RSOZ

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(Redirected from RSOZ for polydisperse systems)

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 $h m m$ because the structure of the medium is unaffected by the presence of fluid particles.

Note: $C f f$ (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.

[edit] 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):

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

$\tilde{\mathbf H}_{fm} = \tilde{\mathbf C}_{fm} + \rho_m \tilde{\mathbf C}_{mm} \tilde{\mathbf H}_{fm} + \rho_f \tilde{\mathbf C}_{fm} \tilde{\mathbf H}_{ff} - \rho_f \tilde{\mathbf C}_{12} \tilde{\mathbf H}_{fm}$

$\tilde{\mathbf H}_{ff} = \tilde{\mathbf C}_{ff} + \rho_m \tilde{\mathbf C}_{fm}^T \tilde{\mathbf H}_{fm} + \rho_f \tilde{\mathbf C}_{ff} \tilde{\mathbf H}{ff} - \rho_f \tilde{\mathbf C}_{12} \tilde{\mathbf H}_{12}$

$\tilde{\mathbf H}_{12} = \tilde{\mathbf C}_{12} + \rho_m \tilde{\mathbf C}_{fm}^T \tilde{\mathbf H}_{fm} + \rho_f \tilde{\mathbf C}_{ff} \tilde{\mathbf H}_{12} + \rho_f \tilde{\mathbf C}_{12} \tilde{\mathbf H}_{ff} -2 \rho_f \tilde{\mathbf C}_{12} \tilde{\mathbf H}_{12}$