RSOZ - Revision history

Carl McBride at 15:06, 12 February 2008

2008-02-12T15:06:06Z

← Older revision		Revision as of 17:06, 12 February 2008
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	In the limit of <math>s \rightarrow 0</math> these equations from the [[~~ROZ~~]] equations (see Eq.s 2.12 --2.16 Ref. 2):		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):

	:<math>h_{mm} = c_{mm} + \rho_m c_{mm} \otimes h_{mm}</math>		:<math>h_{mm} = c_{mm} + \rho_m c_{mm} \otimes h_{mm}</math>
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	==Polydisperse systems==		==Polydisperse systems==
	For a polydisperse fluid, composed of <math>n_f</math> components, in a polydisperse matrix,		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 space]] (see Eq. 18 of Ref. 5):		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>\tilde{\mathbf H}_{mm} = \tilde{\mathbf C}_{mm} + \rho_m \tilde{\mathbf C}_{mm} \tilde{\mathbf H}_{mm}

Nice and Tidy: /* Polydisperse systems */

2007-07-10T15:45:01Z

Polydisperse systems

← Older revision		Revision as of 17:45, 10 July 2007
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	:<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}_{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} +		:<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} +
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	Note: <math>{\mathbf c}_{fm} = {\mathbf c}_{mf}^T</math> and <math>{\mathbf h}_{fm} = {\mathbf h}_{mf}^T</math>.		Note: <math>{\mathbf c}_{fm} = {\mathbf c}_{mf}^T</math> and <math>{\mathbf h}_{fm} = {\mathbf h}_{mf}^T</math>.

	==References==		==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/10.1063/1.463883 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)]

Nice and Tidy at 15:35, 10 July 2007

2007-07-10T15:35:25Z

← Older revision		Revision as of 17:35, 10 July 2007
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	:<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>		:<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>


	:<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>		:<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>
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	describes correlations between fluid particles in the same cavity and the		describes correlations between fluid particles in the same cavity and the
	function <math>h_b(x)</math> 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 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==		==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/10.1063/1.463883 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)]
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	#[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/10.1007/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/10.1063/1.462726 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]]		[[Category: Integral equations]]

Nice and Tidy at 16:22, 28 February 2007

2007-02-28T16:22:19Z

← Older revision		Revision as of 18:22, 28 February 2007
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	In the limit of <math>s \rightarrow 0</math> these equations from the ROZ equations (see Eq.s 2.12 --2.16 Ref. 2):		In the limit of <math>s \rightarrow 0</math> these equations from the [[ROZ]] equations (see Eq.s 2.12 --2.16 Ref. 2):

	:<math>h_{mm} = c_{mm} + \rho_m c_{mm} \otimes h_{mm}</math>		:<math>h_{mm} = c_{mm} + \rho_m c_{mm} \otimes h_{mm}</math>

Nice and Tidy at 16:21, 28 February 2007

2007-02-28T16:21:41Z

← Older revision		Revision as of 18:21, 28 February 2007
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	Given and Stell (Refs 1 and 2) provided '''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 (Refs 3 and 4).		based on the original work of Madden and Glandt (Refs 3 and 4).
	For a two-species system, for the <math>(s+1)</math> replicated system one has (see Eq.s 2.7 --2.11 Ref. 2):		For a two-species system, for the <math>(s+1)</math> replicated system one has (see Eq.s 2.7 --2.11 Ref. 2):

Nice and Tidy at 14:59, 27 February 2007

2007-02-27T14:59:07Z

← Older revision		Revision as of 16:59, 27 February 2007
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	#[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/10.1007/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/10.1063/1.462726 William G. Madden, "Fluid distributions in random media: Arbitrary matrices", Journal of Chemical Physics '''96''' pp. 5422 (1992)]

			[[Category: Integral equations]]

Carl McBride at 13:49, 26 February 2007

2007-02-26T13:49:35Z

← Older revision		Revision as of 15:49, 26 February 2007
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	==References==		==References==
	#[http://dx.doi.org/ 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/10.1063/1.463883 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/ ~~JJames~~ A. Given and George R. Stell "The replica Ornstein-Zernike equations and the structure of partly quenched media",Physica A '''209''' pp. 495 (1994)]		#[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/ JW. G. Madden and E. D. Glandt "Distribution functions for fluids in random media", J. Stat. Phys. '''51''' pp. 537- (1988)]		#[http://dx.doi.org/10.1007/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/ ~~JWilliam~~ G. Madden, "Fluid distributions in random media: Arbitrary matrices", Journal of Chemical Physics '''96''' pp. 5422 (1992)]		#[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)]

Carl McBride at 13:44, 26 February 2007

2007-02-26T13:44:10Z

← Older revision		Revision as of 15:44, 26 February 2007
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	==References==		==References==
	#[~~JCP_1992_97_04573~~]		#[http://dx.doi.org/ 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)]
	#[~~PA_1994_209_0495~~]		#[http://dx.doi.org/ JJames A. Given and George R. Stell "The replica Ornstein-Zernike equations and the structure of partly quenched media",Physica A '''209''' pp. 495 (1994)]
	#[~~JSP_1988_51_0537_nolotengoSpringer~~]		#[http://dx.doi.org/ JW. G. Madden and E. D. Glandt "Distribution functions for fluids in random media", J. Stat. Phys. '''51''' pp. 537- (1988)]
	#[~~JCP_1992_96_05422~~]		#[http://dx.doi.org/ JWilliam G. Madden, "Fluid distributions in random media: Arbitrary matrices", Journal of Chemical Physics '''96''' pp. 5422 (1992)]

Carl McBride at 14:45, 21 February 2007

2007-02-21T14:45:23Z

Show changes

Carl McBride: 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...

2007-02-21T14:31:16Z

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...

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_0537_nolotengoSpringer,JCP_1992_96_05422}.\\
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}):\\
\begin{equation}
h_{mm} = c_{mm} + \rho_m c_{mm} \otimes h_{mm} + s\rho_f c_{mf} \otimes h_{mf}
\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}
\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}
\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}
\end{equation}
\begin{equation}
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 $s \rightarrow 0$ these equations from the ROZ equations (see Eq.s 2.12 --2.16 \cite{PA_1994_209_0495}):\\
\begin{equation}
h_{mm} = c_{mm} + \rho_m c_{mm} \otimes h_{mm}
\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}
\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}
\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}
\end{equation}
\begin{equation}
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
language of percolation theory.
\begin{equation}
h_{mm} = c_{mm} + \rho_m c_{mm} \otimes h_{mm}
\end{equation}
\begin{equation}
h_{fm} = c_{fm} + \rho_m c_{fm} \otimes h_{mm} + \rho_f c_c \otimes h_{fm}
\end{equation}
\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
\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)
\end{equation}
and the total correlation function is also split into
\begin{equation}
h_{ff}(12)= h_c (12) + h_b(12)
\end{equation}
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.