Rogerio Manica

Film drainage and coalescence between deformable drops and bubbles

The interaction between deformable drops or bubbles encompasses a number of distinguishing characteristics not present in the interaction between solid bodies. The drops can entrap a thin liquid film of the continuous phase that can lead to a stable film or coalescence. But before leading to either of these outcomes, the film must drain under the influence of an external driving force. This drainage process exhibits all the characteristic features of dynamic interactions between soft materials. For example, the spatial and temporal variations of forces and geometric deformations, arising from hydrodynamic flow, surface forces and variations in material properties, are all inextricably interconnected. Recent measurements of time-varying deformations and forces between interacting drops and bubbles confirmed that dynamic forces and geometric deformations are coupled and provide the key to understand novel phenomena such as the “wimple” in mechanically perturbed films. The counter-intuitive phenomenon of coalescence triggered by separating proximal drops or bubbles can also be elucidated using the same theoretical framework. One approach to modelling such systems is to use a fluid mechanics formulation of two-phase flow for which a number of parametric numerical studies have been made. Another popular approach focuses on describing the thin film between the interacting drops or bubbles with a flat film model upon which a phenomenological film drainage and rupture mechanism has been developed. While both models have a similar genesis, their predictions of the fate of the draining film are quite different. Furthermore, there have been few quantitative comparisons between results obtained from many different experimental approaches with either theory. One reason for this is perhaps due to difficulties in matching experimental parameters to model conditions. A direct attempt to model dynamic behaviour in many experimental studies is challenging as the model needs to be able to describe phenomena spanning six orders of magnitude in length scales. However, with the recent availability of accurate experimental studies concerning dynamic interaction between drops and bubbles that use very different, but complementary approaches, it is timely to conduct a critical review to compare such results with long-accepted paradigms of film stability and coalescence. This topic involves the coupling of behaviour on the millimetre–micrometre scale familiar to readers with an engineering and fluid mechanics background to phenomena on the micrometre–nanometre scale that is the domain of the interfacial science and nanotechnology community.

Dynamic interactions between microbubbles in water

The interaction between moving bubbles, vapor voids in liquid, can arguably represent the simplest dynamical system in continuum mechanics as only a liquid and its vapor phase are involved. Surprisingly, and perhaps because of the ephemeral nature of bubbles, there has been no direct measurement of the time-dependent force between colliding bubbles which probes the effects of surface deformations and hydrodynamic flow on length scales down to nanometers. Using ultrasonically generated microbubbles (∼100 μm size) that have been accurately positioned in an atomic force microscope, we have made direct measurements of the force between two bubbles in water under controlled collision conditions that are similar to Brownian particles in solution. The experimental results together with detailed modeling reveal the nature of hydrodynamic boundary conditions at the air/water interface, the importance of the coupling of hydrodynamic flow, attractive van der Waals–Lifshitz forces, and bubble deformation in determining the conditions and mechanisms that lead to bubble coalescence. The observed behavior differs from intuitions gained from previous studies conducted using rigid particles. These direct force measurements reveal no specific ion effects at high ionic strengths or any special role of thermal fluctuations in film thickness in triggering the onset of bubble coalescence.

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.

Hydrodynamic forces involving deformable interfaces at nanometer separations

A model is developed to describe the dynamic forces acting between two deformable drops, or between one drop and a solid surface, when they are in relative axisymmetric motion at separations of ≲100nm in a Newtonian liquid. Forces arise from hydrodynamic pressure in the draining liquid film that separates the interfaces and from disjoining pressure due to repulsive or attractive surface forces. Predictions of the model are successfully compared with recent experimental measurements of the force between two micrometer-scale surfactant stabilized decane drops in water in an atomic force microscope [S. L. Carnie, D. Y. C. Chan, C. Lewis, R. Manica, and R. R. Dagastine, Langmuir 21, 2912 (2005); R. R. Dagastine, R. Manica, S. L. Carnie, D. Y. C. Chan, G. W. Stevens, and F. Grieser, Science 313, 210 (2006)] and with subnanometer resolution measurements of time-dependent deformations of a millimeter-scale mercury drop approaching a flat mica surface in a modified surface force apparatus [J. N. Connor and R. G. Horn, Faraday Discuss. 123, 193 (2003); R. G. Horn, M. Asadullah, and J. N. Connor, Langmuir 22, 2610 (2006)]. Special limits of the model applicable to small and moderate deformation regimes are also studied to elucidate the key physical ingredients that contribute to the characteristic behavior of dynamic collisions involving fluid interfaces.A model is developed to describe the dynamic forces acting between two deformable drops, or between one drop and a solid surface, when they are in relative axisymmetric motion at separations of ≲100nm in a Newtonian liquid. Forces arise from hydrodynamic pressure in the draining liquid film that separates the interfaces and from disjoining pressure due to repulsive or attractive surface forces. Predictions of the model are successfully compared with recent experimental measurements of the force between two micrometer-scale surfactant stabilized decane drops in water in an atomic force microscope [S. L. Carnie, D. Y. C. Chan, C. Lewis, R. Manica, and R. R. Dagastine, Langmuir 21, 2912 (2005); R. R. Dagastine, R. Manica, S. L. Carnie, D. Y. C. Chan, G. W. Stevens, and F. Grieser, Science 313, 210 (2006)] and with subnanometer resolution measurements of time-dependent deformations of a millimeter-scale mercury drop approaching a flat mica surface in a modified surface force apparatus [J. N. Connor and R. G. ...

