Spatially Heterogeneous Dynamics in the Density Scaling Regime

Letter pubs.acs.org/JPCL

Spatially Heterogeneous Dynamics in the Density Scaling Regime: Time and Length Scales of Molecular Dynamics near the Glass Transition A. Grzybowski,* K. Koperwas, K. Kolodziejczyk, K. Grzybowska, and M. Paluch Institute of Physics, University of Silesia, Uniwersytecka 4, 40-007 Katowice, Poland S Supporting Information *

ABSTRACT: A fundamental problem of glass transition physics is to find a proper relation between length and time scales of molecular dynamics near the glass transition. Until now, this relation has been usually expected as a single variable function, for instance, as a consequence of the suggested direct relation between the structural relaxation time τ and the correlation volume defined by the maximum of the four-point correlation function max χmax 4 . Based on high pressure data analyses, we show that it is not the case, because χ4 evaluated from its estimate based on the enthalpy fluctuations cannot be, in general, a single variable function of τ. For a wide class of real and model supercooled liquids, the molecular dynamics of which obeys a density scaling law at least to a good approximation, we argue that the important relation between the length and time scales that characterize molecular motions near the glass transition is controlled by a density factor, the exponent of which is a measure of the observed decoupling between τ and χmax 4 . This finding substantially changes our understanding of molecular dynamics near the glass transition. SECTION: Glasses, Colloids, Polymers, and Soft Matter

A

conditions. Depending on the assumed representation, the height of this peak, χmax 4 , informs us of a typical correlation volume or a typical number of dynamically correlated molecules, Nc, at a given temperature T and pressure p. In 2−6 general, Nc ∼ χmax and even an approximate correspond4 , ence between these quantities has been considered3,9−12 in the case of many systems, including bulk samples of simple van der Waals liquids. In the volumetric interpretation, one can expect2 that a corresponding characteristic correlation length obeys the ψ proportionality, χmax ∼ (ξmax 4 4 /a) , where a is a molecular size. For instance, according to the simplest approximation for bulk systems, ψ = 3 and a = υ1/3 m , where υm is the molecular volume. Nevertheless, in our further discussion, we focus mainly on the of the characteristic correlation volume, over measure χmax 4 which structural relaxation processes are correlated.4 In this way, we maintain a general level of our analysis, making it max independent of particular relationships between χmax 4 , ξ4 , and Nc . The crucial achievement of the dynamic susceptibility formalism is the reliable determination of the temperature evolution of χmax for systems approaching the glass transition. 4 For both simulation and experimental data, it has been found increases with decreasing temperature.2−4 This result that χmax 4 is really meaningful, because it confirms that the correlation length scale grows when a liquid upon isobaric cooling is

prominent trend in the study of supercooled liquids invokes the assumption that their molecular motions have a heterogeneous character. This point of view has been initiated by Adam and Gibbs,1 who suggested that interacting molecules of supercooled liquids form cooperatively rearranging regions, the size of which should increase upon approaching the glass transition. Recently, this idea has been considerably developed2−7 by using a powerful formalism of a four-point timedependent correlation function, χ4(t), suggested to study the heterogeneous dynamics near the glass transition by Kirkpatrick and Thirumalai.8 It has been achieved despite the fact that the function χ4(t) for real glass formers is usually only estimated. Since the experimental measurements of χ4(t) are very complex due to a need for detecting a nonlinear response of the examined sample, it has been suggested2,3 to perform a derivative analysis of a two-point correlation function Φ(t) like the stretched exponential function, i.e.,Φ(t) = exp[−(t/τ)β], which involves temporal correlations in the interval between 0 and t, where τ is the structural relaxation time and the stretching exponent, 0< β ≤ 1. In theoretical and simulation studies, the dynamic susceptibility χ4(t) that involves both temporal and spatial correlations in the interval between 0 and t arising at two points, 0 and r, is regarded as a precise measure of the dynamic heterogeneity. This measure quantifies a correlation volume, i.e., the volume that is characteristic for the correlated motions, and consequently the four-point time dependent correlation length ξ4(t) via the relation, χ4(t) ≅ A(t)(ξ4(t))ψ.6 The peak of the dynamic susceptibility emerges at a time scale on the order of the relaxation time τ, which is typical for a given material in considered thermodynamic © XXXX American Chemical Society

