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# Universality of jamming of nonspherical particles

Edited by Pablo G. Debenedetti, Princeton University, Princeton, NJ, and approved October 8, 2018 (received for review July 19, 2018)

## Significance

The jamming transition is a key property of granular materials, including sand and dense suspensions. In the generic situation of nonspherical particles, its scaling properties are not completely understood. Previous empirical and theoretical work in ellipsoids and spherocylinders indicates that both structural and vibrational properties are fundamentally affected by shape. Here we explain these observations using a combination of marginal stability arguments and the replica method. We unravel a universality class for particles with internal degrees of freedom and derive how the structure of packings and their vibrations scale as the particles evolve toward spheres.

## Abstract

Amorphous packings of nonspherical particles such as ellipsoids and spherocylinders are known to be hypostatic: The number of mechanical contacts between particles is smaller than the number of degrees of freedom, thus violating Maxwell’s mechanical stability criterion. In this work, we propose a general theory of hypostatic amorphous packings and the associated jamming transition. First, we show that many systems fall into a same universality class. As an example, we explicitly map ellipsoids into a system of “breathing” particles. We show by using a marginal stability argument that in both cases jammed packings are hypostatic and that the critical exponents related to the contact number and the vibrational density of states are the same. Furthermore, we introduce a generalized perceptron model which can be solved analytically by the replica method. The analytical solution predicts critical exponents in the same hypostatic jamming universality class. Our analysis further reveals that the force and gap distributions of hypostatic jamming do not show power-law behavior, in marked contrast to the isostatic jamming of spherical particles. Finally, we confirm our theoretical predictions by numerical simulations.

Upon compression, an athermal system consisting of purely repulsive particles suddenly acquires finite rigidity at a certain jamming transition density *i*) the power-law behaviors of the elastic modulus and contact number as a function of the proximity to *ii*) the emergence of excess soft modes in the vibrational density of states *iii*) the power-law divergence of the gap distribution function

However, a system of spherical particles is an idealized model and, in reality, constituent particles are, in general, nonspherical. In this case, one should specify the direction of each particle in addition to the particle position. The effects of those extra degrees of freedom have been investigated in detail in the case of ellipsoids (2, 3, 9, 27⇓–32). Notably, the contact number at the jamming point continuously increases from the isostatic value of spheres, as

In this work, we propose a theoretical framework to describe the universality class of hypostatic jamming. As a first example of universality, we map ellipsoids into a model of “breathing” spherical particles (BP), recently introduced in ref. 43. Based on the mapping, we show that the two models indeed have the same critical exponents by using a marginal stability argument. Next, we propose a generalization of the random perceptron model that mimics the BP and can be solved analytically using the replica method. We confirm that this model is in the same universality class of ellipsoids, BP, and other nonspherical particles that display hypostatic jamming. This analysis further predicts the scaling behavior of

## BP Model

The BP model (43) was originally introduced to understand the physics of the swap Monte Carlo algorithm (44), but here we focus on its relation with the jamming of ellipsoids. The model consists of N spherical particles with positions

Because the BP model has

## Mapping from Ellipsoids to BP

We now construct a mapping from a system of ellipsoids to the spherical BP model introduced above. Ellipsoids are described by their position **4** and keeping terms up to **3**. Hence, if we identify

## Marginal Stability

The distinctive feature of both BP and ellipsoids is that the total potential, and thus the Hessian matrix, can be split into two parts: one having finite stiffness and the second one having vanishing stiffness **8** holds when **8** and **9** imply that p and Δ have the same scaling dimension and the following scaling holds:**10** reduces to Eq. **9**, which requires **8**, which requires **10** is confirmed by numerical simulations (43). Assuming that **8**, the shear modulus G behaves as

## Vibrational Spectrum

The marginal stability argument suggests that

Then, *i*) The lowest band corresponds to the *ii*) an intermediate band corresponding to the extra (rotational or radial) degrees of freedom *iii*) the highest band corresponding to the

Numerical results for

## Mean-Field Model

The universality class of isostatic jamming is well understood: It can be described analytically by particles in

We now introduce a mean-field model which describes the universality class of hypostatic jamming in the BP, ellipsoids, and many other models of nonspherical particles. The model, which is a generalization of the perceptron, can be solved analytically and, as we show, the solution reproduces all of the critical exponents of hypostatic jamming. It consists of one tracer particle with coordinate x on the surface of the N-dimensional hypersphere of radius

Because the model can be solved by the same procedure as that of the standard perceptron model, here we give just a brief sketch of our calculation. The free energy of the model at temperature

An important observable to characterize jamming is the gap distribution

The situation is completely different if **8** and **11**.

The simplicity of the model allows us to derive the analytical form of the density of states **12**, w.r.t.

As a final check of universality, we test the prediction for the Δ dependence of the gap distribution function **14**. In Fig. 5, we show numerical results (obtained as in ref. 43) for **14**.

## Conclusions

Using a marginal stability argument, we derived the scaling behavior of the contact number z and the density of states

One of the most surprising outputs of our theory is the universality of the density of states

## Acknowledgments

We thank B. Chakraborty, A. Ikeda, J. Kurchan, S. Nagel, and S. Franz for interesting discussions. We thank the authors of refs. 40 and 32 for sharing their data used in Figs. 1 and 2, respectively. This project received funding from the European Research Council under the European Union’s Horizon 2020 Research and Innovation program (Grant 723955-GlassUniversality). This work was supported by Grants 689 454953 (to M.W.) and 454955 (to F.Z.) from the Simons Foundation and by a public grant from the “Laboratoire d’Excellence Physics Atoms Light Mater” (LabEx PALM) overseen by the French National Research Agency (ANR) as part of the “Investissements d’Avenir” program (reference no. ANR-10-LABX-0039-PALM; to P.U.).

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: harukuni.ikeda{at}lpt.ens.fr.

Author contributions: C.B., H.I., P.U., M.W., 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 is a PNAS Direct Submission.

Published under the PNAS license.

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