Precision AFM Measurements of Dynamic Interactions between Deformable Drops in Aqueous Surfactant and Surfactant-Free Solutions

The atomic force microscope (AFM) has provided unprecedented opportunities to study velocity-dependent interactions between deformable drops and bubbles under a range of solution conditions. The challenge is to design an experimental system that enables accurate force spectroscopy of the interaction between deformable drops and thus the extraction of accurate quantitative information about the physically important force-separation relation. This step requires very precise control and knowledge of the interfacial properties of the interacting drops, the drive conditions of the force-sensing cantilever, the disposition of the interacting drops on the substrate and on the cantilever, and transducer calibrations of the instrument in order to quantify the effects of approach velocities and interfacial deformation. This article examines and quantifies in detail all experimental conditions that are necessary to facilitate accurate processing of dynamic force spectroscopy data from the AFM using the well-defined system of tetradecane drops in aqueous solutions under surfactant and surfactant-free conditions over a range of force magnitudes that has not been attained before. The ability of drops to deform and increase the effective area of interaction instead of decreasing the distance of closest approach when disjoining pressure exceeds the Laplace pressure means that the DLVO paradigm of colloidal stability as being determined by a balance of kinetic energy against the height of the primary maximum is no longer valid. The range of interfacially active species present in alkane-aqueous systems investigated provides insight into the applicability of the tangentially immobile boundary condition in colloidal interactions.

Homo- and hetero-interactions between air bubbles and oil droplets measured by atomic force microscopy

The atomic force microscope was used to analyse the interactions between bubbles and oil droplets in surfactant-free aqueous solutions. Both homo- (bubble-bubble and drop-drop) and hetero- (bubble-drop) interactions were examined to elucidate the role of colloidal and hydrodynamic forces which, together with interfacial deformations dictate the stability in these systems. It is shown that electrical double-layer forces can be rendered attractive within a small pH range, and that the Van der Waals force can be switched from attractive to repulsive by material choice and ionic strength through salt effects on the so-called ‘zero-frequency’ term of the Lifshitz theory. By measuring interaction events between bubbles and drops at higher velocities, it is seen that deformation of the bodies and lubrication in the film generated between them can be predicted with a continuum hydrodynamic theory. These results suggest that solution pH and droplet material choice can be used to enhance or inhibit coalescence in such multi-component and multi-phase systems, and this may prove useful in controlling the behaviour of systems in microfluidics, as well as dispersion and formulation science.

An extended Bretherton model for long Taylor bubbles at moderate capillary numbers

When (long) bubbles are transported in tubes containing a fluid, the presence of a thin film of fluid along the tube walls causes the velocity of the bubble to be different from the average fluid velocity. Bretherton [“The motion of long bubbles in tubes,” J. Fluid Mech. 10, 166 (1961)] derived a model to describe this phenomenon for pressure driven flows based on a lubrication approach coupled with surface deformation of the bubble. Bretherton found that the parameter governing the physics involved is the capillary number (Ca) which expresses the relationship between speed of the bubble, surface tension, and viscosity of the liquid. The results of Bretherton are here re-derived and analyzed in a slightly more perspicuous manner. Incorporating the condition that the bubble-film combination should fit inside the tube results in an expression very similar to the one found empirically by Aussillous and Quere [“Quick deposition of a fluid on the wall of a tube,” Phys. Fluids 12, 2367 (2000)] of the Taylor [“D...

Viscosity effects on hydrodynamic drainage force measurements involving deformable bodies.

Dynamic force measurements have been made between an oil drop and a silica particle in surfactant and sucrose solutions with viscosities that range up to 50 times that of water. These conditions provide variations in the shear rate and the relative time scales of droplet deformation and hydrodynamic drainage in a soft matter system. The results obtained indicate that soft deformable boundaries have a natural response that limits the maximum shear rate that can be sustained in thin films compared to shear rates that can be attained in films bounded by rigid boundaries. In addition, to extend boundary slip studies on rigid surfaces, we use a smooth deformable droplet surface to probe the dependence of the boundary slip on fluid viscosity without the added complications of surface roughness or heterogeneity. Imposing a Navier slip model to characterize possible slip at the deformable oil-sucrose solution interface gives results that are consistent with a slip length of no larger than 10 nm over the range of solution viscosity studied, although an immobile (zero slip length) condition at the oil-sucrose solution interface is perfectly adequate. In high viscosity solutions, cantilever motion at high scan rates induces a significant cantilever deflection. A method has been developed to account for this effect in order to extract the correct dynamic force between the deformable drop and the particle.

Dynamic interactions between drops—a critical assessment

Accurate measurements and modelling of time-dependent deformations of colliding liquid drops demonstrate the need to account for drop deformation and film drainage with appropriate initial and boundary conditions and reveal significant quantitative differences with the Stefan-Reynolds flat film model of drop coalescence.

Dynamic interactions between deformable drops in the Hele–Shaw geometry

A model has been developed to describe the collision and possible coalescence of two driven deformable drops in the Hele–Shaw cell. The interdependence between hydrodynamic effects and interfacial deformations is characterised by a film capillary number: Caf = (μv/σ)(Ro/ho)3/2 as revealed by an analytic perturbation solution of the governing equations for a system with continuous phase viscosity μ, interfacial tension σ, drop radius Ro, characteristic relative velocity v and separation ho between the drops. Numerical solutions of the model demonstrate the importance of the full dynamic history of the interacting drops in determining stability or coalescence. The geometry of the Hele–Shaw cell allows for the possibility of using the model to infer the time-dependent force between colliding drops by measuring their separation.