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# Quantitative relations between cooperative motion, emergent elasticity, and free volume in model glass-forming polymer materials

Edited by Pablo G. Debenedetti, Princeton University, Princeton, NJ, and approved January 28, 2015 (received for review September 30, 2014)

## Significance

Diverse viewpoints have been developed to understand the scientifically fascinating and universal dynamics of glass-forming fluids. Currently, there are several prevailing models in the scientific literature based on seemingly different physical conceptions of glass formation, a fact that limits both theoretical and technological development in many scientific fields. We address this fundamental problem by simulating polymer glass-forming materials having a wide variation in the temperature dependence of structural relaxation (“fragility”), and we show by direct comparison that existing models equally describe our data, revealing deep relations between them. In this way, we achieve a greater theoretical unity of understanding glass-forming materials that should aid many applications in materials development and biology, the preservation and aesthetic properties of food, and medical science.

## Abstract

The study of glass formation is largely framed by semiempirical models that emphasize the importance of progressively growing cooperative motion accompanying the drop in fluid configurational entropy, emergent elasticity, or the vanishing of accessible free volume available for molecular motion in cooled liquids. We investigate the extent to which these descriptions are related through computations on a model coarse-grained polymer melt, with and without nanoparticle additives, and for supported polymer films with smooth or rough surfaces, allowing for substantial variation of the glass transition temperature and the fragility of glass formation. We find quantitative relations between emergent elasticity, the average local volume accessible for particle motion, and the growth of collective motion in cooled liquids. Surprisingly, we find that each of these models of glass formation can equally well describe the relaxation data for all of the systems that we simulate. In this way, we uncover some unity in our understanding of glass-forming materials from perspectives formerly considered as distinct.

There are numerous theoretical approaches aiming to describe the universal liquid dynamics approaching the glass transition. One class of theories emphasizes the importance of the congested nature of the local atomic environment in cooled liquids, focusing on the amount of “free volume” available to facilitate molecular rearrangement (1). This free-volume approach is also linked to the more modern jamming model of glass formation (2). Older treatments of glass formation based on this perspective can be traced back to Batchinski (3), Doolittle (4), and Hildebrand (5) for small liquids, and to Williams and coworkers (6) and Duda and Vrentas (7, 8) for polymer materials. There is also more recent work based on the free-volume perspective, for example, positron lifetime measurements (9) that probe the cavity structure of glass-forming (GF) liquids. Debye–Waller measurements (9, 10), based on neutron, X-ray, or other scattering measurements, emphasize another type of free volume that is associated with the volume explored by particles as they rattle about their mean positions in a condensed material. This type of free-volume modeling has also been refined to take into account the shape of these “rattle” volumes (11, 12).

Another family of glass-formation models emphasizes the emergent elasticity in glassy materials (13). These approaches build on the idea that the solid-like nature of glasses is one of their most conspicuous, and perhaps defining, properties. Dyre (13) and Nemilov (14) have argued that the activation energy for transport should grow in proportion to the shear modulus. The models of Hall and Wolynes (15) and Leporini and coworkers (10, 16) can also be included in this class if the Debye–Waller factor is taken as a measure of local material stiffness.

Approaches emphasizing the underlying complex potential energy surface have also found considerable phenomenological success (17). The venerable Adam–Gibbs (AG) theory of glass formation (18), and the more recent random first-order transition theory (19), emphasize the temperature dependence of the configurational entropy in cooled liquids and its relation to collective motion, although these theories do not explicitly define the form of the “cooperatively rearranging regions” (CRRs). This approach can be extended by identifying these CRRs with string-like clusters of cooperative particle exchange motion (20⇓–22) and analytic calculation of the configurational entropy (23). In addition to these approaches to glass formation, the mode-coupling theory (24), and dynamic facilitation models (25) postulate a “dynamical” glass transition that is unrelated to any underlying thermodynamic transition.

The diverse range of models for glass formation reminds us of the story of the blind men and the elephant, where they grasp at the elephant and describe its attributes in terms of the different parts of which they have happened to take hold. In this respect, all these various approaches to understand glass formation may be “valid,” but are simply focusing on different manifestations of a larger beast.

