Latest revision |
Your text |
Line 1: |
Line 1: |
| The '''ramp model''', proposed by Jagla | | The '''ramp model''' is described by: |
| <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> | | :<math> |
| \Phi_{12}(r) = \left\{ | | \Phi(r) = \left\{ |
| \begin{array}{ll} | | \begin{array}{ll} |
| \infty & {\rm if} \; r < \sigma \\ | | \infty & {\rm if} \; r < \sigma \\ |
Line 13: |
Line 11: |
| </math> | | </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>. | | where <math>\Phi(r)</math> is the [[intermolecular pair potential]], <math>W_r > 0</math> and <math>W_a < 0</math>. |
|
| |
|
| Graphically, one has: | | Graphically, one has: |
| [[Image:Ramp_potential.png|350px|center]] | | [[Image:Ramp_potential.png|center]] |
| where the red line represents an attractive implementation of the model, and the green line a repulsive implementation. | | 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 total 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== |
| <references />
| | #[http://dx.doi.org/10.1103/PhysRevLett.24.1284 P. C. Hemmer and G. Stell "Fluids with Several Phase Transitions", Physical Review Letters '''24''' pp. 1284 - 1287 (1970)] |
| '''Related literature'''
| | #[http://dx.doi.org/10.1063/1.1677857 G. Stell and P. C. Hemmer, "Phase Transitions Due to Softness of the Potential Core", Journal of Chemical Physics' '''56''', pp. 4274-4286 (1972)] |
| *[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/10.1063/1.480241 E. A. Jagla "Core-softened potentials and the anomalous properties of water", Journal of Chemical Physics' '''111''', pp. 8980-8986 (1999)] |
| *[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/10.1063/1.2748043 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/10.3390/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:models]] |
| [[category:Polyamorphic systems]] | | [[category:Polyamorphic systems]] |