# Nonlinear Dynamics and Pattern Formation

## Dendritic Pattern Formation

Snowflakes are examples of crystals which develop instability during growth. The resulting spikes form side branches, which due to their similarity to trees, are named “dendrites”. The solidification of most alloys occurs through dendritic growth. The growth-rate and size of these dendrites (typically just a few microns in length) are largely responsible for the mechanical and chemical properties of the alloy. The three-dimensional dendrite shown here is the result of an analytical calculation, which was confirmed in detail experimentally.

(E.A. Brener and V.I. Mel'nikov : Adv. in Physics 40 (1991) 53 ; E.A. Brener : Phys Rev. Lett. 71 (1993) 3653) .

Morphology Diagram of Growth Patterns

The front between a solid and the melt from which it grows during solidification is usually corrugated. We have formulated a theory for the possible corrugation patterns, shown here as a morphology diagram within the parameters driving-force (supersaturation) versus crystalline anisotropy. The lower right-hand corner corresponds to dendritic growth, the upper left-hand corner to seaweed growth (doublons or triplons), and near the origin, fractals are present.

( T. Abel, E. Brener, H. Mueller-Krumbhaar, Threedimensional Growth Morphologies in Diffusion-Controlled Channel Growth, Phys. Rev. E 55, 7789 (1997); E.Brener, H. Mueller-Krumbhaar, D. Temkin, T. Abel, Morphology Diagram of Possible Structures in Diffusional Growth Physica A 249, 73 ( 1998). )

Dynamics of Polymers in Flow

At present, the dynamics of dilute polymer solutions are largely based on empiricisms which lack a sound microscopic foundation. This level of understanding suffices for many engineering applications but is incapable of predicting intricate phenomena such as turbulent drag reduction. In order to improve on this situation, a profound understanding of polymer-flow interaction is needed. Of special interest are the effects of hydrodynamic interaction and the flow perturbation caused by the polymer. The case of a single tethered polymer in uniform flow provides the simplest non-trivial problem from which important information can be gathered. The picture shows the average shape of a polymer-chain which is tethered in a flow coming in from the left. Furthermore we have calculated relaxation spectra for nonlinear polymer models.

(Rzehak, R., Kromen, W., Kawakatsu, T. and Zimmermann, W.: Deformation of a tethered polymer in uniform flow; European Phys. J. E 2, 3 (2000)).

Fractal Layer Growth with Elastic Interaction

Our studies centre on the growth of fractal layers from a lattice gas of a given density in the presence of elastic interaction. While the evolving structures are fractal on short length scales, the morphology of the layer is compact on large scales. With increasing elastic repulsion, the structures tend to look more compact. This can be characterized by crossover-effects in the fractal dimension.

(F. Gutheim, H. Müller-Krumbaar, E. Brener, and C. Misbah. In: J. A. Freund, T. Pöschel (eds.), " Stochastic Processes in Physics, Chemistry, and Biology", Lecture Notes in Physics, Vol. 557, Springer (2000)).

Fracture Mechanics

Why are cracks jagged rather than straight when they occur in everyday materials? We found out that the surface of a propagating crack is morphologically unstable with respect to mass transport. The Grinfeld instability even deforms the surfaces of an already existing crack. In particular, we investigate its influence on cracks as they spread and its relevance for tip oscillations and splitting. Apart from these detailed descriptions of a single crack, we also analyse the collective growth of many interacting cracks.

(E. Brener and V. Marchenko, Surface Instabilities in Cracks, Phys. Rev. Lett. 81, 5141 (1998); E. Brener, H. Müller-Krumbhaar, and R. Spatschek, Coarsening of Cracks in a Uniaxally Strained Solid, Phys. Rev. Lett. 86, 1291 (2001)).

Just for fun: why does toast always fall buttered-side down?

A popular, nonlinear dynamical problem is the question, whether and under what circumstances a piece of toast (buttered, for simplicity, on the top side) would fall with the buttered side down. Earlier theories on this subject seemed slightly unsatisfactory, since they contained arbitrary free parameters. Our investigations led to the following result, under the assumption, that the piece of toast is thicker than the finger of the person eating it. The toast will fall with the buttered side down if the product of the thickness d of the toast and the initial height h above ground is less than the area L*L of the buttered side: h*d<L*L. - Unfortunately, both quantities tend to be roughly of the same order. So much for 'Murphys law' from the theorist’s point of view.

(The Theory III Coffee Club)