Stable drop shapes under disjoining pressure. III. Sharp contact lines

The shape and stability of a drop with a sharp contact line are analyzed. The disjoining pressure is included. The choice made for the latter is special in that it admits a sharp contact angle as appropriate in this case. The stability analysis is considered for two scenarios, one where the contact line cannot be moved which mode turns out to be stable to small perturbations, and the other where it can be moved but the contact angle is fixed at the equilibrium value. This second case turns out to be unstable to small perturbations due to the curvature effect of the drop.     

Stable drop shapes under disjoining pressure. II. Multiplicity and stability

The stability of the contact line region as affected by the disjoining pressure has been analyzed by solving the augmented Young-Laplace equation. Because of the results in Part I (Zhang, X., Neogi, P., and Ybarra, R. M., J. Colloid Interface Sci.), we have concentrated on obtaining multiple solutions for the same set of conditions. As many as five solutions were obtained: drops that end in a thin film with uniform thickness and where the film shape oscillates, drops that end with microscopic contact angles, as well as uniform thin films of two different thicknesses. The results of linear stability analysis were used to show that most cases were unstable to infinitesimal disturbances. Only two stable drop shapes for the particular disjoining pressure investigated are stable, a thin film of constant thickness and a thin drop that ends in a film of same thickness. Both multiplicity and stability have been discussed here for the first time and shed considerable light on the role of the attractive and repulsive forces.     

Wetting and spreading of nanofluids on solid surfaces driven by the structural disjoining pressure: statics analysis and experiments

The wetting and spreading of nanofluids composed of liquid suspensions of nanoparticles have significant technological applications. Recent studies have revealed that, compared to the spreading of base liquids without nanoparticles, the spreading of wetting nanofluids on solid surfaces is enhanced by the structural disjoining pressure. Here, we present our experimental observations and the results of the statics analysis based on the augmented Laplace equation (which takes into account the contribution of the structural disjoining pressure) on the effects of the nanoparticle concentration, nanoparticle size, contact angle, and drop size (i.e., the capillary and hydrostatic pressure); we examined the effects on the displacement of the drop-meniscus profile and spontaneous spreading of a nanofluid as a film on a solid surface. Our analyses indicate that a suitable combination of the nanoparticle concentration, nanoparticle size, contact angle, and capillary pressure can result not only in the displacement of the three-phase contact line but also in the spontaneous spreading of the nanofluid as a film on a solid surface. We show here, for the first time, that the complete wetting and spontaneous spreading of the nanofluid as a film driven by the structural disjoining pressure gradient (arising due to the nanoparticle ordering in the confined wedge film) is possible by decreasing the nanoparticle size and the interfacial tension, even at a nonzero equilibrium contact angle. Experiments were conducted on the spreading of a nanofluid composed of 5, 10, 12.5, and 20 vol % silica suspensions of 20 nm (geometric diameter) particles. A drop of canola oil was placed underneath the glass surface surrounded by the nanofluid, and the spreading of the nanofluid was monitored using an advanced optical technique. The effect of an electrolyte, such as sodium chloride, on the nanofluid spreading phenomena was also explored. On the basis of the experimental results, we can conclude that a nanofluid with an effective particle size (including the electrical double layer) of about 40 nm, a low equilibrium contact angle (<3°), and a high effective volume concentration (>30 vol %) is desirable for the dynamic spreading of a nanofluid system with an interfacial tension of 0.5 mN/m. Our experimental observations also validate the major predications of our theoretical analysis.     

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.     

A robust algorithm for the simultaneous parameter estimation of interfacial tension and contact angle from sessile drop profiles

The pendant and sessile drop profile analysis using the finite element method (PSDA-FEM) is an algorithm which allows simultaneous determination of the interfacial tension (gamma) and contact angle (theta(c)) from sessile drop profiles. The PSDA-FEM algorithm solves the nonlinear second-order spherical coordinate form of the Young-Laplace equation. Thus, the boundary conditions at the drop apex and contact position of the drop with the substrate are required to solve for the drop profile coordinates. The boundary condition at the position where the drop contacts the substrate may be specified as a fixed contact line or fixed contact angle. This paper will focus on the fixed contact angle boundary condition for sessile drops on a substrate and how this boundary condition is used in the PSDA-FEM curve-fitting algorithm. The PSDA-FEM algorithm has been tested using simulated drop shapes with and without the addition of random error to the drop profile coordinates. The random error is varied to simulate the effect of camera resolution on the estimates of gamma and theta(c) values obtained from the curve-fitting algorithm. The error in the experimental values for gamma from sessile drops of water on acrylic and Mazola corn oil on acrylic falls within the predicted range of errors obtained for gamma values from simulated sessile drop profiles with randomized errors that are comparable in magnitude to the resolution of the experimental setup.     

