Ramp model

The ramp model is described by:

Φ (r) = {\begin{matrix} \infty & r < σ \\ W_{r} - (W_{r} - W_{a}) \frac{r - σ}{d_{a} - σ} & σ \leq r \leq d_{a} \\ W_{a} - W_{a} \frac{r - d_{a}}{d_{c} - d_{a}} & d_{a} < r \leq d_{c} \\ 0 & r > d_{c} \end{matrix}

where $Φ (r)$ is the intermolecular pair potential, $W_{r} > 0$ 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

ρ_{c} σ^{3} = 0.103 \pm 0.001

p_{c}^{*} ≃ 0.042

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

T_{c}^{*} ≃ 0.378 \pm 0.003

ρ_{c} σ^{3} ≃ 0.380 \pm 0.002

p_{c}^{*} / T_{c}^{*} ≃ 0.49 \pm 0.01