The Dynamics of Marangoni-Driven Local Film Drainage between Two Drops

A study of Marangoni-driven local continuous film drainage between two drops induced by an initially nonuniform interfacial distribution of insoluble surfactant is reported. Using the lubrication approximation, a coupled system of fourth-order nonlinear partial differential equations was derived to describe the spatio-temporal evolution of the continuous film thickness and surfactant interfacial concentration. Numerical solutions of these governing equations were obtained using the Numerical Method of Lines with appropriate initial and boundary conditions. A full parametric study was undertaken to explore the effect of the viscosity ratio, background surfactant concentration, the surface Péclet number, and van der Waals interaction forces on the dynamics of the draining film for the case where surfactant is present in trace amounts. Marangoni stresses were found to cause large deformations in the liquid film: Thickening of the film at the surfactant leading edge was accompanied by rapid and severe thinning far upstream. Under certain conditions, this severe thinning leads directly to film rupture due to the influence of van der Waals forces. Time scales for rupture, promoted by Marangoni-driven local film drainage were compared with those associated with the dimpling effect, which accompanies the approach of two drops, and implications of the results of this study on drop coalescence are discussed. Copyright 2001 Academic Press.     

Effect of Insoluble Surfactants on Drainage and Rupture of a Film between Drops Interacting under a Constant Force

The deformation, drainage, and rupture of an axisymmetrical film between colliding drops in the presence of insoluble surfactants under the influence of van der Waals forces is studied numerically at small capillary and Reynolds numbers and small surfactant concentrations. Constant-force collisions of Newtonian drops in another Newtonian fluid are considered. The mathematical model is based on the lubrication equations in the gap between drops and the creeping flow approximation of Navier–Stokes equations in the drops, coupled with velocity and stress boundary conditions at the interfaces. A nonuniform surfactant concentration on the interfaces, governed by a convection–diffusion equation, leads to a gradient of the interfacial tension which in turn leads to additional tangential stress on the interfaces (Marangoni effects). The mathematical problem is solved by a finite-difference method on a nonuniform mesh at the interfaces and a boundary-integral method in the drops. The whole range of the dispersed to continuous-phase viscosity ratios is investigated for a range of values of the dimensionless surfactant concentration, Peclét number, and dimensionless Hamaker constant (covering both “nose” and “rim” rupture). In the limit of the large Peclét number and the small dimensionless Hamaker constant (characteristic of drops in the millimeter size range) a fair approximation to the results is provided by a simple expression for the critical surfactant concentration, drainage being virtually uninfluenced by the surfactant for concentrations below the critical surfactant concentration and corresponding to that for immobile interfaces for concentrations above it.     

Effect of mass transfer on the film drainage between colliding drops

The influence of mass transfer on the drainage behaviour of the thin liquid film between two drops immersed in another liquid colliding at constant approach velocity has been studied experimentally. The liquid-liquid system used is glycerol in silicone oil. The transferred solute is acetone and the volume concentration difference across the interface ranges from 1 to 5%. The film thickness evolution has been measured using a laser interferometry technique. The direction of mass transfer (from the drops towards the film phase and inversely) has been investigated and the results compared to the case with no mass transfer. When the solute transfers from the drops towards the continuous phase, the drainage rate is significantly higher than in the case with no mass transfer. This result is interpreted as a consequence of the mass transfer induced surface mobility in the film region (the so-called Marangoni effect) due to localized surface tension differences. This effect has been demonstrated by the visualization of the flow patterns in the drops and in the film phase (using a particle tracer technique). In this case, the slope of the film height as a function of time seems to be independent of the approach velocity condition imposed on the drop and appears to be controlled by the interfacial tension gradient. In the opposite case, when the solute transfers from the continuous phase towards the drops, the film drainage rate is lowered with respect to the case of no mass transfer, goes to zero or even changes its sign depending on the mass transfer intensity. The results also show that in the range of solute concentration studied, the effect of mass transfer on the film drainage process takes place at large distances compared to the scales at which lubrication theory is valid.     

