The line adsorption equation: the one-dimensional counterpart of the Gibbs adsorption equation

The thermodynamic development for multiphase contact lines is analogous to that for surfaces or interfaces. However, for one of the most important equations in surface thermodynamics, the Gibbs adsorption equation, the one-dimensional analogue is missing. This paper derives such an analogue, the line adsorption equation. Similarly to the Gibbs adsorption equation, the line adsorption equation is derived from Gibbsian thermodynamics. For a three-phase, three-component contact line system (e.g. an oil lens on the surface of an aqueous solution), the line concentrations (excesses) of two immiscible solvents can be made vanish by appropriately placing the dividing line. Consequently, the line concentration of the solute can be evaluated through the line tension change with the volume concentration of the solute. Such an evaluation provides information about molecular adsorption at the contact line, which is important in physical chemistry of lines, but difficult to obtain by any other means.     

Shape of the nematic-isotropic interface in conditions of partial wetting

A nematic liquid crystal in contact with a solid substrate is studied in the partial wetting regime. Both a mesoscopic Landau-de Gennes theory and a macroscopic effective interface Hamiltonian approach are considered. A generalized Young equation for the balance of forces at the three-phase contact line is derived, which takes into account corrections due to distortions of the nematic director field. It is also shown that the asymptotic form of the separation of the nematic-isotropic interface from the substrate has a logarithmic correction to the usual linear behaviour. The characteristic length scale of this correction is given by the ratio K/(2σ_NI), where K and σ_NI are the average elastic constant and the nematic-isotropic surface tension, respectively, and is of the order of a few hundred angstroms. Then, a simple form of an effective interface Hamiltonian is proposed, and results consistent with the predictions of the Landau-de Gennes theory are obtained. It is shown, in the framework of this macroscopic approach, that the line tension associated with the contact line remains finite, when the thermodynamic limit is taken, if the anchoring at both the nematic-substrate and the nematic-isotropic interfaces is homeotropic. However, in the case of different anchoring directions, the line tension diverges logarithmically with the system size.     

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.     

Generalized theory of capillarity for axisymmetric menisci

A high-curvature generalization of the Laplace equation of capillarity and the Young equation of capillarity (including line tension) is developed for an axisymmetric solid-liquid-fluid system. The most general expressions for the Laplace and Young equations do not assume a particular form for the specific surface free energy. However, when a particular form, i.e., ω^(A) = γ^(A)_∞+ C_JJ+ C_kK, which is related to Gibbs' expression for a highly curved menisci,¹ is assumed to hold for the specific surface free energy then we are able to recover the expected simplified form of the Laplace equation. The corresponding high-curvature Young equation includes a couple which balances the surface moments at the contact line. Unfortunately, the effect of this couple could be confused with the effect of line tension in experiments which attempt to measure the dependence of the contact angle on the contact line radius.     

Measurement of the adsorption at solid-liquid interfaces from the pressure dependence of contact angles

Earlier studies have indicated that in an isothermal three-phase system, the liquid-phase pressure at the three-phase line, xL3, may be viewed as the independent variable of the contact angle, theta, and that adsorption at the solid-liquid interface is the mechanism relating them. When the liquid-vapor interface is axi-symmetric, we show that theta can be predicted as a function of xL3 and that by measuring theta(xL3), the amount adsorbed at the solid-liquid interface can be determined. We consider water in differently sized borosilicate glass cylinders. For progressively larger cylinders, xL3 increases with cylinder radius, but when a particularly sized cylinder is rotated about it longitudinal axis, xL3 is decreased. The observed value of theta in each case is found to be in close agreement with that predicted. A Gibbs model of the interphase is used, and the Gibbs adsorption at the solid-liquid interface is found to be negative. As xL3 increases above its value at wetting, the amount adsorbed at the solid-liquid interface becomes progressively more negative. Negative adsorption is shown to mean that the concentration of the fluid component is greater in the bulk liquid than in the interphase and that the difference in concentration increases as xL3 is increased. The data is used to investigate the hypothesis that the curvature of the three-phase line affects theta through line tension, but we find no relation between line tension and theta. There is an apparent relation between the curvature of the liquid-vapor interface, CLV and theta, but this is shown to be because CLV affects xL3.     

