Effect of interfacial mobility on rupture of thin stagnant films on a solid surface due to random mechanical perturbations

Previous analysis of Narsimhan [G. Narsimhan, J. Colloid Interface Sci. 287 (2005) 624-633] for the evaluation of rupture of a nondraining thin film on a solid support due to imposed random mechanical perturbations modeled as a Gaussian white noise has been extended for partially mobile gas-liquid interfaces. The average rupture time of film is evaluated by first passage time analysis (as the mean time for the amplitude of perturbation to become equal to film thickness). The interfacial mobility is accounted for through surface viscosity as well as Marangoni effect. The mean rupture time for partially mobile gas-liquid interface, as characterized by two dimensionless groups, dimensionless surface viscosity and Marangoni number, lies between the two extreme limits for fully mobile and immobile films. The critical wavenumber for minimum rupture time is shown to be insensitive to interfacial mobility. However, the critical dimensionless surface viscosity and critical Marangoni number at which the behavior of thin film deviates from that of fully mobile film and the behavior approaches that of fully immobile film are smaller for higher Hamaker constants, smaller film thickness and smaller surface potentials.     

A Nonlinear Rupture Theory of Thin Liquid Films with Soluble Surfactant

The effects of soluble surfactant on the dynamic rupture of thin liquid films are investigated. A nonlinear coupling evolution equation is used to simulate the motion of thin liquid films on free surfaces. A generalized Frumkin model is adopted to simulate the adsorption/desorption kinetics of the soluble surfactant between the surface and the bulk phases. Numerical simulation results show that the liquid film system with soluble surfactant is more unstable than that with insoluble surfactant. Moreover, a generalized Frumkin model is substituted for the Langmuir model to predict the instability of liquid film with soluble surfactant. A numerical calculation using the generalized Frumkin model shows that the surfactant solubility increases as the values of parameters of absorption/desorption rate constant (J), activation energy desorption (nu(d)), and bulk diffusion constant (D(1)) increase, which consequently causes the film system to become unstable. The surfactant solubility decreases as the rate of equilibrium (lambda) and interaction among molecules (K) are increased, which therefore stabilizes the film system. On the other hand, an increase of relative surface concentration (the index of a power law), beta(n), will initially result in a decrease of corresponding shear drag force as beta and n increase from 0 to 0.3 and 0.85, respectively. This will enhance the Marangoni effect. However, a further increase of beta and n to greater than 0.3 and 0.85, respectively, will conversely result in an increase of the corresponding shear drag force. This will weaken the Marangoni effect and thus result in a reduction of interfacial stability. Copyright 2000 Academic Press.     

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.     

Surfactant-Influenced Gas-Liquid Interfaces: Nonlinear Equation of State and Finite Surface Viscosities

A canonical flow geometry was utilized for a fundamental study of the coupling between bulk flow and a Newtonian gas-liquid interface in the presence of an insoluble surfactant. We develop a Navier-Stokes numerical model of the flow in the deep-channel surface viscometer geometry, which consists of stationary inner and outer cylinders, a floor rotating at a constant angular velocity, and an interface covered initially by a uniformly distributed surfactant. Here, the floor of the annular channel is rotated fast enough so the flow is nonlinear and drives the film toward the inner cylinder. The boundary conditions at the interface are functions of the surface tension, surface shear viscosity, and surface dilatational viscosity, as described by the Boussinesq-Scriven surface model. A physical surfactant system, namely hemicyanine, an insoluble monolayer on an air-water interface, with measured values of surface tension and surface shear viscosity versus concentration, was used in this study. We find that a surfactant front can form, depending on the Reynolds number and the initial surfactant concentration. The stress balance in the radial direction was found to be dominated by the Marangoni stress, but the azimuthal stress was only due to the surface shear viscosity. Numerical studies are presented comparing results of surfactant-influenced interface cases implementing the derived viscoelastic interfacial stress balance with those using a number of idealized stress balances, as well as a rigid no-slip surface, providing added insight into the altered dynamics that result from the presence of a surfactant monolayer. Copyright 2000 Academic Press.     

Role of surfactants on the approaching velocity of two small emulsion drops

Here we present the exact solution of two approaching spherical droplets problem, at small Reynolds and Peclet numbers, taking into account surface shear and dilatational viscosities, Gibbs elasticity, surface and bulk diffusivities due to the presence of surfactant in both disperse and continuous phases. For large interparticle distances, the drag force coefficient, f, increases only about 50% from fully mobile to tangentially immobile interfaces, while at small distances, f can differ several orders of magnitude. There is significant influence of the degree of surface coverage, θ, on hydrodynamic resistance β for insoluble surfactant monolayers. When the surfactant is soluble only in the continuous phase the bulk diffusion suppresses the Marangoni effect only for very low values of θ, while in reverse situation, the bulk diffusion from the drop phase is more efficient and the hydrodynamic resistance is lower. Surfactants with low value of the critical micelle concentration (CMC) make the interfaces tangentially immobile, while large CMC surfactants cannot suppress interfacial mobility, which lowers the hydrodynamic resistance between drops. For micron-sized droplets the interfacial viscosities practically block the surface mobility and they approach each other as solid spheres with high values of the drag coefficient.     

