Ramp model

The ramp model, proposed by Jagla and sometimes known as the Jagla model, is described by:

$\Phi _{12}(r)=\left\{{\begin{array}{ll}\infty &{\rm {if}}\;r<\sigma \\W_{r}-(W_{r}-W_{a}){\frac {r-\sigma }{d_{a}-\sigma }}&{\rm {if}}\;\sigma \leq r\leq d_{a}\\W_{a}-W_{a}{\frac {r-d_{a}}{d_{c}-d_{a}}}&{\rm {if}}\;d_{a}<r\leq d_{c}\\0&{\rm {if}}\;r>d_{c}\end{array}}\right.$

where $\Phi _{12}(r)$ 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 (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^*=3.5; W_a^*=-1.0, d_a^*=1.72, d_c^*=3.0 } , the liquid-vapour critical point is located at (Ref. 4)

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}

$\rho _{c}\sigma ^{3}=0.103\pm 0.001$

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 is located at (Ref. 4)

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}

See also

• Polyamorphism: Ramp model

References

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)
• 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)
• 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)