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Entropy drives the phase behavior of colloids ranging from dense suspensionsof 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 has been demonstrated recently in computer simulationsand experiments. The ordering of shapes appears to arise from the emergence ofdirectional entropic forces (DEFs) that align neighboring particles, but theseforces have been neither rigorously defined nor quantified in generic systems.Here, we show quantitatively that shape drives the phase behavior of systems ofanisotropic particles upon crowding through DEFs. We define DEFs in genericsystems, and compute them for several hard particle systems. We show that theyare on the order of a few kT at the onset of ordering, placing DEFs on par withtraditional depletion, van der Waals, and other intrinsic interactions. Inexperimental systems with these other interactions, we provide directquantitative evidence that entropic effects of shape also contribute toself-assembly. We use DEFs to draw a distinction between self-assembly andpacking behavior. We show that the mechanism that generates directionalentropic forces is the maximization of entropy by optimizing local particlepacking. We show that this mechanism occurs in a wide class of systems, and wetreat, in a unified way, the entropy-driven phase behavior of arbitrary shapesincorporating the well-known works of Kirkwood, Onsager, and Asakura andOosawa.

Understanding shape entropy through local dense packing