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# Cooperative strings and glassy interfaces

Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved May 28, 2015 (received for review February 13, 2015)

## Significance

According to Philip Anderson, the deepest and most interesting unsolved problem in solid-state physics is probably the glass transition. By extension, this includes the highly debated confinement effects in glassy polymer films. The present article introduces a minimal analytical model, which invokes only the ideas of molecular crowding and string-like cooperative rearrangement, before addressing the key effects of interfaces. The validity and simplicity of the approach make it ideal for application to various systems and geometries, and suggest that dynamics in glass-forming materials might be understood from elementary arguments.

## Abstract

We introduce a minimal theory of glass formation based on the ideas of molecular crowding and resultant string-like cooperative rearrangement, and address the effects of free interfaces. In the bulk case, we obtain a scaling expression for the number of particles taking part in cooperative strings, and we recover the Adam–Gibbs description of glassy dynamics. Then, by including thermal dilatation, the Vogel–Fulcher–Tammann relation is derived. Moreover, the random and string-like characters of the cooperative rearrangement allow us to predict a temperature-dependent expression for the cooperative length *ξ* of bulk relaxation. Finally, we explore the influence of sample boundaries when the system size becomes comparable to *ξ*. The theory is in agreement with measurements of the glass-transition temperature of thin polymer films, and allows quantification of the temperature-dependent thickness *h*_{m} of the interfacial mobile layer.

Glassy materials are ubiquitous in nature (1), and discussions about the glass transition involve many areas of physics, from molecular and spin glasses to hard-sphere jamming (2⇓⇓⇓⇓–7). Despite the intense interest in the dynamical slowing that accompanies glass formation, a single microscopic theory has yet to emerge (8⇓⇓⇓⇓–13). Nevertheless, the phenomenological approach of free volume (14) and the Doolittle ansatz (15) have been used to support the Vogel–Fulcher–Tammann (VFT) relation (16⇓–18), which describes so many of the observed behaviors. Fundamental to glass formation are the suggestions that particles are increasingly crowded, and relaxation requires the cooperative participation of a growing number of particles. The hypothesis of a cooperatively rearranging region, as introduced by Adam and Gibbs (19), is appealing and has been observed in computational studies (20, 21).

The existence of a length scale *ξ* for cooperative rearrangement (22) has led to tremendous interest in confined glass formers, as initiated by ref. 23. Perhaps, the most active example of attempts to probe *ξ* is the study of glassy polymer films (24⇓–26), where fascinating observations have been made. For the most studied case of polystyrene, reductions in the measured glass-transition temperature have been almost uniformly reported as the film thickness is reduced, both experimentally (27) and numerically (28). It has been further suggested that this apparent anomaly is linked to the observed existence of a more mobile interfacial layer (29⇓⇓–32). As a consequence, there have been many theoretical attempts to understand the thin-film glass transition, with varying degrees of complexity and success (33⇓⇓⇓⇓–38).

In this article, we present a simple analytical model for relaxation in glass-forming materials. First, from a microscopic molecular picture, the nature of the cooperative mechanism is explicitly defined and characterized as a function of density, and the Adam–Gibbs phenomenology is recovered. Then, by including thermal expansivity, we derive the VFT relation for the temperature dependence of the relaxation time in bulk materials. Finally, to address the effects of interfaces, the theory is applied to the case of thin films.

Beyond any formulation, there are two main ingredients that a microscopic cooperative theory must contain: (*i*) “more cooperative is easier,” and (*ii*) “more cooperative is rarer.” The first one means that to redistribute a given amount of volume in a crowded environment, a cooperative rearrangement is energetically more favorable than a solitary one––because the former is the sum of *N* small displacements, which is easier to satisfy than a single large displacement. This effect tends to maximize *N*. The second ingredient relies on the fact that glass-forming materials are made of independent particles that move incoherently due to thermal fluctuations. Therefore, it is relatively rare to have motions that are coherent in time and space and form collective objects: the larger *N*, the rarer the event. This effect tends to minimize *N*. Taken together, these two ingredients suggest a most probable value of *N*. Changing the temperature may change the crowding constraints and/or the coherence penalty, which results in the temperature dependence of this most probable value and thus the glassy behavior of interest. From numerical simulations (20, 21, 39) comes another important feature: it has been observed that the cooperative regions often have a fractal dimension close to 1. This was also reported in experimental studies of repulsive colloids (40). Therefore, we add a third ingredient to the list above: (*iii*) “cooperative rearrangement is string-like.” In the following, our goal is to build the simplest mean-field toy model that contains *i*, *ii*, and *iii*.

As shown in Fig. 1, we consider an assembly of particles of effective radius *r*, average intermolecular distance *λ*, and volume fraction

As a reference, we consider the liquid-like case of a particle escaping from its cage by solitary motion (Fig. 1*A*). On average, such motion is allowed if

When *δ* can be locally and temporarily made available, thus allowing for a rearrangement, as observed in bidisperse hard disks (43). Inspired by computational studies (20, 21, 39) and experiments (40), we consider cooperative regions in the form of string-like random chains (Fig. 1*B*). Because the gate between particles has an average length *L*, the additional length created by the cooperative move is **1** could have been assumed, independently of any details on the string-like microscopic picture, because it is the simplest expression having the expected properties: it equals 1 at the cooperative onset *ε* is the elementary coherence probability to be determined below, and the termination factor *B*). Finally, the two features above can be combined to express the probability density of a cooperative relaxation process involving *N* particles:*ϕ*.

