Ramp model: Difference between revisions
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:<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> | ||
== Repulsive Ramp Model == | |||
In the repulsive ramp case: <math> W_a = 0 </math>; neither liquid-vapor nor liquid-liquid stable equilibria occur | |||
(See Lomba et al. at the list of the references). However, for this model, it has been found a low density crystalline phase, | |||
This solid phase presents reentrant melting, i.e. It melts into the fluid phase at high pressure. | |||
Recently, the same behavior has ben found in a three-dimensional [[lattice gas|lattice gas]] ramp 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'' Mol. Phys 2009 ] </ref> | |||
==See also== | ==See also== | ||
Revision as of 20:25, 4 March 2009
The ramp model, proposed by Jagla [1] 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi_{12}(r)} is the intermolecular pair potential, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r := |\mathbf{r}_1 - \mathbf{r}_2|} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_r > 0} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 [2]:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_c^* = 1.487 \pm 0.003}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho_c \sigma^3 = 0.103 \pm 0.001}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_c^* \simeq 0.042}
and the liquid-liquid critical point:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_c^* \simeq 0.378 \pm 0.003}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho_c \sigma^3 \simeq 0.380 \pm 0.002}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_c^*/T_c^* \simeq 0.49 \pm 0.01}
Repulsive Ramp Model
In the repulsive ramp case: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_a = 0 } ; neither liquid-vapor nor liquid-liquid stable equilibria occur (See Lomba et al. at the list of the references). However, for this model, it has been found a low density crystalline phase, This solid phase presents reentrant melting, i.e. It melts into the fluid phase at high pressure.
Recently, the same behavior has ben found in a three-dimensional lattice gas ramp model [3]
See also
References
- ↑ E. A. Jagla "Core-softened potentials and the anomalous properties of water", Journal of Chemical Physics' 111 pp. 8980-8986 (1999)
- ↑ 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)
- ↑ 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 Mol. Phys 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)