Soft Matter and Complex Systems Seminar

Laplacian growth is one of the fundamental mechanisms of patternformation, driving such natural processes like electrodeposition,dielectric breakdown or viscous fingering. A characteristic feature ofthese processes is a strong instability of the interface motion: ifthe interface is an isoline of the harmonic field and the growth rateis proportional to the gradient of the field, small perturbations ofthe interface have a tendency to grow and eventually get transformedinto fingers. At short wavelengths, the interface growth is stabilizedby regularization mechanisms such as surface tension or kineticundercooling, but the longer wavelengths are generally unstable.

In this communication, we show that - if the fingers produced as aresult of the instability are long and thin - then globally thepattern continues to grow in a stable way, with its envelope expanding with nearly a constant velocity thus forming highly regular geometric shapes.We show that the stabilizing mechanism here is connectedwith the splitting of the fingers. Whenever the growth velocitybecomes too large, the finger splits in two branches. In this way thesystem can absorb an increased flux and thus damp the instability.The quantitative analysis of these effect is provided by means of theLoewner equation, which allows us to reduce the problem of theinterface motion to that of the evolution of the conformal mappingonto the complex plane. This allows us for an effective analysis ofthe the multi-fingered growth in a variety of different geometries. Weshow how the geometry impacts the shape of the envelope of the growingpattern and compare the results with those observed in naturalsystems.