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.     

毛细管口气泡的曲率半径和附加压力变化

The profiles of the curved surface at critical status during formation of a bubble are drawn schematically, and the dependence of the radius of curvature on time is obtained in this work. The curve is useful for the students to better understand the evolution of radius of curvature and excess pressure in the laboratory experiment of surface tension measurements by the bubble pressure method, and helpful to interpret the exercises regarding excess pressure.     

First correction to surface tension for the curvature of an interface

Tolman parameter δ_T, which determines the first correction to surface tension for the interface curvature, is calculated in molecular-dynamic experiments performed for Lennard-Jones fluid within the temperature range from the critical to the triple point (and slightly below). It is shown that parameter δ_T is positive and slightly depends on temperature; its absolute magnitude is no larger than 0.1–0.2 molecular diameters and does not coincide with the distance between the equimolecular dividing surface and the surface of tension in a flat interfacial layer, as calculated through the first moment of the pressure tensor. The results of the moleculardynamic experiments are compared with the δ_T values calculated in terms of the extended version of the van der Waals theory of capillarity. It is established that taking into account terms of higher orders than the squared density gradient in the expansion of the free energy of an inhomogeneous system does not reverse the negative sign at δ_T and slightly affects its value.     

Scaled equations for the coexistence curve,the capillary constant and the surface tension of n-alkanes

A review of experimental data of several fluids shows that their coexistence curve follows a power law in reduced temperature at the approach of the critical point, with an universal exponent equal to 0.325, their capillary constant a power law with an universal exponent equal to 0.925 and their surface tension a power law with an universal exponent equal to 1.26. In the critical region, the concept of two-scale-factor universality was used to predict the density difference amplitude, the capillary constant amplitude, and the surface tension amplitude between near critical vapor and liquid phases. A comparison with amplitudes determined from experimental data is given. In order to extend this universality all along the liquid–gas coexistence curve from the triple point to the critical point for n-alkanes, a mean field approximation was used far away from T_C. We show that the density difference, the capillary constant and the surface tension can be calculated with a reasonable accuracy by generalized scaled equations adding only two empirical constants. A comparison between calculated and experimental data is presented.     

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.     

Isotherms of phospholipid monolayers measured by a pendant drop technique

The Axisymmetric Drop Shape Analysis (ADSA) has been used to study the surface pressure/area isotherms of insoluble surfactant monolayers. The continuous measurement of surface tension as a function of surface area by increasing and decreasing the drop volume allows to investigate the phase transitions in monolayers. The isotherms of two phospholipids, dipalmitoyl phosphatidyl choline (DPPC) and dimyristoyl phosphatidyl ethanolamine (DMPE), show good agreement with those measured by using a conventional Langmuir-Blodgett film balance, except in the coexistence region. The observed disagreements are discussed in terms of differences in compression rate, curvature of the surface and effect of impurities. Evidence of possible geometric effects on monolayer domain formation and growth is given on the basis of BAM images.Due to the small total surface area, the ADSA technique provides advantages as regards homogeneity of temperature, surface pressure, surface concentration and the symmetry of area changes.     

Variations of surfactant monolayer surface density in induced steady wave regime

Fluctuations of an insoluble surfactant concentration along the free liquid surface induced by steady surface waves are considered theoretically. The energy of a waved surface is assumed to consist of surface tension, curvature, and van der Waals energy components. Dependencies of the surface tension and the bending stiffness versus the surfactant concentration are assumed to be linear relative to some reference level. The van der Waals energy is taken in the form of interaction term for a thin film. Minimization of the total energy allows the expression for the deviations of concentration to be obtained. The distribution of a surfactant concentration relative to some reference level has been found to be periodic, with a period that is half of the wave period, and the amplitude of oscillations is a function of a wave number that is very similar to the Landau expansion of the free-energy near the critical point in phase transitions.     

Computation of constant mean curvature surfaces: Application to the gas-liquid interface of a pressurized fluid on a superhydrophobic surface

The interface shape separating a gas layer within a superhydrophobic surface consisting of a square lattice of posts from a pressurized liquid above the surface is computed numerically. The interface shape is described by a constant mean curvature surface that satisfies the Young-Laplace equation with the three-phase gas-liquid-solid contact line assumed pinned at the post outer edge. The numerical method predicts the existence of constant mean curvature solutions from the planar, zero curvature solution up to a maximum curvature that is dependent on the post shape, size and pitch. An overall force balance between surface tension and pressure forces acting on the interface yields predictions for the maximum curvature that agree with the numerical simulations to within one percent for convex shapes such as circular and square posts, but significantly over predicts the maximum curvature for non-convex shapes such as a circular post with a sinusoidal surface perturbation. Changing the post shape to increase the contact line length, while maintaining constant post area, results in increases of 2 to 12% in the maximum computable curvature for contact line length increases of 11 to 77%. Comparisons are made to several experimental studies for interface shape and pressure stability.     

