Marangoni Stresses Research Articles

Numerical investigation of the effect of surface viscosity on droplet breakup and relaxation under axisymmetric extensional flow

In this study, we perform boundary-integral simulations to investigate the role of interfacial viscosity in the deformation and breakup of a single droplet suspended in an axisymmetric extensional flow under the Stokes flow regime. We model the insoluble surfactant monolayer using the Boussinesq–Scriven constitutive relationship for a Newtonian interface. We compare the deformation and breakup results from our boundary-element simulations with results from small deformation perturbation theories. We observe that the surface shear/dilational viscosity increases/decreases the critical capillary number beyond which the droplet becomes unstable and breaks apart by reducing/increasing the droplet deformation at a given capillary number compared with a clean droplet. We present the relative importance of surface shear and dilational viscosity on droplet stability based on their measured values reported in experimental studies on surfactants, lipid bilayers and proteins. In the second half of the paper, we incorporate the effect of surfactant transport by solving the time-dependent convection–diffusion equation and consider a nonlinear equation of state (Langmuir adsorption isotherm) to correlate the interfacial tension with the changes in surfactant concentration. We explore the coupled influence of pressure-dependent surface viscosity and Marangoni stresses on droplet deformation and breakup. In the case of a droplet with pressure-dependent surface shear viscosity, we find that a droplet with pressure-thinning/thickening surfactant is less/more deformed than a droplet with pressure-independent surfactant. We conclude by discussing the combined impact of surface viscosity and surfactant transport on the relaxation of an initially extended droplet in a quiescent external fluid.

Droplet microfluidics for studying surfactant-rich interfaces found in aerosols, emulsions and foams

<h2>Abstract</h2> The presence of droplets suspended in a liquid or gas is known to play an essential role in fields ranging from materials to atmospheric science. Often, the dispersed phase is stabilized by surface active compounds and surfactants, resulting in complex chemical composition and material properties at the fluid-fluid interface. In this talk, recent advancements in use of microscale flow fields will be highlighted for measuring properties of aqueous multiphase, interface-rich systems. Systems studied will include atmospheric aerosol droplets and liquid-liquid emulsions. Microfluidic contractions, traps, and wells are used to measure surface-bulk partitioning and temperature-dependent liquid-liquid phase separation of aqueous droplets, towards better understanding of suspensions of aerosol droplets in our atmosphere. Advanced measurements of droplet shape deformations and coalescence will also be presented for both water-in-fuel and oil-in-water emulsions, for improved treatment and separation of emulsions. Dynamic interfacial tensions measurements were performed using a microfluidic tensiometer, demonstrating a dependence on if the surfactant approaches the interface from inside (dispersed) versus outside (continuous), implying phase dependent surfactant transport to curved interfaces at the microscale. Droplet coalescence and film drainage experiments are also performed in a microfluidic Stokes trap across a range of viscosity ratios and surfactant concentrations. Results are used to explore the influence of interfacial mobility and Marangoni stresses on film stability with soluble surfactants.

Linear instability of a surfactant-laden shear imposed falling film over an inclined porous bed

The influence of externally imposed shear on a surfactant-laden gravity-driven fluid flow over an inclined porous substrate is studied using the linear perturbation theory. The hydrodynamic instability of the flow system corresponding to infinitesimal disturbances is examined in the framework of the Orr–Sommerfeld (OS) boundary value problem. Furthermore, the generalized OS model is obtained by including the Marangoni stress and external shear on the flow dynamics. The formulated stability problem is solved as an eigenvalue problem by the Chebyshev spectral collocation technique. The analysis encounters the existence of different classes of unstable modes, namely, the surface, surfactant, and shear modes. The surface mode instability occurs in the low range of Reynolds number and is the dominant mode of instability in particular parameter ranges. The imposed shear at the top surface along and opposite to the flow direction induces possible destabilization and stabilization of the flow, respectively. The permeability and porosity of the porous medium have a mixed impact on the surface mode instability. The temporal growth rate of the surface mode enhances for a thicker porous medium. The surface mode of the flow contaminated by an insoluble surfactant is less unstable than that of the clean free surface flow. This is due to the co-existence of the damped surfactant mode together with the unstable surface mode. On the other hand, the shear mode instability is identified at higher Reynolds numbers for a very small inclination angle, and the shear mode propagates faster for stronger imposed shear in the downstream direction. This trend is reversed for the upstream imposed shear. Moreover, the Marangoni effects exhibit the stabilizing influence on the shear mode. Conclusively, the external shear force would be helpful in regulating the instability of the surfactant-laden film flow down a porous medium.