Stability of thin stagnant film on a solid surface with a viscoelastic air-liquid interface

Linear stability analysis for a film on a solid surface with a viscoelastic air-liquid interface is presented. The interfacial dilatational and shear viscoelastic properties were described by Maxwell models. Dilatational and shear interfacial elasticity and viscosity were shown to improve film stability. When the interfacial rheological properties are extremely large or small, the maximum perturbation growth coefficient is shown to reduce to those for immobile and mobile interfaces respectively. Calculated values of maximum growth coefficient for thin film stabilized by 0.5% beta-lactoglobulin approached those of mobile films for thick (>2000 nm) and those for immobile films for thin (<100 nm) films respectively with the values lying between the two limits for intermediate film thicknesses.     

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.     

What is the mechanism of soap film entrainment?

Classical Frankel's law describes the formation of soap films and their evolution upon pulling, a model situation of film dynamics in foams (formation, rheology, and destabilization). With the purpose of relating film pulling to foam dynamics, we have built a new setup able to give an instantaneous measurement of film thickness, thus allowing us to determine film thickness profile during pulling. We found that only the lower part of the film is of uniform thickness and follows Frankel's law, provided the entrainment velocity is small. We show that this is due to confinement effects: there is not enough surfactant in the bulk to fully cover the newly created surfaces which results in immobile film surfaces. At large velocities, surfaces become mobile and then Frankel's law breaks down, leading to a faster drainage and thus to a nonstationary thickness at the bottom of the film. These findings should help in understanding the large dispersion of previous experimental data reported during the last 40 years and clarifying the pulling phenomenon of thin liquid films.     

Plate coating: influence of concentrated surfactants on the film thickness

We present a large range of experimental data concerning the influence of surfactants on the well-known Landau-Levich-Derjaguin experiment where a liquid film is generated by pulling a plate out of a bath. The thickness h of the film was measured as a function of the pulling velocity V for different kinds of surfactants (C(12)E(6), which is a nonionic surfactant, and DeTAB and DTAB, which are ionic) and at various concentrations near and above the critical micellar concentration (cmc). We report the thickening factor α = h/h(LLD), where h(LLD) is the film thickness obtained without a surfactant effect, i.e., as for a pure fluid but with the same viscosity and surface tension as the surfactant solution, over a wide range of capillary numbers (Ca = ηV/γ, with η being the surfactant solution viscosity and γ its surface tension) and identify three regimes: (i) at small Ca α is large due to confinement and surface elasticity (or Marangoni) effects, (ii) for increasing Ca there is an intermediate regime where α decreases as Ca increases, and (iii) at larger (but still small) Ca α is slightly higher than unity due to surface viscosity effects. In the case of nonionic surfactants, the second regime begins at a fixed Ca, independent of the surfactant concentration, while for ionic surfactants the transition depends on the concentration, which we suggest is probably due to the existence of an electrostatic barrier to surface adsorption. Control of the physical chemistry at the interface allowed us to elucidate the nature of the three regimes in terms of surface rheological properties.     

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.     

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.     

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.     

Effects of surfactants on wave-induced drainage of foam and emulsion films

It is well established that the plane-parallel models of foam and emulsion films underestimate the velocity of film thinning by up to several orders of magnitude and show an incorrect dependence of thinning velocity on film radius. A new theory of film thinning has been previously formulated for tangentially immobile films [12, 13], and shows that the reason for this discrepancy is the neglect of experimentally observed finite amplitude surface waves. For thin films of relatively large radii (> 1o^–2 cm), the pumping of the fluid generated by oscillations of the surface waves, provides the dominant contribution to film thinning velocity. The present hydrodynamic model includes the effects of surfactants (Marangoni-Gibbs-effect, surface viscosity and surface diffusion) and surface waves on thinning velocity. As in the case of a tangentially immobile film, it is concluded that the thinning velocity varies inversely with less than the first power of the film radius, and not with the square of the film radius, as predicted by the plane-parallel models of thin film. Also, the velocity of thinning is found to be up to several orders of magnitudes larger than that evaluated from the plane-parallel models. The influence of waves in enhancing the thinning velocity is found to be most significant for a tangentially immobile film and this effect decreases by a factor of up to 3, with a decrease in surface elasticity and surface viscosity.     

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.     

