Crossover between vapor-liquid and consolute critical phenomena

Thermodynamic modeling of fluid polyamorphism in hydrogen at extreme conditions

Fluid polyamorphism, the existence of multiple amorphous fluid states in a single-component system, has been observed or predicted in a variety of substances. A remarkable example of this phenomenon is the fluid–fluid phase transition (FFPT) in high-pressure hydrogen between insulating and conducting high-density fluids. This transition is induced by the reversible dimerization/dissociation of the molecular and atomistic states of hydrogen. In this work, we present the first attempt to thermodynamically model the FFPT in hydrogen at extreme conditions. Our predictions for the phase coexistence and the reaction equilibrium of the two alternative forms of fluid hydrogen are based on experimental data and supported by the results of simulations. Remarkably, we find that the law of corresponding states can be utilized to construct a unified equation of state combining the available computational results for different models of hydrogen and the experimental data.

Behavior of the Widom line in critical phenomena

Using linear scaling theory, we study the behavior of response functions extrema in the vicinity of the critical point. We investigate how the speed of convergence of the loci of response function extrema to the Widom line depends on the parameters of the linear scaling theory. We find that when the slope of the coexistence line is near zero, the line of specific heat maxima does not follow the Widom line but instead follows the coexistence line. This has relevance for the detection of liquid-liquid critical points, which can exhibit a near-horizontal coexistence line. Our theoretical predictions are confirmed by computer simulations of a family of spherically symmetric potentials.

Thermodynamics of supercooled water

We review the available experimental information on the thermodynamic properties of supercooled water and demonstrate the possibility of modeling these thermodynamic properties on a theoretical basis. We show that by assuming the existence of a liquid-liquid critical point in supercooled water, the theory of critical phenomena can give an accurate account of the experimental thermodynamic-property data up to a pressure of 150 MPa. In addition, we show that a phenomenological extension of the theoretical model can account for all currently available experimental data in the supercooled region, up to 400 MPa. The stability limit of the liquid state and possible coupling between crystallization and liquid-liquid separation are also discussed. It is concluded that critical-point thermodynamics describes the available thermodynamic data for supercooled water within experimental accuracy, thus establishing a benchmark for further developments in this area.

Comparison of complete scaling and a field-theoretic treatment of asymmetric fluid criticality

We investigate the connection between the theory of complete scaling and a field-theoretic (FT) treatment of asymmetric fluid criticality. To facilitate the comparison, we develop an equation of state from a simplified form of the complete scaling transformations and systematically compare this equation of state with the equation of state generated by a FT treatment of an asymmetric Landau-Ginzburg-Wilson Hamiltonian. We find, with care in interpretation, that these two approaches may be read as equivalent up to terms involving an independent higher-order asymmetric correction-to-scaling exponent.

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.

Dielectric constant of fluids and fluid mixtures at criticality

The behavior of the dielectric constant epsilon of pure fluids and binary mixtures near liquid-gas and liquid-liquid critical points is studied within the concept of complete scaling of asymmetric fluid-fluid criticality. While mixing of the electric field into the scaling fields plays a role, pressure mixing is crucial as the asymptotic behavior of the coexistence-curve diameter in the epsilon-T plane is concerned. Specifically, it is found that the diameters, characterized by a |T-Tc|1-alpha singularity in the previous scaling formulation [J. V. Sengers, D. Bedeaux, P. Mazur, and S. C. Greer, Physica A 104, 573 (1980)], gain a more dominant |T-Tc|2beta term, whose existence is shown to be supported by literature experimental data. The widely known |T-Tc|1-alpha singularity of epsilon along the critical isopleth in the one-phase region is found to provide information on the effect of electric fields on the liquid-liquid critical temperature: from experimental data it is inferred that Tc usually decreases as the magnitude of the electric field is enhanced. Furthermore, the behavior of mixtures along an isothermal path of approach to criticality is also analyzed: theory explains why the observed anomalies are remarkably higher than those associated to the usual isobaric path.

Phase Separation in Solutions of Room Temperature Ionic Liquids in Hydrocarbons

The room temperature ionic liquids (RTIL) trihexyl-tetradecyl phosphonium chloride (P666 14Cl) and the bromide (P666 14Br) are soluble in hydrocarbons. The investigated solutions in heptane, octane, nonane and decane show liquid–liquid phase separation with an upper critical solution point at ambient temperatures at molar fractions near 0.03 of the salt. Phase diagrams are reported and analysed presuming Ising criticality. The critical temperatures and the critical densities increase with the chain length of the hydrocarbons, where the figures corresponding to the bromides are above that of the chlorides. Scaled by the critical data the phase diagrams show corresponding state behaviour. In accordance with the prediction of the restricted primitive model (RPM), which is a model fluid of equal sized, charged hard spheres in a dielectric continuum, the critical points are located at low temperature and low concentration, when the corresponding state variables of this model are used. However, the critical temperature T c * and the critical density ρc * are well below the figures of the RPM prediction. Comparison is made with the phase diagrams of alcohol solutions of imidazolium ionic liquids and with simulation results of the RPM.

