Measurement of dislocation density in [100]- and [111]- copper single crystals with high relative accuracy

Scripta

METALLURGICA

Vol. 14, pp. 3 2 5 - 3 2 9 , 1980 P r i n t e d in the U.S.A.

P[EASUREMENT COPPER

OF D I S L O C A T I O N

SINGLE

CRYSTALS

P. A m b r o s i ,

DENSITY

WITH

HIGH

W. H o m e i e r

IN

P e r g a m o n P r e s s Ltd. All r i g h t s r e s e r v e d .

[100]- A N D

RELATIVE

and

Ch.

JIll)-

ACCURACY

Schwink

I n s t i t u t A f~r P n y s i k , T e c n n i s c h e O n i v e r s i t ~ t , D - 3 3 0 0 B r a u n s c h w e i g , Fed. Rep. G e r m a n y ( R e c e i v e d D e c e m b e r 12, 1979) ( R e v i s e d J a n u a r y 17, 1980) Introduction In any t h e o r y of p l a s t i c d e f o r m a t i o n the d i s l o c a t i o n d e n s i t y p is one of the b a s i c p a r a m e t e r s w i t h r e s p e c t b o t h to the a b s o l u t e v a l u e and to the s t r e s s d e p e n d e n c e . F o r m e r r e s u l t s on c o p p e r s i n g l e c r y s t a l s are s h o w n in t a b l e i as f u n c t i o n of T, all c o n v e r t e d to v o l u m e d e n s i t i e s . Table

1

F o r m e r m e a s u r e m e n t s of d i s l o c a t i o n on c o r n e r o r i e n t e d c o p p e r s i n g l e (p in m / m 3 , T in MPa, as t h r o u g h o u t

p(T)

Author

p = 2.1.1010

m2

Livingston

(1)

p = 3.5.1010

T2

van

and

Saimoto

p = 7.3.1010

T2

Ambrosi

et

al.

(3)

p = 8.1.1010

m2

G~ttler

(4)

Drunen

densities crystals this p a p e r ) Axis

(2)

]~iethod

[111]

etch

pit

[100]

etch

pit

[111]

TEM

[lOO]

TEH

2 All a u t h o r s m e a s u r e d an i n c r e a s e of p p r o p o r t i o n a l to z , but the a b s o l u t e v a l u e s are q u i t e u n c e r t a i n . B e c a u s e of the d i f f e r e n c e s of b o t h m e t h o d s if a p p l i e d to the s a m e c r y s t a l o r i e n t a t i o n and b e c a u s e of the fact that u n t i l now p has not b e e n m e a s u r e d for the two o r i e n t a t i o n s b y o n e i n v e s t i g a t o r u s i n g the same m e t h o d it is not c e r t a i n w h e t h e r the d i s l o c a t i o n d e n s i t y at a f i x e d T is g r e a t e r in the [ l O 0 ] - c r y s t a l t h a n in the [ l l l ] - c r y s t a l , as i n d i c a t e d in t a D l e 1 T h i s q u e s t i o n is i m p o r t a n t for the u n d e r s t a n d i n g of the d i f f e r e n t i n i t i a l w o r k h a r d e n i n g r a t e s of b o t h t y p e s of c r y t a l s (5), of the d e v e l o p m e n t of t~e cell s t r u c t u r e and the d e n s i t y of the cell w a l l s , and of such p h e n o m e n a as the s t o r e d e n e r g y (6). T h e r e f o r e we h a v e m e a s u r e d the d i s l o c a t i o n d e n s i t y ~ a g a i n a p p l y i n g i d e n t i c a l m e t h o d s (as far as p o s s i b l e ) to b o t h o r i e n t a t i o n s . T h e c e l l s t r u c t u r e in c o p p e r s i n g l e c r y s t a l s w i t h c o r n e r o r i e n t a t i o n was systematically i n v e s t i g a t e d b y A m b r o s i et al. (3) for the [ l l l ] - c r y s t a l and by G ~ t t l e r (4) for the [ l O O ] - c r y s t a l . We are i n t e r e s t e d in p - v a l u e s at s t r e s s e s w h e r e this c e l l s t r u c t u r e is f u l l y d e v e l o p e d , i.e. a b o v e a b o u t T = 15 M P a (3,4) up to the b r e a k i n g s t r e s s ( 7 0 M P a to 1 0 0 M P a ) . The d i s l o c a t i o n d e n s i t y in this r a n g e is too h i g h for the e t c h pit t e c h n i q u e . So we use transmission electron microscopy (TEM).

