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Ramp model

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

\Phi_{12}(r) = \left\{ 
\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:

Ramp potential.png

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 [2]:

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

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  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:

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


Related literature