The Excluded Area of Superellipse Sector Particles

TL;DR

This paper studies the geometric constraints and excluded areas of superellipse sector particles (SeSPs), revealing how their shape parameters influence particle interactions and phase transitions in granular systems.

Contribution

It introduces a detailed analysis of the allowable configurations and excluded areas of SeSPs, linking particle shape parameters to phase transition behavior.

Findings

Excluded area decreases sigmoidally with aperture size.

Sharper corners lead to sharper decrease in excluded area.

Topological changes suggest a phase transition in configuration space.

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 investigate the space of allowable (non-overlapping) configurations of two SeSPs, which 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. The excluded area is a straightforward integration over excluded states; for arbitrary relative orientation this decreases sigmoidally with increasing opening aperture, with sharper SeSP corners resulting in a sharper decrease. Together, this work offers a path towards a unified theory for particle shape-control of bulk material properties.