Received: September 24, 2013 Accepted: November 25, 2013

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Figure 1. The combined power law density scaling of τ and χmax 4 for real van der Waals supercooled liquids. (a,b) Plots of dielectric structural relaxation times versus density for BMPC and OTP. The insets in panels (a) and (b) present the scaling τ(ργ/T) with the values of γ found from the fitting shown in these panels. (c,d) Plots of the degree of the dynamic heterogeneity versus density for BMPC and OTP evaluated at each (T,ρ) at γz which τ(T,ρ) is determined from experimental measurements. The insets in panels (c) and (d) present the scaling χmax 4 (ρ /T) with the values of γχ found from the fitting shown in these panels. (e,f) Isochronal pressure dependences of the degree of the dynamic heterogeneity for BMPC and OTP. In panels (e) and (f), open symbols denote the evaluated values, and the other symbols indicate values predicted from eq 5b. In panels a−d, the 2 measure of the quality of the nonlinear fits of τ andχmax 4 respectively to eqs 3 and 4 is the adjusted R coefficient defined in the caption to Figure S.1 in the Supporting Information.

is smaller than that caused by isobaric cooling.12,13 A consequence of the latter is nontrivial, because it implies that the degree of the dynamic heterogeneity, χmax 4 , decreases with

approaching the glass transition. Very recently, we have shown that the isothermal squeezing of the liquid also increases its degree of the dynamic heterogeneity, χmax 4 , although this effect 4274

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power law density scaling idea for τ, i.e., τ = g(ργ/T) with a material constant γ implies the scaling condition,26,39 ργ/T = Cτ at τ = const, where Cτ depends only on τ. According to this max manner, the scaling law for χmax = h(ργz/T) with a 4 , i.e., χ4 material constant γχ leads to the condition, ργz/T = Cχ at χmax 4 = const, where Cχ depends only on χmax 4 . Thus, we can establish a criterion for the combined power law density scaling of τ and Δγ χmax = Cτ/Cχ with Δγ = γ − γχ, where Cτ = const and Cχ 4 , ρ varies with changing density along an isochrone. Then, exploiting the equations, τ = g(ργ/T) and ργz/T = h−1(χmax 4 ), we arrive at the criterion for the combined power law density scaling of τ and χmax 4 , and we arrive at the nontrivial relation between the characteristic time scale and the measure of the characteristic correlation volume,

