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m (New page: Given and Stell \cite{JCP_1992_97_04573,PA_1994_209_0495} provided {\bf exact} OZ equations for two-phase random media based on the original work of Madden and Glandt \cite{JSP_1988_51_053...)
 
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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''' OZ equations 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}
 
h_{12} = c_{12} + \rho_m c_{fm} \otimes h_{mf} + \rho_f  c_{ff}  \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} </math>
\rho_f  c_{12}  \otimes h_{ff} + (s-2) \rho_f c_{12}   \otimes h_{12}
 
\end{equation}
 
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}):\\
In the limit of <math>s \rightarrow 0</math> these equations from the ROZ equations (see Eq.s 2.12 --2.16 Ref. 2):
\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}
 
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}</math>
\end{equation}
 
\begin{equation}
 
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}</math>
\end{equation}
 
\begin{equation}
 
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}</math>
\end{equation}
 
\begin{equation}
 
h_{12} = c_{12} + \rho_m c_{fm} \otimes h_{mf} + \rho_f  c_{ff}  \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}</math>
\rho_f  c_{12}  \otimes h_{ff} -2 \rho_f c_{12}  \otimes h_{12}
 
\end{equation}
When written in the `percolation terminology'
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.
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
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
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
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,
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
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  
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.
 
==References==
#[JCP_1992_97_04573]
#[PA_1994_209_0495]
#[JSP_1988_51_0537_nolotengoSpringer]
#[JCP_1992_96_05422]

Revision as of 15:45, 21 February 2007

Given and Stell (Refs 1 and 2) provided exact OZ equations 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (s+1)} replicated system one has (see Eq.s 2.7 --2.11 Ref. 2):

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h_{mm} = c_{mm} + \rho_m c_{mm} \otimes h_{mm} + s\rho_f c_{mf} \otimes h_{mf}}


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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}}


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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}}


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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}}


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s \rightarrow 0} these equations from the ROZ equations (see Eq.s 2.12 --2.16 Ref. 2):

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h_{mm} = c_{mm} + \rho_m c_{mm} \otimes h_{mm}}


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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}}


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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}}


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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}}


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} terms connected and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} blocking are adapted from the language of percolation theory.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h_{mm} = c_{mm} + \rho_m c_{mm} \otimes h_{mm}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h_{fm} = c_{fm} + \rho_m c_{fm} \otimes h_{mm} + \rho_f c_c \otimes h_{fm}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h_c = c_c + \rho_f c_c \otimes h_c}

where the direct correlation function is split into

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left.c_{ff}(12)\right. = c_c (12) + c_b (12)}

and the total correlation function is also split into

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left.h_{ff}(12)\right.= h_c (12) + h_b(12)}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} denotes the matrix and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} denotes the fluid. The blocking function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h_{mm}} because the structure of the medium is unaffected by the presence of fluid particles.

  • Note: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_{ff}} (Madden and Glandt) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =h_c} (Given and Stell)
  • Note: fluid: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} (Madden and Glandt), `1' (Given and Stell)
  • Note: matrix: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h_c (x)} describes correlations between fluid particles in the same cavity and the function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h_b(x)} describes correlations between particles in different cavities.

References

  1. [JCP_1992_97_04573]
  2. [PA_1994_209_0495]
  3. [JSP_1988_51_0537_nolotengoSpringer]
  4. [JCP_1992_96_05422]
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