# Assessing the role of static length scales behind glassy dynamics in polydisperse hard disks

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Edited by James S. Langer, University of California, Santa Barbara, CA, and approved April 24, 2015 (received for review January 29, 2015)

## Significance

The origin of dynamical slowing down toward glass transition is a fundamental unsolved problem in condensed matter physics. A crucial question is whether this slowing down has a structural origin. Recently, a method to detect hidden order within the fluid was proposed, based on the idea that freezing a fraction of the particles in a system causes a transition akin to glass transition. Here, we show that a glass former, polydisperse hard disks, has a strong increase of structural order, well correlated with slow dynamics, which goes undetected by the pinning method. This casts doubt on the order-agnostic qualities of the pinning length scale and keeps static length scales in the race for plausible explanations of the glass transition problem.

## Abstract

The possible role of growing static order in the dynamical slowing down toward the glass transition has recently attracted considerable attention. On the basis of random first-order transition theory, a new method to measure the static correlation length of amorphous order, called “point-to-set” (PTS) length, has been proposed and used to show that the dynamic length grows much faster than the static length. Here, we study the nature of the PTS length, using a polydisperse hard-disk system, which is a model that is known to exhibit a growing hexatic order upon densification. We show that the PTS correlation length is decoupled from the steeper increase of the correlation length of hexatic order and dynamic heterogeneity, while closely mirroring the decay length of two-body density correlations. Our results thus provide a clear example that other forms of order can play an important role in the slowing down of the dynamics, casting a serious doubt on the order-agnostic nature of the PTS length and its relevance to slow dynamics, provided that a polydisperse hard-disk system is a typical glass former.

When we supercool a liquid while avoiding crystallization, dynamics becomes heterogeneous (1, 2) and slows down significantly toward the glass transition, below which a system becomes a nonergodic state. Now there is a consensus that this slowing down accompanies the growth of dynamical correlation length (3). Several different physical scenarios have been proposed, yet the origin is still a matter of serious debate: although some scenarios describe the glass transition as a purely kinetic phenomenon (4), others posit a growing static order (5) or a loss of configurational entropy (6) behind dynamical slowing down. Among this last category, we will focus here on two distinct approaches. The first one is random first-order transition (RFOT) theory (7⇓–9), which is based on a finite dimensional extension of mean-field models with an exponentially large number of metastable states. The second approach, recently proposed by some of us (10, 11), ascribes the growth of the dynamical correlation length with the corresponding growth of the static correlation length. Here, we focus on these two scenarios based on static order and consider which is more relevant to the origin of glassy slow dynamics, using a simple model glass former, 2D polydisperse hard disks (12, 13).

In RFOT, metastable states are thought to have amorphous order, whose correlation length diverges toward the ideal glass transition point. It was recently suggested that the so-called point-to-set (PTS) length, which is the correlation length of amorphous order, can be extracted by pinning a finite fraction of particles and studying the dependence of the overlap function on the pinning particle concentration. According to the RFOT theory, amorphous order develops in any glass-forming liquids and this method is thought to be able to pick up the static correlation length whatever the order is, i.e., the method is claimed to be order agnostic (14). Thus, the use of pinning fields has been considered to be a promising new direction in the study of the glass transition. Within the RFOT theory, it was shown that freezing the positions of a finite concentration of particles shifts the ideal glass transition to higher temperatures, potentially granting access to the glass state in equilibrium (15⇓⇓⇓⇓⇓⇓–22). Moreover, the average distance between pinned particles at the liquid-to-glass transition represents a direct measure of the PTS correlation length. PTS correlation lengths aim at measuring hidden static length scales by looking at the extent of the perturbation induced by frozen particles on the rest of the liquid. It is intuitively defined as the average distance between pinned particles that forces the system to stay in an amorphous configuration with a vanishing configurational entropy. The reasons behind the popularity of PTS correlation lengths in the study of the glass transition are at least twofold: (*i*) they are expected to provide an “order-agnostic” method to measure static correlations (23, 24); (*ii*) it is theoretically established that no divergence of the relaxation time of a glass at finite temperature can occur without the concomitant divergence of the static correlation length (25).

On the other hand, it was recently noted (10, 11, 13) that, for moderately polydisperse hard disks, an increase in the area fraction of particles *ϕ*, hexatic (or sixfold bond orientational) order grows and its correlation length

The role of local order on the dynamics of glassy systems remains controversial at least for two reasons. The first problem is that the local order that one needs to measure is system dependent, and up to now the relevance of bond-orientational order was demonstrated only for polydisperse particle systems and a spin liquid, and not for bidisperse systems (10, 11). The second problem is conceptual: are static correlations really responsible for the dynamical slowing down? The PTS correlation length is often described as a remedy to both problems, because it should be able to detect static correlations without a detailed knowledge of the local order involved in these correlations. In studies of binary mixtures of hard spheres, the PTS correlation length was shown to grow only modestly in the regime accessible to computer simulation (14, 23), differently from the dynamical correlation length, which grows much more rapidly. These results suggested that no link exists between a single static length scale and the dynamical slowing down (the only exception would be close to a possible ideal glass transition temperature) (5, 14). On the other hand, measures of the PTS correlation length (27) have shown that it correlates well with the average dynamics of the system, and with dynamic heterogeneities (28). The PTS correlation length is in agreement with measures of the density of plastic modes (29), providing support for the idea of a fundamental length scale controlling the dynamics of the supercooled liquids.

