RSOZ

Given and Stell \cite{JCP_1992_97_04573,PA_1994_209_0495} provided {\bf exact} OZ equations for two-phase random media based on the original work of Madden and Glandt \cite{JSP_1988_51_0537_nolotengoSpringer,JCP_1992_96_05422}.\\ For a two-species system, for the $(s+1)$ replicated system one has (see Eq.s 2.7 --2.11 \cite{PA_1994_209_0495}):\\ \begin{equation} h_{mm} = c_{mm} + \rho_m c_{mm} \otimes h_{mm} + s\rho_f c_{mf} \otimes h_{mf} \end{equation} \begin{equation} h_{mf} = c_{mf} + \rho_m c_{mm} \otimes h_{mf} + \rho_f c_{mf} \otimes h_{ff} + (s-1) \rho_f c_{mf} \otimes h_{12} \end{equation} \begin{equation} h_{fm} = c_{fm} + \rho_m c_{fm} \otimes h_{mm} + \rho_f c_{ff} \otimes h_{fm} + (s-1) \rho_f c_{12} \otimes h_{fm} \end{equation} \begin{equation} h_{ff} = c_{ff} + \rho_m c_{fm} \otimes h_{mf} + \rho_f c_{ff} \otimes h_{ff} + (s-1) \rho_f c_{12} \otimes h_{12} \end{equation} \begin{equation} h_{12} = c_{12} + \rho_m c_{fm} \otimes h_{mf} + \rho_f c_{ff} \otimes h_{12} +

\rho_f  c_{12}  \otimes h_{ff} + (s-2) \rho_f c_{12}   \otimes h_{12}

\end{equation} In the limit of $s \rightarrow 0$ these equations from the ROZ equations (see Eq.s 2.12 --2.16 \cite{PA_1994_209_0495}):\\ \begin{equation} h_{mm} = c_{mm} + \rho_m c_{mm} \otimes h_{mm} \end{equation} \begin{equation} h_{mf} = c_{mf} + \rho_m c_{mm} \otimes h_{mf} + \rho_f c_{mf} \otimes h_{ff} - \rho_f c_{mf} \otimes h_{12} \end{equation} \begin{equation} h_{fm} = c_{fm} + \rho_m c_{fm} \otimes h_{mm} + \rho_f c_{ff} \otimes h_{fm} - \rho_f c_{12} \otimes h_{fm} \end{equation} \begin{equation} h_{ff} = c_{ff} + \rho_m c_{fm} \otimes h_{mf} + \rho_f c_{ff} \otimes h_{ff} - \rho_f c_{12} \otimes h_{12} \end{equation} \begin{equation} h_{12} = c_{12} + \rho_m c_{fm} \otimes h_{mf} + \rho_f c_{ff} \otimes h_{12} +

\rho_f  c_{12}  \otimes h_{ff} -2 \rho_f c_{12}   \otimes h_{12}

\end{equation} When written in the `percolation terminology' where $c$ terms `{\it connected}' and $b$ `{\it blocking}' are adapted from the language of percolation theory. \begin{equation} h_{mm} = c_{mm} + \rho_m c_{mm} \otimes h_{mm} \end{equation} \begin{equation} h_{fm} = c_{fm} + \rho_m c_{fm} \otimes h_{mm} + \rho_f c_c \otimes h_{fm} \end{equation} \begin{equation} h_{ff} = c_{ff} + \rho_m c_{fm} \otimes h_{mf} + \rho_f c_c \otimes h_{ff} + \rho_f c_b \otimes h_c \end{equation} \begin{equation} h_c = c_c + \rho_f c_c \otimes h_c \end{equation} where the direct correlation function is split into \begin{equation} c_{ff}(12) = c_c (12) + c_b (12) \end{equation} and the total correlation function is also split into \begin{equation} h_{ff}(12)= h_c (12) + h_b(12) \end{equation} where $m$ denotes the matrix and $f$ denotes the fluid. The blocking function $h_b(x)$ accounts for correlations between a pair of fluid particles ``blocked" or separated from each other by matrix particles.\\ IMPORTANT NOTE: Unlike an equilibrium mixture, there is only one convolution integral for $h_{mm}$ because the structure of the medium is unaffected by the presence of fluid particles.\\ Note: $C_{ff}$ (Madden and Glandt) $=h_c$ (Given and Stell)\\ Note: fluid: $f$ (Madden and Glandt), `1' (Given and Stell)\\ Note: matrix: $m$ (Madden and Glandt), `0' (Given and Stell)\\ At very low matrix porosities, i.e. very high densities of matrix particles, the volume accessible to fluid particles is divided into small cavities, each totally surrounded by a matrix. In this limit, the function $h_c (x)$ describes correlations between fluid particles in the same cavity and the function $h_b(x)$ describes correlations between particles in different cavities.