Interfacial Surfactant Concentration Research Articles

Surfactant Partitioning Dynamics in Freshly Generated Aerosol Droplets.

Aerosol droplets are unique microcompartments with relevance to areas as diverse as materials and chemical synthesis, atmospheric chemistry, and cloud formation. Observations of highly accelerated and unusual chemistry taking place in such droplets have challenged our understanding of chemical kinetics in these microscopic systems. Due to their large surface-area-to-volume ratios, interfacial processes can play a dominant role in governing chemical reactivity and other processes in droplets. Quantitative knowledge about droplet surface properties is required to explain reaction mechanisms and product yields. However, our understanding of the compositions and properties of these dynamic, microscopic interfaces is poor compared to our understanding of bulk processes. Here, we measure the dynamic surface tensions of 14-25 μm radius (11-65 pL) droplets containing a strong surfactant (either sodium dodecyl sulfate or octyl-β-D-thioglucopyranoside) using a stroboscopic imaging approach, enabling observation of the dynamics of surfactant partitioning to the droplet-air interface on time scales of 10s to 100s of microseconds after droplet generation. The experimental results are interpreted with a state-of-the-art kinetic model accounting for the unique high surface-area-to-volume ratio inherent to aerosol droplets, providing insights into both the surfactant diffusion and adsorption kinetics as well as the time-dependence of the interfacial surfactant concentration. This study demonstrates that microscopic droplet interfaces can take up to many milliseconds to reach equilibrium. Such time scales should be considered when attempting to explain observations of accelerated chemistry in microcompartments.

Thin film modelling of magnetic soap films

Magnetic soap films under the forcing of an inhomogeneous magnetic field are governed by a wide range of interconnected physics. The study of magnetic soap films requires the development of comprehensive models to support experimental observations. In this study, the thin film approximation is applied to the Navier–Stokes equations to derive a model for the film thickness of magnetic soap films that incorporates the effects of interfacial mobility, surfactant transport and magnetite nanoparticle (NP) transport. This derived model consists of a coupled system of equations for the film thickness, interfacial velocity, interfacial surfactant concentration and magnetite NP concentration. Simulations are performed for both soap films and magnetic soap films by solving the system of equations using the Galerkin finite element method, and results are compared with experiments. Simulation results highlight that interfacial flows can dominate the rate of film thinning and that accounting for the dependence of the magnetisation on the local magnetite NP concentration can influence the predicted speed of magnetically driven flows. Furthermore, simulation results demonstrate that the model is able to predict marginal regeneration in qualitative agreement with the experiments for soap films; the model also predicts the same flow pattern as seen in the experiments for magnetic soap films. Overall, this study advances the state of soap film and magnetic soap film modelling and will contribute to acquiring control over the drainage and stability of magnetic soap films in the long term.

Surface coverage ratio of contaminated Taylor bubbles in a square microchannel

Experiments on contaminated Taylor flows in a microchannel were carried out to investigate the surface coverage ratio for different surfactant properties. Nitrogen and water were used for the gas and liquid phases, respectively. Triton X-100, sodium dodecyl sulfate (SDS) of 90% purity and SDS of 99% purity were used as surfactant. The cross section of the microchannel was square and its width was 200μm. The gas and liquid volumetric fluxes were 0.54 and 0.50 m/s, respectively. Taylor flows were formed by mixing the two phases at the T-junction. Shapes of Taylor bubbles were observed using a microscope and a high-speed video camera. The liquid film thickness changed in the film region, demonstrating that a steep gradient of the interfacial surfactant concentration was formed there. The interface was therefore separated into the clean interface region and the immobilized region by detecting the change in the liquid film thickness. The surface coverage ratio estimated from bubble shapes agreed with that measured from the reduction in the bulk concentration due to surfactant separation by bubbles. The surface coverage ratio increased with increasing the bulk concentration and the increasing rate was smaller than that for the adsorption–desorption equilibrium because of the presence of interfacial advection. The data, however, could be fitted by modifying the functional form for the ad-desorption equilibrium. The bulk concentration normalized by the concentration for the breakdown of the logarithmic relation between the surface tension and the concentration given by the isotherm worked for correlating the surface coverage ratio for the three different surfactants, which can be understood from the order of magnitude of the Marangoni stress.

