Ramp model

The ramp model, proposed by Jagla (Ref. 1) and sometimes known as the Jagla model, 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)
3. Limei Xu, Sergey V. Buldyrev, Nicolas Giovambattista, C. Austen Angell, and H. Eugene Stanley "A monatomic system with a liquid-liquid critical point and two distinct glassy states", Journal of Chemical Physics 130 054505 (2009)