We develop a mathematical model for the spreading of a thin volatile liquid droplet on a uniformly heated surface. The model accounts for the effects of surface tension, evaporation, thermocapillarity, gravity and disjoining pressure for both perfectly wetting and partially wetting liquids. Previous studies of non-isothermal spreading did not include the effects of disjoining pressure and therefore had to address the difficult issue of imposing proper boundary conditions at the contact line where the droplet surface touches the heated substrate. We avoid this difficulty by taking advantage of the fact that dry areas on the heated solid surface are typically covered by a microscopic adsorbed film where the disjoining pressure suppresses evaporation. We use a lubrication-type approach to derive a single partial differential equation capable of describing both the time-dependent macroscopic shape of the droplet and the microscopic adsorbed film; the contact line is then defined as the transition region between the two. In the framework of this model we find that both evaporation and thermocapillary stresses act to prevent surface-tension-driven spreading. Apparent contact angle, defined by the maximum interfacial slope in the contact-line region, decays in time as a droplet evaporates, but the rate of decay is different from that predicted in earlier studies of evaporating droplets. We attribute the difference to nonlinear coupling between different physical effects contributing to the value of the contact angle; previous studies used a linear superposition of these effects. We also discuss comparison of our results with experimental data available in the literature.

We review mathematical models of confined bubbles, emphasizing physical mechanisms as expressed in simple geometries. Molecular interactions between liquid, gas, and the confining solid are all important and are described through the disjoining pressure concept. Methods for finding static shapes are considered. The static solution is a springboard for discussing pressure-driven and surface-tension-driven flows, both of which involve viscous effects and macroscopic films entrained near apparent contact lines. We next discuss vapor bubbles produced by thermal effects. Vaporization localized near contact lines and condensation distributed in colder parts of the interface lead to steady vapor bubbles. Their size is determined through global constraints. Unsteady vapor bubbles are discussed and we end with thoughts on open problems.

We consider the evaporation of volatile liquid droplets deposited on a heated substrate in a pure saturated vapour environment. A mathematical model is developed that incorporates the effects of surface tension, evaporation, thermocapillarity, gravity, disjoining pressure, as well as unsteady heat conduction in the solid substrate. The apparent contact line is treated mathematically as a transition region between the macroscopic droplet shape and the adsorbed film of liquid on the heated substrate. Theoretical parametric studies are conducted to clarify the effects of thermocapillarity and wetting properties on the droplet dynamics. An experimental study is conducted in a closed container with de-ionized water droplets on a stainless steel foil heated by an electric current. The interface shapes are recorded together with the temperature profiles under the droplets, measured using thermochromic liquid crystals. Experiment and theory are in very good agreement as long as the conditions of applicability of our lubrication-type mathematical model are satisfied.

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We consider fluid flow in thin films of molten metal resulting from irradiation by a Gaussian laser beam. Surface tension gradients due to nonuniform heating induce a flow of the molten liquid away from the center of the irradiated area, leading to formation of dry areas on the substrate. We develop a mathematical model of the flow under the assumption of the large ratio of laser beam radius to film thickness. The model extends the standard lubrication-type analysis to include the highly nonlinear dependence of evaporative flux on local interfacial temperature, unsteady heat conduction in the substrate, and positive disjoining pressure due to unbalanced contributions from the kinetic energy of free electrons in the metal. The latter is proportional to the inverse square of the film thickness. We identify thermocapillary stresses as the main mechanism of rapid removal of liquid metal from the irradiated area. Characteristic times of the process, as well as shapes of the molten region surface, agree with experimental observations. We investigate rupture of the molten film and find two different rupture scenarios. The melt surface can either touch the substrate at a point (point rupture) or along a line at a certain radial distance away from the center of the irradiated area (ring rupture). Nondimensional criteria for these two mechanisms are identified. In particular, we show that positive disjoining pressure promotes ring rupture.

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.

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.

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