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| 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>: | | The '''Lebowitz and Percus''' mean spherical approximation (MSA) (1966) (Ref. 1) closure is given by |
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| | :<math>c(r) = -\beta \omega(r), ~~~~ r>\sigma.</math> |
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| :<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 |
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| 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>
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| <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>:
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| :<math>{\rm g}_{ij}(r) \equiv h_{ij}(r) +1=0 ~ ~ ~ ~ ~ ~ ~ ~ r < \sigma_{ij} = (\sigma_i + \sigma_j)/2</math>
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| | :<math>{\rm g}_{ij}(r) \equiv h_{ij}(r) +1=0 ~~~~~~~~ r < \sigma_{ij} = (\sigma_i + \sigma_j)/2</math> |
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| and | | and |
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| :<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> |
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| 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
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| molecules of <math>i</math> and <math>j</math> species, <math>\sigma_i</math> is the diameter of <math>i</math> species of molecule.
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| 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
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| | 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 \cite{JCP_1995_103_02625}) write the MSA approximation as |
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| :<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> |
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| | where $\Phi_1$ and $\Phi_2$ comes from the WCA division of the LJ potential.\\ |
| | By introducing the definition (Eq. 10 \cite{JCP_1995_103_02625}) |
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| where <math>\Phi_1</math> and <math>\Phi_2</math> comes from the
| | :<math>s(r) = h(r) -c(r) -\beta \Phi_2 (r)</math> |
| [[Weeks-Chandler-Andersen perturbation theory | Weeks-Chandler-Andersen division]]
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| of the [[Lennard-Jones model | Lennard-Jones]] potential.
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| By introducing the definition (Eq. 10 in <ref name="Duh and Haymet"> </ref>)
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| :<math>\left.s(r)\right. = h(r) -c(r) -\beta \Phi_2 (r)</math> | |
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| one can arrive at (Eq. 11 in <ref name="Duh and Haymet"> </ref>)
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| | one can arrive at (Eq. 11 \cite{JCP_1995_103_02625}) |
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| :<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> |
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| 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>. |
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| ==Thermodynamic consistency==
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| <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>
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| ==References== | | ==References== |
| <references/>
| | #[PR_1966_144_000251] |
| | | #[JSP_1978_19_0317_nolotengoSpringer] |
| | | #[JSP_1980_22_0661_nolotengoSpringer] |
| [[Category:Integral equations]] | |