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.     

Histogram analysis as a method for determining the line tension of a three-phase contact region by Monte Carlo simulations

A method is proposed for determining the line tension, which is the main physical characteristic of a three-phase contact region, by Monte Carlo (MC) simulations. The key idea of the proposed method is that if a three-phase equilibrium involves a three-phase contact region, the probability distribution of states of a system as a function of two order parameters depends not only on the surface tension, but also on the line tension. This probability distribution can be obtained as a normalized histogram by appropriate MC simulations, so one can use the combination of histogram analysis and finite-size scaling to study the properties of a three phase contact region. Every histogram and results extracted therefrom will depend on the size of the simulated system. Carrying out MC simulations for a series of system sizes and extrapolating the results, obtained from the corresponding series of histograms, to infinite size, one can determine the line tension of the three phase contact region and the interfacial tensions of all three interfaces (and hence the contact angles) in an infinite system. To illustrate the proposed method, it is applied to the three-dimensional ternary fluid mixture, in which molecular pairs of like species do not interact whereas those of unlike species interact as hard spheres. The simulated results are in agreement with expectations.     

Nanoporous surface wetting behavior: the line tension influence

The aim of this work is to develop a physical model to describe the evolution of the apparent contact angle for four different liquids on nanotextured alumina surfaces with different pore radius. The nanoporous alumina templates were fabricated by anodization of Al foil in a 0.3 M oxalic acid solution. Scanning electron microscopy was used to characterize the morphology of the surfaces. The templates are approximately 400 nm in thickness and consist of a well-ordered hexagonal array of uniform radius pores spaced 105 nm apart with pore radii from 12 to 42 nm. The wettability of nanoporous alumina templates was investigated using contact-angle measurements. We measured the contact angles using four liquids: water, ethylene glycol, aniline, and a mixture of ethylene glycol and aniline. We developed a new theoretical model for the contact angle on nanoporous surfaces as a function of the pore radius. This model is based on energy considerations and involves liquid penetration into the nanopores driven by the capillarity (Laplace's law). Because the air is compressed inside the pores, this model also includes the effect of the line tension. This is important because the three-phase line length is greatly enhanced in our nanoporous structures. For example: for a millimeter-sized droplet, the three-phase line around the perimeter of the droplet is a few millimeters long, whereas the total three-phase line within the pores can reach several tens of meters. Using our model, the line-tension value for our nanopore samples is positive and ranges from 4 to 13 × 10(-9) N, which falls within the wide interval from 10(-11) to 10(-5) N quoted in the literature. Nanoporous surfaces may allow the effect of line tension to be visible for micro- to macrodroplets.     

Experimental investigation of contact angle, curvature, and contact line motion in dropwise condensation and evaporation

Image-analyzing interferometry is used to measure the apparent contact angle and the curvature of a drop and a meniscus during condensation and evaporation processes in a constrained vapor bubble (CVB) cell. The apparent contact angle is found to be a function of the interfacial mass flux. The interfacial velocity for the drop during condensation and evaporation is a function of the apparent contact angle and the rate of change of radius of curvature. The dependence of velocity on the apparent contact angle is consistent with Tanner's scaling equation. The results support the hypothesis that evaporation/condensation is an important factor in contact line motion. The main purpose of this article is to present the experimental technique and the data. The equilibrium contact angle for the drop is found experimentally to be higher than that for the corner meniscus. The contact angle is a function of the stress field in the fluid. The equilibrium contact angle is related to the thickness of the thin adsorbed film in the microscopic region and depends on the characteristics of the microscopic region. The excess interfacial free energy and temperature jump were used to calculate the equilibrium thickness of the thin adsorbed film in the microscopic region.     