325 0036-9748/80/030325-05502.00/0 C o p y r i g h t (c) 1980 P e r g a m o n P r e s s

Ltd.

326

MEASUREMENT

OF D I S L O C A T I O N

Description

DENSITY

of m e a s u r i n q

Vol.

14,

No.

3

methods

We call the u s u a l m e t h o d d e s c r i b e d by S m i t h and G u t t m a n (7) the " l i n e m e t h o d " . Its p r i n c i p l e is that the n u m b e r N of i n t e r s e c t i o n s b e t w e e n a c i r c l e of r a d i u s R d r a w n on the TEM p i c t u r e s (this r e p r e s e n t s a c y l i n d e r in the foil) and the visible dislocations is c o u n t e d . If T is the foil t h i c k n e s s , m e a s u r e d on the pictures with magnification M, it f o l l o w s that N is the n u m b e r of d i s l o c a t i o n s intersecting the s u r f a c e S = 2 7JR T /(M 2) of the c y l i n d e r . C o n s i d e r i n g that the true n u m b e r of d i s l o c a t i o n s is twice the v i s i b l e n u m b e r on a ( 1 1 1 ) - f o i i b e c a u s e of c o n t r a s t c o n d i t i o n s (4), we get for the s u r f a c e d e n s i t y X of d i s l o c a t i o n s intersecting a unit area 2 N N M2 S - ~ R T "

AC o n v e r s i o n of X into r e q u i r e s a f a c t o r of

the v o l u m e 2 (8)

density

p

(dislocation

length

per u n i t

volume)

2 N M2 p

=

2 X

-

~

R

T

"

Uncertainties in this line m e t h o d a r i s e f r o m the d e t e r m i n a t i o n of foil t h i c k ness T, w h i c h can lend to e r r o r s of 50%, e s p e c i a l l y if two d i f f e r e n t p e r s o n s e v a l u a t e the TEM p i c t u r e s . So one of us (W.H.) p e r f o r m e d all the m e a s u r e m e n t s to avoid any s y s t e m a t i c e r r o r s b e c a u s e of p e r s o n a l e f f e c t s . The s e c o n d m e t h o d is c a l l e d the " p o i n t m e t h o d " . H e r e one c o u n t s the n u m b e r P of end p o i n t s of d i s l o c a t i o n s w i t h i n a c i r c u l a r a r e a of r a d i u s R on the TEM p i c t u r e s , a g a i n w i t h m a g n i f i c a t i o n M. B e c a u s e we see a p r o j e c t i o n of the foil we m e a s u r e t w i c e the s u r f a c e d e n s i t y I. A g a i n t a k i n g into a c c o u n t that o n l y one half of all d i s l o c a t i o n s are v i s i b l e ~ we find p M2 R2 " In the c a s e of the [ 1 1 1 ] - c r y s t a l the c o n v e r s i o n of i into p is m o r e c o m p l i c a t e d than it is in the c a s e of the line m e t h o d . In the l a t t e r m e t h o d we m e a s u r e on the s u r f a c e of a c y l i n d e r w h i c h leads a u t o m a t i c a l l y to a m e a n v a l u e of s e v e r a l g l i d e p l a n e s . The p o i n t m e t h o d g i v e s A on a f i x e d p l a n e ( ~ s u a l l y a ( 1 1 1 ) - p l a n e ) and we may not e x p e c t that the d i s l o c a t i o n d e n s i t y p p e r on the (111)-glide-plane perpendicular to the [ 1 1 1 ] ; c r y s t a l - a x i s (the a p p l i e d s t r e s s on this p l a n e is zero) w i l l be the same as pobl that one on the t h r e e o t h e r oblique (111)-glide-planes. We i n t r o d u c e the r a t i o f = Pper / Pobl