Analysis of tear film rupture: effect of non-Newtonian rheology

We investigate the rupture mechanism of a precorneal thin mucus coating sandwiched between the aqueous tear film and the corneal epithelial surface with a monolayer of surfactant overlying the aqueous layer. The Ostwald constitutive relation is employed to model mucus and a linear equation of state describing the relationship between surface tension and surfactant concentration is adopted. Three nonlinear coupled evolution equations governing the transport of surfactant, mucus, and total liquid layer thicknesses, based on lubrication theory and a perturbation expansion technique, have been derived. The resulting equations are solved numerically in order to explore the influence of the rheological properties of mucus, aqueous-mucus thickness ratio, aqueous-mucus interfacial tension, Marangoni number, and surfactant concentration on both the onset of instability and tear film evolution in the presence of van der Waals interactions, which could rupture the tear film. Our results reveal that the influence of rheological properties, aqueous-mucus thickness ratio, and interfacial tension on the time required for film rupture can be significant and varies considerably, depending on the magnitude of the Hamaker constants governing the strength of the van der Waals forces.     

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.     

Thinning of drying latex films due to surfactant

Lateral non-uniformities in surfactant distribution in drying latex films induce surface tension gradients at the film surface and lead to film thinning through surfactant spreading. Here we investigate the influence of the surfactant driven to the air-water interface, during the early stages of latex film drying, on the film thinning process which could possibly lead to film rupture. A film height evolution equation is coupled with conservation equations for particles and surfactant, within the lubrication approximation, and solved numerically, to obtain the film height, particle volume fraction, and surfactant concentration profiles. Parametric analysis identifies the effect of drying rate, dispersion viscosity and initial particle volume fraction on film thinning and reveals the conditions under which films could rupture. The results from surface profilometry conform qualitatively to the model predictions.     

The effect of surfactant on the efficiency of shear-induced drop coalescence

The volume-averaged shear-induced drop-coalescence efficiency epsilonv is measured by in situ videomicroscopy of blends of poly(propylene glycol) and poly(ethylene glycol), emulsified with poly(ethyleneglycol-b-propyleneoxide-b-ethyleneglycol) block copolymer surfactant. Adsorption of copolymer to the immiscible blend interface is indicated by a reduction in the interfacial tension, measured by the drop retraction method. The effects of temperature, copolymer molecular weight, copolymer concentration, and capillary number Ca are explored. At small Ca, epsilonv is essentially independent of shear rate and drop size, and depends mainly on the solubility, diffusivity, and surface pressure of the surfactant, indicating that drop trajectories during flow are perturbed by surfactant Marangoni stresses that are controlled by the diffusion-limited sorption of surfactant. At larger Ca, epsilonv approaches zero. This arrest of coalescence is associated with the onset of slight deformation of the drops during their collision, and drainage of a film of continuous fluid between them. The effect of the surfactant, though significant, saturates even while the amount of surfactant adsorbed to the interface is quite small. Governing dimensionless parameters, associated material parameters and the behavior of more insoluble surfactants are discussed.     

Hydrodynamic interaction between spheres coated with deformable thin liquid films

In this article, we considered the hydrodynamic interaction between two unequal spheres coated with thin deformable liquids in the asymptotic lubrication regime. This problem is a prototype model for drop coalescence through the so-called "film drainage" mechanism, in which the hydrodynamic contribution comes dominantly from the lubrication region apart from the van der Waals interaction force. First, a general formulation was derived for two unequal coated spheres that experienced a head-to-head collision at a very close proximity. The resulting set of the evolution equations for the deforming film shapes and stress distributions was solved numerically. The film shapes and hydrodynamic interaction forces were determined as functions of the separation distance, film thickness, viscosity ratios, and capillary numbers. The results show that as the two spheres approach each other, the films begin to flatten and eventually to form negative curvature (or a broad dimple) at their forehead areas in which high lubrication pressure is formed. The dimple formation occurs earlier as the capillary number increases. For large capillary numbers, the film liquids are drained out from their forehead areas and the coated liquid films rupture before the two films "touch" each other. Meanwhile, for small capillary numbers, the gap liquid is drained out first and the two liquid films eventually coalesce.     

