Theory of non-equilibrium force measurements involving deformable drops and bubbles

Over the past decade, direct force measurements using the Atomic Force Microscope (AFM) have been extended to study non-equilibrium interactions. Perhaps the more scientifically interesting and technically challenging of such studies involved deformable drops and bubbles in relative motion. The scientific interest stems from the rich complexity that arises from the combination of separation dependent surface forces such as Van der Waals, electrical double layer and steric interactions with velocity dependent forces from hydrodynamic interactions. Moreover the effects of these forces also depend on the deformations of the surfaces of the drops and bubbles that alter local conditions on the nanometer scale, with deformations that can extend over micrometers. Because of incompressibility, effects of such deformations are strongly influenced by small changes of the sizes of the drops and bubbles that may be in the millimeter range. Our focus is on interactions between emulsion drops and bubbles at around 100 μm size range. At the typical velocities in dynamic force measurements with the AFM which span the range of Brownian velocities of such emulsions, the ratio of hydrodynamic force to surface tension force, as characterized by the capillary number, is ~ 10^{− 6} or smaller, which poses challenges to modeling using direct numerical simulations. However, the qualitative and quantitative features of the dynamic forces between interacting drops and bubbles are sensitive to the detailed space and time-dependent deformations. It is this dynamic coupling between forces and deformations that requires a detailed quantitative theoretical framework to help interpret experimental measurements. Theories that do not treat forces and deformations in a consistent way simply will not have much predictive power. The technical challenges of undertaking force measurements are substantial. These range from generating drop and bubble of the appropriate size range to controlling the physicochemical environment to finding the optimal and quantifiable way to place and secure the drops and bubbles in the AFM to make reproducible measurements. It is perhaps no surprise that it is only recently that direct measurements of non-equilibrium forces between two drops or two bubbles colliding in a controlled manner have been possible. This review covers the development of a consistent theory to describe non-equilibrium force measurements involving deformable drops and bubbles. Predictions of this model are also tested on dynamic film drainage experiments involving deformable drops and bubbles that use very different techniques to the AFM to demonstrate that it is capable of providing accurate quantitative predictions of both dynamic forces and dynamic deformations. In the low capillary number regime of interest, we observe that the dynamic behavior of all experimental results reviewed here are consistent with the tangentially immobile hydrodynamic boundary condition at liquid–liquid or liquid–gas interfaces. The most likely explanation for this observation is the presence of trace amounts of surface-active species that are responsible for arresting interfacial flow.     

Experimental determination of dissociation constant,Henry's constant,heat of reactions,SO2 absorbed and gas bubble–liquid interfacial area for dilute sulphur dioxide absorption into water

The experimental results for dissociation constant, Henry's constant, heat of reactions and gas bubble–liquid interfacial area for absorption of dilute SO₂ (SO₂ partial pressure upto 0.963 kPa) into water in bubble column are presented. The relations between Henry's constant versus temperature and dissociation constant versus temperature have been proposed based on the present experimental investigation. The temperature dependence on gas bubble–liquid interfacial area per unit volume of liquid has examined. Result shows a very small decrease in interfacial area with increase in temperature. The experimental results are used to correlate total SO₂ absorbed at saturation as function of temperature and SO₂ partial pressure. Comparisons between experimental results and literature data have also been made at different temperatures. The reasonable fittings between the data obtained from correlation and literature data shows that the proposed correlation can be successfully used for predicting the absorption equilibria of dilute SO₂ in distilled water at different temperatures.     

Effects of interface mobility on the dynamics of colliding bubbles

At its core, the outcome of the collision between air bubbles is determined by the hydrodynamic interaction forces, which in turn are strongly dependent on the tangential mobility of the gas–liquid interfaces. A clean gas–liquid interface is tangentially mobile, whereas the presence of surfactant contaminants can immobilise the interface. Bubbles with mobile surfaces coalescence much easier because of the low hydrodynamic resistance to drainage of the thin liquid film separating the colliding bubbles. In this opinion, we highlight recent experimental and numerical simulations demonstrating that in addition to the expected faster coalescence, mobile-surface bubbles can produce a much stronger rebound from a mobile liquid interface compared to an immobile one. The stronger rebound is explained by the lower viscous dissipation during collisions involving mobile surfaces. The role of the surface mobility in controlling the stability of gas or liquid emulsion should be reassessed in the light of these new findings.     

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.     

