RSOZ: Difference between revisions
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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> | ||
In the limit of <math>s \rightarrow 0</math> these equations from the [[ | 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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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 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== | ==References== | ||
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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]] | ||
Latest revision as of 16:06, 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 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 replica Ornstein-Zernike (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 (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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.
Polydisperse systems[edit]
For a polydisperse fluid, composed 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 n_f} components, in a polydisperse matrix, composed 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 n_m} components, written in matrix form in Fourier space (see Eq. 18 of Ref. 5):
- 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 \tilde{\mathbf H}_{mm} = \tilde{\mathbf C}_{mm} + \rho_m \tilde{\mathbf C}_{mm} \tilde{\mathbf 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 \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} }
- Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\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}}
- 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 \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: 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 {\mathbf c}_{fm} = {\mathbf c}_{mf}^T} 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 {\mathbf h}_{fm} = {\mathbf h}_{mf}^T} .
References[edit]
- 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)
- 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)
- W. G. Madden and E. D. Glandt "Distribution functions for fluids in random media", J. Stat. Phys. 51 pp. 537- (1988)
- William G. Madden, "Fluid distributions in random media: Arbitrary matrices", Journal of Chemical Physics 96 pp. 5422 (1992)
- 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)