where ε is a measure of the interface thickness.
Equation 11-20 describes the evolution of the phase field parameter:
where ftot (SI unit: J/m
3) 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 11-20 aims to minimize the total free energy with a relaxation time controlled by the mobility
γ (SI unit: m
3·s/kg).
where 
is the dimensionless phase field variable, defined such that the volume fraction of the components of the fluid are (1+ 
)/2 and (1− 
)/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
where G (SI unit: Pa) is the chemical potential and
γ (SI unit: m
3·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:
The Cahn–Hilliard equation forces 
to take a value of 1 or
−1 except in a very thin region on the fluid-fluid interface. The Phase Field Thin-Film Flow interface breaks
Equation 11-13 up into two second-order PDEs:
: 
