ANALYSIS AND CRITIQUE OF THE "PEAKING EFFECT" MODELS : I

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ANALYSIS AND CRITIQUE OF THE ”PEAKING EFFECT” MODELS : I

Gilbert Fantozzi, C. Esnouf

To cite this version:

Gilbert Fantozzi, C. Esnouf. ANALYSIS AND CRITIQUE OF THE ”PEAKING EFFECT” MOD- ELS : I. Journal de Physique Colloques, 1983, 44 (C9), pp.C9-557-C9-562. �10.1051/jphyscol:1983982�.

�jpa-00223432�

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JOURNAL DE PHYSIQUE

Colloque C9, suppl6rnent au n012, Tome 44, dgcernbre 1983 page C9-557

G. Fantozzi and C. Esnouf

Groupe d'Etudes de Me'taZZurgie Physique et de Physique des Mate'riaux (LA 3411, Institut Nationaz des Sciences AppZique'es de Lyon, B6t. 502, 20, Avenue A . Einstein, 69621 ViZZeurbanne Cedex, France

Resume : De nombreux modPles theoriques o n t 6t.e proposes pour i n t e r p r e t e r l e Peaking e f f e c t observe generalement l o r s d'une i r r a d i a t i o n e l e c t r o n i - que ou Y dans l e s metaux c . f . c . ou h.c. En p a r t i c u l i e r , Caro e t Mondino o n t propose recemment un mod&le q u i semble d e c r i r e correctement l e s recents r e s u l t a t s experimentaux. Apres une p r e s e n t a t i o n de ces r e s u l t a t s une c r i t i q u e du mod&le de Caro e t Mondino montre que ce modele ne p e u t d e c r i r e correctement l e Peaking E f f e c t .

A b s t r a c t : Several t h e o r e t i c a l models have been proposed f o r i n t e r p r e - t i n g t h e Peaking E f f e c t observed g e n e r a l l y d u r i n g e l e c t r o n o r

Y

i r r a d i a - t i o n i n f.c.c. o r h.c.p. metals. I n p a r t i c u l a r , Caro and Mondino have r e c e n t l y proposed a model which seems t o e x p l a i n a l l t h e c a r a c t e r i s t i c s of t h e PE. A f t e r a review o f these c a r a c t e r i s t i c s , a c r i t i q u e o f t h e model o f Caro and Mondino showsthat t h i s model cannot g i v e a c o r r e c t des- c r i p t i o n o f the PE.

I

-

INTRODUCTION

The peaking e f f e c t (PE) was f i r s t observed i n Cu by Simpson e t a1 ( 1 ) i n 1972. During e l e c t r o n i r r a d i a t i o n , t h e i n t e r n a l f r i c t i o n ( I F ) f i r s t increases, reaches a maximum then decreases g i v i n g a peak o f IF. A t t h e same t i m e

,

t h e modulus increases continuously. Almost s i n c e t h e i n i t i a l observations, t h e e x i s - tence o f t h e PE has been t h e s u b j e c t o f controversy. Since then i t has been observed i n a number o f f.c.c. metals and i n h.c.p. Mg e i t h e r d u r i n g e l e c t r o n o r Y i r r a d i a t i o n o r w i t h i n c r e a s i n g s o l u t e c o n c e n t r a t i o n i n d i l u t e s u b s t i t u t i o n a l a l - l o y s (1-11). The PE i s o n l y observed when t h e a p p l i e d amplitude o f o s c i l l a t i o n i s h i g h enough, as shown c l e a r l y by Lauzier e t a1 ( 7 ) , G i r a r d e t a1 ( 3 ) and Caro ( 5 ) . Several models o f the PE have. been proposed and a r e reviewed by M i n i e r e t a1 ( 1 1 ) . M i n i e r e t a1 show t h a t t h e models proposed up t o now do n o t g i v e a c o r r e c t d e s c r i p - t i o n o f t h e PE. Consequently, we proposed a model (12) where t h e I F i s due t o d i s - l o c a t i o n r e t a r d a t i o n by p o i n t s d e f e c t s encountered by t h e d i s l o c a t i o n d u r i n g i t s o s c i l l a t i n g motion. Two types o f immobile p o i n t s d e f e c t s are considered : p i n n i n g p o i n t s i n the core of t h e d i s l o c a t i o n and d e f e c t s c l o s e enough t o t h e g l i d e plane t o c r e a t e a drag e f f e c t on the d i s l o c a t i o n thereby r e t a r d i n g i t s motion. These l a t t e r d e f e c t s increase i n number as t h e s t r a i n amplitude increases because o f t h e increased area of t h e s l i p plane swept by t h e d i s l o c a t i o n . Thus t h e PE i s amplitude dependent. This model has been c r i t i c i z e d by Caro ( 5 ) , mainly because o f the f r e - quency dependence. Caro and Mondino (5, 6, 13) have proposed a s i m i l a r model based on t h e existence o f two types of defects. However, t h e develooment i s d i f f e r e n t . I n p a r t i c u l a r , t h e amplitude dependence o f the PE i s a t t r i b u t e d o n l y t o t h e a m p l i t u - de dependence o f the background I F . I n t h i s a r t i c l e , we present an a n a l y s i s and c r i - t i q u e o f t h e l a s t two models which take i n t o account the r e c e n t experimental r e s u l t s r e p o r t e d by ! $ i n i e r e t a1 ( 1 1 ) .

