## New Research In

### Physical Sciences

### Social Sciences

#### Featured Portals

#### Articles by Topic

### Biological Sciences

#### Featured Portals

#### Articles by Topic

- Agricultural Sciences
- Anthropology
- Applied Biological Sciences
- Biochemistry
- Biophysics and Computational Biology
- Cell Biology
- Developmental Biology
- Ecology
- Environmental Sciences
- Evolution
- Genetics
- Immunology and Inflammation
- Medical Sciences
- Microbiology
- Neuroscience
- Pharmacology
- Physiology
- Plant Biology
- Population Biology
- Psychological and Cognitive Sciences
- Sustainability Science
- Systems Biology

# Hopping and the Stokes–Einstein relation breakdown in simple glass formers

Contributed by Giorgio Parisi, September 7, 2014 (sent for review June 9, 2014)

## Significance

Like crystals, glasses are rigid because of the self-caging of their constituent particles. The key difference is that crystal formation is a sharp first-order phase transition at which cages form abruptly and remain stable, whereas glass formation entails the progressive emergence of cages. This loose caging complicates the description of the glass transition. In particular, an important transport mechanism in this regime, hopping, has thus far been difficult to characterize. Here we develop a completely microscopic description of hopping, which allows us to clearly assess its impact on transport anomalies, such as the breakdown of the Stokes–Einstein relation.

## Abstract

One of the most actively debated issues in the study of the glass transition is whether a mean-field description is a reasonable starting point for understanding experimental glass formers. Although the mean-field theory of the glass transition—like that of other statistical systems—is exact when the spatial dimension *d* may not be smooth. Finite-dimensional effects could dramatically change what happens in physical dimensions,

Glasses are amorphous materials whose rigidity emerges from the mutual caging of their constituent particles—be they atoms, molecules, colloids, grains, or cells. Although glasses are ubiquitous, the microscopic description of their formation, rheology, and other dynamical features is still far from satisfying. Developing a more complete theoretical framework would not only resolve epistemological wrangles (1), but also improve our material control and design capabilities. However, such a research program remains fraught with challenges. Conventional paradigms based on perturbative expansions around the low-density, ideal gas limit (for moderately dense gases and liquids) or on harmonic expansions around an ideal lattice (for crystals) fail badly. Because dense amorphous materials interact strongly, low-density expansions are unreliable, whereas harmonic expansions lack reference equilibrium particle positions. These fundamental difficulties must somehow be surmounted to describe the dynamical processes at play in glass formation.

A celebrated strategy for studying phase transitions is to consider first their mean-field description, which becomes exact when the spatial dimension *d* of the system goes to infinity (2), before including corrections to this description. In that spirit, we open with the *D* vanish as a power-law *i*) a sharp dynamical glass transition associated with perfect caging, (*ii*) a power-law divergence of *η*, and (*iii*) the SER being obeyed.

As observed in ref. 12, the phenomenology of finite-dimensional systems is, however, quite different from the iRFOT scenario. In particular, it does not recapitulate elementary experimental observations, such as Vogel–Tammann–Fulcher (VTF) viscosity scaling in fragile glasses,

Part of the difficulty of clarifying the situation in finite *d*, where the iRFOT description is only approximate and the dynamical transition is but a crossover, lies in the shear number of different contributions one has to take into account. From a purely field-theoretic point of view, one has to include finite-dimensional corrections to critical fluctuations. A Ginzburg criterion gives *i*) In the iRFOT picture, caging is perfect, and hence in the glass phase each particle is forever confined to a finite region of space delimited by its neighbors (6). However, it has been theoretically proved (21) and experimentally observed (22) that in low-dimensional systems the diffusivity is never strictly zero. Single particles can indeed hop between neighboring cages (23⇓⇓–26), and the free space they leave behind can facilitate the hopping of neighboring particles. Facilitation can thus result in cooperative hopping and avalanche formation (27⇓–29). (*ii*) For some glass formers, activated crystal nucleation cannot be neglected and interferes with the dynamical arrest, leading to a glass composed of microscopic geometrically frustrated crystal domains (30). (*iii*) In the iRFOT scenario, the dynamical arrest is related to the emergence of a huge number of distinct metastable glass states whose lifetime is infinite. In finite dimensions, however, a complex glass–glass nucleation process gives a finite lifetime to these metastable states (5, 12, 31). The dynamics of glass-forming liquids are then profoundly affected. Including glass–glass nucleation into iRFOT leads to the complete RFOT scenario (12), in which the mean-field dynamical glass transition becomes but a crossover (12), and both the VTF scaling and facilitation can be recovered (32, 33).

