Theory of Amorphous Packings of Binary Mixtures of Hard Spheres

Abstract

We extend our theory of amorphous packings of hard spheres to binary mixtures and more generally to multicomponent systems. The theory is based on the assumption that amorphous packings produced by typical experimental or numerical protocols can be identified with the infinite pressure limit of long lived metastable glassy states. We test this assumption against numerical and experimental data and show that the theory correctly reproduces the variation with mixture composition of structural observables, such as the total packing fraction and the partial coordination numbers.Amorphous packings of hard spheres are ubiquitous in physics: they have been used as models for liquids, glasses, colloidal systems, granular systems, and powders. They are also related to important problems in mathematics and information theory, such as digitalization of signals, error correcting codes, and optimization problems. Moreover, the structure and density (or porosity) of amorphous multicomponent packings is important in many branches of science and technology, ranging from oil extraction to storage of grains in silos.Despite being empirically studied since at least sixty years, amorphous packings still lack a precise mathematical definition, due to the intrinsic difficulty of quantifying "randomness" [1]. Indeed, even if a sphere packing is a purely geometrical object, in practice dense amorphous packings always result from rather complicated dynamical protocols: for instance, spheres can be thrown at random in a box that is subsequently shaken to achieve compactification [2], or they can be deposited onto a random seed cluster [3]. In numerical simulations, one starts from a random distribution of small spheres and inflates them until a jammed state is reached [4,5]; alternatively, one starts from large overlapping spheres and reduces the diameter in order to eliminate the overlaps [6][7][8][9]. In principle, each of these dynamical prescriptions produces an ensemble of final packings that depends on the details of the procedure used. Still, very remarkably, if the presence of crystalline regions is avoided, the structural properties of amorphous packings turn out to be very similar. This observation led to the proposal that "typical" amorphous packings should have common structure and density; the latter has been denoted Random Close Packing (RCP) density. The definition of RCP has been intensively debated in the last few years, in connection with the progresses of numerical simulations [1,10].Nevertheless, the empirical evidence, that amorphous packings produced according to very different protocols have common structural properties, is striking and call for an explanation. This is all the more true for binary or multicomponent mixtures, where in addition to the usual structural observables, such as the structure factor, one can investigate other quantities such as the coordination between spheres of different type, and study their variation with the composition of the mixture.In earlier attempts to build statistical models of ...

Quantitative approximation schemes for glasses

“…Thus, it is possible that the numerical result effectively corresponds to ϕ > ϕ d . A more detailed comparison requires repeating the replica computation for a polydisperse system, following [88].…”

Microstructure and macroscopic properties of polydisperse systems of hard spheres

“…3.7(b). The two distinct regimes of ν max of bidisperse systems with moderate to large size ratios are predicted by a number of recently published theories [163][164][165][166]. A few checks of our bidisperse data with theoretical curves shown in Figure 6 in Ref.…”

“…Several authors proposed theories to compute the amorphus jamming density of binary and polydisperse hard sphere mixtures, see Refs. [163][164][165][166][167] and references therein. We construct a simple but physically reasonable model that relates the behavior of different hard sphere mixtures, even for metastable states.…”

Note: A replica liquid theory of binary mixtures

“…Thermodynamic properties near T K are deduced by computing the free energy of a liquid consisting of m-atomic replica molecules and then taking the limit of m → 1 at the end of the calculation. The RLT was later extended to binary systems 2,3 . However, it has been known that the binary RLT is inconsistent with its one-component counterpart; In the limit that all atoms are identical, the configurational entropy, S c , and thus T K calculated by the binary RLT differ from those obtained by the one-component RLT 2,3 .…”

“…The RLT was later extended to binary systems 2,3 . However, it has been known that the binary RLT is inconsistent with its one-component counterpart; In the limit that all atoms are identical, the configurational entropy, S c , and thus T K calculated by the binary RLT differ from those obtained by the one-component RLT 2,3 . More specifically, an extra composition-dependent term, or the mixing entropy, remains finite in S c computed by the binary RLT.…”