RSOZ

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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}


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

\left.c_{ff}(12)\right. = c_c (12) + c_b (12)

and the total correlation function is also split into

\left.h_{ff}(12)\right.= 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[edit]

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[edit]

  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)