The effect of gravity on the drainage of a thin liquid film between a solid sphere and a liquid/fluid interface

A study has been made of the influence of gravitational forces on the thinning of the liquid film which forms as a solid sphere comes to rest on a liquid/fluid interface. It is found that rates of drainage can be dramatically affected by the ratio of gravity to surface tension forces within the film. At long times a secondary film can possibly be formed which spreads out radially from the apex of the sphere.     

Dynamic aspects of small bubble and hydrophilic solid encounters

The capture of solid particles suspended in aqueous solution by rising gas bubbles involves hydrodynamic and physicochemical processes that are central to colloid science. Of the collision, attachment and aggregate stability aspects to the bubble-particle interaction, the crucial attachment process is least understood. This is especially true of hydrophilic solids. We review the current literature regarding each component of the bubble-particle attachment process, from the free-rise of a small, clean single bubble, to the collision, film drainage and interactions which dominate the attachment rate. There is a particular focus on recent studies which employ single, very small bubbles as analysis probes, enabling the dynamic bubble-hydrophilic particle interaction to be investigated, avoiding complications which arise from fluid inertia, deformation of the liquid-vapour interface and the possibility of surfactant contamination.     

Measurement and modeling on hydrodynamic forces and deformation of an air bubble approaching a solid sphere in liquids

The interaction between bubbles and solid surfaces is central to a broad range of industrial and biological processes. Various experimental techniques have been developed to measure the interactions of bubbles approaching solids in a liquid. A main challenge is to accurately and reliably control the relative motion over a wide range of hydrodynamic conditions and at the same time to determine the interaction forces, bubble–solid separation and bubble deformation. Existing experimental methods are able to focus only on one of the aspects of this problem, mostly for bubbles and particles with characteristic dimensions either below 100 μm or above 1 cm. As a result, either the interfacial deformations are measured directly with the forces being inferred from a model, or the forces are measured directly with the deformations to be deduced from the theory. The recently developed integrated thin film drainage apparatus (ITFDA) filled the gap of intermediate bubble/particle size ranges that are commonly encountered in mineral and oil recovery applications. Equipped with side-view digital cameras along with a bimorph cantilever as force sensor and speaker diaphragm as the driver for bubble to approach a solid sphere, the ITFDA has the capacity to measure simultaneously and independently the forces and interfacial deformations as a bubble approaches a solid sphere in a liquid. Coupled with the thin liquid film drainage modeling, the ITFDA measurement allows the critical role of surface tension, fluid viscosity and bubble approach speed in determining bubble deformation (profile) and hydrodynamic forces to be elucidated. Here we compare the available methods of studying bubble–solid interactions and demonstrate unique features and advantages of the ITFDA for measuring both forces and bubble deformations in systems of Reynolds numbers as high as 10. The consistency and accuracy of such measurement are tested against the well established Stokes–Reynolds–Young–Laplace model. The potential to use the design principles of the ITFDA for fundamental and developmental research is demonstrated.     

Drainage of the air-water-quartz film: experiments and theory

Experimental results of the kinetics of drainage of the trapped water film between an approaching air bubble and a quartz plate have been analysed using recent theoretical advances in formulating and solving the flow problem in deformable films. Excellent agreement is obtained between experimental data and a model that assumes the bubble-water interface is tangentially immobile in its hydrodynamic response. The coupling between hydrodynamic pressure, disjoining pressure and film deformation is critical in determining the dynamic behaviour of the drainage process. The Reynolds parallel film model that omits the effects of film deformation predicts results that are qualitatively incorrect.     

Dynamic forces between bubbles and surfaces and hydrodynamic boundary conditions

A bubble attached to the end of an atomic force microscope cantilever and driven toward or away from a flat mica surface across an aqueous film is used to characterize the dynamic force that arises from hydrodynamic drainage and electrical double layer interactions across the nanometer thick intervening aqueous film. The hydrodynamic response of the air/water interface can range from a classical fully immobile, no-slip surface in the presence of added surfactants to a partially mobile interface in an electrolyte solution without added surfactants. A model that includes the convection and diffusion of trace surface contaminants can account for the observed behavior presented. This model predicts quantitatively different interfacial dynamics to the Navier slip model that can also be used to fit dynamic force data with a post hoc choice of a slip length.     

Simultaneous thermal and surfactant-induced Marangoni effects in thin liquid films

The deformation of a thin liquid film in the presence of a surfactant monolayer, varying temperature distributions, and limited mass flux is considered. Use of lubrication theory yields a coupled pair of partial differential equations for the film height and surfactant surface monolayer concentration. The long-wave stability of the isothermal film is examined over a wide range of parameter values. It is shown that droplet patterns are obtained under certain thermal conditions for both an isothermal and nonisothermal underlying substrate. For the case of a localized thermal gradient initially imposed at the air-liquid interface, severe film thinning beneath the heat source was observed, which was not accompanied by droplet formation; pseudo steady states are observed in this case. In all situations the surfactant is found to rigidify the air-liquid interface, retarding thermally driven flow, while evaporation (condensation) acts to destabilize (stabilize) the film.     

