Influence of Cohesive Energy on Relaxation in a Model Glass-Forming

Article pubs.acs.org/Macromolecules

Influence of Cohesive Energy on Relaxation in a Model GlassForming Polymer Melt Wen-Sheng Xu,*,† Jack F. Douglas,*,§ and Karl F. Freed*,†,‡ †

James Franck Institute and ‡Department of Chemistry, The University of Chicago, Chicago, Illinois 60637, United States § Materials Science and Engineering Division, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, United States

ABSTRACT: The wide range of chemical compositions exhibited by polymers enables the fabrication of materials having highly tunable cohesive energy strength ϵ, and many of the properties that make polymers so useful as structural and responsive materials in both manufacturing and living systems derive from the variability of this basic property. The design and characterization of polymer materials then inevitably leads to a consideration of how ϵ impacts the thermodynamic and relaxation properties of polymer liquids. Our prior paper uses molecular dynamics simulations of a model coarse-grained polymer melt to systematically investigate the dependence of commonly measured thermodynamic properties on ϵ, while the present work focuses on the relaxation dynamics of the same molecular model. After demonstrating, as expected, that ϵ greatly influences the segmental relaxation time, we obtain a universal reduction of all our data for relaxation in terms of an activated transport model in which the activation free energy is increased from its high temperature value by a factor precisely determined by the average extent of the cooperative motion of monomers in the polymer liquid. This data reduction is consistent with the recently developed string model of glass formation, as well as with the assumptions of the generalized entropy theory of glass formation derived from a combination of the classical Adam−Gibbs model with a statistical mechanical model of polymer melts. In addition to providing firm observational data facilitating the development of improved theories of polymer glass formation, our results also yield insights into the molecular origin of particular thermodynamic and relaxation properties of polymers, insights that should aid in the design of polymer materials. that the diffusion coefficient of molecules and the fluid viscosity generally exhibit an Arrhenius temperature dependence,3−5 as in chemical reaction processes in gases and in atomic diffusion in crystals. A strong parallelism between data for the vaporization of liquids and for the viscosities of many liquids6,7 led to realization that the activation energies for diffusion and the fluid viscosity are correlated with the heat of vaporization of liquids.3−5 Consequently, the strength of the cohesive interactions between the constituent molecules must be an important component in understanding the dynamics of molecular fluids. Eyring and others3−5 have formulated a simple semiempirical version of transition state theory (TST) to explain the dynamics of condensed fluids. This has proven to be a quantitative approach for describing the dynamic properties

1. INTRODUCTION It has long been appreciated that glass formation is a Janusfaced phenomenon, involving changes in thermodynamic properties with temperature and associated dramatic changes in the transport properties of cooled liquids. While kinetic theory provides a rigorous foundation for understanding the relationship between dynamic and thermodynamic properties of low-density fluids,1 no corresponding fundamental theory for the dynamics of condensed fluids enables the elucidation of how dynamic and thermodynamic properties are interrelated. Progress with this theoretically challenging endeavor has mainly been deduced from using simple theoretical models and devising empirical correlations. Recent advancements in deducing empirical correlations, in turn, motivate the development of more unified theories.2 The absence of a fundamental theoretical understanding for the dynamics of glass-forming (GF) liquids appears even at elevated temperatures where a large body of evidence indicates © XXXX American Chemical Society

Received: July 12, 2016 Revised: October 6, 2016

A

DOI: 10.1021/acs.macromol.6b01504 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules of condensed fluids at elevated temperatures, even though the actual development of Eyring’s absolute rate theory employs a rather idealized view of the dynamics of real fluids based on a formal transformation of a description of gas phase dynamics into a model of the dynamics of condensed liquids. Of course, TST, in general, does not require the many simplified assumptions of Eyring,3−5 but the implementation of this approach is limited to molecular dynamics simulations8 rather than direct analytical calculations.9 Thus, the treatment of the dynamics of condensed fluids remains largely semiempirical and requires experimental data for parameters characterizing the free energy of activation for fluids with very different chemical natures. Hence, the semiempirical version of TST provides a formal framework10 for organizing observations and for assessing models of glass formation. Although a large body of evidence indicates the presence of an Arrhenius temperature dependence of the dynamics of relaxation and diffusion in liquids at elevated temperatures,3−5 this description normally breaks down as liquids are cooled to low temperatures. The term “fragility” has been introduced to quantify the degree to which the dynamics deviates from the Arrhenius behavior.11 The structural relaxation time provides an important quantity whose change in magnitude is often on the order of 6(1015) between the Arrhenius regime and temperatures where materials become glassy, i.e., acquire a solidlike rheological behavior. The apparently universal and practical nature of this phenomenon has naturally prompted much speculation as to its cause. Early experimental studies attempting to rationalize the nonArrhenius temperature dependence of the shear viscosity and other fluid transport properties emphasized that the large changes in the viscosity correlate with corresponding changes in the fluid entropy,12,13 and some influential scientists have inferred from this trend that these thermodynamic changes originate from the emergence of collective motion within cooled liquids.13 A first step forward transforming these qualitative ideas into a quantitative relationship was taken by Bestul and Chang,14 who first noticed a direct relationship between the fluid entropy and the structural relaxation time of GF liquids. Soon afterwards, Adam and Gibbs (AG)15 rationalized the seminal observations of Bestul and Chang with formal arguments that remain highly influential to the present. The central idea of the AG model15 is that the slowing down of the dynamics of cooled liquids arises from the presence of dynamical clusters, hypothetical “cooperatively rearranging regions” (CRRs), that grow upon cooling, causing an increase in the activation energy. These clusters were correspondingly assumed to be responsible for the decrease in the fluid configurational entropy (i.e., the entropy devoid of its vibrational contributions), a quantity that could be roughly estimated from the “excess entropy”, Sexc, the entropy of the liquid relative to that of the solid (often taken to be the crystalline form of the material, but sometimes the glassy state of a polymer material is used to define this quantity). The first assumption of AG posits that the free energy ΔG0 of activation of TST at high temperatures should be multiplied by a factor z, corresponding to the number of molecules or segments participating in the abstract CRRs. Second, they assume that the size of the CRRs should scale inversely to the configurational entropy Sc of the fluid, z = S*c /Sc, where S*c is the high temperature limit of Sc, a quantity that is also assumed to exist by AG.15 In other words, AG basically postulate that the free

