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The particle size dependence of the CRSS of precipitation hardened Cu-1.9 wt. % Co
Scripta METALLURGICA
Vol. 4, pp. 81-84~ 1970 Printed in the United States
THE PARTICLE
SIZE D E P E N D E N C E
PRECIPITATION
HARDENED
Pergamon
OF THE
Press,
Inc.
CRSS OF
Cu-l. 9 wt.% Co*
B. G. Koepke ~ and T. E. Scott I n s t i t u t e for A t o m i c R e s e a r c h and D e p a r t m e n t of M e t a l l u r g y Iowa State U n i v e r s i t y , A m e s , Iowa 50010
{ R e c e i v e d November 2 8 ,
1969)
T h e o r e t i c a l e x p r e s s i o n s for the s t r e n g t h e n i n g p r o v i d e d by c o h e r e n c y s t r a i n fields of s p h e r i c a l p r e c i p i t a t e s have b e e n d e r i v e d by G l e i t e r (I) and O e r o l d and H a b e r k o r n (2), h e r e a f t e r r e f e r r e d to as G and GH, r e s p e c t i v e l y .
The l a t t e r s ~ d e r i v a t i o n e m p l o y e d the a s s u m p -
t i o n of r i g i d d i s l o c a t i o n s while the f o r m e r allowed for t h e i r f l e x i b i l i t y .
Recently, experi-
m e n t a l d a m i n t e n d e d to s u b s t a n t i a t e the GH a r g u m e n t was p r e s e n t e d (3). The p u r p o s e of this p a p e r is to p r e s e n t a d d i t i o n a l s i n g l e c r y s t a l and p o l y c r y s t a l data for c o m p a r i s o n with the t h e o r i e s and to point out a s o u r c e of i n d e f i n i t e n e s s which c o m p l i c a t e s a t t e m p t s to c o r r o b o r a t e e i t h e r t h e o r y . Single c r y s t a l s and p o l y c r y s t a l l i n e s a m p l e s of C u - l . 9 wt. % Co w e r e t e s t e d at r o o m t e m p e r a t u r e , at 4 . 2 x 10 -4 s e c ' l ,
and at 0 . 8 to 1.7 x 10 - 4 s e c ' l ,
respectively.
The s i n g l e
c r y s t a l t e n s i l e s p e c i m e n s , o r i e n t e d for easy glide, w e r e 7 . 6 cm long with a 5. l - c r n e l e c t r o p o l i s h e d gage s e c t i o n 0. g2 c m i n d i a m e t e r .
T h e s e w e r e s o l u t i o n t r e a t e d for ode week at
1025"C, q u e n c h e d in iced b r i n e , and aged at 600°C.
A v e r a g e p a r t i c l e s i z e and v o l u m e
f r a c t i o n s of p r e c i p i t a t e w e r e m e a s u r e d by m a g n e t i c t e c h n i q u e s s i m i l a r to t h o s e u s e d by L i v i n g s t o n and B e c k e t (4, 5).
The p a r t i c l e s i z e s of the aged m a t e r i a l c o r r e l a t e d well with
t h o s e d e t e r m i n e d by B o n a r (6) u s i n g t r a n s m i s s i o n m i c r o s c o p y . . c o n s t a n t at 0 . 0 l 8 in good a g r e e m e n t with p r i o r data (4).
The v o l u m e f r a c t i o n , f, was
T e n s i l e s t r e s s e s of the p o l y c r y s t a l -
line s a m p l e s w e r e c o n v e r t e d to s h e a r s t r e s s e s for c o m p a r i s o n with the s i n g l e c r y s t a l data using a Taylor factor of 3.06 (7). Writing the dislocation line tension as T = r~Gb 2, the strengthening increments predicted by the two theories, G and GH, respectively are
*Work was p e r f o r m e d i n the A m e s L a b o r a t o r y of the U. S. A t o m i c E n e r g y C o m m i s s i o n . C o n t r i b u t i o n No. 2641. ~Present Address: 55343.
H o n e y w e l l , Inc.
