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

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}\left( r \right) = \left\{ \begin{array}{ccc} \infty & ; & r < \sigma \\ \epsilon_1 & ; &\sigma \le r < \lambda_1 \sigma \\ - \epsilon_2 & ; &\lambda_1\sigma \le r < \lambda_2 \sigma \\ 0 & ; & r \ge \lambda_2 \sigma \end{array} \right. }

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_{12}(r)} is the intermolecular pair 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 \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, $r$ 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.s 1-5.

Liquid-liquid transition

For certain parameter sets the square shoulder + square well model has a liquid-liquid transition. See Ref.s 1-4.

References

1. Giancarlo Franzese, Gianpietro Malescio, Anna Skibinsky, Sergey V. Buldyrev, and H. Eugene Stanley "Generic mechanism for generating a liquid-liquid phase transition", Nature 409, pp. 692-695 (2001)
2. Giancarlo Franzese, Gianpietro Malescio, Anna Skibinsky, Sergey V. Buldyrev, and H. Eugene Stanley, "Metstable liquid-liquid phase transition in a single-component system with only one crystal phase and no density anomaly", Physical Review E 66, 051206 (2002)
3. Anna Skibinsky, Sergey V. Buldyrev, Giancarlo Franzese, Gianpietro Malescio, and H. Eugene Stanley, "Liquid-liquid phase transitions for soft-core attractive potentials", Physical Review E 69, 061206 (2004)
4. Gianpietro Malescio, Giancarlo Franzese, Anna Skibinsky, Sergey V. Buldyrev, and H. Eugene Stanley, "Liquid-liquid phase transition for an attractive isotropic potential with wide repulsive range", Physical Review E 71, 061504 (2005)
5. 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)
6. Shiqi Zhou, A. Lajovic, and A. Jamnik "Local structures of fluid with discrete spherical potential: Theory and grand canonical ensemble Monte Carlo simulation", Journal of Chemical Physics 129 124503 (2008)