increasing pressure at a constant structural relaxation time. What is more, this finding leads to a fundamental question, What is the proper relationship between the structural relaxation time and the degree of the dynamic heterogeneity? In this Letter, we show that such relations can be found for real and model supercooled liquids in the density scaling regime. In recent years, much effort has been put into finding a proper relation between the characteristic scales of time and length for molecular dynamics (MD) near the glass transition. For simple binary fluids, the relation τ ∼ ξz has been discussed14 for wavelengths longer than the correlation length ξ. Similarly, for kinetically constrained models, it has been suggested15−17 that τ ∼ Ξζ, where Ξ is the spacing between mobile elements. Thus, simple models of supercooled liquids like the Kob−Andersen binary Lennard-Jones (KABLJ) mixture,18 we can expect that ln τ = f(ln ξ). However, for real glass formers, different models of the temperature dependence of the structural relaxation time are considered, which are characterized by a general assumption that the energy barrier is proportional to ξψ.5,6,17,19−23 According to another quite common point of view, the structural relaxation time can simply be a function of a single variable such as the characteristic correlation volume or its corresponding length scale. For instance, comparing χmax with τ for several glass 4 formers at ambient pressure, Berthier et al.3,4 has even observed a crossover from algebraic, χ4 ∼ τz, to logarithmic, χ4 ∼ exp(τψ), growth of dynamic correlations with increasing τ. However, 12,13 from the mentioned high pressure study of χmax we can 4 , draw a conclusion that some previous results validated by using experimental data at ambient pressure cannot be held if we consider the complete thermodynamic space. This is because no single variable function, neither f(χ4) nor f(ξ4), is able to describe the decrease in the degree of the dynamic heterogeneity with increasing pressure at τ = const. Hence, the sought after relation between τ and χmax should be more 4 complex. To maintain a sufficiently unified level of theory in search of 24,25 a proper dependence of τ on χmax the 4 , it is worth invoking density scaling of MD near the glass transition. In general, this concept relies on many observations for important classes of glass formers, including mainly van der Waals supercooled liquids and polymer melts, which show that primary relaxation times (or viscosities) can be plotted onto one master curve versus a scaling variable ; (ρ)/T.26,27 In the most tempting case, ; (ρ) = ργ, where the scaling exponent γ is a material constant independent of thermodynamic conditions. The exponent γ has been suggested28−35 to be straightforwardly related to the exponent, m = 3γ, of the repulsive inverse power law (IPL) term that constitutes the main part of an effective short-range intermolecular potential, which involves attractive interactions as a small background. The authors of previous combined studies of the power law density scaling and the dynamic heterogeneity have been trying to argue9,36 that these quantities for both experimental and simulation data can be scaled with the same value of γ but in terms of different scaling functions τ = g(ργ/T) and χmax = h(ργ/T). This point of view 4 should be revisited in context of the change in χmax 4 (p) in isochronal conditions, which has been observed11−13,37,38 for data obtained from different experimental techniques, because a composition of the function g and the inverse function h−1 results in a single variable function τ(χmax 4 ). Thus, we postulate can be scaled with the scaling exponent γχ which in that χmax 4 general differs from γ that scales structural relaxation times. The

τ = g (ρΔγ f (χ4max ))

with

Δγ = γ − γχ

(1)

where the scaling exponents γ and γχ are material constant independent of thermodynamic conditions, and f = h−1 is a single variable function of χmax 4 . In the general case of the density scaling, any constant exponent γ does not enable us to achieve the power law density scaling of τ, and we need to employ other usually more complex functions of density to do that.40,41 It is reasonable to assume that the same problem concerns the density scaling of χmax in the extremely wide pressure range. Then, the criterion 4 ρΔγ = Cτ/Cχ for the combined density scaling of τ and χmax 4 should be extended to 9 (ρ) = Cτ/Cχ with a density function, 9 (ρ) = ; τ(ρ)/; χ(ρ), where ; τ(ρ)/T and ; χ(ρ)/T are the density scaling variables for τ and χmax 4 , respectively. Thus, eq 1 can be generalized as follows: τ = g (9(ρ)f (χ4max ))

with

9(ρ) = ;τ(ρ)/; χ (ρ)

(2)

χmax 4 (; χ(ρ)/T),

if the density scaling laws, τ(; τ(ρ)/T) and are obeyed. Nevertheless, we limit the further tests to the case of the simple scaling variables, ργ/T and ργz/T, because experimental data that comply with the power law density scaling rule are commonly accessible in contrast to those measured at very high pressures sufficient to reveal the more complex density scaling behavior. The suggested relation can be verified by checking whether the observed decrease in χmax 4 (p) at τ = const can be reproduced by using eq 1. To perform the test we first exploit experimental data for two typical van der Waals liquids BMPC and OTP (see Materials and Methods). It is already known26,27 that structural relaxation times of BMPC and OTP obey the power law density scaling, which is illustrated in Figures 1a,b and described by using the ργ/T-scaling version of the Avramov entropic model,42,43 τ = τ0 exp[(Aρ γ /T )D ]

