Faraday Ripples, Parametric Resonance, and the Marangoni Effect

In general, the combined actions of two destabilizing mechanisms do not simply add to each other. Here we show that there is a subtle interplay between parametric excitation and thermal gradients leading to interfacial instability, overstability, and generation of surface waves. The case studied refers to the stability of a liquid layer with an open free surface subjected to a transverse temperature gradient (with the Marangoni effect) and also subjected to the simultaneous action of periodic vibrations normal to the layer. Stability is examined in the weak viscosity approximation by applying a multiscale method. To a first approximation, whatever the imposed thermal gradient, vibrations with fairly large amplitude are responsible for excitation of ripples with half the imposed vibration frequency, but their amplitude depends on the Marangoni number. However, as the Marangoni number increases, the critical amplitude decreases from the excitation threshold of Faraday ripples, and after passing through a minimum it monotonically increases with increasing thermal gradient. Another salient finding is that the threshold of the Marangoni overstability is found to be independent of the imposed vibration frequency and amplitude. Copyright 2001 Academic Press.     

Role of Marangoni instability in fabrication of axially and internally grooved hollow fiber membranes

Hollow fiber membranes (HFMs) are extensively used in different industrial applications. Under some controlled fabrication conditions, axially aligned grooves can be formed on the HFM inner surface during typical immersion precipitation-based phase inversion fabrication processes. Such grooved HFMs are found to be promising for nerve repair and regeneration. The axially aligned grooves appearing on the inner surface of the membrane are considered as hydrodynamic instability patterns. During the immersion precipitation process, a transfer of solvent takes place across the interface between a polymer solution and a nonsolvent. This solvent transfer induces gradients of interfacial tension that are considered to be the driving mechanism for Marangoni instability. The onset of the stationary instability is studied by means of a linear instability theory, and the critical and maximum wavenumbers are determined and discussed in terms of the dimensionless groups characterizing the system: viscosity ratio, diffusivity ratio, Schmidt number, crispation number, adsorption number, Marangoni number, and the polymer bulk concentration. A good agreement is found between the predicted wavelength of the most dangerous wave and the experimental groove width. Consequently, solutal Marangoni instability can explain the groove formation mechanism in HFM fabrication.     

Performance and stability of supported liquid membranes using LIX 984N for copper transport

The transport of copper through supported liquid membranes (SLM) using Celgard and Accurel membranes as supports and a novel commercial extractant LIX 984N as a carrier was investigated. LIX 984N provides good overall transport performance for copper from the acidic and dilute solution to the concentrated copper sulfuric acid. The instability of the supported liquid membranes using Celgard 2500 as the membrane support has been studied. It has been demonstrated that initially the surface shear forces due to stirring are a main cause for membrane liquid loss leading to SLM instability. However, during long term permeation no single instability mechanism dominants. The instability of long term operation involves a complex interaction of a number of factors, including surface shear forces, Marangoni effects, changes to membrane morphology, Bernard instabilities and membrane preparation protocal.     

Thermophoretic Motion of a Sphere Parallel to an Insulated Plane

An analytical study is presented for the thermophoresis of a sphere in a constant applied temperature gradient parallel to an adiabatic plane. The Knudsen number is assumed to be small so that the fluid flow can be described by a continuum model with a thermal creep and a hydrodynamic slip at the particle surface. A method of reflections is used to obtain the asymptotic formulas for the temperature and velocity fields in the quasisteady situation. The thermal insulated plane may be a solid wall (no-slip) and/or a free surface (perfect-slip). The boundary effect on the thermophoretic motion is found to be weaker than that on the axisymmetric thermophoresis of a sphere normal to a plane with constant temperature. In comparison with the motion driven by gravitational force, the interaction between the particle and the boundary is less significant under thermophoresis. Even so, the interaction between the plane and the particle can be very strong when the gap thickness approaches zero. For the thermophoretic motion of a particle parallel to a solid plane, the effect of the plane surface is to reduce the translational velocity of the particle. In the case of particle migration parallel to a free surface due to thermophoresis, the translating velocity of a particle can be either greater or smaller than that which would exist in the absence of the plane surface, depending on the relative thermal conductivity and the surface properties of the particle and its relative distance from the plane. Not only the translational velocity but also the rotational velocity of the thermophoretic sphere near the plane boundary is formulated analytically. The rotating direction of the particle is strongly dominated by its surface properties and the internal-to-external thermal conductivity. Besides the particle motion, the thickness of the thermophoretic boundary layer is evaluated by considering the thermophoretic mobility. Generally speaking, a free surface exerts less influence on the particle movement than a solid wall. Copyright 2000 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.      

