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.     

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.     

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.     

Thermal conductivity and interfacial energy of solid Bi solution in the Bi–Al–Zn eutectic system

The equilibrated grain boundary groove shapes for solid Bi solution (Bi–6.1 at.%Zn–0.38 at.%Al) in equilibrium with the Bi–Al–Zn eutectic liquid have been observed from quenched sample with a radial heat flow apparatus. The Gibbs–Thomson coefficient, solid–liquid interfacial energy and grain boundary energy of solid Bi solution have been determined from the observed grain boundary groove shapes. The variations of thermal conductivity with temperature for solid Bi solution (Bi–6.1 at.%Zn–0.38 at.%Al) has been measured up to five degree below the melting temperature by using radial heat flow technique. The ratio of thermal conductivity of equilibrated Bi–Al–Zn eutectic liquid phase to solid Bi solution (Bi–6.1 at.%Zn–0.38 at.%Al) phase has also been measured with a Bridgman type growth apparatus at the melting temperature.     

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.     

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.     

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.     

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.     

Decay of standing foams: drainage,coalescence and collapse

A summary of recent theoretical work on the decay of foams is presented. In a series of papers, we have proposed models for the drainage, coalescence and collapse of foams with time. Each of our papers dealt with a different aspect of foam decay and involved several assumptions. The fundamental equations, the assumptions involved and the results obtained are discussed in detail and presented within a unified framework.Film drainage is modeled using the Reynolds equation for flow between parallel circular disks and film rupture is assumed to occur when the film thickness falls below a certain critical thickness which corresponds to the maximum disjoining pressure. Fluid flow in the Plateau border channels is modeled using a Hagen-Poiseuille type flow in ducts with triangular cross-section.The foam is assumed to be composed of pentagonal dodecahedral bubbles and global conservation equations for the liquid, the gas and the surfactant are solved to obtain information about the state of the decaying foam as a function of time. Homogeneous foams produced by mixing and foams produced by bubbling (pneumatic foams) are considered. It is shown that a draining foam eventually arrives at a mechanical equilibrium when the opposing forces due to gravity and the Plateau-border suction gradient balance each other. The properties of the foam in this equilibrium state can be predicted from the surfactant and salt concentration in the foaming solution, the density of the liquid and the bubble radius.For homogeneous foams, it is possible to have conditions under which there is no drainage of liquid from the foam. There are three possible scenarios at equilibrium: separation of a single phase (separation of the continuous phase liquid by drainage or separation of the dispersed phase gas via collapse), separation of both phases (drainage and collapse occurs) or no phase separation (neither drainage nor collapse occurs). It is shown that the phase behavior depends on a single dimensionless group which is a measure of the relative magnitudes of the gravitational and capillary forces. A generalized phase diagram is presented which can be used to determine the phase behavior.For pneumatic foams, the effects of various system parameters such as the superficial gas velocity, the bubble size and the surfactant and salt concentrations on the rate of foam collapse and the evolution of liquid fraction profile are discussed. The steady state height attained by pneumatic foams when collapse occurs during generation is also evaluated.Bubble coalescence is assumed to occur due to the non-uniformity in the sizes of the films which constitute the faces of the polyhedral bubbles. This leads to a non-uniformity of film-drainage rates and hence of film thicknesses within any volume element in the foam. Smaller films drain faster and rupture earlier, causing the bubbles containing them to coalesce. This leads to a bubble size distribution in the foam, with the bubbles being larger in regions where greater coalescence has occurred.The formation of very stable Newton black films at high salt and surfactant concentrations is also explained.     

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.     

Effect of cosurfactant on the free-drainage regime of aqueous foams

We report results of drainage in aqueous foams of small bubble size D (D=180 microm) prepared with SDS-dodecanol solutions. We have performed free-drainage experiments in which local drainage rates are measured by electrical conductivity and by light scattering techniques. We have investigated the role of the surfactant-cosurfactant mass ratio on the drainage regime. The results confirm that a drainage regime corresponding to a high surface mobility can indeed be found for such small bubbles, and show that an increase in the cosurfactant content can induce a transition to a low surface mobility drainage regime. We show that the transition is not linked to variations of the bulk properties, but rather to variations of the interfacial properties. However, the results show that the added amount of dodecanol to trigger the transition is quite high, evidencing that the relevant control parameter for drainage regimes includes both bubble size and interfacial contributions.     

Dynamic surface and interfacial tensions of surfactant and polymer solutions

Dynamic surface and interfacial tensions are the most frequently measured non-equilibrium properties of adsorption layers at liquid interfaces. The review presents the theoretical basis of adsorption kinetics, taking into consideration different adsorption mechanisms, and specific experimental conditions, such as liquid flow and interfacial area changes. Analytical solutions, if available, approximations as well as numerical procedures for direct solution of the physical models are presented.Several experimental techniques are discussed frequently used in studies of the dynamic adsorption behaviour of surfactants and polymers at liquid interfaces: drop volume, maximum bubble pressure, and pendent drop technique, drop pressure tensiometry, pulsating bubble and elastic ring method. Experimental results, most of all obtained with different technique on one and the same surfactant system, are then discussed on the basis of current theories.Finally, the role of dynamic interfacial properties in several practical applications is discussed: foam and emulsion film formation and stabilisation, rising of bubbles and drops in a surfactant solution.