Thermodynamic response functions in a cell fluid model with Curie-Weiss interaction. I. Supercritical region

TL;DR

This work derives explicit, analytic expressions for key thermodynamic response functions of a many-particle system with a Curie-Weiss-type interaction in the supercritical region, using an exact parametric equation of state obtained in the grand canonical ensemble. By expressing the state in dimensionless variables and employing Gauss-Poisson moments $M_1$ and $M_2$ through the functions $K_j$, the authors obtain closed forms for the isothermal compressibility $\kappa_T^*$, thermal pressure coefficient $\beta_V^*$, isochoric heat capacity $c_V^*$, thermal expansion coefficient $\alpha_P^*$, and isobaric heat capacity $c_P^*$, all as functions of temperature, density, and chemical potential. The results show smooth, finite behavior throughout the supercritical region, with pronounced maxima near critical points governed by $M_1$ and $M_2$, providing a comprehensive thermodynamic portrait and establishing a foundation for the forthcoming subcritical analysis with its cascade of phase transitions. The analysis highlights how the competition between global Curie-Weiss attraction and local repulsion shapes response functions and phase behavior, offering exact benchmarks for theory and simulation in supercritical fluids.

Abstract

Thermodynamic response functions, including the isothermal compressibility, the thermal pressure coefficient, and the thermal expansion coefficient, isochoric and isobaric heat capacities are explicitly derived for a many-particle system interacting through a Curie-Weiss-type potential. These calculations are based on an exact equation of state previously obtained for a cell fluid model in the grand canonical ensemble. The resulting response functions are presented graphically as functions of temperature, density, and chemical potential within the supercritical region.

Figures (8)

• Figure 1: The pressure $P^*$ as a function of the density $\rho^*$ (figure (a)) and the chemical potential $\mu^*$ (figure (b)) at different values of temperature $T^*$ in the supercritical region.
• Figure 2: The isothermal compressibility $\kappa^*_T$ as a function of the particle number density $\rho^*$ (figure (a)) and the chemical potential $\mu^*$ (figure (b)) at different values of temperature $T^*$ ($T > T_c$). Parameters taken as $f=1.5$, $v^*= 5.0$.
• Figure 3: The thermal pressure coefficient $\beta^*_V$ as a function of the particle number density $\rho^*$ (figure (a)) and the chemical potential $\mu^*$ (figure (b)) at different values of temperature $T^*$ in the supercritical region. Parameters taken as $f=1.5$, $v^*= 5.0$.
• Figure 4: The isochoric heat capacity $c^*_V$ as a function of the particle number density $\rho^*$ (figure (a)) and the chemical potential $\mu^*$ (figure (b)) at different values of temperature $T^*$ in the supercritical region.
• Figure 5: The thermal expansion coefficient $\alpha^*_P$ as a function of the particle number density $\rho^*$ (figure (a)) and the chemical potential $\mu^*$ (figure (b)) at different values of temperature $T^*$ in the supercritical region.Parameters taken as $f=1.5$, $v^*= 5.0$.