Film drainage between two surfactant-coated drops colliding at constant approach velocity

The drainage of the intervening continuous phase film between two drops approaching each other at constant velocity under the influence of insoluble surfactant is investigated. The mathematical model to be solved is a coupled pair of fourth-order nonlinear partial differential equations which arise from the relationships governing the evolution of the film thickness and the surfactant interfacial concentration in the lubrication approximation. We adopt a simplified approach which uses lubrication theory to describe the flow within the drop, marking a departure from the conventional framework in which Stokes flow is assumed. When the model is solved numerically together with the relevant initial and boundary conditions, the results obtained are compared with those found in the literature using the "boundary integral" method to solve for the flow in the drop phase. The close agreement between the results inspires confidence in the predictions of the simplified approach adopted. The analysis on the effect of insoluble surfactant indicates that its presence retards the drainage of the film: The fully immobile interface limit is recovered even in the presence of a small amount of surfactant above a critical concentration; film rupture is either prolonged or prevented. The retardation of the film was attributed to gradients of interfacial tension which gave rise to the Marangoni effect. A study of the influence of various system parameters on the drainage dynamics was conducted and three regimes of drainage and possible rupture were identified depending on the relative magnitudes of the drop approach velocity and the van der Waals interaction force: Nose rupture, rim rupture, and film immobilization and flattening. Finally, the possibility of forming secondary droplets by encapsulating the continuous phase film into the coalesced drop at rupture was examined and quantified in light of these regimes.     

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.     

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.     

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．     

Surfactant effects on thermocapillary interactions of deformable drops

A three-dimensional boundary-integral algorithm is used to study thermocapillary interactions of two deformable drops in the presence of bulk-insoluble, non-ionic surfactant. The primary effect of deformation is to slow down the rate of film drainage between drops in close approach and prevent coalescence in the absence of van der Waals forces. Both linear and non-linear models are used to describe the relationship between interfacial tension and surfactant surface concentration. In the linear model, non-monotonic behavior of the minimum separation between the drops as a function of the surface Peclet number Pe(s) is observed for equal drop and external medium viscosities and thermal conductivities. For bubbles with zero drop-to-medium viscosity and thermal conductivity ratios, however, the minimum separation increases with Pe(s). There is a nearly linear relationship between the minimum drop separation and elasticity E. In the simplest non-linear equation of state, the product of the temperature and the surfactant concentration is retained by allowing non-zero values of the dimensionless gas constant Lambda. For Lambda=O(0.05), it is possible for the smaller drop to move faster than the larger drop. In the Langmuir adsorption framework, the tendency of the smaller drop to catch up to the larger one decreases as the ratio of the equilibrium to maximum surfactant surface concentration increases. Finally, in the Frumkin model, a minimum in the drop separation occurs as a function of the interaction parameter lambda(F) for trajectories with all other parameters held constant.     

Nonlinear evolution of thin liquid films dewetting near soft elastomeric layers

The nonlinear evolution of thin liquid films dewetting near soft elastomeric layers is examined in this work. Evolution equations are derived by applying the lubrication approximation and assuming that van der Waals forces in the liquid cause the dewetting and that the solid can be described as a linear viscoelastic material. Two cases are examined: (i) a liquid layer resting on an elastomer bounded from below by a rigid substrate, and (ii) an elastomer overlying a thin liquid film bounded from below by a rigid substrate. Linear stability analysis is carried out to obtain asymptotic relations which are then compared against solutions of the full characteristic equations. In the liquid-on-solid case, numerical solutions of the evolution equations show that van der Waals forces cause thinning of the liquid film and thickening of the elastomeric solid beneath film depressions. Inclusion of a short-range repulsive force suggests that regular patterns may form in which ridges of fluid rest on depressions in the solid. In the solid-on-liquid case, the van der Waals forces cause the solid layer to break up before the liquid film can dewet. The results presented here support the idea that the dewetting of thin liquid films might be exploited to create topographically patterned surfaces on soft polymeric solids.     