Drop manipulation and surgery using electric fields

We study the dynamics of a slender drop sandwiched between two electrodes using lubrication theory. A coupled system of evolution equations for the film thickness and interfacial charge density is derived and simplified for the case of a highly conducting fluid. The contact line singularity is relieved by postulating the existence of a wetting precursor film, which is stabilised by intermolecular forces. We examine the motion of the drop as a function of system parameters: the electrode separation, beta, an electric capillary number, C, and a spatio-temporally varying bottom electrode potential. The possibility of drop manipulation and surgery, which include drop spreading, translation, splitting and recombination, is demonstrated using appropriate tuning of the properties of the bottom potential; these results could have potential implications for drop manipulation schemes in various microfluidic applications. For relatively small beta and/or large C values, the drop assumes cone-like structures as it approaches the top electrode; the latter stages of this approach are found to be self-similar and a power-law exponent has been extracted for this case.     

Instability of confined thin liquid film trilayers

The instability of a system in which three stratified thin liquid films are confined in a channel with parallel walls and the interior film is subject to van der Waals-driven breakup is examined in this work. We derive a model based on lubrication theory and consisting of a pair of nonlinear partial differential equations describing the position of the two liquid interfaces. A linear stability analysis is carried out to show that the effects of varying the boundary film thicknesses can be understood in terms of several known limits, including a supported monolayer, confined bilayer, and supported bilayer. Variation of the boundary film viscosities is shown in many cases to eliminate the supported-bilayer limit. The parameter regimes in which squeezing and bending modes dominate the initial growth are determined, and nonlinear simulations are used to show that the mode always switches to squeezing near rupture. It is also found that a multi-modal dispersion relation may be created by asymmetries in thickness ratio, but not viscosity ratio, even in the absence of asymmetric interfacial tensions. The results of this study are expected to be relevant to multiphase microfluidic systems and the lithographic printing process.     

界面流变性质对小液滴聚并过程的影响

对表面活性剂溶液中两个小液滴的聚并现象进行理论分析，并考虑相界面上质量传递对该过程的影响，得到聚并时间与界而张力和界面张力梯度、界面粘度、表面活性剂界面扩散系数、连续相和分散相的主体性质、范德华力及液滴半径的关系．     

Zooming in on the role of surfactants in droplet coalescence at the macroscale and microscale

Coalescence of dispersed micrometer-scale droplets is an essential step toward the separation of emulsions. The thin film between droplets must form, drain, and rupture for coalescence to occur. In surfactant-stabilized emulsions, the film drainage and droplet coalescence processes are known to be hindered because of reduced interfacial mobility. However, a clear correlation between this mobility and the underlying surfactant transport and interfacial response to shear and dilatational deformations is undercharacterized. For microscale droplets, the effect of surfactant transport to the interface and along the interface is often difficult to isolate from other bulk effects on emulsion stability. In this work, we review surfactant-mitigated coalescence in both macroscale and microscale experiments, highlighting the importance of interfacial curvature and length scales when establishing a correlation between coalescence theory and film mobility.     

Liquid drainage in single plateau borders of foam

This paper reports on an investigation of the influence of the interfacial shear viscosity on the liquid drainage in single Plateau borders of foam. The simplified Navier-Stokes equation governing the liquid flow is solved for the liquid velocity by the numerical computational method. The numerical results show significant influence of the interfacial shear viscosity on the liquid velocity in the Plateau border. Comparison of the numerical results for the average velocity over the cross-section area of the Plateau border to the available analytical solution shows that the available analytical solution underestimates the average velocity. New, simple yet accurate correlations for the dependence of the average velocity on the radius of the cross section of the Plateau border, the pressure gradient, and the interfacial shear viscosity are obtained using the asymptotic analysis and the numerical data.     

