Shape Universality Classes in the Random Sequential Adsorption of Nonspherical Particles
AbstractRandom sequential adsorption (RSA) of particles of a particular shape is used in a large variety of contexts to model particle aggregation and jamming. A key feature of these models is the observed algebraic time dependence of the asymptotic jamming coverage ∼t-ν as t→∞. However, the exact value of the exponent ν is not known apart from the simplest case of the RSA of monodisperse spheres adsorbed on a line (Renyi’s seminal “car parking problem”), where ν=1 can be derived analytically. Empirical simulation studies have conjectured on a case-by-case basis that for general nonspherical particles, ν=1/(d+d˜), where d denotes the dimension of the domain, and d˜ the number of orientational degrees of freedom of a particle. Here, we solve this long-standing problem analytically for the d=1 case—the “Paris car parking problem.” We prove, in particular, that the scaling exponent depends on the particle shape, contrary to the original conjecture and, remarkably, falls into two universality classes: (i) ν=1/(1+d˜/2) for shapes with a smooth contact distance, e.g., ellipsoids, and (ii) ν=1/(1+d˜) for shapes with a singular contact distance, e.g., spherocylinders and polyhedra. The exact solution explains, in particular, why many empirically observed scalings fall in between these two limits.