(3)

where τ0, A, D, γ are fitting parameters. In this way, we choose an acknowledged representation26 of the scaling function g in eq 1. To verify whether the degree of the dynamic heterogeneity χmax complies with a similar power law density 4 scaling rule with a scaling exponent γχ, we postulate that the scaling function for χmax can be given by 4 χ4max = (χ4max )0 (A χ ρ γχ /T )Dχ

(4)

where Aχ, Dχ, and γχ are fitting parameters, the number of which can be diminished by introducing the auxiliary 1/Dz parameter bχ = (χmax Aχ. It is worth noting that the 4 )0 assumed form of eq 4 is chosen to extend the logarithmic (χmax 4 )0,

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relation between τ and χmax previously suggested3,4 by using 4 experimental data at ambient pressure near the glass transition. γz We apply the function χmax 4 (ρ /T) to fit the degrees of the dynamic heterogeneity evaluated at each (T,p) at which dielectric measurements of BMPC and OTP have been carried out. As a result, we find that the fits of χmax 4 to eq 4 are of a very γz high quality (Figures 1c,d). Then, the scaling χmax 4 (ρ /T) for real glass formers is achieved with γχ < γ (the insets in Figures 1c,d). This means that the density factor ρΔγ in eq 1 cannot be reduced to unity, because Δγ > 0. It should be noted that if we evaluate χmax 4 and perform the scaling in terms of eqs 3 and 4 using τ in reduced units (see Materials and Methods), the established values of Δγ remain unchanged, i.e., 4.26 ± 0.10 (4.30 ± 0.10) for BMPC, and 1.04 ± 0.05 (1.05 ± 0.04) for OTP, where the values in brackets come from the analysis in the reduced units, which ensure that both the NVE and NVT Newtonian dynamics is isomorph invariant and the scaling behavior is more clearly revealed.33,34,44 This finding confirms cannot be our supposition that the relation between τ and χmax 4 generally described by a single variable function. This result is especially striking in the case of BMPC, which shows that the decoupling between τ and χmax is rather independent (at least 4 qualitatively) of the used estimation method. Nevertheless, to complete our test, we need to check whether the found values of the scaling exponents γ and γχ actually enable us to reproduce the decrease in χmax 4 (p) at τ = const. For this purpose, we can apply the used scaling functions, given by eqs 3 and 4, to construct the corresponding function g in eq 1, and consequently, to formulate two equivalent equations, τ = τ0 exp[(ρΔγ (χ4max /(χ4max )0 )1/ Dχ A /A χ )D ]

(5a)

χ4max = (χ4max )0 (ρ−Δγ (ln(τ /τ0))1/ D A χ /A)Dχ

(5b)

From our studies of representatives of supercooled van der Waals liquids and polymer melts measured at ambient and high pressures, we can conclude that the degree of the dynamic heterogeneity χmax (estimated as described in Materials and 4 Methods) cannot be, in general, a single variable function of the structural relaxation time τ (or the segmental relaxation time in case of polymers), but it requires an additional factor. If MD obeys the density scaling law at least to a good approximation, this factor depends only on density. For the density scaling γz described by the power laws, τ(ργ/T) and χmax 4 (ρ /T), we show max that the proper relation between τ and χ4 requires the power density factor ρΔγ. Its exponent, Δγ = γ − γχ, can be regarded as a measure of the decoupling between τ and χmax 4 , which can be caused by different impacts of attractive forces on τ and χmax 4 (see Section S.2 in the SI). However, MD NVT simulations of the KABLJ liquid in the relatively wide T−ρ range indicate (see Section S.3 in the SI) that this prototypical model of supercooled liquids does not reproduce the decoupling between τ and χmax 4 , after using reduced units to ensure the isomorph invariance of MD. Thus, MD of real glass formers needs more complex models to describe this intriguing isochronal behavior of χmax 4 , which seems to be an intrinsic feature of real glass formers and cannot be omitted in the further study of the glass transition and related phenomena. The combined density scaling of τ and χmax 4 constitutes an important tool to investigate the time and length scales of MD, which both increase on approaching the glass transition in isobaric or isothermal conditions, but they cannot be in general considered as equivalent representations for the dynamic properties of glass-forming materials, because the observed decoupling between τ and χmax suggests that a given time scale 4 (τ = const) involves various length scales (related to χmax 4 ) depending on thermodynamic conditions. This intriguing picture described by us in the density scaling regime provides a strong motivation to develop the high pressure nonlinear spectroscopy techniques toward the direct experimental investigations of the higher order dynamic susceptibility near the glass transition as well as to modify the recent sophisticated theory44 of the density scaling, which seems not to be able to explain the different values of the scaling exponent for τ and χmax 4 .

