Details

The Cahn–Hilliard equation describes phase separation, for example, of elements in an alloy. It is given by:

,

where is the concentration, with values and representing the two different species; is the diffusion constant; and the parameter relates to the transition region between domains. The differential equations are discretized using finite differences and solved on a 30×30 grid with periodic boundary conditions starting from a random initial condition of or at each node. The equations are solved from time 0 to time 4, with the distances between compartments in the discretization defined as 1.

Snapshot 1: example of domains formed

Snapshot 2: example of a random initial condition

Snapshot 3: example of large and homogeneous domains formed at high diffusion constants and gamma parameter value

Reference

[1] J. W. Cahn and J. E. Hilliard, "Free Energy of a Nonuniform System. I. Interfacial Free Energy," The Journal of Chemical Physics, 28(2), 1958 pp. 258–267. doi:10.1063/1.1744102.