# Mean spherical approximation

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]

- ↑ 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) - ↑ 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) - ↑ 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) - ↑
^{4.0}^{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) Cite error: Invalid`<ref>`

tag; name "Duh_and_Haymet" defined multiple times with different content Cite error: Invalid`<ref>`

tag; name "Duh_and_Haymet" defined multiple times with different content - ↑ 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)