General methods for free-volume theory

Free-volume theory for understanding depletion phenomena in mixtures of two species is generally derived using scaled-particle theory for those specific entities. Here we first give a general scaled-particle method for convex bodies in terms of the characteristic geometrical measures of the depletion agent, i.e., its volume, surface area, and integrated mean curvature, in mixtures with hard spheres. Second, we show that similar results can be derived from fundamental-measure theory. This different approach allows us to get a deep insight into the meaning of the various contributions to the theory from a geometrical point of view. From these two methods we arrive at a generalized "recipe" to free-volume theory. This recipe can be based on a desired equation of state for any convex shape of the depletion agents and is also valid for (polydisperse) mixtures of those. This is illustrated by mixtures of spheres with ellipsoids, spheres with several geometries as models for disklike mesogens, e.g., gibbsite, as well as depletion of spheres due to bar-shaped colloids, e.g., goethite.