Completing the replica glass-state theory of amorphous packings of binary hard-sphere mixtures: the asymmetric large-$r$ regime, partial contacts at large size ratio, and the microscopic origin of the protocol parameter $\Sigma_j$

The replica (glass-state) theory of Biazzo, Caltagirone, Parisi, and Zamponi [Phys. Rev. Lett. 102, 195701 (2009)] identifies amorphous packings of binary hard-sphere mixtures with the infinite-pressure limits of metastable glassy states, and predicts jamming densities and partial coordination numbers as functions of the diameter ratio $r$ and the small-particle volume fraction $\eta$ using a single fitted protocol parameter $\Sigma_j$ . The authors acknowledged three specific gaps, which together define one objective: a parameter-free, first-principles theory of binary-mixture jamming valid over the full composition plane. (1) In the very asymmetric regime (large $r$ , small $\eta$ ) the small spheres remain mobile in the pores of the jammed large-sphere network, and the frozen-cage theory fails there: its domain must be restricted to $\eta$ above a threshold $\eta_{th}\approx 0.265$ in the $r\to\infty$ limit, below which it violates an exact large-r bound on the jamming density. (2) The theory shows an unexplained discrepancy with simulation in the partial coordination numbers of large particles at large $r$ . (3) The parameter $\Sigma_j$ — the configurational-entropy value at which a compressed system falls out of equilibrium — is fitted, not derived from the compression protocol. All three gaps remain open in the literature: the asymmetric regime has been mapped numerically but has no first-principles theory; the contact discrepancy has never been directly re-tested; and protocol-to-state mapping exists only as state following in infinite spatial dimension with equilibrium-prepared parents. The adjacent foundational question — whether an ideal equilibrium glass transition exists in hard spheres — also remains open, but the source paper explicitly sets it aside as beyond the theory's needs.

Background

Dense packings of hard spheres — particles interacting only through excluded volume — are the reference model for random packing in granular media, colloids, and powders. When a hard-sphere fluid is compressed fast enough to avoid crystallization, it arrests in an amorphous jammed configuration: a mechanically rigid packing in which each particle is blocked by its neighbors, reached at a 'random close packing' density around 0.64 for equal-sized spheres. Binary mixtures add two control parameters: the diameter ratio $r$ (large over small) and the composition, conveniently encoded as $\eta$ , the fraction of the total particle volume carried by the small spheres. Experiments and simulations show that the jamming density varies nontrivially with $(r,\eta)$ , which is why mixtures matter for applications from colloids to concrete.

Jamming protocols are inherently dynamical. A standard generator is the Lubachevsky–Stillinger algorithm: spheres grow at a fixed rate while event-driven molecular dynamics lets them rearrange, so the packing jams at a density set by how fast the compression outruns the dynamics. The final packing contains rattlers — particles with too few contacts to be mechanically blocked, which rattle inside cages formed by their neighbors — and after removing them, the contact network of a frictionless packing is isostatic, i.e. it has exactly the number of contacts needed for mechanical equilibrium ( $z=6$ per particle in three dimensions).

The framework behind this problem is the replica, or glass-state, theory of hard-sphere packings (Parisi and Zamponi, Rev. Mod. Phys. 82, 789 (2010)). Its central assumption is that amorphous packings produced by typical protocols are the infinite-pressure limit of metastable glassy states of the equilibrium liquid. Glassy states are counted by a configurational entropy, or complexity, $\Sigma(\varphi,s)$ : the logarithm, per particle, of the number of metastable states with single-state vibrational entropy $s$ at packing fraction $\varphi$ . The computation uses replicas: one couples $m$ copies of the system, constrained to sit in the same metastable state, and the replicated free energy yields $\Sigma(\varphi,s)$ and the cage size. During compression the liquid stays equilibrated until its relaxation time exceeds the compression rate; it then falls out of equilibrium inside a glassy state whose complexity is $\Sigma_j$ , and further compression brings that state to infinite pressure (jamming) at a density $\varphi_j$ determined by $\Sigma(\varphi_j)=\Sigma_j$ . Because different protocols fall out of equilibrium at different points, different protocols jam at different densities — a distinctive and confirmed prediction of the theory.

Biazzo, Caltagirone, Parisi, and Zamponi (Phys. Rev. Lett. 102, 195701 (2009)) extended this framework from monodisperse to binary (and, more generally, multicomponent) mixtures, computing the jamming density $\varphi_j(r,\eta)$ and the partial coordination numbers $z_{\mu\nu}$ (the average number of contacts with species $\nu$ around a particle of species $\mu$ , with $\mu,\nu\in\{small,large\}$ ) from a multicomponent equation of state, and testing them against their own Lubachevsky–Stillinger simulations and classical experiments. With a single, strongly constrained fitting parameter $\Sigma_j=1.7$ (the theory requires $\Sigma_j$ close to 1), the theory reproduces the variation of the total density and of the partial coordinations with composition over the explored range of size ratios $r\le 4$ .

The framework's deepest assumption — that metastable glassy states of arbitrarily long lifetime exist, i.e. that the metastable liquid eventually undergoes an ideal (Kauzmann) glass transition where the complexity vanishes — is itself unresolved: mean-field theory predicts such a transition, but whether it survives in three dimensions at asymptotically long times, or is avoided (for example by nucleation or activated events), is a major open problem of the glass literature. The 2009 paper deliberately sidesteps this question, treating only long-lived metastable states on accessible time scales.

Problem Statement

Complete the equilibrium replica (glass-state) theory of amorphous packings of binary hard-sphere mixtures formulated by Biazzo, Caltagirone, Parisi, and Zamponi (Phys. Rev. Lett. 102, 195701 (2009)), by closing the three gaps its authors acknowledged. The three gaps are scientifically linked components of one objective: a parameter-free, first-principles statistical-mechanical theory of amorphous jamming that is quantitatively accurate over the full $(r,\eta)$ composition plane of binary mixtures.

Asymmetric mixtures. For large diameter ratio $r$ and small small-sphere volume fraction $\eta$ , the small spheres need not be blocked at all: the large spheres form a rigid (jammed) network while the small spheres remain mobile in its pores, so jamming of the mixture loses its usual meaning. The 2009 theory assumes every particle is caged in a frozen structure and therefore cannot describe this regime; a follow-up by the same authors (J. Chem. Phys. 132, 176101 (2010)) showed that in the $r\to\infty$ limit the theory's domain must be restricted to $\eta$ above a threshold $\eta_{th}\approx 0.265$ , below which its predictions violate the exact large- $r$ bound on the jamming density,

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Completing the replica glass-state theory of amorphous packings of binary hard-sphere mixtures: the asymmetric large- $r$ regime, partial contacts at large size ratio, and the microscopic origin of the protocol parameter $\Sigma_j$

Background

Problem Statement