C o r p o r a t e R e s e a r c h C e n t e r , Hopkins, M i n n e s o t a
81
82
CRSS OF P R E C I P I T A T I O N
HARDENED
Cu-l.9
S. Z6A
]~|
whereA
=
wt.%
Co
3/2fs/6 ~:0
Vol.
4, No.
2
(1)
edge 13 s c r e w
and
~y_ = 2,1ZS
le I 31zfi12
where B
(2)
_~I. 0 -\0.
edge 33 s c r e w .
U s i n g the v a l u e s f = volume fraction = 0.018
and Icl--
unrelaxed misfit = 0. 013,
the a b o v e e x p r e s s i o n s r e d u c e to
20.2x 10"SA 2 ~__.)
(la)
and C¢ O u r data plus the data of o t h e r s h a s b e e n p l o t t e d in F i g . 1 in a c c o r d with the f o r m of Eqs.
l a and 2a w h e r e A 7 is the s t r e n g t h e n i n g i n c r e m e n t (CRSS or 0.2% o f f s e t yield} due to
the p r e c i p i t a t e s , G is the s h e a r m o d u l u s of c o p p e r t a k e n as 4100 k g / m m 2, R is the p a r t i c l e r a d i u s and b (= 2. 55 ~) is t h e B u r g e r s v e c t o r f o r c o p p e r .
In o r d e r to obtain the s t r e n g t h e n -
ing i n c r e m e n t f r o m the data, the v a l u e 0 . 4 k g / m m 2 u s e d by Witt and G e r o t d (3) f o r the CRSS of the s o l i d s o l u t i o n a p p r o p r i a t e to the 600"C a g i n g t r e a t m e n t was s u b t r a c t e d f r o m a l l the e x p e r i m e n t a l d at a . According to the Eqs.
la and 2a, the data should fall on ~. straight line passing through
the o r i g i n w h e n p l o t t e d as in F i g . l ,
It is e v i d e n t that o u r d a t a f o r s i n g l e c r y s t a l s and p o l y -
crystalline material and the data of Bonar (6) and Livingston (5) are described adequately by the algebraic f o r m of the theoretical expressions out to particle sizes corresponding to R/b 25.
In order to fit the theoretical expressions to the experimental data the values of 0~ m u s t
be: Eq. la: ~
= ~ 2 " 5 (edge) ~0.05 (screw)
Eq. 2a: ~
=
Z. 5 (edge) 0.27 (screw)"
A l l of t h e s e v a l u e s a r e r e a s o n a b l e in view of the u n c e r t a i n t y a s s o c i a t e d with the p r e s e n t s t a t e of k n o w l e d g e of d i s l o c a t i o n l i n e t e n s i o n .
B e c a u s e the p r o p o r t i o n a l i t y c o n s t a n t s of E q s .
la
and 2a v a r y i n v e r s e l y with c~, it would be v i r t u a l l y i m p o s s i b l e to d i s t i n g u i s h b e t w e e n the t h e o r i e s on the b a s i s of e x p e r i m e n t a l r e s u l t s now a v a i l a b l e . In spite of the uncertainties involved, it s e e m s apparent that precipitation hardening can
Vol.
4, No.
2
CRSS 0F P R E C ~ Z T A T I O N
5OO
I
I
I
H A R D E N E D C u ~ l , 9 ~t,% Co
I
SINGLE POLY" CRYSTALS CRYSTALS PRESENT 0 & REF. 6 + R F. 3 DASHED LINE
I
I
B3
I
7"
400
oo 3OO 0 m
"~
O4 ""
(,.9
200
A + oti
~
___-I.
<] "-'
I00
0
5
0
l i
i
i
i
I
I,
I0
15
20
25
30
35
4O
R/b C o m p a r i s o n between experimental data for Cu-l. 9 wt. % Co aged at 600°C and theory for precipitation hardening according to Eqs. la and Za (line D) withc~ = 0.05 (G) andcz = 0.27 (GH) for screws and ~ = 2. 5 (G and GH) for edges. Curves A, B, and C represent the theoretical expressions for a modified O r o w a n m e c h a n i s m with ro = 5b according to Ref. Ii for edges, Ref. I0 for screws and to Ref. I0 for edges or Ref. ii for screws, respectively. be attributed,
in Cu-Co at least,
to t h e c o h e r e n c y
strain fields at small particle
sizes.
