Equilibrium contact angles of liquid droplets on ideal rough solids

This work proposes a theoretical model for predicting the apparent equilibrium contact angle of a liquid on an ideal rough surface that is homogeneous and has a negligible body force, line tension, or contact angle hysteresis between solid and liquid. The model is derived from the conservation equations and the free-energy minimization theory for the changes of state of liquid droplets. The work of adhesion is expressed as the contact angles in the wetting process of the liquid droplets. Equilibrium contact angles of liquid droplets for rough surfaces are expressed as functions of the area ratios for the solid, liquid, and surrounding gas and the roughness ratio and wetting ratio of the liquid on the solid for the partially and fully wet states. It is found that the ideal critical angle for accentuating the contact angles by the surface roughness is 48°. The present model is compared with existing experimental data and the classical Wenzel and Cassie-Baxter models and agrees with most of the experimental data for various surfaces and liquids better than does the Wenzel model and accounts for trends that the Wenzel model cannot explain.     

Modeling contact angle hysteresis of a liquid droplet sitting on a cosine wave-like pattern surface

A liquid droplet sitting on a hydrophobic surface with a cosine wave-like square-array pattern in the Wenzel state is simulated by using the Surface Evolver to determine the contact angle. For a fixed drop volume, multiple metastable states are obtained at two different surface roughnesses. Unusual and non-circular shape of the three-phase contact line of a liquid droplet sitting on the model surface is observed due to corrugation and distortion of the contact line by structure of the roughness. The contact angle varies along the contact line for each metastable state. The maximum and minimum contact angles among the multiple metastable states at a fixed viewing angle correspond to the advancing and the receding contact angles, respectively. It is interesting to observe that the advancing/receding contact angles (and contact angle hysteresis) are a function of viewing angle. In addition, the receding (or advancing) contact angles at different viewing angles are determined at different metastable states. The contact angle of minimum energy among the multiple metastable states is defined as the most stable (equilibrium) contact angle. The Wenzel model is not able to describe the contact angle along the three-phase contact line. The contact angle hysteresis at different drop volumes is determined. The number of the metastable states increases with increasing drop volume. Drop volume effect on the contact angles is also discussed.     

Physical significance of contact angles

Summary The physical significance of contact angles has been interpreted on the basis of a model derived from known surface energy relationships. The degrees of non-spreading and spreading have been expressed in terms of the magnitude of contact angles. On the basis of the physical picture, hysteresis of contact angle has been calculated from the experimental values of equilibrium contact angle and surface tension of the liquid. It has been suggested that it is not necessary to assume that hysteresis of contact angles is due to surface roughness of solids. The picture also explains why apparent contact angle on a non-flat solid surface is more than that on a flat solid.With 2 figures and 1 table     

Thickness, stability and contact angle of liquid films on and inside nanofibres, nanotubes and nanochannels

While the stability of liquid films on substrates is a classical topic of colloidal science, the availability of nanostructured materials, such as nanotubes, nanofibres and nanochannels, has raised the question of how the stability of liquid films and their wetting behaviour is affected by nanoscale confinement. This paper will present the conditions for the stability of liquid films on and inside cylindrical solid substrates with nanometre scale characteristic dimensions. It is shown that the stability is determined by an effective disjoining/conjoining pressure isotherm which differs from the corresponding disjoining/conjoining pressure isotherm of flat liquid films on flat solid substrates. From the former, the equilibrium contact angles of drops on an outer or inner surface of a cylindrical capillary have been calculated as a function of surface curvature, showing that the expressions for equilibrium contact angles vary for different geometries, in view of the difference in thickness of the film of uniform thickness with which the bulk liquid (drops or menisci) is at equilibrium. These calculations have been extended to the case of glass nanocapillaries and carbon nanotubes, finding good agreement with experimental results in the literature.     

Interactions between particles with an undulated contact line at a fluid interface: capillary multipoles of arbitrary order

A colloidal particle adsorbed at a fluid interface could have an undulated, or irregular contact line in the presence of surface roughness and/or chemical inhomogeneity. The contact-line undulations produce distortions in the surrounding liquid interface, whose overlap engenders capillary interaction between the particles. The convex and concave local deviations of the meniscus shape from planarity can be formally treated as positive and negative "capillary charges," which form "capillary multipoles." Here, we derive theoretical expressions for the interaction between two capillary multipoles of arbitrary order. Depending on the angle of mutual orientation, the interaction energy could exhibit a minimum, or it could represent a monotonic attraction. For undulation amplitudes larger than 5 nm, the interaction energy is typically much greater than the thermal energy kT. As a consequence, a monolayer from capillary multipoles exhibits considerable shear elasticity, and such monolayer is expected to behave as a two-dimensional elastic solid. These theoretical results could be helpful for the understanding of phenomena related to aggregation and ordering of particles adsorbed at a fluid interface, and for the interpretation of rheological properties of particulate monolayers. Related research fields are the particle-stabilized (Pickering) emulsions and the two-dimensional self-assembly of microscopic particles.     