Linear stability of a contaminated two-layer films flowing down a slippery inclined plane: Parametric study and physical mechanism

A linear stability analysis of isothermal two-layer surfactant-laden films flowing down a slippery inclined plane is carried out, with a focus on the shear mode instabilities. The intention is to extend our previous study [Bhat and Samanta, “Linear stability for surfactant-laden two-layer film flows down a slippery inclined plane,” Chem. Eng. Sci. 220, 115611 (2020)] in the high Reynolds number regime. The stability analysis is performed under the umbrella of the Orr–Sommerfeld-type boundary value problem. The method of energy budget is employed to figure out the proper physical mechanisms responsible for the growth of the shear modes under the influence of various flow parameters. Numerical results reveal that the flow configuration under consideration is susceptible to two distinct shear modes in the high Reynolds number regime. The shear mode associated with the lower fluid layer is stabilized if the viscosity ratio, wall slip, and interfacial surfactant concentration are increased. However, increasing density ratio has a destabilizing impact on the lower-layer shear mode. On the other hand, the shear mode associated with the upper fluid layer is destabilized if the viscosity ratio and interfacial surfactant concentration are increased. However, the density ratio exhibits a non-monotonic impact, but the wall slip exhibits a stabilizing influence on the upper-layer shear mode. The energy budget analysis predicts that the primary energy source terms for the growth of the shear mode instabilities are base shear stresses, which transfer energy to the disturbance through the Reynolds stress term.

Effect of soluble surfactant on the interface dynamics of a rising droplet

In this study, we present numerical simulations examining the impact of soluble surfactant on the interface dynamics of a rising droplet. To achieve this, the droplet interface is tracked using an arbitrary Lagrangian–Eulerian approach, and the bulk and interfacial surfactant concentration evolution equations fully coupled with the incompressible Navier–Stokes equations are solved. We systematically evaluate the boundary of interfacial dynamics evolution by varying certain dimensionless parameters. Specifically, we study the effects of changes in parameters such as the Langmuir number, the Biot number, the Damkohler number, the bulk Peclet number, and the elastic number on interfacial tangential velocity, interfacial concentration and its gradient, interfacial viscous shear stress, and droplet rising velocity. Our findings confirm the validity of the stagnant-cap model for describing the interfacial fluidity of a surfactant-laden rising droplet. Increasing the Langmuir number and decreasing the Damkohler number can inhibit interface fluidity, but there is a threshold for the Damkohler number. Additionally, the overall increase in interface tension may mask the hindering effect of the locally increased concentration gradient on the interfacial fluidity. The Biot number has no impact on the steady state of the interface, but a slow adsorption rate may result in a bimodal retardation before the interface reaches a steady state. A clear threshold exists for the Peclet number to hinder the interface velocity, and a too high Peclet number leads to strong nonlinearity in the interface physical quantities. Variations in the elastic number significantly affect the evolution of the interface, causing the interface velocity to pass through several states, ranging from almost no retardation, uniform retardation, stagnant-cap retardation to complete retardation.

A new three dimensional cumulant phase field lattice Boltzmann method to study soluble surfactant

Surfactants play a critical role in the physics of paint and coating formulations, affecting key rheological properties such as viscosity, yield stress, and thixotropy. This paper proposes a new three-dimensional phase-field model that uses the cumulant lattice Boltzmann method (LBM) to simulate soluble surfactants. Although current phase-field models commonly use Langmuir's relationship, they cannot calculate interfacial tension analytically, or the LBM models used are unstable when viscosities are low. However, the proposed method overcomes these limitations through two main features. First, the main parameters for modeling and controlling the surfactant's strength and interaction with other phases are directly obtained from a given initial interfacial tension and bulk surfactant, eliminating the need for trial-and-error simulations. Second, a new equilibrium distribution function in the moment space that includes diagonal and off diagonal elements of the pressure tensor is used to minimize Galilean invariance violation. Additionally, there is no need to use an external force to recover multiphase flows, which could break mass conservation. Furthermore, this method has significant potential for parallelization since only one neighbor's cell is used for discretization. The method shows Langmuir relation behavior and is validated with analytical solutions for various interfacial tensions and surfactant concentrations. Moreover, the paper demonstrates the influence of interfacial tension and surfactants on spurious velocities, indicating the method's stability at low viscosities. The dynamics of droplets in the presence of the surfactants is studied in spinodal decomposition and under various external forces. The method accurately simulates the breaking-up and coalescence for these cases. Furthermore, the method successfully simulates the breakage of a liquid thread at a high viscosity ratio.

