Scaling Analysis of Thermodynamic Properties in the Critical Region of Fluids

Critical behavior of the dielectric constant in asymmetric fluids

By applying a thermodynamic theory that incorporates the concept of complete scaling, we derive the asymptotic temperature dependence of the critical behavior of the dielectric constant above the critical temperature along the critical isochore and below the critical temperature along the coexistence curve. The amplitudes of the singular terms in the temperature expansions are related to the changes of the critical temperature and the critical chemical potential upon the introduction of an electric field. The results of the thermodynamic theory are then compared with the critical behavior implied by the classical Clausius-Mossotti approximation. The Clausius-Mossotti approximation fails to account for any singular temperature dependence of the dielectric constant above the critical temperature. Below the critical temperature it produces an apparent asymmetric critical behavior with singular terms similar to those implied by the thermodynamic theory, but with significantly different coefficients. We conclude that the Clausius-Mossotti approximation only can account for the observed asymptotic critical behavior of the dielectric constant when the dependence of the critical temperature on the electric field is negligibly small.

Asymmetric criticality in weakly compressible liquid mixtures

The thermodynamics of asymmetric liquid-liquid criticality is updated by incorporating pressure effects into the complete-scaling formulation earlier developed for incompressible liquid mixtures [C. A. Cerdeirina et al., Chem. Phys. Lett. 424, 414 (2006); J. T. Wang et al., Phys. Rev. E 77, 031127 (2008)]. Specifically, we show that pressure mixing enters into weakly compressible liquid mixtures as a consequence of the pressure dependence of the critical parameters. The theory is used to analyze experimental coexistence-curve data in the mole fraction-temperature, density-temperature, and partial density-temperature planes for a large number of binary liquid mixtures. It is shown how the asymmetry coefficients in the scaling fields are related to the difference in molecular volumes of the two liquid components. The work resolves the question of the so-called "best order parameter" discussed in the literature during the past decades.

Master crossover functions for one-component fluids

By introducing three well-defined dimensionless numbers, we establish the link between the scale dilatation method able to estimate master (i.e., unique) singular behaviors of the one-component fluid subclass and the universal crossover functions recently estimated [Garrabos and Bervillier, Phys. Rev. E 74, 021113 (2006)] from the bounded results of the massive renormalization scheme applied to the Phi(d)(4)(n) model of scalar order parameter (n=1) and three dimensions (d=3), representative of the Ising-like universality class. The master (i.e., rescaled) crossover functions are then able to fit the singular behaviors of any one-component fluid without adjustable parameter, using only one critical energy scale factor, one critical length scale factor, and two dimensionless asymptotic scale factors, which characterize the fluid critical interaction cell at its liquid-gas critical point. An additional adjustable parameter accounts for quantum effects in light fluids at the critical temperature. The effective extension of the thermal field range along the critical isochore where the master crossover functions seems to be valid corresponds to a correlation length greater than three times the effective range of the microscopic short-range molecular interaction.

Precise simulation of criticality in asymmetric fluids

Extensive grand canonical Monte Carlo simulations have been performed for the hard-core square-well fluid with interaction range b=1.5 sigma. The critical exponent for the correlation length has been estimated in an unbiased fashion as nu=0.63+/-0.03 via finite-size extrapolations of the extrema of properties measured along specially constructed, asymptotically critical loci that represent pseudosymmetry axes. The subsequent location of the critical point achieves a precision of five parts in 10(4) for Tc and about 0.3% for the critical density rhoc. The effective exponents gamma+(eff) and beta(eff) indicate Ising-type critical-point values to within 2% and 5.6%, respectively, convincingly distinguishing the universality class from the "nearby" XY and n=0 (self-avoiding walk) classes. Simulations of the heat capacity CV(T,rho) and d2psigma/dT2, where psigma is the vapor pressure below Tc, suggest a negative but small Yang-Yang anomaly, i.e., a specific-heat-like divergence in the corresponding chemical potential derivative (d2 musigma/dT2) that requires a revision of the standard asymptotic scaling description of asymmetric fluids.

The yang-yang anomaly in fluid criticality: experiment and scaling theory

Yang and Yang proved that the divergence of C(V)(T) at a gas-liquid critical point implies that either d(2)p/dT(2) identical withp(")(sigma) or d(2)&mgr;/dT(2) identical with&mgr;(")(sigma) or both diverge when T-->T(c)- on the phase boundary sigma. They queried the lattice-gas prediction that &mgr;(")(sigma) remains finite. Analysis of two-phase heat-capacity data provides, for the first time, evidence for such a Yang-Yang anomaly (&mgr;(")(sigma)-->+/-infinity) in propane and suggests an anomaly of opposite sign in CO (2). A revision of standard scaling theory for fluid criticality is demanded: specifically, p-p(c) must appear in the ordering field. The coexistence diameter hence gains a |T-T(c)|(2beta) term.

Implementation of Scaling and Extended Scaling Equations of State for the Critical Point of Fluids

An explicit, practical procedure is suggested for transforming from the laboratory variables density (ρ) and temperature (T) into the parametric variables r and θ, which occur in various scaled representations of equations of state and of transport properties of fluids near critical points. A reasonably efficient and versatile computer program illustrating this procedure is provided. With this program, the parametric equations of state which occur in several formulations of simple, extended, and/or revised scaling are as easy to use as any other equation of state for which T and ρ are the independent variables.

An Improved State Equation in the Vicinity of the Critical Point

An improved state equation for the vicinity of the critical point is proposed. An analysis of the experimental data on helium and xenon has been carried out in order to investigate the influence of the number of constants in the equation and the PρT range on the critical constants T c and ρ c and on the critical exponents α, β, γ, and δ. No such influence has been detected. The model for the critical point, recently proposed by Widom. has been checked regarding its consequences for the rectilinear diameter. No definite confirmation but indications for its correctness have been found.

Expanded Formulation of Thermodynamic Scaling in the Critical Region

A description of the thermodynamic properties in the critical region of a physical system is obtained from a scaled expression for the free-energy F(ρ, T). In general, a nonsymmetric coexistence curve is predicted, with the symmetric case (e.g., magnets) included as a special example. For fluids, deviations from symmetry give rise to an expression for the average density below the critical point nonlinear in the temperature near T c (in contrast to the usual "law of rectilinear diameter"); these asymmetries also contribute to the discontinuity in the specific heat along the critical isochore. To lowest order, the formulation reduces to Widom's homogeneous scaling; the classical equations of state of the van der Waals type are incorporated as a special case.

Scaling Analysis of Thermodynamic Properties in the Critical Region of Fluids

A review of the scaled equation of state proposed for the critical region of fluids and magnets is given using the language appropriate for fluids. The experimental evidence for the validity of the basic hypothesis underlying this equation of state is discussed in detail. Experimental data in the critical regions of CO2, Xe. and He4 are then analyzed using a closed-form expression for the chemical potential as a function of density and temperature, based on scaling ideas. Agreement between the proposed equation and the experimental data is found for the three substances. The results of the scaling of Δμ, Δρ, t data are shown not to be in contradiction with the analysis, also based on scaling ideas, of independent experimental measurements of both specific heat and vapor pressure.