On the universal behavior of some thermodynamic properties along the whole liquid-vapor coexistence curve

When thermodynamic properties of a pure substance are transformed to reduced form by using both critical- and triple-point values, the corresponding experimental data along the whole liquid-vapor coexistence curve can be correlated with a very simple analytical expression that interpolates between the behavior near the triple and the critical points. The leading terms of this expression contain only two parameters: the critical exponent and the slope at the triple point. For a given thermodynamic property, the critical exponent has a universal character but the slope at the triple point can vary significantly from one substance to another. However, for certain thermodynamic properties including the difference of coexisting densities, the enthalpy of vaporization, and the surface tension of the saturated liquid, one finds that the slope at the triple point also has a nearly universal value for a wide class of fluids. These thermodynamic properties thus show a corresponding apparently universal behavior along the whole coexistence curve.     

Parachors of liquid/vapor systems: A set of critical amplitudes

In light of the available experimental data and of our current understanding of liquid–vapor critical phenomena, we examine the values of the parachors and of the parachor exponent, which are commonly used to estimate surface tension from the density difference between coexisting liquid and vapor phases. This is a controversial issue, as values for the parachor exponent ranging from 3.5 to 4 have been proposed in the literature. The parachor exponent and parachors can be viewed as a critical exponent and critical amplitudes, respectively. The Ising value, equal to 3.88, should be observed for the exponent “close enough” to the liquid/vapor critical point, i.e., for “low enough” tensions and densities. However, a review of experimental data for several fluids suggests an effective value in the range of 3.6, in line with the effective values observed for the exponents that describe the vanishing of the density difference and capillary length with the distance to the critical temperature. In fact, the asymptotic Ising regime is not reached experimentally, as confirmed by an estimation of the parachors very near the critical point. Those (Ising) parachors can be inferred from other critical amplitudes corresponding to bulk properties by using two-scale factor universality. Their values exceed those deduced from off-critical tension and density data by more than 10%, corresponding to surface tension differences larger than 50%. We argue that effective parachors (i.e., corresponding to an exponent in the range of 3.6) can be utilized in combination with two-scale-factor universality for determining the critical behavior of fluid systems in an extended range around their liquid/vapor critical point.     

Phase diagram and universality of the Lennard-Jones gas-liquid system

The gas-liquid phase transition of the three-dimensional Lennard-Jones particles system is studied by molecular dynamics simulations. The gas and liquid densities in the coexisting state are determined with high accuracy. The critical point is determined by the block density analysis of the Binder parameter with the aid of the law of rectilinear diameter. From the critical behavior of the gas-liquid coexisting density, the critical exponent of the order parameter is estimated to be β = 0.3285(7). Surface tension is estimated from interface broadening behavior due to capillary waves. From the critical behavior of the surface tension, the critical exponent of the correlation length is estimated to be ν = 0.63(4). The obtained values of β and ν are consistent with those of the Ising universality class.     

Static and dynamic thermal quantities near the consolute point of the binary liquid mixture aniline-cyclohexane studied with a photopyroelectric technique and adiabatic scanning calorimetry

We studied the thermal conductivity, thermal effusivity, and specific heat capacity at constant pressure of the critical binary liquid mixture aniline-cyclohexane near the consolute point, using a photopyroelectric (PPE) technique and adiabatic scanning calorimetry (ASC). According to recent theoretical predictions based on renormalization group theory calculations, a substantial (but not diverging) enhancement in the thermal conductivity in the homogeneous phase near the critical temperature was expected for this binary system near the consolute point. However, within an experimental precision of 0.05%, we found no deviation from linear behavior in the range of 5 K above Tc down to Tc. The specific heat capacity calculated from the results for the thermal conductivity and effusivity is in good agreement with that measured by ASC. For the ASC results, the theoretical power law expression with the Ising critical exponent was fitted to the specific heat capacity both above and below the transition temperature. Good agreement with theory was found both for the amplitude ratio and the two-scale universality.     

Universality in eight-arm star polystyrene and methylcyclohexane mixtures near the critical point

Measurements of the coexistence curve and turbidity were made on different molecular mass samples of the branched polymer-solvent system eight-arm star polystyrene in methylcyclohexane near its critical point. We confirmed that these systems belong in the Ising universality class. The location of the critical temperature and composition as well as the correlation length, susceptibility, and coexistence curve amplitudes were found to depend on molecular mass and the degree of branching. The coexistence curve diameter had an asymmetry that followed a "complete scaling" approach. All the coexistence curve data could be scaled onto a common curve with one adjustable parameter. We found the coexistence curve amplitude to be about 12% larger for branched than linear polystyrenes of the same molecular mass in either solvent cyclohexane or methylcyclohexane. The two-scale-factor universality ratio R was found to be independent of molecular mass or degree of branching.     

