Mean spherical approximation

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

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

The {\bf Blum and H$\o$ye} mean spherical approximation (MSA) (1978-1980) \cite{JSP_1978_19_0317_nolotengoSpringer,JSP_1980_22_0661_nolotengoSpringer} closure is given by

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 {\rm g}_{ij}(r) \equiv h_{ij}(r) +1=0 ~~~~~~~~ r < \sigma_{ij} = (\sigma_i + \sigma_j)/2}

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 c_{ij}(r)= \sum_{n=1} \frac{K_{ij}^{(n)}}{r}e^{-z_nr} ~~~~~~ \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 \cite{JCP_1995_103_02625}) write the MSA approximation as

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 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 LJ potential.\\ By introducing the definition (Eq. 10 \cite{JCP_1995_103_02625})

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

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]