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.     

两个液滴之间的聚并

对于系统中不含杂质时两个液滴在不互溶液体中的聚并过程进行理论分析，得到聚并所需时间与两相物理性质一范德华力的关系，该结果也适用于气泡在液体中的聚并，只要知道系统的物性数据和液滴半径，就可以计算聚并时间，理论预测与实验结果符合较好。     

Pattern formation in thin liquid films with charged surfactants

The dynamic evolution of positively charged surfactants at the interface of a thin film resting atop a negatively charged base is investigated. The lubrication approximation is used to develop coupled equations governing the dynamic evolution of such a system in the presence of charge effects coupled with van der Waals forces. The equations are investigated numerically and analytically, and, for certain parameter ranges, pattern formation is observed reminiscent of that accompanying thermocapillary-driven thin films. Spatial nonuniformities in the charge of the underlying substrate are also studied as a possible tool for film rupture wavelength selection.     

Rupture of wetting films caused by nanobubbles

It is now widely accepted that nanometer sized bubbles, attached at a hydrophobic silica surface, can cause rupture of aqueous wetting films due to the so-called nucleation mechanism. But the knowledge of the existence of such nanobubbles does not give an answer to how the subprocesses of this rupture mechanism operate. The aim of this paper is to describe the steps of the rupture process in detail: (1) During drainage of the wetting film, the apex of the largest nanobubble comes to a distance from the wetting film surface, where surface forces are acting. (2) An aqueous "foam film" in nanoscale size is formed between the bubble and the wetting film surface; in this foam film different Derjaguin-Landau-Verwey-Overbeek (DLVO) forces are acting than in the surrounding wetting film. In the investigated system, hydrophobized silica/water/air, all DLVO forces in the wetting film are repulsive, whereas in the foam film the van der Waals force becomes attractive. (3) The surface forces over and around the apex of the nanobubble lead to a deformation of the film surfaces, which causes an additional capillary pressure in the foam film. An analysis of the pressure balance in the system shows that this additional capillary pressure can destabilize the foam film and leads to rupture of the foam film. (4) If the newly formed hole in the wetting film has a sufficient diameter, the whole wetting film is destabilized and the solid becomes dewetted. Experimental data of rupture thickness and lifetime of wetting films of pure electrolyte and surfactant solutions show that the stabilization of the foam film by surfactants has a crucial effect on the stability of the wetting film.     

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.     

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.     

Variations of surfactant monolayer surface density in induced steady wave regime

Fluctuations of an insoluble surfactant concentration along the free liquid surface induced by steady surface waves are considered theoretically. The energy of a waved surface is assumed to consist of surface tension, curvature, and van der Waals energy components. Dependencies of the surface tension and the bending stiffness versus the surfactant concentration are assumed to be linear relative to some reference level. The van der Waals energy is taken in the form of interaction term for a thin film. Minimization of the total energy allows the expression for the deviations of concentration to be obtained. The distribution of a surfactant concentration relative to some reference level has been found to be periodic, with a period that is half of the wave period, and the amplitude of oscillations is a function of a wave number that is very similar to the Landau expansion of the free-energy near the critical point in phase transitions.     

Dewetting of thin liquid films near soft elastomeric layers

Thin liquid film instabilities driven by van der Waals forces and in the proximity of soft elastomeric layers are considered in this work through two model problems: (i) a liquid film resting on an elastomeric layer and (ii) a liquid film bounded from one side by a rigid substrate and from the other side by an elastomeric layer. The elastomeric layers are modeled as linear viscoelastic solids, van der Waals forces are assumed to act only in the liquid, and lubrication theory and linear stability analysis are applied. For a liquid film resting on an elastomeric layer, substrate deformability has a destabilizing effect, as evidenced by an increase in the maximum growth rate and range of unstable wavenumbers. The destabilization worsens for thicker solid layers and is due to a lowering of the effective liquid-air interfacial tension. For an elastomeric layer resting on a liquid film, layer deformability has a stabilizing effect for thin layers but a destabilizing effect for thicker layers, with the former due to an enhancement and the latter due to a reduction of the effective solid-air interfacial tension. The results presented here suggest the possibility of exploiting the dewetting of thin liquid films to create topographically patterned surfaces on soft polymeric solids.     