Numerical study of surfactant effects on the rise of a single bubble and two coaxial bubbles

A hybrid lattice Boltzmann and finite difference method is presented for two-phase flows with unequal densities and insoluble surfactants. This method simulates two-phase flows through an improved lattice Boltzmann color-gradient model, which is developed from the previous density ratio model but considers surfactant effect by incorporating dynamic surface tension and Marangoni stress, and solves surfactant transport by a finite difference method. We first use it to simulate a single bubble rise and explore the surfactant roles on bubble motion and deformation for varying Bond (Bo) and Galilei (Ga) numbers. Results show that the addition of surfactants always retards bubble motion and deformation but the retarding effect decreases with increasing Bo. As Ga increases, the retarding effect of surfactants increases due to the increased Marangoni stress and a wake vortex would arise behind the surfactant-laden bubble, which opposes the surfactant accumulation at bubble bottom. We then simulate the rise of two coaxial bubbles and study the surfactant roles on bubble coalescence for varying Ga and radius ratios of leading to trailing bubbles. It is found that the presence of surfactants delays bubble coalescence when Ga or radius ratio is relatively low, but promotes coalescence when Ga or radius ratio is high.

Influence of surfactant concentration on drop production by bubble bursting

Bubble bursting at the surface of the sea water produce drops and is the main source of sea spay aerosol. The mechanisms underlying the drops production from a single bubble bursting event have been intensively studied and the influence of the bubble size and liquid parameters (density, viscosity and surface tension) has been unified. However, despite the diversity of the surfactant molecules present in the oceans, their influence has been overlooked. In this paper we experimentally explore the influence of the surfactant concentration (Sodium Dodecyl Surfate - SDS) in a water solution on a single bubble collapse and subsequent drop production. We show that these surfactant molecules have an astonishing effect. In particular, we quantitatively show that they modify the bubble collapse, they induce less, smaller and faster drops and they can even completely prevent the drop production for a particular concentration. These results are presented in the existing dimensionless framework and they allow us to affirme that these effects are mainly a consequence of the surface tension gradients (Marangoni stresses) more than just the surface tension lowering. Finally, this study shows that the role of water contamination by surface-active agents is important and needs to be understood to improve the prediction of the sea spray aerosol.

Surfactant-laden film lining an oscillating cap: problem formulation and weakly nonlinear analysis

A surfactant-laden liquid film that lines the inside of an oscillating spherical cap is considered as a model of lung alveoli. Pulmonary surfactant solubility is described by Langmuir adsorption kinetics, modified by incorporating the intrinsic compressibility of the adsorbed monolayer. A novel boundary condition, supported by experimental data and scaling arguments, is applied at the rim. The condition enforces mass conservation of water and surfactant by matching the ‘large-scale’ dynamics of the alveolus to ‘small-scale’ equilibrium over mid-alveolar septa of small but finite thickness. Linear and weakly nonlinear analysis around the conditions in a non-oscillating cap indicates that the occurrence of shearing motion in the liquid is related to the non-zero film thickness over the rim, and shearing velocity at the interface is predicted an order-of-magnitude lower than the velocity of radial oscillation. Marangoni stresses dominate the interfacial dynamics, but capillary stresses affect significantly the interior flow field. In particular, they produce spatial modulations in flow rate, surface concentration of surfactant and wall shear stress, whose length scale varies with $Ca^{-1/3}$ , i.e. is determined by a balance between capillary and viscous forces. Non-zero adsorption kinetics modifies at first order only the amplitude and phase of surface concentration, but affects all other variables at second order. In particular, it sets a steady drift of surfactant away from the alveolus and towards the rim. Finally, an attempt is made to relate the present predictions to physiological findings about air flow and particle deposition inside alveoli, and about shear stress-inflicted damage in diseased lungs.