Nonequilibrium bubbles in a flowing langmuir monolayer

We investigate the nonequilibrium behavior of two-dimensional gas bubbles in Langmuir monolayers. A cavitation bubble is induced in liquid expanded phase by locally heating a Langmuir monolayer with an IR-laser. At low IR-laser power the cavitation bubble is immersed in quiescent liquid expanded monolayer. At higher IR-laser power thermo capillary flow around the laser-induced cavitation bubble sets in. The thermo capillary flow is caused by a temperature dependence of the gas/liquid line tension. The slope of the line tension with temperature is determined by measuring the thermo capillary flow velocity. Thermodynamically stable satellite bubbles are generated by increasing the surface area of the monolayer. Those satellite bubbles collide with the cavitation bubble. Upon collision the satellite bubbles either coalesce with the cavitation bubble or slide past the cavitation bubble. Moreover we show that the satellite bubbles can also be produced by the emission from the laser-induced cavitation bubbles.     

Surface Tension of the Vapor–Liquid Interface with Finite Curvature

Based on the division of particles into internal and surface particles, the expression is derived closing the system of equations of classical thermodynamics for curvature-dependent surface tension, equimolar radius, and radius of tension surface. A solution to this system allows one to find the surface tension of new phase nucleus of any size (including minimal) and any sign of surface curvature. The obtained results indicate the weak size dependence of thermodynamic parameters that are the functions of surface tension; it is shown that Tolman's length cannot be determined using experimental determination of these parameters. It is shown that the work of nucleus formation strongly depends on its size and is the function of effective rather than true surface tension. Numerical simulation of clusters by the molecular dynamics method indicates that the pressure inside a fairly small cluster is described by Laplace's formula with the coefficient of surface tension for the plane surface of a liquid that agrees with the proposed theory.     

Effects of solid-liquid interface on the interfacial tension measured by micropipet technique

Measurement of interfacial tension (IFT) using the micropipet technique involves the solid-liquid interface. At equilibrium, oil-water interfacial tension is determined from the interface curvature and the critical pressure, according to the Young-Laplace equation. This paper aims to examine the possible contribution of the solid-liquid interface on IFT measurement. Three different experimental configurations are used to examine the sought effect. The three configurations are straight, concentric, and tapered pipets with diameters ranging from 2.5 to 30 microm. For all three configurations, the critical pressure is found to depend only on the pipet diameter. However, when the Young-Laplace equation is applied to determine the IFT, a significant error was noticed at small pipet diameters. The IFT error was described by an exponential function whose asymptote approached the independently determined IFT value with a sufficiently large pipet diameter. The IFT error is anticipated to arise from the layerlike effect of an "ultrastructured" liquid near the solid surface. The solid-induced error in oil-water IFT is noted to fade away at lowered IFT by the addition of surfactant.     

Effect of temperature and composition on the density,viscosity, surface tension,and thermodynamic properties of binary mixtures of N-octylisoquinolinium bis{(trifluoromethyl)sulfonyl}imide with alcohols

Density and viscosity were determined for the binary mixtures containing the ionic liquid N-octylisoquinolinium bis{(trifluoromethyl)sulfonyl}imide ([C₈iQuin][NTf₂]) and 1-alcohol (1-butanol, 1-hexanol, and 2-phenylethanol) at five temperatures (298.15, 308.15, 318.15, 328.15, and 338.15) K and ambient pressure. The density and viscosity correlations for these systems were tested by an empirical second-order polynomial and by the Vogel–Fucher–Tammann equation. Excess molar volumes were described by the Redlich–Kister polynomial expansion. The density and viscosity variations with compositions were described by polynomials. Viscosity deviations were calculated and correlated by the Redlich–Kister polynomial expansions. The surface tensions of pure ionic liquid and binary mixtures of [C₈iQuin][NTf₂] with 1-hexanol were measured at atmospheric pressure at three temperatures (298.15, 308.15, and 318.15) K. The surface tension deviations were calculated and correlated by the Redlich–Kister polynomial expansion. The surface thermodynamic functions such as surface entropy and enthalpy were derived from the temperature dependence of the surface tension values. The critical temperature, parachor, and speed of sound for pure ionic liquid were described. A qualitative analysis on these quantities in terms of molecular interactions is reported. The obtained results indicate that ionic liquid interactions with alcohols are strong dependent on the special trend of packing effects and hydrogen bonding of this ionic liquid with hydroxylic solvents. As previously observed, an increase by a 1-alcohol carbon chain length leads to lower interactions on mixing.     

Surface tension of water droplets upon homogeneous droplet nucleation in water vapor

A method has been proposed for determining interfacial free energy from the data of molecular dynamics simulation. The method is based on the thermodynamic integration procedure and is distinguished by applicability to both planar interfaces and those characterized by a high curvature. The workability of the method has been demonstrated by the example of determining the surface tension for critical nuclei of water droplets upon condensation of water vapor. The calculation has been performed at temperatures of 273–373 K and a pressure of 1 atm, thus making it possible to determine the temperature dependence of the surface tension for water droplets and compare the results obtained with experimental data and the simulation results for a “planar” vapor–liquid interface.     