'

w h i c h can be d e t e r m i n e d f r o m A o b I and lper, the s u r f a c e d e n s i t y on an o b l i q u e g l i d e p l a n e and the one on the p e r p e n d i c u l a r g l i d e plane, r ~ s p e c t i v e l y . Dislocations on one ( 1 1 1 ) - g l i d e - p l a n e with volume density p c o n t r i b u t e to the s u r f a c e d e n s i t y i by (~/~/~).p~ ( S c h o e c k (8)). So we have:

It f o l l o w s

lobl

= 24

• Pobl

lper

= 3 q~ E Pobi

+ ~~ P pm e r

=

(2 + f ) ~ ~ @obl

that f = ~

= Pobl

3 X o b I - 2 :)'per Zper

Vol.

14,

F,~ow we

No.

can

3

MEASUREMENT

determine

the v o l u m e

~ii~

+

3

: @per

density

~

@obl

(For this s p e c i a l c a s e t h e r e c o n c e r n i n g the r a t i o p / ~).

p for

3 + f

=

3

T h e c a l c u l a t i o n for the [ l O O ] - c r y s t a l are e q u i v a l e n t (i.e. f:1, ~ g l is the ~1OO

OF D I S L O C A T I O N

: 4 ~

and

[111)-crystal:

~per

= 2 + f~

lobl

is s i m p l e ~ b e c a u s e h e r e s u r f a c e d e n s i t y on s u c h

_

g4~ Izg in

527

3 + f ~

~.

~

is an e r r o r

Results

the

DENSITY

all g l i d e p l a n e s a glide plane):

I

the p a p e r

of S c h o e c k

(8)

Discussion

As m e n t i o n e d a b o v e t h e r e e x i s t s a c e l l s t r u c t u r e in o u r c r y s t a l s . T h e r e f o r e , w i t h i n e a c h c r y s t a l two t y p e s of d i s l o c a t i o n v o l u m e d e n s i t i e s c a n be d e f i n e d : @ t o t a v e r a g e d o v e r the t o t a l c r y s t a l v o l u m e and @ w a l l a v e r a g e d o v e r the cell w a l l s a l o n e , w h e r e the d i s l o c a t i o n s a r e a l m o s t c o m p l e t e l y c o n c e n t r a t e d . It w a s a n o t h e r i n t e n t i o n of this w o r k to t e s t w h e t h e r t h e r e e x i s t s a d i f f e r e n c e in the s t r e s s d e p e n d e n c e of t h e s e d i s l o c a t i o n d e n s i t i e s . S u c h a d i f f e r e n c e is s u g g e s t e d by e a r l i e r r e s u l t s on the s t r e s s d e p e n d e n c e of d e n s i t i e s in the c e l l w a l l s (3). A d d i t i o n a l l y , t h e o r e t i c a l w o r k c o n s i d e r i n g the c o n t r i b u t i o n of the d i s l o c a t i o n f o r e s t to the f l o w s t r e s s (9) and to the f o r e s t h a r d e n i n g in fcc m e t a l s (10) led to the c o n c l u s i o n t h a t in this c a s e the d i s l o c a t i o n d e n s i t y ( a s s u m e d as h o m o g e n e o u s t h r o u g h o u t the c r y s t a l ) is not a p u r e q u a d r a t i c f u n c t i o n of the s t r e s s , p ~ T 2, b u t a m o r e s t e e p l y i n c r e a s i n g one, @ ~ T 2 - 2 7 (9) resp. T 2-35 (10), w i t h i n the t y p i c a l r a n g e of s t r e s s e s . Now, in o u r m o d e l for the i n i t i a l w o r k h a r d e n i n g r a t e of [ 1 0 0 ] - and [ 1 1 1 ) - c r y s t a l s (5) the s t r e s s d e t e r m i n i n g p r o c e s s is the c u t t i n g of g l i d e d i s l o c a t i o n s t h r o u g h the c e l l w a l l s , not the a c t i v a t i o n of the l o n g e s t s e g m e n t as in the m e s h l e n g t h t h e o r y (ii) w h i c h o t h e r w i s e is f o r m a l l y r a t h e r s i m i l a r (12) to the f o r e s t h a r d e n i n g m o d e l ( 1 3 - 16). T h e r e f o r e , the d e t e r m i n a t i o n of the e x p o n e n t of T in the @ w a l l - T r e l a t i o n s h i p e n a b l e s us to e x a m i n e a b a s i c f e a t u r e of o u r m o d e l . T h e p r o p e r m e t h o d to d e t e r m i n e @ t o t is the line m e t h o d , w h e r e a s ~ w a l l c a n be b e t t e r d e t e r m i n e d b y the p o i n t m e t h o d . T h e r e s u l t s are s h o w n in Fig. 1 and Fig. 2. F r o m t h e s e f i g u r e s we d e d u c e an a c c u r a c y of the e x p o n e n t of ± 0. I. In a g r e e m e n t w i t h o u r m o d e l the d i s l o c a t i o n d e n s i t y in the c e l l w a l l s i n c r e a s e s p r o p o r t i o n a l to @ w a l l ocT 2 - 3 ± O - 1 . C o n c e r n i n g the [ 1 1 1 ] - c r y s t a l w e f o u n d for the r a t i o of the d i s l o c a t i o n d e n s i t y on the p e r p e n d i c u l a r ( 1 1 1 ) - p l a n e to the d e n s i t y on a ( 1 1 1 ) - g l i d e - p l a n e f = @ p e r / P o~ b l