energy of activation is the temperature-dependent quantity, z(T)ΔG0. Within this physically attractive picture of glass formation, collective motion simply renormalizes the activation energy by a factor equal to the number of particles involved in the collective motion, z, basically rationalizing the empirical relation of Bestul and Chang.14 Since the AG model is evidently not so much a theory as series physically motivated hypotheses, validation and modification of these physically attractive ideas had to await the development of molecular dynamic simulations and experimental measurements capable of probing the actual nature of the molecular dynamics in cooled liquids and its relation to the observed relaxation dynamics of the material as a whole. The use of simulations to test these models and assumptions represents the main focus of the present work. Originally, AG15 argued for the rough approximation Sc ≈ Sexc to make contact with the Vogel−Fulcher−Tammann (VFT) equation and rationalize the correlation of Bestul and Chang,14 while Angell and co-workers16 emphasize the use of Sexc as a thermodynamic predictor of the dynamics of GF liquids in the spirit of the AG model. No effort was made by AG to calculate Sc from theoretical considerations. The development of the lattice cluster theory of polymer thermodynamics17,18 enables the direct statistical mechanical calculations of Sc in polymer fluids, so the aforementioned approximation Sc ∼ Sexc is unnecessary. The combination of the lattice cluster theory with the AG relation provides a precise theoretical framework for predicting the glass formation of polymer fluids in terms of molecular parameters governing their thermodynamics.19 This uniquely predictive theory, however, still suffers the weaknesses of the lack of a description of TST at elevated temperatures. Moreover, a number of assumptions must be validated and clarified (e.g., just what form do the CRRs take?) before the assumptions of AG can be accepted, regardless of any phenomenological successes of this model in agreement with experiments. Simulations offer invaluable information regarding the development of a theory of glass formation based on firm information, thereby alleviating the need of unwarranted assumptions. The preceding paper20 addresses the most basic thermodynamic properties of a model GF polymer melt using molecular dynamics simulations in a range of temperatures where equilibrium simulations can be performed to determine if there are clearly identifiable thermodynamic signatures of glass formation. We show that signatures of both the “onset” and “end” of glass formation can be identified from thermodynamic features of the static structure factor, S(q). In particular, the Hansen−Verlet freezing criterion21 involving a critical peak height Sp in the first peak of S(q) fairly well identifies the onset of particle localization, as corresponding to the onset of glass formation. In physical terms, this criterion indicates that the fluids exhibit “local jamming” by neighboring molecules, leading to a deviation from simple Arrhenius relaxation at low temperatures. We also find that the dimensionless isothermal compressibility, as determined from the limiting value of S(q) as q → 0 normalized by Sp, approaches a critical range of values upon approaching the glassy state. This ratio, as explained by Torquato and co-workers,22−24 provides a measure of the degree to which the material is “jammed” in a global sense. Using molecular dynamics simulations, we investigate the influence of cohesive interaction strength on the glass formation of a model coarse-grained unentangled polymer melt under both constant volume and constant pressure B


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Macromolecules conditions. Our prior work20 uncovers universal relations between a range of basic dimensionless thermodynamic quantities, such as the cohesive energy density, number density, thermal expansion coefficient, isothermal compressibility, and surface tension, and a reduced temperature defined in terms of a measure of the local cohesive interaction strength, ϵ. However, the present paper shows that this reduced variable description does not apply to the dynamics of our model polymer melt. Varying ϵ is thus not “equivalent” to varying T in the dynamics of polymer fluids. This nontrivial influence of ϵ on the polymer dynamics originates from the fact that generally more than one energy scale affects the properties of polymer systems.20 The “fragility” of the model polymer melt is determined from the temperature dependence of the structural relaxation time τα, and this engineering property of polymer materials is found to be strongly influenced by thermodynamic path but not by ϵ in our molecular model. In order to quantify the dynamic heterogeneity of the polymer melt, we explore how ϵ affects the non-Gaussian parameter, α2, the four-point susceptibility, χ4, and the extent of stringlike cooperative motion, L. The analysis further reveals an interesting link between L and the peak height of α2 in our model polymer melt. Finally, we demonstrate that our simulation data for τα for various ϵ can successfully be described by the string model of glass formation,25 which predicts that structural relaxation in GF liquids is governed by the high temperature activation free energy and L.

directions. For constant V conditions, all simulations are performed at a fixed number density of ρ = N/V = 1 in the NVT ensemble, where the temperature is maintained by the Nosé−Hoover thermostat,35,36 which is implemented in HOOMD-blue via the Martyna−Tobias−Klein equations of motion.37 For constant P conditions, our results are obtained along an isobaric path with P = 0 by employing the procedure described below. At the first stage, the simulations are performed for a given T at P = 0 in the NPT ensemble, where the temperature and pressure is maintained by the Martyna−Tobias−Klein barostat−thermostat37 and where the integration is performed in a cubic box at constant hydrostatic pressure by simultaneously rescaling all three lengths of the simulation box. These NPT simulations then allow for the determination of the desired density for that T at P = 0. The simulations are then performed at the target density in the NVT ensemble, thereby generating the results approximately at P = 0. We use a time step of Δt = 0.002 in all simulations. We ensure that the properties of interest are obtained in our simulations after the polymer fluid has been fully equilibrated. In addition, four independent runs are performed for each state point to improve the statistics.

3. RESULTS AND DISCUSSION This section begins by exploring the structural relaxation of polymer melts upon cooling, followed by a discussion of the characteristic temperatures and fragility of glass formation. This section then investigates the dynamic heterogeneity of polymer melts by analyzing the non-Gaussian parameter, the four-point susceptibility, and the stringlike cooperative motion. 3.1. Structural Relaxation Time. The structural relaxation time τα is a basic quantity that allows for the determination of some basic properties of glass formation, such as the characteristic temperatures and fragility of glass formation. Our analysis for the dynamics thus begins with the determination of the structural relaxation time. To this end, we first calculate the self-part of the intermediate scattering function

2. MODEL AND SIMULATION DETAILS We briefly describe the model and simulation details here; more details can be found in the preceding paper.20 Our results are based on a coarse-grained model of GF polymer melts composed of 200 linear chains. Each chain contains 16 beads, so our results are not complicated by the entanglement effects.26,27 The attractive nonbonded interactions are described by the standard truncated-and-shifted LJ potential28 12 ⎧ ⎪ 4ϵε[(σ / r ) − (σ /r )6 ] + C(rcut), r < rcut ULJ(r ) = ⎨ ⎪ r ≥ rcut ⎩ 0,

N

Fs(q , t ) =

(1)

where r is the distance between two beads, σ is the effective diameter of the beads, ε sets the energy scale of the system, and the constant C(rcut) ensures that ULJ goes smoothly to zero at the cutoff distance rcut, which is taken as 2.5σ. The parameter ϵ thus controls the strength of the nonbonded attractive interactions between the beads. The preceding paper20 motivates calling ϵ a “cohesive energy parameter”, a term that is used in the present paper. Bond connectivity along the neighboring beads is maintained by employing a harmonic spring potential Uharm(r ) =

1 k b(r − r0)2 2

1 ⟨∑ exp{−iq·[rj(t ) − rj(0)]}⟩ N j=1

(3)

where i = −1 , q = |q| is the wavenumber, and rj(t) is the position of particle j at the time t. The wavenumber is chosen to be q = 7.1 in this work, which is close to the first peak of the static structure factor S(q), a property that is discussed extensively in the preceding paper.20 Figure 1 displays the time dependence of Fs(q, t) for various T for ϵ = 1 at P = 0. While the time dependence of Fs(q, t) is found to be nearly exponential at high temperatures, it becomes highly “stretched” and develops a two-step decay upon cooling, where the time characterizing the process grows rapidly upon decreasing T. The two-step decay is typical of GF liquids and reflects the rattling motion of particles being trapped in cages formed by their nearest neighbors at short times (β-relaxation) and the motion of particles escaping from the cages at long times (α-relaxation). In particular, the long time decay of Fs(q, t) is generally described by a stretched exponential function, Fs(q, t) = A exp[−(t/τ)β], where A is a prefactor, τ is a time scale characterizing the α-relaxation process, and the exponent β can be much smaller than 1 at low T, as shown in the inset to Figure 1. Hence, the behavior of Fs(q, t) is typical of glassy dynamics, as expected. Following previous work,2,25,38−42 the

