Fractals in the natural sciences - Diffusion-controlled growth

• Published in: Proceedings of the Royal Society of London. Series A, Mathematical and physical sciences Vol. 423; no. 1864; pp. 123 - 132
• Main Authors: Ball, R. C., Blunt, M. J., Spivack, O. Rath
• Format: Journal Article
• Language: English
• Published: London The Royal Society 08-05-1989
• ISSN: 0080-4630; 2053-9169
• DOI: 10.1098/rspa.1989.0045

The conditions of diffusion-controlled growth are outlined and the observed importance of anisotropy is discussed through a tentative flow diagram. A crucial role is played by the forwardmost tips, which lead to growth. The nature of the singularity in their growth rate determines the overall fractal dimension. This has been estimated in two dimensions from effective cone-angle models, which work well for the most extreme anisotropic growth and can be augmented into a self-consistent approximation for the isotropic fractal case. The way in which the tip growth rate singularity is limited by finite tip radius is also a key ingredient. For diffusion-limited solidification where it is set by competition with surface tension, this significantly changes the form of the equivalent model with a fixed (e. g. lattice spacing) imposed tip scale. The full distribution of growth rates everywhere provides a much richer problem. We show new data and examine the consistency of how sites can evolve from the regions of high growth rate where they are born, into well-screened regions devoid of further growth.