= 0.45

in the s t r e s s r a n g e f r o m T = 30 M P a to T = 50 MPa. In p r i n c i p l e it is p o s s i b l e to m e a s u r e @ t o t by the p o i n t m e t h o d , b u t o n e h a s to e v a l u a t e l a r g e a r e a s to o b t a i n the m e a n v a l u e of c e l l w a l l s and d i s l o c a t i o n f r e e c e l l a r e a s . T h i s l e a d s to e x t r e m e l y h i g h n u m b e r s of p o i n t s . We d i d this for o n e c r y s t a l and got a @ t o t v a l u e of o n l y a b o u t 2/3 of the o n e u s i n g the l i n e m e t h o d . We b e l i e v e t h a t this is d u e to the v e r y s h o r t d i s l o c a t i o n s e g m e n t s in the foil w h i c h h a v e the t e n d e n c y to a r r a n g e p e r p e n d i c u l a r to the foil s u r f a c e . In this c a s e it is i m p o s s i b l e to d i s t i n g u i s h t h e m f r o m the p o i n t d e f e c t c l u s t e r s w h i c h are c a u s e d b y the n e u t r o n i r r i d i a t i o n u s e d for d i s l o c a t i o n s t a b i l i s a t i o n (17). T h e e n d p o i n t s of t h e s e d i s l o c a t i o n s are t h e r e f o r e not c o u n t e d and the c a l c u l a t e d d e n s i t y is too low. T h e l i n e m e t h o d is not a f f e c t e d b y t h e s e s h o r t s e g m e n t s and for t h a t r e a s o n w e t h i n k t h a t the a b s o l u t e v a l u e s of the line m e t h o d are m o r e r e l i a b l e , w h e r e a s for r e l a t i v e m e a s u r e m e n t s and for the d e t e r m i n a t i o n of the s t r e s s d e p e n d e n c e b o t h m e t h o d s are u s e f u l .

328

MEASUREMENT

OF D I S L O C A T I O N

DENSITY

1013 ,.10,.~2 // 40 m2 111 30 Ptot:ll.JlU l. //3 .

.

.

130 I

.

100J

111

Pwa[[= 9.1'

101o2.3

7ot

20

°

50" 40-

10

30-

7~

20"

/~82

V

1010"K2

3

lOO

PwQtt= 5.5' 1010

7 2

110

14, No.

