Closed-loop liquid-vapor equilibrium in a one-component system

We report Monte Carlo simulations that show a closed-loop liquid-vapor equilibrium in a pure substance. This finding has been achieved on a two-dimensional lattice model for patchy particles that can form network fluids. We have considered related models with a slightly different patch distribution in order to understand the features of the distribution of patches on the surface of the particles that make possible the presence of the closed-loop liquid-vapor equilibrium, and its relation to the phase diagram containing so-called empty liquids. Finally we discuss the likelihood of finding the closed-loop liquid-vapor equilibria on related models for three-dimensional models of patchy particles in the continuum, and speculate on the possible relationship between the mechanism behind the closed-loop liquid-vapor equilibrium of our simple lattice model and the salt-induced reentrant condensation found in complex systems.

Figure 1

Sketch of the patchy particles in the three lattice models: (a) o-2A4B model, ${\alpha }_{AA}={60}^{\circ }$; (b) m-2A4B model, ${\alpha }_{AA}={120}^{\circ }$; (c) p-2A4B model, ${\alpha }_{AA}={180}^{\circ }$. Thin lines represent the underlying triangular lattice. Solid circles represent particles (occupied sites). Thick segments indicate the orientation of $A$ patches.

Figure 2

Temperature-density phase diagrams for the p-2A4B model, and different values of $y$ (see the legends). Solid symbols indicate critical points. $T^{*}$ is the reduced temperature, $T^{*}=k_{B}T/{\epsilon }_{AA}$ (where $k_B$ is the Boltzmann constant), and the density $\rho$ corresponds to a fraction of occupied sites.

Figure 3

Temperature-density phase diagrams for the m-2A4B model. See the caption of Fig. 2 for notation details.

Figure 4

Liquid-vapor equilibria phase diagrams for the m-2A4B model with an alternative temperature scaling.

Figure 5

Configuration for the liquid phase at the binodal for the $p$-2A4B model with $y=0.40$, $T^{*}=0.07$, $\rho \simeq 0.375$.

Figure 6

Configuration for the $m$-2A4B model with $y=0.44$, $\rho =1/3$, $T^{*}=0.03$ (below the lower critical temperature).

Figure 7

Configuration for the $m$-2A4B model with $y=0.44$, $\rho =1/3$, $T^{*}=0.10$. At these conditions there is LVE phase separation.