Because the treatment of these different processes has thus far been mostly qualitative, their relative importance cannot be easily evaluated. A controlled first-principle, quantitative treatment is for the moment limited to the exact solution for *d* (6, 8, 36) completely ignores the nonperturbative effects mentioned above. This approach therefore cannot, on its own, cleanly disentangle the various corrections. Systematic studies of glass formation as a function of *d* have encouragingly shown that these corrections are limited, even down to *d* the distribution of particle displacements (the self-van Hove function) loses its second peak associated with hopping (16), the critical power-law regimes lengthen (41), and the SER breakdown weakens (15, 16, 40), which motivates investigating corrections to iRFOT in a controlled way.

Here we develop a way to isolate the simplest of these corrections, i.e., hopping, by studying a finite-dimensional mean-field model. Through the use of the cavity reconstruction methodology developed in the context of spin glass and information theory (42), we carefully describe caging, using self-consistent equations that can be solved numerically. We can thus compute the cage width distribution and isolate hopping processes. Our results provide an unprecedentedly clear view of the impact of hopping on the dynamical transition and on the SER breakdown in simple glass formers.

## MK Model

We consider the infinite-range variant of the hard sphere(s) (HS)-based model proposed by Mari and Kurchan (MK) for simple structural glass formers (43⇓–45) (details in *SI Text*, section I.A). The key feature of the MK model is that, even though each sphere has the same diameter *σ*, pairs of spheres interact via an additional constant shift that is randomly selected over the full system volume. This explicit quenched disorder eliminates the possibility of a crystal state, suppresses coherent activated barrier crossing that leads to glass–glass nucleation (44), and diminishes the possibility of facilitated hopping (as we discuss below). However, at finite densities the number of neighbors that interact with a given particle is finite and therefore finite-dimensional corrections related to hopping remain, in principle, possible.

MK liquids have a trivial structure. Even in the dense, strongly interacting regime, the pair correlation in the liquid phase is simply *i* and *k* are nearby particle *j*, they need not be close neighbors, and hence all higher-order structural correlations are perfectly factorizable. Because only two-body correlations contribute, the virial series can be truncated at the second virial coefficient (44), and hence the equation of state for pressure is trivially *d*-dimensional hard spheres, *d*-dimensional ball of radius *R*, *ρ* is the number density [the packing fraction *β* is set to unity (43⇓–45) (*SI Text*, section I.A). Note that these structural features hold for the liquid phase of the MK model in all *d* and for standard HS liquids in the limit *d*, however, MK liquids are structurally more similar to their

For the MK model, one can easily construct equilibrated liquid configurations at all *φ*, even for *SI Text*, section I.B). It is thus possible to study MK liquids arbitrarily close to, both above and below, the dynamical glass transition at *SI Text*, section II.A), a method adapted from the statistical physics of random networks (42).

## Caging

The MK model dynamics are studied by event-driven MD simulations of planted initial configurations with *SI Text*, section I.B) (37, 38). The mean square displacement (MSD) *SI Text*, section I.C). Simply put, after a few collisions with its neighbors, a particle becomes confined to a small region of space of linear size

In the *SI Text*, section II.A). One can also estimate *A*). As expected from the suppression of various finite *d* corrections, the critical power-law regime is much longer for the MK model than for standard finite-dimensional HS (Fig. 1*B*) (44). Marked qualitative discrepancies from the iRFOT predictions are nonetheless observed: (*i*) Numerical estimates for *C*), even though the two quantities grow closer with dimension. (*ii*) The diffusion time *SI Text*, section I.C) follow the SER, *d* (Fig. 1*D*). With increasing *d*, however, the timescale for this crossover, *D*), and thus *C*). (*iii*) Even above *A*), but the magnitude of this effect diminishes with increasing *d*.