Unlike previous studies, in this work we measure the PTS correlation length in a system for which a growing local order was previously found, i.e., polydisperse hard disks. This will allow us to compare the growth of PTS correlation lengths with that of bond-orientational lengths. We will consider several pinning strategies (random pinning, uniform pinning, and cavity pinning) (Fig. 1 *A–C*), and then look for the underlying structural features that are captured by the PTS correlation length. In principle, each different pinning geometry probes a different length scale (30). The PTS static length scale

## Results: Unpinned Case

The system studied is composed of *Materials and Methods*). We start by considering the case without an external pinning field, *SI Appendix*, we extract the correlation length for bond-orientationally ordered regions (*SI Appendix*, Fig. S1) by fitting the exponential decay of the peaks of the correlation function *SI Appendix*, Fig. S2). We use an exponential function instead of a 2D Ornstein–Zernike function to avoid a priori assumptions on the origin of the growth of the correlation length. The correlation length *D*, together with other length scales that we will derive later. The two-body correlation function *SI Appendix*, Fig. S3). The results of this fit are also summarized in Fig. 1*D*. These results confirm that, for polydisperse glass-forming systems, the growth of many-body correlations associated with bond-orientational order is much faster than the growth of two-body correlations (11, 13, 32).

As was shown in refs. 11 and 13, there is a link between dynamic heterogeneities and regions of high hexatic order in the fluid. We provide here further support to this scenario. We measure dynamical length scales by performing event-driven molecular dynamics simulations and extracting the four-point density correlator, with the procedure described in *SI Appendix*, Figs. S4 and S5. In Fig. 1*D*, we plot the dynamical correlation length *E* and *F*, we plot snapshots of configurations prepared at *Materials and Methods*), where 200 trajectories are started from the same initial configuration but with different initial velocities. The degree of structural order is investigated by taking the average hexatic field over these *SI Appendix*, Fig. S6). The dynamics is instead investigated through the isoconfigurational average of the displacement *SI Appendix*, Fig. S4). All disks in the snapshots of Fig. 1 *E* and *F* are then grouped into sets of high and low mobility/order, depending on whether their mobility/order is higher or lower than the 50th percentile. We can then identify four different sets of particles: low mobility and high order (white); high mobility and low order (black); low mobility and low order (orange); and high mobility and high order (magenta). Our results show that, for all densities considered in this work, 66% of particles are in the first two sets (33% in each), demonstrating a high degree of correlation between structural ordered regions and immobile regions (or, vice versa, between disordered regions and mobile regions) even at a particle level. Moreover, the remaining two sets (each accounting for the 17% of particles) are located at the interface between mobile and immobile extended regions. Magenta disks are located on the surface and in between black clusters, whereas orange disks are located on the surface and in between white clusters. In other words, disks that are next to a low/high mobility region will also have low/high mobility. Here, we note that the embedded fractal nature of order parameter fluctuations is characteristic of critical fluctuations. In *SI Appendix*, we also note that a higher degree of correlation can be obtained if the relative displacement *i* with respect to its *M* neighbors, *SI Appendix*, Fig. S7).

## Results: Pinning

Having characterized the static and dynamic properties of the unperturbed system, we now introduce the pinning field. First, simulations are fully equilibrated with cluster-moves algorithms (*Materials and Methods*). A representation of the pinning fields is given in Fig. 1 *A–C*. In the “random-pinning” geometry, *R* are pinned. Because the static length scales currently accessible to simulations are expected to be small, well within

Because the pinning field is applied to equilibrium configurations, the static properties should be unchanged with respect to the *c*. All results are consistent with the *c* is represented with the error bars for *D*. Correlation lengths are extracted from all pinning geometries, following the procedure outlined in *SI Appendix*, Figs. S8–S12. In all cases, the physical idea is to detect the characteristic length (average distance between pinned particles in the random- and uniform-pinning geometries, or the size of the cavity in the cavity geometry), which produces a high localization of the mobile particles, as measured by overlap functions. We plot the random-pinning correlation length *D* (see *SI Appendix*, Figs. S8–S12, on the details of its estimation). We see that the growth of the PTS length scale, irrespective of the pinning strategy, is significantly slower than the growth of bond-orientational order *Inset* of Fig. 1*D* shows the scaling between the *D* is a measure of the fragility of the system (10). This scaling also supports a direct connection between the growth of structural correlation and slow dynamics, but its origin is still unclear. A justification of this relation in the context of a theory of topological ordered clusters has been given in ref. 26.

