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ULTRASONIC INVESTIGATION OF THE
DISLOCATION STRUCTURE IN PLASTICALLY
DEFORMED COPPER
H. Schmidt, D. Lenz, E. Drescher, K. Lücke
To cite this version:
CoZZoque C5, suppZ4ment au nO1O, Tome 42, octobre 1981 page C5-339
ULTRASONIC INVESTIGATION O F THE DISLOCATION STRUCTURE IN PLASTICALLY
DEFORMED COPPER
H. Schmidt, D. Lenz, E. ~ r a s c h e r * and K. Liicke
I n s t i t u t ffir AZZgerneine MetaZZkunde und MetaZZphysik RWTH Aachen, F. R. G. * ~ m t i t u t r of FundamentaZ TeehnoZogicaZ Research, Academy o f Science, Warzawa,
PO Zand
Abstract.- We have simultaneously measured t h e d i s l o c a t i o n c o n t r i b u t i o n s t o u l t r a s o n i c a t t e n u a t i o n aD and r e d u c t i o n o f sound v e l o c i t y A v ~ / v between 10 and 200 MHz i n deformed Cu. The samples were y - i r r a d i a t e d i n o r d e r t o reduce t h e a t t e n u a t i o n and extend t h e e x p e r i m e n t a l l y usable frequency range t o high- e r frequencies. The r e s u l t s have been evaluated i n terms o f d i s l o c a t i o n r e - sonance damping. Q u a n t i t a t i v e data f o r t h e dislocation d e n s i t y and f r e e d i s l o - c a t i o n l o o p l e n g t h as f u n c t i o n o f E have been d e r i v e d b o t h f o r d i s l o c a t i o n s
i n t a n g l e s and i n t h e c e l l i n t e r i o r .
1. I n t r o d u c t i o n . - D i s l o c a t i o n resonance (DR) i s a f r e q u e n t l y s t u d i e d and t h e o r e t i - c a l l y we1
l
described i n t e r n a l f r i c t i o n phenomenon /l/. The p r e d i c t e d frequency (f )-c h a r a c t e r i s t i c s o f overdamped OR has been observed by MHz-attenuation a measure- ments /2/ and Hz- and kHz-experiments a r e expected t o show t h e low f DR behaviour
3
f o r t h e modulus d e f e c t MD
-
AL" ( A = d i s l o c a t i o n d e n s i t y , L = d i s l o c a t i o n l o o p4
l e n g t h ) and f o r t h e damping decrement 6
-
a / f-
BAL f (B = damping f o r c e constant). There i s considerable experimental / 3 , 4 / and t h e o r e t i c a l /5,6/ support f o r t h e Granato Liicke DR-theory /7/ based on Koehlers v i b r a t i n g s t r i n g model /g/.However, several problems w i t h respect t o the i n t e r p r e t a t i o n o f 6 and MD data have accumulated /g/. One problem i s t h a t 6
-
f as p r e d i c t e d by t h e GL-theory i s o f t e n n o t observed i n low f experiments. Also i n MHz experiments a p u z z l i n g d e v i a t i o n from t h e p r e d i c t e d f-dependence has been observed a t low f, b u t o n l y i n pre-defomed samples. To our o p i n i o n an e x p l a n a t i o n o f t h i s e f f e c t may c o n t r i b u t e t o t h e c l a r i - f i c a t i o n o f o t h e r low f problems.We i n v e s t i g a t e d t h e i n f l u e n c e o f deformation on 6 and MD f o l l o w e d by y - i r r a d i - a t i o n induced d i s l o c a t i o n pinning. MHz pulse-echo experiments o f f e r t h e unique chan- ce t o study t h e most r e v e a l i n g f-dependence o f DR over a broad f-range ( t y p i c a l l y
5 t o 200 MHz) i n one and t h e same w e l l d e f i n e d sample volume i n d i f f e r e n t p i n n i n g s t a t e s .
2. Experimental.- A s e t o f i d e n t i c a l h i g h p u r i t y (RRR 2 1000) Cu s i n g l e c r y s t a l s was
deformed a t room temperature by compression (0.19% 2 E 6 27%) along cl l l > . Sample
preparation, y - i r r a d i a t i o n and measuring techniques f o r a t t e n u a t i o n a and sound ve- l o c i t y v a r e described i n / I Q / .
