Ramp model

The ramp model is described 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(r) = \left\{ \begin{array}{ll} \infty & {\rm if} \; r < \sigma \\ W_r - (W_r-W_a) \frac{r-\sigma}{d_a-\sigma} & {\rm if} \; \sigma \leq r \leq d_a \\ W_a - W_a \frac{r-d_a}{d_c-d_a} & {\rm if} \; d_a < r \leq d_c \\ 0 & {\rm if} \; r > d_c \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(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 W_r > 0} 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 W_a < 0} .

Graphically, one has:

where the red line represents an attractive implementation of the model, and the green line a repulsive implementation.

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)
3. E. A. Jagla "Core-softened potentials and the anomalous properties of water", Journal of Chemical Physics' 111, pp. 8980-8986 (1999)
4. E. Lomba, N. G. Almarza, C. Martin, C. McBride "Phase behaviour of attractive and repulsive ramp fluids: integral equation and computer simulation studies", Journal of Chemical Physics 126 244510 (2007)