In contrast, spinodal decomposition is spontaneous. Rain, acid rain, air pollution, vapor-phase synthesis of materials, and industrial crystallization are but a few of the important natural and technical phenomena and processes in which nucleation plays a key role. Yet, despite nucleation's ubiquity and importance, the accurate prediction of nucleation rates is only possible for the simplest of physical systems. These expressions extend and generalize the classic work of Gibbs to the case where the embryo is not in unstable equilibrium with the mother phase. The structure of dense liquids is governed by repulsive interactions, while attractions can, for most practical purposes, be thought of as providing a uniform background cohesive potential.
Hard spheres are therefore the natural reference system in descriptions of the liquid state.
Because of this, hard spheres continue to play a central role in the theory of liquids. Geometry and thermodynamics are indistinguishable in hard sphere systems. In other words, one can calculate the equation of state and chemical potential of a hard sphere system from knowledge of such quantities as the mean cavity volume and the mean interface area between cavities and spheres. This information is important in flow through porous media, protein solvation, and the characterization of random media, in addition to the thermodynamics of hard spheres.