A model of bubble coalescence in the presence of a nonionic surfactant with a low bubble approach velocity

AbstractA bubble coalescence model for a solution with a nonionic surfactant and with a low bubble approach velocity was developed, in which the mechanism of how coalescence is hindered by Marangoni stress was quantitatively analyzed. The bubble coalescence time calculated for ethanol–water and MIBC–water systems were in good agreement with experimental data. At low surfactant concentrations, the Marangoni stress and bubble coalescence time increased with bulk concentration increase. Conversely, in the high concentration range, the Marangoni stress and coalescence time decreased with bulk concentration. Numerical results showed that the nonlinear relationship between coalescence time and surfactant concentration is determined by the mass transport flux between the film and its interface, which tends to diminish the spatial concentration variation of the interface, that is, it acts as a “damper.” This damping effect increases with increased surfactant concentration, therefore decreasing the coalescence time at high concentrations.

Numerical and experimental analysis of temperature distribution and melt flow in fiber laser welding of Inconel 625

In these days, laser is a useful and valuable tool. Low input heat, speed, accuracy, and high controllability of laser welding have led to widespread use in various industries. Nickel-based superalloys are creep-resistant materials used in high-temperature conditions. Also, these alloys have high strength, fatigue, and suitable corrosion resistance. Inconel 625 is a material that is strengthened by a complex deposition mechanism. Therefore, the parameters related to laser welding affect the microstructure and mechanical properties. Therefore, in this study, the effect of fiber laser welding parameters on temperature distribution, weld bead dimensions, melt flow velocity, and microstructure was investigated by finite volume and experimental methods. In order to detect the temperature history during continuous laser welding, two thermocouples were considered at a distance of 2 mm from the welding line. The heat energy from the laser beam was modeled as surface and volumetric heat flux. The results of numerical simulation showed that Marangoni stress and buoyancy force are the most important factors in the formation of the flow of liquid metal. Enhancing the laser power to 400 W led to the expansion of the width of the molten pool by 1.44 mm, which was in good agreement with the experimental results. Experimental results also showed that increasing the temperature from 500 °C around the molten pond leads to the formation of a coarse-grained austenitic structure.

Self-avoidant memory effects on enhanced diffusion in a stochastic model of environmentally responsive swimming droplets.

Enhanced diffusion is an emergent property of many experimental microswimmer systems that usually arises from a combination of ballistic motion with random reorientations. A subset of these systems, autophoretic droplet swimmers that move as a result of Marangoni stresses, have additionally been shown to respond to local, self-produced chemical gradients that can mediate self-avoidance or self-attraction. Via this mechanism, we present a mathematical model constructed to encode experimentally observed self-avoidant memory and numerically study the effect of this particular memory on the enhanced diffusion of such swimming droplets. To disentangle the enhanced diffusion due to the random reorientations from the enhanced diffusion due to the self-avoidant memory, we compare to the widely used active Brownian model. Paradoxically, we find that the enhanced diffusion is substantially suppressed by the self-avoidant memory relative to that predicted by only an equivalent reorientation persistence timescale in the active Brownian model. We attribute this to transient self-caging that we propose is novel for self-avoidant systems. Additionally, we further explore the model parameter space by computing emergent parameters that capture the velocity and reorientation persistence, thus finding a finite parameter domain in which enhanced diffusion is observable.

Surfactant spreading on a deep subphase: Coupling of Marangoni flow and capillary waves

HypothesisSurfactant-driven Marangoni spreading generates a fluid flow characterized by an outwardly moving “Marangoni ridge”. Spreading on thin and/or high viscosity subphases, as most of the prior literature emphasizes, does not allow the formation of capillary waves. On deep, low viscosity subphases, Marangoni stresses may launch capillary waves coupled with the Marangoni ridge, and new dependencies emerge for key spreading characteristics on surfactant thermodynamic and kinetic properties. Experiments and modelingComputational and physical experiments were performed using a broad range of surfactants to report the post-deposition motion of the surfactant front and the deformation of the subphase surface. Modeling coupled the Navier-Stokes and advective diffusion equations with an adsorption model. Separate experiments employed tracer particles or an optical density method to track surfactant front motion or surface deformation, respectively. FindingsMarangoni stresses on thick subphases induce capillary waves, the slowest of which is co-mingled with the Marangoni ridge. Changing Marangoni stresses by varying the surfactant system alters the surfactant front velocity and the amplitude – but not the velocity – of the slowest capillary wave. As spreading progresses, the surfactant front and its associated surface deformation separate from the slowest moving capillary wave.

