# Ramp model

The ramp model, proposed by Jagla [1] and sometimes known as the Jagla model, is described by:

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

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

Graphically, one has:

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

## Critical points

For the particular case ${\displaystyle W_{r}^{*}=3.5;W_{a}^{*}=-1.0,d_{a}^{*}=1.72,d_{c}^{*}=3.0}$, the liquid-vapour critical point is located at [2]:

${\displaystyle T_{c}^{*}=1.487\pm 0.003}$
${\displaystyle \rho _{c}\sigma ^{3}=0.103\pm 0.001}$
${\displaystyle p_{c}^{*}\simeq 0.042}$

and the liquid-liquid critical point:

${\displaystyle T_{c}^{*}\simeq 0.378\pm 0.003}$
${\displaystyle \rho _{c}\sigma ^{3}\simeq 0.380\pm 0.002}$
${\displaystyle p_{c}^{*}/T_{c}^{*}\simeq 0.49\pm 0.01}$

While this liquid-liquid critical point was long held to be in the stable region of the phase diagram, a high density double-network structure was found to be thermodynamically more stable than the high-density liquid under any conditions.[3]:

## Repulsive Ramp Model

In the repulsive ramp case, where ${\displaystyle W_{a}=0}$, neither liquid-vapor nor liquid-liquid stable equilibria occur [2]. 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 [4] 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:

${\displaystyle U=\epsilon \sum _{[ij]}S_{i}S_{j}}$

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