# Mathematics Colloquia and Seminars

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### Mathematical Modeling and Adaptive Numerical Simulation of Microstructured Materials

**Colloquium**

Speaker: | John Lowengrub, School of Math, U. Minnesota |

Location: | 693 Kerr |

Start time: | Mon, Mar 4 2002, 4:10PM |

Microstructured materials, such as emulsions and polymer blends, crystals and metallic alloys, blood and biological tissues, are fundamental to many industrial and biomedical applications. These diverse materials share the common feature that the microscale and macroscale are linked. The phenomena at microscopic scale, such as the morphological instability of crystalline precipitates and drop deformation, break-up and coalescence determine the microstructure and its time evolution; thus affecting the rheology and mechanical properties of the materials on the macroscale.

In this talk, I will focus on mathematical and numerical modeling at the microscale. In particular, I will consider the quasi-steady evolution of growing crystals in 3-d. A re-examination of this fundamental problem in materials science reveals that the Mullins-Sekerka shape instability associated to volume growth of the crystal may be suppressed by appropriately varying the undercooling (far-field temperature) in time. For example, in 3 dimensions, by imposing the far-field temperature flux (rather than a temperature condition), a class of asymptotically self-similar, non-spherical growing crystals can be found. Simulations show that this class of solutions is robust with respect to perturbations and anisotropies and is well-predicted by solutions of the linearized equations. To simulate the problem numerically, we use a boundary element method with a fully adaptive surface triangulation. This enables us to simulate three dimensional crystals stably and accurately well into the nonlinear regime. Simulations of both stable and unstable crystal growth will be presented. This work has important implications for shape control in processing applications.

Finally, I will demonstrate how these models and numerical techniques originally developed for crystal growth may be adapted to also investigate the behavior of complex fluids and biological systems.