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# Ramp model

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

where is the intermolecular pair potential, , and .

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 ,
the liquid-vapour critical point is located at
^{[2]}:

and the liquid-liquid critical point:

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 , 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:

where ; refers to all the pairs of sites that are second neighbors, and indicates the occupation of site (0 indicates an empty site, 1 indicates an occupied site).

## See also

## References

- ↑ E. A. Jagla "Core-softened potentials and the anomalous properties of water", Journal of Chemical Physics'
**111**pp. 8980-8986 (1999) - ↑
^{2.0}^{2.1}E. Lomba, N. G. Almarza, C. Martin, C. McBride "Phase behaviour of attractive and repulsive ramp fluids: integral equation and computer simulation studies", Journal of Chemical Physics**126**244510 (2007) Cite error: Invalid`<ref>`

tag; name "lomba" defined multiple times with different content - ↑
A. P. Bartók, G. Hantal, L. B. Pártay "Insight into Liquid Polymorphism from the Complex Phase Behavior of a Simple Model", Physical Review Letters
**127**015701 (2021) - ↑
Johan Skule Hoye, Enrique Lomba, and Noe Garcia Almarza, "One- and three-dimensional lattice models with two repulsive ranges: simple systems with complex phase behaviour", Molecular Physics
**107**, 321-330 (2009)

**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) - Jiayuan Luo, Limei Xu, C. Austen Angell, H. Eugene Stanley and Sergey V. Buldyrev "Physics of the Jagla model as the liquid-liquid coexistence line slope varies", Journal of Chemical Physics
**142**224501 (2015)