Fractal analysis of a nucleation and growth process
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Fractal Analysis of a Nucleation and Growth Process M. YAWORSKI and J. G. BYRNE Many interesting applications of factal analysis have recently been made to the earth and mineral sciences. These have sought to provide a more general description of shapes, perimeters, and areas than available heretofore. In summary, fractal geometrv, although formulated about a century ago, was only recently m recognized as applying to nature and to materials in particular. 1231 J. G. BYRNE is Ivor D. Thomas Professor of Physical Metallurgy and Chairman of the Department of Metallurgy and Metallurgical Engineering. University of Utah, Salt Lake City. UT 84112. M. YAWORSKI is a Senior Thesis Student, University of Utah. Sail Lake City. UT 84112, Manuscript submitted August 3. 1987. METALLURGICAL TRANSACTIONS A
The fractal dimension, D, relates the length, L , of a perimeter such as that of an island, to the length, s . of the scale to measure L by the relation [41 [1]
L = K s D-I
where K is a constant. Finer and finer scales reveal more detail and lead to larger perimeters. If one plots log~ L v s log~0 s and obtains a clearly defined slope (negative) of magnitude (D - 1), then the shape enclosed by the perimeter may be described by the characteristic fractal dimension D and will be comparable to any other physical shape of the same D independent of what that other physical shape is. Not infrequently, a common fractal dimension between things which upon first appraisal seem quite distinct leads to the realization that they are in fact related. For example, a satellite photograph of the Red Sea and a profile view of a crack in a metallic sample 14r both have a fractal dimension of 1.09. Mecholsky I41 pointed out, however, that these two entities are in fact mechanistically related since the Red Sea is most likely the result of geologic fracture. In the present instance we were interested in determining if a fractal dimension could be associated with a secondphase interface obtained from a nucleation and growth process in condensed matter, since this had not been attempted heretofore. We chose arbitrarily the fairly irregular shape of the/3 phase which forms peritectically during the freezing of copper-37 pct zinc alloy. This /3 phase tends to form mostly around a skeleton of excess a phase, although some /3 crystals also are found within the a grains. Etching in an ammonia/hydrogen peroxide mixture develops the/3 coring of the a matrix. The test of the fractal nature of the result is done by measuring the perimeter of each of a number of selected particles at a number of magnifications and plotting the results as log~0 L v s log~0 s as prescribed by Eq. [1]. The sample used was one of a set of annotated metallographic samples prepared by Metallurgical Services Laboratories Ltd. to illustrate various microstructures in alloys. Five different/3 regions were observed using four magnifications from 200x to 1000• in an IBAS Image Analyzing System coupled with an optical microscope and television monitoring capability. Coordinate data are
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