If we consider the latter equation with the values of its parameters taken from fitting τ and χmax 4 to the scaling eqs 3 and 4, respectively, we should be able to reproduce the decreasing degree of the dynamic heterogeneity with increasing pressure in isochronal conditions. One can see in Figure 1e,f that it can be indeed achieved for the examined van der Waals real supercooled liquids BMPC and OTP with Δγ > 0. Another large class of materials, the molecular dynamics of which obeys the power law density scaling, includes polymer melts. In this case, a question arises how to properly employ the estimate2,3 of χmax (presented in the Materials and Methods). 4 Since the segmental relaxation is the primary relaxation process in polymers related to the glass transition, it is reasonable to assume that the characteristic correlation volume defined by χmax should be calculated taking into account not all polymer 4 chains, but rather polymer segments. In other words, χmax 4 evaluated in this way is expected to be equal or at least proportional to the average number of dynamically correlated polymer segments, Ns. On this assumption, a representative of polymer melts PVAc (see Materials and Methods) has been analyzed (see Section S.1 in the SI) in the way we have earlier examined BMPC and OTP. As a result, a decoupling has been established between the segmental relaxation time τ and χmax 4 of PVAc, which is characterized by the value of Δγ = 0.61 ± 0.03 (0.53 ± 0.03 in the reduced units) that is smaller than those obtained for the examined van der Waals liquids, but corresponds to a significant decrease in χmax within the range 4 0.1−730 MPa at τ = const (see Figure S.1(c) in the SI).

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MATERIALS AND METHODS To perform the test of eq 1, we exploit experimental dielectric, volumetric, and heat capacity data for two typical van der Waals l iq uid s , 1, 1′ - b i s ( p - m e t h o x y p h e n y l ) c y c l o h e x a n e (BMPC)45,46,12 and o-terphenyl (OTP),47−49 previously considered by us12 to discuss the effect of temperature and pressure changes on the degree of the dynamic heterogeneity, but herein we introduce another dielectric isobar of OTP measured50 at ambient pressure in a considerably wider temperature range than that in ref 47 and a recent parametrization51 of PVT data for OTP in terms of the same equation of state52 we have exploited to describe PVT data of BMPC in ref 12. Nevertheless, values of χmax 4 (T,p) for BMPC and OTP are evaluated here in the same way we have used and described in detail in ref 11, which is based on the estimate 2 proposed by Berthier et al.,2,3 χmax ≈ kBT2(χmax 4 T ) /Δcp. In this formula, which has its origin in enthalpy fluctuations, Δcp is the change in the isobaric heat capacity between liquid and glassy states and χmax ≈ (∂Φ(x)/∂x)x=1(∂ ln τα/∂ ln T)T−1, where x 4 (= t/τα) equals 1 at t = τα, Φ′(1) = β/e, because the Kohlrausch−Williams−Watts (KWW) function Φ(t) = exp4276