Critical behaviour of binary mixture of {xC6H5CN + (1 − x)CH3(CH2)7CH3}: Measurements of coexistence curves,light scattering,and heat capacity

Liquid + liquid coexistence, light scattering, and isobaric heat capacity per unit volume for the critical solutions of (benzonitrile + n-nonane) have been measured. The critical exponents relating to the coexistence curve β, the osmotic compressibility γ, the correlation length ν, and the heat capacity α have been deduced and the values are consistent with the 3D-Ising values in the range close to the critical point. The experimental results of the liquid + liquid coexistence were analyzed to examine the Wegner correction terms and the behaviour of the diameter of the coexistence curves. The light scattering data were well described by the crossover model proposed by Anisimov and Sengers, and showed a tendency of monotonic crossover of the critical exponents γ and ν from the 3D-Ising values to the mean-field values as the temperature departures from the critical point. From calorimetric measurements, the amplitude A^± and the critical background B_cr of the heat capacity in the critical region have been deduced and some universal ratios are tested.     

(Liquid + liquid) phase equilibrium and critical behavior of binary solution {heavy water + 2,6-dimethylpyridine}

The (liquid + liquid) coexistence curves, the isobaric heat capacities per unit volume and the turbidities for the binary solution of {heavy water + 2,6-dimethylpyridine} have been precisely measured. The values of the critical exponents were obtained, which confirmed the 3D-Ising universality. It was found that the critical temperature dropped by 5.9 K and the critical amplitude of the coexistence curve significantly increased as compared to the binary solution of {water + 2,6-dimethylpyridine}. The complete scaling theory was applied to well describe the asymmetric behavior of the diameter of the coexistence curve as the heat capacity contribution was considered. Moreover, the values of the critical amplitudes of the correlation length and the osmotic compressibility were deduced, which together with the critical amplitudes of the coexistence curve and the heat capacity to test universal amplitude ratios.     

On approximate mathematical modeling of the vapor–liquid coexistence curves by the Van der Waals equation of state and non-classical value of the critical exponent

Van der Waals equation of state as well as power laws and critical exponent theories are prototypes to study the cubic shape, asymmetries and “flatness” of the vapor–liquid equilibrium curves near the critical point. In this work we study two similar methods to determine the phase curves in analytical form, which differ from each other by simplicity of mathematical calculation. We analyze temperature dependence of the coexistence curves asymptotically close to the vapor–liquid critical point. We explain the novelty of our method with respect to the standard thermodynamic limit discussed in the literature. Therefore we show that the shape of the coexistence curves can strongly influence the accepted value of the critical exponent. The results of theoretical studies have been compared with the ones obtained by experimental methods.     

Critical Behavior of Binary Mixtures of Nitrobenzene?+?n-Undecane and Nitrobenzene?+?n-Dodecane

Turbidities and isobaric heat capacities per unit volume, in a wide temperature range, have been measured for critical binary solutions of {nitrobenzene?+?n-undecane} and {nitrobenzene?+?n-dodecane}. The critical exponents and the system-dependent critical amplitudes were deduced. The non-critical and the critical-fluctuation induced contributions to the background heat capacities were determined. We also obtained the coupling constant $ \bar{u} $ by analyzing the coexistence curve data with crossover theory. These parameters were used to test some universal amplitude ratios and together with the coexistence-curve data to test the complete scaling theory. It is shown that the contribution from heat capacity plays an important role in describing the asymmetric criticality of the coexistence curve.     

Monte Carlo simulations of squalane in the Gibbs ensemble

We investigate the liquid–vapour coexistence curve of 2,6,10,15,19,23-hexamethyltetracosane (squalane) near the critical point with a new Lennard–Jones parameter set and compare our results to existing simulation data as well as to recent experimental vapour pressure data. Comparison of the liquid–vapour coexistence curve to previous simulation data reveals that this new force field, which includes tail corrections to the truncation of the non-bonded interactions increases the liquid density. We determine the critical temperature to 829 K and 825 K (with roughly 1% error) for two different system sizes, 72 and 108 molecules, and the critical density to 0.211 g/cm³ and 0.228 g/cm³, respectively. We extrapolate experimental vapour pressure data by use of Antoine's law to the temperature range covered by simulation and yield good agreement between simulation and experiment. We note that the vapour pressure in simulation is essentially governed by the ideal vapour pressure.     