(2)

where the parameters are taken as kb = 2000ε/σ and r0 = σ so that crystallization can be avoided.29−31 All beads have the same mass m. Length and time are reported in units of σ and 2

mσ 2/ε , respectively. We perform molecular dynamics (MD) simulations in three dimensions under both constant volume (V) and constant pressure (P) conditions using the HOOMD-blue simulation package.32−34 Periodic boundary conditions are employed in all C


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Macromolecules

If this is the case, τα for all ϵ can likewise be understood by considering a reduced temperature given by T/ϵ. Figure 3

Figure 1. Self-intermediate scattering function Fs(q, t) as a function of the time t for various temperatures T for ϵ = 1 at a fixed pressure of P = 0. The wavenumber is chosen to be q = 7.1, which is close to the position of the first peak of the static structure factor S(q). Circles indicate the positions of Fs(q, t = τα) = 0.2, a common convention for defining the structural relaxation time τα. The inset depicts the T dependence of the exponent β, as determined from the fits of Fs(q, t) to a stretched exponential function (see text).

structural relaxation time τα is defined as Fs(q, t = τα) = 0.2. The resultant τα should be proportional to τ as fitted from the stretched exponential function, so we report only τα in the present paper for convenience. Figures 2a and 2b display the Arrhenius plots of τα for various ϵ at both constant V and constant P. The results in Figures 2a and 2b clearly show that τα is strongly influenced by the cohesive energy parameter ϵ in the entire regime of T. One might wonder whether this is a mere quantitative effect that can be accommodated by renormalizing the temperature scale, as found for the dimensionless thermodynamic properties in the preceding paper.20 To test this hypothesis, we explore whether the T dependences of τα for various ϵ collapse onto a single curve when T is scaled by ϵ.

Figure 3. Structural relaxation time τα as a function of ϵ/T for various ϵ at (a) constant V and (b) constant P. Evidently, data for τα do not reduce to a universal scaling form involving ϵ/T, in contrast to what we have shown for the thermodynamic properties in the preceding paper.20 These results clearly indicate that changing ϵ is not equivalent to changing T in the polymer dynamics.

Figure 2. (a) and (b) show the Arrhenius plots of the structural relaxation time τα for various ϵ at constant V and constant P, respectively. Pluses and crosses denote the positions of the localization temperatures Tl and the Arrhenius temperatures TA, respectively. Dashed, dotted, and dash-dotted lines indicate the Arrhenius, Vogel−Fulcher−Tammann (VFT), and power-law fits to τα, respectively. (c) and (d) show the VFT collapse of the temperature dependence of τα for various ϵ at constant V and constant P, respectively. Dotted lines indicate the VFT relation, ln(τα/τ0) = DT0/(T − T0). D


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Macromolecules displays τα as a function of ϵ/T for various ϵ under both constant V and constant P conditions. This simple analysis reveals the absence of a master curve between τα and ϵ/T for various ϵ. Since the preceding paper20 illustrates that universal scaling relationships emerge only between the dimensionless thermodynamic properties and T/ϵ, we further test whether or not master curves can be obtained when τα is properly reduced. In particular, we consider a reduced relaxation time defined naturally by τ̃α = ρ1/3(kBT/m)1/2τα. We find that τ̃α for various ϵ likewise do not collapse onto a master curve (data not shown), although the differences in τ̃α among various ϵ appear to be diminished as compared to those in τα. Figure 4 displays the structural relaxation time τα as a function of ϵ for various T at constant V and constant P. It is

Specially, these temperatures prescribe the onset at TA, the middle (or crossover) at Tc, and the termination at T0 of glass formation as well as the common glass transition temperature Tg, below which the material falls out of equilibrium. In addition, previous simulations2,25,38−42 identify a “localization” or “caging” temperature Tl at which particle caging starts to emerge. The preceding paper20 describes in detail how to determine Tl, Tc, Tg, and T0 from simulations by following established methods.2,25,38−42 Below, we provide a brief summary for these methods and additionally describe how TA is estimated in our simulations. The localization temperature Tl is determined in terms of a feature evidenced in the logarithmic derivative [i.e., ∂ ln ⟨r2(t)⟩/∂ ln(t)] of the mean-squared displacement (MSD), which begins to develop a minimum around 1 ps at Tl. These characteristic temperatures are shown as pluses in Figures 2a and 2b. The crossover temperature Tc of glass formation is estimated by fitting τα to a power-law equation51 τα = τc(T − Tc)−γc

(5)

where τc, Tc, and γc are fitting parameters. The ideal glass transition temperature T0 is obtained by fitting τα to the VFT equation43−45 ⎛ DT0 ⎞ τα = τ0 exp⎜ ⎟ ⎝ T − T0 ⎠

where τ0, T0, and D denote the high T limit of τα, the temperature at which τα diverges, and the VFT kinetic fragility parameter that quantifies the strength of the T dependence of τα. The fitted curves from the power-law and VFT equations are shown in Figures 2a and 2b as dotted and dash-dotted lines, respectively. Figures 2c and 2d show the VFT collapse of the temperature dependence of τα for various ϵ at constant V and constant P, respectively. The glass transition temperature Tg is then estimated from the VFT fits to data for τα using the common empirical definition of τα(Tg) = 100 s. One may also estimate the fragility parameter m (or “steepness index”) from the VFT fits proposed by Angell,11 a widely reported metric for fragility that can readily be calculated from the GET.19 However, results for m are not considered in the present paper since this quantity concerns the dynamics near Tg, while our simulation data are restricted to a temperature range above Tc. We now focus on TA, a characteristic temperature that is only briefly mentioned in our preceding paper.20 Figures 2a and 2b clearly indicate that while data for τα at low T exhibit superArrhenius behavior, they follow the Arrhenius equation at sufficiently high T, as expected. Hence, in order to determine TA, data for τα at high T are first fitted for each ϵ to the Arrhenius equation, expected from TST of Eyring3−5

Figure 4. VFT collapse of the ϵ dependence of τα for various T at (a) constant V and (b) constant P. Dashed lines indicate the VFT-like equation, ln(τα/τ0) = Dϵϵ/(ϵ0 − ϵ). The insets depict τα as a function of ϵ for various T. Dashed lines indicate the fits of data for τα to the VFT-like equation, τα = τ0 exp[Dϵϵ/(ϵ0 − ϵ)].

evident that the ϵ dependence of τα follows a functional form, similar to the well-known VFT equation43−45 ⎛ Dϵ ⎞ τα = τ0 exp⎜ ϵ ⎟ ⎝ ϵ0 − ϵ ⎠

(6)

(4)

τα = τ∞ exp(ΔE /T )

We find that the fitted Dϵ varies very weakly with T, but ϵ0 increases with T in a nearly linear manner. Our simulations indicate that the cohesive energy parameter ϵ, which is strongly variable in ionic and polar materials,46−50 is a highly relevant variable in the dynamics of GF polymer melts. 3.2. Characteristic Temperatures and Fragility of Glass Formation. GF materials exhibit multiple characteristic temperatures. This universal behavior is emphasized in the generalized entropy theory (GET) of glass formation,19 which describes the glass transition as a broad underlying thermodynamic transition with multiple characteristic temperatures.

