RSOZ: Difference between revisions

Latest revision as of 17:06, 12 February 2008

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_{m m} = c_{m m} + ρ_{m} c_{m m} \otimes h_{m m} + s ρ_{f} c_{m f} \otimes h_{m f}

h_{m f} = c_{m f} + ρ_{m} c_{m m} \otimes h_{m f} + ρ_{f} c_{m f} \otimes h_{f f} + (s - 1) ρ_{f} c_{m f} \otimes h_{12}

h_{f m} = c_{f m} + ρ_{m} c_{f m} \otimes h_{m m} + ρ_{f} c_{f f} \otimes h_{f m} + (s - 1) ρ_{f} c_{12} \otimes h_{f m}

h_{f f} = c_{f f} + ρ_{m} c_{f m} \otimes h_{m f} + ρ_{f} c_{f f} \otimes h_{f f} + (s - 1) ρ_{f} c_{12} \otimes h_{12}

h_{12} = c_{12} + ρ_{m} c_{f m} \otimes h_{m f} + ρ_{f} c_{f f} \otimes h_{12} + ρ_{f} c_{12} \otimes h_{f f} + (s - 2) ρ_{f} c_{12} \otimes h_{12}

In the limit of $s \to 0$ these equations from the replica Ornstein-Zernike (ROZ)equations (see Eq.s 2.12 --2.16 Ref. 2):

h_{m m} = c_{m m} + ρ_{m} c_{m m} \otimes h_{m m}

h_{m f} = c_{m f} + ρ_{m} c_{m m} \otimes h_{m f} + ρ_{f} c_{m f} \otimes h_{f f} - ρ_{f} c_{m f} \otimes h_{12}

h_{f m} = c_{f m} + ρ_{m} c_{f m} \otimes h_{m m} + ρ_{f} c_{f f} \otimes h_{f m} - ρ_{f} c_{12} \otimes h_{f m}

h_{f f} = c_{f f} + ρ_{m} c_{f m} \otimes h_{m f} + ρ_{f} c_{f f} \otimes h_{f f} - ρ_{f} c_{12} \otimes h_{12}

h_{12} = c_{12} + ρ_{m} c_{f m} \otimes h_{m f} + ρ_{f} c_{f f} \otimes h_{12} + ρ_{f} c_{12} \otimes h_{f f} - 2 ρ_{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_{m m} = c_{m m} + ρ_{m} c_{m m} \otimes h_{m m}

h_{f m} = c_{f m} + ρ_{m} c_{f m} \otimes h_{m m} + ρ_{f} c_{c} \otimes h_{f m}

h_{f f} = c_{f f} + ρ_{m} c_{f m} \otimes h_{m f} + ρ_{f} c_{c} \otimes h_{f f} + ρ_{f} c_{b} \otimes h_{c}

h_{c} = c_{c} + ρ_{f} c_{c} \otimes h_{c}

where the direct correlation function is split into

c_{f f} (12) = c_{c} (12) + c_{b} (12)

and the total correlation function is also split into

h_{f f} (12) = 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.

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{H}}_{m m} = {\tilde{C}}_{m m} + ρ_{m} {\tilde{C}}_{m m} {\tilde{H}}_{m m}

{\tilde{H}}_{f m} = {\tilde{C}}_{f m} + ρ_{m} {\tilde{C}}_{m m} {\tilde{H}}_{f m} + ρ_{f} {\tilde{C}}_{f m} {\tilde{H}}_{f f} - ρ_{f} {\tilde{C}}_{12} {\tilde{H}}_{f m}

{\tilde{H}}_{f f} = {\tilde{C}}_{f f} + ρ_{m} {\tilde{C}}_{f m}^{T} {\tilde{H}}_{f m} + ρ_{f} {\tilde{C}}_{f f} \tilde{H} f f - ρ_{f} {\tilde{C}}_{12} {\tilde{H}}_{12}

{\tilde{H}}_{12} = {\tilde{C}}_{12} + ρ_{m} {\tilde{C}}_{f m}^{T} {\tilde{H}}_{f m} + ρ_{f} {\tilde{C}}_{f f} {\tilde{H}}_{12} + ρ_{f} {\tilde{C}}_{12} {\tilde{H}}_{f f} - 2 ρ_{f} {\tilde{C}}_{12} {\tilde{H}}_{12}

Note: $c_{f m} = c_{m f}^{T}$ and $h_{f m} = h_{m f}^{T}$ .

References[edit]