General correlation model for some physical properties of saturated pure fluids

In this work, we use a general expression to accurately correlate the liquid density, the vaporization enthalpy, the surface tension, and the isobaric heat capacity of a saturated liquid versus temperature along the whole coexistence curve. The general expression used is the same for the four thermodynamic properties, and uses both critical and triple point values as reference. As representative examples of the use of the model, results are given for a set of 22 pure substances. We find that this general expression correlates the data with smaller or similar overall deviations when compared with other published models whose number of coefficients are the same or greater.     

Critical behavior of 2,6-dimethylpyridine-water: measurements of specific heat, dynamic light scattering, and shear viscosity

The specific heat C(p) at constant pressure, the shear viscosity eta(s), and the mutual diffusion coefficient D of the 2,6-dimethylpyridine-water mixture of critical composition have been measured in the homogeneous phase at various temperatures near the lower critical demixing temperature T(c). The amplitude of the fluctuation correlation length xi(0)=(0.198+/-0.004) nm has been derived from a combined evaluation of the eta(s) and D data. This value is in reasonable agreement with the one obtained from the amplitude A(+)=(0.26+/-0.01) J(g K) of the critical term in the specific heat, using the two-scale-factor universality relation. Within the limits of error the relaxation rate Gamma of order parameter fluctuations follows power law with the theoretical universal exponent and with the amplitude Gamma=(25+/-1)x10(9) s(-1). No indications of interferences of the critical fluctuations with other elementary chemical reactions have been found. A noteworthy result is the agreement of the background viscosity eta(b), resulting from the treatment of eta(s) and D data, with the viscosity eta(s)(nu=0) extrapolated from high-frequency viscosity data. The latter have been measured in the frequency range of 5-130 MHz using a novel shear impedance spectrometer.     

Critical dynamics of the binary system nitroethane/3-methylpentane: relaxation rate and scaling function

Shear viscosity and dynamic light scattering measurements as well as ultrasonic spectrometry studies of the nitroethane/3-methylpentane mixture of critical composition have been performed at various temperatures near the critical temperature, T(c). A combined evaluation of the shear viscosity and mutual diffusion coefficient data yielded the amplitude, xi(0), of the fluctuation correlation length, xi, assumed to follow power law, and the relaxation rate, Gamma, or order parameter fluctuations. The latter was found to follow power law with the theoretical universal exponent. The amplitudes xi(0) = 0.23 +/- 0.02 nm and Gamma(0) = (125 +/- 5) x 10(9) s(-1) nicely agree with literature values. Using the relaxation rates resulting from the viscosity and diffusion coefficient data, the scaling function has been calculated assuming the ultrasonic spectra to be composed of a critical part and a noncritical background contribution. The experimental scaling function fits well to the predictions of the Bhattacharjee-Ferrell dynamic scaling model with scaled half-attenuation frequency, Omega(BF)1/2= 2.1. The amplitude of the sonic spectra yields the amount |g| = 0.26 of the adiabatic coupling constant, g, in fair agreement with -0.29 from another thermodynamic relation.     

The liquid–liquid coexistence curves of {x dimethyl adipate + (1 − x) n-octane} and {x dimethyl adipate + (1 − x) n-nonane} in the critical region

The liquid–liquid coexistence curves of (dimethyl adipate + n-octane) and (dimethyl adipate + n-nonane) have been determined within about 10 K from the critical temperatures, from which the critical amplitudes and the critical exponents are deduced. The critical exponents corresponding to the coexistence curve β are consistent with the 3D-Ising values. The experimental results have been analyzed to determine Wegner-correction terms and to discuss the asymmetric behaviour of the diameters of the coexistence curves by the complete scaling theory. Molar mass-dependences of the critical amplitude and the critical volume fraction have been shown to be consistent with the theoretical prediction.     