(7)

where τ∞ is the usual vibrational prefactor and ΔE is the activation energy of TST. TA is then identified as the temperature available in our simulations at which τα − τα(Arrhenius) > a, where τα(Arrhenius) is the expected structural relaxation time from Arrhenius fits and a is a threshold. We use a = 10−2 and 10−3 for constant V and constant P conditions, respectively. (The choice of a does not strongly affect the estimated TA.) The resultant estimates for TA are shown as crosses in Figures 2a and 2b. To avoid confusion with Tl, TA is termed the “Arrhenius temperature” in the E


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Macromolecules present paper. For completeness, Figure 5 shows the fitted Arrhenius parameters τ∞ and ΔE as a function of ϵ. While ΔE

that KVFT increases as the fragility increases. The ϵ dependence of KVFT is shown in Figure 7. As can be seen, the fragility of

Figure 5. Arrhenius parameters ΔE (filled symbols) and τ∞ (open symbols) as a function of ϵ at constant V and constant P. Dashed lines are a guide to the eye. Solid lines indicate the fits of data for ΔE to the linear equation ΔE = e1ϵ, where the fitting parameters e1 is determined to be 2.55 at constant V and 1.95 at constant P.

Figure 7. VFT fragility parameter KVFT (filled symbols) and the ratio T0/Tl (open symbols) as a function of ϵ at constant V and constant P. Lines indicate averages of each data set over ϵ.

glass formation becomes larger when shifting the thermodynamic path from the isochoric conditions with ρ = 1 to the isobaric conditions with P = 0, a trend that is in good agreement with experimental measurements for GF polymer melts.53 While the GET predicts that the fragility decreases with increasing the cohesive interaction strength,54−58 our simulations indicate that the fragility depends very weakly on ϵ for both constant V and constant P conditions in the present model. We emphasize that the determination of KVFT in our sifgmulations is based on the data for τα in the high T regime (Tc < T < TA), while the GET instead estimates the fragility of glass formation using data in the low T regime (Tg < T < Tc). We also point out that our simulations employ a model of polymer melts composed of fully flexible chains, while all calculations based on the GET have been performed for models of semiflexible polymers so far.54−58 The dependence of fragility on ϵ remains to be investigated for semiflexible polymer melts in simulations. We also mention a previous simulation study of a GF liquid composed of particles with only power-law soft repulsive interactions,59 where KVFT is found to be invariant to the potential power over a large range of exponent values. The activation energy and characteristic temperatures of glass formation, however, are affected by the exponent when ϵ is varied. Figure 7 examines how ϵ influences the breadth of glass formation, a property that has long be advocated as a complementary measure of the fragility of glass formation.19,60−64 In particular, the GET calculations even indicate universal relationships between the breadth of glass formation and the common fragility parameter m (or KVFT) in polymer melts.56,57 While the preceding paper20 indicates that all ratios of the characteristic temperatures of glass formation are nearly independent of ϵ, we particularly consider the ratio T0/Tl as a measure of the breadth of glass formation since Tl and T0 describe the onset and end of glass formation, respectively. Similar to KVFT, Figure 7 shows that T0/Tl is nearly independent of ϵ, and again, T0/Tl at constant P is larger than that at constant V. Interestingly, T0/Tl is nearly identical to KVFT for constant V conditions. The observation that KVFT is nearly independent of ϵ implies that the T dependence of τα remains similar in the low T regime when ϵ is varied, suggesting a universal behavior for the

evidently increases linearly with ϵ, τ∞ decreases with ϵ in a nonlinear fashion. A linear growth of ΔE with ϵ is also observed in two- and three-dimensional LJ GF liquids.52 The preceding paper20 indicates that the characteristic temperatures Tl, Tc, Tg, and T0 all increase nearly linearly with ϵ. Figure 6 shows that similar linear scalings also hold for

Figure 6. Localization temperature Tl (filled symbols) and Arrhenius temperature TA (open symbols) as a function of ϵ at constant V and constant P. Lines indicate the fits of data to the equation Tx = Ix + Jxϵ (x = l or A), where the fitting parameters are determined to be (Il, Jl) = (0.33, 1.02) at constant V and (Il, Jl) = (−0.02, 0.71) at constant P and (IA, JA) = (0.29, 0.76) at constant V and (IA, JA) = (0.27, 0.97) at constant P.

TA. The ϵ dependence of Tl is also displayed in Figure 6 in order to quantitatively compare them with another onset temperature of glass formation, T A . Previous simulations2,25,41,42 show that the two onset temperatures, Tl and TA, are often close to each other. However, Figure 6 shows that TA is generally different from Tl in our simulations. In particular, TA is lower than Tl for each ϵ at constant V but becomes higher than Tl at constant P. We thus consider both Tl and TA in the following discussion. Now we explore how the cohesive energy parameter ϵ influences the fragility of glass formation in the present model, which is only briefly discussed in the preceding paper.20 In particular, we consider the fragility parameter KVFT ≡ 1/D so F


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Macromolecules T dependence of τα when both T and τα are properly reduced. To explore this possible scenario, we take advantage of the Arrhenius behavior of τα at high T and examine the relation between τα/τ∞ and ΔE/T. While this procedure ensures that data for τα collapse onto a single curve in the high T Arrhenius regime, Figure 8 reveals the presence of a master curve between

expressed to a different degree in each system. The present work focuses on both the mobile and immobile particles. Starr et al.40 and Zhang et al.65 discuss the problem of the dynamic heterogeneity in considerable detail in the context of polymeric and metallic GF liquids, respectively, but here we note some basic features of these different types of dynamic cluster types having basic significance for structural relaxation in cooled liquids. The lifetime t* of the mobile particle clusters can be determined from the peak of the non-Gaussian parameter α2, a measure of fluctuations in the diffusion coefficient defined in precise terms in terms of the second and fourth moments of the van Hove function describing the displacement of the particles in the fluid (chain segments in the present context of polymeric fluids). In more technical terms, α 2 derives from a four-point velocity autocorrelation function66−68 extending the two-point velocity correlation function from which the self-diffusion constant D is defined. These details are not important here, but it is easy to appreciate that dynamic molecular clustering should naturally lead to mobility fluctuations and that α2 is a measure of the variance of the resulting D fluctuations. It is then an indirect measure of “dynamic heterogeneity” that is heavily weighted toward the mobile particles leading to particle diffusion. Correspondingly, the lifetime of immobile particle clusters can be determined from another four-point function, χ4.40,69 This autocorrelation function is directly related to the variance in the density autocorrelation function Fs(q, t),69−71 and χ4 exhibits a peak at a time tχ near the structural relaxation time τα, defining the lifetime of the immobile particle clusters.40 The lifetime of the immobile particles is generally longer than that of the mobile particles and the gap between t* and tχ grows upon cooling, a general phenomenon termed “decoupling”. This is a basic and universal property of GF liquids that we quantify below for our model polymer melt. We also mention that being immobile or mobile implies nothing about whether particles are exhibiting collective motion. The van Hove distinct correlation function72 allows for the quantification of collective motion at the level of two particles, but different methods are needed when the collective motion involves a large collection of particles exhibiting concerted movement, as hypothesized by AG.15 Below, we will review how these particles exhibiting coordinate movement are precisely identified. Not surprisingly, the particle clusters involved concerted particle irreversible displacement are also “mobile particles” in the sense defined before. Indeed, mobile particles are clusters of more primary clusters exhibiting cooperative rearrangement (CRRs in the jargon of AG15). We next characterize α2 and χ4 to determine the lifetimes of mobile and immobile particles, respectively. We then determine the size of the clusters exhibiting cooperative rearrangement motion, which happen to have a stringlike form, regardless of the type of GF materials. As already mentioned, GF liquids exhibit a significant fraction of particles with extremely high or low mobility relative to the mean upon cooling, whose positions are spatially correlated. It is then interesting to determine how ϵ influences the non-Gaussian parameter

Figure 8. Scaled structural relaxation time τα/τ∞ as a function of ΔE/ T for various ϵ at constant V and constant P. The dashed line indicate τα/τ∞ = exp(ΔE/T). This figure presents clear evidence that relaxation occurs by a simple thermally activated process at elevated temperatures, consistent with simple transition state theory.