Influence of surface topography on the interactions between nanostructured hydrophobic surfaces

Nanostructured particle coated surfaces, with hydrophobized particles arranged in close to hexagonal order and of specific diameters ranging from 30 nm up to 800 nm, were prepared by Langmuir-Blodgett deposition followed by silanization. These surfaces have been used to study interactions between hydrophobic surfaces and a hydrophobic probe using the AFM colloidal probe technique. The different particle coated surfaces exhibit similar water contact angles, independent of particle size, which facilitates studies of how the roughness length scale affects capillary forces (previously often referred to as "hydrophobic interactions") in aqueous solutions. For surfaces with smaller particles (diameter < 200 nm), an increase in roughness length scale is accompanied by a decrease in adhesion force and bubble rupture distance. It is suggested that this is caused by energy barriers that prevent the motion of the three-phase (vapor/liquid/solid) line over the surface features, which counteracts capillary growth. Some of the measured force curves display extremely long-range interaction behavior with rupture distances of several micrometers and capillary growth with an increase in volume during retraction. This is thought to be a consequence of nanobubbles resting on top of the surface features and an influx of air from the crevices between the particles on the surface.     

Facile method to fabricate raspberry-like particulate films for superhydrophobic surfaces

A facile method using layer-by-layer assembly of silica particles is proposed to prepare raspberry-like particulate films for the fabrication of superhydrophobic surfaces. Silica particles 0.5 microm in diameter were used to prepare a surface with a microscale roughness. Nanosized silica particles were then assembled on the particulate film to construct a finer structure on top of the coarse one. After surface modification with dodecyltrichlorosilane, the advancing and receding contact angles of water on the dual-sized structured surface were 169 and 165 degrees , respectively. The scale ratio of the micro/nano surface structure and the regularity of the particulate films on the superhydrophobic surface performance are discussed.     

An investigation of the stable orientations of orthorhombic particles in a thin film and their effect on its critical failure pressure

The effects of shape and contact angle on the behaviour of orthorhombic particles at an interface and in thin films were investigated using Surface Evolver. It is shown that the energetically stable orientations of the particle change with its aspect ratio. Long, wide, flat particles with low contact angles are more stable in flat orientations, i.e. with two faces parallel to the flat film surface. More cubic particles with higher contact angles are more stable in twisted orientations, where the opposite sides of the film can be drawn together at the sharp edges of the particle. The combination of contact angle and orientation has been found to have a large effect on the capillary pressure required to rupture the film. A film containing a particle in a flat orientation will rupture at a capillary pressure up to three times greater than one containing an identical particle in a twisted orientation. Wider, flatter particles with low contact angles stabilise thin liquid films to a greater extent than cubic particles with high contact angles.     

Wetting and wetting transitions on copper-based super-hydrophobic surfaces

Rough and patterned copper surfaces were produced using etching and, separately, using electrodeposition. In both of these approaches the roughness can be varied in a controlled manner and, when hydrophobized, these surfaces show contact angles that increase with increasing roughness to above 160 degrees . We show transitions from a Wenzel mode, whereby the liquid follows the contours of the copper surface, to a Cassie-Baxter mode, whereby the liquid bridges between features on the surface. Measured contact angles on etched samples could be modeled quantitatively to within a few degrees by the Wenzel and Cassie-Baxter equations. The contact angle hysteresis on these surfaces initially increased and then decreased as the contact angle increased. The maximum occurred at a surface area where the equilibrium contact angle would suggest that a substantial proportion of the surface area was bridged.     

Nonlinear penetration of liquid drops into radial capillaries

The nonlinear dynamics of the impact and penetration of a liquid droplet in a radial capillary is studied numerically. The radial capillary is formed by two parallel plates at a distance of delta(g). The top plate has an orifice at its center. A droplet impacting on the orifice-plate partly spreads over the top plate, and the rest penetrates into the capillary gap between the two plates. The rate of spread of the fluid on the orifice plate, xi(out), is governed by the contact angle, beta, between the liquid and the orifice plate and the droplet initial momentum, whereas the rate of fluid spread inside the capillary gap, xi(in), is decided by the contact angles with both plates and the plate gap, delta(g).     