Rheology of high internal phase emulsions

The mechanical dispersion technology used in this study employs rotor-stator mixers that produce water-continuous high internal phase emulsions (HIPEs) with narrow drop size distributions and small drop sizes, even when the internal phase (oil) viscosity is quite high. Analysis of these HIPEs reveals trends that are consistent with formation by a capillary instability mechanism in which a shear deformation produces highly elongated drops that rupture to form uniform, small droplets. In the search for a predictive tool to aid in the manufacture and use of HIPEs, rheology data for these shear-thinning HIPEs have been compared to data for models in the literature. Existing models do not correctly account for the effect of a high internal phase viscosity on the rheological properties of the HIPE. Another shortcoming is failure to correctly address the shear-thinning exponent. Whereas internal phase viscosity does not seem to affect the shear-thinning exponent, the surfactant apparently plays an important role, possibly through its modification of the interfacial tension and continuous phase rheology.     

Influence of dynamic interfacial tension on droplet formation during membrane emulsification

Membrane emulsification is a promising and relatively new technique for producing emulsions. The purpose of this study was to better understand the influence of interfacial tension on droplet formation during membrane emulsification. Droplet formation experiments were carried out with a microengineered membrane; the droplet diameter and droplet formation time were studied as a function of the surfactant concentration in the continuous phase. These experiments confirm that the interfacial tension influences the process of droplet formation; higher surfactant concentrations lead to smaller droplets and shorter droplet formation times (until 10 ms). From drop volume tensiometer experiments we can predict the interfacial tension during droplet formation. However, the strong influence of the rate of flow of the to-be-dispersed phase on the droplet size cannot be explained by the predicted values. This large influence of the oil rate of flow is clarified by the hypothesis that snap-off is rather slow in the studied regime of very fast droplet formation.     

Rupture of draining foam films due to random pressure fluctuations

A generalized formalism for the rupture of a draining foam film due to imposed random pressure fluctuations, modeled as a Gaussian white noise, is presented in which the flow inside the film is decomposed into a flow due to film drainage and a flow due to imposed perturbation. The evolution of the amplitude of perturbation is described by a stochastic differential equation. The rupture time distribution is calculated from the sample paths of perturbation amplitude as the time for this amplitude to equal one-half the film thickness and is calculated for different amplitudes of imposed perturbations, film thicknesses, electrostatic interactions, viscosities, and interfacial mobilities. The probability of film rupture is high for thicker films, especially at smaller times, as a result of faster growth of perturbations in a thick film due to a smaller disjoining pressure gradient. Larger viscosity, larger surface viscosity, higher Marangoni number, and smaller imposed pressure fluctuation result in slower growth of perturbation of a draining film, thus leading to larger rupture time. It is shown that a composite rupture time distribution combining short time simulation results with equilibrium distribution is a good approximation.     

Stability of macroemulsions

The rupture of the thin films separating emulsion droplets has long been considered to be triggered by the long-range, attractive van der Waals forces; however, this study conducted in a shearfield coalescer shows that systems containing surfactant may, in some cases, yield trends contrary to predictions based on this hypothesis. These trends can be understood in terms of a new mechanism leading to film rupture which is called here percolation-enhanced coalescence.Those regimes in which this new mechanism dominates can be determined based on simple equilibrium phase behavior studies of the surfactant, oil, and water mixtures using the percolation model proposed here.     

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.     

Dilational viscoelasticity of anionic polyelectrolyte/surfactant adsorption films at the water–octane interface

In this article, the interfacial tension and interfacial dilational viscoelasticity of polystyrene sulfonate/surfactant adsorption films at the water–octane interface have been studied by spinning drop method and oscillating barriers method respectively. The experimental results show that different interfacial behaviors can be observed in different type of polyelectrolyte/surfactant systems. Polystyrene sulfonate sodium (PSS)/cationic surfactant hexadecanetrimethyl–ammonium bromide systems show the classical behavior of oppositely charged polyelectrolyte/surfactant systems and can be explained well by electrostatic interaction. In the case of PSS/anionic surfactant sodium dodecyl sulfate (SDS) systems, the coadsorption of PSS at interface through hydrophobic interaction with alkyl chain of SDS leads to the increase of interfacial tension and the decrease of dilational elasticity. For PSS/nonionic surfactant TX100 systems, PSS may form a sub-layer contiguous to the aqueous phase with partly hydrophobic polyoxyethylene chain of TX100, which has little effect on the TX100 adsorption film and interfacial tension.     

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