C5-340 JOURNAL DE PHYSIQUE
3 . R e s u l t s . - F i g . l a s h o w s a ( ~ ) m e a s u r e d a t 1 0 M H z . Curve D represents t h e a values 30 min a f t e r d e f o r -
m l
mation (time f o r c o n t r o l o f sample q u a l i t y and f o r transducer bonding) : a increases w i t h E t o a maximum
-
6
a t
=
0.4%, decreases then and becomes n e a r l y con- 01s t a n t a t E > 5%. An ( I h , 373K)-anneal reduces a c o n - 5
s i d e r a b l y (curve R). The y - i r r a d i a t i o n (TIRR = 320K,
f o l l o w e d by an ( I h, 353K) -anneal ) f u r t h e r reduces 001 005 01 05 1 5 10 l5 Deformatton E I%I
a towards t h e minimum value clB ("non d i s l o c a t i o n
background") which c o u l d n o t be reduced f u r t h e r by FIG l a a d d i t i o n a l i r r a d i a t i o n . For c 1% t h e ag values
a r e t h e same as f o r background i r r a d i a t e d standard cu long wave lDMHl
samples /I I/. The aB increase a t E > 1% i s due t o
zuOj
V,""'
I
sound wave s c a t t e r i n g by deformation induced inhomo- 25100
v , a
g e n e i t i e s ( k i n k bands, as shown by meta: lographic
g
a n a l y s i s ) and i s n o t due t o d i s l o c a t i o n s (as proved fmOby a d d i t i o n a l i r r a d i a t i o n ) . F i g . l a demonstrates t h a t 84900/
t h e a-maximum around E = 0.4% i s caused by d i s l o - l l
0 5 10 15 -20 25 30 _ - _ l
c a t i o n s which become pinned w i t h i n c r e a s i n g y-dose Deformat~on E 1x1
( 0 ). F o r comparison f i g . l b shows t h e sound v e l o c i t y
Y FIG I b
v 30 rnin a f t e r deformation (D) and a f t e r t h e "back-
ground" dose (curve vo). The V ( E ) data a f t e r i n t e r - l a y b Influence of deform-
a t i o n (curve D), reco- mediate gy ( o m i t t e d f o r sake o f c l a r i t y ) approach very (R: l h,373K) and w i t h i n c r e a s i n g
fly
t h e v. curve (which as aB cannot y - i r r a d i a t i o n ( 0 ont h e a t t e n u a t i o n ?fig. be changed by f u r t h e r i r r a d i a t i o n ) . At E < 1% v. i s l a ) and sound v e l o c i t y given by t h e l o n d i t u d i n a l sound v e l o c i t y v ~ < o f ~ ~ ~ > ( f i g . l b ) .
t h e i d e a l ( " d i s l o c a t i o n f r e e " ) Cu c r y s t a l /10/. The v -decrease a t 0 E > 1% i s due t o i n c r e a s i n g deforma-
t i o n induced m i s o r i e n t a t i o n ( " c r y s t a l l o g r a p h i c modulus d e f e c t " ) as shown by x-ray measurements. Since v
L<111> represents t h e maximum f c c sound v e l o c i t y any d e v i a t i o n from <l1 l > r e s u l t s i n a v e l o c i t y r e d u c t i o n .
4. General Discussion.- We consider t h e e v a l u a t i o n o f a ( f ) and v ( f ) ( f i g . l a , b shows o n l y t h e 10 MHz r e s u l t s ) t o be t h e key t o f u r t h e r understanding. Fig.2 shows t h e complete data s e t f o r t h e decrement 6 (open symbols) and t h e modulus d e f e c t MD
( s o l i d symbols)
U)
3
I
102 103Frequency f [ M H t ]
FIG 2 Frequency dependence o f d i s -
-
l o c a t i o n modulus d e f e c t (so-l i d symbols) and decrement (open symbols) a f t e r deform- a t i o n (D), recovery (R) and y - i r r a d i a t i o n s (gy) f i t t e d by t h e o r e t i c a l KGL-curves.
