Continuous theory of phase separation:The Cahn-Hilliard equation

Abstract

In this paper, we present a detailed deduction of the Cahn-Hilliard equation. It is a non-linear partial differential equation that continuously describes the dynamics of spinodal decomposition, i.e., spontaneous phase separation, of a binary mixture. From such an equation we can model systems that undergo phase separation toward the equilibrium, i.e., passive systems, and systems that are kept out of the equilibrium by the presence of either external forces or chemical reactions, i.e., active systems. To accomplish our main goal, we present the main concepts of non-equilibrium thermodynamics and from that, we obtain the Cahn-Hilliard equation by defining a general equation for the transport of matter and calculating the free energy for a homogeneous and non-homogeneous mixture of real fluids. By the end, we perform simulations of the Cahn-Hilliard equation and present two examples of passive and active systems that (quasi) suppress Ostwald ripening.

Keywords

• Non-equilibrium thermodynamics
• liquid-liquid phase separation
• passive and active systems