On
the other hand, Fig. 1 reveals that the data depart from the theoretical line at about R/b = 25, the lower values of departure being for polycrystalline samples.
Anticipating that the
departure was due to the onset of the O r o w a n bow-out mechanism, w e included the corresponding theoretical curves in Fig. i. A n important parameter in the O r o w a n m e c h a n i s m is ~, the appropriate average spacing of particles along a dislocation. T w o expressions for the strengthening increment due to the O r o w a n m e c h a n i s m have recently appeared which consider the statistics involved in determining ~. used above, t a k i n g ~
A~ and
The modifications, in terms of the volume fraction
= (217/3f)I/ZR (9), are (Ref. i0)
~K
84
CRSS OF PREC]~P~rAT]~0N HARDENED Cu~l,~ Rl:,t CO
G~T (Ref. 11)
Y01~
4~ No,
2
= (_.R.~ l n ( _ ~ Z ~ ~ ° A2R
with
~0K
= {0.022 = edge 0. 018 screw
~A
= ~ 0 . 0 1 3 = edge 0. 022 screw
and
U s i n g r . = 5b as a r e a s o n a b l e e s t i m a t e of the d i s l o c a t i o n c o r e r a d i u s , the m o d i f i e d O r o w a n e x p r e s s i o n s w e r e p l o t t e d i n the f o r m
2 v e r s u s ~ as shown i n F i g . 1.
The i n t e r s e c t i o n
of t h e s e c u r v e s with the s t r a i g h t line b a s e d on the t h e o r i e s of G and GH i n d i c a t e s the p a r t i c l e s i z e s a t which the O r o w a n b o w i n g - o u t m e c h a n i s m should b e c o m e the c o n t r o l l i n g m e c h a n i s m . C l e a r l y , the p a r t i c l e s i z e s at which the e x p e r i m e n t a l data d e v i a t e f r o m the t h e o r e t i c a l l i n e a r e i n the r a n g e of t h o s e w h e r e the O r o w a n m e c h a n i s m is f e a s i b l e . Acknowledgements T h e a u t h o r s a r e i n d e b t e d to R. J. Stokes for h e l p f u l d i s c u s s i o n s and to U. F . K o c k s for h i s c o m m e n t s a n d for p r o v i d i n g a copy of his m a n u s c r i p t (Ref. 10) p r i o r to p u b l i c a t i o n . References 1.
H. G t e i t e r ,
Z. Angew. P h y s . 23, 108 (1907).
2. V. Gerold and H. Haberkorn, Phys. Star. Sol. 16, 675 (1966). 3. I~. Witt and V. Gerold, Scripta Met. 3, 371 (1969}. 4. 5. D. Livingston, Trans. AIIVIE 215, 566 (1959). 5. 5. D. Livingston and J. J. Becket, Trans. A I M E 212, 316 (1958). 6. L. G. Bonar, Ph.D. Thesis, Cambridge University (1962). 7. U. F. Kocks, Acta Met. 6, 85 (1958). 8. Iv[. F. Ashby and L. Iv[. Brown: Proc. 5th Intern. Congress for Electron Microscopy, 1 K. 5, A c a d e m i c P r e s s , New York (1962). 9. U. F. Kocks, Acta Met. 14, 1629 (1966). 10.
U. F . Kocks, P h y s i c s Of S t r e n g t h a n d P l a s t i c i t y , The O r o w a n A n n i v e r s a r y V o l u m e , ed. by A. S. A r g o n , to be p u b l i s h e d by M . I . T . P r e s s , C a m b r i d g e , Iv[ass.
II.
/Vi. F . A s h b y , Oxide D i s p e r s i o n S t r e n g t h e n i n g , P r o c . ed. by G. S. A n s e l l , T. D. C o o p e r and F . V. L e n e l . (1968).