Impact of droplets onto surfactant-laden thin liquid films

We study the effect of insoluble surfactants on the impact of surfactant-free droplets onto surfactant-laden thin liquid films via a fully three-dimensional direct numerical simulation approach that employs a hybrid interface-tracking/level-set method, and by taking into account surfactant-induced Marangoni stresses due to gradients in interfacial surfactant concentration. Our numerical predictions for the temporal evolution of the surfactant-free crown are validated against the experimental work by Che & Matar (Langmuir, vol. 33, 2017, pp. 12140–12148). We focus on the ‘crown-splash regime’, and we observe that the crown dynamics evolves through various stages: from the growth of linear modes (through a Rayleigh–Plateau instability) to the development of nonlinearities leading to primary and secondary breakup events (through droplet shedding modulated by an end-pinching mechanism). We show that the addition of surfactants does not affect the wave selection via the Rayleigh–Plateau instability. However, the presence of surfactants plays a key role in the late stages of the dynamics as soon as the ligaments are driven out from the rim. Surfactant-induced Marangoni stresses delay the end-pinching mechanisms to result in longer ligaments prior to their capillary singularity. Our results indicate that Marangoni stresses bridge the gap between adjacent protrusions promoting the adjacent protrusions' collision and the merging of ligaments. Finally, we demonstrate that the addition of surfactants leads to surface rigidification and consequently to the retardation of the flow dynamics.

Effects of fluid–fluid interfacial properties on the dynamics of bounded viscoelastic thin liquid films

The influence of fluid–fluid interfacial properties on the dynamics of viscoelastic thin liquid films bounded above by liquid–liquid interface and below by a horizontal solid plate is studied in this work. The present study allows for integration of both chemical and physical properties of the thin liquid film within the long wave approximation of the evolution equation for viscoelastic thin films. The effects of the contact angle, slippage and exposure time on interfacial film dynamics are verified. We demonstrate the importance of the interfacial tension due to surfactant concentration and the Marangoni number to responses of thin films to prescribed perturbations. The linear stability analysis shows that the rupture time is dependent on the initial tension, Marangoni number, slippage and contact angle, whereas the cut off wavenumber is only affected by the contact angle. Results of the numerical simulations of the nonlinear regime show that the film dynamics is faster with slippage and exposure time. We confirm that viscoelasticity increases the number of droplets formed after the film rupture. However, the linear relationship between interfacial tension and surfactant concentration fails to describe the thin film dynamics when the Weissenberg number exceeds the Reynolds number. • Effects of fluid-fluid properties on the dynamics of viscoelastic films is studied. • Both capillary and disjoining pressures are influenced by Interfacial tension (IFT). • Increased Marangoni number leads to slower film dynamics. • Domain of stability is only affected by contact angle.

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>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<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.

Effect of Janus particles and non-ionic surfactants on the collapse of the oil-water interface under compression

HypothesisJanus particles (JPs) and surfactants express different behaviors at the oil–water interface under compression. When both are present at the interface, their synergies result in a different collapse mechanism than when present individually depending on the concentration of the JPs and surfactants. ExperimentsCoarse-grained modeling methods were used to probe the synergies between Janus nanoparticles and nonionic surfactants on the stability of an oil–water interface under compression. When both JPs and surfactants were present, the interface was covered at 0–55% area by JPs and contained surfactants at 0–40% of the interfacial surfactant concentration corresponding to the critical micelle concentration (CMC). FindingsCompression of the interface with only surfactants resulted in the expulsion of surfactant molecules to the water phase once the interfacial concentration of surfactant molecules reached the CMC value. Compression of a Janus particle-laden interface past the closed-packing point led to a buckled interface, so that the total interfacial area remained constant upon further compression. When both surfactants and JPs were present on the interface, JPs still caused buckling, which helped retain the surfactant molecules on the interface. The interface exhibited a higher level of deformation in presence of surfactants. When the surfactant concentration was high, under compression, the surfactants partitioned into the water phase, but the buckling of the interface persisted.

