Ramp model

The ramp model, proposed by Jagla 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

For the particular case \( W_r^*=3.5; W_a^*=-1.0, d_a^*=1.72, d_c^*=3.0 \), the liquid-vapour critical point is located at :

\[T_c^* = 1.487 \pm 0.003\] \[\rho_c \sigma^3 = 0.103 \pm 0.001\] \[p_c^* \simeq 0.042\]

and the liquid-liquid critical point:

\[T_c^* \simeq 0.378 \pm 0.003\] \[\rho_c \sigma^3 \simeq 0.380 \pm 0.002\] \[p_c^*/T_c^* \simeq 0.49 \pm 0.01\]

Repulsive Ramp Model

In the repulsive ramp case, where \( W_a = 0 \), neither liquid-vapor nor liquid-liquid stable equilibria occur . However, for this model a low density crystalline phase has been found. This solid phase presents re-entrant melting, i.e. this solid melts into the fluid phase as the pressure is increased.

Lattice gas version

Recently, similar behaviour has been found in a three-dimensional Repulsive Ramp Lattice Gas model The system is defined on a simple cubic lattice. The interaction is that of a lattice hard sphere model with exclusion of nearest neighbours of occupied positions plus a repulsive interaction with next-to-nearest neighbours. The total potential energy of the system is then given by:

\[ U = \epsilon \sum_{[ij]} S_i S_j \]

where \( \epsilon > 0 \) ; \( [ij] \) refers to all the pairs of sites that are second neighbors, and \( S_k \) indicates the occupation of site \( k \) (0 indicates an empty site, 1 indicates an occupied site).

See also

• Triangular lattice ramp model
• Polyamorphism: Ramp model

References

Related literature

• Limei Xu, Sergey V. Buldyrev, C. Austen Angell, and H. Eugene Stanley "Thermodynamics and dynamics of the two-scale spherically symmetric Jagla ramp model of anomalous liquids", Physical Review E 74 031108 (2006)
• 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)
• Limei Xu, Sergey V. Buldyrev, Nicolas Giovambattista, and H. Eugene Stanley "Liquid-Liquid Phase Transition and Glass Transition in a Monoatomic Model", International Journal of Molecular Sciences 11 pp. 5184-5200 (2010)
• Limei Xu, Nicolas Giovambattista, Sergey V. Buldyrev, Pablo G. Debenedetti, and H. Eugene Stanley "Waterlike glass polyamorphism in a monoatomic isotropic Jagla model", Journal of Chemical Physics 134 064507 (2011)