2nd Bolton L a n d i n g Conf. (1966), G o r d o n and B r e a c h , New York
Vol. 4, pp. 81-84~ 1970 Printed in the United States
THE PARTICLE
SIZE D E P E N D E N C E
PRECIPITATION
HARDENED
Pergamon
OF THE
Press,
Inc.
CRSS OF
Cu-l. 9 wt.% Co*
B. G. Koepke ~ and T. E. Scott I n s t i t u t e for A t o m i c R e s e a r c h and D e p a r t m e n t of M e t a l l u r g y Iowa State U n i v e r s i t y , A m e s , Iowa 50010
{ R e c e i v e d November 2 8 ,
1969)
T h e o r e t i c a l e x p r e s s i o n s for the s t r e n g t h e n i n g p r o v i d e d by c o h e r e n c y s t r a i n fields of s p h e r i c a l p r e c i p i t a t e s have b e e n d e r i v e d by G l e i t e r (I) and O e r o l d and H a b e r k o r n (2), h e r e a f t e r r e f e r r e d to as G and GH, r e s p e c t i v e l y .
The l a t t e r s ~ d e r i v a t i o n e m p l o y e d the a s s u m p -
t i o n of r i g i d d i s l o c a t i o n s while the f o r m e r allowed for t h e i r f l e x i b i l i t y .
Recently, experi-
m e n t a l d a m i n t e n d e d to s u b s t a n t i a t e the GH a r g u m e n t was p r e s e n t e d (3). The p u r p o s e of this p a p e r is to p r e s e n t a d d i t i o n a l s i n g l e c r y s t a l and p o l y c r y s t a l data for c o m p a r i s o n with the t h e o r i e s and to point out a s o u r c e of i n d e f i n i t e n e s s which c o m p l i c a t e s a t t e m p t s to c o r r o b o r a t e e i t h e r t h e o r y . Single c r y s t a l s and p o l y c r y s t a l l i n e s a m p l e s of C u - l . 9 wt. % Co w e r e t e s t e d at r o o m t e m p e r a t u r e , at 4 . 2 x 10 -4 s e c ' l ,
and at 0 . 8 to 1.7 x 10 - 4 s e c ' l ,
respectively.
The s i n g l e
c r y s t a l t e n s i l e s p e c i m e n s , o r i e n t e d for easy glide, w e r e 7 . 6 cm long with a 5. l - c r n e l e c t r o p o l i s h e d gage s e c t i o n 0. g2 c m i n d i a m e t e r .
T h e s e w e r e s o l u t i o n t r e a t e d for ode week at
1025"C, q u e n c h e d in iced b r i n e , and aged at 600°C.
A v e r a g e p a r t i c l e s i z e and v o l u m e
f r a c t i o n s of p r e c i p i t a t e w e r e m e a s u r e d by m a g n e t i c t e c h n i q u e s s i m i l a r to t h o s e u s e d by L i v i n g s t o n and B e c k e t (4, 5).
The p a r t i c l e s i z e s of the aged m a t e r i a l c o r r e l a t e d well with
t h o s e d e t e r m i n e d by B o n a r (6) u s i n g t r a n s m i s s i o n m i c r o s c o p y . . c o n s t a n t at 0 . 0 l 8 in good a g r e e m e n t with p r i o r data (4).
The v o l u m e f r a c t i o n , f, was
T e n s i l e s t r e s s e s of the p o l y c r y s t a l -
line s a m p l e s w e r e c o n v e r t e d to s h e a r s t r e s s e s for c o m p a r i s o n with the s i n g l e c r y s t a l data using a Taylor factor of 3.06 (7). Writing the dislocation line tension as T = r~Gb 2, the strengthening increments predicted by the two theories, G and GH, respectively are
*Work was p e r f o r m e d i n the A m e s L a b o r a t o r y of the U. S. A t o m i c E n e r g y C o m m i s s i o n . C o n t r i b u t i o n No. 2641. ~Present Address: 55343.
H o n e y w e l l , Inc.