Extended version of the van der Waals capillarity theory

An extended version of the van der Waals capillarity theory describing the liquid-vapor interface in the temperature range from the triple to the critical point is suggested. A model functional of thermodynamic potential for a two-phase Lennard-Jones system taking into account the effect of the highest degree terms of gradient expansion has been constructed. The identity of the thermodynamic and the mechanical definition of Tolman's length has been proved in the framework of the adopted form of functional. The properties of nuclei of the liquid and the vapor phase are described. The paper determines: the work of formation of a nucleus, density profiles, size dependences of the surface tension, and the parameter delta in the Gibbs-Tolman-Koenig-Buff equation.     

Wetting Study of Hydrophobic Membranes via Liquid Entry Pressure Measurements with Aqueous Alcohol Solutions

A new concept of liquid entry pressure measurements is applied to study the hydrophobicity of microporous membranes for aqueous alcohol solutions. The effects of alcohol concentration, type of alcohol, and temperature on liquid entry pressure of the membrane have been studied. Two theoretical equations for the determination of membrane pore size have been proposed. The former equation was developed taking into account the deviation from the Laplace–Young equation due to the membrane structure by means of the structure angle. The latter equation was established considering only the range of alcohol concentration in which the dispersion component of liquid surface tension remains practically constant. Hydrophobicity has been expressed in terms of wetting surface tension, γ_L^w. Based on these measurements, the maximum concentration before the spontaneous wetting occurs would be predicted.     

The dynamics of a bubble ensemble in a cavitation field

A system of equations was obtained to describe the dynamics of bubbles in a cavitation cloud taking into account the interaction of pulsating bubbles involved in translational motion. The kinetics of cavitation bubble concentration changes, changes in the compressibility of the liquid, and phase transitions within a cavitation bubble and in the neighboring volume of the liquid were taken into account. The role played by bubble deformation in a cavitation cloud was considered. The Bernoulli pressure effect was shown to be negligible. The interaction of cavitation bubbles was a substantial factor that strongly influenced the dynamics of bubbles. It was suggested that there was at least one more mechanism that reduced sonoluminescence intensity from the multiple-bubble cavitation field, namely, a fairly high efficiency of sonoluminescence quenching could additionally be related to the arrival of a cumulative liquid stream at the central cavitation bubble region, where the concentration of active species was high. The dynamics of bubbles in the cavitation field is not only related to the expansion and compression of cavitation bubbles in the acoustic field, but also governed to a great extent by their interaction, translational motion, deformation, and the influence of cumulative streams penetrating the bubbles.     

Homogeneous nucleation: classical formulas as asymptotic limits of the Cahn-Hilliard approach

Classical expressions for the critical cluster work of formation approximate the nonclassical expressions based in the density functional theory of capillarity for the limit of low supersaturation degrees. However, the ratio between classical and nonclassical expressions for nucleation rates grows as the supersaturation degree decreases. Here, with the aim to obtain simple and more accurate expressions that approximate the modern nucleation rate formulas, an asymptotic expansion of the Cahn-Hilliard expression of the critical work of formation is developed within the limit of low supersaturation. In such asymptotic expansion, terms up to third order are retained. The ratios between the corrected classical expressions and the nonclassical ones are now decreasing for supersaturation degrees tending to zero. However, the corrected approximate formulas are as difficult to handle as the exact Cahn-Hilliard expressions. When only the two first low-order terms of the asymptotic expansion are retained, a simpler corrected classical expression is obtained but it can only approximate nonclassical expressions up to order unity. Finally, using a Becker-Doring model of nucleation, the kinetic prefactor of the critical nuclei rate of formation is modeled consistently with the Cahn-Hilliard approach to the critical work of formation.     

Equilibrium shapes of liquid bridges under gravity: Symmetry breaking and imperfect bifurcations of two-dimensional bridges

The equilibrium shapes of a two-dimensional liquid bridge are constructed via a shooting and continuation numerical solution of the Laplace—Young equation which focuses on the bifurcation points of the solution branches. For the neutrally buoyant case, two new asymmetric shapes bifurcate from the point of maximum excess pressure where circular profiles with reflective symmetry have been shown to be unstable with respect to constant pressure disturbances. This occurs when the profile is exactly at right angles to the support. With the introduction of gravitational effects, this codimension 5 singularity is retained although the critical contact angle decreases slightly below 90° as one increases the difference between the interior and the exterior densities. However, when a slight tilt angle is introduced to the bridge, the maximum pressure singularity degenerates into two turning points (folds) and the solution branches are isolated from each other. This indicates that hysteretic jumps in the surface curvature and excess pressure exist with respect to changes in diameter/length ratio or liquid volume. Our numerical solution also reveals the existence of a maximum bridge volume that can be sustained by a nonneutrally buoyant bridge. No equilibrium shapes, symmetric or asymmetric, exist for bridge volumes beyond this critical value.     

Kelvin方程的一种理论推导

从液滴平衡条件推导出严格意义的Kelvin方程, 验证了其在宏观尺度可以转化为经典形式. 利用Tolman方程, 在考虑表面张力与曲率半径关系的条件下, 给出在液体压缩性可忽略时, 饱和蒸气压、蒸气密度、蒸气摩尔体积和曲率半径等关系; 液体压缩性不可忽略时, 得出以等温压缩系数和Tolman长度表示的饱和蒸气压与液滴半径的关系.