damped resonance maximum, arrows i n f i g . 2 ) and a MD-branch which a r e i n t e r l i n k e d ( l a r g e open c i r c l e s i n f i g . 2 ) because o f t h e Kramers-Kronig r e l a t i o n /12/. We use t h e GLP f o r t h e exponential l o o p l e n g t h d i s t r i b u t i o n which i s i n v a r i a n t a g a i n s t ran- dom p i n n i n g /10/. As f i g . 2 shows t h e GLP f i t s our h i g h gy data very w e l l . However, f o r t h e fly=800uAh d a t a and e s p e c i a l l y f o r t h e s t a t e R and D data an i n c r e a s i n g
dis-
crepancy e x i s t s f o r t h e decrement a t low f: The measured 6 values c o n s i d e r a b l y ex- ceed t h e t h e o r e t i c a l curves (e.g. by t h e f a c t o r 1.6 a t 10 MHz i n s t a t e D) whereas t h e MD data a r e much b e t t e r described by t h e r e s p e c t i v e GLP. According t o t h e GL- t h e o r y any change o f t h e mean l o o p l e n g t h L due t o p i n n i n g causes t h e GLP t o move downwards a s l o p e - l l i n e . Therefore t h e present GLP f i t s a r e s t r o n g l y supported by t h e observed s l o p e - l s h i f t (dashed d o t t e d l i n e through (arrows) i n f i g . 2 ) be- tweenfly
= 800 and 15000 pAh. Accordingly, we f i t our R a n d D data ( f o r s t a t e D on- l y 10 MHz data a r e a v a i l a b l e because.of extremely h i g h a ( c . f . f i g . l a ) ) w i t h t h e GLP s h i f t e d upwards t h esame
s l o p e - l l i n e . The observed low f discrepancy cannot be r e - moved by a m o d i f i c a t i o n o f t h e exponential L - d i s t r i b u t i o n f u n c t i o n ( s i n c e t h i s would d e t e r i o r a t e t h e good (&,MD) f i t s a t h i g hfly
and a l s o t h e f a i r MD f i t a t low f i n s t a - t e R and D. The discrepancy i s observed a f t e rl l
A areas ( c e l l i n t e r i o r ) /13,14/. Consequent- d e f o m e d sample, RT deformations between 0.1 and 10%; i t i s
s t i l l observed a f t e r standard annealing a t
-
z"
923K (fig.3;c.f. a l s o f i g . 3 i n / 1 0 / ) . However, as f i g . 3 shows t h e discrepancy i s
not
obser- EE
ved i n as grown c r y s t a l s . Thus we conclude
:
t h a t t h e discrepancy i s caused by deforma-2
E = M 2 % , L h 6 5 O o C
t i o n .
loo
10' 102 103Frequency f [MHz] It has been shown by TEM t h a t p o l y s l i p
o r i e n t e d (e.g. <l1 l > ) c r y s t a l s a l r e a d y a t
-
FIG 3 6 ( f ) b e f o r e ( 0 ) and a f t e r low E develop a d i s l o c a t i o n s t r u c t u r e con- standard treatment ( 0 ) com-C5-342 JOURNAL DE PHYSIQUE
l y we base t h e f u r t h e r e v a l u a t i o n on such a s t r u c t u r e : In a d d i t i o n t o t h e (61 ,MD1) component (which
is
described by t h e s o l i d GLP i n f i g . 2 ) r e s p o n s i b l e f o r most of t h e observed 6 and MD a second d i s l o c a t i o n resonance component (62,MD2) with 62 = d-
(e.g. dashed GLP f o r s t a t e D
w i t h
fMAX2 2 1 MHz) accounts f o r t h e 6-discrepancy a t low f .As
seen by t h e dashed GLP i n f i g . 2 t h e accompanying MD2 i s hardly observable (MD2 << MD1 ). Furthermore f i g . 2 shows t h a t (62 ,MD2) r a p i d l y decrease during recovery and i r r a d i a t i o n (800 M A ~ ) , i . e . t h e component 2 becomes r a p i d l y reduced by pinning2 2
i n comparison with component 1. Since GNAX
-
A L and fMAX-
1/L /7/ i t follows from f i g . 2 t h a tL 2
-r 10L1 and A 2 = 0.01 Al. Consequently we a t t r i b u t e t h e component 2 (cau- sed by long d i s l o c a t i o n s of low A) t o t h e c e l l i n t e r i o r d i s l o c a t i o n s and component l (caused by s h o r t d i s l o c a t i o n s of high h ) t o t h e d i s l o c a t i o n s i n t h e t a n g l e s . The ra- pid pinning of component 2 i s a consequence of t h e small Aand longL
of t h e c e l l i n t e r i o r d i s l o c a t i o n s competing f o r t h e deformation- and i r r a d i a t i o n induced point d e f e c t s ( t h e d i s l o c a t i o n / p o i n t d e f e c t drainage volume V D per loop l e n g t hL
i s given3
by V D - - L / A i . e . VD2/VD1 2 10 i n s t a t e D). The r e s i s t a n c e of t h e observed d i s c r e -
pancy a g a i n s t heat treatment i n d i c a t e s t h a t t h e tangled s t r u c t u r e i s not removed (e.g. by 4h, 923°C a n n e a l i n g ) . This i s i n agreement with TEM r e s u l t s which show t h a t f o r E < 0.5% t h e deformation induced d i s l o c a t i o n s t r u c t u r e i s not s i g n i f i c a n t l y changed by our standard anneal /15/.