C o r p o r a t e R e s e a r c h C e n t e r , Hopkins, M i n n e s o t a
81
82
CRSS OF P R E C I P I T A T I O N
HARDENED
Cu-l.9
S. Z6A
]~|
whereA
=
wt.%
Co
3/2fs/6 ~:0
Vol.
4, No.
2
(1)
edge 13 s c r e w
and
~y_ = 2,1ZS
le I 31zfi12
where B
(2)
_~I. 0 -\0.
edge 33 s c r e w .
U s i n g the v a l u e s f = volume fraction = 0.018
and Icl--
unrelaxed misfit = 0. 013,
the a b o v e e x p r e s s i o n s r e d u c e to
20.2x 10"SA 2 ~__.)
(la)
and C¢ O u r data plus the data of o t h e r s h a s b e e n p l o t t e d in F i g . 1 in a c c o r d with the f o r m of Eqs.
l a and 2a w h e r e A 7 is the s t r e n g t h e n i n g i n c r e m e n t (CRSS or 0.2% o f f s e t yield} due to
the p r e c i p i t a t e s , G is the s h e a r m o d u l u s of c o p p e r t a k e n as 4100 k g / m m 2, R is the p a r t i c l e r a d i u s and b (= 2. 55 ~) is t h e B u r g e r s v e c t o r f o r c o p p e r .
In o r d e r to obtain the s t r e n g t h e n -
ing i n c r e m e n t f r o m the data, the v a l u e 0 . 4 k g / m m 2 u s e d by Witt and G e r o t d (3) f o r the CRSS of the s o l i d s o l u t i o n a p p r o p r i a t e to the 600"C a g i n g t r e a t m e n t was s u b t r a c t e d f r o m a l l the e x p e r i m e n t a l d at a . According to the Eqs.
la and 2a, the data should fall on ~. straight line passing through
the o r i g i n w h e n p l o t t e d as in F i g . l ,
It is e v i d e n t that o u r d a t a f o r s i n g l e c r y s t a l s and p o l y -
crystalline material and the data of Bonar (6) and Livingston (5) are described adequately by the algebraic f o r m of the theoretical expressions out to particle sizes corresponding to R/b 25.
In order to fit the theoretical expressions to the experimental data the values of 0~ m u s t
be: Eq. la: ~
= ~ 2 " 5 (edge) ~0.05 (screw)
Eq. 2a: ~
=
Z. 5 (edge) 0.27 (screw)"
A l l of t h e s e v a l u e s a r e r e a s o n a b l e in view of the u n c e r t a i n t y a s s o c i a t e d with the p r e s e n t s t a t e of k n o w l e d g e of d i s l o c a t i o n l i n e t e n s i o n .
B e c a u s e the p r o p o r t i o n a l i t y c o n s t a n t s of E q s .
la
and 2a v a r y i n v e r s e l y with c~, it would be v i r t u a l l y i m p o s s i b l e to d i s t i n g u i s h b e t w e e n the t h e o r i e s on the b a s i s of e x p e r i m e n t a l r e s u l t s now a v a i l a b l e . In spite of the uncertainties involved, it s e e m s apparent that precipitation hardening can
Vol.
4, No.
2
CRSS 0F P R E C ~ Z T A T I O N
5OO
I
I
I
H A R D E N E D C u ~ l , 9 ~t,% Co
I
SINGLE POLY" CRYSTALS CRYSTALS PRESENT 0 & REF. 6 + R F. 3 DASHED LINE
I
I
B3
I
7"
400
oo 3OO 0 m
"~
O4 ""
(,.9
200
A + oti
~
___-I.
<] "-'
I00
0
5
0
l i
i
i
i
I
I,
I0
15
20
25
30
35
4O
R/b C o m p a r i s o n between experimental data for Cu-l. 9 wt. % Co aged at 600°C and theory for precipitation hardening according to Eqs. la and Za (line D) withc~ = 0.05 (G) andcz = 0.27 (GH) for screws and ~ = 2. 5 (G and GH) for edges. Curves A, B, and C represent the theoretical expressions for a modified O r o w a n m e c h a n i s m with ro = 5b according to Ref. Ii for edges, Ref. I0 for screws and to Ref. I0 for edges or Ref. ii for screws, respectively. be attributed,
in Cu-Co at least,
to t h e c o h e r e n c y
strain fields at small particle
sizes.