τα/τ∞ and ΔE/T in the entire regime of T at both constant V and constant P, supporting our earlier analysis that ϵ has a strong influence on the characteristic temperatures of glass formation but barely affects the fragility of glass formation in the present model. The reader is warned that the presence of an angle bending potential or a torsional potential for bond rotation may alter the simple scaling that we see in the present polymer model. 3.3. Dynamic Heterogeneity in Glass-Forming Liquids. “Dynamic heterogeneity” is often invoked to explain deviations from Arrhenius relaxation, along with other general, but poorly understood, aspects of GF liquids such as the temperature dependence of β shown in the inset of Figure 1, the breakdown of the Stokes−Einstein relation, etc. The specific nature of this heterogeneity has often been vague, and a wide range of types of heterogeneity have been invoked to rationalize particular measurements. The atomic resolution of simulations has helped to resolve the nature of this heterogeneity, but many questions still remain. One important finding of significance for the present work is that there are two major types of dynamic heterogeneity that each contributes in its own way to the dynamics of GF liquids, such as our model polymer melt. In particular, there are excessively “mobile” particles in comparison to Brownian particles having the same average diffusion coefficient, and there are by complementarity excessively “immobile” particles.40,65 Both particles exhibit the property of forming diffuse polymeric clusters of particles,40,65 regardless of the molecular bond connectivity of individual molecules, reminiscent of behaviors in self-assembly process25 that remains to be fully understood. The mobile particles have been a preoccupation since, as we shall see below, they are related to the CRRs of AG.15 The immobile particles have their importance for understanding β(T), decoupling, and other aspects of glass formation and deserve their own study. Dynamic heterogeneity then refers to more than one kind of clustering phenomenon where each mobility cluster type is

α2(t ) =

3 r 4(t ) 5 r 2(t )

2

−1 (8)

This quantity vanishes for Gaussian processes in which the motion of particles is uncorrelated and mobility fluctuations are vanishingly small. G


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Macromolecules Figure 9 displays the t dependence of α2(t) for various ϵ at a fixed temperature of T = 1.45 under constant V conditions. We

Figure 9. Non-Gaussian parameter α2(t) as a function of the time t for various ϵ at a fixed temperature of T = 1.45 at constant V. Circles highlight the maxima in α2(t).

see that α2(t) exhibits a peak for each ϵ with the peak time t* defining a characteristic time scale of the spatially heterogeneous motion.40 Both the peak amplitude α2* and t* are found to significantly grow with increasing ϵ for fixed T, a common observation in GF liquids. In line with previous work,39,40 t* increases less rapidly than τα upon cooling for all ϵ under both constant V and constant P conditions, implying a decoupling of these characteristic time scales. Previous work39 has also established a fractional power law t* ∼ (τα)2/3 linking these characteristic time scales for a similar model of GF polymer melts. Figure 10 indicates that the same relation remains valid for all ϵ considered and independent of the thermodynamic path. This is the “decoupling” phenomenon described in general terms above. Decoupling is extremely important for the present work because it means that the relative change in the activation free energy for diffusion and the relaxation time are both determined by the same factor.40 This fundamental quantity, as intuited by AG,15 is the number z of particles in a group of cooperatively rearranging particles (statistical segments or “beads” in the case of our model polymer melt). To characterize the fluctuations of the immobile particles, we consider the four-point susceptibility χ4, a widely studied property for the quantification of the dynamic heterogeneity in GF liquids that is first systematically formulated by Glotzer and co-workers.69 Again, our discussion is restricted to the self-part of χ4. First, a time-dependent self-overlap order parameter Qs(t) is defined as

Figure 10. Relationship between the peak time t* of the non-Gaussian parameter α2(t) and the structural relaxation time τα for various ϵ at (a) constant V and (b) constant P. Dashed lines indicate the fits of data for T ≤ Tl to the power law t* = b(τα)δα, where the fitting parameters (b, δα) are determined to be (0.89, 0.68) at constant V and (1.10, 0.68) at constant P, respectively.

Figure 11. Self-part of the four-point susceptibility χ4,s(t) as a function of the time t for various ϵ at a fixed temperature of T = 1.45 at constant V. Circles highlight the maxima in χ4,s(t).

N

exhibits a peak at an intermediate time tχ that defines the lifetime of the immobile particles, as discussed earlier. Both the peak height χ*4,s of χ4,s(t) and tχ increase as ϵ grows at a fixed temperature of T = 1.45. Figure 12 compares tχ with τα for various ϵ. We find that a master curve between these two time scales appears under both constant V and constant P conditions. This master curve can be described by a power law tχ ∼ (τα)δχ. Our numerical fits yield δχ ≈ 0.96 using the low T data, a value that is equal to 1 within numerical uncertainty, so that the approximation tχ ∼ τα provides a good description of all data under both constant V and constant P conditions over the entire range of T investigated (see dash-dotted lines in Figure 12). Starr et al.40

Q s(t ) = ⟨∑ w(|rj(t ) − rj(0)|)⟩ j=1

(9)

with w = 1 (0) for |rj(t) − rj(0)| < (≥) 0.3. The mean-square variance of Qs(t) then defines the self-part of the four-point susceptibility χ4, s (t ) =

V [⟨Q s(t )2 ⟩ − ⟨Q s(t )⟩2 ] N2

(10)

which measures the degree of the cooperativity of structural relaxation. Figure 11 presents the t dependence of χ4,s(t) for various ϵ at a fixed temperature of T = 1.45 at constant V. Consistent with previous results,69 we observe that χ4,s(t) H


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where P(s) is the probability of finding a string of length s, and ∑∞ s=1P(s) = 1 by construction. As an illustration, Figure 13

Figure 13. Number-averaged string length ⟨s(t)⟩ as a function of the time t for various ϵ at a fixed temperature of T = 1.45 at constant V. Circles highlight the maxima in ⟨s(t)⟩.

exhibits ⟨s(t)⟩ for various ϵ at a fixed temperature of T = 1.45 under constant V conditions. We see that ⟨s(t)⟩ displays a maximum L ≡ ⟨s(tL)⟩ at a characteristic time tL, where L defines the characteristic string length.2,25,38−42 (The four-point length ξ4 characterizing the size of the immobile particles is correspondingly defined at the peak time of χ4, defining fluctuations in the density autocorrelation function.69) As expected from our earlier analysis for τα and α2(t), both L and tL are strongly influenced by ϵ. Again, the influence of ϵ on L and tL cannot be understood by simply renormalizing T by ϵ (data not shown). Previous work has shown visualizations of the strings in model polymer melts25,40 and metallic GF liquids65 so that we avoid this exercise here. Since the formation of stringlike clusters concerns mobile particles that play a dominant role in determining molecular diffusion, we examine how the string length ⟨s(t)⟩ relates to α2(t). To this end, we first compare tL with t* in Figure 14, which indicates the presence of a master curve between these two time scales for all ϵ under both constant V and constant P conditions. Apparently, this master curve can be described by a power law tL ∼ (t*)δL. Our numerical fits yield δL ≈ 1.05 using the low T data, a value that is quite close to 1, suggesting a coupling of the characteristic time scales arising from the formation of stringlike clusters and the non-Gaussian behavior. Indeed, the simple approximation tL ∼ t* provides a good description of all data under both constant V and constant P conditions over the entire range of T investigated (see dashdotted lines in Figure 14). While previous work40 has established that α2(t) ≡ ⟨s(t)⟩ peak at a similar time, the generality of a power-law relation between t* and tL is currently unclear. Figure 15 further examines the relation between the characteristic string length L and the peak value α2* of the non-Gaussian parameter. Again, we find a universal relation between these two quantities, which applies for all ϵ under both constant V and constant P conditions. In particular, a power law with α2* ∼ Lθ emerges at low temperatures, where the exponent is determined to be around θ = 3.3 using data with τα > 50. Therefore, the above analysis implies the presence of an interesting link between L and α*2 . However, this interesting result is not well understood presently and remains to be confirmed in other GF liquids.