Single Contaminated Drops Falling through Stagnant Liquid at Low Reynolds Numbers

Numerical simulations of contaminated spherical drops falling through a stagnant liquid at low Reynolds numbers are carried out using the finite difference method. The numerical results are used to describe the behavior of the surfactant concentrations and to understand the surfactant effects on the fluid motions in detail. The predicted interfacial surfactant concentration, Γ, is almost zero for angles, θ, below a certain value (the stagnant-cap angle, θcap), whereas it steeply increases and reaches a large value for θ&gt;θcap (the stagnant-cap region). The increase in the initial surfactant concentration, C0, in the drop enhances the adsorption from the drop to the interface, which results in the increase in Γ and the decrease in θcap. Peaks appear in the predicted Marangoni stresses around θcap, which causes similar peaks in the pressure distribution. The high-pressure spots prevent the fluid motion along the interface, which results in the formation of the stagnant-cap region and the attenuation of the tangential velocity in the continuous phase. The surfactant flux from the bulk to the interface decreases C in the vicinity of the interface for θ&lt;θcap and the weak diffusion cannot compensate for the reduction in C by adsorption, which results in C at the interface smaller than C0. The pattern of the low C region is determined by the advection and does not smear out because of a small diffusive flux.

Swimming dynamics of a self-propelled droplet

Chemically active droplets often show intriguing self-propulsion behaviour in a surfactant solution. The drop motion is controlled by the nonlinear coupling among chemical transport in the bulk fluid, consumption of surfactant at the drop surface, and the fluid flow driven by the self-generated Marangoni stress. To quantify the underlying hydrodynamics, this work investigates the swimming motion of a two-dimensional drop that is determined by two dimensionless parameters, the Péclet number ($Pe$) and Damköhler number ($Da$). The weakly nonlinear analysis shows that near the instability threshold, the drop undergoes a supercritical bifurcation with velocity $U\sim \sqrt {Pe-Pe_c}$, where $Pe_c$ is the critical Péclet number for the onset of dipole mode. In the highly nonlinear regime, the drop transits from steady translation of pusher swimming to unsteady motion of mixed pusher–puller swimming along zigzaging trajectories of quadrangle and/or triangle waves. Mode decomposition shows that the zigzag motion is directly related to the interaction between the secondary wake of low surfactant concentration and the primary wake.

Quantifying the interacting mechanisms in shape evolution of sessile volatile droplets

Non-uniform distribution of interfacial mass flux across an evaporating droplet will subsequently induce a temperature gradient across the liquid-air interface, and result in thermal Marangoni stress that reforms the flow field inside the droplet. Recent study (Shiri, et al. Phys. Rev. Lett. 2021, Yang, et al. Langmuir 2022) confirmed the role of thermal Marangoni effect on the shape of evaporating single component droplets on completely wetting substrates. Nevertheless, a comprehensive evaluation of the interacting mechanisms is still lacking due to the intrinsic limitation of experimental techniques to decompose the different influence factors. To this end, we formulate a mathematical model to simulate the evaporation of volatile droplets on completely wetting substrates based on the lubrication theory. With this model, we elucidate the interacting physical mechanisms that governs the droplet kinetics during phase change, which includes the capillary effect, the thermal Marangoni effect, the interface motion due to evaporation, as well as the removal of energy barriers by precursor film. A phase diagram is summarized in terms of the Knudsen number and liquid-to-solid Biot number, which quantifies the prevalence of the interacting effects.

