Mean spherical approximation: Difference between revisions

From SklogWiki
Jump to navigation Jump to search
No edit summary
m (Updated internal link + tidy + Cite references format)
 
(9 intermediate revisions by 3 users not shown)
Line 1: Line 1:
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>
:<math>c(r) = -\beta \omega(r), ~ ~ ~ ~ r>\sigma.</math>




The '''Blum and Høye''' mean spherical approximation (MSA) (1978-1980) (Refs 2 and 3) closure is given by
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 and the direct correlation functions for two spherical
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 ''i'' and ''j'' species, <math>\sigma_i</math> is the diameter of '''i'' species of molecule.
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 Ref. 4) write the MSA approximation as
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




Line 23: Line 24:




where <math>\Phi_1</math> and <math>\Phi_2</math> comes from the [[WCA division]] of the [[Lennard-Jones model | Lennard-Jones]] potential.
where <math>\Phi_1</math> and <math>\Phi_2</math> comes from the  
By introducing the definition  (Eq. 10 Ref. 4)  
[[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>)  




Line 30: Line 33:




one can arrive at  (Eq. 11 in Ref. 4)
one can arrive at  (Eq. 11 in <ref name="Duh and Haymet"> </ref>)




Line 38: Line 41:
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==
#[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)]
<references/>
#[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)]
 
#[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)]
#[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)]


[[Category:Integral equations]]
[[Category:Integral equations]]

Latest revision as of 13:07, 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 species of molecule. Duh and Haymet (Eq. 9 in [4]) write the MSA approximation as



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



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



The Percus Yevick approximation may be recovered from the above equation by setting .

Thermodynamic consistency[edit]

[5]

References[edit]