Limiting conditions for applying the spherical section assumption in contact angle estimation

The shape of liquid drops on solid surfaces deviates from the spherical as tension decreases and gravity effects start affecting the drop shape. This paper attempts to define this deviation and estimates the dimensionless Eotvos number limits above which the deviation becomes "significant." The use of these limiting values can facilitate estimation of contact angle in the following manner. It is well known that the equilibrium contact angle made by a liquid drop on a solid surface can be estimated from measurements of two drop parameters. These parameters can be any two chosen from the drop volume, height, and wetted radius. In case the effect of gravity on the drop shape is negligible, simple algebraic relations derived from the spherical section assumption exist, from which the contact angle can be estimated. In systems where the "spherical section" assumption is invalid, the Laplace equation for the drop shape has been solved numerically with any two of the above parameters as the constraints, to obtain the contact angle. In this paper, Eotvos numbers at which the deviation of the drop profile from the spherical is significant enough to result in contact angle deviation of 1 degrees are estimated. The limiting values of Eotvos number, expressed as a function of the original contact angle made by the spherical profile, are obtained by solving the Laplace equation for the drop shape with the drop volume and wetted radius constraints for decreasing values of Interfacial tension. These limiting values are also estimated for different drop sizes and for cases where the drop phase is heavier (sessile) and lighter (buoyant) than the surrounding fluid. The independence of the Eotvos number estimates from the sign of the density difference as well as the drop size is shown. These Eotvos number limits can be used to check if the spherical section assumption, with the resulting simple algebraic relations, can be used for contact angle estimation and other shape-related analysis for a system.     

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.     

Thin film drainage: hydrodynamic and disjoining pressures determined from experimental measurements of the shape of a fluid drop approaching a solid wall

Accurate measurements of the shape of a mercury drop separated from a smooth flat solid surface by a thin aqueous film reported recently by Connor and Horn (Faraday Discuss. 2003, 123, 193-206) have been analyzed to calculate the excess pressure in the film. The analysis is based on calculating the local curvature of the mercury/aqueous interface, and relating it via the Young-Laplace equation to the pressure drop across the interface, which is the difference between the aqueous film pressure and the known internal pressure of the mercury drop. For drop shapes measured under quiescent conditions, the only contribution to film pressure is the disjoining pressure arising from double-layer forces acting between the mercury and mica surfaces. Under dynamic conditions, hydrodynamic pressure is also present, and this is calculated by subtracting the disjoining pressure from the total film pressure. The data, which were measured to investigate the thin film drainage during approach of a fluid drop to a solid wall, show a classical dimpling of the mercury drop when it approaches the mica surface. Four data sets are available, corresponding to different magnitudes and signs of disjoining pressure, obtained by controlling the surface potential of the mercury. The analysis shows that total film pressure does not vary greatly during the evolution of the dimple formed during the thin film drainage process, nor between the different data sets. The hydrodynamic pressure appears to adjust to the different disjoining pressures in such a way that the total film pressure is maintained approximately constant within the dimpled region.     

Spreading of nanofluids driven by the structural disjoining pressure gradient

This paper discusses the role of the structural disjoining pressure exerted by nanoparticles on the spreading of a liquid film containing these particles. The origin of the structural disjoining pressure in a confined geometry is due to the layering of the particles normal to the confining plane and has already been traced to the net increase in the entropy of the system in previous studies. In a recent paper, Wasan and Nikolov (Nature, 423 (2003) 156) pointed out that the structural component of the disjoining pressure is strong enough to move a liquid wedge; this casts a new light on many applications-most notably, detergency. While the concept of spreading driven by the disjoining pressure is not new, the importance of the structural disjoining pressure arises from its long-range nature (as compared to the van der Waals' force), making it an important component of the overall force balance near the contact line. In this paper, we report on a parametric study of the spreading phenomena by examining the effects of nanoparticle size, concentration and polydispersity on the displacement of an oil-aqueous interface with the aqueous bulk containing nanoparticles. The solution of the extended Laplace-Young equations for the profile of the meniscus yields the position of the nominal contact line under the action of the structural disjoining pressure. Simulations show that the displacement of the contact line is greater with a high nanoparticle volume fraction, small particles for the same volume fraction, monodispersed (in size) particles rather than polydispersed particles and when the resisting capillary pressure is small, i.e., when the interfacial tension is low and/or the radius of the dispersed phase drop/bubble is large.     

