Research

I use a wide variety of tools to understand phenomenon related to nonlinear partial differential equations and interfaces, including asymptotic analysis and numerical simulation. Some application areas include:

  • Materials science: block copolymers, grain boundaries, phase field modelling
  • Fluid Dynamics: thin films, contact line dynamics, microfluidics
  • Dynamics of interfaces: computation, coarsening, pattern formation
  • Optics and nonlinear waves: broadband laser pulses, self-focusing and singular behavior

Research highlights

Block copolymers and their dynamics

Block copolymers are chain molecules composed of two or more types of subchains. Unlike regular mixtures (e.g. water and oil) that separate completely at low enough temperatures, distinct components of copolymer mixtures are molecularly bound to one another, so only ``microphase" segregation can occur. Because of this, a vast of array of patterned microstructures are observed.

Segregation does not have to lead to space-filling patterns. The transition between homogeneous mixtures and complete segregation is complicated, and is characterized by localized states which undergo various instabilities and self-replicating processes [1,2].

Some publications:
[1] Characterising the disordered state of block copolymers: Bifurcations of localised states and self-replication dynamics , Euro. J. Appl. MAth., Vol. 23 (2012).
[2] Spatially locallized structures in diblock copolymer mixtures , SIAM J. Appl. Math, Vol. 70 (2010).
[3] Coarsening and self-organization in dilute diblock copolymer melts and mixtures , with R. Choksi, Physica D, Vol. 238 (2009).

Pattern dynamics

Many physical problems can ultimately be understood in terms of pattern morphology and evolution. This approach often leads to ``canonical" prototype equations which describe whole classes of phenomena. One such equation is the Swift-Hohenberg model. Our efforts have focused on dynamics of curved-like domains and their analogy to elastic bilayers [1]. Another remarkable set of phenomena arises in the Cross-Newell phase diffusion equation (a simulation is here ). We have characterized the long time dynamics in terms of motion of grain boundary networks [2].

Some papers:
[1] The Stability and Evolution of Curved Domains Arising from One-Dimensional Localized Patterns , w/ Alan Lindsay,SIAM Journal on Applied Dynamical Systems 12.2 (2013): 650-673.
[2] Grain boundary motion arising from the gradient flow of the Aviles-Giga functional , Physica D,Volume 215, pp. 80-98 (2006).

Fluid droplets and contact lines

Huh and Scriven noted in 1971 that liquids shouldn't slide on solid surfaces -- at least if one applies the traditional rules of fluid mechanics. The motion of the contact line (where liquid, solid and vapor meet) has since been one of the most studied and controversial subjects in fluid mechanics. Research has largely focused on which additional physical mechanisms should be included. The problem is that there is not just one - many physical effects regulate contact line motion.

The crux of the contact line "paradox" is infinite energy dissipation. In reality, energy dissipation is finite, and we could therefore characterize the manifestation of microscopic physics that regulate contact line motion in terms of the energy dissipated. This can be done in terms of a variational formulation [2].

An important dynamic limit is the quasisteady case, where the fluid is nearly at mechanical equilibrium except near the contact lines. This leads to a substantial reduction of the equations of motion which can be solved by a numerically efficient boundary integral method [1].

Quasistatic droplet motion also occurs in settings where there are forces other than capillarity, such as disjoining pressure and gravity. We have given a simplified description of the dynamics in these cases as well [3-8].

References:
[1] A boundary integral formulation of quasi-steady fluid wetting , J. Comp. Phys., 207, pp. 529-541 (2005).
[2] Variational models for moving contact lines and the quasi-static approximation, Euro. J. Appl. Math, 16, pp. 1-28 (2005).
[3] Spreading of droplets under the influence of intermolecular forces, Phys. of Fluids, Vol.15, pp. 1837-1842 (2003).
[4] Homogenization of contact line dynamics, Int. Free Bound., (2006).
[5] Viscosity solutions for a model of contact line motion , with I. Kim, Interfaces and Free boundaries (2009).
[6] Ostwald Ripening in Thin Film Equations , SIAM J. Appl. Math., Vol. 69 (2008).
[7] Ostwald Ripening of Droplets: The Role of Migration , w/ F. Otto, T. Rump. D. Slepcev, Euro. J. Appl. Math. (2008).
[8] The dynamics of pendant droplets on a one dimensional surface, Phys. Fluids, Vol. 19, (2007).
[9] A Diffuse Interface Model for Electrowetting Droplets in a Hele-Shaw Cell , w/ H.-W. Lu, A. Bertozzi, C.-J. Kim, JFM, Vol. 590, 411-435 (2007).
[10] Homogenization of contact line dynamics, Int. Free Bound., Volume 8, (2006).