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
replicated system one has (see Eq.s 2.7 --2.11 Ref. 2):





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





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




where the direct correlation function is split into

and the total correlation function is also split into

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.
Polydisperse systems[edit]
For a polydisperse fluid, composed of
components, in a polydisperse matrix,
composed of
components, written in matrix form in Fourier space (see Eq. 18 of Ref. 5):




Note:
and
.
References[edit]
- 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)
- 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)