Thermal Marangoni Convection of a Fluid Film Coating a Deformable Membrane

The thermal Marangoni instability of a fluid film coating a deformable membrane has been investigated by taking into account the deformation of the fluid free surface. Numerical calculations for different thermal boundary conditions are presented. The prestressed membrane is supposed to be very thin and therefore its behavior is similar to that of an isothermal fluid free surface with a surface tension but with a different mechanical boundary condition; that is, the fluid should stick on its surface and thus the fluid velocity is zero. An important assumption is that the membrane has no temperature dependence and therefore that only one Marangoni number exists for the free surface of the fluid. Numerical results are presented for stationary and oscillatory thermocapillary instability in both the sinuous and the varicose modes. It is shown that membrane deformation has important implications on the Marangoni instability of the fluid layer for positive and negative Marangoni numbers. Copyright 2001 Academic Press.     

Spontaneous oscillations due to solutal Marangoni instability: air/water interface

Systems far from equilibrium are able to self-organize and often demonstrate the formation of a large variety of dissipative structures. In systems with free liquid interfaces, self-organization is frequently associated with Marangoni instability. The development of solutal Marangoni instability can have specific features depending on the properties of adsorbed surfactant monolayer. Here we discuss a general approach to describe solutal Marangoni instability and review in details the recent experimental and theoretical results for a system where the specific properties of adsorbed layers are crucial for the observed dynamic regimes. In this system, Marangoni instability is a result of surfactant transfer from a small droplet located in the bulk of water to air/water interface. Various dynamic regimes, such as quasi-steady convection with a monotonous decrease of surface tension, spontaneous oscillations of surface tension, or their combination, are predicted by numerical simulations and observed experimentally. The particular dynamic regime and oscillation characteristics depend on the surfactant properties and the system aspect ratio.      

Effect of ionic surfactants on drainage and equilibrium thickness of emulsion films

This paper presents new theoretical and experimental results that quantify the role of surfactant adsorption and the related interfacial tension changes and interfacial forces in the emulsion film drainage and equilibrium. The experimental results were obtained with plane-parallel microscopic films from aqueous sodium dodecyl sulphate solutions formed between two toluene droplets using an improved micro-interferometric technique. The comparison between the theory and the experimental data show that the emulsion film drainage and equilibrium are controlled by the DLVO interfacial forces. The effect of interfacial viscosity and interfacial tension gradient (the Marangoni number) on the film drainage is also significant.     

Marangoni flow of Ag nanoparticles from the fluid-fluid interface

Fluid flow is observed when a volume of passivated Ag nanoparticles suspended in chloroform is mixed with a water/ethanol (v/v) mixture containing acidified 11-mercaptoundecanoic acid. Following mechanical agitation, Ag nanoparticles embedded in a film are driven from the organic-aqueous interface. A reddish-brown colored film, verified by transmission electron microscopy to contain uniformly dispersed Ag nanoparticles, is observed to spontaneously climb the interior surface of an ordinary, laboratory glass vial. This phenomenon is recorded by a digital video recorder, and a measurement of the distance traveled by the film front versus time is extracted. Surface (interfacial) tension gradients due to surfactant concentration, temperature, and electrostatic potential across immiscible fluids are known to drive interface motion; this well-known phenomenon is termed Marangoni flow or the Marangoni effect. Experimental results are presented that show the observed mass transfer is dependent on an acid surfactant concentration and on the volume fraction of water in the aqueous phase, consistent with fluid flow induced by interfacial tension gradients. In addition, an effective desorption rate constant for the Marangoni flow is measured in the range of approximately 0.01 to approximately 1 s(-1) from a fit to the relative film front distance traveled versus time data. The fit is based on a time-dependent expression for the surface (interface) excess for desorption kinetics. Such flow suggests that purposeful creation of interfacial tension gradients may aid in the transfer of 2- and 3-dimensional assemblies, made with nanostructures at the liquid-liquid interface, to solid surfaces.     

Spreading of Oil Droplets Containing Surfactants and Pesticides on Water Surface Based on the Marangoni Effect

Oil droplets containing surfactants and pesticides are expected to spread on a water surface, under the Marangoni effect, depending on the surfactant. Pesticides are transported into water through this phenomenon. A high-speed video camera was used to measure the movement of Marangoni ridges. Gas chromatography with an electron capture detector was used to analyze the concentration of the pesticide in water at different times. Oil droplets containing the surfactant and pesticide spread quickly on the water surface by Marangoni flow, forming an oil film and promoting emulsification of the oil–water interface, which enabled even transport of the pesticide into water, where it was then absorbed by weeds. Surfactants can decrease the surface tension of the water subphase after deposition, thereby enhancing the Marangoni effect in pesticide-containing oil droplets. The time and labor required for applying pesticides in rice fields can be greatly reduced by using the Marangoni effect to transport pesticides to the target.