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[−(t/τα)β] has been assumed to parametrize the two-point time dependent correlator Φ and find its response χT(t) = ∂Φ(t)/∂T to a change of temperature at constant pressure. The dependence χmax 4 (T,ρ) has been established using ρ(T,p) found from the recently reported equation of state.52 Additionally, the same test of eq 1 is conducted by using experimental dielectric,53,54 volumetric,55 and heat capacity56 data for a polymer melt polyvinyl acetate (PVAc) as well as a recent parametrization52 of PVT data for PVAc. To test eq 1 we have also performed57 NVT molecular dynamics simulations of a large system consisting of 8000 particles of the typical KABLJ liquid18 in the relatively wide temperature-density range (0.75 ≤ T ≤ 3.0 and 1.2 ≤ ρ ≤ 1.6 in LJ units). In our analysis, the structural relaxation times τ are calculated from simulation data in the usual manner,34,35 i.e., by using incoherent intermediate self-scattering functions (τ = t if FS(q,t)=e−1)18 determined at the wave vector q of the first peak of the structure factor for the particle species that dominates the binary content of the KABLJ system, whereas the variance of the fluctuations of Fs(q,t) suggested58 to be a well-defined direct measure of the dynamic heterogeneity is used to establish the four-point dynamic susceptibility function χ4(t), the maximum of which yields the value of χmax 4 . Besides typical units, we have also used reduced units in the density scaling analyses for both the real glass formers and the KABLJ model. The reduced units (e.g., the structural relaxation time in the reduced units, τ∼ = τρ1/3(kBT/M)1/2, for real materials, where M is the average particle mass (or the average mass of polymer unit if τ is the segmental relaxation time of polymer) and kB is the Boltzmann constant, and τ∼ = τρ1/3T1/2 for the KABLJ model if the Boltzmann constant and the particle mass are assumed to equal 1 in LJ units) ensure that NVE and NVT Newtonian dynamics are isomorph invariant and the scaling behavior is more clearly revealed.33,34,44

