Polarization of disk electrodes in high-conductivity electrolyte solutions.

We investigate the polarization of disk electrodes immersed in an electrolyte solution and subjected to a small external AC voltage over a wide range of frequencies. A mathematical model is developed based on the Debye-Falkenhagen approximation to the coupled Poisson-Nernst-Planck equations. Analytical techniques are used for predicting the spatial distribution of the electric potential and the complex impedance of the system. Scales for impedance and frequency are identified, which lead to a self-similar behavior for a range of frequencies. Experiments are conducted with gold electrodes of sizes in the range 100-350 µm immersed in a high-conductivity KCl solution over five orders of magnitude in frequency. A collapse of data on impedance magnitude and phase angle onto universal curves is observed with scalings motivated by the mathematical model. A direct comparison with the approximate analytical formula for impedance is made without any fitting parameters, and a good agreement is found for the range of frequencies where the analytical model is valid.

Evaporation and fluid flow near the boundary of a stationary dry patch.

We consider an isolated circular dry patch formed in an evaporating liquid layer and investigate local viscous flows in both liquid and air near the contact line which is the boundary of the dry patch. Flow patterns in the liquid deviate significantly from the predictions of single-phase models even when the air-to-liquid dynamic viscosity ratio is small. In particular, the separatrices in the liquid flow patterns at large contact angles disappear completely for a range of realistic viscosity ratios when the shear stress on the air side of the interface is accounted for. Experimentally observed motion of microdroplets near the contact line under combined action of gravity and moist air flow is described using our local flow model. We demonstrate that analysis of droplet trajectories leads to unambiguous determination of the local evaporative flux profile. A numerical finite-element approach for the steady diffusion equation is then used to extract the same flux profile from the global solution for concentration field in the limiting case of very thin liquid layer and small contact angle. The local evaporation rate is underpredicted by the numerical method, most likely due to the neglect of convective mass transfer in the air.

Levitation conditions for condensing droplets over heated liquid surfaces.

We report experimental studies and develop mathematical models of levitation of microscale droplets over an evaporating liquid layer. The maximum size of droplets is estimated from the balance between gravity and Stokes force due to the action of upward Stefan flow generated by evaporation. Mathematical models of diffusion around levitating droplets allow us to determine Stefan flow velocity at the liquid layer surface. These results are then used to determine the dependence of levitation height on droplet size. Experimental data for a range of conditions are shown to collapse onto a single curve predicted from the model.

Interaction of advancing contact lines with defects on heated substrates.

We consider an advancing contact line traveling over a region of locally modified wetting or thermal substrate properties. A lubrication-type model is developed to account for coupling of viscous flow, evaporation, surface tension, and disjoining pressure. Stick-slip-type behavior is found for a range of conditions as the contact line passes over the defect and explained by a temporary increase in the local stresses disrupting the liquid supply into the contact line region. A simple estimate of the degree of contact line slowdown is obtained and compared with the numerical simulation results. Tangential stresses arising from the action of the electric field on the interfacial changes are accounted for in our model; neglecting them would lead to an overprediction of the time of interaction between the contact line and the defect. Increasing the substrate temperature uniformly has little effect on contact line motion, but local increase of the temperature enhances the tendency of the contact line to be pulled back by the defect, an effect explained by the Marangoni stresses.

Levitation and Self-Organization of Liquid Microdroplets over Dry Heated Substrates.

Levitating droplets of liquid condensate are known to organize themselves into ordered arrays over hot liquid-gas interfaces. We report experimental observation of similar behavior over a dry heated solid surface. Even though the lifetime of the array is shorter in this case, its geometric characteristics are remarkably similar to the case of droplets levitating over liquid-gas interfaces. A simple model is developed that predicts the mechanisms of both droplet levitation and interdroplet interaction leading to pattern formation over a dry surface; the model is shown to be in good agreement with the experimental data. Using the insights from the new experiments, we are able to resolve some long-standing controversies pertaining to the mechanism of levitation of droplets over liquid-gas interfaces.

Stability and break-up of thin liquid films on patterned and structured surfaces.

Solid surfaces with chemical patterning or topographical structure have attracted attention due to many potential applications such as manufacture of flexible electronics, microfluidic devices, microscale cooling systems, as well as development of self-cleaning, antifogging, and antimicrobial surfaces. In many configurations involving patterned or structured surfaces, liquid films are in contact with such solid surfaces and the issue of film stability becomes important. Studies of stability in this context have been largely focused on specific applications and often not connected to each other. The purpose of the present review is to provide a unified view of the topic of stability and rupture of liquid films on patterned and structured surfaces, with particular focus on common mathematical methods, such as lubrication approximation for the liquid flow, bifurcation analysis, and Floquet theory, which can be used for a wide variety of problems. The physical mechanisms of the instability discussed include disjoining pressure, thermocapillarity, and classical hydrodynamic instability of gravity-driven flows. Motion of a contact line formed after the film rupture is also discussed, with emphasis on how the receding contact angle is expected to depend on the small-scale properties of the substrate.

