Dendritic patterns are branching forms that appear in many biological and physical processes, such as neuron formation, bacterial and fungal growth, electrical breakdown, crystallization from supersaturated solution or supercooled melts, and particle aggregation. The form of these dendrites are often important in determining the properties of such systems. For instance, the spacing of the branches in certain crystallizing alloys controls phase separation and many resulting properties[1]. Dendrites may also form by electrochemical precipitation in electronic circuits, thus shorting them[2]. In this case, the large scale structure of the dendrite is responsible for determining whether the percolation threshold for conduction is reached. A similar effect has been reported in the breakdown of conducting polymers during dope-undope cycles[3]. Substantial theoretical work has addressed the growth rate at the tip of dendrites[4]. However, the large scale structures often show chaotic patterns that are impossible to predict by deterministic methods.
Active walker models may be used to reproduce the large scale structure of dendritic patterns. An active walker, in this case, the dendrite tip, advances under a bias potential, known as a landscape. The walker modifies this potential as it advances via some landscaping function[5]. The walker may choose a direction to walk either deterministically or probabilistically. For dendritic crystal growth, the value of the landscape may be considered as the concentration of solute in a supersaturated solution or the degree of undercooling in a supercooled melt. The landscaping function accounts for the diffusion of solute or thermal energy during the crystal growth process.
Particle aggregation may also result in dendritic patterns and may be modeled via a random walker which represents a diffusing particle. The particle may become stationary, with some sticking probability, when it contacts another stationary particle. Dendritic pattern have been shown to emerge from this type of simulation[6].
We have produced dendritic patterns by both active and random walker methods, and have attempted to characterize their structures using bulk properties such as fractal dimension and radius of gyration. Comparisons have been made between the behavior of the two models with respect to their salient parameters.
1. Galenko, P.K. and V.A. Zhuravlev, Physics of Dendrites: Computational Experiments. 1994, Singapore: World Scientific.
2. Presser, N. and G.W. Stupian. Formation of Dendrites:Experimental Observations. in 1994 IEEE Aerospace Applications Conference. 1994. Vail, CO: IEEE.
3. Fujii, M., K. Arii, and K. Yoshino, The Growth of Dendrites of Fractal Pattern on a Conducting Polymer. Journal of Physics: Condensed Matter, 1991. 3: p. 7207-7211.
4. Moir, S.A. and D.M. Herlach, Observation of Phase Selection from Dendrite Growth in Undercooled Fe-Ni-Cr Melts. Acta Materilia, 1997. 45(7): p. 2827-2837.
5. Lam, L. and R. Pochy, Active-Walker Models: Growth and Form in Nonequilibrium Systems. Computers in Physics, 1993. 7(5 (Sept./Oct.)): p. 534-541.
6. Witten, T.A.J. and L.M. Sander, Diffusion-Limited Aggregation, a Kinetic Critical Phenomenon. Physical Review Letters, 1981. 47(19): p. 1400-1403.