Adsorption of acicular particles at liquid-fluid interfaces and the influence of the line tension

In this paper, the adsorption energy of an acicular (prolate and cylindrical) particle onto a liquid-fluid interface and the effect of the line tension are investigated. The results show that, without line tension, acicular particles always prefer to lie flat in the plane of the interface. However, line tension plays a significant role in determining the adsorption of an acicular particle. First, the line tension creates an energy barrier for the adsorption of particles onto an interface. The planar configuration has a larger energy barrier due to the longer contact line. Therefore, the particles prefer to enter the interface in a homeotropic configuration and then rearrange to a planar configuration or an oblique configuration with a small tilt angle. Second, for prolate particles, an energy maximum occurs at some tilt angles when the line tension is large. Therefore, once the prolate particle is adsorbed on the interface in a homeotropic configuration or with a larger tilt angle, it must conquer an energy barrier to rearrange to a planar configuration. For cylindrical particles, when the line tension is higher, the planar configuration will not be the most energy-favorable configuration. The cylindrical particles prefer to stay in the interface with a small tilt angle.     

Optical trapping forces for colloids at the oil-water interface

We calculate the optical trapping forces exerted by a single laser beam strongly focused on a dielectric sphere located at a two-dimensional (2D) oil-water interface. The calculated lateral trapping forces, based on the geometrical optics approximation (GOA), agree with experimental measurements of the trapping force. Importantly, the calculations verify that the radiation force exerted on particles perpendicular to the interface is not sufficient to induce capillary interactions between particle pairs, which could be mistaken for particle-particle interactions. Finally, we find that the trapping forces depend on the three-phase contact angle of the particle at the interface.     

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.     

Force balance of particles trapped at fluid interfaces

We study the effective forces acting between colloidal particles trapped at a fluid interface which itself is exposed to a pressure field. To this end, we apply what we call the "force approach," which relies solely on the condition of mechanical equilibrium and turns to be in a certain sense less restrictive than the more frequently used "energy approach," which is based on the minimization of a free energy functional. The goals are (i) to elucidate the advantages and disadvantages of the force approach as compared to the energy approach, and (ii) to disentangle which features of the interfacial deformation and of the capillary-induced forces between the particles follow from the gross feature of mechanical equilibrium alone, as opposed to features which depend on the details of, e.g., the interaction of the interface with the particles or the boundaries of the system. First, we derive a general stress-tensor formulation of the forces at the interface. On that basis we work out a useful analogy with two-dimensional electrostatics in the particular case of small deformations of the interface relative to its flat configuration. We apply this analogy in order to compute the asymptotic decay of the effective force between particles trapped at a fluid interface, extending the validity of the previous results and revealing the advantages and limitations of the force approach compared to the energy approach. It follows the application of the force approach to the case of deformations of a nonflat interface. In this context, we first compute the deformation of a spherical droplet due to the electric field of a charged particle trapped at its surface and conclude that the interparticle capillary force is unlikely to explain certain recent experimental observations within such a configuration. We finally discuss the application of our approach to a generally curved interface and show as an illustrative example that a nonspherical particle deposited on an interface forming a minimal surface is pulled to regions of larger curvature.     

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.     

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.     

Effect of Surface Depressions on Wetting and Interactions between Hydrophobic Pore Array Surfaces

The surface structure is known to significantly affect the long-range capillary forces between hydrophobic surfaces in aqueous solutions. It is, however, not clear how small depressions in the surface will affect the interaction. To clarify this, we have used the AFM colloidal probe technique to measure interactions between hydrophobic microstructured pore array surfaces and a hydrophobic colloidal probe. The pore array surfaces were designed to display two different pore spacings, 1.4 and 4.0 μm, each with four different pore depths ranging from 0.2 to 12.0 μm. Water contact angles measured on the pore array surfaces are lower than expected from the Cassie-Baxter and Wenzel models and not affected by the pore depth. This suggests that the position of the three-phase contact line, and not the interactions underneath the droplet, determines the contact angle. Confocal Raman microscopy was used to investigate whether water penetrates into the pores. This is of importance for capillary forces where both the movement of the three-phase contact line and the situation at the solid/liquid interface influence the stability of bridging cavities. By analyzing the shape of the force curves, we distinguish whether the cavity between the probe and the surfaces was formed on a flat part of the surface or in close proximity to a pore. The pore depth and pore spacing were both found to statistically influence the distance at which cavities form as surfaces approach each other and the distance at which cavities rupture during retraction.     

