Pecelerowicz Michal, Szymczak Piotr
Institute of Theoretical Physics, Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland.
Phys Rev E. 2016 Dec;94(6-1):062801. doi: 10.1103/PhysRevE.94.062801. Epub 2016 Dec 19.
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.
诸如电沉积、介质击穿或粘性指进等图案形成过程通常由不稳定性驱动。因此，所产生的生长图案通常是高度分支的分形结构。然而，在一些不稳定生长过程中，结构的包络以高度规则的方式生长，扰动会随着时间的推移而平滑化。在本文中，我们表明包络生长的规律性可以与导致结构前进前沿处指状物尖端分裂的小尺度不稳定性相关联。每当生长速度变得过大时，指状物就会分裂成两个分支。通过这种方式，它可以吸收增加的通量，从而抑制不稳定性。因此，有点违反直觉的是，小尺度的不稳定性会导致大尺度的稳定性。这些效应的定量分析是通过洛厄纳方程进行的，该方程可用于将界面运动问题简化为复平面上共形映射的演化问题。这使得能够有效地分析各种不同几何形状中的多指生长。我们展示了几何形状如何影响生长图案包络的形状，并将结果与在自然系统中观察到的结果进行比较。
