Excluded area of superellipse sector particles

Abstract

Superellipse sector particles (SeSPs) are segments of superelliptical curves that form a tunable set of hard-particle shapes for granular and colloidal systems. SeSPs allow for continuous parameterization of corner sharpness, aspect ratio, and particle curvature; rods, circles, rectangles, and staples are examples of shapes SeSPs can model. We measure the excluded area of SeSPs as a function of opening aperture size, beginning with the special case of semi-circular particles and then generalizing to more angular shapes. The excluded area decreases sigmoidally with increasing opening aperture, with sharper SeSP corners resulting in a sharper decrease. Whether two particles can be placed without overlap depends on both the center-of-mass separation and relative orientation. Radial correlation plots of the allowed configurations reveal circular regions centered at each of the particle's two endpoints that indicate configurations of mutually-entangled particle interactions. Simultaneous entanglement with both endpoints is geometrically impossible; the overlap of these two regions therefore represents an excluded area in which no particles can be placed regardless of orientation. The regions' distinct boundaries indicates a translational frustration with implications for the dynamics of particle rearrangements (e.g. under shear). Representing translational and rotational degrees of freedom as a hypervolume, we find a topological change that suggests geometric frustration arises a phase transition in this space. Together, this work offers a path towards a unified theory for particle shape-control of bulk material properties.