Summing the *τ* and Eq. **3**, through

Having described the effect of crowding on the string-like cooperativity, we now study the glass transition of bulk systems. In particular, we characterize the relaxation time as a function of temperature *T*, by coupling the previous density-based picture to the thermal expansion coefficient **1** and **4**, one directly derives the VFT relation (16⇓–18), or equivalently the Williams–Landel–Ferry (WLF) relation (48), respectively:*τ* reaches a given large experimental time scale

Because our model leads to the VFT and WLF time–temperature superpositions, it captures well the so-called fragile-glass phenomenology (50), and links thermal expansion and fragility as observed in metallic glasses (51). Strong-glass phenomenology can be recovered as well when

The bulk relaxation process presented above consists of random cooperative strings and is entirely determined by *ξ* of the cooperative regions reads **1** and thermal expansivity, one obtains:*ξ* diverges with exponent *ξ* (23, 24, 26).

We now turn to the case of a thin film of thickness *h*, supported on an inert substrate. Our model predicts that the key effect of the free interface is to favor a higher surface mobility, as observed in experiments (27, 29⇓⇓–32). Indeed, the cooperative strings can now be truncated by the cage-free boundary (Fig. 3), leading to a lower surface cooperativity. Therefore, we introduce the average local cooperativity *z* from the free interface. The natural length scale of the problem is the bulk cooperative length *ξ*. When *z* over the bulk cooperative length *ξ*, and is expected to have the following asymptotic self-similar form in the vicinity of *f* is a continuous and monotonic function satisfying

Because our description of the cooperative process involves random strings of particles, Eq. **7** can be supported at large *g* of first passage at the interface with **7** with *f* is not crucial when comparing to the experimental data below, as other sufficiently sharp functions provide similar results.

Following the derivation of Eq. **4** and assuming that one can replace **7**, one obtains the local relaxation time:*f* acts as a local exponent, ranging from 0 to 1, on the normalized bulk relaxation time. This formula generalizes the Adam–Gibbs phenomenology (19) by accounting for the effect of a free interface.

Finally, we compare our theory to dilatometric measurements of reduced glass-transition temperatures in thin polystyrene films supported on silicon substrates. In the experiments, the thickness-dependent glass-transition temperature *f* is monotonic, this translates to the apparent transition occurring when, at the middle **5** and **8**, *f*. The solution of this equation is plotted in Fig. 4, and compared with measurements on polystyrene (61, 62). The literature data encompass a wide variety of techniques and protocols (61), and for purposes of fitting we consider the restricted data of ref. 62, where the annealing conditions and atmosphere have been carefully controlled and documented. We see from Fig. 4 that our model provides excellent agreement with the experiments. The two adjustable parameters are the critical interparticle distance **6** and **9**, one would obtain larger, smaller, or even immeasurable reductions in

As one notices in Fig. 4, and because *ξ* does not vary much around **6**, the cross-over thickness at which the measured glass-transition temperature first shows deviations from the bulk value is a few

Because the comparison between theory and experiments (Fig. 4) provides an estimate of the critical interparticle distance *ξ* increases with reducing temperature and diverges at *ξ* and *ξ*, or does it introduce another independent length scale through the mobile-layer thickness **5** and **8**, this implies:**10** and Fig. 5, *ξ* is a bulk quantity whereas

To conclude, we have developed a cooperative-string model that connects in a predictive manner essential ingredients of the glass transition, in bulk systems and near interfaces. The theory is based only on the idea of cooperativity required by increasing molecular crowding, and introduces a string-like cooperative mechanism that is motivated by recent studies. An outcome of our idealized microscopic description is to recover the Adam–Gibbs picture, as well as the VFT relation, without the need for the Doolittle ansatz to link free volume and relaxation. In particular, we derive explicit scaling expressions for the cooperativity and associated relaxation probability. Furthermore, the simplicity of the model enables application to reported anomalies in the glass transition of thin polymer films. Specifically, the free interface truncates the cooperative strings and thus enhances the mobility in its vicinity. Agreement between the present theory and reported dilatometric measurements of the glass-transition temperature of supported polystyrene films is excellent. The two adjustable parameters are the critical interparticle distance at kinetic arrest, which is found to be similar to the persistence length of polystyrene; and the Vogel temperature, which is found to be close to literature values. Finally, the model provides a way to distinguish between purely finite-size and surface effects, and to clarify the existing link between cooperative length and mobile-layer thickness. Importantly, the success of the theory applied to the thin-film data suggests that thin-film experiments are indeed relevant probes of the length scale of bulk cooperative dynamics that may exist independently of any structural length scale in the material. This approach may be refined with additional cooperative processes, and could be adapted to the cases of attractive substrates, freestanding films, or other geometric confinements, whose effects on the measured glass-transition temperature may be crucial.

## Acknowledgments

The authors thank Yu Chai, Olivier Dauchot, Jack Douglas, Mark Ilton, Florent Krzakala, and James Sharp for insightful discussions. This research was supported in part by the Natural Sciences and Engineering Research Council of Canada and the Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Economic Development

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. Email: thomas.salez{at}espci.fr or jforrest{at}perimeterinstitute.ca.

Author contributions: T.S., K.D.-V., E.R., and J.A.F. designed research; T.S., J.S., K.D.-V., E.R., and J.A.F. performed research; and T.S., K.D.-V., E.R., and J.A.F. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

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