I 1

-

RECENT EXPERIMENTAL RESULTS

Recent experimental r e s u l t s a r e reviewed by M i n i e r e t a1 (11). We j u s t

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1983982

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JOURNAL

DE

PHYSIQUE

d e t a i 1 some i m p o r t a n t p o i n t s .

Simultaneous measurements o f modulus and I F d u r i n g t h e PE show t h a t t h e generation o f p o i n t d e f e c t s provokes a decrease i n v i b r a t i n g l e n g t h o f t h e d i s l o - c a t i o n motion which increases t h e f r i c t i o n c o e f f i c i e n t . The r e l a t i v e h e i g h t o f

t h e PE depends on t h e i n i t i a l p r e i r r a d i a t i o n value o f t h e I F background Q-l0.

Caro ( 5 ) has proposed t h e f o l l o w i n g e m p i r i c a l r e l a t i o n s h i p t o describe t h i s e f f e c t :

Q"

,,

, and Q-ln a r e measured a t t h e same s t r a i n amplitude sn and A i s a constant o f t h e o r d e r

07

⁵^x

lo-'.

Numerous experimental r e s u l t s r e p o r t e d i n t h e l i t e r a t u r e are described r a t h e r w e l l by t h i s expression (13) w i t h t h e p r e c i s e value of A depending on the mi r o s t r u c t r a l s t a t e o f t h e sample. I n Cu, L a u z i e r ( 4 ) has shown f o r example t h a t Q-?max

-

Q-Yo i s n o t constant and t h a t

A

decreases as t h e amount o f p l a s t i c deformation increases. V e r i f i c a t i o n o f equation (1) f o r p o l y c r i s t a l l i n e Cu ( 5 ) i s shown i n Fig. 1. I n a d d i t i o n , Caro e t a1 ( 5 , 6, 13) have shown t h a t t h e dependence o f t h e PE on amplitude E i s 1 in k e d t o t h e ampli tude dependence of t h e I F background. The f o l l o w i n g e m p i r i c a l r e l a t i o n i s proposed :

1

Q- max A +

B lo)^

(T) = 1 - B +

E -1

Q 0 ( Q O ) En

This describes t h e s t r a i n amplitude dependence very approximately and asserts t h a t t h e h e i g h t o f t h e PE i s amplitude dependent o n l y when t h e background i t s e l f i s amplitude dependent. A number o f experimental r e s u l t s a r e a t variance w i t h t h i s a s s e r t i o n . An example o f PE dependence o f E i n Cu i s shown by F i g . 2, where the background i s almost E

-

independent ( a c t u a l l y i t decreases s l i g h t l y w i t h E ) . According t o equ. ( 2 ) t h e PE should n o t change w i t h ^E

,

which i s c l e a r l y n o t t h e case as seen i n F i g . 2 . A s i m i l a r r e s u l t has been observed f o r Mg (10). Therefore t h e E - dependence o f t h e PC-cannot be a t t r i b u t e d t o an ~ d e p e n d ~ n c e o f t l i e background I F alone. The E

-

dependence o f t h e background I F can o n l y induce some secondary e f f e c t s .