To clarify the physical origin of the above discrepancies, we first determine whether the mismatch between *i* to self-consistently determine the overall cage size and/or shape distribution *SI Text*, section II.A). The process involves placing Poisson-distributed neighbors *j* that are randomly assigned a cage size *i*, which is the probability density of the particle being at position **r** (Fig. 2*A*). The existence of a cage centered around *i* is guaranteed by the cavity reconstruction procedure. The variance *i* within this cage, which can be computed through simple Monte Carlo sampling, provides the posterior caging radius *B*) and that *C*). The average cage size *D*), including its characteristic square-root singularity upon approaching *SI Text*, section II.A).

It follows that deviations from the *A*). Because the above calculations solely consider single-cage forms, a fixed-point distribution *SI Text*, section II.A). Not only does *SI Text*, section II.C). This equivalence between cavity reconstruction and void percolation sheds light on the single-cage assumption. In the iRFOT description, the MSD of each particle should remain finite when

From MD simulations of the MK model, we detect the first hopping event of each particle (details in *SI Text*, section III.A). Around *φ*. Although the hopping of a particle does not leave an empty void in the MK model, it can nonetheless unblock a channel for a neighboring particle to leave its cage and hence facilitate its hopping. Facilitation is thus present, but weaker than in standard finite-dimensional HS, especially at high densities. Weakened facilitation is notably signaled by the fact that the distribution of hopping times computed from a regular MD simulation largely coincides with the distribution obtained in the cavity procedure, where a single particle hops in an environment where neighboring particles are forbidden to do so (Fig. 3*B*, *Inset*). We find the cumulative distribution of hopping times over the accessible dynamical range to be well described by a power law *B*), with the characteristic hopping time *d* (Fig. 3*D*). This Arrhenius-like scaling form is consistent with a gradual and uncorrelated narrowing of the hopping channels with *φ*. Note that similar phenomenological power-law distributions have recently been reported for other glass-forming systems, such as the bead-spring model for polymer chains (51). We get back to this point in *Conclusions*.

## Finite-Dimensional Phase Diagram

A clear scenario for hopping in the MK model follows from this analysis (Fig. 4). Dynamically, the system becomes increasingly sluggish upon increasing *φ* above *A*), although the fitting parameter

The dynamics can also be understood from the organization of cages. The critical density *B*). Based on this analysis, in the absence of facilitation the dynamical arrest should take place at *d*. Hopping is then infinitely suppressed because both the width and the number of hopping channels between cages vanish. However, even if hopping interferes with caging, well above *SI Text*, section III.A).

As expected from the exactness of the iRFOT description in *d*. Both *γ* also appear to converge to the *B*) (34). Because *d*, the suppression of hopping with increasing *d* (Fig. 1D, *Inset*) ought to be ascribed either to the narrowing of the hopping channels or to topological changes to the cage network. Because the pressure at the dynamical transition increases only slowly with dimension *φ* because the loss of the cage network fractality takes place through the single-point inclusion of nonpercolating clusters (52). The network topology is therefore such that the hopping channels (even assuming that their cross section remains constant) cover a vanishingly small fraction of the cage surface as *d* increases. The limited number of ways out of a local cage thus entropically suppresses hopping.

## SER Breakdown

With hopping events clearly identified, it becomes possible to isolate the pure critical iRFOT (or mode-coupling) regime. Within this regime, we obtain a power-law scaling that is consistent with *SI Text*, section II.B), and the SER is followed. Deviations from the extrapolated critical scaling coincide with the SER breakdown in all *d*. Although *d*,

By modifying the cavity reconstruction analysis, a self-consistent caging determination of *d* increases (Fig. 5), whereas

Contrasting Figs. 1*D* and 5*A* suggests that near *ω* is similar for HS and the MK model. In this regime, HS hopping is consistent with MK-like hopping. In HS, however, single-particle hopping leaves an actual structural void that enhances the correlation (and hence the facilitation) of hopping events (27⇓–29). As HS become more sluggish, cooperativity plays a growing role. As a result, a pronounced difference between HS and MK hopping for *ω* as *d* increases to a delayed onset of hopping.

## Conclusions

We have numerically and theoretically studied a model glass former in which it is possible to isolate hopping from the critical mode-coupling dynamical slowing down and in which no other dynamical effects are present besides these two. The results illuminate the key role played by hopping in suppressing the iRFOT dynamical transition in finite *d* and in breaking the SER scaling. The MK model gives an example where single-particle hopping is sufficient to cause the SER breakdown, but in HS facilitation likely amplifies the effect, which may explain the dependence of *ω* on density (Fig. 5) (57).