Figure 12. Relationship between the peak time tχ of the self-part of four-point susceptibility χ4,s(t) and the structural relaxation time τα for various ϵ at (a) constant V and (b) constant P. Dotted and dashdotted lines indicate the fits of data for T ≤ Tl to the power law tχ = c(τα)δχ and to the linear equation tχ = dτα, where the fitting parameters (c, δχ, d) are determined to be (0.67, 0.97, 0.60) at constant V and (0.69, 0.96, 0.60) at constant P, respectively.

have previously showed that both these characteristic times scale with the lifetime of the immobile particles so that τα and tχ can be identified as measures of the lifetime of the immobile particles. We further characterize the mobile particles by analyzing the stringlike cooperative motion of these particles. It is widely appreciated that atoms or molecules of extreme mobility (or immobility) tend to cluster in GF liquids and that the most mobile clusters can be further divided into groups of atoms or molecules that move cooperatively in a roughly collinear, or stringlike, fashion.72,73 Recent extensive analysis by Douglas and co-workers2,25,38−42 indicates that the size of stringlike clusters provides the most consistent measure of the CRRs in the AG model.15 The identification of stringlike cooperative motion follows the previously established procedure for polymer systems.40,73 First, mobile particles are defined as the f 0 = 6.5% of particles with the greatest displacement over any chosen interval. (This fraction of particles maximizes the scale of the mobile particles40 and is somewhat variable with the type of fluid.) Two mobile particles j and k are then considered to be in the same string if min[|rj(t ) − rk(0)| , |rk(t ) − rj(0)|] < δ

(11)

We use δ = 0.55 in this work. Then, the number-averaged string length ⟨s(t)⟩ is calculated according to ∞

⟨s(t )⟩ ≡

∑s = 1 ⟨sP(s)⟩ ∞

∑s = 1 P(s)

(12) I


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Figure 15. Relationship between the peak height α2* of the nonGaussian parameter α2(t) and the average string length L for various ϵ at (a) constant V and (b) constant P. Dashed lines indicate the fits of data with τα > 50 to the power law α*2 = gLθ, where the fitting parameters (g, θ) are determined to be (0.08, 3.42) at constant V and (0.09, 3.26) at constant P, respectively.

Figure 14. Relationship between the peak time tL of the numberaveraged string length ⟨s(t)⟩ and the peak time t* of the non-Gaussian parameter α2(t) for various ϵ at (a) constant V and (b) constant P. Dotted and dash-dotted lines indicate the fits of data for T ≤ Tl to the power law tL = e(t*)δL and to the linear equation tL = f t*, where the fitting parameters (e, δL, f) are determined to be (1.33, 1.06, 1.52) at constant V and (1.39, 1.04, 1.55) at constant P, respectively.

3.4. Quantification of the Extent of Stringlike Cooperative Motion. We now focus on the T dependence of L. Figure 16 displays L as a function of 1/T for various ϵ under both constant V and constant P conditions. In the high T regime (T > Tl), L grows with 1/T in a nearly linear fashion L = L∞ + ΔL /T

(13)

where L∞ and ΔL are constants that characterize the variation of L above Tl for each ϵ. This linear behavior is mathematically equivalent to the Ferry−Bässler relation,74−76 which indicates that τα ∼ exp(B/T2) with B being an empirical constant. The dashed lines in Figure 16 show the fits to eq 13, and the fitted values of L∞ and ΔL are summarized in Figure 17, where we see that L∞ depends very weakly on ϵ and that ΔL increases nearly linearly with ϵ. A more complicated variation of L arises at lower T. The string model of glass formation25 indicates that the strings can quantitatively be modeled as equilibrium polymeric structures,77 which is rather compatible with the AG theory15 but more precise at the same time in its description of the character of the collective motion in GF liquids. In particular, L in the low T regime (T < Tl) can be described well by the relation25 ⎫ ⎛ ΔGp ⎞⎤⎪ ⎪ ⎛ Φ ⎞⎧ Φl ⎡ L ⎢1 + ϕ0 exp⎜ − ⎥⎬ 1 = ⎜1 − l ⎟⎨ + ⎟ ⎪ Ll ⎝ 2 ⎠⎩ 2 ⎢⎣ ⎝ kBT ⎠⎥⎦⎪ ⎭

Figure 16. Arrhenius plots of the average string length L for various ϵ at (a) constant V and (b) constant P. Crosses and pluses denote the positions of the Arrhenius temperatures TA and the localization temperatures Tl, respectively. Dashed, dotted, and dash-dotted lines indicate the fits of data for L to eqs 13−15, respectively.

thermodynamics of string polymerization rather than the activation free energy of the fluid. Actually, eq 14 is a simplified high T expansion for L. The full form for L, described in ref 25, exhibits a plateau at low T, indicating a return to Arrhenius relaxation in the glassy state, albeit with a higher effective activation free energy. While the difference between the simplified expression and the full form is evident at low T, eq

(14)

where Ll ≡ L(Tl), Φl is the extent of polymerization (given by the fraction of monomers participating in polymeric structures) at Tl, and ϕ0 is the volume fraction of mobile particles. The free energy ΔGp = ΔHp − TΔSp in eq 14 describes the J


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Figure 17. ΔL (filled symbols) and L∞ (open symbols), determined from eq 13 for the T dependence of L in the high T regime, as a function of ϵ at constant V and constant P. Dashed lines indicate averages of each data set over ϵ. Solid lines indicate the fits of data to the equation ΔL = l1ϵ, where the fitting parameter l1 is determined to be 0.24 at constant V and 0.36 at constant P, respectively.

