Hemmer and Stell model

The Hemmer and Stell model intermolecular pair potential, $\Phi _{12}(r)$, is given by (Ref. 1 Eq. 2, Ref. 2 Eq. 1-3):

$\Phi _{12}(r)=\Phi _{12}(r)_{\mathrm {repulsive} }+\Phi _{12}(r)_{\mathrm {attractive} }$

where a short range piece-wise potential

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \Phi _{12}(r)_{\mathrm {repulsive} }=\left\{{\begin{array}{ll}\infty &{\rm {if}}\;r<d\\V_{0}\left({\frac {1-(r-d)}{\lambda d}}\right)&{\rm {if}}\;d<r<d(1+\lambda )\\0&{\rm {if}}\;d(1+\lambda )<r\end{array}}\right.}

where $r:=|\mathbf {r} _{1}-\mathbf {r} _{2}|$, $d$ is the hard core radius, along with a long-range attractive potential;

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_{12}(r)_{\mathrm{attractive}} = -a\gamma \exp (-\gamma 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 a} is a positive constant.

References

1. P. C. Hemmer and G. Stell "Fluids with Several Phase Transitions", Physical Review Letters 24 pp. 1284 - 1287 (1970)
2. G. Stell and P. C. Hemmer, "Phase Transitions Due to Softness of the Potential Core", Journal of Chemical Physics' 56, pp. 4274-4286 (1972)