Ramp model

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

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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}<r\leq d_{c}\\0&{\rm {if}}\;r>d_{c}\end{array}}\right.}

where $\Phi _{12}(r)$ is the intermolecular pair potential, $r:=|\mathbf {r} _{1}-\mathbf {r} _{2}|$, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle W_{r}>0} and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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 :

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle T_{c}^{*}=1.487\pm 0.003}

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \rho _{c}\sigma ^{3}=0.103\pm 0.001}

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle p_{c}^{*}\simeq 0.042}

and the liquid-liquid critical point:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle T_{c}^{*}\simeq 0.378\pm 0.003}

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \rho _{c}\sigma ^{3}\simeq 0.380\pm 0.002}

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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 $\epsilon >0$ ; Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle [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

• 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)