Significance of electrically induced shear stress in drainage of thin aqueous films.

We develop a novel model of drainage of microscale thin aqueous film separating a gas bubble and a solid wall. In contrast to previous studies, the electrostatic effects are accounted for not only in the normal but also in the shear stress balance at the liquid-gas interface. We show that the action of the tangential component of the electric field leads to potentially strong spatially variable shear stress at the deforming charged interface. This previously overlooked effect turns out to be essential for correctly estimating the long-time drainage rates. Comparison of time-dependent fluid interface shapes predicted by our model with the experimental data is discussed.

Effect of charge regulation on the stability of electrolyte films.

The stability of a thin liquid film of an electrolyte on a solid substrate is investigated. In the framework of the Debye-Hückel approximation, we show that the commonly used approximation of fixed potential at the solid-liquid interface does not lead to predictions of film rupture. To reconcile the model with experimental observations, we consider the constant charge density approximation for the solid substrate and then proceed to systematically investigate the effects of charge regulation based on a linear relationship between charge density and potential. Stability criteria are formulated in terms of charge regulation parameters and electrolyte properties, resulting in different types of stability diagrams. Critical thickness below which the film ruptures is shown to decrease as the charge regulation at the solid-liquid interface becomes stronger.

Rupture of thin liquid films on structured surfaces.

We investigate stability and breakup of a thin liquid film on a solid surface under the action of disjoining pressure. The solid surface is structured by parallel grooves. Air is trapped in the grooves under the liquid film. Our mathematical model takes into account the effect of slip due to the presence of menisci separating the liquid film from the air inside the grooves, the deformation of these menisci due to local variations of pressure in the liquid film, and nonuniformities of the Hamaker constant which measures the strength of disjoining pressure. Both linear stability and strongly nonlinear evolution of the film are analyzed. Surface structuring results in decrease of the fastest growing instability wavelength and the rupture time. It is shown that a simplified description of film dynamics based on the standard formula for effective slip leads to significant deviations from the behavior seen in our simulations. Self-similar decay over several orders of magnitude of the film thickness near the rupture point is observed. We also show that the presence of the grooves can lead to instability in otherwise stable films if the relative groove width is above a critical value, found as a function of disjoining pressure parameters.

Static and dynamic contact angles of evaporating liquids on heated surfaces.

We studied both static and dynamic values of the apparent contact angle for gravity-driven flow of a volatile liquid down a heated inclined plane. The apparent contact line is modeled as the transition region between the macroscopic film and ultra-thin adsorbed film dominated by disjoining pressure effects. Four commonly used disjoining pressure models are investigated. The static contact angle is shown to increase with heater temperature, in qualitative agreement with experimental observations. The angle is less sensitive to the details of the disjoining pressure curves than in the isothermal regime. A generalization of the classical Frumkin-Derjaguin theory is proposed to explain this observation. The dynamic contact angle follows the Tanner's law remarkably well over a range of evaporation conditions. However, deviations from the predictions based on the Tanner's law are found when interface shape changes rapidly in response to rapid changes of the heater temperature. The Marangoni stresses are shown to result in increase of the values of apparent contact angles.

Ripples in a wetting film formed by a moving meniscus.

We carry out a theoretical investigation of the evolution of a wetting film formed by pressing a bubble against a solid substrate. Our model incorporates the effects of capillarity and Derjaguin-Landau-Verwey-Overbeek (DLVO) (van der Waals and electrostatic) components of the disjoining pressure. Rapid changes in the relative position of the bubble and the substrate are shown to result in surprisingly rich dynamics of wetting film deformations. Even for stable films, we find transient rippled deformations with several points of local maximum of wetting film thickness and discuss how their evolution depends on changes in the meniscus position relative to the substrate and the disjoining pressure parameters. A connection is made to the recently reported experimental observations of one such rippled deformation, the so-called wimple, characterized by a local minimum of the thickness in the center, surrounded by a ring of greater film thickness and bounded at the outer edge by the barrier rim. Guidelines are provided for experimental detection of more complex rippled deformations in stable wetting films. Development of instability is studied in a situation when the electrostatic component of disjoining pressure is destabilizing, with particular emphasis on the nonlinear evolution and rupture of the film. Potential applications of our findings to small-scale mixing and deposition of nanoparticles are discussed.