As a step toward bringing together some of these seemingly disparate ideas, the present paper explores the extent to which the thermodynamic perspectives of glass formation in terms of elasticity, collective motion, and vibrational free volume represent complementary perspectives of the same complex object, i.e., GF liquids. In particular, we consider the potential correspondence among perspectives through the direct computation of the relationship between the structural relaxation time determined from the density correlations and molecular free volume, defined in terms of the Debye–Waller factor *L* of string-like molecular displacements. We find that all these approaches offer an accurate description of our relaxation time data for a model bulk polymer melt, polymer–nanoparticle composites, and supported polymer films—immediately implying quantitative relationships between the scale of collective motion, Debye–Waller factor, and emergent elasticity in cooled liquids. The differing models of GF liquids indeed involve different perspectives on essentially the same phenomenon. Several recent papers have sought to establish quantitative relations between emergent elasticity, free volume, and configurational entropy theories of glass formation, but with inconclusive results (26⇓–28).

## Results

### Free-Volume Model of Relaxation.

It has long been appreciated, both intuitively and theoretically, that the equilibrium and transport properties of fluids depend on the space available for molecular motion, but the lack of methods to accurately compute or measure free volume has limited the development of this perspective. Batchinski noticed that the viscosity of many simple fluids is nearly independent of temperature at constant volume at elevated temperatures (3), suggesting the applicability of a free-volume description of molecular transport in liquids. Hildebrand (5) and Hildebrand and Lamoreaux (29) developed Batchinski’s phenomenological relation further by introducing a critical reference volume *η* prompted theoretical efforts to quantify free volume and to rationalize Doolittle’s observations. The free-volume model was also greatly influenced by Fox and Flory (31⇓–33), who interpreted the Doolittle expression as implying that the glass transition corresponds to a vanishing of “sufficient” free volume for molecular movement and implying a physical interpretation of Doolittle’s free-volume parameter,

The Batchinski–Doolittle relation (Eq. **1**) also offers one possible explanation for the widely used, empirical Vogel–Fulcher–Tammann (VFT) relation. In particular, if the specific volume is reasonably taken to vary linearly with temperature in the range of glass formation, **1** becomes the VFT equation,*D* is a dimensionless constant that quantifies the strength of the *T* dependence of *η*. The reciprocal of *D* offers one definition of the fragility of glass formation (34). The same expression is normally argued to apply to the diffusion coefficient, structural relaxation time *τ*, and other transport properties.

Fig. 1 shows the applicability of the VFT equation to all our simulation data for the relaxation time *τ* of the coherent intermediate scattering function (density–density correlations), including data for the pure polymer melt, polymer nanocomposites (35), and thin polymer films (36, 37). This description of our data is uniformly excellent from a numerical standpoint, but the free-volume model provides little insight into the magnitude of *D*, and

### Emergent Elasticity and Relaxation.

To address questions relating to elasticity and relaxation, we first must identify an appropriate and physically accessible measure of material “stiffness.” Both experiments and simulations have recently emphasized that the Debye–Waller factor

We next explore the quantitative relation between *τ* and **1**) suggests a proportionality of

The localization model of relaxation (12) also starts from a free-volume perspective for relating τ and *α* is a measure of free-volume anisotropy, and *α*. Simmons et al. treated the parameters *α* as fit parameters (12), and our data can be well described by Eq. **3**, where these parameters are allowed to vary freely. Similarly, the model of Leporini and coworkers (10, 16) fits just as well if the same number of parameters is allowed to vary, so it is clearly desirable to reduce the number of free parameters to better understand their physical origin and have a more predictive relationship.

To do so, we take the localization model (Eq. **3**) further by defining the parameters *Materials and Methods*. Accordingly, **3** requires that *e* is Euler’s constant). With these definitions, Eq. **3** becomes*α*, because

We now test the validity of Eq. **4** to quantitatively describe our simulation results. Fig. 2 shows the scaled relaxation data **4**. For both cases, nanocomposites and thin films, the data nearly collapse to a master curve, described by Eq. **4** where *SI Text*, we also consider fixing

