11 Critical Region

Perturbation theories for fluids with short-ranged attractive forces: A case study of the Lennard-Jones spline fluid

It is generally not straightforward to apply molecular-thermodynamic theories to fluids with short-ranged attractive forces between their constituent molecules (or particles). This especially applies to perturbation theories, which, for short-ranged attractive fluids, typically must be extended to high order or may not converge at all. Here, we show that a recent first-order perturbation theory, the uv-theory, holds promise for describing such fluids. As a case study, we apply the uv-theory to a fluid with pair interactions defined by the Lennard-Jones spline potential, which is a short-ranged version of the LJ potential that is known to provide a challenge for equation-of-state development. The results of the uv-theory are compared to those of third-order Barker–Henderson and fourth-order Weeks–Chandler–Andersen perturbation theories, which are implemented using Monte Carlo simulation results for the respective perturbation terms. Theoretical predictions are compared to an extensive dataset of molecular simulation results from this (and previous) work, including vapor–liquid equilibria, first- and second-order derivative properties, the critical region, and metastable states. The uv-theory proves superior for all properties examined. An especially accurate description of metastable vapor and liquid states is obtained, which might prove valuable for future applications of the equation-of-state model to inhomogeneous phases or nucleation processes. Although the uv-theory is analytic, it accurately describes molecular simulation results for both the critical point and the binodal up to at least 99% of the critical temperature. This suggests that the difficulties typically encountered in describing the vapor–liquid critical region are only to a small extent caused by non-analyticity.

Accurate first-order perturbation theory for fluids: uf-theory

We propose a new first-order perturbation theory that provides a near-quantitative description of the thermodynamics of simple fluids. The theory is based on the ansatz that the Helmholtz free energy is bounded below by a first-order Mayer-f expansion. Together with the rigorous upper bound provided by a first-order u-expansion, this brackets the actual free energy between an upper and (effective) lower bound that can both be calculated based on first-order perturbation theory. This is of great practical use. Here, the two bounds are combined into an interpolation scheme for the free energy. The scheme exploits the fact that a first-order Mayer-f perturbation theory is exact in the low-density limit, whereas the accuracy of a first-order u-expansion grows when density increases. This allows an interpolation between the lower "f"-bound at low densities and the upper "u" bound at higher liquid-like densities. The resulting theory is particularly well behaved. Using a density-dependent interpolating function of only two adjustable parameters, we obtain a very accurate representation of the full fluid-phase behavior of a Lennard-Jones fluid. The interpolating function is transferable to other intermolecular potential types, which is here shown for the Mie m-6 family of fluids. The extension to mixtures is simple and accurate without requiring any dependence of the interpolating function on the composition of the mixture.

Finite-size scaling study of dynamic critical phenomena in a vapor-liquid transition

Via a combination of molecular dynamics (MD) simulations and finite-size scaling (FSS) analysis, we study dynamic critical phenomena for the vapor-liquid transition in a three dimensional Lennard-Jones system. The phase behavior of the model has been obtained via the Monte Carlo simulations. The transport properties, viz., the bulk viscosity and the thermal conductivity, are calculated via the Green-Kubo relations, by taking inputs from the MD simulations in the microcanonical ensemble. The critical singularities of these quantities are estimated via the FSS method. The results thus obtained are in nice agreement with the predictions of the dynamic renormalization group and mode-coupling theories.

Study of critical dynamics in fluids via molecular dynamics in canonical ensemble

With the objective of understanding the usefulness of thermostats in the study of dynamic critical phenomena in fluids, we present results for transport properties in a binary Lennard-Jones fluid that exhibits liquid-liquid phase transition. Various collective transport properties, calculated from the molecular dynamics (MD) simulations in canonical ensemble, with different thermostats, are compared with those obtained from MD simulations in microcanonical ensemble. It is observed that the Nosé-Hoover and dissipative particle dynamics thermostats are useful for the calculations of mutual diffusivity and shear viscosity. The Nosé-Hoover thermostat, however, as opposed to the latter, appears inadequate for the study of bulk viscosity.