Droplet morphologies on particles with macroscopic surface roughness

The equilibrium configuration of liquid droplets on the surface of macroscopically rough solid particles was determined by numerical simulations using the volume-of-fluid (VOF) method. The fractional surface coverage of the particle as a function of the droplet size, equilibrium contact angle, and the particle surface roughness amplitude and correlation length has been systematically investigated. Droplet size and contact angle were found to generally have a stronger effect on surface coverage than particle surface roughness. Because of droplet coalescence, a relatively large variation in surface coverage was observed for any given total liquid volume, particularly for larger values of the equilibrium contact angle.     

Fabrication of hydrophobic surfaces by coupling of Langmuir-Blodgett deposition and a self-assembled monolayer

A novel method coupling the Langmuir-Blodgett (LB) deposition of silica particles and the formation of a self-assembled monolayer (SAM) of alkylsilane is proposed for fabricating hydrophobic surfaces. The LB deposition and the SAM are supposed to confer the substrate surface roughness and low surface energy, respectively. By controlling the hydrophobic-hydrophilic balance of the silica particle surface through the adsorption of surfactant molecules, deposition of monolayers consisting of hexagonally close-packed arrays of particles on a glass substrate can then be successfully conducted in a Langmuir trough. LB particulate films with a particle layer number up to 5 were thereby prepared. A sintered and hydrophobically finished particulate film with roughness factor of 1.9 was finally fabricated by sintering and surface silanization. Effects of particle size and particle layer number on the wetting behavior of the particulate films were systematically studied by measuring static and dynamic water contact angles. The experimental results revealed that a static contact angle of about 130 degrees resulted from the particulate films regardless of the particle size and particle layer number. This is consistent with the predictions of both the Wenzel model and the Cassie and Baxter model in that roughness of a hydrophobic surface can increase its hydrophobicity and a switching of the dominant mode from Wenzel's to Cassie and Baxter's. In general, an advancing contact angle of about 150 degrees , a receding contact angle of about 110 degrees , and a contact angle hysteresis of about 40 degrees were exhibited by the particulate films fabricated.     

Kinetics of wetting of liquid on a solid surface

To consider a sessile drop on an ideal solid surface in equilibrium with a vapor phase, the classic Young equation was given. The derivation of the Young equation was based on both the mechanics and the energy knowledge. According to the constant volume of the liquid in the wetting process of the liquid on a smooth and homogeneous solid surface and the low energy law, Young equation was ob-tained through the mathematic method in this paper. The previous work indicated that the contact angle θ was a function...     

Modeling Pendular Liquid Bridges with a Reducing Solid-Liquid Interface

Liquid bridges formed between particles of dissimilar surface properties are important in many processes involving the handling of powders in mixtures. For instance, growth kinetic models for wet granulation frequently incorporate the evolution and resistance to breakage of individual liquid bridges between particles in a statistical form. These models generally propose a confusing definition of liquid-to-solid contact angles. Taken as a single thermodynamic value, they typically neglect the influence of wetting hysteresis on the liquid bridge. In this paper, a simple model based on the interfacial energies is proposed for the evolution of liquid bridges when one solid-liquid interface reduces. This receding process is well described by a balance between the adhesion energy of the bridge liquid on the particle surface and the capillary energy stored by the liquid free surface. The extent of solid-liquid interfacial area reduction can hence be predicted from the initial liquid bridge configuration. The liquid bridge shape is approximated by a parabolic curve, which is validated from the good agreement between measured and calculated contact angles or liquid-vapor interfacial area. Copyright 2000 Academic Press.     

Thermodynamic modeling of contact angles on rough, heterogeneous surfaces

Theoretical modelling for contact angle hysteresis carried out to date has been mostly limited to several idealized surface configurations, either rough or heterogeneous surfaces. This paper presents a preliminary study on the thermodynamics of contact angles on rough and heterogeneous surfaces by employing the principle of minimum free energy and the concept of liquid front. Based on a two-dimensional regular model surface, a set of relations were obtained, which correlate advancing, receding and system equilibrium contact angles to surface topography, roughness and heterogeneity. It was found that system equilibrium contact angles (theta(ES)) can be expressed as a function of surface roughness factor (delta) and the Cassie contact angle (theta(C)): costheta(ES) = deltacostheta(C). This expression can be reduced to the classical Wenzel equation.: theta(ES) = theta(W) for rough but homogeneous surfaces, and the classical Cassie equation theta(ES) = theta(C) for heterogeneous but smooth surfaces. A non-dimensional parameter called surface feature factor (omega) was proposed to classify surfaces into three categories (types): roughness-dominated, heterogeneity-dominated and mixed-rough-heterogeneous. The prediction of advancing and receding contact angles of a surface is dependent on which category the surface belongs to. The thermodynamic analysis of contact angle hysteresis was further extended from the regular model surface to irregular surfaces; consistent results were obtained. The current model not only agrees well with the models previously studied by other researchers for idealized surfaces, but also explores more possibilities to explain the reported experimental results/observations that most existing theories could not explain.     

