Document Type


Date of Award

Spring 5-31-2008

Degree Name

Doctor of Philosophy in Mathematical Sciences - (Ph.D.)


Mathematical Sciences

First Advisor

Lou Kondic

Second Advisor

Demetrius T. Papageorgiou

Third Advisor

Michael Siegel

Fourth Advisor

Pushpendra Singh

Fifth Advisor

Yuan N. Young


We report on instabilities during spreading of volatile liquids, with emphasis on the novel instability observed when isopropyl alcohol (IPA) is deposited on a monocrystalline silicon (Si) wafer. This instability is characterized by emission of drops ahead of the expanding front, with each drop followed by smaller, satellite droplets, forming the structures which we nickname “octopi” due to their appearance. A less volatile liquid, or a substrate of larger heat conductivity, suppress this instability. In addition, we examine the spreading of drops of water (DJW)-JPA mixtures on both Si wafers and plain glass slides, and describe the variety of contact line instabilities which appear. We find that the decrease of IPA concentration in mixtures leads to transition from “octopi” to mushroom-like instabilities. Through manipulation of our experimental set up, we also find that the mechanism responsible for these instabilities appears to be mostly insensitive to both the external application of convection to the gas phase, and the doping of the gas phase with vapor in order to create the saturated environment.

In order to better understand the “octopi” instability, we develop a theoretical model for evaporation of a pure liquid drop on a thermally conductive solid substrate. This model includes all relevant physical effects, including evaporation, thermal conductivity in both liquid and solid, (thermocapillary) Marangoni effect, vapor recoil, disjoining pressure, and gravity. The crucial ingredient in this problem is the evaporation model, since it influences both the motion of the drop contact line, and the temperature profiles along the liquid-solid and liquid-gas interfaces. We consider two evaporation models: the equilibrium “lens” model and the non-equilibrium one-sided (NEOS) model. Along with the assumption of equilibrium at the liquid-gas interface, the “lens” model also assumes that evaporation proceeds in a (vapor) diffusion-limited regime, therefore bringing the focus to the gas phase, where the problem of vapor mass diffusion is to be solved, which invokes analogy with the problem of lens-shaped conductor from electrostatics. On the other hand, NEOS model assumes non-equilibrium at the liquid-gas interface and a reaction-limited regime of evaporation; the liquid and gas phases are decoupled using the one-sided assumption, and hence, the problem is to be solved in the liquid phase only. We use lubrication approximation and derive a single governing equation for the evolution of drop thickness, which includes both models. An experimental procedure is described next, which we use in order to estimate the volatility parameter corresponding to each model. We also describe the numerical code, which we use to solve the governing equation for drop thickness, and show how this equation can be used to predict which evaporation model is more appropriate for a particular physical problem.

Next, we perform linear stability analysis (LSA) of perturbed thin film configuration. We find excellent agreement between our numerical results and LSA predictions. Furthermore, these results indicate that the IPA/Si configuration is the most unstable one, in direct agreement with experimental results. We perform numerical simulations in the simplified 2d geometry (cross section of the drop) for both planar and radial symmetry and show that our theoretical model reproduces the main features of the experiment, namely, the formation of “octopus” -like features ahead of the contact line of an evaporating drop. Finally, we perform quasi-3d numerical simulations of evaporating drops, where stability to azimuthal perturbations of the contact line is examined. We recover the “octopi” instability for IPA/Si configuration, similarly as seen in the experiments.

Included in

Mathematics Commons



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