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
replicated system one has (see Eq.s 2.7 --2.11 Ref. 2):
![{\displaystyle h_{mm}=c_{mm}+\rho _{m}c_{mm}\otimes h_{mm}+s\rho _{f}c_{mf}\otimes h_{mf}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac25ed9b2b26a6ec7262524a2f8069db2badd1ab)
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e70a5f2486d504fbf35844ab4d58ca72bef1e5ef)
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a0e10534c56c3b1e6093f1d16c1a14eba440a568)
![{\displaystyle 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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/44163a6925709debd6e3fc448a519eb3d16d71c2)
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c4a3c943eba0486153233ed543b8251949101c)
In the limit of
these equations from the ROZ equations (see Eq.s 2.12 --2.16 Ref. 2):
![{\displaystyle h_{mm}=c_{mm}+\rho _{m}c_{mm}\otimes h_{mm}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9dc2d8bd0710d5ffd4c2b3044c65c7dfa7edc003)
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/20cf15b872c20a52716e17a4f3957706d9caaf42)
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da74d1b968e5ba9369f20856e4f999911b1c3e67)
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/79127b7f1e4f2e55038b8df6d591afb2a3c0e173)
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0eef70aa464cc8e53059f94fd7aba789d31a7f28)
When written in the `percolation terminology'
where
terms connected and
blocking are adapted from the
language of percolation theory.
![{\displaystyle h_{mm}=c_{mm}+\rho _{m}c_{mm}\otimes h_{mm}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9dc2d8bd0710d5ffd4c2b3044c65c7dfa7edc003)
![{\displaystyle h_{fm}=c_{fm}+\rho _{m}c_{fm}\otimes h_{mm}+\rho _{f}c_{c}\otimes h_{fm}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6b5edb407f10d94918c5d251597215a281f3203)
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d394fade3da9039e80e0c1d88f86a93bb9326deb)
![{\displaystyle h_{c}=c_{c}+\rho _{f}c_{c}\otimes h_{c}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/733c08be4825826ea393b07ff44b2bcc8d9a00a3)
where the direct correlation function is split into
![{\displaystyle \left.c_{ff}(12)\right.=c_{c}(12)+c_{b}(12)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a1640e47934fb097b7f126b3129ef5a93edd4bef)
and the total correlation function is also split into
![{\displaystyle \left.h_{ff}(12)\right.=h_{c}(12)+h_{b}(12)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3cba36e5f2b504e6ed6e65f026bdccad78afa2a6)
where
denotes the matrix
and
denotes the fluid.
The blocking function
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
because the structure of the medium is
unaffected by the presence of fluid particles.
- Note:
(Madden and Glandt)
(Given and Stell)
- Note: fluid:
(Madden and Glandt), `1' (Given and Stell)
- Note: matrix:
(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
describes correlations between fluid particles in the same cavity and the
function
describes correlations between particles in different cavities.
References
- 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)
- 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)
- W. G. Madden and E. D. Glandt "Distribution functions for fluids in random media", J. Stat. Phys. 51 pp. 537- (1988)
- William G. Madden, "Fluid distributions in random media: Arbitrary matrices", Journal of Chemical Physics 96 pp. 5422 (1992)