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Next: Growth from a Line Up: Results Previous: Two-Dimensional Radial Geometry

3D radial geometry

Experiments such as those depicted in Fig. 1 have three-dimensional analogs which have been studied [13, 14, 15] almost as extensively as their more easily interpreted two-dimensional variants. Recent advances in admittance spectroscopy [14] and image analysis [15] make it possible to analyze the shapes of evolving three-dimensional patterns. High speed magnetic resonance imaging also has been applied to the study of fluid flow in porous media. Three dimensional radial pattern formation in which branches grow outward from the end of a conduit constitute another class of systems to which our growth model should pertain.

In the full three-dimensional radial geometry model, the unperturbed spherical interface advances with velocity

equation115

An infinitesimal perturbation to the growth front with a spherical harmonic

equation122

grows at the relative rate

  eqnarray124

where tex2html_wrap_inline679 . Figure 2 shows a plot of the growth rate as a function of the perturbation harmonic for a range of dissipation values. The growth rate is negative for values of l less than a critical mode number tex2html_wrap_inline571 which depends on the degree of dissipation in the growth channels. Such long wavelength perturbations shrink as the aggregate grows. The greater tex2html_wrap_inline571 , therefore, the more nearly spherical the evolving pattern appears. In Fig. 3 this marginally stable harmonic number is plotted as a function of the amount of dissipation ( tex2html_wrap_inline573 ) and anisotropy ( tex2html_wrap_inline559 ) in the system. As seen from the plot, dissipation and current confinement act in concert to stabilize the dense branching morphology in the three dimensional radial geometry.

In the limit tex2html_wrap_inline691 , we can solve for the critical aggregate radius beyond which the envelope will be stable against perturbations of harmonic number l:

equation135

In the anisotropic limit ( tex2html_wrap_inline695 ), the expression simplifies to

  equation139

For tex2html_wrap_inline697 the DBM is stable against perturbations of harmonic number greater than l. Even for tex2html_wrap_inline701 , an initially disordered core pattern can cross over into a regime where densely branched growth will be stable, provided the enveloping equipotential still is reasonably smooth by the time the interface has advanced to a mean position tex2html_wrap_inline703 . The size at which the envelope of the growing pattern becomes stable scales with the size of the inner boundary, tex2html_wrap_inline603 , and depends strongly on the amount of dissipation in the system. The position at which such a crossover to stable dense radial growth might occur for tex2html_wrap_inline707 and tex2html_wrap_inline709 appears in Fig. 4 as a rapid increase in tex2html_wrap_inline571 with tex2html_wrap_inline657 . Under certain conditions, tex2html_wrap_inline571 decreases again for tex2html_wrap_inline657 very near R and a second crossover from stable DBM to unstable growth is possible.


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Next: Growth from a Line Up: Results Previous: Two-Dimensional Radial Geometry

David G. Grier
Mon May 20 13:07:47 CDT 1996