Evolution of dry patches in evaporating liquid films.

We investigate evolution of dry patches in a thin film of a volatile liquid on a heated plate in the framework of a model that accounts for the effects of surface tension, evaporation, thermocapillarity, and disjoining pressure. Dry areas on the plate are modeled by isothermal microscopic films which are in thermodynamic equilibrium with the vapor. For nonpolar liquids such equilibrium is achieved due to van der Waals forces, well-defined capillary ridges are formed around growing dry patches, contact line speed increases with time. For polar liquids the microscopic film is formed by combined action of van der Waals and electrical double layer forces, capillary ridge is small, and contact line speed quickly approaches a constant value. Thermocapillary stresses tend to increase the height of the capillary ridges formed around expanding patches. Numerical simulations demonstrate that the proposed model is capable of describing a number of complicated phenomena observed in experimental studies of evaporating films including fingering instabilities and merger of growing dry patches.

Viscous flow of a volatile liquid on an inclined heated surface.

We investigate the effects of evaporation on a gravity-driven flow of a viscous liquid on a heated solid surface. Vapor molecules are adsorbed on the dry areas of the solid and form a microscopic adsorbed film. The thickness of this film is calculated from the formulas for disjoining pressure and the principles of equilibrium thermodynamics. A lubrication-type approach is used to derive an evolution equation capable of describing both the macroscopic shape of the vapor-liquid interface and the adsorbed film on the vapor-solid interface. Under the conditions of negligible evaporation, the numerical solution of the evolution equation predicts translational motion and formation of capillary ridge, in agreement with previous investigations. Moderate evaporation is shown to slow down the flow and decrease the height of the capillary ridge, which implies a stabilizing effect of evaporation on the well-known instability observed in gravity-driven thin film flows. We also study the combined effects of evaporation and thermocapillary stresses and show that the latter act to reduce the velocity of the downward motion, but increase the height of the capillary ridge. Apparent contact angles are found from the solution and shown to increase with evaporation and contact line speed. For strong evaporation, steady state solutions are found such that evaporation balances the downward motion of the interface under the action of gravity.

Dynamic response of geometrically constrained vapor bubbles.

We consider a two-dimensional model of a vapor bubble between two horizontal parallel boundaries held at different temperatures. When the temperatures are constant, a steady state can be achieved such that evaporation near the contact lines at the hot bottom plate is balanced by condensation in colder areas of the interface near the top. The dynamic response of the bubble is probed by treating the case of time-dependent wall temperatures. For periodic modulations of the wall temperature the bubble oscillates about the steady state. In order to describe such time-dependent behavior we consider the limit of small capillary number, in which the effects of heat and mass transfer are significant only near the contact lines at the bottom plate and in a small region near the top. When the bottom temperature is modulated and the top temperature is held fixed, the amplitude of forced oscillations is constant for low-frequency modulations and then rapidly decays in the high-frequency regime. When the top temperature is modulated with fixed bottom temperature, the dynamic-response curve is flat in the low-frequency regime as well, but it also flattens out when the frequency is increased. This shape of the response curve is shown to be the result of the nonmonotonic behavior of the thickness of the liquid film between the bubble interface and the top plate: when the temperature is decreased, the film thickness increases rapidly, but then slowly decays to a value which is smaller than the initial thickness. The dynamic response is also studied as a function of the forcing amplitude.

Steady Vapor Bubbles in Rectangular Microchannels.

We consider vapor bubbles in microchannels in which the vapor is produced by a heater element and condenses in cooler parts of the interface. The free boundary problem is formulated for a long steady-state bubble in a rectangular channel with a heated bottom. Lubrication-type equations are derived for the shape of the liquid-vapor interface in a cross-sectional plane and in the regime for which the vapor phase fills most of the cross section. These equations are then solved numerically over a range of parameter values with given temperature profiles in the walls and subject to a global integral condition requiring evaporation near the heater to balance condensation in colder areas of the interface. Our results show that depending on the temperature, the side walls can be either dry or covered with a liquid film and we identify criteria for these two different regimes. The asymptotic method breaks down in the limit when capillary condensation becomes important near the bubble top and a different approach is used to determine the shape of the bubble in this limit. Solutions here involve localized regions of large mass fluxes, which are asymptotically matched to capillary-statics regions where the heat transfer is negligible. Copyright 2001 Academic Press.