Hausdorff Forum - December 18, 2015 - 14h

Location: Lipschitz-Saal

Michael Ortiz (Caltech, Computational Mechanics): A model of facetted crystal interfaces

We develop an explicit model for the interfacial energy in crystals which emphasizes the geometric origin of the cusps in the energy profile. We start by formulating a general class of interatomic energies that are reference-configuration-free but explicitly incorporate the lattice geometry of the ground state. In particular, away from the interface the energy is minimized by a perfect lattice. We build these attributes into the energy by locally matching, as best as possible, a perfect lattice to the atomic positions and then quantifying the local energy in terms of the inevitable remaining mismatch, hence the term lattice-matching used to describe the resulting interatomic energy. Based on this general energy, we formulate a simpler rigid-lattice model in which the atomic positions on both sides of the interface coincide with perfect, but misoriented, lattices. The energy density of crystal interfaces exhibits a characteristic ’cusp’ structure that renders it non-convex, corresponding to faceting of crystal interfaces. We study the relaxation of our model and show that the effective, or relaxed, interfacial energy density, with all possible interfacial morphologies accounted for, corresponds to the convexification of the bare or unrelaxed interfacial energy density, and that the requisite convexification can be attained by means of a faceting construction. We validate the approach by means of comparisons with experiment and molecular dynamics simulations including symmetric and asymmetric tilt boundaries in face-centered cubic (FCC) and body-centered cubic (BCC) crystals. By comparison with simulated and experimental data, we show that this simple model interfacial energy combined with a general microstructure construction based on convexification is able to replicate complex interfacial morphologies, including thermally-induced morphological transitions.

Joachim Schultze (Bonn, LIMES): Upcoming opportunities in the life sciences by genomic analyses on the single cell level

The most important biologically unit is the single cell, which is made of lipids, proteins, ribonucleotides and other large molecules located in several subcellular compartments. For the first time, we can measure all ribonucleotides of an individual cell and this process can be performed in parallel in several thousands of cells at the same time. These single cell genomics technologies allow us in the future to study cell population heterogeneity, cell-to-cell communication, cellular dynamics over time, or cellular functions in changing environments at an unprecedented level. The life sciences and medicine will be transformed by such technological developments. I will also discuss the enormous necessity for novel biomathematical approaches to leverage this treasure of novel biological data.