Closed Solubility Loops in Liquid Mixtures

Based on the principle of isomorphic critical behavior of fluid mixtures we derive a simple equation for closed solubility loops in a temperature–concentration diagram at constant pressure. The validity of the simple equation is tested by comparisons with experimental solubility data for liquid mixtures at atmospheric pressure and at elevated pressures. We also elucidate the nature of liquid–liquid phase separation in the vicinity of a double critical point. Specifically, we show how from an expansion around the critical double point one can develop a description of solubility loops as a function of pressure.

Thermodynamic consistency near the liquid-liquid critical point

The thermodynamic consistency of the isobaric heat capacity per unit volume at constant composition C(p,x) and the density rho near the liquid-liquid critical point is studied in detail. To this end, C(p,x)(T), rho(T), and the slope of the critical line (dT/dp)(c) for five binary mixtures composed by 1-nitropropane and an alkane were analyzed. Both C(p,x)(T) and rho(T) data were measured along various quasicritical isopleths with a view to evaluate the effect of the uncertainty in the critical composition value on the corresponding critical amplitudes. By adopting the traditionally employed strategies for data treatment, consistency within 0.01 K MPa(-1) (or 8%) is attained, thereby largely improving the majority of previous results. From temperature range shrinking fits and fits in which higher-order terms in the theoretical expressions for C(p,x)(T) and rho(T) are included, we conclude that discrepancies come mainly from inherent difficulties in determining the critical anomaly of rho accurately: specifically, to get full consistency, higher-order terms in rho(T) are needed; however, the various contributions at play cannot be separated unambiguously. As a consequence, the use of C(p,x)(T) and (dT/dp)(c) for predicting the behavior of rho(T) at near criticality appears to be the best choice at the actual experimental resolution levels. Furthermore, the reasonably good thermodynamic consistency being encountered confirms that previous arguments appealing to the inadequacy of the theoretical expression relating C(p,x) and rho for describing data in the experimentally accessible region must be fairly rejected.

Bubble and droplet motion in binary mixtures: Evaporation-condensation mechanism and Marangoni effect

Bubble and droplet motion in binary mixtures is studied in weak heat and diffusion fluxes and in gravity by solving the linearized hydrodynamic equations supplemented with appropriate surface boundary conditions. Without gravity, the velocity field is induced by evaporation and condensation at the interface and by the Marangoni effect due to a surface-tension gradient. In pure fluids, the latter nearly vanishes since the interface temperature tends to the coexistence temperature T_{cx}(p) even in heat flow. In binary mixtures, the velocity field can be much enhanced by the Marangoni effect above a crossover concentration c;{ *} inversely proportional to the radius R of the bubble or droplet. Here c;{ *} is usually very small for large R for non-azeotropic mixtures. The temperature and concentration deviations are also calculated.

Principle of isomorphism and complete scaling for binary-fluid criticality

The extension of the principle of critical-point universality to binary fluid mixtures, known as isomorphism of critical phenomena, has been reformulated in terms of complete scaling, a concept that properly matches asymmetric fluid-phase behavior with the symmetric Ising model. The controversial issue of the proper definition of the order parameter in binary fluid mixtures is clarified. We show that asymmetry of liquid-liquid coexistence in terms of mole fractions originates from two different sources: one is associated with a correlation between concentration and entropy fluctuations, whereas the other source is the correlation between concentration and density fluctuations. By analyzing the coexistence curves of liquid solutions of nitrobenzene in a series of hydrocarbons (from n -pentane to n -hexadecane), we have separated these two sources of asymmetry and found that the leading nonanalytical contribution to the asymmetry correlates linearly with the solute-solvent molecular-volume ratio. Other thermodynamic consequences of complete scaling for binary mixtures, such as an analog of the Yang-Yang anomaly in the behavior of the heat capacity and a curvature correction to the interfacial tension, are also discussed.

