Mean spherical approximation

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

$c(r)=-\beta \omega (r),~~~~r>\sigma .$

The Blum and Høye mean spherical approximation (MSA) (1978-1980) (Refs 2 and 3) closure is given by

${\rm {g}}_{ij}(r)\equiv h_{ij}(r)+1=0~~~~~~~~r<\sigma _{ij}=(\sigma _{i}+\sigma _{j})/2$

and

$c_{ij}(r)=\sum _{n=1}{\frac {K_{ij}^{(n)}}{r}}e^{-z_{n}r}~~~~~~\sigma _{ij}<r$

where $h_{ij}(r)$ and $c_{ij}(r)$ are the total and the direct correlation functions for two spherical molecules of i and j species, $\sigma _{i}$ is the diameter of 'i species of molecule. Duh and Haymet (Eq. 9 Ref. 4) write the MSA approximation as

$g(r)={\frac {c(r)+\beta \Phi _{2}(r)}{1-e^{\beta \Phi _{1}(r)}}}$

where $\Phi _{1}$ and $\Phi _{2}$ comes from the WCA division of the Lennard-Jones potential. By introducing the definition (Eq. 10 Ref. 4)

$\left.s(r)\right.=h(r)-c(r)-\beta \Phi _{2}(r)$

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

$B(r)\approx B^{\rm {MSA}}(s)=\ln(1+s)-s$

The Percus Yevick approximation may be recovered from the above equation by setting $\Phi _{2}=0$.

References

1. 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)
2. 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)
3. 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. 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)
5. 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)