Finite-size scaling study of shear viscosity anomaly at liquid-liquid criticality

We study the equilibrium dynamics of a symmetrical binary Lennard-Jones fluid mixture near its consolute criticality. Molecular dynamics simulation results for the shear viscosity, η, from a microcanonical ensemble are compared with those from a canonical ensemble with various thermostats. It is observed that the Nosé-Hoover thermostat is a good candidate for this purpose, and is therefore adopted for the quantification of the critical singularity of η, to avoid the temperature fluctuations (or even drifts) that are often encountered in microcanonical simulations. Via a finite-size scaling analysis of our simulation data we have been able to confirm that the shear viscosity exhibits a weak critical singularity in agreement with the theoretical predictions.

Two-state thermodynamics of the ST2 model for supercooled water

Thermodynamic properties of the ST2 model for supercooled liquid water exhibit anomalies similar to those observed in real water. A possible explanation of these anomalies is the existence of a metastable, liquid-liquid transition terminated by a critical point. This phenomenon, whose possible existence in real water is the subject of much current experimental work, has been unambiguously demonstrated for this particular model by most recent simulations. In this work, we reproduce the anomalies of two versions of the ST2 model with an equation of state describing water as a non-ideal "mixture" of two different types of local molecular order. We show that the liquid-liquid transition in the ST2 water is energy-driven. This is in contrast to another popular model, mW, in which non-ideality in mixing of two alternative local molecular orders is entropy-driven, and is not sufficiently strong to induce a liquid-liquid transition.

Simulation of transport around the coexistence region of a binary fluid

We use Monte Carlo and molecular dynamics simulations to study phase behavior and transport properties in a symmetric binary fluid where particles interact via Lennard-Jones potential. Our results for the critical behavior of collective transport properties, with particular emphasis on bulk viscosity, is understood via appropriate application of finite-size scaling technique. It appears that the critical enhancements in these quantities are visible far above the critical point. This result is consistent with an earlier report from computer simulations where, however, the authors do not quantify the critical singularity.

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.

Liquid-liquid phase separation in solutions of ionic liquids: phase diagrams, corresponding state analysis and comparison with simulations of the primitive model

Phase diagrams of ionic solutions of the ionic liquid C(18)mim(+)NTF(2)(-) (1-n-octadecyl-3-methyl imidazolium bistrifluormethylsulfonylimide) in decalin, cyclohexane and methylcyclohexane are reported and compared with that of solutions of other imidazolium ionic liquids with the anions NTF(2)(-), Cl(-) and BF4(-) in arenes, CCl(4), alcohols and water. The phase diagrams are analysed presuming Ising criticality and taking into account the asymmetry of the phase diagrams. The resulting parameters are compared with simulation results for equal-sized charged hard spheres in a dielectric continuum, the restricted primitive model (RPM) and the primitive model (PM) that allows for ions of different size. In the RPM temperature scale the critical temperatures vary almost linearly with the dielectric permittivity of the solvent. The RPM critical temperatures of the solutions in non-polar solvents are very similar, somewhat below the RPM value. Correlations with the boiling temperatures of the solvents and a dependence on the length of the side chain of the imidazolium cations show that dispersion interactions modify the phase transition, which is mainly determined by Coulomb forces. Critical concentrations, widths of the phase diagrams and the slopes of the diameter are different for the solutions in protic and aprotic solvents. The phase diagrams of the solutions in alcohols and water get a lower critical solution point when represented in RPM variables.

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.

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.

A general perturbation approach for equation of state development: Applications to simple fluids, ab initio potentials, and fullerenes

A new perturbation scheme based on the Barker-Henderson perturbation theory [J. Chem. Phys. 47, 4714 (1967)] is proposed to predict the thermodynamic properties of spherical molecules. Accurate predictions of second virial coefficients and vapor-liquid coexistence properties are obtained for a large variety of potential functions (square well, Yukawa, Sutherland, Lennard-Jones, Buckingham, Girifalco). New Gibbs ensemble Monte Carlo simulations of the generalized exp-m Buckingham potential are reported. An extension of the perturbation approach to mixtures is proposed, and excellent predictions of vapor-liquid equilibria are obtained for Lennard-Jones mixtures. The perturbation scheme can be applied to complex potential functions fitted to ab initio data to predict the properties of real molecules such as neon. The new approach can also be used as an auxiliary tool in molecular simulation studies, to efficiently optimize an intermolecular potential on macroscopic properties or match force fields based on different potential functions.