On the coalescence of dissolving bubbles in surfactant presence

In any real system with dispersed bubbles, dissolution and impurity/surfactant presence will affect the bubble dynamics to a certain extent. This work aims to determine the quantitative conditions under which these mechanisms significantly change the coalescence behavior and time in a dispersed system. To achieve this, coalescence of two gently colliding, deformable, dissolving bubbles is investigated through a film drainage study in the surfactant presence. The drainage model is analyzed in the thin film limit and coupled with the boundary integral method. The dissolution is found to slow down the drainage and increase the coalescence time. When the film saturation is relatively fast, i.e., low solubility and high collision velocity, however, the change in the coalescence time appears to be insignificant. The non-uniform distribution of the surfactants along the interface creates Marangoni stresses, which in turn delays the coalescence by immobilizing the particle interfaces. The contribution of the immobilization in delaying the coalescence is found to be much bigger than that of the dissolution, even for very low surfactant concentrations. The conditions under which the interfaces are completely immobilized due to Marangoni stresses are presented. However, these conditions do not seem to be easily achievable in typical systems, indicating that other factors, such as the interfacial viscosity, may play significant roles in immobilization.

Measurements of Static and Dynamic Bubble Surface Tension Using a Deformation-Based Microfluidic Tensiometer.

The surface tension of bubbles is critical for processes involving mixed liquid-gas systems, from sea spray aerosol generation to firefighting foam aspiration. In particular, the size- and surfactant-dependent time scales of dynamic surface tension decay due to adsorption of surface-active chemicals at the curved interface significantly dictate the multiphase system dynamics. While size-dependent surfactant adsorption and interfacial dynamics have been well characterized for liquid-liquid systems using microfluidic platforms, application of microfluidic methods to liquid-gas systems has received less attention. This work uses a high-throughput microfluidic tensiometer to measure the static and dynamic surface tension of microscale bubbles compared with millimeter bubbles characterized by pendant drop. It is shown that the static surface tension measurements for surfactant-free interfaces with microfluidics show good agreement with pendant drop for most systems. At the same time, its accuracy can be affected by bubble pressure, inertia force at high Re, drag force, bubble expansion, and image processing limitation. In the presence of surfactants, the dynamic surface tension measurements show that both smaller bubbles and higher surfactant concentrations can lead to a much shorter time to reach equilibrium compared with pendant drop, similar to the observation for liquid-liquid interfaces. This work shows the potential of a microfluidic tensiometer to capture early time surface tension decay and accurately measure surface tension even in the presence of Marangoni stress tangential to the interface.

On the free surface boundary of moving particle semi-implicit method for thermocapillary flow

The moving particle semi-implicit (MPS) method has great potential in dealing with free surface flow due to its Lagrangian nature. In most cases, the free surface boundary is simply served as the pressure boundary condition. In this paper, an improved MPS method is presented for thermocapillary driven free surface flow. A series of surface nodes explicitly represent the free surface boundary. The normal stress on the free surface provides the Dirichlet pressure boundary condition, while the velocity boundary condition, i.e., Marangoni stress, is enforced through the Taylor series expansion and least squares method. Meanwhile, a quasi-Lagrangian formulation is introduced to avoid particle clustering and the corresponding numerical instability by slightly modifying the advection velocity. The upwind scheme is employed for the convection term to obtain accurate and stable results. A novel constraint scheme with the divergence of provisional velocity is developed for the pressure gradient to enhance stability further. The consistency of the derived generalized boundary condition is firstly verified with a simple convergence test. Then, several numerical tests, including square patch rotation, lid-driven and square droplet oscillation, are simulated to show the improvements. Finally, thermocapillary driven flows in an open cavity without and with buoyancy effect are studied. Good agreements are obtained by comparing with reference simulations taken from literature. Heat transfer characteristics are further investigated for different dimensionless numbers, including the Rayleigh number and Marangoni number.<PE-FRONTEND>

Marangoni instabilities of drops of different viscosities in stratified liquids

For an immiscible oil drop immersed in a stably stratified ethanol–water mixture, a downwards solutal Marangoni flow is generated on the surface of the drop, owing to the concentration gradient, and the resulting propulsion competes against the downwards gravitational acceleration of the heavy drop. In prior work of Li et al. (Phys. Rev. Lett., vol. 126, issue 12, 2021, 124502), we found that for drops of low viscosity, an oscillatory instability of the Marangoni flow is triggered once the Marangoni advection is too strong for diffusion to restore the stratified concentration field around the drop. Here we experimentally explore the parameter space of the concentration gradient and drop radius for high oil viscosities and find a different and new mechanism for triggering the oscillatory instability in which diffusion is no longer the limiting factor. For such drops of higher viscosities, the instability is triggered when the gravitational effect is too strong so that the viscous stress cannot maintain a stable Marangoni flow. This leads to a critical drop radius above which the equilibrium is always unstable. Subsequently, a unifying scaling theory that includes both the mechanisms for low and for high viscosities of the oil drops is developed. The transition between the two mechanisms is found to be controlled by two length scales: the drop radius $R$ and the boundary layer thickness $\delta$ of the Marangoni flow around the drop. The instability is dominated by diffusion for $\delta &lt; R$ and by viscosity for $R&lt;\delta$ . The experimental results for various drops of different viscosities can well be described with this unifying scaling theory. Our theoretical description thus provides a unifying view of physicochemical hydrodynamic problems in which the Marangoni stress is competing with a stable stratification.