14 should provide an adequate approximation in the T range where equilibrium simulations and measurements are normally performed. Indeed, Mauro and co-workers78 have recently claimed that the inverse configurational entropy 1/Sc, or the mass of the cooperatively rearranging regions in the AG model,15 should have an Arrhenius temperature dependence, 1/ Sc = C exp(F/kBT), where C and F are adjustable parameters. As a matter of curve fitting, this functional form has been found to fit relaxation time data for many GF liquids better than the VFT equation. A number of liquids, such as water and silica, and a number of metallic GF liquids,79 however, show a “fragile-to-strong” transition upon sufficient cooling, corresponding to a return to Arrhenius relaxation, as predicted by the string model in its generalized form.25 The interesting thing about this transition is that it can occur in a T regime where the liquid is fully at equilibrium. The position of the fragile−strong transition in the full string model depends on Φl. Following the method described in ref 25, the volume fraction of mobile particles can be estimated as ϕ0 = f 0(π/6) ≈ 0.034 in our simulations. While Φl can be calculated in simulations,25 we treat Φl here as an adjustable parameter so that eq 14 contains three fitting parameters (Φl, ΔHp, and ΔSp). Figure 16 indicates that eq 14 provides an excellent description of our simulation data for both constant V and constant P conditions (see the dotted lines in Figure 16). We summarize the fitted parameters in Figure 18, which indicates that ΔHp decreases linearly with ϵ, but Φl and ΔSp both depend only weakly on ϵ. The interpretation of the variations of these energetic parameters will be left to future work. We obtain a much more mathematically “economical” expression for L inspired by the VFT equation. In particular, L is rather well described below Tl by the relation DLT0 L = L0 + , Ll T − T0

T > Tg

Figure 18. (a) Φl and (b) ΔHp and ΔSp, determined from eq 14 for the T dependence of L in the low T regime, as a function of ϵ at constant V and constant P. Dashed lines in (a) and (b) indicate averages of each data set over ϵ. Solid lines in (b) indicate the fits of data for ΔHp to the equation ΔHp = −h1ϵ, where the fitting parameters h1 are determined to be 1.48 at constant V and 3.12 at constant P, respectively.

constant P. We emphasize that eq 15 should only be applied for a temperature range above Tg, so that a good fit to this expression does not literally imply that L diverges to infinity at T0. The general string model expression25 for L/Ll, applicable at all temperatures below Tl, cannot be expressed in closed analytic form, which explains the absence of such a relation in our discussions above. The need for this more general expression is normally only required at low T where conceptually important new features emerge and where equilibrium measurements are generally difficult or impossible. In general, L/Ll is predicted to vary from unity at Tl to a constant (typically around 4−5) at low T that depends on the high temperature activation energy ΔHa and Φl.25 (This prediction accords with the Doremus model of glass formation.82,83 The ratio of the high T to the low T activation free energy can be taken as a more fundamental measure of “fragility” since this ratio relates to only the change of scale of collective motion from high to low T.) This means that the reduced activation energy L/Ll varies in a roughly sigmoidal fashion upon cooling, and eq 15 is a useful approximation in a middle temperature range of glass formation. We reiterate that the string theory25 implies that there is no divergence in τα at any finite T and that relaxation should return to be Arrhenius in the glassy state, provided that the material can come into equilibrium after a long period of aging. We emphasize that neither the AG model nor its string model extension predicts the presence of any phase transition accompanying glass formation. Measurements by McKenna and co-workers84 on amber samples, arguably aged over many millions of years, would seem to support a return to Arrhenius relaxation at low T. Many other measurements of glasses aged on long time

(15)

where L0 describes the high T limit of L/Ll, the parameter DL quantifies the strength of the T dependence of L, and T0 is the VFT temperature determined from τα. Miller80,81 previously suggested an expression similar to eq 15 as an approximation for z of the AG model.15 The fitted curves to eq 15 are displayed as dash-dotted lines in Figure 16. The fitted values for DL are found to be approximately independent of ϵ, and the average of DL is determined to be 0.43 at constant V and 0.24 at K


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The results are summarized in Figure 19, which shows that the string model quantitatively describes all simulation data in this

scales under more controlled conditions but on more human activity related time scales have also suggested a return to Arrhenius relaxation, but this limiting behavior is difficult to prove absolutely because of the experimental difficulties involved. 3.5. String Model of Relaxation in Glass-Forming Liquids. As noted before, AG15 proposed a rationale for relaxation in GF liquids that the activation free energy barrier for molecular relaxation grows in proportion to the number of particles involved in hypothetical CRRs. While this conceptual picture is quite appealing, no precise molecular definition for the CRRs is provided in the AG theory. The string model of Douglas and co-workers25 provides a quantitative realization of the AG theory where the stringlike clusters provides a concrete realization of the CRRs. The string model is further justified by a recent theoretical analysis by Freed,85 where transition state theory is extended to account for stringlike cooperative barrier crossing events in GF liquids. Predictions from the string model have been confirmed in a range of GF polymer systems2,25,41,42 and metallic GF liquids.65 Here, we test whether the string model can describe our simulation data over the range of ϵ considered. The central prediction of the string model of glass formation is that the activation free energy ΔG for structural relaxation is proportional to the average string length L, where the proportionality factor becomes unity at an onset temperature. If the onset temperature is taken to be Tl, the string model implies the relation ΔG(T ) = ΔG0(L /Ll )

Figure 19. Relationship between the natural logarithm of the reduced structural relaxation time ln(τα/τ0) and ΔG0(L/Ll)/kBT for various ϵ at (a) constant V and (b) constant P. Dashed lines indicate the prediction of the string model, ln(τα/τ0) = ΔG0(L/Ll)/kBT.

(16)

where ΔG0 = ΔH0 − TΔS0 is the activation free energy at high T with ΔH0 and ΔS0 designating the enthalpy and entropy of activation. (In both the AG model15 and the GET,19 ΔG0 is denoted as Δμ. For economy of notation, we term ΔH0 and ΔS0 as the high temperature activation enthalpy and entropy, respectively.) The structural relaxation time then takes the form

⎛ ΔG0 L ⎞ τα = τ0 exp⎜ ⎟ ⎝ kBT Ll ⎠

work. Another feature not anticipated by AG15 is that collective motion still occurs in the high T Arrhenius regime, but the scale of collective motion varies mildly with T above Tl. The relative scale of collective motion replaces z in the AG model. The dependence of Ll on ϵ is examined in Figure 20, where Ll is shown to be nearly independent of ϵ for both constant V and constant P conditions but to be influenced by the thermodynamic path. Figure 21 displays the variations of ΔH0 and ΔS0 with ϵ. We see that ΔH0 grows approximately linearly with ϵ for both constant V and constant P conditions, a trend that is also observed in two- and three-dimensional Lennard-Jones GF

(17)

where τ0 can be determined from a knowledge of τα at the onset temperature2,42 so that the high temperature activation energies ΔH0 and ΔS0 are the only adjustable parameters. Note that the entropic contribution ΔS0 to ΔG0 is assumed to be neglected in the AG model15 as well as in the GET,19 an assumption that is not warranted since ΔS0 can be appreciable in some polymer systems, as discussed in previous work.2,25,41,42 Evidently, the onset temperature plays a key role in the string model. Previous analysis2,25,41,42 indicates that using either TA or Tl leads to an equally good description of simulation data for a variety of systems since TA and Tl are often found to be close to each other. However, we find from the analysis above that the onset temperatures TA and Tl of glass formation, determined from two independent methods, can differ considerably from each other, so we consider both of them in order to test the string model. For constant V conditions, we found that the string model provides an excellent description of our simulation data using either TA or Tl. However, a satisfactory description of all our data at constant P only emerges when Tl is taken as the onset temperature of GF liquids. Therefore, we use Tl as the “onset temperature” in the string model for both constant V and constant P conditions.

Figure 20. Average string length Ll at the localization temperature Tl as a function of ϵ at constant V and constant P. Dashed lines indicate averages of each data set over ϵ. L


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Figure 21. String model parameters ΔH0 and ΔS0 as a function of ϵ at constant V and constant P. Dashed lines indicate averages of each data set over ϵ. Solid lines indicate the fits of data for ΔH0 to the equation ΔH0 = H1ϵ, where the fitting parameter H1 is determined to be 2.20 at constant V and 3.31 at constant P.