Drops on microstructured surfaces coated with hydrophilic polymers: Wenzel's model and beyond

In the experiments described in this paper, micromachined silicon post surfaces were coated with thin films of a polymer for which the contact angle on the smooth material was around 70 degrees . Drops wetted these surfaces in the Wenzel or "penetration" mode. We have determined the advancing and receding angles as a function of the roughness geometry and quantitatively compared our results to the contact angles predicted by Wenzel's model. We discuss reasons for discrepancies and propose a model for the motion of the meniscus where we take into account the precise shape of the liquid front during its movement through the post structure.     

The Removal of Particles from Oils by Nonionic Surfactants

The separation of particulate material from a dispersion of particles in an oil can be effected by transport of the particles through the bounding oil interface in a process mediated by surfactant. When the oil phase containing particles is contacted with an aqueous phase, the transport process through the interface is resisted by a capillary force which depends on the interfacial tension and on the contact angle for the three-phase system. It is shown how the addition of a judiciously-chosen nonionic surfactant to a model system of carbon particles in n-alkanes can bring about near-complete transfer of the particles to the aqueous phase. The underlying mechanisms for the transport process are discussed. Some implications for systems of practical importance are discussed.     

Theoretical evaluation of nano- or microparticulate contact angle at fluid/fluid interfaces: analysis of the excluded area behavior upon compression

A novel method for the determination of the particle contact angle at the liquid/gas or liquid/liquid interface based on the excluded area concept revealed, in spite of its simplicity, some serious difficulties connected with the exact quantitative particle deposition at the interface and with changes in the particulate contact angle upon binary monolayer compression. The comprehensive theoretical consideration of the contact angle behavior made for such a system allowed considerable improvements: firstly, the prediction of direction of the particles' displacement at surface pressure increase is now possible and hence an unambiguous identification of particle hydrophobicity can be done. Secondly, the analytical relation describing the dependence of the particulate contact angle on the surface tension (surface pressure) was derived, allowing the prediction of whether or not particles of a given hydrophobicity will be expelled from the monolayer at certain surface pressure and of the area relinquished by the displaced particles. Thirdly, the transformation of this relation upon taking into consideration the initial conditions led to a linear dependence between excluded area and normalized surface tension allowing the determination of the particle contact angle and the exact number of deposited particles simultaneously and independently of each other. Finally, the application of the improved approach to the previously collected experimental data yielded reasonable values for both particle contact angle and number of deposited particles.     

Disjoining pressure,contact angles and line tension in free thin liquid films

Three are the main thermodynamic quantities which characterize the physical behaviour of the thin liquid films: the disjoining pressure, the macroscopic contact angle between the film and its adjacent bulk liquid phase, and the line tension of the film contact perimeter. All these quantities manifest the action of the long-range interaction surface forces which take place in any small capillary system. The rigorous introduction of such quantities is closely related to the Gibbs concept of surface of tension. For the case of a thin plane parallel liquid film there exist three surfaces of tension. By making use of them the thickness of the film would also be defined. There are several experimental methods for determining the tension of film, the contact angles. the line tension. Some important details of these methods. some experimental results together with important features of the thermodynamic theory of the thin liquid films are the subject of the present paper.     

Influence of the Type of Contact between Particles Joined by a Liquid Bridge on the Capillary Cohesive Forces

The influence of the particle dimensions and type of interparticle contact on the magnitude of the capillary forces between the powder particles is studied on the basis of a model describing a capillary interaction of two particles joined by a liquid bridge. Various contact types were implemented using combinations of different particle shapes: spherical, conical, or plane. The meniscus of the bridge is described using a circular approximation; experimental results confirm that its use is justified. A method is developed for calculating the capillary forces and the amount of the liquid in the bridge with allowance for various parameters of the powder. The calculated results show that the dimensions of the particles and the type of their contact significantly affect the magnitude of the capillary forces.