Optimal shapes of disk assembly in saturated random packings

Particle morphology is one of the most significant factors influencing the packing structures of granular materials. With certain targeted properties or optimization criteria, inverse packing problems have drawn extensive attention in terms of their adaptability to many material design tasks. An important question hard to answer is which particle shape, especially within given shape families, forms the densest (loosest) random packing? In this paper, we address this issue for the disk assembly model in two dimensions with an infinite variety of shapes, which are simulated in the random sequential adsorption process to suppress crystallization. Via a unique shape representation method, particle shapes are transformed into genotype sequences in the continuous shape space where we utilize the genetic algorithm as an efficient shape optimizer. Specifically, we consider three representative species of disk assembly, i.e., congruent tangent disks, incongruent tangent disks, and congruent overlapping disks, and carry out shape optimization on their packing densities in the saturated random state. We numerically search optimal shapes in the three species with a variable number of constituent disks which yield the maximal and minimal packing densities. We obtain an isosceles circulo-triangle and an unclosed ring for the maximal and minimal packing density in saturated random packings, respectively. The perfect sno-cone and isosceles circulo-triangle are also specifically investigated which give remarkably high packing densities of around 0.6, much denser than those of ellipses. This study is beneficial for guiding the design of particle shapes as well as the inverse design of granular materials.