Cahn-Hilliard Equation for Phase Separation

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The Cahn–Hilliard equation describes phase separation, for example, of elements in an alloy. Starting from a random distribution of and (representing two species), the concentration evolves in time. To obtain solutions of the differential equations, use the "" slider to adjust the diffusion constant and the "" slider to adjust the gamma parameter. Use the "time step" slider to visualize the formation of domains.

Contributed by: Oliver K. Ernst (September 2019)
Open content licensed under CC BY-NC-SA


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.


Snapshots



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