Understanding shape entropy through local dense packing
Significance Many natural systems are structured by the ordering of repeated, distinct shapes. Understanding how this happens is difficult because shape affects structure in two ways. One is how the shape of a cell or nanoparticle, for example, affects its surface, chemical, or other intrinsic properties. The other is an emergent, entropic effect that arises from the geometry of the shape itself, which we term “shape entropy,” and is not well understood. In this paper, we determine how shape entropy affects structure. We quantify the mechanism and determine when shape entropy competes with intrinsic shape effects. Our results show that in a wide class of systems, shape affects bulk structure because crowded particles optimize their local packing. Entropy drives the phase behavior of colloids ranging from dense suspensions of hard spheres or rods to dilute suspensions of hard spheres and depletants. Entropic ordering of anisotropic shapes into complex crystals, liquid crystals, and even quasicrystals was demonstrated recently in computer simulations and experiments. The ordering of shapes appears to arise from the emergence of directional entropic forces (DEFs) that align neighboring particles, but these forces have been neither rigorously defined nor quantified in generic systems. Here, we show quantitatively that shape drives the phase behavior of systems of anisotropic particles upon crowding through DEFs. We define DEFs in generic systems and compute them for several hard particle systems. We show they are on the order of a few times the thermal energy (kBT) at the onset of ordering, placing DEFs on par with traditional depletion, van der Waals, and other intrinsic interactions. In experimental systems with these other interactions, we provide direct quantitative evidence that entropic effects of shape also contribute to self-assembly. We use DEFs to draw a distinction between self-assembly and packing behavior. We show that the mechanism that generates directional entropic forces is the maximization of entropy by optimizing local particle packing. We show that this mechanism occurs in a wide class of systems and we treat, in a unified way, the entropy-driven phase behavior of arbitrary shapes, incorporating the well-known works of Kirkwood, Onsager, and Asakura and Oosawa.