Wetting behavior of pentane on water. The analysis of temperature dependence

The temperature dependence of wetting behavior for pentane on water is analyzed from the standpoint of the Derjaguin-Frumkin theory. The joint action of two mechanisms of surface forces, the van der Waals and the image charge interactions, are considered to calculate the isotherms of the disjoining pressure. To analyze the temperature influence on the magnitude of van der Waals forces, we have used the exact Dzyaloshinsky, Lifshitz, and Pitaevsky equation. It is shown that image forces, arising due to the restricted solubility of water in pentane, decay much faster with increasing the film thickness and can be considered as short ranged in comparison to the van der Waals forces. The competitive action of the image charge and the van der Waals forces provides the plausible explanation of the temperature dependence of wetting in the system under consideration.     

长程范德华力导向作用下胶体凝聚的计算机模拟

采用计算机模拟方法研究了长程范德华力在胶体凝聚过程中的作用, 发现由于胶粒间的范德华力是长程力, 它对胶粒或团簇运动将产生导向作用. 与不考虑导向作用的扩散控制团簇凝聚(DLCA)模型比较, 这种导向作用不仅加速了胶体的凝聚过程, 而且形成了更致密、分形维数更大的结构体. 研究还发现, 长程范德华力导向作用对胶粒的初始浓度非常敏感, 不论是在凝聚物的结构还是凝聚速率方面, 只有在胶粒初始浓度较低时, 该导向作用效应才明显. 其可能的原因是,在胶粒初始浓度较高时, 由于胶粒布朗运动的平均自由程很短而且位阻效应大, 从而使导向作用效应未能反映出来.     

Interfacial profiles in fluid/liquid systems: a description based on the storing of elastic energy

The formation and stability of drops in the presence of nanoparticles was studied in a microfluidic device to directly observe the early stages of Pickering emulsification (low interfacial coverage). We observed several key differences between oil droplet necking and rupture in aqueous phases of nanoparticles (methylated silica) and well-characterised surfactant systems. The presence of particles did not influence drop formation dynamics and thus the size of the drops generated. In addition, observations of in-channel drop stability shortly after formation (several milliseconds) indicated that particles in the aqueous phase slow film thinning processes, but do not prevent coalescence. In contrast, downstream collection and densification (at the microchannel outlet), showed that particle-stabilised drops do not coalesce for several weeks, above a critical particle concentration. The implications of our results for droplet microfluidics and our understanding of conventional emulsification systems are discussed.     

Different surface corrugations occurring during drainage of axisymmetric thin liquid films

This paper deals with the different surface corrugations observable during the thinning of axisymmetric thin and large aqueous films, stabilized by saponin. The films are observed using a thin film balance under a constant driving pressure. This device allows measurement of the thicknesses of the film surface shapes arising all along the drainage, as well as the following-up of their evolution before equilibrium is attained. Depending on the electrolyte (NaCl) concentration, three different sorts of corrugation were originally observed in such suspended thin liquid films. At the lowest NaCl concentrations, corresponding to repulsive potential between film walls, only the hydrodynamic corrugations deformed the film surfaces. Regarding the higher NaCl concentrations, when van der Waals forces become predominant, and following the thickness of the first-established thin film, the experiments disclose either that the thinner films are broken up by spinodal decomposition, or that the thicker ones are broken by nucleation and growth of black film. In addition, an original aspect of these works appears in the fact that these observations of the spontaneous decomposition of suspended thin films are relatively similar to those usually described for dewetting experiments on solid substrates, and are well fitted by the existing theoretical models.