Polymer drag reduction in surfactant-contaminated turbulent bubbly channel flows

Polymer additives are commonly utilized to manipulate bubbly flows in various applications. Here, we investigate the effects of clean and contaminated bubbles driven upwards (upflow) in Newtonian and viscoelastic turbulent channel flows. Interface-resolved direct numerical simulations are performed to examine sole and combined effects of soluble surfactant and viscoelasticity using an efficient 3D finite-difference/front-tracking method. The incompressible flow equations are solved fully coupled with the FENE-P viscoelastic model and the equations governing interfacial and bulk surfactant concentrations. The latter coupling is accomplished by a non-linear equation of state that relates the surface tension to the surfactant concentration. For Newtonian turbulent bubbly flows, the effects of Triton X-100 and 1-Pentanol surfactant are examined. It is observed that the sorption kinetics highly affect the dynamics of bubbly flow. A minute amount of Triton X-100 is found to be sufficient to prevent the formation of bubble clusters restoring the single-phase behavior while even two orders of magnitude more 1-Pentanol surfactant is not adequate to prevent the formation of layers. For viscoelastic turbulent flows, it is found that the viscoelasticity promotes formation of the bubble-wall layers and thus the polymer drag reduction is completely lost for the surfactant-free bubbly flows, while the addition of small amount of surfactant (Triton X-100) in this system restores the polymer drag reduction resulting in $25\%$ drag reduction for the $Wi=4$ case.

Influence of interfacial characteristics and antioxidant polarity on the chemical stability of β-carotene in emulsions prepared using non-ionic surfactant blends

To evaluate the effects of characteristics of the interfacial region and physical location of antioxidants on the chemical stability of β-carotene in emulsions, β-carotene-loaded emulsions were prepared with the blends of surfactants with different hydrophilic head sizes (ethylene oxide (EO) unit number). In addition, antioxidants having the different solubilities, tert-butylhydroquinone (TBHQ) and lauryl gallate as lipophilic and amphiphilic antioxidants, respectively, were incorporated into emulsions. Decrease in the average EO unit number at interfacial region by surfactant blending did not affect the stability of β-carotene in emulsions at neutral pH but it had a positive effect on the β-carotene stability at acidic pH. Decrease in the average EO unit number at interfacial region reduced the ability of antioxidants, regardless of their polarity, to protect β-carotene in emulsions at neutral and acidic pH. TBHQ was more effective in retarding β-carotene degradation than lauryl gallate, independent of the interfacial composition and surfactant concentration.

Influence of Blending of Nonionic Emulsifiers Having Various Hydrophilic Head Sizes on Lipid Oxidation: Investigation of Antioxidant Polarity Interfacial Characteristics Relationship.

The purpose of this study was to deliver insights into the effect of interfacial composition and antioxidant polarity on the lipid oxidation of emulsions. Emulsions were created using blends of nonionic ethoxylated fatty acid alcohol surfactants with different hydrophilic head sizes, and lipophilic (TBHQ) and amphiphilic (lauryl gallate) antioxidants were incorporated into the emulsions. At the same surfactant concentration, emulsion stabilized with surfactant with a smaller hydrophilic head was more susceptible to lipid oxidation than that stabilized with surfactant with a larger hydrophilic head. When surfactants with a similar hydrophilic head size were used, lipid oxidation in emulsion containing more surfactant was slightly faster than that containing less surfactant. When emulsions were created with a 1:1 molar ratio mixture of surfactants with small and large hydrophilic heads, surfactant concentration (1.00 and 2.932 mM) had little effect on lipid peroxide generation rate. However, the concentration of thiobarbituric acid-reactive substances (TBARSs) in the emulsion prepared at 1.00 mM increased faster than that prepared at 2.93 mM. Alteration of interfacial composition and surfactant concentration did not affect antioxidant ability, regardless of antioxidant polarity, to inhibit lipid peroxide generation. However, the ability of lauryl gallate and TBHQ to prevent TBARS generation was elevated by mixing surfactants with small and large hydrophilic heads and by decreasing surfactant concentration. In most emulsions, lauryl gallate showed a more effective antioxidant ability than TBHQ.

