Interaction Potential

In order to model the phase separation of fluids, an interaction between the fluids is needed to drive them apart. Here a force, $\frac{d{\bf p}^i}{dt}({\bf x})$, between the two fluids is introduced that effectively perturbs the equilibrium velocity[1,2] for each fluid so that they have a tendency to phase separate:

$$n^i({\bf x}){\bf v}^{'}({\bf x})=n^i {\bf v}({\bf x})+ \tau_i \frac{d{\bf p}^i}{dt}({\bf x})$$ (7)

where v' is the new velocity used in Eqs. [3] and [4]. We use a simple interaction that depends on the density of each fluid, as follows[1,2]:

$$\frac{d{\bf p}^i}{dt}({\bf x})=-n^i({\bf x})\sum_{i'}^ {S}\sum_{a}G_{ii'}^{a} n^{i'}({\bf x}+{\bf e}_a) {\bf e}_a$$ (8)

with G_ii'^a= 2G for $\vert{\bf e}^a\vert = 1$; G_ii'^a= G for $\vert{\bf e}^a\vert = \sqrt{2}$; and G_ii'^a= 0 for i=i'. G is a constant that controls the strength of the interaction. Clearly, the forcing term is related to the density gradient of the fluid. It has been shown that the above forcing term can drive the phase separation process and naturally produce an interfacial surface tension effect consistent with the Laplace law boundary condition [3].

In this model, phase separation takes place when the mutual diffusivity of the binary mixture becomes negative. An analytical expression for the mutual diffusivity has been determined in a previous work[6]. For the case of a critical composition the condition for the system studied to undergo phase separation is $G \geq \frac{T}{12(n^1+n^2)}$.