Estimating random close packing in polydisperse and bidisperse hard spheres via an equilibrium model of crowding

Abstract

We show that an analogy between crowding in fluid and jammed phases of hard spheres captures the density dependence of the kissing number for a family of numerically generated jammed states. We extend this analogy to jams of mixtures of hard spheres in $d=3$ dimensions, and thus obtain an estimate of the random close packing (RCP) volume fraction, $\phi_{\textrm{RCP}}$, as a function of size polydispersity. We first consider mixtures of particle sizes with discrete distributions. For binary systems, we show agreement between our predictions and simulations, using both our own and results reported in previous works, as well as agreement with recent experiments from the literature. We then apply our approach to systems with continuous polydispersity, using three different particle size distributions, namely the log-normal, Gamma, and truncated power-law distributions. In all cases, we observe agreement between our theoretical findings and numerical results up to rather large polydispersities for all particle size distributions, when using as reference our own simulations and results from the literature. In particular, we find $\phi_{\textrm{RCP}}$ to increase monotonically with the relative standard deviation, $s_{\sigma}$, of the distribution, and to saturate at a value that always remains below 1.A perturbative expansion yields a closed-form expression for $\phi_{\textrm{RCP}}$ that quantitatively captures a distribution-independent regime for $s_{\sigma} < 0.5$. Beyond that regime, we show that the gradual loss in agreement is tied to the growth of the skewness of size distributions.

Dynamics and Time Scales of Higher-Order Correlations in Supercooled Colloidal Systems

“…A fit of the VFT law to the data is shown in Figure b, resulting in a ϕ VFT of 0.65 ± 0.04. This almost matches the random close packing concentration of ϕ RCP = 0.68 for hard spheres with a size dispersity of δ = 15% and has been observed previously. − …”

Molecular-Level Relation between the Intraparticle Glass Transition Temperature and the Stability of Colloidal Suspensions

“…As noticed in refs and , for these systems, the critical packing fraction ϕ c occupied by the monomers at the glass transition can be reasonably expressed as ϕ normalc = ϕ normalc * − normalΛ z co where z co = 2(1 – 1/ n ) is the average connectivity due to intrachain covalent bonds, ϕ c * is the maximum packing fraction occupied by the monomers at the glass transition in the absence of covalent bonds (i.e., in case z co = 0), and Λ is a parameter expressing the effect of topological constraints due to covalent bonds on ϕ c . It follows that, when the already mentioned relation ln(1/ϕ) = α T T + c is evaluated at the glass transition ( T g , z c , and ϕ c ), after linearization, one correctly obtains the Fox–Flory-type relation between T g , thermal expansion, and molecular weight α T T normalg ≈ ( 1 − c − ϕ c * + 2 Λ ) − 2 normalΛ n When the values n ≈ 200, c ≈ 0.48, Λ ≈ 0.1, α T = 2 × 10 –4 K –1 , and ϕ c * ≈ 0.64 (i.e., ϕ c * coinciding with the random close packing of a system of hard spheres − ) as found in polymer glass ,, and T g = 383 K are considered, a G ( T ) profile in agreement with experimental data of ref follows from the insertion of eq into eqs and and of the resulting expressions, in turn, into eq (see, e.g., Figure 4 of ref ).…”

A geometric probabilistic approach to random packing of hard disks in a plane

“…The data shows a convex Z ( φ ) relation, so a power ζ > 1. Also, the model by Zaccone et al 26 and Anzivino et al 39 yields a convex equation. This can be attributed to their assumption that Z is proportional to the product of φ and the semi-empirical Carnahan-Starling (CS) expression.…”