5. Q u a n t i t a t i v e e v a l u a t i o n and discussion.- The described (6 ,MD) f i t procedure was
done f o r t h e samples with E
<
6.2% /16/ ( f o r --E > 6.2% t h e measured 6 and MD values a r e t o o small f o r re1 i a b l e q u a n t i t a t i v e evaluation
( c . f . f i g . l a , b ) ) . From t h e GL-theory A and
L
7a r e c a l c u l a t e d according t o
2 P
A = 15.66 &M*X-fMAXB/QGb ( 3 )
Dislocation d e n s i t y .
-
With equ.(3) we ob- 9t a i n ~ ~ [ c m - ~ ] = 1.5-10 E[$] ( f i g . 4 ) . This r e -
s u l t (being i n q u a n t i t a t i v e agreement with TEM 10:1
L
Deformation-
E [%l 101
( 4 )L
= ~ ' 0 . 1 1 3 C/fMAXB .- C3
loa with 6MAX,fb,AX coordinates of t h e 6-maximum,2
B = 6 . 5 - 1 0 - ~ ~ s / m ~ , R = 0.088 o r i e n t a t i o n f a c - t o r , G = 4 . 0 8 . 1 0 ' ~ ~ / m ~ s h e a r modulus, b =measurements /13,14/) proves t h a t component 1
FIG 4 i s caused by t h e tangled d i s l o c a t i o n s . In con-
-
-
t r a s t t o P t h e c e l l i n t e r i o r d e n s i t y P 2
i s
i n -Dislocation density
i n tangles 1dependent of E f o r small E and c l o s e t o t h e ( k l ) and i n t h e c e l l i n t e r i o r d i s l o c a t i o n d e n s i t y o f our undef ormed c r y s t a l s ( A2 )
as
Of10
2.55.10-
m
Burgers v e c t o r and C = 5 . 5 - 1 0 - ' ~ ~ l i n e tension (values from / 2 / ) .f i n e d t o h i g h d i s l o c a t i o n d e n s i t y areas.
D i s l o c a t i o n l o o p length.- Since L i s determined by d i f f e r e n t p i n n i n g e f f e c t s we d i s - cuss t h e p i n n i n g i n terms o f ni t h e number o f p i n n i n g p o i n t s (PP) per cm d i s l o c a - t i o n n = 1/L = Cni. I n a w e l l annealed c r y s t a l we expect n =Anc+nc+nA w i t h
n, =
ETA
t h e network PP (network l e n g t h LN shortened by mutual d i s l o c a t i o n c u t t i n g ) ; nc t h e s t a t i s t i c a l i m p u r i t y PP i n t h e l a t t i c e ; Anc t h e enrichment o f i m p u r i t y PP a t d i s l o c a t i o n s (due t o e.g. C o t t r e l l i n t e r a c t i o n ) . F o r t h e deformed c r y s t a l s we t a k e Anc = 0 ( i t seems u n l i k e l y t h a t i n I h a t 373K C o t t r e l l clouds a r e formed around f r e s h d i s l o c a t i o n s ) , and a d d i t i o n a l l y account f o r t h e deformation produced PP nE and t h e i r r a d i a t i o n induced PP ny:n = nc
+
n A + nE + ny ( 5 ) 4We assume ( i ) nc t o be independent o f E and given by nc = (/Lc = 2.5-10 c i 1 / 1 7 / ; ( i i ) n A = (network model); ( i i i ) nE = KN(e)/A; w i t h N(E) = qLA2 t h e l a t t i c e con- c e n t r a t i o n o f deformation induced p o i n t d e f e c t s /18/ we g e t nE = K ~ = ~
nA
A; ( i v ) ny = o*oy/A w i t h o* t h e e f f e c t i v e cross-section f o r i r r a d i a t i o n PP. The l / A f a c t o r i n ( i i i ) and ( i v ) accounts f o r t h e drainage c o m p e t i t i o n (which reduces nE and ny w i t h i n c r e a s i n g A i n t h e case o f complete drainage). Then f o r t h e deformed and i r r a d i a t e d c r y s t a l s we o b t a i nn = nc
