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# Force distribution affects vibrational properties in hard-sphere glasses

Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved October 21, 2014 (received for review August 8, 2014)

## Significance

How a liquid becomes rigid at the glass transition is a central problem in condensed matter physics. In many scenarios of the glass transition, liquids go through a critical temperature below which minima of free energy appear. However, even in the simplest glass, hard spheres, what confers mechanical stability at large density is highly debated. In this work we show that to quantitatively understand stability at a microscopic level, the presence of weakly interacting pairs of particles must be included. This approach allows us to predict various nontrivial scaling behavior of the elasticity and vibrational properties of colloidal glasses that can be tested experimentally. It also gives a spatial interpretation to recent, exact calculations in infinite dimensions.

## Abstract

We theoretically and numerically study the elastic properties of hard-sphere glasses and provide a real-space description of their mechanical stability. In contrast to repulsive particles at zero temperature, we argue that the presence of certain pairs of particles interacting with a small force *f* soften elastic properties. This softening affects the exponents characterizing elasticity at high pressure, leading to experimentally testable predictions. Denoting *i*) the density of states has a low-frequency peak at a scale *ω* is the frequency, (*ii*) shear modulus and mean-squared displacement are inversely proportional with *iii*) continuum elasticity breaks down on a scale *z* is the coordination and *d* the spatial dimension. We numerically test (*i*) and provide data supporting that

The emergence of rigidity near the glass transition is a fundamental and highly debated topic in condensed matter and is perhaps most surprising in hard-sphere glasses where rigidity is purely entropic in nature. The rapid growth of relaxation time around a packing fraction *z* (as already discussed by Maxwell in ref. 11), and the applied compressive strain *e* (10). As one may intuitively expect, increasing coordination is stabilizing, whereas increasing pressure at fixed coordination is destabilizing. Second, within a long-lived metastable state the vibrational free energy of a hard-sphere system can be approximated as a sum of local interaction terms between pairs of colliding particles, which are said to be in contact. On a time scale that contains many collisions, at high packing fraction the interaction follows approximately *h* is the time-averaged distance between two adjacent particles (6, 7). This directly leads to an effective force law

Very recently a replica calculation (17, 18) predicted *γ* was argued and numerically shown to be related to the force distribution exponents *Hard Spheres*). Here we propose a resolution of these issues: Heterogeneity in contact strength was neglected in refs. 6 and 7, but the prevalence of weak forces in hard-sphere systems corrects scaling exponents and leads to the scaling relation

This work is organized as follows. In *Elastic Networks*, we present a variational argument for the density of vibrational modes in weakly coordinated networks with stiffness heterogeneity. We also use scaling arguments to compute the shear modulus and the mean-squared displacement. In *Effective Medium Theory*, we confirm these predictions using a standard mean-field approximation, and furthermore predict the length scale below which continuum elasticity breaks down in such systems. In *Hard Spheres*, we show how these results apply to colloidal glasses and discuss the subtle issue associated with the existence of two kinds of contacts at low forces in sphere packings. We also present numerical results supporting our views. In *Comparison with Replica Theory in d = ∞* and *Conclusion*, we compare our results with replica calculations and discuss prospects for experimental tests in colloidal systems.

## Elastic Networks

We consider an elastic network of *N* points of mass *m*, connected by *d*. The quadratic expansion of the elastic energy for an imposed displacement field *β*. Here *β*, i.e., *β*.

We assume that **1** defines the stiffness matrix *ω* are the frequencies of vibrational modes, of density

### Variational Argument.

First we consider the springs at rest length, so that all **1** is present. Let **1** that there are at least *β*. We assume that the shape of the stiffness distribution *z*, and wish to compute the scaling properties of *Supporting Information*.

Our strategy to estimate *L*, as shown in Fig. 1*A*, one cuts a fraction *q*. These modes can be distorted to lead to trial modes of frequency *ω*,

If *q* of bonds are cut, the density of induced floppy modes is

We now show that if the distribution of stiffnesses is broad enough, then the above bound is not saturated. In this case, we can improve the variational argument by creating a different set of trial modes (illustrated in Fig. 1*B*); we cut a fraction *q* of the weakest links, and use the induced floppy modes. We then make the key assumption that these floppy modes do not decay appreciably with distance from the broken bonds, but extend in the entire system, displacing particles by some characteristic amplitude. We shall see below that for hard spheres, our assumption only holds for a fraction of the contacts at low force.

We assume that the distribution of stiffnesses follows *q* of the weakest extended bonds then have a characteristic stiffness *q* of weak springs of characteristic stiffness *Supporting Information*. This leads to a characteristic frequency**[3]** applies. These are our central results: At the Maxwell threshold (**[3]**, holds above the characteristic frequency

For

Assuming harmonic dynamics, we can obtain from **[3]** a bound on the particles’ mean-squared displacement

Using previous results on floppy mode displacements, the shear modulus can also be estimated. As discussed in *Supporting Information*, one finds

### Role of Prestress.

So far we have considered stress-free elastic networks. The presence of a compressive force in the bonds reduces the modes’ frequency, as implied by Eq. **1**, and can lead to an elastic instability. It was argued and checked numerically in ref. 10 that the strongly scattered modes that appear above **1**, this implies that some soft modes will be shifted to a frequency *A* is a numerical constant. Stability requires **[4]**, this becomes

## Effective Medium Theory

All of the above predictions can be derived and extended with effective medium theory (EMT), a mean-field approximation that treats disorder in a self-consistent way (27⇓⇓⇓⇓–32). EMT has been shown to give quantitatively correct values for scaling exponents related to the vibrational spectrum and heat transport properties of frictionless packings (29, 31). In EMT, a random elastic network, such as depicted in Fig. 1, is modeled by a regular lattice with effective frequency-dependent spring constants. Here we follow the EMT developed in ref. 31, which includes the effect of forces in Eq. **1**. In ref. 31, the randomness in the interaction between two nodes was limited to the presence or absence of a spring; when a spring was present, its stiffness was always identical. Here we relax this assumption and allow a full distribution of stiffnesses, behaving as *k*, and allow a distribution of contact forces, *f*. Details of the EMT are presented in *Supporting Information*.

