Branched crack patterns in layers of Laponite® dried under electric fields: Evidence of power-laws and fractal scalin>.

Abstract

In the present work we report crack patterns formed in aqueous Laponite® gel in a rectangular box, while exposed to a uniform static electric field. The crack pattern shows a very interesting tree-like geometry extending from the positive to the negative electrode. At the positive electrode a large number of cracks appear at first and merge with each other in stages thus forming tree-like fractal structures. These structures are reminiscent of the Bethe lattice or Cayley tree. The "trees" divide the system into peds of varying size, with numerous smaller ones on the positively charged end, gradually increasing in size, and decreasing in number towards the negative end. If the cumulative distribution of the number of peds exceeding a certain area in size, is plotted against that area, a power-law relation is obtained. This implies a scale-invariant fractal character of the pattern. For a given system size, the exponent of the power-law has a nearly constant value for different applied voltages. We present an experimental study demonstrating this behaviour and discuss how it compares with similar distributions of river-basin areas and viscous fingers in a Hele-Shaw cell.