RESEARCH

Tim Schulze
Professor and Associate Head
Director of Graduate Studies
Department of Mathematics
University of Tennessee

I am a professor of mathematics at the University of Tennessee. In 1995 I received my Ph.D. in Applied Mathematics at Northwestern University under the direction of Professor Stephen H. Davis. Before coming to UTK in 1999, I held post doctorial positions at the Dept. of Applied Mathematics and Theoretical Physics, University of Cambridge, and at the Courant Institute of Mathematics, NYU.

Research
My research centers upon modeling, analysis of interactions between fluid flow and phase change processes, and simulation of crystal growth on both atomistic and continuum length-scales using Kinetic Monte Carlo methods. This research can be broken down into three principal areas described below. Other research interests include game theory and quantum mechanics.

Solidification
This is the study of phase transformation of liquid into solid. Frequently the solidification process involves fluid flow as well. Solidification problems are examples of free-boundary problems, where the mathematical model must specify both bulk fields and interfacial quantities, including the location of the solid-liquid interface. The phenomenon of interest in solidification are interfacial instability and the growth of dendrites (snowflake like structures).

Epitaxial Growth
When crystal growth occurs at very slow rates, usually from a vapor growth or molecular beam process, the instabilities that occur during solidification of liquids are avoided. This results in the production of materials with uniform crystal structures. Since this process is slow, it produces only small amounts of material which are typically used to coat another material with a thin film. Thin film growth is studied using a variety of simulation and modeling techniques, depending on the length scale of interest. These include continuum approaches, Kinetic Monte Carlo models, molecular dynamics, and fundamental models relying on quantum mechanics.

Mushy Layers
When the instabilities during solidification are highly pronounced, there are a large number of dendrites that form a "mushy layer''. The mushy layer is often modeled as a homogenized porous medium with solid-fraction dependent permeability. Convection in mushy zones leads to interesting flow phenomena and bifurcation structure.

Selected Publications
D. M. Anderson, T. P. Schulze, and B. N. Wahl, "Not Playing with a Full Deck?", accepted to Recreational Mathematics Magazine (2023) PDF

‍J. Hicks and T. P. Schulze, “Examining Saddle Point Searches in the Context of Off-Lattice Kinetic Monte Carlo,” Communications In Computational Physics 30 (2021) pp. 749 - 770 PDF

T. P. Schulze and P. Smereka, "Kinetic Monte Carlo Simulation of Heteroepitaxial Growth: Wetting Layers, Quantum Dots, Capping, and NanoRings," Phys Rev B 86 (2012) Art. no. 235313 PDF

T. P. Schulze, "Efficient Kinetic Monte Carlo Simulation," Journal of Computational Physics 227 (2008) pp. 2455 - 2462 PDF