On
the other hand, Fig. 1 reveals that the data depart from the theoretical line at about R/b = 25, the lower values of departure being for polycrystalline samples.
Anticipating that the
departure was due to the onset of the O r o w a n bow-out mechanism, w e included the corresponding theoretical curves in Fig. i. A n important parameter in the O r o w a n m e c h a n i s m is ~, the appropriate average spacing of particles along a dislocation. T w o expressions for the strengthening increment due to the O r o w a n m e c h a n i s m have recently appeared which consider the statistics involved in determining ~. used above, t a k i n g ~
A~ and
The modifications, in terms of the volume fraction
= (217/3f)I/ZR (9), are (Ref. i0)
~K
84
CRSS OF PREC]~P~rAT]~0N HARDENED Cu~l,~ Rl:,t CO
G~T (Ref. 11)
Y01~
4~ No,
2
= (_.R.~ l n ( _ ~ Z ~ ~ ° A2R
with
~0K
= {0.022 = edge 0. 018 screw
~A
= ~ 0 . 0 1 3 = edge 0. 022 screw
and
U s i n g r . = 5b as a r e a s o n a b l e e s t i m a t e of the d i s l o c a t i o n c o r e r a d i u s , the m o d i f i e d O r o w a n e x p r e s s i o n s w e r e p l o t t e d i n the f o r m
2 v e r s u s ~ as shown i n F i g . 1.
The i n t e r s e c t i o n
of t h e s e c u r v e s with the s t r a i g h t line b a s e d on the t h e o r i e s of G and GH i n d i c a t e s the p a r t i c l e s i z e s a t which the O r o w a n b o w i n g - o u t m e c h a n i s m should b e c o m e the c o n t r o l l i n g m e c h a n i s m . C l e a r l y , the p a r t i c l e s i z e s at which the e x p e r i m e n t a l data d e v i a t e f r o m the t h e o r e t i c a l l i n e a r e i n the r a n g e of t h o s e w h e r e the O r o w a n m e c h a n i s m is f e a s i b l e . Acknowledgements T h e a u t h o r s a r e i n d e b t e d to R. J. Stokes for h e l p f u l d i s c u s s i o n s and to U. F . K o c k s for h i s c o m m e n t s a n d for p r o v i d i n g a copy of his m a n u s c r i p t (Ref. 10) p r i o r to p u b l i c a t i o n . References 1.
H. G t e i t e r ,
Z. Angew. P h y s . 23, 108 (1907).
2. V. Gerold and H. Haberkorn, Phys. Star. Sol. 16, 675 (1966). 3. I~. Witt and V. Gerold, Scripta Met. 3, 371 (1969}. 4. 5. D. Livingston, Trans. AIIVIE 215, 566 (1959). 5. 5. D. Livingston and J. J. Becket, Trans. A I M E 212, 316 (1958). 6. L. G. Bonar, Ph.D. Thesis, Cambridge University (1962). 7. U. F. Kocks, Acta Met. 6, 85 (1958). 8. Iv[. F. Ashby and L. Iv[. Brown: Proc. 5th Intern. Congress for Electron Microscopy, 1 K. 5, A c a d e m i c P r e s s , New York (1962). 9. U. F. Kocks, Acta Met. 14, 1629 (1966). 10.
U. F . Kocks, P h y s i c s Of S t r e n g t h a n d P l a s t i c i t y , The O r o w a n A n n i v e r s a r y V o l u m e , ed. by A. S. A r g o n , to be p u b l i s h e d by M . I . T . P r e s s , C a m b r i d g e , Iv[ass.
II.
/Vi. F . A s h b y , Oxide D i s p e r s i o n S t r e n g t h e n i n g , P r o c . ed. by G. S. A n s e l l , T. D. C o o p e r and F . V. L e n e l . (1968).
2nd Bolton L a n d i n g Conf. (1966), G o r d o n and B r e a c h , New York