RSOZ

Given and Stell (Refs 1 and 2) provided exact OZ equations for two-phase random media based on the original work of Madden and Glandt (Refs 3 and 4). For a two-species system, for the Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (s+1)} replicated system one has (see Eq.s 2.7 --2.11 Ref. 2):

$h_{mm}=c_{mm}+\rho _{m}c_{mm}\otimes h_{mm}+s\rho _{f}c_{mf}\otimes h_{mf}$

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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}}

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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}}

$h_{ff}=c_{ff}+\rho _{m}c_{fm}\otimes h_{mf}+\rho _{f}c_{ff}\otimes h_{ff}+(s-1)\rho _{f}c_{12}\otimes h_{12}$

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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}}

In the limit of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s \rightarrow 0} these equations from the ROZ equations (see Eq.s 2.12 --2.16 Ref. 2):

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h_{mm} = c_{mm} + \rho_m c_{mm} \otimes h_{mm}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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}}

When written in the `percolation terminology' where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} terms connected and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} blocking are adapted from the language of percolation theory.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h_{mm} = c_{mm} + \rho_m c_{mm} \otimes h_{mm}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h_{fm} = c_{fm} + \rho_m c_{fm} \otimes h_{mm} + \rho_f c_c \otimes h_{fm}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h_c = c_c + \rho_f c_c \otimes h_c}

where the direct correlation function is split into

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left.c_{ff}(12)\right. = c_c (12) + c_b (12)}

and the total correlation function is also split into

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left.h_{ff}(12)\right.= h_c (12) + h_b(12)}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} denotes the matrix and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} denotes the fluid. The blocking function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h_{mm}} because the structure of the medium is unaffected by the presence of fluid particles.

• Note: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_{ff}} (Madden and Glandt) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =h_c} (Given and Stell)
• Note: fluid: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} (Madden and Glandt), `1' (Given and Stell)
• Note: matrix: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h_c (x)} describes correlations between fluid particles in the same cavity and the function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h_b(x)} describes correlations between particles in different cavities.

References

1. [JCP_1992_97_04573]
2. [PA_1994_209_0495]
3. [JSP_1988_51_0537_nolotengoSpringer]
4. [JCP_1992_96_05422]