Mechanism and Kinetics of Hexamethyldisilazane Reaction with a Fumed Silica Surface

An analytical study is presented on the thermocapillary migration of a fluid sphere within a constant applied temperature gradient in an arbitrary direction with respect to a plane surface. The Peclet and Reynolds numbers are assumed to be small so that the Laplace and Stokes equations, respectively, govern the temperature distributions and fluid velocities inside and outside the droplet. The asymptotic formulas for the temperature and the velocity fields in the quasi-steady situation are obtained by using a method of reflections. The plane surface can be a no-slip solid wall and/or a perfect-slip free surface. The boundary effect on the thermocapillary migration is found to be weaker than that on the motion driven by a body force. Even so, the interaction between the plane and the droplet can be very significant when the gap thickness approaches zero. For the motion of a droplet normal to a solid wall, the effect of the plane surface reduces the translational velocity of the droplet; however, this solid wall can be an enhancement factor on the particle migration as it is translating parallel to the wall. On the other hand, in case of a droplet migrating close to a free surface due to thermocapillarity, the droplet velocity can be either greater or smaller than that which would exist in the absence of the plane surface, depending on the relative thermal conductivity and the surface properties of the particle and its relative distance from the plane. Furthermore, the interacting thickness of the affected region by the presence of the plane is discussed by considering the droplet mobility. Generally speaking, a free surface exerts less influence on the particle movement than does a solid surface. Copyright 2000 Academic Press.     

Dewetting at soft viscoelastic interfaces

The dewetting transition of thin liquid films (approximately 100 nm) at soft viscoelastic interfaces is analyzed theoretically. It is shown that viscoelastic losses in the soft material can drastically increase the time to complete the dewetting. Thus, the influence of the thinning of the liquid film, due to the hydrodynamic drainage caused by the external applied pressure, has to be considered too. The squeezing pressure coupled with the hydrodynamic drainage may slow down the dewetting to almost zero growth rate of the dry zone; in this case a trapped rim of fluid should be observed.     

Hydrodynamic interactions of evaporating droplet with droplet of nonvolatile substance

The hydrodynamic interactions of freely evaporating or growing droplet (suspended in gaseous medium) in the supersaturated vapor with the droplet of nonvolatile substance or spherical solid particle are theoretically studied with allowance for effects that are linear with respect to the Knudsen number. The process of interaction between the volatile droplet and the infinite plane surface of nonvolatile liquid or solid is considered as a limiting case. Numerical estimates of the velocities of the steady motion of evaporating droplets of water and castor oil are reported. For the droplet of water and spherical solid particle, the effect of the heat conductivity of the latter on the velocity of particle motion is considered. Analogous estimates are obtained for a water droplet that evaporates near the infinite solid surface of castor oil or solid. The effects of the droplet size and the heat conductivity of wall on the rate of the evaporation of water droplet are analyzed.     

The effect of interfacial viscosity on the droplet dynamics under flow field

The effects of interfacial viscosity on the droplet dynamics in simple shear flow and planar hyperbolic flow are investigated by numerical simulation with diffuse interface model. The change of interfacial viscosity results in an apparent slip of interfacial velocity. Interfacial viscosity has been found to have different influence on droplet deformation and coalescence. Smaller interfacial viscosity can stabilize droplet shape in flow field, while larger interfacial viscosity will increase droplet deformation, or even make droplet breakup faster. Different behavior is found in droplet coalescence, where smaller interfacial viscosity speeds up film drainage and droplet coalescence, but larger interfacial viscosity postpones the film drainage process. This is due to the change of film shape from flat‐like for smaller interfacial viscosity to dimple‐like for larger interfacial viscosity. The film drainage time still scales as Ca⁰ at smaller capillary number (Ca), and Ca^1.5 at higher capillary number when the interfacial viscosity changes. The interfacial viscosity only affects the transition between these limiting scaling relationships. © 2008 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 46: 1505–1514, 2008     

Deformation and breakup of micro- and nanoparticle stabilized droplets in microfluidic extensional flows

Using a microfluidic flow-focusing device, monodisperse water droplets in oil were generated and their interface populated by either 1 μm or 500 nm amine modified silica particles suspended in the water phase. The deformation and breakup of these Pickering droplets were studied in both pure extensional flow and combined extensional and shear flow at various capillary numbers using a microfluidic hyperbolic contraction. The shear resulted from droplet confinement and increased with droplet size and position along the hyperbolic contraction. Droplet deformation was found to increase with increasing confinement and capillary number. At low confinements and low capillary numbers, the droplet deformation followed the predictions of theory. For fully confined droplets, where the interface was populated by 1 μm silica particles, the droplet deformation increased precipitously and two tails were observed to form at the rear of the droplet. These tails were similar to those seen for surfactant covered droplets. At a critical capillary number, daughter droplets were observed to stream from these tails. Due to the elasticity of the particle-laden interface, these drops did not return to a spherical shape, but were observed to buckle. Although increases in droplet deformation were observed, no tail streaming occurred for the 500 nm silica particle covered droplets over the range of capillary numbers studied.     