Polymer-induced depletion interaction between weakly attractive plates

Lattice Monte Carlo simulations have been employed to calculate depletion interaction of excluded volume chains in a weakly attractive slit, particularly in the region around the critical point of adsorption. The simulations were performed under full equilibrium conditions where a dilute solution in a slit was in contact with the reservoir. The free energy of confinement deltaA, the force f, and the relative pressurepI/pE on the slit walls were calculated as a function of slit width D and the attraction strength epsilon. The depletion region in the pressure profile pI/pE vs D is reduced by an increase in the attraction potential epsilon in a manner resembling the influence of polymer concentration. At the critical point of adsorption epsilonc the depletion interaction vanishes both in the pressure pI/pE and in the intraslit concentration profile phiI(x). The parameters used to assess the stability of colloidal dispersions such as the depletion potential W(D) (an integral of the net pressure deltap) reach a unique value at the critical condition. A monotonic repulsive profilepI vs D was found for chains trapped in the slit at restricted equilibrium. The mean dimensions (R2) of chains compressed in attractive slits feature a distinct minimum at intermediate slit widths.     

Forces between a Rigid Probe Particle and a Liquid Interface

The effect of disjoining pressure between a rigid spherical probe particle (attached to an AFM cantilever) and a liquid interface (e.g., oil/water or air/water) is treated in an analytic manner to describe the total force F exerted on the probe as a function of the distance X of the probe from the rigid substrate (AFM stage) on which the liquid interface resides. Two cases (i) a flat interface under gravity and (ii) a drop whose size is sufficiently small that gravity can be neglected have been examined. A simple numerical algorithm is given for computing F(X) (the AFM observable) from a given form for the disjoining pressure. Numerical results are displayed for electrostatic probe/interface interactions which reveal the linear compliance regime experimentally observed in AFM experiments on these systems. The slope of the linear compliance regime is shown to be a function of the properties of the interface (capillary length, particle radius, drop size, contact angle of drop on rigid substrate etc.). Copyright 2001 Academic Press.     

Contribution of the surface dipole moment and the contact potential-induced disjoining pressure to the stress balance at a three-phase contact

The contact between three insulators results in a set up of contact potentials related to the adsorbed dipole moment at each surface. The produced electric field applies force (disjoining pressure) on each interface. This disjoining pressure is a long-ranged force (1/distance²) which is proportional to the difference between the dielectric permittivities of the phases on the two sides of the interface and, for small angles, to the square of the contact angle. The contact potential leads to a logarithmic perturbation of the profile of the three-phase contact zone.     

Can contact-angle measurements determine the disjoining pressure in liquid nanofilms on rigid substrates?

The disjoining pressure of lubricant nanofilms used in the magnetic recording industry controls the equilibrium wetting, the dynamics of film restoration, and the evaporation kinetics of the film. It has been claimed that by measuring the contact angle of nonpolar and polar liquids on lubricant films, the disjoining pressure can be extracted using the method of Girafalco and Good, and such analyses have appeared in the literature. The approximations underlying the method have been discussed before in the literature. In view of the importance of measuring disjoining pressure in nanofilms of lubricants, it is timely to revisit these assumptions to understand the validity of the contact-angle method presently in use. We re-derive the relevant equations using a thermodynamic-interaction-energy approach which is free of the problems inherent in the original derivation and make explicit the assumptions which must be made in the derivation. General interaction energy arguments are then invoked to demonstrate that it does not appear possible to obtain the disjoining pressure in the film from contact-angle measurements in an unambiguous manner.     

Thermodynamic studies on thin liquid films. II. Foam film stabilized by decyl methyl sulfoxide

Thermodynamic equations in Part I of this series were extended so as to be applicable to electrolyte mixtures and the resultant equations were applied to the experimental results of a NaCl-decyl methyl sulfoxide (DeMS) mixture. Film thickness and contact angle of the black foam film stabilized by DeMS were measured as a function of the total molality of NaCl and DeMS at constant mole fraction of DeMS in the mixture under constant disjoining pressure. Newton black film was observed only above a certain DeMS concentration and the phase transition between common black and Newton black films took place twice as NaCl concentration increased at constant DeMS concentration. The surface densities of NaCl and DeMS at the film surface and the differences in the surface densities between the adsorbed films at the film surface and bulk one coexisting at equilibrium were numerically evaluated by applying the thermodynamic equations to the film tension obtained from the contact angle. The film states and phase transitions were clarified in terms of the film thickness and surface densities.     

