On a new example of ramified electrodeposits
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We have grown extremely flat (2D) electrodeposited copper leaves that exhibit ramified patterns. These patterns are stable and can be withdrawn from the solution. A fractal analysis based on correlation function and branch counting was carried out. Also, x-ray and SEM pictures give insight into the fractal structure of this new type of aggregates and help to find the lower crossover to nonfractal compact substructure.
I. INTRODUCTION Metal electrodeposition has been widely studied in the past decade.1"3 These works and many others in related fields (dielectric breakdown and viscous fingering, for example) were triggered by the evidence of fractal behavior in the numerical simulations of Diffusion Limited Aggregation (DLA)4'5 which exhibit patterns that are strikingly similar to those obtained experimentally. The Witten and Sander algorithm shows that when one grows a cluster by successively launching random walkers from a source and have them wander around before attaching to the cluster, the typical pattern one obtains is fractal with a fractal dimension of Df ~ 1.7. The formal problem can be put this way: A V = 0 (field out of the cluster) (1) J = — n-WV (growth kinetic on the cluster border)
where V is a probability field that describes the probability of reaching a given site of the underlying lattice (in the case of an on-lattice simulation) without returning to the source, and n stands for the normal to the cluster surface. This "potential" V is zero on the aggregate since the aggregate is an absorbing surface and is a constant on the source. The problem is then nonlinear since every time a new particle reaches the cluster, the whole field outside is changed and so is the growth probability on the cluster. This process is known to give rise to ramified fractal structures. The formalism of the DLA problem seems to resemble the formalism of the electrodeposition in electrolytic cells (generally in a very thin layer bounded by two electrodes that can be either two metallic strips or a needle and a circle) without supporting electrolyte. In this case one sets a potential difference between the electrodes and grows electrodeposits from a salt (copper acetate, copper sulfate, and zinc sulfate, for example). The electrochemical mechanism at the surface of the aggregate is, in a zeroth order approximation: Mp+ + pe~
II. ABOUT THE DIMENSIONALITY OF THE DEPOSITS Let us forget most of the problems that arise and that might cause discrepancies between the DLA for-
M
J. Mater. Res., Vol. 6, No. 6, Jun 1991
http://journals.cambridge.org
where M is the metallic atom involved in the salt and e~ the electron. If the reaction is carried out in nonequilibrium conditions, it is known that the deposit is very ramified or dendritic. If the deposit is assumed to be equipotential, then the potential field outside the deposit abides by the well-known Laplace, Ohm's, and Poisson's laws, showing a seemingly one-to-one correspondence with the DLA formalism. Of course, one does not expect the actual movement of the ions in the sol
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