Deposition of poly(styrene/alpha-tert-butoxy-omega-vinyl-benzyl-polyglycidol) microspheres on mica plates crossing the liquid-air interface: formation of stripe pattern

Formation of stripelike assemblies of poly(styrene/alpha-tert-butoxy-omega-vinyl-benzyl-polyglycidol) microspheres adsorbed on nonpatterned mica plates moving perpendicularly from suspension of particles through the water-air interface has been observed. It was found that ordered assemblies were formed by capillary forces acting on particles crossing the water-air boundary. At sufficiently high rates of plate movement (i.e., at appropriate dynamic loading conditions) the adsorbed microspheres approaching the water surface begin sliding on the plate, due to capillary forces, in the direction opposite to the plate movement and are kept below the water surface. Plate movement brings new adsorbed particles to the water-air interface, where particles are assembled into aggregates. When particle aggregates are large, the capillary forces cannot overcome shearing forces and the particle assemblies are withdrawn on the plate above the water surface. This process repeated during continuous movement of the plate results in the formation of the quite regularly distributed stripes of adsorbed microspheres. Formation of the regularly distributed particle assemblies depends on concentration of microspheres in suspension.     

Adsorption of microstructured particles at liquid-liquid interfaces

The solid particles are adsorbed at interfaces and form self-assembled structures when the particles have suitable wettability to both liquids. Here, we show theoretically how the microstructure on the particle surface affects their adsorption properties. The physical properties of the interface adsorbing a particle will be described by taking into account the surface roughness due to the microstructure. The microstructure on the surface changes drastically the wettability and the equilibrium position of the adsorbed particle. Therefore, the contact angle of the particle at the three-phase contact line shifts with the particle surface area, because the surface roughness enhances the interfacial properties of the particle surface. Moreover, the range of the interfacial tensions at which the particle is adsorbed becomes narrower with the increase of the surface roughness. The effect of the particle shape on the adsorption properties is also studied. In the case of disk-shaped particles, the energy changes discontinuously when the plane surface of the particle contacts the liquid-liquid interface. The adsorbing position does not change with the surface roughness. The orientation of a parallelepiped particle at the liquid-liquid interface is governed by the aspect ratio and the surface area of the particle. On the other hand, the particle which is partially covered with the microstructured surface is adsorbed firmly at the interface in an oriented state. We should consider not only the interfacial tensions but also the surface structure and the particle shape to control the adsorption behavior of the particle.     

Small solid particles and liquid lenses at fluid/fluid interfaces

The behaviour of small solid particles and liquid droplets at fluid interfaces is of wide interest, in part because of the roles they play in the stability of foams and emulsions. Here we focus on solid particles at liquid interfaces, both singly and in highly structured monolayers. We briefly mention small oil lenses on water in connection with the determination of line tension, τ. Particles are surface-active in the sense that they often adhere quite strongly to liquid surfaces, although of course they are not usually amphiphilic. The three-phase contact line around a particle at an interface is associated with an excess free energy resulting in a tendency of the line to contract (positive τ, which is a 1D analogue of surface tension) or to expand (negative τ). Positive line tension acts so as to push the contact angle of a particle with the fluid interface further away from 90°, i.e. to force the particle towards the more “wetting” of the two bulk phases. It also leads to activation barriers to entry and departure of particles from an interface. The behaviour of particle monolayers at octane/water interfaces is also discussed . It is found that, for monodisperse spherical polystyrene particles containing ionisable sulphate groups at the surface, highly ordered monolayers are formed. This appears to result from very long range electrostatic repulsion mediated through the oil phase. Surface pressure–surface area isotherms are discussed for particle monolayers and it is shown, using light microscopy, that at monolayer “collapse” particles are not expelled from the monolayers but rather the monolayer folds, remaining intact. This has an important bearing on methods, involving the use of the Langmuir trough, for the experimental determination of contact angles and line tensions in particulate systems. Received: 18 July 1999/Accepted: 30 August 1999     

