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| ==Critical points== | | ==Critical points== |
| For the particular case <math> W_r^*=3.5; W_a^*=-1.0, d_a^*=1.72, d_c^*=3.0 </math>, | | 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 | | the liquid-vapour critical point (for the particular case is located at (Ref. 4) |
| <ref name="lomba">
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| [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)]
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| </ref>:
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| :<math>T_c^* = 1.487 \pm 0.003</math> | | :<math>T_c^* = 1.487 \pm 0.003</math> |
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| :<math>p_c^* \simeq 0.042</math> | | :<math>p_c^* \simeq 0.042</math> |
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| and the [[Polyamorphic systems |liquid-liquid]] critical point: | | and the liquid-liquid critical point is located at (Ref. 4) |
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| :<math>T_c^* \simeq 0.378 \pm 0.003</math> | | :<math>T_c^* \simeq 0.378 \pm 0.003</math> |
| :<math>\rho_c \sigma^3 \simeq 0.380 \pm 0.002</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> | | :<math>p_c^*/T_c^* \simeq 0.49 \pm 0.01</math> |
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| 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">
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| [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)]
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| </ref>:
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| == Repulsive Ramp Model ==
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| In the repulsive ramp case, where <math> W_a = 0 </math>, neither liquid-vapor nor liquid-liquid stable equilibria occur
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| <ref name="lomba"> </ref>.
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| However, for this model a low density crystalline phase has been found.
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| This solid phase presents re-entrant melting, i.e. this solid melts into the fluid phase as the pressure is increased.
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| ==== Lattice gas version ====
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| Recently, similar behaviour has been found in a three-dimensional Repulsive
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| Ramp [[lattice gas|Lattice Gas]] model
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| <ref>
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| [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)]
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| </ref>
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| The system is defined on a simple cubic lattice. The interaction is that of a [[lattice hard spheres|lattice
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| hard sphere]] model with exclusion of nearest neighbours of occupied positions plus a repulsive interaction
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| with next-to-nearest neighbours.
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| The total potential energy of the system is then given by:
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| :<math>
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| U = \epsilon \sum_{[ij]} S_i S_j
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| </math>
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| where <math> \epsilon > 0 </math> ; <math> [ij] </math> refers to all the pairs of sites that are
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| second neighbors, and <math> S_k </math> indicates the occupation of site <math> k </math>
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| (0 indicates an empty site, 1 indicates an occupied site).
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| ==See also== | | ==See also== |
| *[[Triangular lattice ramp model]]
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| *[[Polyamorphism: Ramp model]] | | *[[Polyamorphism: Ramp model]] |
| *[[Fermi-Jagla model]]
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| ==References== | | ==References== |
| <references /> | | <references /> |
| '''Related literature''' | | '''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/10.1103/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/10.1063/1.2748043 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)] |
| *[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/10.1063/1.3043665 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)]
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| *[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)]
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| *[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)]
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| [[Category:models]] | | [[Category:models]] |
| [[category:Polyamorphic systems]] | | [[category:Polyamorphic systems]] |