All static length scales can also be obtained by considering coarse-grained variables, dividing the simulation box into smaller boxes of side length *i* in configuration α, and *Left* to *Right* in the figure), the amount of localization progressively increases. Localization occurs when the occupancy field becomes strongly peaked in discrete locations in space, surrounded by regions of negligible probability occupancy. For *a* are always bigger than the pinning correlation length (Fig. 1*D*), and the occupancy field *SI Appendix*, Fig. S13, we show that, at the level of undercooling that we can reach, the transition from a homogenous state to a localized state is continuous.

A visual inspection of Fig. 2 (bottom row) shows that localized particles have a high degree of hexatic order (*SI Appendix*, Fig. S13), already suggesting that *D*, we know that the pinning length scale *D* thus indicate that the localization occurs when the average distance between pinned particles is comparable to the range of two-body correlations,

The results thus show that the growth of the pinning correlation length is similar to the growth of two-body correlations. On the other hand, the growth of bond-orientational order is much faster and clearly decoupled from the pinning correlation length. We also checked that the same is true for coarse-grained quantities. This strongly indicates that the PTS correlation length is not order agnostic but targets the growth of a particular order in the system, that is, the size of the regions where particles are localized due to the pinning field. The growth of these regions follows the growth of two-body correlations: particles are localized due to the perturbation that pinned particles introduce to the

## Discussion and Conclusions

In this article, we have extracted several static length scales from systems of polydisperse hard disks with polydispersity

The PTS length captures a localization transition that occurs in presence of pinned particles. This localization transition originates when the occupancy field,

The localization transition that occurs with increasing concentration of pinned particles happens continuously in the density range we could access in equilibrium. The results do not rule out the possibility that strong nonlinearities will produce a localization transition that extends beyond pair correlations for higher (but yet unreachable) densities.

To summarize, the PTS length measured by particle pinning simply reflects pair correlation and fails in detecting the correlation of bond-orientational order (more precisely, hexatic order), which intrinsically originates from many-body interactions. Although the PTS length is decoupled from the dynamical correlation length, the hexatic order correlation is strongly coupled to it. This implies that slow dynamics in our system is controlled by the development of hexatic ordering, and not by translational order detected by the PTS correlation. Although the generality of this conclusion needs to be checked carefully, our study suggests that the PTS length is not order agnostic and the growth of the PTS length is not responsible for glassy slow dynamics, at least for our system.

Differently from the fluid-hexatic transition, the results in refs. 11 and 13 have provided evidence that the glass transition in polydisperse hard disks involves a power-law growth of the hexatic correlations that follows the Ising universality class, but, differently from ordinary critical phenomena, is accompanied by a strong divergence [

## Materials and Methods

We study 2D polydisperse hard disks with Monte Carlo simulations. The diameter σ of the disks is extracted from a Gaussian distribution, and the polydispersity is defined as the SD of the distribution,

All simulations are run at fixed densities

The connection between static and dynamic length scale shown in Fig. 1*D* was obtained with event-driven simulations (37) in the isoconfigurational ensemble (38), where 200 trajectories are started from an equilibrated configuration at

Details are given in the *SI Appendix*.

## Acknowledgments

We are grateful to the following individuals for valuable comments and constructive criticisms: Ludvic Berthier, Gulio Biroli, Patrick Charbonneau, Walter Kob, Jim Langer, David Reichman, Gilles Tarjus, and Sho Yaida. This study was partly supported by Grants-in-Aid for Scientific Research (S) and Specially Promoted Research from the Japan Society for the Promotion of Science.

## Footnotes

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^{1}To whom correspondence may be addressed. Email: tanaka{at}iis.u-tokyo.ac.jp or russoj{at}iis.u-tokyo.ac.jp.

Author contributions: H.T. designed research; J.R. performed research; J.R. analyzed data; and J.R. and H.T. 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.1501911112/-/DCSupplemental.

## References

- ↵
- ↵.
- Yamamoto R,
- Onuki A

- ↵
- ↵.
- Hedges LO,
- Jack RL,
- Garrahan JP,
- Chandler D

- ↵.
- Royall CP,
- Williams SR

- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵.
- Cammarota C,
- Biroli G

- ↵
- ↵
- ↵.
- Karmakar S,
- Parisi G

- ↵.
- Chakrabarty S,
- Karmakar S,
- Dasgupta C

- ↵.
- Ozawa M,
- Kob W,
- Ikeda A,
- Miyazaki K

- ↵.
- Charbonneau P,
- Tarjus G

- ↵
- ↵
- ↵.
- Langer JS

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

- ↵
- ↵
- ↵
- ↵
- ↵.
- Jack RL,
- Fullerton CJ

- ↵.
- Kob W,
- Coslovich D

- ↵
- ↵
- ↵
- ↵

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