In a planar geometry version of the dissipative quasistatic model, the growth rate of a sinusoidal perturbation of wavenumber k is given by
where is the location of the unperturbed interface along the growth direction, . The growth rate is positive for all wave numbers in all regions of parameter space, so that the flat envelope is unstable against perturbations at all wavelengths. This result reflects the planar geometry's lack of a characteristic length scale to play the role played by in the radial geometry. Without such a reference against which to distinguish perturbations of different sizes, they are either all stable or all unstable. The branches' transport properties as modeled in Eq. (1) do not introduce new length scales themselves.
The quasistatic approximation to Eq. (2) requires the diffusion length to be larger than any other characteristic sizes in the system. If we relax this requirement, then may influence the stability of the DBM. Thus outside the patterned aggregate region we seek a solution to the diffusion equation. The similarity transform rescales the problem into the frame moving with the interface. In this frame, the diffusion equation has the form
where and . Equation (20) is an integrable ODE provided that is a constant. This requirement in turn determines the time dependence of the interfacial position and velocity:
Similar time evolution was obtained by Zener in his model of non-dissipative diffusive growth . The solution to Eq. (20) which satisfies the boundary conditions in Eqs. (5)-(7) in the limit of large R is given by
where satisfies the transcendental equation
We are now in a position to show that diffusion without dissipation in the growth channels is not sufficient to stabilize the planar DBM. In the limit , the advancing pattern is an equipotential with and the linear stability calculation for the similarity solution gives
Surprisingly this is the same result as was obtained by Mullins and Sekerka in the quasistatic limit . Although the actual growth rate depends on the diffusion length, the non-dimensional growth rate does not.
Including contributions from dissipation and current confinement in the advancing region requires to satisfy continuity conditions with with the field inside,
at the interface. As before, we perturb the flat interface with a sinusoid of wavenumber k and solve for the growth rate of the perturbation to linear order in the perturbation's amplitude. The linear stability calculations then give
Equations (26) and (27) reduce to the dissipative Laplacian result [Eq. (19)] and non-dissipative short diffusion length result [Eq. (24)] in the limits , and , respectively. These equations also can be solved for , although the result is messy and so not particularly informative. For the sake of clarity, we leave the solution as two coupled equations which we can compare more easily with results from the earlier analyses.
In the limit that is small compared with , q plays the role of a wavenumber whose lower limit is set by the diffusion length, . This is the length scale against which features in the evolving pattern can be compared. The factor of in the numerator of Eq. (26) then provides the offset necessary to achieve negative values of and thus stability at long wavelengths.
Setting in Eqs. (26) and (27) allows us to solve for the marginally stable mode number:
Long wavelength modes with are stable while modes with grow unstably; this is consistent with the overall picture of a large number of branches advancing behind a smooth envelope. The critical mode number depends inversely on the diffusion length through the constant . Since the diffusion length changes as the pattern advances, however, is a more useful control parameter.
It is noteworthy that the conductivity anisotropy, , does not influence in the planar geometry despite its significant role in determining in the radial geometry (see Fig. 3). The extent, , of the central boundary condition sets the characteristic scale for quasistationary growth in the radial geometry and couples to the growth front through the pattern's transport properties. The diffusion length, , on the other hand, sets the scale of the problem outside the advancing interface and its influence depends only indirectly on the disposition of currents within the pattern.
Finally, it would appear from Eq. (28) that DBM growth is inevitable in the planar geometry since diverges when . In fact, the initial transient behavior at the onset of growth is not treated by Eq. (22) and thus is not accounted for in Eq. (28). Furthermore, the value for may evolve as the morphology of the aggregate changes during early growth, and this evolution will affect the interface's stability. The above analysis simply provides a mechanism by which the planar DBM, once formed, can be linearly stable. This analysis also demonstrates that the planar DBM must eventually become unstable since the critical mode number vanishes as the interface advances.