Liquid-liquid phase transition for an attractive isotropic potential with wide repulsive range

Gianpietro Malescio, Giancarlo Franzese, Anna Skibinsky, Sergey V. Buldyrev, and H. Eugene Stanley

Phys. Rev. E 71, 061504 – Published 17 June 2005

Abstract

We investigate how the phase diagram of a repulsive soft-core attractive potential, with a liquid-liquid phase transition in addition to the standard gas-liquid phase transition, changes by varying the parameters of the potential. We extend our previous work on short soft-core ranges to the case of large soft-core ranges, by using an integral equation approach in the hypernetted-chain approximation. We show, using a modified van der Waals equation we recently introduced, that if there is a balance between the attractive and repulsive part of the potential this potential has two fluid-fluid critical points well separated in temperature and in density. This implies that for the repulsive (attractive) energy $U_{R} (U_{A})$ and the repulsive (attractive) range $w_{R} (w_{A})$ the relation $U_{R} ∕ U_{A} \propto w_{R} ∕ w_{A}$ holds for short soft-core ranges, while $U_{R} ∕ U_{A} \propto 3 w_{R} ∕ w_{A}$ holds for large soft-core ranges.

Received 10 November 2004

DOI:https://doi.org/10.1103/PhysRevE.71.061504

Authors & Affiliations

Gianpietro Malescio¹, Giancarlo Franzese^2,3, Anna Skibinsky³, Sergey V. Buldyrev^3,4, and H. Eugene Stanley³

¹Dipartimento di Fisica, Università di Messina and Istituto Nazionale Fisica della Materia, 98166 Messina, Italy
²Departament de Física Fonamental, Facultat de Física, Universitat de Barcelona, Diagonal 647, 08028 Barcelona, Spain
³Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts 02215, USA
⁴Department of Physics, Yeshiva University, 500 W 185th St., New York, New York 10040,USA

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Vol. 71, Iss. 6 — June 2005

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Figure 1
Inset: General shape of the attractive soft-core potential used in this work, with hard-core distance $a$ , soft-core distance $b$ , interaction range $c$ , attractive energy $U_{A}$ , and repulsive energy $U_{R}$ . Panels: Instability line of the HNC equation for the potential in the inset with $w_{A} ∕ a = 0.2$ : (a) $w_{R} ∕ a = 1$ and (from top to bottom) $U_{R} ∕ U_{A} = - 1$ (potential $A$ ), $- 0.5$ , 0, 0.2, 0.4, 0.6, 0.8, 1, $\infty$ (potential $B$ ); (b) $w_{R} ∕ a = 0.8$ , and $U_{R} ∕ U_{A} = 0.2$ , 0.3, 0.4, 0.5.Reuse & Permissions
Figure 2
Instability line of the HNC equation for $w_{R} ∕ a = 1$ and six values of the parameter $w_{A} ∕ a$ : (a) $w_{A} ∕ a = 0.1$ , and $U_{R} ∕ U_{A} = 0.2$ , 0.3, 0.4, 0.5; (b) $w_{A} ∕ a = 0.2$ , and $U_{R} ∕ U_{A} = 0.3$ , 0.4, 0.5, 0.6, 0.7, $\infty$ (potential B); (c) $w_{A} ∕ a = 0.3$ , and $U_{R} ∕ U_{A} = 0.4$ , 0.5, 0.6, 0.7; (d) $w_{A} ∕ a = 0.4$ , and $U_{R} ∕ U_{A} = 0.6$ , 0.7, 0.8, 0.9, 1; (e) $w_{A} ∕ a = 0.5$ , and $U_{R} ∕ U_{A} = 0.7$ , 0.8, 0.85, 0.9, 1; (f) $w_{A} ∕ a = 0.6$ , and $U_{R} ∕ U_{A} = 0.6$ , 0.8, 0.9, 1, 1.1, $\infty$ (potential B).Reuse & Permissions
Figure 3
Instability line of the HNC equation for $w_{R} ∕ a = 1.5$ and four values of the parameter $w_{A} ∕ a$ : (a) $w_{A} ∕ a = 0.5$ , and (from top to bottom) $U_{R} ∕ U_{A} = 0.5$ , 0.7, 0.8, 0.9; (b) $w_{A} ∕ a = 1$ , and $U_{R} ∕ U_{A} = 1.5$ , 1.6, 1.7, 1.8, 2; (c) $w_{A} ∕ a = 1.5$ , and $U_{R} ∕ U_{A} = 2$ , 2.5, 2.75, 3; (d) $w_{A} ∕ a = 2.5$ , and $U_{R} ∕ U_{A} = 4$ , 5, 6, 7.Reuse & Permissions
Figure 4
Instability line of the HNC equation for (a) $w_{R} ∕ a = 0.5$ , $w_{A} ∕ a = 0.5$ , and (from top to bottom) $U_{R} ∕ U_{A} = 1$ , 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7; (b) an enlarged view at low $T$ .Reuse & Permissions
Figure 5
(a) Symbols mark the combinations of the potential’s parameters where the instability line, calculated by the HNC approach, shows two maxima, suggesting the presence of two fluid-fluid critical points. Sets with $w_{R} ∕ a = 0.6$ (circles) were investigated for $0 ⩽ U_{R} ∕ U_{A} ⩽ 2$ and $0 ⩽ w_{A} ∕ a ⩽ 1$ ; sets with $w_{R} ∕ a = 0.8$ (squares) and with $w_{R} ∕ a = 1.0$ (diamonds) for $0 ⩽ U_{R} ∕ U_{A} ⩽ 1.2$ and $0 ⩽ w_{A} ∕ a ⩽ 0.6$ ; sets with $w_{R} ∕ a = 1.5$ (triangles) for $0 ⩽ U_{R} ∕ U_{A} ⩽ 6$ and $0 ⩽ w_{A} ∕ a ⩽ 3$ . Parameters outside these regions have not been investigated. (b) Middle points $U_{R}^{*}$ of intervals of $U_{R}$ in panel (a). Symbols are as in panel (a). Error bars represent the intervals in panel (a). Lines are one-parameter fits with Eq. (5): for set 2 (squares) the fitting parameter is $A ∕ (U_{A} V_{SC}) = 0.31$ (dashed line); for set 3 (diamonds) the fitting parameter is $A ∕ (U_{A} V_{SC}) = - 0.84$ (dot-dashed line); for set 4 (triangles) the fitting parameter is $A ∕ (U_{A} V_{SC}) = 2.23$ (dotted line). Since we only have three points for set 1, to avoid a fit with a large indeterminacy on the parameters we arbitrarily chose $A ∕ (U_{A} V_{SC}) = 1$ (solid line) to show that the data are consistent with Eq. (5). (c) Lines are Eq.(5) evaluated with $A = 0$ for each set. Symbols and lines are as in panel (b).Reuse & Permissions

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