RSOZ

Given and Stell (Refs 1 and 2) provided exact Ornstein-Zernike relations 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 $(s+1)$ replicated system one has (see Eq.s 2.7 --2.11 Ref. 2):

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

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

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

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

$h_{12}=c_{12}+\rho _{m}c_{fm}\otimes h_{mf}+\rho _{f}c_{ff}\otimes h_{12}+\rho _{f}c_{12}\otimes h_{ff}+(s-2)\rho _{f}c_{12}\otimes h_{12}$

In the limit of $s\rightarrow 0$ these equations from the replica Ornstein-Zernike (ROZ)equations (see Eq.s 2.12 --2.16 Ref. 2):

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

$h_{mf}=c_{mf}+\rho _{m}c_{mm}\otimes h_{mf}+\rho _{f}c_{mf}\otimes h_{ff}-\rho _{f}c_{mf}\otimes h_{12}$

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 $b$ 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

$\left.c_{ff}(12)\right.=c_{c}(12)+c_{b}(12)$

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 $h_{mm}$ 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: $f$ (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.

Polydisperse systems

For a polydisperse fluid, composed 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 n_f} components, in a polydisperse matrix, composed of $n_{m}$ components, written in matrix form in Fourier space (see Eq. 18 of Ref. 5):

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 \tilde{\mathbf H}_{mm} = \tilde{\mathbf C}_{mm} + \rho_m \tilde{\mathbf C}_{mm} \tilde{\mathbf 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 \tilde{\mathbf H}_{fm} = \tilde{\mathbf C}_{fm} + \rho_m \tilde{\mathbf C}_{mm} \tilde{\mathbf H}_{fm} + \rho_f \tilde{\mathbf C}_{fm} \tilde{\mathbf H}_{ff} - \rho_f \tilde{\mathbf C}_{12} \tilde{\mathbf 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 \tilde{\mathbf H}_{ff} = \tilde{\mathbf C}_{ff} + \rho_m \tilde{\mathbf C}_{fm}^T \tilde{\mathbf H}_{fm} + \rho_f \tilde{\mathbf C}_{ff} \tilde{\mathbf H}{ff} - \rho_f \tilde{\mathbf C}_{12} \tilde{\mathbf 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 \tilde{\mathbf H}_{12} = \tilde{\mathbf C}_{12} + \rho_m \tilde{\mathbf C}_{fm}^T \tilde{\mathbf H}_{fm} + \rho_f \tilde{\mathbf C}_{ff} \tilde{\mathbf H}_{12} + \rho_f \tilde{\mathbf C}_{12} \tilde{\mathbf H}_{ff} -2 \rho_f \tilde{\mathbf C}_{12} \tilde{\mathbf H}_{12}}

Note: ${\mathbf {c} }_{fm}={\mathbf {c} }_{mf}^{T}$ 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 {\mathbf h}_{fm} = {\mathbf h}_{mf}^T} .

References

1. James A. Given and George Stell "Comment on: Fluid distributions in two-phase random media: Arbitrary matrices", Journal of Chemical Physics 97 pp. 4573 (1992)
2. James A. Given and George R. Stell "The replica Ornstein-Zernike equations and the structure of partly quenched media",Physica A 209 pp. 495-510 (1994)
3. W. G. Madden and E. D. Glandt "Distribution functions for fluids in random media", J. Stat. Phys. 51 pp. 537- (1988)
4. William G. Madden, "Fluid distributions in random media: Arbitrary matrices", Journal of Chemical Physics 96 pp. 5422 (1992)
5. S. Jorge; Elisabeth Schöll-Paschinger; Gerhard Kahl; María-José Fernaud "Structure and thermodynamic properties of a polydisperse fluid in contact with a polydisperse matrix", Molecular Physics 101 pp. 1733-1740 (2003)