Pressure dependence of the contact angle

When a liquid and its vapor contact a smooth, homogeneous surface, Gibbsian thermodynamics indicates that the contact angle depends on the pressure at the three-phase line of an isothermal system. When a recently proposed adsorption isotherm for a solid-vapor interface is combined with the equilibrium conditions and the system is assumed to be in a cylinder where the liquid-vapor interface can be approximated as spherical, the contact-angle-pressure relation can be made explicit. It indicates that a range of contact angles can be observed on a smooth homogeneous surface by changing the pressure at the three-phase line, but it also indicates that the adsorption at the solid-liquid interface is negative, and leads to the prediction that the contact angle increases with pressure. The predicted dependence of the contact angle on pressure is investigated experimentally in a system that has an independent mechanism for determining when thermodynamic equilibrium is reached. The predictions are in agreement with the measurements. The results provide a possible explanation for contact angle hysteresis.     

Movement of fine particles on an air bubble surface studied using high-speed video microscopy

A CCD high-speed video microscopy system operating at 1000 frames per second was used to obtain direct quantitative measurements of the trajectories of fine glass spheres on the surface of air bubbles. The glass spheres were rendered hydrophobic by a methylation process. Rupture of the intervening water film between a hydrophobic particle and an air bubble with the consequent formation of a three-phase contact was observed. The bubble-particle sliding attachment interaction is not satisfactorily described by the available theories. Surface forces had little effect on the particle sliding with a water film, which ruptured probably due to the submicrometer-sized gas bubbles existing at the hydrophobic particle-water interface.     

Rotation and alignment of anisotropic particles on nonplanar interfaces

We study the alignment of micron-scale particles at air-water interfaces with unequal principle radii of curvature by optical microscopy. The fluid interface bends to satisfy the wetting conditions at the three phase contact line where the interface intersects the particle, creating deflections that increase the area of the interface. These deflections decay far from the particle. The far field interface shape has differing principle radii of curvature over length scales large compared to the particle. The deflections create excess area which depends on the angle of the particle with respect to the principle axes of the interface. We show that when particles create surface deflections with quadrupolar modes, the particles rotate to preferred orientations to minimize the free energy. In experiment, we focus on uniform surface energy particles, for which quadrupolar modes are forced by the particle shape. Analytical expressions for the torque and stable states are derived in agreement with experiment and confirmed computationally.     

Connect the drops: using solids as adhesives for liquids

Colloidal particles are shown to be capable of developing adhesion between liquid phases through a bridging mechanism by which intervening, micrometer-scaled, fluid films are stabilized. Particle dynamics leading to the assembly of the stabilizing structure are discussed. Models for the resulting adhesive force are developed from considerations of both interface shape perturbation and the force applied by surface tension on an individual particle. Finally, predictions from these models are compared to direct measurements of the forces that arise during the separation of adhering interfaces. Such comparisons lead to a novel method for determining the three-phase contact angle inherent to particles residing at fluid interfaces.     

Surface functionalization of SBA-15 and a nonordered mesoporous silica with a 1,4-diazabicyclo[2.2.2]octane derivative: study of CuCl2 adsorption from ethanol solution

The equilibrium position of a spherical or prolate spheroidal particle resembling a needle floating at the interface between two immiscible fluids is discussed. A three-dimensional meniscus attached to an a priori unknown contact line at a specified contact angle is established around the particle, imparting to the particle a capillary force due to surface tension that is balanced by the buoyancy force and the particle weight. An accurate numerical solution for a floating sphere is obtained by solving a boundary-value problem, and the results are compared favorably with an approximate solution where the effect of the particle surface curvature is ignored and the elevation of the contact line is computed using an analytical solution for the meniscus attached to an inclined flat plate. The approximate formulation is applied locally around the nearly planar elliptical contact line of a prolate spheroid to derive a nonlinear algebraic equation governing the position of the particle center and the mean elevation of the contact line. The effect of the fluid and particle densities, contact angle, and capillary length is discussed, and the shape of the contact line is reconstructed and displayed from the local solution.     

