Editing Ramp model

The '''ramp model''', proposed by Jagla 
<ref>[http://dx.doi.org/10.1063/1.480241 E. A. Jagla "Core-softened potentials and the anomalous properties of water", Journal of Chemical Physics' '''111''' pp. 8980-8986  (1999)]</ref>
and sometimes known as the '''Jagla model''', is described by:

:<math>
\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.
</math>

where <math>\Phi_{12}(r)</math> is the [[intermolecular pair potential]], <math>r := |\mathbf{r}_1 - \mathbf{r}_2|</math>, <math>W_r > 0</math> and <math>W_a < 0</math>.

Graphically, one has:
[[Image:Ramp_potential.png|350px|center]]
where the red line represents an attractive implementation of the model, and the green line  a repulsive implementation.
==Critical points==
For the particular case <math> W_r^*=3.5; W_a^*=-1.0, d_a^*=1.72, d_c^*=3.0 </math>,
the liquid-vapour critical point   is located at
<ref name="lomba">
[http://dx.doi.org/10.1063/1.2748043  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)]
</ref>:

:<math>T_c^* = 1.487 \pm 0.003</math>
:<math>\rho_c \sigma^3 = 0.103 \pm 0.001</math>
:<math>p_c^* \simeq 0.042</math>

and the [[Polyamorphic systems |liquid-liquid]] critical point:

:<math>T_c^* \simeq 0.378 \pm 0.003</math>
:<math>\rho_c \sigma^3  \simeq 0.380 \pm 0.002</math>
:<math>p_c^*/T_c^* \simeq 0.49 \pm 0.01</math>

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.<ref name="paraty">
[https://doi.org/10.1103/PhysRevLett.127.015701  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)]
</ref>:

== Repulsive Ramp Model ==
In the repulsive ramp case, where <math> W_a = 0 </math>, neither liquid-vapor nor liquid-liquid stable equilibria occur
<ref name="lomba"> </ref>. 
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|Lattice Gas]] model 
<ref>
[http://dx.doi.org/10.1080/00268970902729269  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)]
</ref>
The system is defined on a simple cubic lattice. The interaction is that of a [[lattice hard spheres|lattice
hard sphere]] model with exclusion of nearest neighbours of occupied positions plus a repulsive interaction
with next-to-nearest neighbours.
The potential energy of the system is then given by:

:<math>
U = \epsilon \sum_{[ij]} S_i S_j
</math>

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

==See also==
*[[Triangular lattice ramp model]]
*[[Polyamorphism: Ramp model]]
*[[Fermi-Jagla model]]

==References==
<references />
'''Related literature'''
*[http://dx.doi.org/10.1103/PhysRevE.74.031108 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)]
*[http://dx.doi.org/10.1063/1.3043665 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)]
*[http://dx.doi.org/10.3390/ijms11125184 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)]
*[http://dx.doi.org/10.1063/1.3521486 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)]
*[http://dx.doi.org/10.1063/1.4921559 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)]


[[Category:models]]
[[category:Polyamorphic systems]]