Mean spherical approximation: Difference between revisions

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The '''Lebowitz and Percus''' mean spherical approximation (MSA) (1966) (Ref. 1) closure is given by
The '''mean spherical approximation''' (MSA) [[Closure relations | closure relation]] of Lebowitz and Percus is given by <ref>[http://dx.doi.org/10.1103/PhysRev.144.251  J. L. Lebowitz and J. K. Percus "Mean Spherical Model for Lattice Gases with Extended Hard Cores and Continuum Fluids", Physical Review '''144''' pp. 251-258 (1966)]</ref>:


:<math>c(r) = -\beta \omega(r), ~~~~ r>\sigma.</math>


The '''Blum and Hoye''' mean spherical approximation (MSA) (1978-1980) (Refs 2 and 3) closure is given by
:<math>c(r) = -\beta \omega(r), ~ ~ ~ ~ r>\sigma.</math>
 
 
In the '''Blum and Høye''' mean spherical approximation for [[mixtures]] the closure is given by <ref>[http://dx.doi.org/10.1007/BF01011750 L. Blum and J. S. Høye "Solution of the Ornstein-Zernike equation with Yukawa closure for a mixture", Journal  of Statistical Physics, '''19''' pp. 317-324 (1978)]</ref>
<ref>[http://dx.doi.org/10.1007/BF01013935  Lesser Blum "Solution of the Ornstein-Zernike equation for a mixture of hard ions and Yukawa closure" Journal  of Statistical Physics, '''22''' pp. 661-672 (1980)]</ref>:
 
 
:<math>{\rm g}_{ij}(r) \equiv h_{ij}(r) +1=0 ~ ~ ~ ~ ~ ~ ~ ~ r < \sigma_{ij} = (\sigma_i + \sigma_j)/2</math>


:<math>{\rm g}_{ij}(r) \equiv h_{ij}(r) +1=0 ~~~~~~~~ r < \sigma_{ij} = (\sigma_i + \sigma_j)/2</math>


and
and


:<math>c_{ij}(r)= \sum_{n=1} \frac{K_{ij}^{(n)}}{r}e^{-z_nr} ~~~~~~ \sigma_{ij} < r</math>
:<math>c_{ij}(r)= \sum_{n=1} \frac{K_{ij}^{(n)}}{r}e^{-z_nr} ~ ~ ~ ~ ~ ~ \sigma_{ij} < r</math>
 
where <math>h_{ij}(r)</math> and <math>c_{ij}(r)</math> are the [[Total correlation function |total]] and the [[direct correlation function]]s for two spherical
molecules of <math>i</math> and <math>j</math> species, <math>\sigma_i</math> is the diameter of <math>i</math> species of molecule.
Duh and Haymet (Eq. 9 in <ref name="Duh and Haymet">[http://dx.doi.org/10.1063/1.470724      Der-Ming Duh and A. D. J. Haymet "Integral equation theory for uncharged liquids: The Lennard-Jones fluid and the bridge function", Journal of Chemical Physics '''103''' pp. 2625-2633 (1995)]</ref>) write the MSA approximation as


where <math>h_{ij}(r)</math> and <math>c_{ij}(r)</math> are the total and the direct correlation functions for two spherical
molecules of ''i'' and ''j'' species, <math>\sigma_i</math> is the diameter of '''i'' species of molecule.
Duh and Haymet (Eq. 9 Ref. 4) write the MSA approximation as


:<math>g(r) = \frac{c(r) + \beta \Phi_2(r)}{1-e^{\beta \Phi_1(r)}}</math>
:<math>g(r) = \frac{c(r) + \beta \Phi_2(r)}{1-e^{\beta \Phi_1(r)}}</math>


where <math>\Phi_1</math> and <math>\Phi_2</math> comes from the [[WCA division]] of the [[Lennard-Jones]] potential.
By introducing the definition  (Eq. 10 \cite{JCP_1995_103_02625})


:<math>s(r) = h(r) -c(r) -\beta \Phi_2 (r)</math>
where <math>\Phi_1</math> and <math>\Phi_2</math> comes from the
[[Weeks-Chandler-Andersen perturbation theory | Weeks-Chandler-Andersen division]]
of the [[Lennard-Jones model | Lennard-Jones]] potential.
By introducing the definition  (Eq. 10 in <ref name="Duh and Haymet"> </ref>)
 
 
:<math>\left.s(r)\right. = h(r) -c(r) -\beta \Phi_2 (r)</math>
 
 
one can arrive at  (Eq. 11 in <ref name="Duh and Haymet"> </ref>)


one can arrive at  (Eq. 11 \cite{JCP_1995_103_02625})


:<math>B(r) \approx B^{\rm MSA}(s) = \ln (1+s)-s</math>
:<math>B(r) \approx B^{\rm MSA}(s) = \ln (1+s)-s</math>


The [[Percus Yevick]] approximation may be recovered from the above equation by setting <math>\Phi_2=0</math>.
The [[Percus Yevick]] approximation may be recovered from the above equation by setting <math>\Phi_2=0</math>.


==Thermodynamic consistency==
<ref>[http://dx.doi.org/10.1063/1.2712181 Andrés Santos "Thermodynamic consistency between the energy and virial routes in the mean spherical approximation for soft potentials" Journal of Chemical Physics '''126''' 116101 (2007)]</ref>
==References==
==References==
#[PR_1966_144_000251]
<references/>
#[JSP_1978_19_0317_nolotengoSpringer]
 
#[JSP_1980_22_0661_nolotengoSpringer]
 
#[JCP_1995_103_02625]
[[Category:Integral equations]]

Latest revision as of 14:07, 16 February 2012

The mean spherical approximation (MSA) closure relation of Lebowitz and Percus is given by [1]:



In the Blum and Høye mean spherical approximation for mixtures the closure is given by [2] [3]:



and

where and are the total and the direct correlation functions for two spherical molecules of and species, is the diameter 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 i} species of molecule. Duh and Haymet (Eq. 9 in [4]) write the MSA approximation as


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 g(r) = \frac{c(r) + \beta \Phi_2(r)}{1-e^{\beta \Phi_1(r)}}}


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 \Phi_1} 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 \Phi_2} comes from the Weeks-Chandler-Andersen division of the Lennard-Jones potential. By introducing the definition (Eq. 10 in [4])


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.s(r)\right. = h(r) -c(r) -\beta \Phi_2 (r)}


one can arrive at (Eq. 11 in [4])



The Percus Yevick approximation may be recovered from the above equation by setting 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 \Phi_2=0} .

Thermodynamic consistency[edit]

[5]

References[edit]

  1. ↑ J. L. Lebowitz and J. K. Percus "Mean Spherical Model for Lattice Gases with Extended Hard Cores and Continuum Fluids", Physical Review 144 pp. 251-258 (1966)
  2. ↑ L. Blum and J. S. Høye "Solution of the Ornstein-Zernike equation with Yukawa closure for a mixture", Journal of Statistical Physics, 19 pp. 317-324 (1978)
  3. ↑ Lesser Blum "Solution of the Ornstein-Zernike equation for a mixture of hard ions and Yukawa closure" Journal of Statistical Physics, 22 pp. 661-672 (1980)
  4. ↑ 4.0 4.1 4.2 Der-Ming Duh and A. D. J. Haymet "Integral equation theory for uncharged liquids: The Lennard-Jones fluid and the bridge function", Journal of Chemical Physics 103 pp. 2625-2633 (1995)
  5. ↑ Andrés Santos "Thermodynamic consistency between the energy and virial routes in the mean spherical approximation for soft potentials" Journal of Chemical Physics 126 116101 (2007)
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