Bubble shapes and orientations in low Re simple shear flow

We present measurements of shape and orientation of air bubbles in a viscous Newtonian fluid deformed by simple shear. The apparatus is a variation of the "parallel band" device developed by G. I. Taylor. Previous experimental studies on low viscosity ratio, low Reynolds number (Re < 1) bubble deformation have focussed on either small or large deformations (mostly small deformation) and have only qualitatively examined the orientation of bubbles except for small deformations. Our data set spans both the theoretical small deformation and high deformation limits. With these data we confirm theoretical relationships and assess the range of capillary numbers (Ca) over which theoretical relationships for shape and orientation of bubbles are appropriate. We also examine the geometry of deformed bubbles as they relax to a spherical shape once shear stresses are removed. Our data indicate that for extremely small Reynolds numbers and viscosity ratios, the small deformation theoretical relationship first developed by Taylor, is a good approximation for Ca<0.5. The large deformation results for both shape and bubble orientation derived by Hinch and Acrivos agree with our data for Ca>1 and Ca>0.5, respectively.     

Anomalous Capillary Pressure, Stress, and Stability of Solids-Coated Bubbles

A two-dimensional theoretical model for solids-coated, or "armored," bubbles shows how the armor can support a liquid-vapor interface of reduced or reversed curvature between the particles, giving the bubble zero or even negative capillary pressure. The inward capillary force pulling the particles into the center of the bubble are balanced by large contact forces between the particles in the armor. Thus the bubble is stabilized against dissolution of gas into surrounding liquid, which otherwise would rapidly collapse the bubble. The stresses between particles in such cases are large and could drive sintering of the particles into a rigid framework. Earlier work on solids-coated bubbles assumed that solids can freely enter or leave the bubble surface as the bubble shrinks or expands. In such a case, armored bubbles would not be stable to gas dissolution into surrounding liquid. A new free-energy analysis, however, suggests that a shrunken bubble would not spontaneously expel a solid particle from its armor to relieve stress and allow the bubble to shrink further. Implications and limitations of the theory are discussed. Copyright 1999 Academic Press.     

Measurement of dynamical forces between deformable drops using the atomic force microscope. I. Theory

Recent experimental developments have enabled the measurement of dynamical forces between two moving liquid drops in solution using an atomic force microscope (AFM). The drop sizes, interfacial tension, and approach velocities used in the experiments are in a regime where surface forces, hydrodynamics, and drop deformation are all significant. A detailed theoretical model of the experimental setup which accounts for surface forces, hydrodynamic interactions, droplet deformation, and AFM cantilever deflection has been developed. In agreement with experimental observations, the calculated force curves show pseudo-constant compliance regions due to drop flattening, as well as attractive pull-off forces due mainly to hydrodynamic lubrication forces.     

Do hydration forces play a role in thin film drainage and rupture observed in electrolyte solutions?

The mechanism that controls bubble coalescence in electrolyte solutions remains unresolved. The problem is difficult as sensitive dynamic thin film processes are critical. Here we discuss the relationship between film dynamics, specific-ion effects and the combining rules that codify electrolyte effects on bubble coalescence. The relationship with Hofmeister effects is explored, revealing that these very different manifestations of specific ion effects ultimately have the same origin, being the interfacial positioning of ions, which for the air–water interface correlates with the empirically derived α and β assignments used in the combining rules. Ion hydration is important as it strongly influences the interfacial positioning of ions and therefore ultimately bubble coalescence, however dynamic events determine if a collision results in coalescence and therefore we conclude that hydration forces play no role in bubble coalescence in electrolyte solutions.     

Nonlinear instabilities and pathways of rupture in thin liquid bilayers

A long-wave nonlinear analysis of dewetting of thin (<100 nm) liquid bilayers on solid substrates is presented. The short and the long time dynamics, interfacial morphologies, and the pathways of rupture and dewetting are studied to assess the roles of interfacial energies, film thicknesses, and viscosities. The twin interfaces (liquid-liquid and liquid-air) of bilayers under the influence of attractive van der Waals forces show a variety of dewetting pathways which, depending on the interfacial energies and film thicknesses, initially start with one of the two basic modes of instability--in-phase bending and out-of-phase squeezing. These short time modes of evolution and the extent of relative deformations at the interfaces are predicted from the linear stability analysis and verified by the nonlinear simulations. Simulations also show that in the later nonlinear regime, the intermolecular and viscous forces can profoundly modify the initial mode of instability and its growth rate leading to different pathways of dewetting and late stage morphologies. The complex late time patterns such as embedded droplets, inversion of top and bottom phases, and encapsulation of one fluid into the other are also engendered by tuning the intermolecular forces.     

