# Difference between revisions of "Mean spherical approximation"

Carl McBride (talk | contribs) |
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− | :<math>{\rm g}_{ij}(r) \equiv h_{ij}(r) | + | :<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>g(r) = \frac{c(r) | + | :<math>g(r) = \frac{c(r) \beta \Phi_2(r)}{1-e^{\beta \Phi_1(r)}}</math> |

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

## Revision as of 11:59, 4 July 2007

The **Lebowitz and Percus** mean spherical approximation (MSA) (1966) (Ref. 1) closure relation is given by

In the **Blum and Høye** mean spherical approximation for mixtures (Refs 2 and 3) the closure is given by

and

where and are the total and the direct correlation functions for two spherical
molecules of *i* and *j* species, is the diameter of '*i* species of molecule.
Duh and Haymet (Eq. 9 Ref. 4) write the MSA approximation as

where and comes from the
Weeks-Chandler-Anderson division
of the Lennard-Jones potential.
By introducing the definition (Eq. 10 Ref. 4)

one can arrive at (Eq. 11 in Ref. 4)

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

## Thermodynamic consistency

See Ref. 5.

## References

- 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) - 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) - 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)