Editing Mean spherical approximation
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The '''mean spherical approximation | The '''Lebowitz and Percus''' mean spherical approximation (MSA) (1966) (Ref. 1) [[Closure relations | closure relation]] is given by | ||
:<math>c(r) = -\beta \omega(r), ~ ~ ~ ~ r>\sigma.</math> | :<math>c(r) = -\beta \omega(r), ~~~~ r>\sigma.</math> | ||
In the '''Blum and Høye''' mean spherical approximation for | In the '''Blum and Høye''' mean spherical approximation for mixtures (Refs 2 and 3) the closure is given by | ||
:<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} ~ ~ ~ ~ | :<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 | 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 | molecules of ''i'' and ''j'' species, <math>\sigma_i</math> is the diameter of '''i'' species of molecule. | ||
Duh and Haymet (Eq. 9 | Duh and Haymet (Eq. 9 Ref. 4) write the MSA approximation as | ||
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where <math>\Phi_1</math> and <math>\Phi_2</math> comes from the | where <math>\Phi_1</math> and <math>\Phi_2</math> comes from the | ||
[[Weeks-Chandler- | [[Weeks-Chandler-Anderson perturbation theory | Weeks-Chandler-Anderson division]] | ||
of the [[Lennard-Jones model | Lennard-Jones]] potential. | of the [[Lennard-Jones model | Lennard-Jones]] potential. | ||
By introducing the definition (Eq. 10 | By introducing the definition (Eq. 10 Ref. 4) | ||
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one can arrive at (Eq. 11 in | one can arrive at (Eq. 11 in Ref. 4) | ||
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==Thermodynamic consistency== | ==Thermodynamic consistency== | ||
See Ref. 5. | |||
==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)] | |||
#[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)] | |||
#[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)] | |||
[[Category:Integral equations]] | [[Category:Integral equations]] | ||