Pattern Formation in Screened Electrostatic Fields: Growth in a Channel and in two Dimensions

Abstract

The Diffusion-Limited-Aggregation (DLA)^1,2 and the Dielectric Breakdown (DB)³ models have been very successful in illustrating the possibility of fractal growth⁴ in Laplacian fields. Nature does offer, however, a much richer scenario in which both fractal and non-fractal patterns may grow. The dependence on the boundary conditions in DB discussed in ref. 5 illustrates this point: a change in the shape of electrodes induces drastic changes in the growing patterns which evolve into a rather dense multi-branched structure with fractal dimension D ∼ 2. Although several reasons have been suggested⁵ to explain this dependence on the boundary conditions, among which we mention the existence of a threshold field and the internal resistance of the breakdown pattern (plasma channels in the case of a discharge in a gas), in very few instances their effects have been analysed in any depth. Only the possibility of a different growth law in which the growth rate is assumed to be proportional to a power η of the local field, different in general from unity, has been examined in detail in the DB context³, and by utilizing an equivalent approach in DLA⁶, and used to explain the more diluted than DLA patterns that may occur in Nature. Note, however, that there are microscopic reasons for expecting η = 1 in DB⁷. More recently, the possibility of a crossover from a DLA pattern to a more diluted one has also been investigated by using more complicated growth laws, both in DB⁸ and in the somewhat similar phenomena of mechanical breakdown^9–11. The variety of structures further increases for the growth of metallic aggregates through Electrochemical Deposition (ECD). These may have a fractal character like in DLA, be dendritic crystals, or give rise to dense radial structures^12–16; the stability of the latter has been ascribed to the finite resistivity of the aggregate¹², or to the anion migration between the electrodes^13,16. Besides, a transition from a dense pattern to a more diluted branched structure has been observed^17,18 and referred to as the Hecker transition¹⁷.