CahnHilliardlike

The Cahn-Hilliardlike equation is a mathematical model used in materials science and physics to describe the process of spinodal decomposition and phase separation in binary mixtures. It is a nonlinear partial differential equation that extends the original Cahn-Hilliard equation by incorporating additional terms or modifying existing ones to capture more complex phenomena. These modifications can arise from various physical considerations, such as the inclusion of elastic strain energy, different interfacial energy functionals, or external driving forces.

At its core, the Cahn-Hilliardlike equation describes the evolution of a concentration field, often denoted by c(x,t), over space (x) and time (t). The equation typically involves a free energy functional that includes both a bulk term, representing the energetic cost of forming different phases, and an interfacial term, accounting for the energy associated with the boundary between phases. The diffusion term, proportional to the Laplacian of the chemical potential, drives the system towards equilibrium.

The 'like' in Cahn-Hilliardlike signifies that the equation deviates from the standard Cahn-Hilliard formulation. These deviations are crucial for modeling specific physical systems that exhibit behaviors not fully described by the original model. For instance, modifications might be introduced to study the influence of crystallographic anisotropy on domain coarsening or to investigate phase separation in systems with anisotropic interfacial properties. Applications of Cahn-Hilliardlike equations are found in the study of alloys, polymers, glasses, and other materials undergoing phase transformations. The numerical solution of these equations often requires sophisticated computational techniques due to their nonlinear nature and the complexity of the incorporated physics.