Negative synergistic effects of surfactant and fluid viscoelasticity on hydrodynamic resistance of single droplet in confined microchannel

Polymers and surfactants are often employed simultaneously to control droplet dynamics with higher flexibility in many applications, such as droplet microfluidics and chemical enhanced oil recovery. However, the coupling effects of polymer-induced fluid viscoelasticity and surfactant have not been fully uncovered yet. To facilitate studies in this area, we present a systematic investigation on the transport of a surfactant-laden viscoelastic droplet through a confined microchannel by using our own three-dimensional front-tracking finite-difference methodology. Of particular interest is the droplet-induced additional pressure loss, which is important to deeply understand the flow rate–pressure loss relation of droplet-laden flows. We have found that either the fluid viscoelasticity or surfactant tends to enlarge the additional pressure loss, while their co-occurrence induces a further increase. Notably, negative synergistic effects are indicated between fluid viscoelasticity and surfactant; that is, their combined effect to increase the additional pressure loss is smaller than the sum of their individual effects. This synergistic effect primarily results from mutual inhibition of the viscoelastic stress and the surfactant-induced Marangoni stress to reduce the droplet surface mobility, no matter whether the surfactant is soluble or insoluble. Particularly, when the surfactant is soluble to the viscoelastic fluid phase, its transport and the consequent Marangoni stress is suppressed by the bulk viscoelastic stress via two mechanisms: the weakened surface convection by direct impact of the viscoelastic stress on the droplet surface mobility and the weakened bulk convection by the flow modification effect.

Lattice Boltzmann simulation of drop splitting in a fractal tree-like microchannel

Fractal tree-like microchannel is advantageous to faster emulsification and droplet production in microchannels. Although computer simulation is becoming a powerful tool for design and optimization, precise treatment of a large number of walls in fractal tree-like microchannel is troublesome. The numerical roughness caused by traditional bounce-back boundary condition may accumulate and generate large errors in flow pattern recognization. In this paper, an approach integrating Immersed Boundary Method (IBM), Phase-Field Model (PFM) and Lattice Boltzmann Method (LBM) is developed, aiming to accurately simulate droplet splitting in fractal tree-like microchannel system involving surfactants and wetting boundary walls. Then the operating conditions including capillary number, flow rate ratio and wetting boundary conditions were optimized. We found that the capillary number in 0.02 ∼ 0.05 and flow rate ratio in 1/3.6 – 1/9 can accelerate emulsification. Hydrophobic-lipophilic walls generate a slug-like water drop of bullet shape and thin oil film between the drop and walls, facilitating drop movement in microchannel. The effects of surfactant, Marangoni stress and interfacial tension force on droplet splitting were investigated. The contact walls at the corner and forks resist the surfactant migration, and therefore enrich surfactant at the upstream end of drop interface, leading to uneven distribution of interfacial tension. The breaking interfaces and the interfaces contacting with wall or fork are subjected to opposite reaction of Marangoni stress and tend to be thinner and finally deform or break.

Active control of jet breakup and droplet formation using temperature modulation

Different regimes of temperature modulated breakup were studied using a slender jet model. The instability is induced through capillary and Marangoni stresses and can be used to control droplet formation in terms of intact length and resultant drop size distribution, which is otherwise irregular due to the inevitable presence of background noise. When thermal modulation is weak, the surface tension gradient forces act only as a trigger and curvature-gradient forces soon take over and grow exponentially downstream of the jet due to inertio-capillary growth. When thermal modulation is strong, the surface tension gradient forces not only act as a trigger, but remain significant till breakup.