The Cahn-hilliard Equation

1. Steady states. There are many two component systems in which phase separation can be induced by rapidly cooling the system. Thus, if a two component system, which is spatially uniform at temperature T1, is rapidly cooled to a second sufficiently lower temperature T2, then the cooled system will separate into regions of higher and lower concentration. A phenomenological description of the behavior of such systems can be obtained by energy arguments. The claim would be that there exists a critical temperature Tc, such that for T > Tc the free energy F (c, T ) of the system is a single welled function of the concentration c of one of the species, whereas for T < Tc the free energy is double welled. Referring to Figure 1, a system which was spatially uniform at temperature T1, when cooled to temperature T2, would find it energetically preferrable to separate itself into two systems, one at concentration cA and one at concentration cB. To be more specific consider now the system at temperature T < Tc. Assume that the free energy F (c) per unit volume (Gibbs free energy or Landau-Ginzburg free energy) of the spatially homogeneous system has the convex/concave shape indicated in Figure 2. More precisely F is concave in the spinodal interval cA < c < c s B and convex elsewhere. The points cA and cB where the supporting tangent touches the graph are sometimes referred to as the binodal points. The derivative f(c) = F (c) is depicted in Figure 3. The free energy of a spatially heterogeneous system would then be given by