Shape Universality Classes in the Random Sequential Adsorption of Nonspherical Particles

Abstract

Random 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 non-spherical particles ν = 1/(d +d), where d denotes the dimension of the domain andd 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 that the scaling exponent depends on 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; (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.The question of how particle shape affects the dynamical and structural properties of particle aggregates is one of the outstanding problems in statistical mechanics with profound technological implications [1][2][3]. Jammed systems are particularly challenging, since they are dominated by the geometry of the particles and are not described by conventional equilibrium statistical mechanics [4]. Exploring the effect of shape variation thus relies on extensive computer simulations [5][6][7][8] or mean-field theories whose solutions require similar computational efforts [9,10]. From a theoretical perspective it is striking that so far there has been hardly any insight from exactly solvable analytical models, even though these are most suitable to identify and classify shapes in the infinite shape space.In this letter, we consider the probably simplest nontrivial packing model that takes into account excluded volume effects due to shape anisotropies: random sequential adsorption (RSA). Since Renyi's seminal work on the 'car parking problem' (the RSA of monodisperse spheres on a line) [11,12], RSA models have been widely used to model particle aggregation and jamming in physical, chemical and biological systems [13][14][15]. Their great appeal is the paradigmatic nature of the adsorption mechanism: the particles' positions and orientations are selected with uniform probability and then placed sequentially into the domain if there is no overlap with any previously placed particles. Particles are not able to move or reorient once being placed.Two key features of RSA models are: (i) the existence of a finite jamming density φ in the infinite time limit φ(∞) = lim t→∞ φ(t) and (ii) the algebraic time dependence of the approach to jamming, which has been conj...

Kinetics of random sequential adsorption of two-dimensional shapes on a one-dimensional line

“…Although in general the kinetics of packing fraction growth and particles density growth is different, here, to be consistent with the previous theoretical study [21], we will focus on the second one. We have also checked that for large enough t, both kinetics converge to each other; thus, the presented results should be universal.…”

“…Theoretical arguments also support the explanation of the transition as a finite size effect. In the analytical solution of the growth kinetics [21], it is shown that d is determined by the analytic properties of the function ψ(z, α, β) = r(α, z) + r(z, β) as z approaches the minimum z * of ψ for given α, β. Here, r denotes the contact function and α, β are the orientations of the particles at the left/right end of the interval of length z.…”

“…The dependence of the exponent d near saturation on the width-to-height ratio x for ellipses, spherocylinders and rectangles. Dashed lines correspond to d = 1.5 and d = 2 derived analytically for shapes with analytical and nonanalytical contact function[21].…”

“…Dots correspond to numerical data for shapes of anisotropy x = 2.0. Dashed black lines on the right panel correspond to d = 1.5 and d = 2 derived analytically for shapes with analytical and non-analytical contact function[21].…”

Saturated random packing built of arbitrary polygons under random sequential adsorption protocol

“…Although the parameter d in (5) is simply the space dimension for spheres, Baule has shown recently that when anisotropic particles are placed according to the RSA protocol in such a way that their centers lay on a one-dimensional line, d depends on properties of contact function of packed shapes. Parameter d is noticeably bigger when this function is not analytic [22]. For two-dimensional non-spherical RSA packings, there is numerical evidence that d = 3 for RSA of ellipses [8,17], or rectangles [27,28], however, these studies do not reach saturation limit.…”