Local thermodynamic derivation of Young's equation

A derivation of Young's equation based on the energy balance near the contact line is presented. Our proposal is rigorous and avoids the errors identified in the usual local derivation. It is valid under very general conditions (for any geometry, in a gravitational field and for compressive fluids). Deviations of the contact angle from Young's equation are discussed in several cases: surfaces of high curvature and line tension. Finally, the relationship between surface tensions and surface energies comes as an additional, natural result. Our derivation also provides a new physical insight into the equilibrium of forces acting near the contact line. Its local character makes the recourse to integral analysis unnecessary, which results in a great simplification when compared to other general treatments.     

Instability of the Contact Line between Three Fluid Phases during Mass Transfer Processes

The movable contact line between two liquids and a gas phase sensitively reacts to small disturbances in the force equilibrium. The shape of the contact line and the adjoining interfaces is determined by the interface and surface tensions, the contact angles, the density differences (hydrostatic pressure), and the Laplace capillary pressure. When these change, the three-phase contact line can deform and even become unstable. Interface and surface tension depend on the concentration and temperature. During mass transport processes (concentration changes) various forms of the instability of the contact line can be observed: -Oscillations of a circular contact line (regular expansion and reduction); -Single deformations (bulges) which quickly disappear again; -Deformations (bulges) which run along the boundary line; -Periodically generated and damped deformations with different modes. The behavior of the three-phase contact line is of practical importance for coalescence processes and for spontaneous emulsification on liquid surfaces. Copyright 2000 Academic Press.     

Measurement of the film tension of foam films in dynamic conditions

The method for direct measurement of the film tension of foam films has been developed with a view to measuring the film tension in dynamic conditions. The new method allows the determination of the dynamic film tension when the curvature radius, the contact line radius, and the area of the film increase or decrease with very different rates. Measurements with Newton black films from sodium dodecyl sulphate aqueous solution have been performed. The results show that in a wide range of variation rates of the film geometrical parameters the film tension remains constant.     

Size-dependent lens angles for small oil lenses on water

Line tension is the excess energy associated with unit length of a three-phase contact line and it has long been of interest, in part because if sufficiently large, it can affect various processes of industrial and biological importance. Most recently, interest has centred on the magnitude and sign of experimentally determined values. Reported line tensions in systems with liquid alkanes in contact with aqueous phases include values from about +10⁻¹⁰ to 10⁻⁹ N on the one hand, and −10⁻⁶ N on the other. If the actual values of line tension lie close to the lower end of the spectrum quoted above, the influence on many systems of interest will be negligible. The higher values, however, could lead to pronounced effects. A study to determine line tension in the three-phase contact line around lenses of dodecane resting on a water subphase is presented. The method involves measuring, by interferometry, the variation of lens angle with the contact line radius. In order to bring the angles into a convenient range for measurement (around 2°), small amounts of dodecanol (ca. 2 mmol dm⁻³) have been added to the dodecane. The line tension is found to be +1.6±0.3×10⁻¹¹ N. The magnitude and sign of the line tension is discussed in terms of surface forces.     

Empirical equation to account for the length dependence of line tension

Measured values of the three-phase line tension in the literature are correlated with the spreading parameter and with the radii of the drops or bubbles under investigation. The latter dependence contradicts an assumption of the modified Young equation. We suggest an alternative empirical formulation that describes the data consistently, and we discuss its possible physical significance.     

Surface nanobubbles as a function of gas type

We experimentally investigate the nucleation of surface nanobubbles on PFDTS-coated silicon as a function of the specific gas dissolved in water. In each case, we restrict ourselves to equilibrium conditions (c = 100%, T(liquid) = T(substrate)). Not only is nanobubble nucleation a strong function of gas type, but there also exists an optimal system temperature of ～35 -40 °C where nucleation is maximized, which is weakly dependent on gas type. We also find that the contact angle is a function of the nanobubble radius of curvature for all gas types investigated. Fitting this data allows us to describe a line tension that is dependent on the type of gas, indicating that the nanobubbles sit on top of adsorbed gas molecules. The average line tension was τ ≈ -0.8 nN.     