Liquid-liquid phase transition in solutions of ionic liquids with halide anions: Criticality and corresponding states

Measurements of the liquid-liquid phase diagrams of solutions of the ionic liquids (ILs) 1-dodecyl-3-methylimidazolium chloride (C12mimCl) in arenes (benzene, toluene, o-xylene, tetraline) and 1-tetradecyl-3-methylimidazolium chloride (C14mimCl) in CCl4 are reported and compared with those of solutions of trihexyl-tetradecyl-phosphonium halides (P666 14Cl, P666 14Br) in hydrocarbons and 1-alkyl-3-methylimidazolium tetrafluoroborates (CnmimBF4) in alcohols and water. The phase diagrams of solutions of tetrapentyl-ammonium bromide (N5555Br) in water and KI in SO2 are also discussed. Except for the KI/SO2 system, which features a lower critical solution point (LCSP), all systems have an upper critical solution point (UCSP) and show corresponding-states behavior. The experimental data are compared with results from simulations and theory concerning the model fluid of charged hard spheres in a dielectric continuum, termed restricted primitive model (RPM). The analysis in terms of of RPM variables shows agreement with the location of the critical point (CP) of the model with noticeable systematic deviations. However, for protic solvents, the CP becomes an LCSP, while in aprotic solvents the CP is a UCSP as expected for Coulomb systems. This indicates that in aprotic solvents, the phase transition is essentially determined by the Coulomb interactions, while in the solutions in protic solvents with hydrogen bonds, both Coulomb and solvophobic interactions are important.

Fluid-phase behavior of binary mixtures in which one component can have two critical points

We investigate theoretically the binary fluid-phase behavior of mixtures in which one water-like component can have two critical points. We consider three equal-sized nonpolar solutes that differ in the strength of their dispersive interactions (a1 < a2 < a3, where a denotes the van der Waals attractive parameter). In each case, we compare the phase behavior predicted using two sets of parameters for water: one giving rise to a pure component low-temperature liquid-liquid transition terminating at a critical point (two-critical-point parameter set), and one in which no such second critical point exists (singularity-free parameter set). Regardless of the parameter values used, we find five mixture critical lines. Using the two-critical-point parameter set, we find that a critical line originates at water's second critical point for aqueous mixtures involving solutes 1, 2, or 3. For mixtures involving solutes 1 or 2, this line extends towards low pressures and high temperatures as the solute mole fraction increases, and is closely related to the critical line originating at water's ordinary vapor-liquid critical point: these two critical lines are loci of upper and lower consolute points corresponding to the same liquid-liquid transition. In mixtures involving solute 2, the critical locus emanating from water's second critical point is shifted to higher temperatures compared to mixtures involving solute 1, and extends up to T approximately 310 K at moderate pressures (ca. 200 bars). This suggests the possibility of an experimentally accessible manifestation of the existence of a second critical point in water. For binary mixtures involving solutes 1 or 2, changing the water parameters from the two critical points to the singularity-free case causes the disappearance of a lower consolute point at moderate pressures. For binary mixtures involving solute 3, the differences between two-critical-point and singularity-free behaviors occur only in the experimentally difficult-to-probe low-temperature and high-pressure region.

Nature of vapor-liquid asymmetry in fluid criticality

We have investigated the nature and experimental consequences of vapor-liquid asymmetry in near-critical fluids within the framework of "complete scaling" [M. E. Fisher and G. Orkoulas, Phys. Rev. Lett. 85, 696 (2000); Y. C. Kim, Phys. Rev. E 67, 061506 (2003)]. We used the thermodynamic freedom for a choice of the critical-entropy value to simplify "complete scaling" to a form with only two independent parameters, responsible for two different sources of the asymmetry. We then developed a procedure to obtain these two parameters from mean-field equations of state. By combining accurate liquid-vapor coexistence and heat-capacity data, we have unambiguously separated two nonanalytic contributions from the two sources of vapor-liquid asymmetry and proved the validity of "complete scaling." Since the nonanalytic asymmetry effects in the critical region are fully determined by the Ising critical exponents for the symmetric lattice-gas model, there is no need for a special renormalization-group theoretical treatment of "non-Ising" asymmetry in fluid criticality.

Divergence of Tolman's length for a droplet near the critical point

The application of "complete scaling" [Kim et al., Phys. Rev. E 67, 061506 (2003); Anisimov and Wang, Phys. Rev. Lett. 97, 025703 (2006)] to the interfacial behavior of fluids shows that Tolman's length, a curvature correction to the surface tension, diverges at the critical point of fluids much more strongly than is commonly believed. The amplitude of the divergence depends on the degree of asymmetry in fluid phase coexistence. In highly asymmetric fluids and fluid mixtures the Tolman length may become large enough to significantly affect the interfacial behavior.

