On the Interaction of Capillary Shapes with Solid Surfaces

Control over the interaction of droplets with solid surfaces is commonplace in nature. Famous examples are the water-shedding capabilities of the lotus leaf and the water-harvesting skin of certain types of beetles. To date, this type of control remains a challenge in engineering applications. Consider, for example, droplets on the windscreen of a car that need to be wiped off because they unwantedly stick to the glass, or the dripping of droplets from fog harvesting devices, where one actually wants the droplets to stick. In this thesis, we address the interaction of droplets with solid surfaces that underlies the sticking behavior. In particular, we investigate how droplets stick to chemically and physically heterogeneous surfaces, and what shape droplets take when squeezed by various confining geometries. The results of these investigations can be used to predict the force needed to set a droplet on a given surface into motion, and to calculate the volume of droplets in confining geometries. An important tool in our studies is the principle of energy minimization, by which we determine the (local) equilibrium shape of a droplet under the influence of surface tension, gravity and solid boundaries. It turns out that the interaction of droplets with solid surfaces can largely be captured in terms of two aspects of the system. The first is the contact line of the droplet, the line where the fluid-fluid interface meets the solid surface. The second is the contact angle distribution around this line, where the contact angle is defined as the angle at which the fluid-fluid interface meets the solid surface. A core question in our work is how to predict the shape and sticking force of a droplet on a realistic, thus physically or chemically heterogeneous, inclined surface. The answer to this question constitutes an important step towards a model for the sticking force. We resolve this question by studying the model system of a droplet on an incline, using 2D analytical and 3D numerical energy minimization. An important conclusion from our work is that the maximum sticking force of the droplet is determined by local constraints on repositioning of the contact line, originating from a range of possible contact angles within which the droplet can deform while the contact line remains static. The possible, i.e. accessible, shapes of the droplet during the inclination process are then also determined by constraints on repositioning, which causes the initial shape and deformation history of the droplet to influence the shape of the droplet at roll off and thereby the maximum sticking force. Accounting for inaccessible shapes and droplet shape history hence results in accurate predictions of the sticking force, in contrast to currently available models that allow all shapes and are thus bound to yield incorrect predictions. Based on the insight that local constraints control the global shape of the droplet we developed an approach to analytically predict the deformation of 3D droplets upon inclination and the maximum sticking force of these droplets. This prediction can be used for a number of engineering applications where the sticking force of droplets is an important factor. Examples include the design and positioning of solar panels and windscreens, as well as more industrial applications such as condensors and chemical reactors with small flow channels. The analytical model for 3D droplets builds on the findings on the role of local constraints by explicitly modelling the deformation process of the contact line, in contrast to the current state-of-the-art where contact line shapes are either assumed or predicted without regard for the constraints. To keep the model analytically tractable, we parameterize the deformation of the droplet in terms of the contact line shape and the contact angle distribution around the contact line. In combination with a few physically sound assumptions, based on our earlier findings, this model accurately predicts the deformation and maximum sticking force of a droplet, as compared with experiments and numerical simulations. Unlike currently available models, this model also accounts for the history of the droplet in the prediction of the maximum sticking force. Apart from droplets sticking to surfaces we also use energy minimization to determine the shape of bubbles and droplets in straight microchannels of various cross-sectional geometries. With this, we develop a model to calculate the volume of a droplet on the basis of only the channel geometry and the droplet length as observed from a top view. This determination of volume is for example important to accurately determine the concentration of gaseous species or nutrients in studies of mass transfer rates or growth of cells. The model is based on a physical description of the central part of the droplet, where we calculate the shape of a cross-sectional slice of the droplet based on energy minimization. The volume calculation is completed with two caps on the droplet that smoothly fit onto the central part and mimick the physically realistic shape. We compare our analytical model with 3D numerical energy minimization calculations and find excellent agreement for a large range of droplet lengths and channel geometries, thus validating the model for use in quantitative research. The findings of this research can, with small adaptations, be applied in related systems, such as pinned gas bubbles or droplets in a shear flow. We discuss the necessary adaptations and what open questions remain in the field of droplet pinning, specifically related to cases where assumptions made in our work do not apply. This results in a series of recommendations for future research on the role of viscous and inertial effects and heterogeneities with sizes on the order of the droplet.Chemical EngineeringApplied Science