Triple point in a cell fluid model with effective temperature-dependent attraction

Abstract

We study a cell fluid model of a many-particle system with Curie-Weiss-type interaction potential. It is considered as an open system in a fixed volume partitioned into a large number of congruent cubic cells. The interaction potential comprises two competing components: a global uniform attraction acting between all particle pairs in the volume and a short-range repulsion between particles occupying the same cell. Previous studies have established that this model admits an exact solution, exhibits multiple critical points, and undergoes a sequence of first-order phase transitions. Despite variations in the interaction strengths, no triple point appears as long as these parameters remain fixed. We demonstrate that incorporating effective {temperature-dependent} attractive interactions fundamentally alters the phase behavior of the cell model. This modification preserves the model's exact solvability while resulting in the emergence of a triple point in the phase diagram.

Figures (4)

• Figure 1: Phase diagram in the pressure-temperature plane for the cell fluid model with Curie-Weiss-type interaction potential \ref{['1d1']} and repulsion-to-attraction ratio $f=J_2/J_1=1.5$. The first three phase coexistence lines (blue, green, and red) are shown, which separate four distinct stable phases (I -- IV) and correspond to first-order phase transitions between these phases. Each coexistence line terminates at its critical point, indicated by a solid circle.
• Figure 2: Chemical potential $\mu^*(T^*;\bar{z})$ as a function of $\bar{z}$ given by \ref{['1d10']} at fixed value of subcritical temperature $T^*=0.15$ and repulsion to attraction ratio $f=1.5$. The black, magenta, and orange curves correspond to $\alpha=0$, $\alpha=0.25$, and $\alpha=0.5$ in \ref{['1d2']}, respectively. Open circles denote extrema of $\mu^*$. Black crosses mark geometric midpoints $z_{01}$, $z_{02}$, and $z_{03}$ on segments between nearest maxima and minima pairs. Bold solid sections of plots show the regions of thermodynamically stable phases given by $\mu^*(T^*;\bar{z}_{\rm max})$.
• Figure 3: Parametric plots of $E_{I}(T^*;\bar{z})$ as a function of the chemical potential $\mu^*_{I} (T^*;\bar{z})$ --- blue, $E_{II}(T^*;\bar{z})$ versus $\mu^*_{II} (T^*;\bar{z})$ --- green, and $E_{III}(T^*;\bar{z})$ versus $\mu^*_{III} (T^*;\bar{z})$ --- red, with $\bar{z}$ being a parameter. In Figure (a), the temperature $T^*=0.15$ is in the range $T^*_{tr}<T^*<T^*_c$; Figure (b) shows the special case of the triple-point temperature $T^*=T^*_{tr}=0.135897$; in Figure (c), the temperature $T^*= 0.125$ is lower than at triple point ($T^*<T^*_{tr}$). In each figure, bold curve sections display regions of stable phases corresponding to $E_{\pi}(T^*;\bar{z}_{\rm max})$ and their thin dashed continuations refer to metastable states. The blue, green, and red curves correspond to $\alpha_\pi$ from \ref{['1d19']}. Parameters are taken as $f=1.5$ and $v^*=5.0$.
• Figure 4: Phase diagram of the CFM with temperature-dependent attraction interaction. The plot spans the temperature interval from $T^*=0.117$ up to the supercritical region (white background). The blue, green and purple curves are the first-order phase transition lines between Phases I, II, and III. The Phase I -- II coexistence line is shown in blue, the II --III coexistence line is green, and the I -- III line is purple. In the high-temperature region, the blue and green curves terminate at their critical points indicated by full circles. The red circle marks the triple point where Phases I, II, and III coexist in thermodynamic equilibrium at a single temperature and pressure. The inset shows the magnified region around the triple point. As before, $f=1.5$ and $v^*=5.0$.