Competition of mesoscales and crossover to theta-point tricriticality in near-critical polymer solutions

The approach to asymptotic critical behavior in polymer solutions is governed by a competition between the correlation length of critical fluctuations diverging at the critical point of phase separation and an additional mesoscopic length scale, the radius of gyration. In this paper we present a theory for crossover between two universal regimes: a regime with Ising (fluctuation-induced) asymptotic critical behavior, where the correlation length prevails, and a mean-field tricritical regime with theta-point behavior controlled by the mesoscopic polymer chain. The theory yields a universal scaled description of existing experimental phase-equilibria data and is in excellent agreement with our light-scattering experiments on polystyrene solutions in cyclohexane with polymer molecular weights ranging from 2 x 10(5) up to 11.4 x 10(6). The experiments demonstrate unambiguously that crossover to theta-point tricriticality is controlled by a competition of the two mesoscales. The critical amplitudes deduced from our experiments depend on the polymer molecular weight as predicted by de Gennes [Phys. Lett. 26A, 313 (1968)]. Experimental evidence for the presence of logarithmic corrections to mean-field tricritical theta-point behavior in the molecular-weight dependence of the critical parameters is also presented.

Reaction kinetics and critical phenomena: rates of some first order gas evolution reactions in binary solvents with a consolute point

We have measured the rate of carbon dioxide evolution in the aniline catalyzed decomposition of acetone dicarboxylic acid in a mixture of isobutyric acid + water near its consolute point. Within a temperature interval of 1 degrees C, which included the critical solution temperature, the first-order rate constant oscillated in magnitude by about 10% as it passed through three complete cycles of slowing down followed by speeding up. Whereas we can find no ready explanation for the speeding up, we suggest that, because the mixture contained no inert components, the slowing down should belong to the Griffiths-Wheeler class of strong critical effects [Phys. Rev. A 1970, 2, 1047]. As a check on this conclusion, we have measured the rate of the SN1 decomposition of benzene diazonium tetrafluoroborate in 2-butoxyethanol + water near the lower critical solution temperature and also the rate of the acid-catalyzed decomposition of ethyl diazoacetate in isobutyric acid + water near the upper critical solution temperature. Both of these reactions evolve nitrogen. In the first reaction, 2-butoxyethanol is inert, whereas in the second, isobutyric acid is inert. In both cases, because there was one inert component, we regarded the response of the rate constant to temperature in the critical region to be representative of the Griffiths-Wheeler class of weak critical effects. Within our accuracy of measurement of about 2% in the rate constant and about 1 mK in the temperature, we could detect no effect of the critical point on the rates of either of these reactions, suggesting that a weak effect may be too small to be seen with our experimental apparatus. The successful observation of a critical effect in the rate of decomposition of acetone dicarboxylic acid proves, however, that kinetic critical phenomena are observable in heterogeneous reactions.

Supercritical Water as a Solvent

Water is not restricted to moderate temperatures and low pressures, but can exist up to very high temperatures, far above its critical point at 647 K. In this supercritical regime, water can be gradually compressed from gas-like to liquid-like densities. The resulting dense supercritical states have extraordinary properties which can be tuned by temperature and pressure, and form the basis for innovative technologies. This Review covers the current knowledge of the major properties of supercritical water and its solutions with nonpolar, polar, and ionic compounds, and of the underlying molecular processes.

Collective and self-diffusion coefficients in an ionic critical mixture: 3-methylpyridine+water+NaBr

The dynamics of concentration fluctuations of three critical samples of the 3-methylpyridine (3MP)+water+NaBr system have been measured by photon correlation spectroscopy. The collective-diffusion coefficient D shows the usual Ising behavior near the critical temperature T(c). However, as |T-T(c)| increases, the dynamic correlation length calculated from D, xi, takes values higher than the correlation length of the critical fluctuations calculated from static light scattering, xi(s). At the largest |T-T(c)| measured, xi approaches the value, xi(0,d) approximately equal to 1.13 nm, while the amplitude of xi(s) is xi(0,s)=0.38 nm. Pulsed-gradient NMR spectroscopy points out the existence of two dynamic contributions. One of them is consistent with the existence of molecular entities of hydrodynamic radius 0.31 nm, while the other one indicates the existence of aggregates rich in 3MP of radius 1.16 nm. The existence of the aggregates may explain the apparent anomalous behavior of the dynamic light scattering experiments for this system far from the critical point.