The Equations for the Phase Field Method
The free energy is a functional of a dimensionless phase field parameter, ϕ: 
where ε is a measure of the interface thickness. Equation 11-11 describes the evolution of the phase field parameter:
(11-11)
where ftot (SI unit: J/m3) is the total free energy density of the system, and u (SI unit: m/s) is the velocity field for the advection. The right-hand side of Equation 11-11 aims to minimize the total free energy with a relaxation time controlled by the mobility γ (SI unit: m3·s/kg).
The free energy density of an isothermal mixture of two immiscible fluids is the sum of the mixing energy and elastic energy. The mixing energy assumes the Ginzburg–Landau form:
where ϕ is the dimensionless phase field variable, defined such that the volume fraction of the components of the fluid are (1+ ϕ)/2 and (1− ϕ)/2. The quantity λ (SI unit: N) is the mixing energy density and ε (SI unit: m) is a capillary width that scales with the thickness of the interface. These two parameters are related to the surface tension coefficient, σ (SI unit: N/m), through the equation
(11-12)
The PDE governing the phase field variable is the Cahn–Hilliard equation:
(11-13)
where G (SI unit: Pa) is the chemical potential and γ (SI unit: m3·s/kg) is the mobility. The mobility determines the time scale of the Cahn–Hilliard diffusion and must be large enough to retain a constant interfacial thickness but small enough so that the convective terms are not overly damped. In COMSOL Multiphysics the mobility is determined by a mobility tuning parameter that is a function of the interface thickness γ = χε2. The chemical potential is:
(11-14)
The Cahn–Hilliard equation forces ϕ to take a value of 1 or −1 except in a very thin region on the fluid-fluid interface. The Phase Field in Fluids interface breaks Equation 11-13 up into two second-order PDEs:
(11-15)
(11-16)