The preceding sections demonstrate that diffusive pattern forming systems in which growth currents are confined by resistive branches within the advancing pattern are capable of generating the dense branching morphology. We find that in the two and three dimensional radial geometries, dissipation and current confinement alone are sufficient to stabilize densely branched growth. However, in the flat geometry, both dissipation in the advancing region and a short diffusion length in the displaced region are necessary to account for stable DBM growth. Our simple yet realistic model predicts the region of parameter space where densely branched growth should be seen. This model provides quantitative predictions for the interfacial velocity as well as for the length scales at which the DBM can appear. We have focused our investigation on the long wavelength stability of the DBM's envelope, with the understanding that instabilities at short wavelengths are responsible for the finely branched structure of the DBM. Extensions of this model could include consideration of short wavelength stabilizing mechanisms such as surface tension and terms accounting for the kinetics of attachment. The problem of mode selection for the dense branching morphology then could be addressed.
The photographs in Fig. 1 were produced in collaboration with Len Sander, Roy Clarke, and Nancy Hecker at The University of Michigan. We also are grateful to Peter Garik for pointing out the similarity transform used in section III. This work was supported in part by the MRSEC Program of the National Science Foundation under Award Number DMR-9400379 and in part by the Petroleum Research Fund of the American Chemical Society under Award Number 26873-G.