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.     

An Insoluble Surfactant Model for a Vertical Draining Free Film

A mathematical model is constructed to study the evolution of a vertically oriented thin liquid film draining under gravity when there is an insoluble surfactant with finite surface viscosity on its free surface. Lubrication theory for this free film results in three coupled nonlinear partial differential equations describing the free surface shape, the surface velocity, and the surfactant transport at leading order. We will show that in the limit of large surface viscosity, the evolution of the free surface is that obtained for the tangentially immobile case. For mobile films with small surface viscosity, transition from a mobile to an essentially immobile film is observed for large Marangoni effects. It is verified that increasing surface viscosity and the Marangoni effect retard drainage, thereby enhancing film stability. The theoretical results are compared with experiment; the purpose of both is to act as a model problem to evaluate the effectiveness of surfactants for potential use in foam-fabrication processes. Copyright 2000 Academic Press.     

Convective instability in a liquid-liquid system due to complexation with a crown ether

Periodic convective instability has been observed in a biphasic system during the complexation reaction of alkali picrate and dicyclohexano-18-crown-6 which undergoes mass transfer from the hexane phase into the aqueous phase. The convection was visualized by means of precipitated crystals that are formed in both phases by the complexation reaction. The fluid motion was observed with an optical microscope and further analyzed with the particle image velocimetry (PIV) technique. The partition at the extraction of cesium into the organic phase was followed by means of the radioactive isotope (137)Cs. The type of the hydrodynamic instability is governed by the alkali metal expressed via its stability constants for the complex formed. More stable complexes trigger a higher precipitation, thereby favoring a Raleigh-Taylor instability. Complexes with a lower stability constant induce Marangoni cells which show a pulsating character in a cubic container. Depending on the confinement of the experiment cell the fluid motion can also follow a back-and-forth movement. Possible mechanisms for the occurring oscillations are discussed.     

Mechanism and Kinetics of Hexamethyldisilazane Reaction with a Fumed Silica Surface

An analytical study is presented on the thermocapillary migration of a fluid sphere within a constant applied temperature gradient in an arbitrary direction with respect to a plane surface. The Peclet and Reynolds numbers are assumed to be small so that the Laplace and Stokes equations, respectively, govern the temperature distributions and fluid velocities inside and outside the droplet. The asymptotic formulas for the temperature and the velocity fields in the quasi-steady situation are obtained by using a method of reflections. The plane surface can be a no-slip solid wall and/or a perfect-slip free surface. The boundary effect on the thermocapillary migration is found to be weaker than that on the motion driven by a body force. Even so, the interaction between the plane and the droplet can be very significant when the gap thickness approaches zero. For the motion of a droplet normal to a solid wall, the effect of the plane surface reduces the translational velocity of the droplet; however, this solid wall can be an enhancement factor on the particle migration as it is translating parallel to the wall. On the other hand, in case of a droplet migrating close to a free surface due to thermocapillarity, the droplet velocity can be either greater or smaller than that which would exist in the absence of the plane surface, depending on the relative thermal conductivity and the surface properties of the particle and its relative distance from the plane. Furthermore, the interacting thickness of the affected region by the presence of the plane is discussed by considering the droplet mobility. Generally speaking, a free surface exerts less influence on the particle movement than does a solid surface. Copyright 2000 Academic Press.     

Surface instability in a finite thickness fluid saturated porous layer

The Rayleigh-Taylor (RT) instability at the interface between fluid and fluid saturated sparsely packed porous medium has been investigated making use of boundary layer approximation and Saffmann [8] boundary condition. An analytical solution for dispersion relation is obtained and is numerically evaluated for different values of the parameters. It is shown that RT instability can be controlled by a suitable choice of the thickness of porous layer, ratio of viscosities and the slip parameter.     

