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

${\mathrm{g}}_{ij}(r)\equiv h_{ij}(r)+1=0r<{\sigma }_{ij}=({\sigma }_i+{\sigma }_j)/2$

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

$g(r)=\frac{c(r)+\beta {\mathrm{\Phi }}_2(r)}{1-e^{\beta {\mathrm{\Phi }}_1(r)}}$

where ${\mathrm{\Phi }}_1$ and ${\mathrm{\Phi }}_2$ comes from the WCA division of the Lennard-Jones potential. By introducing the definition (Eq. 10 \cite{JCP_1995_103_02625})

$s(r)=h(r)-c(r)-\beta {\mathrm{\Phi }}_2(r)$

one can arrive at (Eq. 11 \cite{JCP_1995_103_02625})

$B(r)\approx B^{\mathrm{M}\mathrm{S}\mathrm{A}}(s)=\ln (1+s)-s$

The Percus Yevick approximation may be recovered from the above equation by setting ${\mathrm{\Phi }}_2=0$.

References

1. [PR_1966_144_000251]
2. [JSP_1978_19_0317_nolotengoSpringer]
3. [JSP_1980_22_0661_nolotengoSpringer]
4. [JCP_1995_103_02625]