Using Liquid Crystals for In Situ Optical Mapping of Interfacial Mobility and Surfactant Concentrations at Flowing Aqueous-Oil Interfaces.

Flow-induced states of fluid interfaces decorated with amphiphiles underlie phenomena such as emulsification, foaming, and spreading. While past studies have shown that interfacial mass transfer, the kinetics of surfactant adsorption and desorption, interfacial mobility, and surfactant reorganization regulate the dynamic properties of surfactant-laden interfaces, few simple methods permit simultaneous monitoring of this interplay. Here, we explore the optical responses of micrometer-thick films of oils (4-cyano-4'-pentylbiphenyl, 5CB) with a liquid crystalline order in contact with flowing aqueous phases of soluble [e.g., sodium dodecyl sulfate (SDS)] or insoluble (e.g., 1,2-dilauroyl-sn-glycero-3-phosphocholine) amphiphiles. We observe the onset of flow of 0.5 mM SDS solutions within a millifluidic channel (area-average velocity of 200 mm/s) to transform a liquid crystal (LC) film with an alignment along the interface normal into a bright birefringent state (average LC tilt angle of 30°), consistent with an initially mobile interface that shears and thus tilts the LC along the flow direction. Subsequently, we observed the LC film to evolve to a steady state (over ∼10 s) with position-dependent optical retardance controlled by gradients in surfactant concentration and thus Marangoni stresses. For 0.5 mM SDS solutions, by using particle tracking and a simple hydrodynamic model, we reveal that the dominant role of the flow-induced interfacial surfactant concentration gradient is to change the mobility of the interface (and thus shear rate of LC) and not to change the easy axis (equilibrium orientation) or anchoring energy (orientation-dependent interfacial energy) of the LC. At lower surfactant concentrations (0.015 mM SDS), however, we show that the LC directly maps flow-induced interfacial surfactant concentration gradients via a change in the local easy axis of the LC. When combined with additional measurements obtained with simple salts and insoluble amphiphiles, these results hint that LC oils may offer the basis of general and facile methods that permit mapping of both interfacial mobilities and surfactant distributions at flowing interfaces.

Three-dimensional dynamics of falling films in the presence of insoluble surfactants

Abstract

Turbulent bubbly channel flows: Effects of soluble surfactant and viscoelasticity

Interface-resolved direct numerical simulations are performed to examine the combined effects of soluble surfactant and viscoelasticity on the structure of a bubbly turbulent channel flow. The incompressible flow equations are solved fully coupled with the FENE-P viscoelastic model and the equations governing interfacial and bulk surfactant concentrations. The latter coupling is achieved through a non-linear equation of state which relates the surface tension to the surfactant concentration at the interface. The two-fluid Navier-Stokes equations are solved using a front-tracking method, augmented with a very efficient FFT-based pressure projection method that allows for massively parallel simulations of turbulent flows. It is found that, for the surfactant-free case, bubbles move toward the wall due to inertial lift force, resulting in formation of wall layers and a significant decrease in the flow rate. Conversely, a high-enough concentration of surfactant changes the direction of lateral migration of bubbles, i.e., the contaminated bubbles move toward the core region and spread out across the channel. When viscoelasticity is considered, viscoelastic stresses counteract the Marangoni stresses, promoting formation of bubbly wall-layers and consequently strong decrease in the flow rate. The formation of bubble wall-layers for combined case depends on the interplay of the inertial and elastic, and Marangoni forces.

Effects of soluble surfactant on lateral migration of a bubble in a pressure driven channel flow

The effects of soluble surfactant on the lateral migration of a bubble in a pressure-driven channel flow are examined by interface-resolved numerical simulations. The interfacial and bulk surfactant concentration evolution equations are solved fully coupled with the incompressible Navier-Stokes equations. A non-linear equation of state is used to relate interfacial surface tension to surfactant concentration at the interface. Extensive computations are performed to investigate the bubble dynamics for a wide range of parameters. It is found that surfactant dramatically changes the bubble dynamics. In the clean case, the bubble position depends on its deformability, characterized by the Eo¨tvo¨s (Eo) and the capillary (Ca) numbers. The spherical bubble moves towards the wall, while the deformable one migrates away from it. On the other hand, in the presence of the surfactant, even the spherical bubble moves away from the wall. It is also found that the contaminated bubble stays away from the wall for Eo ≤ 0.1 and Eo ≥ 1.5 while it migrates towards the wall for 0.1 < Eo < 1.5. Also, at high Eo, the onset of path instability is observed for both the clean and the contaminated cases. However, adding surfactant to the system triggers the path instability earlier and amplifies the oscillations afterwards.

