Mechanism of steady-state grain growth in aluminum
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out 2 upon a s e r i e s of s p e c i m e n s of 99.99+ pet aluminum that had been cold worked at - 195~ and subjected to i s o t h e r m a l grain growth at 635~ for t i m e s ranging from 1 min to 1 h. This t r e a t m e n t provided a range of grain size, by volume, of a l m o s t t h r e e o r d e r s of m a g nitude. In o r d e r to avoid the intrusion of s u r f a c e effects upon the m e a s u r e m e n t s , that portion of each s p e c imen which was analyzed was taken at a distance of not l e s s than ten grain d i a m e t e r s f r o m any e x t e r n a l s u r face. The exact volume of m a t e r i a l analyzed in each case was m e a s u r e d and the topological p a r a m e t e r s were e x p r e s s e d in t e r m s of number p e r cubic c e n t i m e t e r , thus: N V (grains p e r cm3), F V (faces p e r cm3), E V (edges p e r cmS), and C v ( c o r n e r s p e r cmS). The r e s u l t s of this study provide the e x p e r i m e n t a l b a s i s for the development that follows. The a v e r a g e g r a i n v o l u m e , obtained by taking the r e c i p r o c a l of the number of grains p e r cubic c e n t i m e t e r 1 / N v , p r o v i d e s a m e a s u r e of grain size* *The conventional expression for grain size is the mean intercept, obtained by counting the number of grains crossed by unit length of line laid upon a twodimensional section through the grain structure. This parameter is sensitive to shape. It is a function of the total surface area of the grain boundary, Le., it is not a grain diameter. Neither this nor any other measurement that can be made upon a two-dimensionalsection through the material can be used to determine grain volume. Nv must be measured in three-dimensional space.
that is Independent of grain shape, grain size d i s tribution and shape a s s o r t m e n t . It will be shown in the following that the a v e r a g e grain volume is a lineal function of the time, in s t e a d y - s t a t e grain growth at constant t e m p e r a t u r e ( 1 / N v = let). TOPOLOGICAL NATURE OF THE GRAIN STRUCTURE The g e o m e t r i c f o r m of the grain s t r u c t u r e is defined by the s y s t e m of grain boundary, which c o n s t i tutes a topological network. If the grain c o r n e r s be taken as nodes and the grain edges as b r a n c h e s of a point-line network, Fig. 1, its connectivity in one cubic c e n t i m e t e r of m a t e r i a l will be, by network law: 3 Connectivity = E V - CV + 1
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VOLUME 5, FEBRUARY 1974-413
If, f u r t h e r , t h e g r a i n s t h e m s e l v e s b e r e p r e s e n t e d a s individual points connected by lines representing the grain faces, an interpenetrating network is formed, F i g . 2, a n d i t s c o n n e c t i v i t y i s : Connectivity = F y - N y + 1
energy which is felt as a surface tension* [Quincke 5 and Desch6). This energy provides the driving force for
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T h e c o n n e c t i v i t i e s of t h e two n e t w o r k s a r e i d e n t i c a l , because both networks correspond to opposite retracts* *The corner-edge network
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