The CahnâHilliard equation (after John W. Cahn and John E. Hilliard) [1] is an equation of mathematical physics which describes the process of phase separation, spinodal decomposition, by which the two components of a binary fluid spontaneously separate and form domains pure in each component. If is the concentration of the fluid, with indicating domains, then the equation is written as
where is a diffusion coefficient with units of and gives the length of the transition regions between the domains. Here is the partial time derivative and is the Laplacian in dimensions. Additionally, the quantity is identified as a chemical potential.
Related to it is the AllenâCahn equation, as well as the stochastic AllenâCahn and the stochastic CahnâHilliard equations.
Of interest to mathematicians is the existence of a unique solution of the CahnâHilliard equation, given by smooth initial data. The proof relies essentially on the existence of a Lyapunov functional. Specifically, if we identify
as a free energy functional, then
so that the free energy does not grow in time. 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.
The CahnâHilliard equation finds applications in diverse fields: in complex fluids and soft matter (interfacial fluid flow, polymer science and in industrial applications). The solution of the CahnâHilliard equation for a binary mixture demonstrated to coincide well with the solution of a Stefan problem and the model of Thomas and Windle. [2] Of interest to researchers at present is the coupling of the phase separation of the CahnâHilliard equation to the NavierâStokes equations of fluid flow.
The CahnâHilliard equation (after John W. Cahn and John E. Hilliard) [1] is an equation of mathematical physics which describes the process of phase separation, spinodal decomposition, by which the two components of a binary fluid spontaneously separate and form domains pure in each component. If is the concentration of the fluid, with indicating domains, then the equation is written as
where is a diffusion coefficient with units of and gives the length of the transition regions between the domains. Here is the partial time derivative and is the Laplacian in dimensions. Additionally, the quantity is identified as a chemical potential.
Related to it is the AllenâCahn equation, as well as the stochastic AllenâCahn and the stochastic CahnâHilliard equations.
Of interest to mathematicians is the existence of a unique solution of the CahnâHilliard equation, given by smooth initial data. The proof relies essentially on the existence of a Lyapunov functional. Specifically, if we identify
as a free energy functional, then
so that the free energy does not grow in time. 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.
The CahnâHilliard equation finds applications in diverse fields: in complex fluids and soft matter (interfacial fluid flow, polymer science and in industrial applications). The solution of the CahnâHilliard equation for a binary mixture demonstrated to coincide well with the solution of a Stefan problem and the model of Thomas and Windle. [2] Of interest to researchers at present is the coupling of the phase separation of the CahnâHilliard equation to the NavierâStokes equations of fluid flow.