Caro ( 5 ) s t a t e s t h a t i n Cu s i n g l e c r y s t a l s , t h e PE o n l y e x i s t s i n t h e range o f s t r o n g E dependence o f t h e background. I n fact, more c a r e f u l a n a l y s i s o f t h e r e s u l t s r e p o r t e d by Caro h i m s 6 l f i n d i c a t e s t h a t f o r E -values corresponding t o s t r o n g amplitude dependence o f t h e background, t h e PE i s n o t always observed ( f i g . IV.23 and 24, r e f . 5 ) .

I n conclusion, t h e experimental r e s u l t s show c l e a r l y t h a t E dependence i s i n v o l v e d i n t h e PE and i s n o t d i r e c t l y r e l a t e d t o t h e E dependence o f t h e background.

The h e i g h t o f t h e PE g e n e r a l l y s a t u r a t e s w i t h i n c r e a s i n g E as shown i n f i g . 3 The existence o f s a t u r a t i o n depends on t h e m e t a l l u r g i c a l s t a t e o f t h e specimen b u t does n o t depend upon t h e &dependence o f t h e background as asserted by Caro.

Indeed, a s a t u r a t i o n o f t h e PE w i t h E can be observed i n t h e absence o f a satura- t i o n o f t h e background.

Temperature i s a l s o an i m p o r t a n t parameter f o r t h e observation o f the PE Usually, t h e PE i s o n l y observed above t h e temperature o f t h e Bordoni r e l a x a t i o n as shown by M i n i e r e t a1 (11). Nevertheless, i n some cases, t h e PE has been observed a t very low temperature.

The PE depends upon t h e amount o f p r e i r r a d i a t i o n p l a s t i c deformation.

L a u z i e r (4) has shown t h a t i n Cu t h e PE h e i g h t decreases w i t h t h e amount o f c o l d - work and i t s E

-

dependence i s s t r o n g e r i n w e l l annealed samples.

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F i g . 1

-

V a r i a t i o n of t h e PE h e i g h t with t h e i n i t i a l v a l u e of t h e b a c k g r o u n d ~ o l f o r copper ( a f t e r Caro (5)).

Fig. 2

-

V a r i a t i o n of background ( a ) and o f PE ( b ) w i t h s t r a i n amplitude of v i b r a t i o n E i n Cu ( a f t e r Lauzier e t a l . ( 7 ) ) .

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JOURNAL DE PHYSIQUE

F i n a l l y , t h e e x i s t e n c e and p r o p e r t i e s o f t h e PE are independent o f t h e a p p l i e d frequency throughout t h e range 0.3 t o 30

KHz

( 5 ) .

A l l o f t h e observed c h a r a c t e r i s t i c s o f t h e PE g i v e n above must be accounted f o r i n an acceptable model o f t h e PE.

I 1 1

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ANALYSIS AND CRITIQUE OF THE CAR0 AND MONDINO MODEL

Caro and Mondino (5, 13) assume t h a t t h e energy d i s s i p a t e d p e r c y c l e i s given by :

where b i s t h e Burgers v e c t o r o f t h e d i s l o c a t i o n , A, t h e area i t sweeps o u t and uc t h e s l i d i n g f r i c t i o n s t r e s s . They f u r t h e r assume t h a t A i s t h e t o t a l area swept o u t by t h e d i s l o c a t i o n d u r i n g i t s motion, whereas i t seems more l o g i c a l t o us t o consider t h e area swept o u t up t o t h e s t r e s s uc; then, W becomes p r o p o r t i o n a l t o uc2. The s t r e s s uc i s taken t o be p r o p o r t i o n n a l t o t h e energy a. r a d i a t e d from a broken band, b u t t h e p h y s i c a l meaning o f a. i s n o t s p e c i f i e d i n any more d e t a i l . This r a d i a t e d energy i s assumed t o be p r o p o r t i o n a l t o t h e c o n c e n t r a t i o n of p o i n t s d e f e c t s i n t h e b u l k nL :

a', = a,

+

a l n ~

k i i t h t h e preceding r e l a t i o n s and t a k i n g i n t o account t h e ^Edependence o f t h e background and t h e E dependence o f t h e d i s l o c a t i o n l e n g t h d u r i n g i r r a d i a t i o n (some unpinning occurs), Caro and Mondino o b t a i n t h e f o l l o w i n g expression o f t h e I F :

w i t h

q-lo0

= I F a t t = 0 and E = E

n

A' = AL al/ao n L ( t ) = A L ( t ) : number o f d e f e c t s i n t h e b u l k d u r i n g i r r a d i a t i o n dL00

