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

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 $\nu =1/d_{\mathrm{f}}$ for a particle with $d_{\mathrm{f}}$ degrees of freedom. While the $d_{\mathrm{f}}=1$ result of spheres (Renyi's classical car parking problem) can be derived analytically, evidence for the $1/d_{\mathrm{f}}$ 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.

5 pages, 1 figure

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• "Paris car parking problem" for partially ordered discorectangles on a line
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