Particle motion near a deformable fluid interface

The present paper reports on one aspect of our recent studies of the hydrodynamic interactions of two deformable particles in a viscous fluid. This general hydrodynamics problem represents an initial step toward a fundamental invertigation of particle/ drop or droplet/droplet interactions in processes such as coalescence and flotation where both hydrodynamic and colloidal effects may be important. Here we consider only the limiting problem of translation of a rigid sphere with constant velocity normal to the plane of an initially flat interface. The Reynolds number is assumed to be vanishingly small; however, no restriction is imposed on the magnitude of the interface deformation.A primary focus of our research has been the qualitative dependence of the mode of interface deformation on the viscosity ratio, and on appropriate non-dimensional measures of interfacial tension and the density difference across the interface. In some instances, the deformation is relatively small and a so called “film drainage” configuration is attained as the particle passes across the plane of the undeformed interface. In other cases. however, the particle passes well into the domain of the second fluid while still surrounded by a layer of the first fluid that is connected to its original domain by a thin column (or “tail”) of fluid behind the sphere. In these latter cases, the rate of thinning of the tail is greate than the rate of thinning of the fluid layer around the particle; thus suggesting a second mode of particle “breakingthrough”, in addition to that associated with the film drainage configuration.     

Forces between two oil drops in aqueous solution measured by AFM

The surface and hydrodynamic forces between individual oil droplets in solution can provide insight into both emulsion stability and processes such as drop coalescence in liquid-liquid extraction. We present the first measurements of the interaction forces between alkane droplets in aqueous solution using atomic force microscopy. The radii of the droplets were well below the capillary lengths for the system, thus gravity effects are negligible, and interfacial tension and interaction forces governed the system behavior. The effects of modulating electrostatic double-layer interactions and interfacial tension through the presence of an anionic surfactant are demonstrated. Challenges in interpretation of the force data due to drop deformation are also discussed. A range of drop approach and retract speeds was used to determine the regime where hydrodynamic drainage effects had significant impact on the measurement.     

Multibubble Sonoluminescence from a Theoretical Perspective

In the present review, complexity in multibubble sonoluminescence (MBSL) is discussed. At relatively low ultrasonic frequency, a cavitation bubble is filled mostly with water vapor at relatively high acoustic amplitude which results in OH-line emission by chemiluminescence as well as emissions from weakly ionized plasma formed inside a bubble at the end of the violent bubble collapse. At relatively high ultrasonic frequency or at relatively low acoustic amplitude at relatively low ultrasonic frequency, a cavitation bubble is mostly filled with noncondensable gases such as air or argon at the end of the bubble collapse, which results in relatively high bubble temperature and light emissions from plasma formed inside a bubble. Ionization potential lowering for atoms and molecules occurs due to the extremely high density inside a bubble at the end of the violent bubble collapse, which is one of the main reasons for the plasma formation inside a bubble in addition to the high bubble temperature due to quasi-adiabatic compression of a bubble, where “quasi” means that appreciable thermal conduction takes place between the heated interior of a bubble and the surrounding liquid. Due to bubble–bubble interaction, liquid droplets enter bubbles at the bubble collapse, which results in sodium-line emission.     

The boundary condition at the air–liquid interface and its effect on film drainage between colliding bubbles

The drainage of thin liquid films between colliding bubbles is strongly influenced by the boundary conditions at the air–liquid interface. Theoretically, the interface should not resist any tangential stress (fully mobile) in a clean water system, resulting in very fast film drainage and coalescence between bubbles within milliseconds. In reality, under most experimental and industrial conditions, the presence of impurities or surfactants can immobilize the interface and significantly hinder bubble coalescence by several orders of magnitude. In this opinion, we introduce the recent progress on understanding the boundary conditions at the air–water interface, and how they may affect the outcome of bubble collisions. The transition from mobile to immobile boundary conditions in the presence of contaminations is discussed. Despite the considerable recent progress, there are still experimental and theoretical challenges remaining on this topic, for example, finding the mechanism for hindered bubble coalescence by high salt concentrations.     

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.     