8 ' =

-

A d : c o n s t a n t A : d i s l o c a t i o n d e n s i t y

L o ( € ) = LOO ($): : amplitude dependence o f t h e l o o p l e n g t h a t t = 0

L o ( t , E) = ^E

-M

: t h i s expression i n d i c a t e s t h a t t h e amplitude dependen- 1 + no(t)(,)

m

ce i s n o t t h e same i n t h e i r r a d i a t e d s t a t e . n D ( t ) : number c f d e f e c t s on d i s l o c a - t i o n a t t i m e t; Exponents N and M are obtained from E - dependence o f I F i n the i n i t i a l s t a t e Q - l ( t = O,E) and i n . t h e i r r a d i a t e d s t a t e Q - l ( t = m , E ) , r e s p e c t i v e - l y . Thus equ. (4) shows c l e a r l y t h a t i n t n i s model, t h e PE and I F background 0-1 ( t = 0, E) correspond t o the same phenomenon.

The i n t e r n a l f r i c t i o n background i s g i v e n by :

G = shear modulus

F r ~ m equs. ( 4 ) , ( 5 ) and ( 6 ) , s e v e r a l remarks a r e p e r t i n e n t : - ( I ) - i f t h e l o o p l e n g t h i s E

-

independent ( i . e . N = O), t h e I F background v a r i e i as I/€, which has never been r e p o r t e d - ( 2 ) - t h e background i s E

-

independent when

N

= 1/2 ; thus, simultaneously t h e loop l e n g t h must depend on E . This r e s u l t i s s u r p r i s i n g . Indeed, e x p e r i m e n t a l l y b o t h t h e background and t h e modulus d e f e c t show a s i m i l a r

E

-

dependence. However, the modulus d e f e c t i s p r o p o r t i o n a l t o t h e square of t h e

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loop l e n g t h ; thus the d i s l o c a t i o n l e n g t h cannot vary w i t h E i n t h e range o f

E

-

independent I F o r modulus d e f e c t . Consequently, equ. (5) does n o t account f o r the experimental r e s u l t s . - ( 3 ) - I F i s always p r o p o r t i o n a l t o AL2 regardless o f t h e value o f ^E. Experimentally, numerous authors have shown t h a t the I F tends t o be p r o p o r t i o n a l t o L~ f o r the h i g h E

-

values which g i v e a PE and t o L4 f o r low E

-

values when t h e r e i s no PE. F i g . 4 shows an example o f t h e v a r i a t i o n o f normalized modulus d e f e c t and I F f o r d i f f e r e n t values o f E d u r i n g i r r a d i a t i o n . The slope i s near 1/2 f o r low E

-

values and near 1 f o r ~ > 1 0 - 6 . The value o f 1/2 means t h a t t h e I F i s p r o p o r t i o n a l

tt

L4. Thus, when t h e PE i s absent a t low

E -values, t h e I F i s p r o p o r t i o n a l t o L as p r e d i c t e d by t h e Granato

-

Lucke theory (14). Only f o r h i g h E

-

values, where t h e PE occurs, t h e I F i s propor- t i o n a l t o L2. S i m i l a r observations were r e p o r t e d by Simpson e t a1 ( 1 ) . Once again, t h e Caro

-

Mondino model does n o t adequately describe t h e observed beha- v i our.

- ( 4 ) - The Caro

-

Mondinomodelgives a o u d i t a t i v e v a r i a t i o n o f t h e PE w i t h t h e background as described by equs ( 1 ) and ( 2 ) . From equs ( 4 ) and ( 6 ) , one f i n d s t h a t :

Q-lmax M-N

(TI

= -

Q ^E 4 B ' Q 00

The exponents a r e n o t t h e same as those i n equs ( 1 ) and ( 2 ) b u t q u a l i t a t i v e l y t h e r e i s agreement between t h e model and experimental r e s u l t s

-(5)-The r e l a t i v e PE h e i g h t depends on l o o p l e n g t h ; we have ( 5 ) :

The h e i g h t increases when t h e d i s l o c a t i o n l e n g t h decreases. A f t e r p l a s t i c deforma- t i o n the PE amplitude decreases, which does n o t correspond t o t h e r e l a t i o n ( 9 ) . -9)-The occurence o f a PE depends o n l y on the ^E- dependence o f t h e background.

