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Synopsis Fractals have caught the interest of scientists for more than decade. These complex structures are studied in a broad variety of systems that span through the interdisciplinary spectrum of modern research including biology as well as materials sciences and geology. In the field of physical chemistry, fractals have been observed in various growth processes. In particular, the electrochemical reduction of aqueous metal ions to solid metal can give rise to fractal geometry. This example is not only of great scientific interest, but also of technological relevance, since the same phenomena can impair the electroplating of materials. Our experiments intend to study the formation of fractals in the electrodeposition of zinc under reduced and increased gravity conditions. Since it is known that buoyancy-driven convection occurs in this process, we expect to observe a pronounced dependence of fractal morphology on the gravitational force.

The Mandelbrot Set is One of the Most Famous Fractals

What are Fractals?

The term fractal is derived from the Latin word 'fractus' ("fragmented") and was coined by the Polish-born mathematician Benoit Mandelbrot.In Mathematics any complex geometric shape that exhibits self-similarity is called a fractal. Fractals differ from the simple figures of classical geometry (the square, the circle, the sphere, etc). They are capable of describing many irregularly shaped objects or spatially nonuniform phenomena in nature.

Let us consider the triadic Koch curve on the left. The figure shows the first three iterations and a high-order approximation. The iteration process involves replacing a length l by N=4 lines of length l/3. The question is now: 'How long is the curve after an infinite number of iterations?' If we assume that the initial line has a length of one, we find that after n iterations the curve consists of (4)ⁿ pieces of length (1/3)ⁿ. Therefore, the total length equals (4)ⁿ times (1/3)ⁿ = (4/3)ⁿ and as n approaches infinity, the length of the curve becomes infinitely long! This information is not very useful and it is a good idea to introduce a new measure that characterizes the curve. This measure is the Hausdorff dimension.

The Hausdorff dimension, D_H , is the limit of the ratio between the logarithm of the number of pieces and the logarithm of their inverse lengths. For the Koch curve this ratio is log((4)ⁿ) / log((3)ⁿ) and hence log(4) / log(3) = 1.26.... The fact that D_H lies between one and two is satisfying in the sense that the 'fully-developed' Koch curve is more than a one-dimensional curve, but not quite a real two-dimensional area.

In contrast to deterministic fractals (such as the Koch curve), natural fractals involve an element of randomness. Nevertheless, we can apply the concept of fractal dimension. The following is a list of examples for natural fractals.

Arteries and veins in mammals have a fractal dimension of about 2.7. Also the bronchi of the lung show self-similarity. More simple systems include viscous fingers and solids grown from supercooled or saturated solutions.

Some of the most interesting fractals studied in laboratory experiments are caused by processes that involve diffusion-limited aggegation (DLA). The electrodeposition phenomena in our experiments belong to this particular class of fractals. If you would like to know a little bit more about DLA, read the following paragraph.

DLA and Electrodeposition A simple model for diffusion limited aggregation was suggested by Witten and Sander in 1981. Their original paper has been cited 2000 times and their model is easily implemented on a computer. The model considers single particles that move randomly on a square grid. A seed is centered on the grid. If the particle hits the seed, it gets stuck and becomes part of the immobile seed. Once this happened, a new particle is started from a random place far away from the seed. The process is repeated many thousand times and results in a fractal pattern that grows from the small initial seed.

Simulation of Diffusion Limited Aggregation

(a) Illustration of the growth of a diffusion-limited aggregate on a square lattice showing the outer cut-off limit S1 and the ring S2 on which new particles are started. We show random trajectories for a particle that does contribute to the growing cluster, and a particle that wanders beyond S1. In (b) we show the result of such a growth for 50000 particles on a hexagonal lattice.
Figure taken from: Andrew Harrison "Fractals in Chemistry" Oxford Chemistry Primers (Vol. 22) Oxford University Press, 1995.

An experimental way to create a fractal by diffusion limited aggregation is the electrochemical deposition of a solid. In our experiments, positive zinc ions are playing the role of the Witten-Sander particles and a carbon electrode acts as the immobile seed. The ions diffuse towards the negatively charged electrode where they get deposited as solid metal. The Brownian motion of the diffusing ions corresponds to the random walk of the particles in the Witten-Sander model. Our experiment, however, involves also a weak non-random electromigration of the ions. The typical fractal dimension (1.7) of the resulting zinc leaves, however, is in good agreement with the values expected from typical DLA simulations.

Test Description The experiment consists of an electrical circuit driven by two 6 volt batteries connected in series for a total of 12 volts, and containing three growth systems. The entire set-up is contained in a closed box. The fractals grow between two microscope slides with a zinc anode at one end and a graphite (carbon) cathode at the other. The zinc cations are reduced at the cathode according to the reaction equation
Zn²⁺(aq) + 2 e^- = Zn(s) ; E^o = -0.76 V .
The growth of the fractals is documented by the use of a video camera. The recorded videos are analyzed later by digitizing image sequences employing a PC-based frame grabber (Data Translation). The growth of the fractals during the flight is compared to fractals grown in Earth-bound experiments. The key idea is that the electrodeposition process gives rise to gradients in the concentration of metal ions. Consequently, local variations in the density of the solution are generated. These density gradients are the driving force for hydrodynamic flows that interact with the growing fractal in a complex fashion. While this buoyancy-induced effect is pronounced in the presence of gravitational forces, one expects it to diminish under microgravity. Therefore, significant differences should be observed between the fractal characteristics of metals that are electrodeposited under 1-g from those produced under near 0-g conditions.

Results

The following figures summarize the results of the preliminary analysis of our data. Open circles represent the length of the fractal structure at a given time and closed circles indicate the strength of the gravitational force. Due to the flight pattern the gravitational accelaration was initially 1g and then changed between 2g and 0g in cycles of about 40 s. The length of the fractal is determined by the distance between the graphite electrode and the leading point of the metal leaf.

The left figure shows an experiment in which the electric field was switched off for an intermediate time interval of about 2 min. The growth rate of the fractal is easily determined from the temporal evolution of the length of the structure, since it equals the slope of the open-circle curve. The comparison of the growth rates observed during the initial phase and during the parabola maneuvers (0g-2g) reveals no striking differences (compare linear fits). The right figure shows a more detailed view of the dynamics during the microgravity maneuvers. The presented data might indicate a weak acceleration of the growth rate during the 2g phases. However, more thorough analyses are necessary to clarify this important point.

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