Square shoulder + square well model

The Square shoulder + square well model consists, as its name suggests, of a combination of the square shoulder model and the square well model. The model is given by

$\Phi _{12}\left(r\right)=\left\{{\begin{array}{ccc}\infty &;&r<\sigma \\\epsilon _{1}&;&\sigma \leq r<\lambda _{1}\sigma \\-\epsilon _{2}&;&\lambda _{1}\sigma \leq r<\lambda _{2}\sigma \\0&;&r\geq \lambda _{2}\sigma \end{array}}\right.$

where $\Phi _{12}(r)$ is the intermolecular pair potential, $\epsilon _{1}$ 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 \epsilon_2} are the well depths, 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 r} is the distance between site 1 and site 2 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 r := |\mathbf{r}_1 - \mathbf{r}_2|} , σ is the hard core diameter 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 \lambda_1,\lambda_2>1}

Critical points

See Ref. 2.

Liquid-liquid transition

For certain parameter sets the square shoulder + square well model has a liquid-liquid transition.

References

1. Andreas Lang, Gerhard Kahl, Christos N Likos, Hartmut Löwen and Martin Watzlawek "Structure and thermodynamics of square-well and square-shoulder fluids", Journal of Physics: Condensed Matter 11 pp. 10143-10161 (1999)
2. W. Rzysko, O. Pizio, A. Patrykiejew, and S. Sokolowski "Phase diagram of a square-shoulder, square-well fluid revisited", Journal of Chemical Physics 129 124502 (2008)