Stabilizing effect of tip splitting on the interface motion

TL;DR

This paper demonstrates that small-scale tip splitting instabilities can stabilize the overall growth pattern in interface-driven processes like electrodeposition, using conformal mapping analysis to connect local dynamics with large-scale regularity.

Contribution

It introduces a quantitative analysis linking tip splitting instabilities to envelope regularity in pattern formation, employing the Loewner equation for diverse geometries.

Findings

Tip splitting absorbs increased flux, damping large-scale instability.

Small-scale tip splitting leads to regular, envelope growth.

The geometry influences the shape of the growth pattern's envelope.

Abstract

Pattern-forming processes, such as electrodeposition, dielectric breakdown or viscous fingering are often driven by instabilities. Accordingly, the resulting growth patterns are usually highly branched, fractal structures. However, in some of the unstable growth processes the envelope of the structure grows in a highly regular manner, with the perturbations smoothed out over the course of time. In this paper, we show that the regularity of the envelope growth can be connected to small-scale instabilities leading to the tip splitting of the fingers at the advancing front of the structure. Whenever the growth velocity becomes too large, the finger splits into two branches. In this way it can absorb an increased flux and thus damp the instability. Hence, somewhat counterintuitively, the instability at a small scale results in a stability at a larger scale. The quantitative analysis of these effects is provided by means of the Loewner equation, which one can use to reduce the problem of the interface motion to that of the evolution of the conformal mapping onto the complex plane. This allows an effective analysis of the multifingered growth in a variety of different geometries. We show how the geometry impacts the shape of the envelope of the growing pattern and compare the results with those observed in natural systems.