Effects of surfactant on propagation and rupture of a liquid plug in a tube.

Surfactant-laden liquid plug propagation and rupture occurring in lower lung airways are studied computationally using a front-tracking method. The plug is driven by an applied constant pressure in a rigid axisymmetric tube whose inner surface is coated by a thin liquid film. The evolution equations of the interfacial and bulk surfactant concentrations coupled with the incompressible Navier-Stokes equations are solved in the front-tracking framework. The numerical method is first validated for a surfactant-free case and the results are found to be in good agreement with the earlier simulations of Fujioka et al. (2008) and Hassan et al. (2011). Then extensive simulations are performed to investigate the effects of surfactant on the mechanical stresses that could be injurious to epithelial cells such as pressure and shear stress. It is found that the liquid plug ruptures violently to induce large pressure and shear stress on airway walls and even a tiny amount of surfactant significantly reduces the pressure and shear stress and thus improves cell survivability. However, addition of surfactant also delays the plug rupture and thus airway reopening.

A treatment for contaminated interfaces and its application to study the hydrodynamics of a spherical bubble contaminated by surfactants

A numerical model for the contaminated interface is discussed, and then it is applied to investigate detailedly the interfacial surfactant concentration, the interfacial properties and the wake flow of a contaminated spherical bubble. For the present model, the surfactant molecules adsorbed on the bubble surface are assumed to be distributed over the concentration boundary layer adjacent to the bubble. The thickness of concentration boundary layer is obtained through the revised value from a theoretical formula. Then, the interfacial surfactant concentration is evaluated by coupling the transport equations of interfacial and bulk surfactant concentrations with the Navier-Stokes equations. The connection between the tangential stress and the interfacial surfactant concentration is set through the Marangoni stress and the state equation. The current results show that the present numerical model is reliable. The convention from the main flow, the Marangoni effect and the wake vortex determine the distribution of interfacial surfactant concentration on the bubble surface. The gradient of interfacial surfactant concentration along the bubble surface further affects the interfacial properties which react on the surrounding flow as feedbacks. The gradient of interfacial surfactant concentration at the front edge of the “contaminated” interface (FECI) is closely related to the location of FECI to the strongest convection position in the bulk around the bubble. The peak point on the curves of tangential stress and interfacial vorticity corresponds to the inflection point on the curve of interfacial surfactant concentration. The vortex size is affected by the positive pressure gradient at the bubble rear in bulk and the abrupt change of interfacial vorticity at FECI.

The transition in settling velocity of surfactant-covered droplets from the Stokes to the Hadamard–Rybczynski solution

The exact solution for a small falling drop is a classical result by Hadamard and Rybczynski. But experiments show that small drops fall slower than predicted, giving closer agreement with Stokes’ result for a falling hard sphere. Increasing the drop size, a transition between these two extremes is found. This is due to surfactants present in the system, and previous work has led to the stagnant-cap model. We present here an alternative approach which we call the continuous-interface model. In contrast to the stagnant-cap model, we do not consider a surfactant advection–diffusionequation at the interface. Taking instead the normal and tangential interfacial stresses into account, we solve the Stokes equation analytically for the falling drop with varying interfacial tension. Some of the solutions thus obtained, e.g. the hovering drop, violate conservation of energy unless energy is provided directly to the interface. Considering the energy budget of the drop, we show that the terminal velocity is bounded by the Stokes and the Hadamard–Rybczynski results. The continuous-interface model is then obtained from the force balance for surfactants at the interface. The resulting expressions gives the transition between the two extremes, and also predicts that the critical radius, below which drops fall like hard spheres, is proportional to the interfacial surfactant concentration. By analysing experimental results from the literature, we confirm this prediction, thus providing strong arguments for the validity of the proposed model.