regions above the modecoupling temperature, and correspondingly the dependence of relaxation times on configurational entropy. The predicted dependence of relaxation times on configurational entropy differs from the Adam–Gibbs form, whereas our results strikingly confirm the Adam–Gibbs form. Franz and Montanari (31) have estimated a mosaic length scale in addition to a heterogeneity length scale, and have discussed the crossover in the dominant length scale near the modecoupling temperature. However, this analysis does not contain explicit predictions regarding the relevance of the configurational entropy at temperatures higher than the modecoupling temperature.
Our observation that the configurational entropy predicts the relaxation times in accordance with the AG relation for all of the temperatures and system sizes we study poses serious challenges to current theoretical descriptions based on the analogy with the behavior of meanfield models. Although the relevance of the configurational entropy at high temperatures has been observed in earlier simulation studies and analyses based on the inherent structure approach (24, 33, 34), we emphasize that a theoretical analysis that satisfactorily explains such dependence is not at hand at present, and our results concerning the robustness of the Adam–Gibbs relation in finite systems highlights further the challenge to existing theoretical descriptions. Indeed, earlier work (28, 35) has highlighted the puzzle that aspects of the energy landscape and modecoupling theory descriptions appear to apply over a significant temperature range side by side, rather than in neatly separated temperature regimes as expected from mean field theoretical descriptions. Our results emphasize the importance of understanding such overlap of temperature regimes and relaxation mechanisms, which has recently been addressed in (32). Equally importantly, our results indicate that the length scale that describes the growth of dynamical heterogeneity in IMCT may not play the central role attributed to it in recent analyses, and highlights the necessity to understand the role of other relevant length scales, along the lines of the analysis in ref. 31.
Methods
Simulation Details.
The system we study is a 80:20 (A:B) binary mixture of particles interacting via the Lennard–Jones potential:
where α,β ∈ {A,B} and ε_{AB}/ε_{AA} = 1.5, ε_{BB}/ε_{AA} = 0.5, σ_{AB}/σ_{AA} = 0.80, σ_{BB}/σ_{AA} = 0.88, masses m_{A} = m_{B}. The interaction potential is cutoff at 2.50σ_{αβ}. Length, energy and time are reported in units of σ_{AA}, ε_{AA} and
Dynamics.
Dynamics is studied via a 2 point correlation function, the overlap Q(t), where ρ(r→,t_{0}) etc are spacetime dependent particle densities, w(r) = 1, if r ≤ a and zero otherwise, and averaging over the initial time t_{0} is implied. The use of the window function [a = 0.30] treats particle positions separated due to small amplitude vibrational motion as the same. The second part of the definition is an approximation that uses only the selfterm, which we have verified to be reliable (see ref. 17 for details). The structural relaxation time τ is measured by a stretched exponential fit of the longtime decay of Q(t).
The fluctuations in Q(t) yields the dynamical susceptibility: Ref. 17 shows that χ_{4}(t) reaches a maximum for times τ_{4} which are proportional to the structural relaxation time τ. We report the values of χ_{4}^{p} ≡ χ_{4}(t = τ_{4}).
The Binder cumulant, which we use for finitesize scaling, is defined as B(N,T) = 0, if the distribution P(Q(τ_{4})) is Gaussian, and is a scaling function of ξ/L only (where L is the linear size of the system, and ξ is the correlation length), without any prefactor.
Configurational Entropy.
S_{c}, the configurational entropy per particle, is calculated as the measure of the number of distinct local energy minima, by subtracting from the total entropy of the system the “vibrational” component: Details of the calculation procedure are as given in ref. 24.
Acknowledgments
We thank Jack Douglas, Pablo Debenedetti, Frank Stillinger, Biman Bagchi, Peter Wolynes, Francesco Zamponi and especially Silvio Franz, Giulio Biroli, and David Reichman for useful discussions and comments on the manuscript. This work was supported in part by the Swarnajayanti Fellowship (S.S.), the J. C. Bose Fellowship (C.D.) and the Department of Science and Technology, Government of India through a grant to the Centre for Computational Materials Science, Jawaharlal Nehru Centre for Advanced Scientific Research.
Footnotes
 ^{1}To whom correspondence should be addressed. Email: cdgupta{at}physics.iisc.ernet.in

Author contributions: C.D. and S.S. designed research; S.K. performed research; S.K., C.D., and S.S. analyzed data; and C.D. and S.S. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

↵* We have checked from simulations with a shorter cutoff for the interaction potential that the observed N dependence of τ is not due to the cutoff being larger than L/2.

↵† A static“mosaic length” has recently been estimated in computer simulations (29), whose magnitude of change in a comparable range of temperatures is similar to that of ξ.
References
 ↵
 ↵
 ↵
 ↵
 ↵
 Biroli G,
 Bouchaud JP,
 Miyazaki K,
 Reichman D R
 ↵
 ↵
 ↵
 ↵
 ↵
 Whitelam S,
 Berthier L,
 Garrahan J P
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 Donati C,
 Franz F,
 Parisi G,
 Glotzer SC
 ↵
 Berthier L
 ↵
 Berthier L,
 et al.
 ↵
 DalleFerrier C,
 et al.
 ↵
 ↵
 Privman V
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 Brumer Y,
 Reichman D R
 ↵
 ↵
 ↵
 ↵
 Bhattacharyya SM,
 Bagchi B,
 Wolynes PG
 ↵
 ↵
 ↵
Citation Manager Formats
Sign up for Article Alerts
Article Classifications
 Physical Sciences
 Physics