CahnHilliardGleichungen

The Cahn-Hilliard equations are a set of nonlinear, time-dependent partial differential equations used to model phase separation and pattern formation in binary alloys. They were derived by John W. Cahn and John E. Hilliard in 1958. The equations describe the evolution of the concentration field of one component in a binary mixture, taking into account the effects of diffusion and the free energy of the system.

The Cahn-Hilliard equations can be written as:

∂c/∂t = ∇ · [M(c) ∇ (∂f/∂c)]

where c is the concentration of one component, t is time, M(c) is the mobility, and f is the free energy density. The term ∂f/∂c represents the chemical potential, which drives the diffusion of the components.

These equations are particularly useful in materials science, where they are used to simulate the microstructure evolution during solidification, spinodal decomposition, and other phase separation processes. The Cahn-Hilliard equations have also been applied in other fields, such as biology and chemistry, to model pattern formation and phase separation phenomena.

Numerical methods, such as finite difference and finite element methods, are commonly used to solve the Cahn-Hilliard equations. These methods allow for the simulation of complex systems and the study of the underlying mechanisms of phase separation and pattern formation.