For standard finite-dimensional HS and other structural glass formers, we expect the situation to be made more complex by the other dynamical processes mentioned in the Introduction. One might then conjecture the existence of at least three dynamical regimes for glass formers, upon increasing density: (*i*) an iRFOT/mode-coupling regime below *ii*) a MK-like hopping regime around *γ* changes, and the SER breakdown is incipient [in this regime the hopping timescale increases (exponentially) quickly with density (Fig. 3*D*); we expect this increase to be similar for HS and MK liquids, because the probability of finding a neighboring cage is roughly *iii*) at yet higher densities, hopping becomes too slow and other dynamical effects likely become important. If glass–glass nucleation barriers do not grow as quickly as the hopping barriers, then these processes may eventually become the dominant relaxation mechanism, following the RFOT prediction (5, 12, 31). In this regime (and hence in deeply supercooled liquids much below

We also stress, in line with previous studies, that VTF fits of the structural relaxation time in regimes *i* and *ii* should not be used to extract the putative Kauzmann transition point. In our opinion it makes no sense to test the Adam–Gibbs relation in these dynamical regimes. In the MK model, although the VTF law can be used to fit the dynamical data, there is no associated Adam–Gibbs relation and thus

Finally, we note that the MK model could also serve as a test bench for descriptions of hopping (24, 25, 58), as well as for relating percolation and glassy physics more broadly (59). These studies may further clarify other finite-dimensional effects, such as the correlation observed between local structure and dynamics (30).

## Acknowledgments

We thank G. Biroli, J.-P. Bouchaud, D. Chandler, J.-P. Garrahan, J. Kurchan, D. Reichman, C. Rycroft, and G. Tarjus for stimulating discussions. Financial support was provided by the European Research Council through European Research Council Grants 247328 and NPRGGLASS. P.C. acknowledges support from the Alfred P. Sloan Foundation.

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. Email: jinyuliang{at}gmail.com or giorgio.parisi{at}roma1.infn.it.

Author contributions: P.C., Y.J., G.P., and F.Z. designed research, performed research, contributed new reagents/analytic tools, analyzed data, and wrote the paper.

The authors declare no conflict of interest.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1417182111/-/DCSupplemental.

## References

- ↵.
- Berthier L,
- Biroli G,
- Bouchaud JP,
- Cipelletti L,
- van Saarloos W

- ↵
- ↵
- ↵
- ↵
- ↵.
- Götze W

- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵.
- Eaves JD,
- Reichman DR

- ↵
- ↵
- ↵
- ↵.
- Franz S,
- Jacquin H,
- Parisi G,
- Urbani P,
- Zamponi F

- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵.
- Garrahan JP,
- Chandler D

- ↵
- ↵.
- Keys AS,
- Hedges LO,
- Garrahan JP,
- Glotzer SC,
- Chandler D

- ↵
- ↵.
- Xia X,
- Wolynes PG

- ↵.
- Bhattacharyya SM,
- Bagchi B,
- Wolynes PG

- ↵.
- Wolynes P,
- Lubchenko V

- ↵.
- Kurchan J,
- Parisi G,
- Urbani P,
- Zamponi F

- ↵.
- Charbonneau P,
- Kurchan J,
- Parisi G,
- Urbani P,
- Zamponi F

- ↵.
- Mézard M,
- Parisi G

*Glasses and Replicas*, eds Wolynes PG, Lubchenko V (Wiley, New York) - ↵
- ↵
- ↵.
- Charbonneau B,
- Charbonneau P,
- Tarjus G

- ↵.
- Sengupta S,
- Karmakar S,
- Dasgupta C,
- Sastry S

- ↵.
- Charbonneau P,
- Ikeda A,
- Parisi G,
- Zamponi F

- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵.
- Ikeda A,
- Berthier L,
- Biroli G

- ↵
- ↵
- ↵
- ↵.
- Stauffer D,
- Aharony A

- ↵
- ↵
- ↵.
- Flenner E,
- Staley H,
- Szamel G

- ↵.
- Hocky GM,
- Berthier L,
- Kob W,
- Reichman DR

- ↵
- ↵
- ↵.
- Arenzon JJ,
- Coniglio A,
- Fierro A,
- Sellitto M

## Citation Manager Formats

## Sign up for Article Alerts

## Article Classifications

- Physical Sciences
- Physics