Further comments on the entropy of thermally activated plastic deformation at low temperatures
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R. J , ARSENAULT
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a r e c e n t c o m m u n i c a t i o n~ J o n a s and Lutton s t a t e that when c o n s i d e r i n g the AS for low t e m p e r a t u r e def o r m a t i o n the c r u c i a l t e m p e r a t u r e r a n g e i s below 30 K. T h i s s t a t e m e n t a p p e a r s to be b a s e d on the a s s u m p t i o n that AS i s r e l a t e d to the t e m p e r a t u r e d e p e n d e n c e of the s h e a r m o d u l u s (0/x/a T); for in this t e m p e r a t u r e R. J. ARSENAULTis Professor of Materials, Materials Engineering Group, and Department of Chemical Engineering,Unwersity of Maryland, College Park, Md. Manuscript submitted November 8, 197 I. 1014-VOLUME 3,APRIL 1972
150~ 200 TEMPERATURE, *K
250
500
Fig. 1--Entropy vs temperature and T A S vs temperature as determined by analysis proposed by Sehoeekfl r a n g e , a2p//0 Tz shows a m a x i m u m for m o s t m e t a l s . If Eq. [10] of J o n a s and Lutton i s employed, it then b e c o m e s n e c e s s a r y to d e t e r m i n e the (aH//a T) in the t e m p e r a t u r e r a n g e of 30 K. However, if the equation a s p r o p o s e d by Schoeck 2 is e m p l o y e d t h e n , AS=
1 O# [AG + ZaA(r*)b ] #o a T
[1]
w h e r e /~o i s the s h e a r m o d u l u s at a b s o l u t e z e r o , A ( T * ) is the a c t i v a t i o n a r e a , b is the B u r g e r s v e c t o r , ~'a is the a p p l i e d s t r e s s and AG iS the a p p r o p r i a t e Gibbs f r e e energy. If the v a l u e s for m a g n e s i u m a r e u s e d in the above expression,a,4 then a plot of T A S v s T and AS v s T can be obtained, Fig. 1. The s i g n i f i c a n t point i s that a A S / / a T has its l a r g e s t v a l u e s above 77 K; for below 77 K the c u r v e is confined so that a AS//a T i s z e r o at a b s o l u t e z e r o and not a finite v a l u e as J o n a s and L u t ton have shown in t h e i r F i g . 2. A n o t h e r i n t e r e s t i n g a s p e c t of F i g . 2 (Jonas and Lutton) i s that an e x t r a p o lation f r o m 600 to 300 K was m a d e . P r e s u m a b l y the j u s t i f i c a t i o n for t h i s e x t r a p o l a t i o n i s the m o d i f i c a t i o n of the above equation which d e f i n e s the entropy a s follow s: AS = (~o -- ~ r ) AHo /~oT
Further Comments on the Entropy of Thermally Activated Plastic Deformation at Low Temperatures
115
[2]
w h e r e tt T i s the s h e a r m o d u l u s at the given t e m p e r a t u r e , and AHo i s the a c t i v a t i o n enthalpy at z e r o effective s t r e s s (T*). The f o r m of the c u r v e of AS v s T, which was obt a i n e d for m a g n e s i u m , is the g e n e r a l f o r m of the c u r v e for i r o n , n e u t r o n - i r r a d i a t e d c o p p e r and t a n t a l u m . T h e r e f o r e , a AS//a T has its l a r g e s t value above 77 K and not in the t e m p e r a t u r e r a n g e of 30 K, which in t u r n m e a n s that the t e m p e r a t u r e r a n g e of i m p o r t a n c e in d e t e r m i n i n
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