Karl Glasner

A diffuse-interface model for electrowetting drops in a Hele-Shaw cell

Electrowetting has recently been explored as a mechanism for moving small amounts of fluids in confined spaces. We propose a diffuse-interface model for drop motion, due to electrowetting, in a Hele-Shaw geometry. In the limit of small interface thickness, asymptotic analysis shows that the model is equivalent to Hele-Shaw flow with a voltage-modified Young-Laplace boundary condition on the free surface. We show that details of the contact angle significantly affect the time scale of motion in the model. We measure receding and advancing contact angles in the experiments and derive their influence through a reduced-order model. These measurements suggest a range of time scales in the Hele-Shaw model which include those observed in the experiment. The shape dynamics and topology changes in the model agree well with the experiment, down to the length scale of the diffuse-interface thickness.

A diffuse interface approach to Hele-Shaw flow

A diffuse interface model for the one-phase Hele–Shaw problem is derived from a gradient flow characterization due to Otto (1998 Arch. Rat. Mech. Anal. 141 63). The resulting dynamical model yields a generalized form of Darcys law, and reduces to a degenerate version of the well-known Cahn–Hilliard equation. Formal asymptotics illustrate the connection to the classical Hele–Shaw free boundary problem. Some example computations are carried out to demonstrate the flexibility of the modelling framework.

Ostwald ripening of droplets: The role of migration

A configuration of near-equilibrium liquid droplets sitting on a precursor film which wets the entire substrate can coarsen in time by two different mechanisms: collapse or collision of droplets. The collapse mechanism, i.e., a larger droplet grows at the expense of a smaller one by mass exchange through the precursor film, is also known as Ostwald ripening. As was shown by K. B. Glasner and T. P. Witelski (‘Collision versus collapse of droplets in coarsening of dewetting thin films’, Phys. D 209(1–4), 2005, 80–104) in case of a one-dimensional substrate, the migration of droplets may interfere with Ostwald ripening: The configuration can coarsen by collision rather than by collapse. We study the role of migration in case of a two-dimensional substrate for a whole range of mobilities. We characterize the velocity of a single droplet immersed into an environment with constant flux field far away. This allows us to describe the dynamics of a droplet configuration on a two-dimensional substrate by a system of ODEs. In particular, we find by heuristic arguments that collision can be a relevant coarsening mechanism.

Spreading of droplets under the influence of intermolecular forces

The motion of fluid droplets under the influence of short and long range intermolecular forces is examined using a lubrication model. Surface energies as well as the microscopic contact line structure are identified in the model. A physically constructed precursor film prevents the usual stress singularity associated with a moving contact line. In the quasistatic limit, an analysis of the energy and its dissipation yield an ordinary differential equation for the rate of spreading. Two dimensional and axisymmetric solutions are found and compared to numerical simulations. The motion of the contact line is found to be both a function of the local contact angle and the overall droplet geometry.

Solute trapping and the non-equilibrium phase diagram for solidification of binary alloys

Abstract A phase field model for the solidification of binary alloys is presented and analyzed. A matched asymptotic approach is used to recover the model’s leading order sharp interface motion. Equations for both the solute profile and free energy balance at the interface are derived, demonstrating solute trapping at large growth velocities and leading to a construction of the non-equilibrium phase diagram over a large range of growth conditions. A rigorous understanding of the interfacial conditions is provided, and comparisons are made to existing theories.

The stability and evolution of curved domains arising from one-dimensional localized patterns

In many pattern forming systems, narrow two-dimensional domains can arise whose cross sections are roughly one-dimensional localized solutions. This paper investigates this phenomenon in the variational Swift--Hohenberg equation. Stability of straight line solutions is analyzed, leading to criteria for either curve buckling or curve disintegration. Matched asymptotic expansions are used to derive a two-term expression for the geometric motion of curved domains, which includes both elastic and surface diffusion-type regularizations of curve motion. This leads to novel equilibrium curves and space-filling pattern proliferation. Numerical tests are used to confirm and illustrate these phenomena.

Viscosity solutions for a model of contact line motion

This paper considers a free boundary problem that describes the motion of contact lines of a liquid droplet on a flat surface. The elliptic nature of the equation for droplet shape and the monotonic dependence of contact line velocity on contact angle allows us to introduce a notion of “viscosity” solutions for this problem. Unlike similar free boundary problems, a comparison principle is only available for a modified short-time approximation because of the constraint that conserves volume. We use this modified problem to construct viscosity solutions to the original problem under a weak geometric restriction on the free boundary shape. We also prove uniqueness provided there is an upper bound on front velocity.

Variational models for moving contact lines and the quasi-static approximation

This paper proposes the use of a variational framework to model fluid wetting dynamics. The central problem of infinite energy dissipation for a moving contact line is dealt with explicitly rather than by introducing a specific microscopic mechanism which removes it. We analyze this modelling approach in the context of the quasi-steady limit, where contact line motion is slower than bulk relaxation. We find that global effects enter into Tanner-type laws which relate line velocity to apparent contact angle through the role that energy dissipation plays in the bulk of the fluid. A comparison is made to the dynamics of lubrication equations that include attractive and repulsive intermolecular interactions. A Galerkin-type approximation method is introduced which leads to reduced-dimensional dynamical descriptions. Computations are conducted using these low-dimensional approximations, and a substantial connection to lubrication equation dynamics is found.

Ostwald ripening in thin film equations

Fourth order thin film equations can have late stage dynamics that are analogous to the classical Cahn–Hilliard equation. We undertake a systematic asymptotic analysis of a class of equations that describe partial wetting with a stable precursor film introduced by intermolecular interactions. The limit of small precursor film thickness is considered, leading to explicit expressions for the late stage dynamics of droplets. Our main finding is that exchange of mass between droplets characteristic of traditional Ostwald ripening is a subdominant effect over a wide range of kinetic exponents. Instead, droplets migrate in response to variations of the precursor film. Timescales for these processes are computed using an effective medium approximation to the reduced free boundary problem, and dynamic scaling in the reduced system is demonstrated.

Homogenization of contact line dynamics

This paper considers the effects of substrate inhomogeneity on the motion of the three phase contact line. The model employed assumes the slowness of the contact line in comparison to capillary relaxation. The homogenization of this free boundary problem with a spatially periodic velocity law is considered. Formal multiple scales analysis yields a local, periodic problem whose time-averaged dynamics corresponds to the homogenized front velocity. A rigorous understanding of the long time dynamics is developed using comparison techniques. Computations employing boundary integral equations are used to illustrate the consequences of the analysis. Advancing and receding contact angles, pinning and anisotropic motion can be predicted within this framework. In many realistic circumstances, the static and dynamic wetting properties of liquids are substantially influenced by imperfections in the solid surface. Heterogeneities result in contact lines with a fine scale structure that may lead to pinning of the evolving front and hysteresis of the overall fluid shape. Understanding the role that surface imperfections play is part of larger theoretical effort to