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REFERENCES

(1) Adam, G.; Gibbs, J. H. On the Temperature Dependence of Cooperative Relaxation Properties in Glass-Forming Liquids. J. Chem. Phys. 1965, 43, 139−146. (2) Berthier, L.; Biroli, G.; Bouchaud, J.-P.; Cipelletti, L.; El Masri, D.; L’Hôte, D.; Ladieu, F.; Pierno, M. Direct Experimental Evidence of a Growing Length Scale Accompanying the Glass Transition. Science 2005, 310, 1797−1800. (3) Dalle-Ferrier, C.; Thibierge, C.; Alba-Simionesco, C.; Berthier, L.; Biroli, G.; Bouchaud, J.-P.; Ladieu, F.; L’Hôte, D.; Tarjus, G. Spatial Correlations in the Dynamics of Glassforming Liquids: Experimental Determination of Their Temperature Dependence. Phys. Rev. E 2007, 76, 041510-1−041510-15. (4) Berthier, L. Dynamic Heterogeneity in Amorphous Materials. Physics 2011, 4, 42-1−42-7. (5) Berthier, L.; Biroli, G. Theoretical Perspective on the Glass Transition and Amorphous Materials. Rev. Mod. Phys. 2011, 83, 587− 645. (6) Berthier, L.; Biroli, G.; Bouchaud, J.-P.; Cipelletti, L.; van Saarloos, W., Eds. Dynamical Heterogeneities in Glasses, Colloids, and Granular Media; Oxford University Press: Oxford, U.K., 2011. (7) Capaccioli, S.; Ruocco, G.; Zamponi, F. Dynamically Correlated Regions and Configurational Entropy in Supercooled Liquids. J. Phys. Chem. B 2008, 112, 10652−10658. (8) Kirkpatrick, T. R.; Thirumalai, D. Comparison between Dynamical Theories and Metastable States in Regular and Glassy Mean-Field Spin Models with Underlying First-Order-Like Phase Transitions. Phys. Rev. A 1988, 37, 4439−4448. (9) Fragiadakis, D.; Casalini, R.; Roland, C. M. Density Scaling and Dynamic Correlations in Viscous Liquids. J. Phys. Chem. B 2009, 113, 13134−13137. (10) Roland, C. M.; Fragiadakis, D.; Coslovich, D.; Capaccioli, S.; Ngai, K. L. Correlation of Nonexponentiality with Dynamic Heterogeneity from Four-Point Dynamic Susceptibility χ4(t) and its approximation χT(t). J. Chem. Phys. 2010, 133, 124507-1−124507-4. (11) Fragiadakis, D.; Casalini, R.; Roland, C. M. Comparing Dynamic Correlation Lengths from an Approximation to the Four-Point Dynamic Susceptibility and from the Picosecond Vibrational Dynamics. Phys. Rev. E 2011, 84, 042501-1−042501-4. (12) Grzybowski, A.; Kolodziejczyk, K.; Koperwas, K.; Grzybowska, K.; Paluch, M. Effects of Lowering Temperature and Raising Pressure on the Spatially Heterogeneous Dynamics of Glass-Forming van der Waals Liquids. Phys. Rev. B 2012, 85, 220201(R)-1−220201(R)-5. (13) Koperwas, K.; Grzybowski, A.; Grzybowska, K.; Wojnarowska, Z.; Sokolov, A. P.; Paluch, M. Effect of Temperature and Density Fluctuations on the Spatially Heterogeneous Dynamics of GlassForming van der Waals Liquids under High Pressure. Phys. Rev. Lett. 2013, 111, 125701-1−125701-5. (14) Hohenberg, P. C.; Halperin, B. I. Theory of Dynamic Critical Phenomena. Rev. Mod. Phys. 1977, 49, 435−479. (15) Toninelli, C.; Biroli, G.; Fisher, D. S. Spatial Structures and Dynamics of Kinetically Constrained Models of Glasses. Phys. Rev. Lett. 2004, 92, 185504-1−185504-4. (16) Ritort, F.; Sollich, P. Glassy Dynamics of Kinetically Constrained Models. Adv. Phys. 2003, 52, 219−342. (17) Bouchaud, J.-P.; Biroli, G. On the Adam−Gibbs−Kirkpatrick− Thirumalai−Wolynes scenario for the viscosity increase in glasses. J. Chem. Phys. 2004, 121, 7347−7354. (18) Kob, W.; Andersen, H. C. Scaling Behavior in the β-Relaxation Regime of a Supercooled Lennard-Jones Mixture. Phys. Rev. Lett. 1994, 73, 1376−1379. (19) Kirkpatrick, T. R.; Thirumalai, D.; Wolynes, P. G. Scaling Concepts for the Dynamics of Viscous Liquids near an Ideal Glassy State. Phys. Rev. A 1989, 40, 1045−1054. (20) Wolynes, P. G. Entropy Crises in Glasses and Random Heteropolymers. J. Res. Natl. Inst. Stand. Technol. 1997, 102, 187−194. (21) Xia, X.; Wolynes, P. G. Fragilities of Liquids Predicted from the Random First Order Transition Theory of Glasses. Proc. Natl. Acad. Sci. U.S.A. 2000, 97, 2990−2994.

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information includes: (S.1) the analysis of the decoupling between τ and χmax 4 of PVAc, (S.2) an attempt to in real glass explain different scaling exponents for τ and χmax 4 formers, and (S.3) details of our analysis of the direct measure of the dynamic heterogeneity in the isochronal conditions and the density scaling of τ and χmax 4 in the KABLJ model compared with some previous results of theoretical investigations and simulations.14−17,30,33−36,44,59 This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors gratefully acknowledge the financial support from the Polish National Science Centre within the program MAESTRO 2. K. Koperwas and K. Kolodziejczyk are deeply thankful for the stipend received within the project “DoktoRIS the stipend program for the innovative Silesia”, which is cofinanced by the EU European Social Fund. 4277

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dx.doi.org/10.1021/jz402060x | J. Phys. Chem. Lett. 2013, 4, 4273−4278