Several distinct classes of patterns, or morphologies, can emerge when the interface between two phases is driven out of equilibrium by a diffusive field. Highly branched fractals resembling diffusion limited aggregation (DLA) clusters , snowflake-like dendrites , and dense branching patterns [3, 4] are produced by processes as varied as viscous fluid displacement and electrochemical deposition of metals and polymers. The dense branching morphology (DBM), examples of which appear in Fig. 1, is characterized by a large number of fine branches advancing behind a smooth stable envelope. Unlike ordered dendrites, the individual branches in the DBM are unstable against repeated tip splitting. In this respect, the DBM more closely resembles DLA. Unlike DLA, however, the ensemble of DBM branches fills space uniformly. The underlying interfacial instability responsible for branch formation also might be expected to destabilize the apparent interface enclosing the branch tips. A central challenge for models of densely branched growth, thus, is to explain the stability of the smooth advancing envelope.
Many pattern forming systems such as electrochemical deposition, viscous fingering and dielectric breakdown can be described by a model in which the interface's movement is governed by a scalar field satisfying Laplace's equation at least in the quasistatic limit [5, 6]. The simplest version of the Laplacian growth model treats the moving interface as an equipotential. Under these conditions, the tendency of protrusions to concentrate field gradients, which was first emphasized in this context by Mullins and Sekerka , renders a smooth advancing interface linearly unstable to perturbations at all wavelengths. Corrections to the interfacial potential due to surface tension and growth kinetics can stabilize the interface at wavelengths comparable to the width of a branch, but do not suppress longer-wavelength instabilities. Extension of the Mullins-Sekerka analysis to systems with finite diffusion lengths also results in linear instability at long wavelengths. The existence of these long wavelength instabilities suggest that the DBM cannot form in standard non-dissipative models for diffusive pattern formation.
Previous efforts [4, 8, 9] to extend these models by accounting for the small but non-vanishing resistance to gradient-driven currents in the patterns' branches have been obliged to distinguish between two experimental geometries. In the radial geometry the pattern grows outward from a source of radius centered within a region of radius R, while in the flat geometry it advances between parallel planar boundaries separated by distance R. The distinction arises because the DBM was found to be linearly stable in the two-dimensional radial case, but not in the planar geometry . Thus, these models fail to account for the appearance of structures such as that in Fig. 1(b) which suggests that the DBM can occur in the planar geometry also.
This article is organized as follows. The growth model is presented in section II in the context of pattern formation during electrochemical deposition and viscous fingering. This section also outlines the linear stability analysis used in the following section to investigate morphological stability of the DBM. The central results of this article are presented in section III. Section III.A reviews our previously reported results for the 2D radial geometry in the quasistationary approximation. We extend this analysis in sections III.B and III.C both to the 3D radial geometry and also beyond the quasistationary approximation in the planar geometry. We find a range of nontrivial growth conditions under which the radial DBM is linearly stable in three dimensions. No such conditions are found for growth from a line or a plane in the quasistationary approximation. Stability of the planar DBM is established, however, by including both dissipation and a finite diffusion length. This analysis therefore extends the range of pattern forming systems for which the dissipative growth model accounts for the appearance of the dense branching morphology.