Impact of surface forces on wetting of hierarchical surfaces and contact angle hysteresis

The influence of the long-range surface forces on the wetting of multi-scale partially wetted surfaces is discussed. The possibility of partial wetting is stipulated by a specific form of the Derjaguin isotherm. Equilibrium of a liquid meniscus inside a cylindrical capillary is used as a model. The interplay of capillary and disjoining pressures governs the equilibrium of the liquid in the nano- and micrometrically scaled pores constituting the relief of the surface. It is shown that capillaries with a radius smaller than a critical one will be completely filled by water, whereas the larger capillaries will be filled only partially. Thus, small capillaries will show the Wenzel type of wetting behavior, while the same liquid inside the large capillaries will promote the Cassie-Baxter type of wetting. Consideration of disjoining/conjoining pressure allows explaining of the “rose petal effect”, when a high apparent contact angle is accompanied with a high contact angle hysteresis.     

Contact angle hysteresis of bovine serum albumin (BSA) solution/metal (Au-Cr) coated glass substrate

Bovine serum albumin (BSA) has an extraordinary property to carry biomolecules. An experimental study on the wettability of BSA is presented in this study. The variations in the surface tension and the equilibrium contact angle with the change in BSA concentration are also reported. The surface tension and the contact angle are measured with pendant and sessile drop techniques, respectively. A nonlinear decrement in the surface tension with the increment in the BSA concentration is observed. An equilibrium contact angle of a BSA solution with particular concentration is determined by studying the hysteresis in the contact angle from dynamic contact angle measurements. The needle-in-drop technique is used to study the hysteresis of the contact angle. It is observed that the obtained surface tension and the equilibrium contact angle vary with the BSA concentration. In this reported study, for the considered combination of the BSA concentration and solid surface, the liquid drop does not recede as the drop volume decreases, which represents nonreceding contact angle condition. The increment in the contact angle with the increment in the BSA concentration is observed. Finally, it is observed that the inclusion of the proteins not only changes the surface tension but also changes the contact angle.     

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.     

As-placed contact angles for sessile drops

As-placed contact angle is the contact angle a drop adapts as a result of its placement on a surface. As expected, the as-placed contact angle, thetaAP, of a sessile drop on a horizontal surface decreases with the drop size due to the increase in hydrostatic pressure. We present a theoretical prediction for thetaAP which shows that it is a unique function of the advancing contact angle, thetaA, drop size, and material properties (surface tensions and densities). We test our prediction with published and new data. The theory agrees with the experiments. From the relation of the as-placed contact angle to drop size the thermodynamic equilibrium contact angle is also calculated.     

Extremes of some foam properties and elasticity of thin foam films near the critical micelle concentration

The elasticity of open and closed thin foam films is analyzed. The elasticity modulus of a closed film is shown to be additive with respect to contributions from Gibbs elasticity and disjoining pressure. A detailed expression for the film elasticity modulus explains the pronounced maxima of foaminess and foam stability near the critical micelle concentration observed earlier in many experiments. A theory of transversal elasticity of thin foam films is formulated under conditions excluding the action of Gibbs elasticity. Near the critical micelle concentration, the theory predicts maxima of the transversal elasticity modulus and of the films thickness as functions of concentration at a given disjoining pressure. The prediction has been verified experimentally by measuring the film thickness in equilibrium foam as a function of height.     

Continuous-flow production of polymeric micelles in microreactors: experimental and computational analysis

The dynamics and stability of a thin, viscous film of volatile liquid flowing under the influence of gravity over a non-uniformly heated substrate are investigated using lubrication theory. Attention is focused on the regime in which evaporation balances the flow due to gravity. The film terminates above the heater at an apparent contact line, with a microscopically thin precursor film adsorbed due to the disjoining pressure. The film develops a weak thermocapillary ridge due to the Marangoni stress at the upstream edge of the heated region. As for spreading films, a more significant ridge is formed near the apparent contact line. For weak Marangoni effects, the film evolves to a steady profile. For stronger Marangoni effects, the film evolves to a time-periodic state. Results of a linear stability analysis reveal that the steady film is unstable to transverse perturbations above a critical value of the Marangoni parameter, leading to finger formation at the contact line. The streamwise extent of the fingers is limited by evaporation. The time-periodic profiles are always unstable, leading to the formation of periodically-oscillating fingers. For rectangular heaters, the film profiles after instability onset are consistent with images from published experimental studies.