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Finger Morphology
Motivation
Figure 1
The finger morphology
403 s after the start of the experiment. The solid black line at the bottom of the image is the cathode. Watch a
movie (6.1 MB) of the growing deposit.
After the addition of a small amount of an inert electrolyte, such as sodium sulfate, to a copper sulfate solution, the morphology of copper deposits changes from a typical fractal or dense-branched red copper structure to some fine-meshed texture with a fingerlike envelope. Figure 1 gives an example.
The underlying mechanism is believed to be qualitatively understood. The increase of the electric conductivity due to the inert electrolyte enables alternative reaction paths like the reduction of H20. The resulting increase of the pH value triggers the formation of a copper hydroxide layer (Cum(OH)n(2m-n)+) in front of the advancing deposit. When considering, that the fluid between the copper filaments contains no copper hydroxide, the ensuing situation resembles the Saffman-Taylor instability, where a more viscous fluid is pushed by a less viscous one and their interface develops the same type of fingering. Here we study this explanation by measuring the dispersion relation.
Experimental Setup
Figure 2
The electrodeposition is performed in a cell with two glass plates of 8 x 8 cm2 area as side-walls. Two parallel copper wires separated by a distance of 4 cm serve as electrodes and spacers. Their diameter d ranges between 125 μm and 300 μm. Figure 2(a) shows a sketch of this setup.
Images of the growing deposit were taken with a Kodak Megaplus 6.3i CCD camera with 3070 × 2048 pixel mounted on a Nikkor SLR macro lens. To take full advantage of the spatial resolution of 7.9 μm per pixel it was necessary to employ a Köhler illumination.
Results
Figure 3
Exponential growth of the Fourier modes. Only the data points represented by filled symbols were included in the exponential fits. The data corresponds to the experiment shown in Figure 1.
First we did track down the temporal evolution of the front which was then decomposed in it's Fourier modes A (k,t) where k is the wave number. Then the temporal evolution of each of this Fourier modes was tried to fit with an exponential growth law as in Figure 3.
For a precise measurement of negative growth rates we used a textured electrode. Figure 5 shows the dependency of the growth rates σ on the wave number k for 3 different plate separations d. The dispersion relation of the Saffman-Taylor instability
gives a fair fit with resonable results for the viscosity of the copper hydroxide ηCuOH (equals 2 times the viscosity of water ηH2O) and the surface tension of the interface γ (about 10-5 times the one of an air-water interface).
Figure 4
Dependence of the dispersion relation on the cell thickness. The solid line is a fit of the Saffman-Taylor dispersion relation to the growth rates for d=155μm. All data sets are averaged over three experiments.
However the apparent independence of the measured dispersion relations on the plate separation requires additional assumptions over the influence of the gravity-driven convection roll on the surface tension.
Conclusion
The initial growth of the fingermorphology can be fit with a disperion relation of type σ=ak-bk3 which indicates the existence of some finite surface energy at the interface between deposit and electrolyte.
The Saffman-Taylor instability is a possible candidate for the growth mechanism of the fingering morphology.
Publication
In collaboration with
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Last modified: Fri Mar 18 11:02:09 CST 2005
