The stereology of grain boundary allotriomorphs
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R. BRADLEY and H. I. AARONSON are GraduateStudent and Professor, respectively, Department of Metallurgical Engineering, Michigan Technological University, Houghton, MI49931. Manuscript submitted April 26, 1976.
METALLURGICAL TRANSACTIONS A
ing and thickening r a t e s of g r a i n boundary allotriomorphs. Finally, the relationships obtained for the lengthening and thickening kinetics of allotriomorphs will be applied t o THEEM growth rate data1,2 t o a s c e r tain what portion of the considerable scatter observed may be attributed t o stereological effects. In the companion paper,7 the results oI the present study will be used t o minimize stereological effects upon data obtained by statistical sectioning, thereby enabling a more effective study t o be made of the growth kinetics of g r a i n boundary f e r r i t e allotriomorphs. C. S . Smith,9 u s i n g an equation due t o H a r k e r and P a r k e r ,l° calculated the distribution of apparent dih e d r a l angles f o r m e d by a pair of intersecting planes observed on a randomly intersecting plane of polish. In the present investigation this distribution will be redeveloped for the dihedral a n g l e s associated with lenticular allotriomorphs. Account will be taken of the fact that the observed dihedral a n g l e depends not only upon the true a n g l e but also upon the distance from its center at which the allotriomorph is sectioned. Nonrandom planes of polish will be considered in o r d e r t o adapt the calculations t o currently employed experimental techniques. The "growth path envelope" analysis developed by DeHoff11 provides a rigorous stereological approach t o evaluating b o t h the nucleation and the growth r a t e s of precipitates, all of which are assumed t o have the same shape. Unfortunately, an appreciable proportion of f e r r i t e allotriomorphs is not well formed. Further, even those allotriomorphs whose appearance on the plane of polish approximates satisfactorily a section through an allotriomorph of accurately lenticular shape appear t o be subject t o interference with t h e i r growth by fine-scale faceting, not observable with an optical microscope.? The experimental technique used in the companion paper,7 in which the length of the longest and the thickness of the widest allotriomorph were m e a s u r e d in specimens whose g r a i n boundaries were approximately perpendicular t o the plane of polish, minimizes the latter effect. Hence the more specialized stereological analysis described in this paper appears more appropriate to this particular technique than the elegant g e n e r a l analysis of DeHoff.11
VOLUME 8A, FEBRUARY 1977-317
THE SHAPE OF GRAIN BOUNDARY ALLOTRIOMORPHS
V
As did previous investigators, %s the shape of an idealized g r a i n boundary allotriomorph is assumed t o be that of two spherical caps, with both spheres having the same radius. Consider such an allotriomorph sectioned by a plane of polish passing through its c e n t e r of volume perpendicular t o the g r a i n boundary as shown in Fig. 1.* For t
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