Mechanical stability of micropipet-aspirated giant vesicles with fluid phase coexistence

Micropipet aspiration of phase-separated lipid bilayer vesicles can elucidate physicochemical aspects of membrane fluid phase coexistence. Recently, we investigated the composition dependence of line tension at the boundary between liquid-ordered and liquid-disordered phases of giant unilamellar vesicles obtained from ternary lipid mixtures using this approach. Here we examine mechanical equilibria and stability of dumbbell-shaped vesicles deformed by line tension. We present a relationship between the pipet aspiration pressure and the aspiration length in vesicles with two coexisting phases. Using a strikingly simple mechanical model for the free energy of the vesicle, we predict a relation that is in almost quantitative agreement with experiment. The model considers the vesicle free energy to be proportional to line tension and assumes that the vesicle volume, domain area fraction, and total area are conserved during aspiration. We also examine a mechanical instability encountered when releasing a vesicle from the pipet. We find that this releasing instability is observed within the framework of our model that predicts a change of the compressibility of a pipet-aspirated membrane cylinder from positive (i.e., stable) to negative (unstable) values, at the experimental instability. The model furthermore includes an aspiration instability that has also previously been experimentally described. Our method of studying micropipet-induced shape transitions in giant vesicles with fluid domains could be useful for investigating vesicle shape transitions modulated by bending stiffness and line tension.     

General continuum boundary conditions for miscible binary fluids from molecular dynamics simulations

Molecular dynamics simulations are used to explore the flow behavior and diffusion of miscible fluids near solid surfaces. The solid produces deviations from bulk fluid behavior that decay over a distance of the order of the fluid correlation length. Atomistic results are mapped onto two types of continuum model: Mesoscopic models that follow this decay and conventional sharp interface boundary conditions for the stress and velocity. The atomistic results, and mesoscopic models derived from them, are consistent with the conventional Marangoni stress boundary condition. However, there are deviations from the conventional Navier boundary condition that states that the slip velocity between wall and fluid is proportional to the strain rate. A general slip boundary condition is derived from the mesoscopic model that contains additional terms associated with the Marangoni stress and diffusion, and is shown to describe the atomistic simulations. The additional terms lead to strong flows when there is a concentration gradient. The potential for using this effect to make a nanomotor or pump is evaluated.     

Effect of buoyancy on appearance and characteristics of surface tension repeated auto-oscillations

The effect of buoyancy on spontaneous repeated nonlinear oscillations of surface tension, which appear at the free liquid interface by dissolution of a surfactant droplet under the interface, is considered on the basis of direct numerical simulation of the model system behavior. The oscillations are the result of periodically rising and fading Marangoni instability. The buoyancy force per se cannot lead to the oscillatory behavior in the considered system, but it influences strongly both the onset and decay of the instability and therefore, affects appearance and characteristics of the oscillations. If the surfactant solution density is smaller than the density of the pure liquid, then the buoyancy force leads to a considerable decrease of the induction period and the period of oscillations. The buoyancy force affects also the dependence of the oscillation characteristics on the system dimensions. The results of the simulations are compared with the available experimental data.     

Steady Marangoni flow traveling with chemical fronts

When autocatalytic chemical fronts propagate in thin layers of solution in contact with air, they can induce capillary flows due to surface tension gradients across the front (Marangoni flows). We investigate here such an interplay between autocatalytic reactions, diffusion, and Marangoni effects with a theoretical model coupling the incompressible Navier-Stokes equations to a conservation equation for the autocatalytic product concentration in the absence of gravity and for isothermal conditions. The boundary condition at the open liquid/air interface takes the surface activity of this product into account and introduces the solutal Marangoni number M representing the intensity of the coupling between hydrodynamics and reaction-diffusion processes. Positive and negative Marangoni numbers correspond, respectively, to the cases where the product decreases or increases surface tension behind the front. We show that, in both cases, such coupled systems reach an asymptotic dynamics characterized by a steady fluid vortex traveling at a constant speed with the front and deforming it, with, however, an asymmetry between the results for positive and negative M. A parametric study shows that increased propagation speed, front deformation, and possible transient oscillating dynamics occur when the absolute value of M is increased.     

