Appendix A: Contribution to pressure tensor from fluid/fluid interaction

Here it is useful to discuss the relation between the LB model and other mean-field models of phase separation. First, consider the equation for a single particle distribution function P₁ based on the continuum BBGKY [8,9] formalism which is extended to the case of multiple species,

$$\partial_t P^i_1 + \vec{k}_1 \! \cdot \! \nabla P^i_1 + \vec{F} \! \cdot \! \nabla_{\!\! k} P^i_1 =\Omega^i \! , \\$$ (34)

with $\vec{k}$ is the microscopic momentum, F is the acceleration due to a body force and $\Omega$ is a collision operator. It can be shown, when making a molecular chaos approximation [9], that the collision operator can be written as

$$\Omega^i=-\sum_{j=1}^{s}\frac{\partial P^i_1(\vec{r}_1,\vec{k... ...ec{r}_2,t) \frac{\partial V_{ij}(r_{12})}{\partial \vec{r}_1}.$$ (35)

This approximation of the collision operator is of the form of a body-force term, $F \cdot \nabla_k P_1$. For $\vert\vec{r}_1-\vec{r}_2\vert > d$ where d is of order a few "effective" hard sphere diameters, $g_{ij}(\vec{r}_1,\vec{r}_2) \approx 1$. After expanding $\rho^j$ about r₁, the contribution to the collision operator associated with the attractive intermolecular interaction, $\Omega^i$can be approximated by, $\Omega^i=\sum_{j=1}^s\nabla V^{ij}_m \cdot \nabla_k P^i_1$, where $V^{ij}_m=2a_{ij}\rho^{j} (\vec{r}_1)+\kappa_{ij} \nabla^2 \rho^{j} (\vec{r}_1)$with $a_{ij}=\frac{1}{2}\int_d d^3rV_{ij}(r)$ and $\kappa_{ij}=\frac{1}{6}\int_d d^3rr^2V_{ij}(r)$. V^ij_m can be thought of as a mean-field potential produced by neighboring particles and $-\nabla V^{ij}_m$ is the associated mean-field force.

The pressure tensor can be determined for this system,

This expansion is a counterpart of the Cahn-Hilliard or Landau free energy expansion [12,17]. The forcing used in this paper is for the case where all terms are in the above pressure tensor are zero except that with the coefficient a₁₂, which is proportional to the coupling constant, G, described earlier in the paper. While there is no explicit inclusion of a surface tension term in the model studied in this paper, an effective surface tension force results in the Shan-Chen model due to how the forcing between fluid components is incorporated. This can be seen from the leading term in the expansion of F,

$$F \sim \rho(x+\Delta x)- \rho(x-\Delta x) \approx \frac{\partial \rho}{\partial x}.$$ (36)

where F² corrections to the pressure tensor, a feature in the Shan-Chen model [7], scales with the surface tension as in standard Cahn-Hilliard models. Higher order terms, from Chapman Enskog analysis, also contribute to the effective surface tension, along with contributions that arise from finite difference approximations of the contiuum equations. Quantitative predictions of the surface tension require an understanding of all these terms.