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

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

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi_1} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi_2} comes from the WCA division of the Lennard-Jones potential. By introducing the definition (Eq. 10 \cite{JCP_1995_103_02625})

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s(r) = h(r) -c(r) -\beta \Phi_2 (r)}

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B(r) \approx B^{\rm MSA}(s) = \ln (1+s)-s}

The Percus Yevick approximation may be recovered from the above equation by setting Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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]