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 :

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

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 in ) 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 Weeks-Chandler-Andersen division of the Lennard-Jones potential. By introducing the definition (Eq. 10 in )

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

one can arrive at (Eq. 11 in )

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

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