Multiexponential decay autocorrelation function in dynamic light scattering in near-critical ternary liquid mixture

The dynamic structure factor of a ternary liquid mixture is calculated from the theory of thermodynamic fluctuations with the help of linearized hydrodynamic equations. The theoretical model allows evaluating and classifying the transport properties near a critical solution point of a ternary mixture. In the vicinity of the critical solution point, experimental dynamic light scattering measurements reveal two hydrodynamic relaxation modes with well-separated characteristic relaxation times. From the autocorrelation functions, we can determine two effective diffusivities D(1) and D(2). As theoretically predicted by a model developed in this work, one of these two modes can be associated with thermal diffusion and the other with mass diffusion. In the special case of an incompressible liquid mixture limit, D(1) and D(2) are decoupled, becoming thermodiffusion coefficient D(T) and mutual mass diffusion coefficient D(ij). A possible physical meaning of D(1) and D(2) for a ternary mixture is discussed.

Global Fluid Phase Equilibria and Critical Phenomena of Selected Mixtures Using the Crossover Soft-SAFT Equation

We present here the extension of the crossover soft-statistical associating fluid theory (soft-SAFT) equation of state to mixtures, as well as some illustrative applications of the methodology to mixtures of particular scientific and technological interest. The procedure is based on White's work (White, J. A. Fluid Phase Equilib. 1992, 75, 53) from the renormalization group theory, as for the pure fluids, with the isomorphism assumption applied to the mixtures. The equation is applied to three groups of mixtures: selected mixtures of n-alkanes, the CO2/n-alkane homologous series, and the CO2/1-alkanol homologous series. The crossover equation is first applied to the pure components of the mixtures, CO2 and the 1-alkanol family, while an available correlation is used for the molecular parameters of the n-alkane series (Llovell et al. J. Chem. Phys 2004, 121, 10715). A set of transferable molecular parameters is provided for the 1-alkanols series; these are accurate for the whole range of thermodynamic conditions. The crossover soft-SAFT equation is able to accurately describe these compounds near to and far from the critical point. The theory is then used to represent the phase behavior and the critical phenomena of the selected mixtures. We use binary interaction parameters xi and eta for dissimilar mixtures. These parameters are fitted at some particular conditions (one subcritical temperature or binary critical data) and used to predict the behavior of the mixture at different conditions (other subcritical conditions and/or critical conditions). The equation is able to capture the continuous change in the critical behavior of the CO2/n-alkane and the CO2/1-alkanol homologous series as the chain length of the second compound increases. Excellent agreement with experimental data is obtained, even in the most nonideal cases. The new equation is proved to be a powerful tool to study the global phase behavior of complex systems, as well as other thermodynamic properties of very challenging mixtures.

Determination of fluid-phase behavior using transition-matrix Monte Carlo: Binary Lennard-Jones mixtures

We present a novel computational methodology for determining fluid-phase equilibria in binary mixtures. The method is based on a combination of highly efficient transition-matrix Monte Carlo and histogram reweighting. In particular, a directed grand-canonical transition-matrix Monte Carlo scheme is used to calculate the particle-number probability distribution, after which histogram reweighting is used as a postprocessing procedure to determine the conditions of phase equilibria. To validate the methodology, we have applied it to a number of model binary Lennard-Jones systems known to exhibit nontrivial fluid-phase behavior. Although we have focused on monatomic fluids in this work, the method presented here is general and can be easily extended to more complex molecular fluids. Finally, an important feature of this method is the capability to predict the entire fluid-phase diagram of a binary mixture at fixed temperature in a single simulation.

Griffiths-Wheeler geometrical picture of critical phenomena: experimental testing for liquid-liquid critical points

An experimental approach to the verification of specific relations between thermodynamic properties as predicted from the Griffiths-Wheeler theory of critical phenomena in multicomponent systems is developed for the particular case of ordinary liquid-liquid critical points of binary mixtures. Densities rho(T) , isobaric heat capacities per unit volume C(p)(T) , and previously reported values of the slope of the critical line (dT/dp)c for five critical mixtures are used to check the thermodynamic consistency of C(p) and rho near the critical point. An appropriate treatment of rho (T) data is found to provide the key solution to this issue. In addition, various alternative treatments for C(p)(T) data provide values for both the critical exponent alpha and the ratio between the critical amplitudes of the heat capacity A+/A- that are in agreement with their widely accepted counterparts, whereas two-scale-factor universality is successfully verified in one of the systems studied.

Nature of double critical points in binary solutions

The nature of the reentrant demixing transition in binary solutions with H bonds is studied in the framework of an Ising-like Hamiltonian with effective spin-spin interaction constant. It is taken into account that the internal variables describing the H-bond network are characterized by the spatial time scales essentially shorter than those for the spin variables. Due to this the contribution of H bonds to the effective spin-spin interaction constant is described by thermodynamic methods. With the help of the catastrophe theory the classification of possible types of phase diagrams leading to the double critical point is given. The influence of small quantities of third component (electrolytes, water, and molecules similar to CCl4) is discussed.