+
+
n h + o * ~ ~ / d (6 Fig.5 shows a p l o t o f (n-nC)lfi vsfi
f o r component 1 ( t a n g l e d d i s l o c a t i o n s ) as suggested by equ.(6) f o rfly
=O. We postpone t h e d i s c u s s i o n o f t h e i r r a d i a t i o n p i n n i n g /19/. The obtained l i n e a r f i t ( y i e l d i n g5
= 0.75, Q = 4 - 1 0 - ~ c m ) shows t h a t mutual d i s l o c a t i o n c u t t i n g as w e l l as p i n n i n g bydeformation induced p o i n t d e f e c t s c o n t r i b u t e
5 -
t o L ( € ) . The value f o r
5
i s i n good agreement w i t h5
= 0.6 f o r t h e simple cubic network2
( A L = 3 ) ; from t h e Q-value i t f o l l o w s t h a t
-
3K 2 10 i .e. o n l y 1 O/oo o f t h e deformation
induced d e f e c t s a r e converted i n t o PP /19/. A t ~>0.4% t h e L ( & ) - s h o r t e n i n g overcomes t h e
i n f l u e n c e o f A(€) on 6 and MD ( f i g . l a , b ) . For a s i n g l e component t h e & ( A ) , MD( A )
behaviour can be given f o r t h e low f range
o
2 4 6 8 1 0 ( f < fMAX) w i t h L = l / n and equ.(6,
gY=O)I O - ~ ~ A
[cm-'] we o b t a i n 4-
FIG 5s,,,,,
- A L = A / ( ~ _ + ~ J - A + Q A ) (7) L""3 L F i t f o r e v a l u a t i o n o f E and n (L-shortening c o e f f i c i e n t s ) i n equ.(6).C5-344 JOURNAL DE PHYSIQUE
pass through a maximum (given e.g. f o r t h e MD by
iMD
= nc/q) and decrease a t high3 LIM
n
as;
:
:
6
-
l / A ; MDLOw-
l/A. We expect t h a t egus.(7) and ( 8 ) hold f o r Hz- and kHz experiments. In t h e s e cases ( c . f . f i g . 2 ) t h e measured DR damping i s almost comple- t e l y caused by t h e c e l l i n t e r i o r d i s l o c a t i o n s i . e .6
= 62; however, an excess modu- l u s d e f e c t due t o t h e tangled d i s l o c a t i o n s should be observed MD = MD2+
M D 1 . We-p
-point out t h a t f o r f > fMAX (and i f d i f f e r e n t
DR
components c o n t r i b u t e - t o 6 andMD
a s i n t h e present c a s e ) t h e complete GLP ( i n c l u d i n g s e p a r a t i o n i n t o components) has t o be used f o r t h e e v a l u a t i o n . We note t h a t S c h l i p f 1201 considering d i s l o c a t i o n i n t e r - a c t i o n e f f e c t s (mutual c u t t i n g and point d e f e c t generation) a r r i v e s from d i f f e r e n t arguments a t t h e same A - dependence a s given i n our equ.(6,
flY=O). A more d e t a i l e d discussion of t h e present5
and 0 values and of i r r a d i a t i o n pinning i n deformed Cu w i l l be published elsewhere 1191.Work supported by t h e DFG (SFB 125). References
/ l / A.V.Granato, K.Lucke: i n "Physical Acoustics" (W.P.Mason Ed.) Vol.IV A , p.225 Academic Press, New York 1966
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10,
358 (1962)/3/ D.O.Thompson, V.K. ParP: i n "Physical Acoustics" ( W . ~ . ~ a s o n Ed.) Vol .IIIA,p.293, Academic P r e s s , New York 1966.
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45,
3351 (1974) /7/ A.V.Granato, K.Lucke: J.App1. Phys.-
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