Measurement and analysis of forces in bubble and droplet systems using AFM

The use of atomic force microscopy to measure and understand the interactions between deformable colloids - particularly bubbles and drops - has grown to prominence over the last decade. Insight into surface and structural forces, hydrodynamic drainage and coalescence events has been obtained, aiding in the understanding of emulsions, foams and other soft matter systems. This article provides information on experimental techniques and considerations unique to performing such measurements. The theoretical modelling frameworks which have proven crucial to quantitative analysis are presented briefly, along with a summary of the most significant results from drop and bubble AFM measurements. The advantages and limitations of such measurements are noted in the context of other experimental force measurement techniques.     

THE ROLE OF INTERFACIAL INTERACTIONS IN THE PARTICLE/BUBBLE ATTACHMENT

The Van Oss surface thermodynamic theory of polar and apolar interfacial interactions was extended to the interaction between mineral surfaces and bubbles across liquid media. The acid base (polar) interfacial interactions are supposed to be responsible for the hydration repulsion between a hydrophilic mineral and a bubble as well as for the hydrophobic attraction between a hydrophobic mineral and the bubble.     

Adsorption of heavy metal ions from aqueous solution by polyrhodanine-encapsulated magnetic nanoparticles

The influence of salt concentration on the terminal velocities of gravity-driven single bubbles sliding along an inclined glass wall has been investigated, in an effort to establish whether surface forces acting between the wall and the bubble influence the latter's mobility. A simple sliding bubble apparatus was employed to measure the terminal velocities of air bubbles with radii ranging from 0.3 to 1.5 mm sliding along the interior wall of an inclined Pyrex glass cylinder with inclination angles between 0.6 and 40.1°. Experiments were performed in pure water, 10 mM and 100 mM KCl solutions. We compared our experimental results with a theory by Hodges et al. which considers hydrodynamic forces only, and with a theory developed by two of us which considers surface forces to play a significant role. Our experimental results demonstrate that the terminal velocity of the bubble not only varies with the angle of inclination and the bubble size but also with the salt concentration, particularly at low inclination angles of ～1-5°, indicating that double-layer forces between the bubble and the wall influence the sliding behavior. This is the first demonstration that terminal velocities of sliding bubbles are affected by disjoining pressure.     

Effect of Surfactants on the Film Drainage

Drainage of a partially mobile thin liquid film between two deformed and nondeformed gas bubbles with different radii is studied. The lubrication approximation is used to obtain the influence of soluble and insoluble surfactants on the velocity of film thinning in the case of quasi-steady state approach. The material properties of the interfaces (surface viscosity, Gibbs elasticity, surface diffusivity, and/or bulk diffusivity) are taken into account. In the case of deformed bubbles the influence of the meniscus is illustrated assuming simple approximated shape for the local film thickness. Simple analytical solutions for large and small values of the interfacial viscosity, and for deformed and nondeformed bubbles, are derived. The correctness of the boundary conditions used in the literature is discussed. The numerical analysis of the governing equation shows the region of transition from partially mobile to immobile interfaces. Quantitative explanation of the following effects is proposed: (i) increase of the mobility due to increasing bulk and surface diffusivities; (ii) role of the surface viscosity, comparable to that of the Gibbs elasticity; and (iii) significant influence of the meniscus on the film drainage due to the increased hydrodynamic resistance. Copyright 1999 Academic Press.     

Ion-specific influence of electrolytes on bubble coalescence in nonaqueous solvents

We report the effects of electrolytes on bubble coalescence in nonaqueous solvents methanol, formamide, propylene carbonate, and dimethylsulfoxide (DMSO). Results in these solvents are compared to the ion-specific bubble coalescence inhibition observed in aqueous electrolyte solutions, which is predicted by simple, empirical ion combining rules. Coalescence inhibition by electrolytes is observed in all solvents, at a lower concentration range (0.01 M to 0.1M) to that observed in water. Formamide shows ion-specific salt effects dependent upon ion combinations in a way analogous to the combining rules observed in water. Bubble coalescence in propylene carbonate is also consistent with ion-combining rules, but the ion assignments differ to those for water. In both methanol and DMSO all salts used are found to inhibit bubble coalescence. Our results show that electrolytes influence bubble coalescence in a rich and complex way, but with notable similarities across all solvents tested. Coalescence is influenced by the drainage of fluid between two bubbles to form a film and then the rupture of the film and one might expect that these processes will vary dramatically between solvents. The similarities in behavior we observe show that coalescence inhibition is unlikely to be related to the surface forces present but is perhaps related to the dynamic thinning and rupture of the liquid film through the hydrodynamic boundary condition.