Dynamic scaling of the critical binary mixture methanol-hexane

Acoustical attenuation spectrometry, dynamic light scattering, shear viscosity, density, and heat capacity measurements of the methanol/n-hexane mixture of critical composition have been performed. The critical part in the sonic attenuation coefficients nicely fits to the empirical scaling function of the Bhattacharjee-Ferrell [Phys. Rev. A 24, 1643 (1981)] dynamic scaling model if the theoretically predicted scaled half-attenuation frequency Omega(12) (BF)=2.1 is used. The relaxation rates of order parameter fluctuations, as resulting from the acoustical spectra, within the limits of experimental error agree with those from a combined evaluation of the light scattering and shear viscosity measurements. Both series of data display power law with amplitude Gamma(0)=44x10(9) s(-1). The amplitude of the fluctuation correlation length follows as xi(0)=0.33 nm from the light scattering data and as xi(0)=0.32 nm from the amplitude of the singular part of the heat capacity if the two-scale factor universality relation is used. The adiabatic coupling constant g=0.11 results from the amplitude of the critical contribution to the acoustical spectrum near the critical point, in conformity with g=0.12 as following from the variation of the critical temperature with pressure along the critical line and the thermal expansion coefficient.     

The measurements of coexistence curves and light scattering for {xC6H5CN + (1 − x)CH3(CH2)16CH3} in the critical region

The coexistence curves and light scattering data for a critical solution of benzonitrile + octadecane have been reported. The critical exponents relating to the difference in density variables of two coexisting phases β, the correlation length ν, and the osmotic compressibility γ have been calculated. The experimental results of the coexistence curves have also been analyzed to examine the Wegner correction terms and the behavior of the diameter of the coexistence curves. The data analysis shows that the 3D-Ising behavior is valid in a temperature range close to the critical point. However in a wide temperature range the exponential values of ν and γ change with the temperature significantly, clearly exhibiting the critical crossover from the 3D-Ising universality class to the classical one.     

Spreading of liquid drops over dry surfaces

Spreading of thin, axisymmetric, non-volatile, Newtonian liquid drops over a dry, smooth, flat solid surface is considered both theoretically and experimentally in the case of complete wetting. The drop profile is solved analytically by matching the “outer” solution for large film thicknesses, where only the capillary effects are important, with the “inner” solution for small film thicknesses, where the viscous and disjoining pressure effects are comparable to capillary effects. It is shown that the apparent radius of the wetted spot, the apex height of the drop, and the apparent advancing dynamic contact angle follow different power laws in time and the advancing dynamic contact angle follows a power law in capillary number. Both the prefactor and the exponent of each power law are derived theoretically. Good agreement between the theory predictions and experimental measurements is shown for both the prefactor and exponent of each power law. It is necessary to emphasize that the theory suggested does not include any fitting parameters.     

The liquid–liquid coexistence curves of {x dimethyl adipate + (1 − x) n-hexane} and {x dimethyl adipate + (1 − x) n-heptane} in the critical region

The liquid–liquid coexistence curves for (dimethyl adipate + n-hexane), (dimethyl adipate + n-heptane) have been measured, from which the critical amplitudes and the critical exponents are deduced. The critical exponent β corresponding to the coexistence curves are consistent with the 3D-Ising value. The experimental results have also been analyzed to determine the critical amplitudes of Wegner-correction terms when β and Δ are fixed at their theoretical values, and to examine the asymmetry of the diameters for the coexistence curves.     

The evolution of viscoelasticity near the gel point of end-linking poly(dimethylsiloxane)s

The linear viscoelasticity of polymers near the gel point can be described by two scaling laws. The material at the gel point has a power-law linear viscoelastic relaxation modulus, and the relaxation exponent has been found to vary with the composition of the precursor materials, i.e., it is not universal for gelation. A second scaling law describes the evolution of the linear viscoelastic properties through the gel point. The rate of change of the dynamic mechanical modulus/viscosity is observed to scale as a power-law function of frequency. This power-law function defines a dynamic critical exponent, and this has been found to be independent of precursor composition for end-linking poly(dimethylsiloxane) polymers and equal to κ = 0.21 ± 0.02. This exponent may be a universal measure of gelation. The technique of Time Resolved Mechanical Spectroscopy is used to observe the evolution of linear viscoelastic properties of crosslinking polymers in situ in the rheometer. A stretched exponential relaxation modulus describes the evolution of mechanical properties in the vicinity of the gel point very well. The exponents which characterize the divergence of the zero-shear viscosity and the equilibrium modulus are not universal, since they are related to the relaxation exponent and the dynamic critical exponent.