Excluded area, entanglement, and packing of superellipse sector particles

Superellipse sector particles (SeSPs) are segments of superelliptical curves and 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 specific SeSPs. We measure the excluded area of SeSPs as a function of opening aperture size, starting with the special case of annular particles before generalizing to more angular shapes. The excluded area of annuli decreases sigmoidally with increasing opening aperture; for SeSPs with sharper corners the curve becomes more step-shaped. Whether two particles can be placed without overlap depends on both radial separation and relative orientation. Mapping allowed two-particle configurations reveals circular regions centered at each of a particle's two endpoints that indicate configurations of mutually-entangled particle interactions. Simultaneous mutual entanglement with both endpoints is geometrically impossible: the overlap of these two regions represents an excluded area in which no particles can be placed regardless of orientation. The regions' distinct boundaries reflects translational frustration with implications for the dynamics of particle rearrangements.