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.     

Rheological transition of concentrated emulsions during successive shearing cycles

Rheological investigations have identified a shear viscosity transition from shear thinning to Newtonian at low to moderate shear rates for concentrated polydimethylsiloxane emulsions during successive shearing cycles. The viscosity "flattening" behavior is dependent on the maximum shear rate applied and on droplet deformation. Atomic force microscopy measurements indicate attraction between the "repulsive" emulsion droplets under deformation. The results suggest formation of stable droplet layers due to deformation under high shear hydrodynamic compression. Based on these findings, unique methods to control the post-shear rheology of concentrated emulsions can be envisaged.     

Thermophoretic Motion of a Sphere Parallel to an Insulated Plane

An analytical study is presented for the thermophoresis of a sphere in a constant applied temperature gradient parallel to an adiabatic plane. The Knudsen number is assumed to be small so that the fluid flow can be described by a continuum model with a thermal creep and a hydrodynamic slip at the particle surface. A method of reflections is used to obtain the asymptotic formulas for the temperature and velocity fields in the quasisteady situation. The thermal insulated plane may be a solid wall (no-slip) and/or a free surface (perfect-slip). The boundary effect on the thermophoretic motion is found to be weaker than that on the axisymmetric thermophoresis of a sphere normal to a plane with constant temperature. In comparison with the motion driven by gravitational force, the interaction between the particle and the boundary is less significant under thermophoresis. Even so, the interaction between the plane and the particle can be very strong when the gap thickness approaches zero. For the thermophoretic motion of a particle parallel to a solid plane, the effect of the plane surface is to reduce the translational velocity of the particle. In the case of particle migration parallel to a free surface due to thermophoresis, the translating velocity of a particle can be either greater or smaller than that which would exist in the absence of the plane surface, depending on the relative thermal conductivity and the surface properties of the particle and its relative distance from the plane. Not only the translational velocity but also the rotational velocity of the thermophoretic sphere near the plane boundary is formulated analytically. The rotating direction of the particle is strongly dominated by its surface properties and the internal-to-external thermal conductivity. Besides the particle motion, the thickness of the thermophoretic boundary layer is evaluated by considering the thermophoretic mobility. Generally speaking, a free surface exerts less influence on the particle movement than a solid wall. 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.     

Application of the Christensen‐Lo Model to the Reinforcement of Elastomers by Fractal Fillers

This paper is concerned with the development of a hydrodynamic model for the reinforcement of rubber by colloidal fillers such as silica and carbon black. Each fractal aggregate is replaced by an equivalent effective sphere, and the reinforcing ability of the latter is estimated using the Christensen‐Lo solution of the “three‐phase composite sphere model”. With a single adjustable parameter, the model allows a quantitative interpretation of the small‐strain modulus of rubber loaded with up to 50 PHR of N234 carbon black, which falls just below the filler overlap concentration. No additional contributions to the small‐strain modulus by filler–filler “interactions” are needed to interpret the data.

A branched filler aggregate made up of spherical primary particles (black) and the effective sphere replacing it (gray).     

Structures and dynamics of thermosensitive microgel suspensions studied with three-dimensional cross-correlated light scattering

The structure factors, short- and long-time diffusion coefficients, and hydrodynamic interactions of concentrated poly(N-isopropylacryamide) microgel suspensions were measured with simultaneous static and dynamic three-dimensional cross-correlated light scattering. The data are interpreted through comparison to hard sphere theory. The structure factors are known to be described well by the hard sphere approximation. When the structure factor is fit to an effective hard sphere volume fraction and radius, the diffusion and hydrodynamic interactions are also well described by the hard sphere model. We demonstrate that one single hard sphere volume fraction is sufficient to describe the microgel structures, hydrodynamic interactions, and long- and short-time collective diffusion coefficients. This result is surprising because the particle form of the microgels at these temperatures is not rigid, but rather "fuzzy" spheres with dangling polymer chains.     

Forces between a hard surface and an air–aqueous interface with and without films

The adsorption of particles to air–aqueous interfaces is vital in many applications, such as mineral flotation and the stabilization of food foams. The forces in the system determine whether a particle will attach to an air–aqueous interface. The forces between a particle and an air–aqueous interface are influenced by Derjaguin–Landau–Verwey–Overbeek forces (i.e. van der Waals and electrostatic forces), non–Derjaguin–Landau–Verwey–Overbeek forces (e.g. hydrophobic, hydrodynamic, structural, and capillary forces), liquid drainage, and liquid flow. As an air–aqueous interface can be deformed by a particle, the forces measured between an air–aqueous interface and a particle can differ from those measured between two hard surfaces separated by liquid. The presence of a film at an air–aqueous interface can also change the forces.