Figure 22. Relationship between the Arrhenius parameter ΔE and ΔH0 − Tl × ΔS0 at constant V and constant P. Dashed lines indicate the relation ΔE = ΔH0 − Tl × ΔS0.

4. SUMMARY Building on the results of our previous paper20 investigating the static structure factor and other thermodynamic properties of a model glass-forming polymer melt by molecular dynamics simulation, we systematically investigate the dynamic properties of this same polymer melt through a consideration of the selfintermediate scattering function over a wide range of temperatures and cohesive interaction strengths, ϵ. As opposed to the thermodynamic properties investigated before, we do not find any universal scaling of the dynamic properties, such as the structural relaxation time τα, as a function of ϵ/T, so that changing ϵ is not equivalent to changing temperature when considering polymer relaxation dynamics. We determine the activation enthalpy ΔH0 and entropy ΔS0 in the high temperature regime of glass formation where relaxation is Arrhenius, and we find that ΔH0 increases proportionally with ϵ, consistent with arguments by Eyring and others3−5 that the activation energy should increase with the heat of vaporization, another measure of the cohesive interaction strength that particularly accessible in small molecule liquids. Contrary to assumptions of the Adam−Gibbs model,15 the activation entropy cannot be neglected, and this quantity is found to be surprisingly insensitive to variations in ϵ in our polymer model. Our examination of dynamic heterogeneity in the nonArrhenius relaxation temperature regime focuses on quantifying the extent of cooperative exchange motion with a view of testing the predictions of the string model of glass formation25 for the range of polymer models investigated having variable ϵ. In complete agreement with a former analysis of a different coarse-grained polymer model in which the bond potential of the polymer is represented by a nonlinear rather than an harmonic potential and for a wide range of nanocomposite and thin film systems,2,25,41,42 we find that the string model of glass formation quantitatively describes all relaxation data for both constant volume and constant pressure conditions, providing another striking validation of the string model of glass formation. Such an extensive consideration of glass formation under constant volume and constant pressure conditions was not considered previously, and this analysis reveals that glass formation becomes stronger at constant volume than at constant pressure, a result that accords well with the experimental observations of McKenna and co-workers.53 The string model of glass formation25 leads to predictions of the dynamics of glass formation that are surprisingly consistent with the heuristic relation of AG,15 but the string model goes

liquids.52 On the other hand, the dependence of ΔS0 on ϵ is rather weak. Previous simulation work on polymer glass formation2,41,42 indicated that ΔH0 and ΔS0 both varied proportionally in polymer films and nanocomposite materials, corresponding to an “entropy−enthalpy compensation” relationship. Such a relation is clearly absent in our simulation data, an unexpected effect that likely reflects our use of the fully flexible chains in our coarse-grained polymer model. Figure 21 further indicates that thermodynamic path also has a big influence on both ΔH0 and ΔS0. We need to consider the dependence of the activation free energy parameters on P and how other primary molecular parameters, such as chain length and stiffness, influence ΔH0 and ΔS0. We mention here that Jeong and Douglas8 have recently studied the dependence of these activation free energy parameters on molecular mass in atomistic molecular dynamic simulations of unentangled alkane polymers, where an entropy−enthalpy compensation effect was observed as the chain length was varied. Therefore, our simulations show that the cohesive interaction strength has a significant influence on molecular dynamics even at elevated temperatures so that attractive interactions cannot be neglected. Because the thermally activated transport, conspicuous in the dynamical properties of most fluids, derives from intermolecular attractive interactions, cohesive interaction strength is central to understanding the dynamics of molecular fluids and the occurrence of the condensed fluid state. While the fluids theoretically exist for particles having purely repulsive interactions at a fixed density, e.g. hard-sphere liquids, the activation free energy is then entirely entropic. In general, both enthalpic and entropic contributions to the free energy of activation must be considered. Finally, it is instructive to examine the relationship between the Arrhenius parameter ΔE and the string model parameters ΔH0 and ΔS0, given the fact that while ΔH0 and ΔS0 are determined from the data below Tl, ΔE is instead obtained from the high T data above TA. We particularly consider whether or not the relation ΔE = ΔH0 − Tl × ΔS0 holds for our simulation data. Figure 22 indicates that data for constant V conditions follow this relation to a good approximation. For constant P conditions, deviations from the expected relation are evidently seen, which is understandable since Tl is already located in the non-Arrhenius regime (see Figure 2b). M


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for useful comments on the manuscript. W.-S.X. is grateful to Prof. Juan J. de Pablo and his group members for providing the opportunity to attend their group meeting while working at the University of Chicago, from which the present work has greatly benefited. We are grateful for the support of the University of Chicago Research Computing Center for assistance with the simulations carried out in this work. This work is supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award DE-SC0008631.

much further by prescribing that the geometric form of the collective motion is polymeric, by emphasizing the polydispersity of the “coopeartively rearranging regions” or “strings”, and by identifying the number z of particles in these clusters as being the average size of these precisely defined and observed clusters, the string length L, rather than some ill-defined minimal scale of particle rearrangement suggested by AG.15 Even though there are differences in detail between AG and the string model, previous2,25,41,42 and our simulations remarkably accord with the spirit of the AG model. We clearly observe clusters of cooperatively rearranging particles, where the number of particles in these clusters influences the activation free energy. The string model takes things even further by formulating a statistical mechanical theory of polymerization that allows a formal treatment of relaxation at much lower temperatures than accessible by equilibrium simulations,25 but we do not pursue the results of this theory to extrapolate to the low temperature “glass” regime in the present work. Freed85 has recently formulated a version of transition state that accounts for the collective barrier crossing events seen in the simulations, placing the string model of glass formation on a firmer foundation. The wheel of observation and modeling turns toward an increasingly refined understanding of the nature of glass formation. Since the generalized entropy theory of glass formation is an amalgamation of the AG theory15 and an analytic statistical mechanical theory of the thermodynamic properties of polymer materials, our results also have great significance for advancing this analytic and predictive theory of relaxation in glass forming polymer liquids starting from simulations for a detailed molecular structural formulation. In particular, we find that it is simply incorrect to neglect the entropy of activation, as originally assumed heuristically by AG.15 Including this contribution to the free energy of activation has recently been shown to greatly improve the prediction of relaxation in the string model,2,25,41 and the inclusion of this entropic term of activation is expected to improve the predictive capability of the generalized entropy theory as well. Further work is needed to understand how the high temperature activation energy parameters vary with fluid molecular structure, confinement, mixtures, etc., as well as to further explore the thermodynamic properties of fluids. Knowledge of these relationships should allow more predictive modeling of the dynamics of polymer melts in terms of molecular structural and input thermodynamic information.

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REFERENCES

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AUTHOR INFORMATION

Corresponding Authors

*E-mail [email protected] (K.F.F.). *E-mail [email protected] (W.-S.X.). *E-mail [email protected] (J.F.D.). Present Address

W.-S.X.: Center for Nanophase Materials Sciences, Oak Ridge National Laboratory, Oak Ridge, TN 37831. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Prof. Salvatore Torquato (Princeton University) for helpful discussions on hyperuniformity and valuable comments on the manuscript, Prof. Francis Starr (Wesleyan University) for helpful conversations, and Dr. Alexandros Chremos (NIST) N


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