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.
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 .
- 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)