@@ Line 1: / Line 1: @@
-Given and Stell \cite{JCP_1992_97_04573,PA_1994_209_0495} provided {\bf exact} OZ equations for two-phase random media
+Given and Stell (Refs 1 and 2) provided '''exact''' [[Ornstein-Zernike relation]]s  for two-phase random media
-based on the original work of Madden and Glandt \cite{JSP_1988_51_0537_nolotengoSpringer,JCP_1992_96_05422}.\\
+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 \cite{PA_1994_209_0495}):\\
+For a two-species system, for the <math>(s+1)</math> replicated system one has (see Eq.s 2.7 --2.11 Ref. 2):
-\begin{equation}
-h_{mm} = c_{mm} + \rho_m c_{mm} \otimes h_{mm} + s\rho_f c_{mf}   \otimes h_{mf}
+:<math>h_{mm} = c_{mm} + \rho_m c_{mm} \otimes h_{mm} + s\rho_f c_{mf}   \otimes h_{mf}</math>
-\end{equation}
-\begin{equation}
-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}
+:<math>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}</math>
-\end{equation}
-\begin{equation}
-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}
+:<math>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}</math>
-\end{equation}
-\begin{equation}
-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}
+:<math>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}</math>
-\end{equation}
-\begin{equation}
+:<math>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} </math>
-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}
-\end{equation}
+In the limit of <math>s \rightarrow 0</math> these equations from the [[Replica Ornstein-Zernike relation |replica Ornstein-Zernike]] (ROZ)equations (see Eq.s 2.12 --2.16 Ref. 2):
-In the limit of $s \rightarrow 0$ these equations from the ROZ equations (see Eq.s 2.12 --2.16 \cite{PA_1994_209_0495}):\\
-\begin{equation}
+:<math>h_{mm} = c_{mm} + \rho_m c_{mm} \otimes h_{mm}</math>
-h_{mm} = c_{mm} + \rho_m c_{mm} \otimes h_{mm}
-\end{equation}
-\begin{equation}
+:<math>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}</math>
-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}
-\end{equation}
-\begin{equation}
+:<math>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}</math>
-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}
-\end{equation}
-\begin{equation}
+:<math>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}</math>
-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}
-\end{equation}
-\begin{equation}
+:<math>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}</math>
-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}
-\end{equation}
 When written in the `percolation terminology'
-where $c$ terms `{\it connected}' and $b$  `{\it blocking}' are adapted from the
+where <math>c</math> terms ''connected'' and <math>b</math>  ''blocking'' are adapted from the
 language of percolation theory.
-\begin{equation}
-h_{mm} = c_{mm} + \rho_m c_{mm} \otimes h_{mm}
+:<math>h_{mm} = c_{mm} + \rho_m c_{mm} \otimes h_{mm}</math>
-\end{equation}
-\begin{equation}
+:<math>h_{fm} = c_{fm} + \rho_m c_{fm} \otimes h_{mm} + \rho_f c_c \otimes h_{fm}</math>
-h_{fm} = c_{fm} + \rho_m c_{fm} \otimes h_{mm} + \rho_f c_c \otimes h_{fm}
-\end{equation}
+:<math>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</math>
-\begin{equation}
-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
+:<math>h_c = c_c + \rho_f c_c \otimes h_c</math>
-\end{equation}
-\begin{equation}
-h_c = c_c + \rho_f c_c \otimes h_c
-\end{equation}
 where the direct correlation function is split into
-\begin{equation}
-c_{ff}(12) = c_c (12) + c_b (12)
+:<math>\left.c_{ff}(12)\right. = c_c (12) + c_b (12)</math>
-\end{equation}
 and the total correlation function is also split into
-\begin{equation}
+:<math>\left.h_{ff}(12)\right.= h_c (12) + h_b(12)</math>
-h_{ff}(12)= h_c (12) + h_b(12)
-\end{equation}
+where <math>m</math> denotes the matrix
-where $m$ denotes the matrix
+and <math>f</math> denotes the fluid.
-and $f$ denotes the fluid.
+The blocking function <math>h_b(x)</math> accounts for correlations between a pair of
-The blocking function $h_b(x)$ accounts for correlations between a pair of
+fluid particles ``blocked" or separated from each other by matrix particles.
-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
+integral for <math>h_{mm}</math> because the structure of the medium is
-unaffected by the presence of fluid particles.\\
+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: <math>C_{ff}</math> (Madden and Glandt) <math>=h_c</math> (Given and Stell)
-Note: matrix: $m$  (Madden and Glandt), `0'  (Given and Stell)\\
+*Note: fluid: <math>f</math>  (Madden and Glandt), `1'  (Given and Stell)
+*Note: matrix: <math>m</math>  (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 $h_c (x)$
+totally surrounded by a matrix. In this limit, the function <math>h_c (x)</math>
 describes correlations between fluid particles in the same cavity and the
-function $h_b(x)$ describes correlations between particles in different cavities.
+function <math>h_b(x)</math> describes correlations between particles in different cavities.
+==Polydisperse systems==
+For a polydisperse fluid, composed of <math>n_f</math> components, in a polydisperse matrix,
+composed of <math>n_m</math> components, written in matrix form in [[Fourier analysis |Fourier space]] (see Eq. 18 of Ref. 5):
+:<math>\tilde{\mathbf H}_{mm} = \tilde{\mathbf C}_{mm} + \rho_m \tilde{\mathbf C}_{mm} \tilde{\mathbf H}_{mm}
+</math>
+:<math>\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}
+</math>
+:<math>\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}</math>
+:<math>\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}</math>
+Note: <math>{\mathbf c}_{fm} = {\mathbf c}_{mf}^T</math> and <math>{\mathbf h}_{fm} = {\mathbf h}_{mf}^T</math>.
+==References==
+#[http://dx.doi.org/10.1063/1.463883  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)]
+#[http://dx.doi.org/10.1016/0378-4371(94)90200-3  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)]
+#[http://dx.doi.org/10.1007/BF01028471 W. G. Madden and E. D. Glandt "Distribution functions for fluids in random media", J. Stat. Phys. '''51''' pp. 537- (1988)]
+#[http://dx.doi.org/10.1063/1.462726  William G. Madden, "Fluid distributions in random media: Arbitrary matrices",  Journal of Chemical Physics '''96''' pp. 5422 (1992)]
+#[http://dx.doi.org/10.1080/0026897031000085128 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)]
+[[Category: Integral equations]]

RSOZ: Difference between revisions

Latest revision as of 17:06, 12 February 2008

Polydisperse systems[edit]

References[edit]

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