Spectral analysis of a fluid under thermal constraint

Abstract

A spectral analysis is presented to investigate the onset of the convective instability in a fluid subject to a linear temperature gradient. The hydrodynamic theory is developed for the case of a binary system, where the concentration of one of the components is small. Therefore the present results will be applicable to the case of a Brownian system. We consider exclusively those modes which correspond to the central components of the spectral distribution, i.e., the diffusion mode and the thermal diffusivity mode. One finds that those modes are effected by the presence of the external temperature gradient in such a way that the spectrum of the scattered light should exhibit an important narrowing of the thermal diffusivity peak and a slight narrowing of the diffusion peak when approaching the convective instability critical point. Only the thermal diffusivity mode is affected in the limit of a pure fluid     

Numerical investigation of the effect of insoluble surfactants on drop deformation and breakup in simple shear flow

The effect of insoluble surfactants on drop deformation and breakup in simple shear flow is studied using a combination of a three-dimensional boundary-integral method and a finite-volume method to solve the coupled fluid dynamics and surfactant transport problem over the evolving interface. The interfacial tension depends nonlinearly on the surfactant concentration, and is described by the equation of state for the Langmuir isotherm. Results are presented over the entire range of the viscosity's ratio lambda and the surface coverage x, as well as the capillary number Ca that spans from that for small deformation to values that are beyond the critical one Ca(cr). The values of the elasticity number E, which reflects the sensitivity of the interfacial tension to the maximum surfactant concentration, are chosen in the interval 0.1 < or = E < or = 0.4 and a convection dominated regime of surfactant transport, where the influence of the surfactant on drop deformation is the most significant, is considered. For a better understanding of the processes involved, the effect of surfactants on the drop dynamics is decoupled into three surfactant related mechanisms (dilution, Marangoni stress and stretching) and their influence is separately investigated. The dependence of the critical capillary number Ca(cr)(lambda) on the surface coverage is obtained and the boundaries between different modes of breakup (tip-streaming and drop fragmentation) in the (lambda; x) plane are searched for. The numerical results indicate that at low capillary number, even with a trace amount of surfactant, the interface is immobilized, which has also been observed by previous studies. In addition, it is shown that for large Péclet numbers the use of the small deformation theory to measure the interfacial tension in the case where surfactants are present can introduce a significant error.     

A three-dimensional boundary-integral algorithm for thermocapillary motion of deformable drops

A three-dimensional boundary-integral algorithm has been developed to handle the tangential Marangoni stresses in thermocapillary motion of drops. Depending on whether the integration and observation points are on the same or different drops, singularity or near-singularity subtraction is used in the inhomogeneous term of the boundary-integral formulation. Integration is then performed analytically over flat triangles in the subtracted region. Relative trajectories for two deformable drops are calculated for different values of the drop size ratio, drop-to-medium thermal conductivity ratio, and viscosity ratio and compared to those for spherical and slightly deformable drops. Results indicate that deformation increases the minimum separation and inhibits coalescence but is not important enough for appropriate physical parameters to induce the capture or breakup behaviors observed in buoyancy. Interaction times calculated by artificially continuing spherical drop trajectories yield results accurate to within about 10%.     

Solutal Marangoni effect : I. Pure interfacial transfer

The motion of a nitroethane lens located at the surface of an aqueous solution of dodecyltrimethylammonium bromide is analyzed. This motion is generated by a solutal Marangoni effect, or soluto—capillary instability, induced by the transfer of the surfactant from one phase to the second one, both solvents being mutually saturated; its quasi-periodic character is demonstrated by a statistical and spectral analysis, and its characteristics, amplitude, and frequency are discussed as a function of the physicochemical conditions. The enhancement of the interfacial transfer by the Marangoni effect is evaluated. It is shown that none of the available models of linear stability analysis can account for the present results.