Wetting behavior of spherical nanoparticles at a vapor-liquid interface: a density functional theory study

The wetting behavior of spherical nanoparticles at a vapor-liquid interface is investigated by using density functional theory, and the line tension calculation method is modified by analyzing the total energy of the vapor-liquid-particle equilibrium. Compared with the direct measurement data from simulation, the results reveal that the thermodynamically consistent Young's equation for planar interfaces is still applicable for high curvature surfaces in predicting a wide range of contact angles. The effect of the line tension on the contact angle is further explored, showing that the contact angles given by the original and modified Young's equations are nearly the same within the region of 60° < θ < 120°. Whereas the effect is considerable when the contact angle deviates from the region. The wetting property of nanoparticles in terms of the fluid-particle interaction strength, particle size, and temperature is also discussed. It is found that, for a certain particle, a moderate fluid-particle interaction strength would keep the particle stable at the interface in a wide temperature range.     

Equilibrium of multi-phase systems in gravitational fields

Four necessary conditions for equilibrium of an isolated solid-liquid-vapor system in a gravitational field were derived by Ward and Sasges (1998) in a unified setting, by using an entropy maximization approach, and under the assumption that the liquid-vapor surface tension does not depend on elevation. These are thermal equilibrium, the Laplace and Young equations, and a condition on the chemical potentials of the components present in the system. Gibbs (1876) had obtained the Young equation in a derivation separate from the derivation of the other three conditions and by using an energy minimization approach. However, Gibbs had derived a more general form of the Laplace equation than Ward and Sasges's. Gibbs's equation contained a term expressing the contribution of the variation of surface tension with elevation. This equation has since been neglected by most of the scientific community. In the present paper, the same approach as Ward and Sasges's is used to derive, in an unified setting, the conditions for equilibrium of an isolated solid-liquid-vapor system in a gravitational field but under the assumption that the liquid-vapor surface tension may depend on elevation. The four well-known conditions for equilibrium are obtained, with Gibbs's generalized Laplace equation instead of the classical Laplace equation. The derivations in this paper were carried out for two different system geometries, namely, for a sessile drop and for a conical capillary tube, and similar conditions for equilibrium were obtained.     

Thermodynamic theory of thin liquid films including line tension effects

This paper presents the contemporary state-of-art of the phenomenological thermodynamic theory of the thin foam and emulsion films, both symmetrical and asymmetrical ones. The roots of this theory are in the Gibbs' theory of capillarity. Two basic approaches — with two Gibbs dividing surfaces and with three surfaces of tension, are described. The generalization of the theory for systems with more complex geometry is commented. The ways of determining of the thermodynamic thickness of the film are described. The basic thermodynamic quantities of the thin film: disjoining pressure, tension of the film and surface tension of the film, are defined. The tangential mechanical equilibrium conditions with two types of contact angles, θ_h, and θ₀, are discussed. The effect of line tension of the three-phase contact-line perimeter on the film contact angles is elucidated.     

Determination of line tension for systems near wetting

Contact angle measurements for three n-alkanes, heptane, octane, and nonane, on two different self-assembled surfaces (SAM) are reported as a function of drop size. These liquids all formed low contact angles (below 20 degrees ); the measurements were performed using an accurate method for systems with low contact angle, ADSA-D. The observed drop size dependence of the contact angles was interpreted using the modified Young equation. It was concluded that the observed drop size dependence of contact angles was due to line tension. The choice of systems also provided the opportunity to examine the behavior of the line tension for systems near wetting (i.e., low contact angles). It was determined that the line tension is positive and ranges from below 10(-7) to just below 10(-6) J/m for the systems studied; the observations suggested that the line tension decreases as the contact angle decreases and likely vanishes at complete wetting.