AllenCahnvergelijking

The Allen-Cahn equation is a partial differential equation used to model phase separation and pattern formation in materials science and other fields. It was introduced by John W. Cahn and John E. Hilliard in 1958. The equation is given by:

∂u/∂t = ∇²u + u - u³

where u is a scalar field representing the order parameter, t is time, and ∇² is the Laplacian operator. The term u - u³ is a double-well potential, which drives the system towards two stable states, u = ±1. The Laplacian term ∇²u promotes smoothness in the solution, leading to the formation of interfaces between the two phases.

The Allen-Cahn equation exhibits a rich variety of behaviors, including spinodal decomposition, Ostwald ripening, and the formation of complex patterns such as labyrinths and dendrites. These patterns are often observed in phase-separating systems, such as alloys, polymers, and biological tissues. The equation has been extensively studied both analytically and numerically, and it serves as a fundamental model for understanding pattern formation in non-equilibrium systems.