To observe the PE, two c o n d i t i o n s must be f u l f i l l e d : both t h e p o s i t i o n and h e i g h t o f t h e PE must increase w i t h E. This i m p l i e s t h a t M > N and M

+

N > 1. Moreover, e x p e r i m e n t a l l y , I F g e n e r a l l y increases w i t h ^Eregardess o f t h e t i m e ; thus i n ad- d i t i o n t o t h e preceding c o n d i t i o n s we must add M > 1/2 and N > 1/2.

Therefore, i n t h e Caro and Mondino model, t h e c o n d i t i o n s f o r t h e occurence o f t h e PE are v e r y r e s t r i c t i v e . I n p a r t i c u l a r , i f the l o o p l e n g t h s do n o t depend on ^E (N = M = 0), o n l y an abnormal PE can occur ( i t i s E

-

independent o r decreases w i t h E). Moreover, the c o n d i t i o n N > N means t h a t t h e amplitude dependence o f t h e

I F i s h i g h e r a f t e r i r r a d i a t i o n than before. This i s n o t i n agreement w i t h observed r e s u l t s . I n t h e study b y Caro ( 5 ) , M i s f i x e d t o o b t a i n agreement between t h e model and experimental r e s u l t s . M i s n o t obtained from t h e ^E-dependence o f Q-1 ( t + m). Furthermore, t h e r e s u l t s show t h a t n e i t h e r M n o r N a r e constant b u t vary w i t h E and t and t h a t t h e ^E-dependence i s lower a f t e r i r r a d i a t i o n than befo- r e . This leads t o a more d i f f i c u l t a n a l y s i s .

F i n a l l y , if, i n equ. ( 3 )

4

i s taken t o be p r o p o r t i o n a l t o ac i n s t e a d o f ⁰

,

t h e I F becomes p r o p o r t i o n a l t o c2N-2 so t h a t t h e N value i s equal t o t h e Caro

-

Mondino value p l u s 1/2 and t h e N < M c o n d i t i o n i s no l o n g e r f u l f i l l e d .

- ( 7 ) - For M

+

N > 1, t h e h e i g h t o f t h e PE always increases w i t h E ; i . e . equ. ( 4 ) does n o t p r e d i c t a s a t u r a t i o n w i t h E. Experimentally a s a t u r a t i o n i s observed ( f i g . 3) regardless o f t h e ^E-dependence o f the background.

- ( 8 ) - The Caro Mondino model does n o t e x p l a i n t h e occurence o f t h e PE above t h e temperature range o f t h e Bordoni r e l a x a t i o n . I t o n l y e x p l a i n s t h e e v o l u t i o n o f the PE by p o i n t d e f e c t d i f f u s i o n and d i s l o c a t i o n p i n n i n g .

- ( 9 ) - There i s no frequency dependence i n t h e Caro

-

Mondino model ; t h i s i s because of t h e frequency independence o f t h e d i s s i p a t e d energy. This frequency indepen- i s i n agreement w i t h r e p o r t e d experimental observations.

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C9-562 JOURNAL DE PHYSIQUE

I n conclusion, t h e model proposed by Caro and Mondino, based on our idea o f two l e v e l s o f i n t e r a c t i o n between t h e p o i n t d e f e c t s and t h e d i s l o c a t i o n s , does n o t g i v e an a c c e p t a b l e d e s c r i p t i o n o f t h e experimental r e s u l t s . I n p a r t i c u l a r the model i n c o r r e c t l y p r e d i c t s t h e E -dependence o f t h e PE l i n k e d t o an E

-

dependent background. I n p a r t 11, we show t h a t o u r model gives a good d e s c r i p t i o n o f a l l t h e experimental r e s u l t s .

REFERENCES : I n p a r t I 1

F i g . 3 : E f f e c t o f s t r a i n amplitude ^Eon PE h e i g h t : (1) copper a t 160 K ( 4 ) ( 2 ) : copper a t 115 K ( 4 ) ; ( 3 ) copper a t 350 K ( 5 )

olr

r

^--

L O ~ Mn

i

F i g . 4 : V a r i a t i o n o f normalized modu-

l u s d e f e c t and i n t e r n a l f r i c t i o n f o r d i f f e r e n t values o f E d u r i n g 35 K i r r a d i a t i o n o f Cu ( a f t e r L a u z i e r e t a1 ( 7 ) ) .

0 2 ",d

Log

,-A