Pwalt

Ptot

5 4

Vol.

%2.3

5 4

2'0

30 FIG.

3

10

50 70 MPo

%

,

v

20 30 FIG.

1

l

l

!

C~-

50 70 MPQ

2

D i s l o c a t i o n d e n s i t y averaged over the cell wall volume alone for the [lO0]-crystal (D) and for the [1ill-crystal (V).

D i s l o c a t i o n d e n s i t y averaged over the total v o l u m e for the [lO0]-crystal (O r i from (4)) and for the [111]-crystal (V).

C a l c u l a t i o n of the initial slope of the work h a r d e n i n g curves (5) gives for both c r y s t a l s with great a c c u r a c y the e x p e r i m e n t a l values if the d i s l o c a t i o n d e n s i t y e v a l u a t e d by the line m e t h o d is used (18). This supports as well the absolu t e values of this method as the T exponent. T h e r e f o r e we think that the absolute value of the line method is correct w i t h i n ~ 2 0 % whereas the accuracy of the point m e t h o d is ~ 5 0 % . Summary 1) At a given stress the d i s l o c a t i o n d e n s i t y of the [%11]-crystal is c l e a r l y h i g h e r than the one of the [100]-crystal~ by 3 8 % for Ptot and by 65 % for pwall- This in c o n t r a r y to the s u g g e s t i o n s of table 1. 2) The d i s l o c a t i o n d e n s i t y ptot which is an average over the total volume found to be p r o p o r t i o n a l to T 2 ± 0 . 1 This is in a g r e e m e n t with all former m e a s u r e m e n t s .

is

3) The d i s l o c a t i o n d e n s i t y ~wall in the cell walls is p r o p o r t i o n a l to T2.3 ± 0.1. This is in a c c o r d a n c e with t h e o r e t i c a l c o n s i d e r a t i o n s by Schoeck and F r y d m a n (9) and a s s u m p t i o n s by Basinski (10) and supports a basic feature of the model for the initial work h a r d e n i n g (5).

3

Vol

14,

No.

3

MEASUREHENT OF DISLOCATION DENSITY

329

References (~ (2

(3 (4 (5 (,~ (7 (S (9 (10 (11 ~12 t3 14

~5 16 17 18

Livingston, J.D. (1962). Acta Met., 10, 229. van Drunen, G. and S. Saimoto (1971). Acta Met., 19, 213. Ambrosi, P., E. Gottler and Ch. Schwink (1974). Scripta Met., ~, 1093~ G~ttler, E. (1973). Phil. Mag., 28, 1057. Schwink, Ch. and E. G~ttler (1976). Acta Met., 24, 173. R~nnpagel, D. and Ch. Schwink (1978). Acta Met., 26, 319.Here further ref. Smith, C.S. and L. Guttmann (1953). Trans. Met. Soc. AIME, 197, 81. Schoeck, G. (1962). J. Appl. Phys., 33, 1745. Schoeck, G. and R. Frydman (1972). phys. star. sol. (b), 53, 661. Basinski, Z.S. (1974). Scripta Met., ~, 1301. Kuhlmann-Wi!sdorf, D. (1962). Trans. Met. Soc. AIHE, 224, 1047. Hirsch, P.B. (1975). The Physics of Metals. Vol. 2, ed. by P.B. Hirsch, Cambridge U n i v e r s i t y Press, 189 ff. Basinski, Z.S. (1959). Phil. Hag., ~, 393. Hirsch, P.B. (1959). Internal stresses and fatigue in metals, ed. by G.M. Rassweiler and W.L.Grube, Elsevier, Amsterdam. 139 ff. Saada, G. (1960). Acta Met., ~, 841. Saada, G. (1961). Acta Met., ~, 166. Essmann, U. (1965). phys. star. sol., 12, 707. Ambrosi, P. (1980). Diss. TU Braunschweig. ~cknowledgement

The authors support.

wish

to thank the Deutsche

Forschungsgemeinschaft

for financial