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 $s\to 0$ these equations from the ROZ equations (see Eq.s 2.12 --2.16 Ref. 2):

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

$h_{mf}=c_{mf}+{\rho }_m{c}_{mm}\otimes h_{mf}+{\rho }_f{c}_{mf}\otimes h_{ff}-{\rho }_f{c}_{mf}\otimes h_{12}$

$h_{fm}=c_{fm}+{\rho }_m{c}_{fm}\otimes h_{mm}+{\rho }_f{c}_{ff}\otimes h_{fm}-{\rho }_f{c}_{12}\otimes h_{fm}$

$h_{ff}=c_{ff}+{\rho }_m{c}_{fm}\otimes h_{mf}+{\rho }_f{c}_{ff}\otimes h_{ff}-{\rho }_f{c}_{12}\otimes h_{12}$

$h_{12}=c_{12}+{\rho }_m{c}_{fm}\otimes h_{mf}+{\rho }_f{c}_{ff}\otimes h_{12}+{\rho }_f{c}_{12}\otimes h_{ff}-2{\rho }_f{c}_{12}\otimes h_{12}$

When written in the `percolation terminology' where $c$ terms connected and $b$ blocking are adapted from the language of percolation theory.

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

$h_{fm}=c_{fm}+{\rho }_m{c}_{fm}\otimes h_{mm}+{\rho }_f{c}_c\otimes h_{fm}$

$h_{ff}=c_{ff}+{\rho }_m{c}_{fm}\otimes h_{mf}+{\rho }_f{c}_c\otimes h_{ff}+{\rho }_f{c}_b\otimes h_c$

$h_c=c_c+{\rho }_f{c}_c\otimes h_c$

where the direct correlation function is split into

$c_{ff}(12)=c_c(12)+c_b(12)$

and the total correlation function is also split into

$h_{ff}(12)=h_c(12)+h_b(12)$

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.

Polydisperse systems

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

${\tilde{\mathbf{H}}}_{mm}={\tilde{\mathbf{C}}}_{mm}+{\rho }_m{\tilde{\mathbf{C}}}_{mm}{\tilde{\mathbf{H}}}_{mm}$

${\tilde{\mathbf{H}}}_{fm}={\tilde{\mathbf{C}}}_{fm}+{\rho }_m{\tilde{\mathbf{C}}}_{mm}{\tilde{\mathbf{H}}}_{fm}+{\rho }_f{\tilde{\mathbf{C}}}_{fm}{\tilde{\mathbf{H}}}_{ff}-{\rho }_f{\tilde{\mathbf{C}}}_{12}{\tilde{\mathbf{H}}}_{fm}$

${\tilde{\mathbf{H}}}_{ff}={\tilde{\mathbf{C}}}_{ff}+{\rho }_m{\tilde{\mathbf{C}}}_{fm}^T{\tilde{\mathbf{H}}}_{fm}+{\rho }_f{\tilde{\mathbf{C}}}_{ff}\tilde{\mathbf{H}}ff-{\rho }_f{\tilde{\mathbf{C}}}_{12}{\tilde{\mathbf{H}}}_{12}$

${\tilde{\mathbf{H}}}_{12}={\tilde{\mathbf{C}}}_{12}+{\rho }_m{\tilde{\mathbf{C}}}_{fm}^T{\tilde{\mathbf{H}}}_{fm}+{\rho }_f{\tilde{\mathbf{C}}}_{ff}{\tilde{\mathbf{H}}}_{12}+{\rho }_f{\tilde{\mathbf{C}}}_{12}{\tilde{\mathbf{H}}}_{ff}-2{\rho }_f{\tilde{\mathbf{C}}}_{12}{\tilde{\mathbf{H}}}_{12}$

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

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)