Mean spherical approximation

The mean spherical approximation (MSA) closure relation of Lebowitz and Percus is given by :

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

In the Blum and Høye mean spherical approximation for mixtures the 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 in ) 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 Weeks-Chandler-Andersen division of the Lennard-Jones potential. By introducing the definition (Eq. 10 in )

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

one can arrive at (Eq. 11 in )

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

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