Constitutive modeling of contact angle hysteresis

We introduce a phase field model of wetting of surfaces by sessile drops. The theory uses a two-dimensional non-conserved phase field variable to parametrize the Gibbs free energy of the three-dimensional system. Contact line tension and contact angle hysteresis arise from the gradient term in the free energy and the kinetic coefficient respectively. A significant advantage of this approach is in the constitutive specification of hysteresis. The advancing and receding angles of a surface, the liquid-vapor interfacial energy and three-phase line tension are the only required constitutive inputs to the model. We first simulate hysteresis on a smooth chemically homogeneous surface using this theory. Next we show that it is possible to study heterogeneous surfaces whose component surfaces are themselves hysteretic. We use this theory to examine the wetting of a surface containing a circular heterogeneous island. The contact angle for this case is found to be determined solely by the material properties at the contact line in accord with recent experimental data.     

A model for investigating the behaviour of non-spherical particles at interfaces

This paper introduces a simple method for modelling non-spherical particles with a fixed contact angle at an interface whilst also providing a method to fix the particles orientation. It is shown how a wide variety of particle shapes (spherical, ellipsoidal, disc) can be created from a simple initial geometry containing only six vertices. The shapes are made from one continuous surface with edges and corners treated as smooth curves not discontinuities. As such, particles approaching cylindrical and orthorhombic shapes can be simulated but the contact angle crossing the edges will be fixed. Non-spherical particles, when attached to an interface can cause large distortions in the surface which affect the forces acting on the particle. The model presented is capable of resolving this distortion of the surface around the particle at the interface as well as allowing for the particle's orientation to be controlled. It is shown that, when considering orthorhombic particles with rounded edges, the flatter the particle the more energetically stable it is to sit flat at the interface. However, as the particle becomes more cube like, the effects of contact angle have a greater effect on the energetically stable orientations. Results for cylindrical particles with rounded edges are also discussed. The model presented allows the user to define the shape, dimensions, contact angle and orientation of the particle at the interface allowing more in-depth investigation of the complex phenomenon of 3D film distortion around an attached particle and the forces that arise due to it.     

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.     

On the role of the three-phase contact line in surface deformation

Viscoelastic braking theories developed by Shanahan and de Gennes and by others predict deformation of a solid surface at the solid-liquid-air contact line. This phenomenon has only been observed for soft smooth surfaces and results in a protrusion of the solid surface at the three-phase contact line, in agreement with the theoretical predictions. Despite the large (enough to break chemical bonds) forces associated with it, this deformation was not confirmed experimentally for hard surfaces, especially for hydrophobic ones. In this study we use superhydrophobic surfaces composed of an array of silicon nanostructures whose Young modulus is 4 orders of magnitude higher than that of surfaces in earlier recorded viscoelastic braking experiments. We distinguish between two cases: when a water drop forms an adhesive contact, albeit small, with the apparent contact angle θ < 180° and when the drop-surface adhesion is such that the conditions for placing a resting drop on the surface cannot be reached (i.e., θ = 180°). In the first case we show that there is a surface deformation at the three-phase contact line which is associated with a reduction in the hydrophobicity of the surface. For the second case, however, there cannot be a three-phase contact line associated with a drop in contact with the surface, and indeed, if we force-place a drop on the surface by holding it with a needle, no deformation is detected, nor is there a reduction in the hydrophobic properties of the surface. Yet, if we create a long horizontal three-phase contact line by partially immersing the superhydrophobic substrate in a water bath, we see a localized reduction in the hydrophobic properties of the surface in the region where the three-phase contact line used to be. The SEM scan of that region shows a narrow horizontal stripe where the nanorods are no longer there, and instead there is only a shallow structure that is lower than the nanorods height and resembles fused or removed nanorods. Away from that region, either on the part of the surface which was exposed to bulk water or the part which was exposed to air, no change in the hydrophobic properties of the surface is observed, and the SEM scan confirms that the nanorods seem intact in both regions.     

Colloidal rings in a liquid mixture

We investigate the self-assembly of colloidal particles on microscopic decane droplets in water and show that, by use of paramagnetic colloids, it is possible to assemble ringlike structures that can be controlled with a magnetic field. Moreover, the use of paramagnetic colloids allows us to determine the attractive forces between the colloids located at the three-phase contact line between decane, water, and air. The attractive force is in the femtonewton range and is attributed to capillary interactions due to interface deformations. When the liquid emulsion dries on a glass slide, we observe solid deposits in the form of microscopic rings of varying diameters.