### Cooperative Motion and Relaxation.

AG (18) proposed an intuitively appealing and enduring conceptual picture for relaxation in GF liquids in which the activation free-energy barrier for molecular relaxation is assumed to increase in proportion to the number particles involved in hypothetical CRRs. The random first-order transition (RFOT) theory of Lubchenko and Wolynes (19) is related to the AG theory, in the sense that it also postulates dynamic CRR clusters (“entropic droplets”) whose geometrical size (rather than the number of particles) determines the activation barrier for relaxation. The conception of such dynamic clusters has framed many modern investigations of dynamical heterogeneity in GF fluids, but, unfortunately, neither the AG nor RFOT theories offers a prescription for defining the CRRs. Simulation has led the way in defining the existence and precise nature of cooperative motion in cooled liquids. In particular, several studies have established that the activation free-energy barrier *L* that can be extended to the glass transition (21). Recently, Freed (22) provided an analytic extension of transition state theory that accounts for string-like cooperative barrier crossing events, providing a theoretical basis for the string model extension of the AG description (21). Our analysis of data starts from this fully developed “string model” of relaxation, a quantitative descendant of the AG model that preserves the original AG conception of the physical nature of glass formation.

The central prediction of the string model of glass formation is that the activation free energy for structural relaxation is proportional to the average string length *L*, where the proportionality factor is unity at

Similar to our approach to reduce the number of free parameters in the localization model, we can reduce the number of adjustable parameters in Eq. **5** through the introduction of the reference values **5** then implies,**6**. Conveniently, these basic transition state theory parameters can be determined by simulations at high *T*, and have a definite physical meaning in transition state theory.

We test the validity of Eq. **6** to quantitatively describe our simulation results in Fig. 4, which show a remarkable collapse of all data, supporting the validity of this approach. Fig. 4 (*Insets*) shows that the activation parameters exhibit a linear entropy–enthalpy “compensation” relation, a phenomenon commonly observed in the dynamics and thermodynamics of condensed materials (47⇓–49). Obviously, such a compensation relation cannot be recognized if *β*-relaxation in GF liquids (56⇓–58). Our data reduction in terms of the scale of collective motion is equally as compelling as the relation indicated above between *τ* and

By extension, consistency between these relations for *τ* implies a direct and precise relation between *L*, so that**7**) for *T* dependence of the extent of cooperative motion (21). Combining the present results with the theory of ref. 21, we can then predict the *T* dependence of *L* and the fluid configurational entropy **7** implies a curious relation between

## Conclusions

We have examined well-defined experimental molecular-scale measures of material elasticity, free volume, and the scale of cooperative motion in a class of model polymeric GF liquids whose fragility is varied over a large range by varying the nanoparticle concentration or film thickness. This unified analysis reveals that the description of the structural relaxation time *τ* obtained from the coherent intermediate scattering function can be quantitatively described in terms of each of these perspectives of glass formation. We find that the introduction of mathematical consistency conditions and a definition of the scale of *τ* and *τ* in terms of an apparently distinct relation involving the scale of collective motion *L* and the high-temperature Arrhenius activation parameters, *L*, and a measure of the emergent elasticity of GF liquids,

## Materials and Methods

### Data Analysis.

The structural relaxation time *τ* is obtained by evaluating the coherent intermediate scattering function, *T* dependence than the self part. However, we anticipate our results would not differ in a qualitative way if we used only the self part of the relaxation function for our analysis. The “onset” temperature *τ*(*T*) departs from Arrhenius behavior, **4** and **5**, as described in the main text.

### Computational Model.

Our findings are based on equilibrium molecular dynamics simulations of a common “bead–spring” model for polymer chains. Previous studies (34⇓⇓–37, 61) have shown that we can introduce substantial changes to the polymer relaxation time and its *T* dependence by the addition of nanoparticles to form nanocomposites, or by confinement in supported thin films. Consequently, these systems offer us a way to systematically vary both *ε* and *σ* are the energy and length parameters of the LJ potential, respectively. We use periodic boundary conditions so that we mimic a perfect cubic lattice of nanoparticles (NP) with a variable NP concentration that determines their separation and consider both attractive and nonattractive interactions between NP and the polymer matrix. We simulate a wide range of NP concentration for both systems, because the type of NP interaction and NP concentration alters differently

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. Email: bpazminobeta{at}wesleyan.edu, fstarr{at}wesleyan.edu, or jack.douglas{at}nist.gov. ↵

^{2}Present address: Department of Physics, Boston University, Boston, MA 02215.

Author contributions: B.A.P.B., F.W.S., and J.F.D. designed research; B.A.P.B., P.Z.H., F.W.S., and J.F.D. performed research; B.A.P.B., P.Z.H., and F.W.S. analyzed data; and B.A.P.B., F.W.S., and J.F.D. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1418654112/-/DCSupplemental.

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