  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 has been 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 that they
are on the order of a few kT 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.
