Research

Copolymer Mixtures

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.
Some publications:
  • Spatially locallized structures in diblock copolymer mixtures , SIAM J. Appl. Math, Vol. 70 (2010).
  • Coarsening and self-organization in dilute diblock copolymer melts and mixtures , with R. Choksi, Physica D, Vol. 238 (2009).

    • Grain boundary dynamics

    A simulation of the regularized Cross-Newell equation is here.

    Reference:

    Grain boundary motion arising from the gradient flow of the Aviles-Giga functional , Physica D,Volume 215, pp. 80-98 (2006).

    Dynamics of contact lines

    Huh and Scriven noted in 1971 that liquids shouldn't slide on solid surfaces -- at least if one applies the usual 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 different mechanisms can regulate contact line motion. From a practical perspective, we might ask if it is important to identify the particular mechanisms at work, or is it possible that all mechanisms have the same macroscopic consequences?

    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 which has been recently studied by the author.

    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, as is the basis for numerically efficient boundary integral calculations.
    References:

    A boundary integral formulation of quasi-steady fluid wetting , J. Comp. Phys., 207, pp. 529-541 (2005).

    Variational models for moving contact lines and the quasi-static approxima tion, Euro. J. Appl. Math, 16, pp. 1-28 (2005).

    Spreading of droplets under the influence of intermolecular forces, Phys. of Fluids, Vol.15, pp. 1837-1842 (2003).

    Homogenization of contact line dynamics, Int. Free Bound., (2006).