My general research interest is to understand how complex patterns arise in physical and biological systems that are outside of thermodynamic equilibrium, e.g. my son. One important class of nonequilibrium patterns are those produced naturally and unavoidably by the solidification process, such as the snow-flakes familiar to Bostonians and the dendrites familiar to casting and welding engineers. Thus, understanding how they form is practically relevant. Modeling solidification microstructures quantitatively is rendered extremely difficult by tracking the highly nonlinear evolution of the solid-liquid interface and, moreover, by the fact that the basic lengthscales that control this evolution differ by several orders of magnitude. Surface tension acts on a nanometer scale whereas heat and mass transport by diffusion takes place on a scale of a millimeter.
Convective transport can act on even larger scales.
The current research in my group focuses on further developing a phase-field approach, inspired from continuum models of phase transitions, in order to cope with both of these difficulties. Using a reformulated asymptotic analysis of the phase-field model and a Monte Carlo treatment of the long range diffusion field, we are just reaching the point of making benchmark predictions of dendritic growth that can be compared directly to low undercooling experiments in three dimensions. Movie of 3D dendritic growth in an undercooled melt. The long term partical aim of this research, in terms of human helath, is to render modeling sufficiently realistic and accurate to guide the search for successful antifibrillatory drugs, implented low voltage defibrillators, and improved surgical procedures. A stability analysis.