The EMT confirms that when **[3]**, with a logarithmic correction in

Regarding the shear modulus, EMT confirms the scaling **6**, and in addition we find the dependence on *μ* drops by a finite factor at elastic instability, relative to its unstressed value. Finally, EMT predicts that modes at

## Hard Spheres

The above results on elastic networks can be applied to the free energy of hard spheres within a metastable state, and near maximum packing at *τ*, much larger than the typical interval between collisions, *τ* (6, 7, 10). Using the fact that the contact network at *h* is the time-averaged gap between contacting particles. Hence in link *β* the force *z*.

Although missing from many theoretical approaches (5, 34), the distribution of contact forces at *f*, with *α*.

In ref. 20 it was observed that when contacts are opened from hard-sphere packings at *Supporting Information*, we show that the variational argument is not improved by including the localized contacts, and therefore we want to consider only the extended type. In

We can now present our results for hard spheres. Geometrically, the characteristic gap **[7]** that **[3–6]** and **[8]** we then deduce*Supporting Information, F. Effect of Change of Stiffness Distribution with ϕ*, we argue that these results are not changed if the evolution of **11** relates two experimentally accessible quantities, *κ* that depends on *γ* describing the distribution of gaps between particles, **14** and **15** are satisfied with equality, with numerical values **[15]** was recently proven for certain dynamics (38). Assuming such marginal stability, it follows that **[11]**, **[14]**, and **[15]** lead to a description of jammed packings and glasses based on four exponents, with three scaling relations between them. We have in particular

## Comparison with Numerics

To confirm the prediction that *Supporting Information*). The density of states **[3]**, but larger simulations are needed, preferably in

## Comparison with Replica Theory in d = ∞

A very recent replica computation (17, 18, 39, 40) was used to compute exponents in *Supporting Information*). Our results therefore support that system preparation does not affect the exponent *Supporting Information*.

The value for **14** is not satisfied in *γ* appears to be independent of dimension (20, 35). To resolve this dilemma, note that **[15]** is also exactly satisfied by the **[15]** (where **[11]**, both exactly satisfied in the replica calculation. The scaling description we propose based on the marginality of real-space excitations (both linear and nonlinear) is thus fully consistent with the replica calculation, as these two scaling relations are satisfied.

The fact that localized excitations appear to be absent in large dimension seems plausible, as their existence depends on the presence of local arrangements of particles that are very soft (illustrated in Fig. 3), which may become unlikely when each particle shares many contacts. This situation may be similar to the behavior of rattlers, i.e., particles which are trapped in a packing but do not contribute to mechanical stability. The fraction of rattlers is observed to decay exponentially with *d* (35), so that in large dimension, it is extremely rare to find a gap that is large enough to hold a particle. A similar decay may occur for localized excitations. This could be checked by explicit enumeration of localized contacts, as described in ref. 33.

## Conclusion

We have shown that the stability of hard-sphere glasses is affected by heterogeneity in contact strengths. Our numerics on the force distribution exponent

If localized excitations are absent in large dimension, then our results are fully consistent with the replica theory valid for **16** may change in their final digit.

Our scaling predictions on *μ*, Eqs. **10** and **11**, may be tested experimentally in colloidal systems. From the covariance matrix of particle displacements, **[9]**. Given sufficient temporal resolution, one can also extract the effective density of states from displacement autocorrelations (42). This procedure has been carried out in simulations (6, 7, 43) and experiments (13⇓–15), confirming the presence of a peak in

When *ε* needed to overlap particles by the characteristic gap in the system, thus facilitating rearrangement (12). For commonly studied harmonic soft spheres, *k* is a stiffness, this gives the hard-sphere regime as *T*. This expected crossover from hard- to soft-sphere behavior corresponds to departures from **[9]**, and will be discussed elsewhere.

Overall, our approach leads to a description of jamming in finite dimensions based on the marginal stability of three distinct types of excitations, both linear and nonlinear. It remains to be seen if plastic flow under shear and thermally activated process near the glass transition can be expressed in terms of the relaxation of these excitations.

## Acknowledgments

We thank the authors of ref. 17 for sharing their preprint and for discussions; and Jie Lin, Le Yan, Gustavo Düring, Colm Kelleher, and Marija Vucelja for discussions. This work was supported primarily by the Materials Research Science and Engineering Center (MRSEC) Program of the National Science Foundation under Award number DMR-0820341, with additional support from National Science Foundation Grants CBET-1236378 and DMR-1105387.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: ed87{at}nyu.edu.

Author contributions: E.D., E.L., C.B., and M.W. designed research, performed research, analyzed data, and 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.1415298111/-/DCSupplemental.

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