Marangoni flows induced by non-uniform surfactant distributions

The spreading dynamics of surfactants is of crucial importance for numerous technological applications ranging from printing and coating processes, pulmonary drug delivery to crude oil recovery. In the area of inkjet printing surfactants are necessary for lowering surface tension of water-based ink solutions, but can considerably delay the leveling and redistribution of inkjet-printed lines. In the context of oil recovery, up to about 60% of the originally present crude oil remains in a reservoir after the so-called primary and secondary recovery phases. Injection of surfactant solutions is considered a potential means for extracting a larger fraction of the oil owing to two different mechanisms. Surfactant-induced reductions of interfacial tension facilitate deformations of oil-brine interfaces and therefore oil extraction. Furthermore, a non-uniform surfactant distribution at fluid-fluid interface gives rise to interfacial tension gradients and associated Marangoni stresses, which generate flow from regions of lower to regions of higher interfacial tension. This can be utilized to transport surfactant along dead-end pores that are inaccessible to pressure-driven transport. This dissertation is mainly dedicated to the modeling of liquid transport induced by non-uniform surfactant distributions at liquid-air and liquid-liquid interfaces. Several types of material systems were investigated for surfactant-induced flows in the presence of geometrical confinement and constraints. We monitor the evolution of the liquid height profile after non-uniform surfactant deposition at the interface. The most prominent morphological feature of the spreading process is the formation of a local maximum in film thickness, called rim, and its propagation away from the deposition region. The rim dynamics can be well approximated by power-law ...??, where ?? is called spreading exponent. We conducted an extensive numerical study of surfactant spreading on thin films and quantitatively compared our results with experimental data. In addition, we also investigated how surfactant-induced flow phenomena affect leveling and redistribution dynamics of inkjet-printed lines. Chapter 2 presents the spreading dynamics of insoluble surfactants on thin liquid films with initially uniform thickness. Numerical simulations based on the lubrication approximation of the far-field axisymmetric spreading dynamics compare very favorably with the experimental results reported in the literature for oleic acid spreading on glycerol. The corresponding non-linear equation of state, which provides an excellent fit for the experimental dependence of surface tension on surfactant concentration, was shown to influence spreading rates considerably compared with a linear one. A fingering instability was observed, which is induced by the temporary entrapment of sub-phase liquid in the surfactant deposition area and its subsequent release. The expulsion has a direct effect by increasing the spreading exponent. Chapter 3 is dedicated to insoluble surfactant spreading at curved liquid-air interfaces. Using a numerical model based on the lubrication approximation, we monitored the evolution of the liquid height profile after deposition of an insoluble surfactant monolayer at rivulet-air interface. Continuous, i.e. unlimited surfactant supply led to higher exponents and increased the influence of the rivulet aspect ratio as compared to the case of limited supply. The spreading exponents determined from a model implementing a continuous supply of surfactant compare favorably with experimental data for oleic acid on glycerol. The initial film thickness has little effect on the spreading exponents for surfactant spreading at rivulet interfaces. The lateral confinement induces non-uniform height- and surface velocity profiles, which manifest themselves in a pronounced transition of the evolving rivulet morphology. Chapter 4 contains the study of soluble surfactant spreading on curved liquidair interfaces. Our numerical model was based on the lubrication approximation and the assumption of vertically uniform concentration profiles. A proper choice of initial and boundary conditions in the numerical models resulted in spreading exponents that are in excellent agreement with the experimental results for sodium dodecyl sulfate spreading on narrow glycerol rivulets. The influence of fingering instabilities, commonly observed during the spreading process, on the rim shape and propagation rate was studied. The rim height profiles deduced from experiments were in excellent agreement with numerical data at early times, but systematically lower at later stages. The origin of this discrepancy resides in vertical concentration non-uniformities, caused by slow vertical diffusion of the surfactant, which is not accounted for in the lubrication model. Chapter 5 deals with the spreading at liquid-air interfaces of a soluble, slowly diffusing surfactant. The spreading was studied by employing two different numerical models. The first model is based on the full Navier-Stokes equation and convection-diffusion equations for bulk- and surface surfactant transport. It accounts for domain deformability and allows for vertically non-uniform concentration profiles. The second model is based on the lubrication approximation and the assumption of vertically parabolic concentration profiles. The two studied models gave considerably different results with respect to film thickness evolution for diffusion coefficients below a certain value. The difference originates in vertical concentration non-uniformities in the sub-phase liquid that are not accurately represented by the parabolic profile upon which the lubrication model was based. In addition, Chapter 6 analyzes the spreading of a soluble surfactant at liquidliquid interfaces between parallel solid plates. We consider an interface with initially uniform height as well as narrow rivulets with initially curved interface. A model accounting for vertically non-uniform concentration profiles was used for slowly diffusing surfactants, while a model based on the lubrication approximation corresponding to vertically uniform concentration profiles was applied in the case of sufficiently large diffusion coefficients. We conclude that two liquids of a similar viscosity result in lower spreading rates and rim heights compared to the liquids with large difference in viscosities. A comparison of rivulet height profiles from experiments for a material system characterized by a very small viscosity ratio with the lubrication model data yielded similar conclusions as for spreading at liquid-air interfaces. Good quantitative agreement can be obtained with the lubrication model with combination of continuous surfactant supply at the interface and finite supply in the bulk. Finally, Chapter 7 investigates leveling and redistribution dynamics of inkjetprinted lines in the presence of soluble and insoluble surfactants. We present numerical results as well as scaling relations for both the leveling and redistribution times of sinusoidal ripples and multi-lines as a function of the number of individual lines, the lateral pitch, the average line height and width. Our results were obtained for a non-linear equations of state and in the case of soluble surfactants for surfactant adsorption kinetics described by a Langmuir equation, which provides a basis for estimating optimal process conditions. Surfactants cause surface tension gradients that oppose capillary leveling of ripples initially present at the surface, and thus can slow down the leveling process. The retarding effect of surfactants depends on rates of bulk-surface exchange.