Shape universality classes in the random sequential addition of non-spherical particles

TL;DR

This paper analytically investigates the asymptotic behavior of the random sequential addition (RSA) of non-spherical particles constrained to a line, revealing two universality classes of scaling exponents based on contact function smoothness.

Contribution

It provides an analytical solution for RSA of non-spherical particles with moderate aspect ratios and classifies universality classes of scaling exponents based on particle shape and contact function properties.

Findings

Scaling exponent depends on particle shape and contact function smoothness.

Two universality classes identified: smooth contact functions and singular contact functions.

Analytical explanation for empirical scalings observed in 2D and 3D RSA studies.

Abstract

Random sequential addition (RSA) models are used in a large variety of contexts to model particle aggregation and jamming. A key feature of these models is the algebraic time dependence of the asymptotic jamming coverage as $t\to\infty$. For the RSA of monodisperse non-spherical particles the scaling is generally believed to be $~t^{-\nu}$, where $\nu=1/d_{\rm f}$ for a particle with $d_{\rm f}$ degrees of freedom. While the $d_{\rm f}=1$ result of spheres (Renyi's classical car parking problem) can be derived analytically, evidence for the $1/d_{\rm f}$ scaling for arbitrary particle shapes has so far only been provided from empirical studies on a case-by-case basis. Here, we show that the RSA of arbitrary non-spherical particles, whose centres of mass are constrained to fall on a line, can be solved analytically for moderate aspect ratios. The asymptotic jamming coverage is determined by a Laplace-type integral, whose asymptotics is fully specified by the contact distance between two particles of given orientations. The analysis of the contact function $r$ shows that the scaling exponent depends on particle shape and falls into two universality classes for generic shapes with $\tilde{d}$ orientational degrees of freedom: (i) $\nu=1/(1+\tilde{d}/2)$ when $r$ is a smooth function of the orientations as for smooth convex shapes, e.g., ellipsoids; (ii) $\nu=1/(1+\tilde{d})$ when $r$ contains singularities due to flat sides as for, e.g., spherocylinders and polyhedra. The exact solution explains in particular why many empirically observed scalings in $2d$ and $3d$ fall in between these two limiting values.