Ramp model

The ramp model, proposed by Jagla, is described by:

$\Phi _{12}(r)=\left\{{\begin{array}{ll}\infty &{\rm {if}}\;r<\sigma \\W_{r}-(W_{r}-W_{a}){\frac {r-\sigma }{d_{a}-\sigma }}&{\rm {if}}\;\sigma \leq r\leq d_{a}\\W_{a}-W_{a}{\frac {r-d_{a}}{d_{c}-d_{a}}}&{\rm {if}}\;d_{a}<r\leq d_{c}\\0&{\rm {if}}\;r>d_{c}\end{array}}\right.$

where $\Phi _{12}(r)$ is the intermolecular pair potential, $r:=|\mathbf {r} _{1}-\mathbf {r} _{2}|$, $W_{r}>0$ and $W_{a}<0$.

Graphically, one has:

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

Critical points

The liquid-vapour critical point is located at (Ref. 4)

$T_{c}^{*}=1.487\pm 0.003$

$\rho _{c}\sigma ^{3}=0.103\pm 0.001$

$p_{c}^{*}\simeq 0.042$

The liquid-liquid critical point is located at (Ref. 4)

$T_{c}^{*}\simeq 0.378\pm 0.003$

$\rho _{c}\sigma ^{3}\simeq 0.380\pm 0.002$

$p_{c}^{*}/T_{c}^{*}\simeq 0.49\pm 0.01$

See also

• Polyamorphism: Ramp model

References

1. E. A. Jagla "Core-softened potentials and the anomalous properties of water", Journal of Chemical Physics' 111, pp. 8980-8986 (1999)
2. 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)