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Cahn-Hilliard equation
From Wikipedia, the free encyclopedia
The Cahn-Hilliard equation is an equation of mathematical physics which describes the process of phase separation, by which the two components of a binary fluid spontaneously separate and form domains pure in each component. If c is the concentration of the fluid, with 
indicating domains, then the equation is written as
where D is a diffusion coefficient with units of Length2 / Time and 
gives the length of the transition regions between the domains. Here 
is the partial time derivative and 
is the Laplacian in n dimensions. Additionally, the quantity 
is identified as a chemical potential.
Of interest to mathematicians is the existence of a unique solution to the Cahn-Hilliard equation, given smooth initial data. The proof relies essentially on the existence of a Lyapunov functional. Specifically, if we identify
![F[c]=\int d^n x \left[\frac{1}{4}\left(c^2-1\right)^2+\frac{\gamma}{2}\left|\nabla c\right|\right],](../../../math/6/3/c/63cf676758a1d468fcbc51fb3a291284.png)
as a free energy functional, then
so that the free energy decays to zero. This also indicates segregation into domains is the asymptotic outcome of the evolution of this equation.
In real experiments, the segregation of an initially mixed binary fluid into domains is observed. The segregation is characterized by the following facts.
- There is a transition layer between the segregated domains, with a profile given by the function
and hence a typical width . This is due to the fact that this function is an equilibrium solution of the Cahn-Hilliard equation.
- Of interest also is the fact that the segregated domains grow in time as a power law. That is, if L(t) is a typical domain size, then
. This is the Lifshitz-Slyozov law, and has been proved rigorously for the Cahn-Hilliard equation and observed in numerical simulations and real experiments on binary fluids.
- Owing to the form equation, the Cahn-Hilliard phase separation process conserves the total concentration
.
The Cahn-Hilliard equations finds applications in diverse fields: in interfacial fluid flow, polymer science and in industrial applications. Of interest to researchers at present is the coupling of the phase separation of the Cahn-Hilliard equation to the Navier-Stokes equations of fluid flow.
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- J. Zhu, L. Q. Shen, J. Shen, V. Tikare, and A. Onuki, “Coarsening kinetics from a variable mobility Cahn–Hilliard equation: Application of a semi-implicit Fourier spectral method,” Phys. Rev. E 60, 3564 (1999).
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- T. Hashimoto, K. Matsuzaka, and E. Moses, “String phase in phase-separating fluids under shear flow,” Phys. Rev. Lett. 74, 126 (1995).