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

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

C. Esnouf, Gilbert Fantozzi

To cite this version:

C. Esnouf, Gilbert Fantozzi. ANALYSIS AND CRITIQUE OF THE ”PEAKING EFFECT” MOD- ELS : II. Journal de Physique Colloques, 1983, 44 (C9), pp.C9-563-C9-568. �10.1051/jphyscol:1983983�.

�jpa-00223433�

C. Esnouf and G. Fantozzi

Groupe dlEtudes de Me'taZZurgie Physique e t de Physique des Mate'riaux

(LA 3 4 1 / , I n s t i t u t NationaZ des Sciences AppZique'es de Lyon, BGt. 502,

2 0 , Avenue A. Einstein, 69621 ViZZeurbanne Cedex, France

R@sum@

-

A b s t r a c t - I n t h e preceding a r t i c l e , we have shown t h a t t h e Caro and Mondino model cannot g i v e a good d e s c r i p t i o n o f t h e Peaking E f f e c t . Now, we analyse more I n d e t a i l o u r own model and show t h a t i t e x p l a i n s w e l l e l l the characte-

r i s t i c s o f the PE.

I - INTRODUCTION

Our model presented i n Manchester (12) i s based on t h e e x i s t e n c e o f two types o f immobile p o i n t d e f e c t s which i n t e r a c t d i f f e r e n t l y w i t h d i s l o c a t i o n s . There are d e f e c t s l o c a l i z e d on d i s l o c a t i o n s which p i n them and d e f e c t s s i t u a t e d near t h e g l i d e plane responsible f o r t h e i r r e t a r d a t i o n . The l a t t e r d e f e c t s provoke an i n c r e a s e AT o f the r e l a x a t i o n time o f t h e d i s l o c a t i o n . This increase depends on ^E because t h e number o f defects met i s p r o p o r t i o n a l t o t h e a r e a A swept o u t by t h e d i s l o c a t i o n ( t h e defor-

mation ~ d due t o the d i s l o c a t i o n motion i s equal t o u b / L where A i s t h e d i s l o c a t i o n d e n s i t y and L t h e l o o p l e n g t h - T h e P E i s dueontheone hand t o t h e d i s l o c a t i o n p i n n i n g which reduces the i n t e r n a l f r i c t i o n andmodulus d e f e c t and on t h e o t h e r hand t o an i n c r e - ase o f tne d i s s i p a t e d energy due t o a d i s l o c a t i o n r e t a r d a t i o n by t h e second type of p o i n t s defects. The increase AT o f t h e r e l a x a t i o n time i s given by (12) :

AT =

CA

where C i s 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 p o f p o i n t s d e f e c t s c r e a t e d by i r r a d i a - t i o n ( P = Potm). Ed depends on. the s t r a i n amplitude E by t h e r e l a t i o n (12) :

Ed = E A i,2;2 /I/

2 2

w i t h A = AL Gb / 1 2 Y = r e l a x a t i o n s t r e n g t h , G = shear modulus, y = d i s l o c a t i o n energy per u n i t l e n g t h ~ b ' / 2 ) . Thus, we o b t a i n :

(Co = constant)

The d i s l o c a t i o n p i n n i n g can be describe by the f o l l o w i n g law :

L = Lo ( 1 + a 9 t ) - n w i t h a and n : constants, @ : i r r a d i a t i o n f l u x .

So,

-

=X,

A0 Lo K

,lo = r e l a x a t i o n s t r e n g t h a t t = 0, K = l i n e t e n s i o n term = 12 y/L 2 ( 4 ) .

The modulus d e f e c t AM/M and the i n t e r n a l f r i c t i o n Q-I are c a l c u l a t e d by

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1983983

C9-564 JOURNAL DE PHYSIQUE

c o n s i d e r i n g t h a t t h e r e l a x a t i o n time T i s t h e sum o f t h r e e c o n t r i b u t i o n s due t o t h e d i f f e r e n t processes which i n t e r v e n e :

TO i s the term associated w i t h a thermal 1 y a c t i v a t e d phenomenon (Bordoni peak f o r instance). Bo/K i s an atherrnal term where t h e v i s c o s i t y c o e f f i c i e n t Bo i s r e l a t e d t o phonon e f f e c t s . These two terms ( T ~ and Bo/K) determine the i n t e r n a l f r i c t i o n before i r r a d i a t i o n . The modulus d e f e c t and inter.na1 f r i c t i o n are given by the c l a s s i c a l equations :

-

M l + w T _{l + u} 2 2

T

(AT i s a f u n c t i o n o f E and so the equation of motion o f the d i s l o c a t i o n i s non- l i n e a r ; b u t we can consider a l i n e a r e q u a t i o n w i t h a mean value f o r AT f o r c a l c u l a - t i n g Q-1. Rigorously, t h e equation should be r e s o l v e d n u m e r i c a l l y b u t t h e conclu- sions should be s i m i l a r ) .

I1

-

The t o t a l r e l a x a t i o n time i s given by t h e f o l l o w i n g r e l a t i o n (15) :

w i t h k =-

0 6b

An approximate expression f0r-Q-' and AM/M can be obtained when u~ i s << 1, which i s t r u e when T~ i s = 0 ( i .e. f a r from a r e l a x a t i o n peak) and when Bo/K i s low ( l o w i n t e r n a l f r i c t i o n background).

With t h i s hypothesis, we o b t a i n :

ko KO ko EWA 0

where x =

-

-

Bo 9 - l 0

The subscr-tpt 0 i n d i c a t e s t = 0.

(However, even when i s n o t << 1, the same conclusions as t h e f o l l o w i n g can be obtained).

The p r i n c i p a l f e a t u r e s o f o u r model are as f o l l o w s :

i) The PE existence and h e i g h t depend on t h e term x : when x i s t o o low t h e r e i s no PE and o n l y a decrease o f Q-1. So, we can s p e c i f y the e f f e c t o f :

-

f e r e n t values of the i n i t i a l background and F i g . 6b presents t h e same v a r i a t i o n w i t h Q - l o f o r d i f f e r e n t amplitudes ^E. :de can note t h a t t h e v a r i a t i o n of the PE h e i g h t i s almost l i n e a r as a f u n c t i o n o f E o r Qo (except f o r low values).

- The p l a s t i c deformation : t h e PE depends on the p a s t i c deformation

A

on account of t h e dependence o f t h e term k w i t h Lo

(ko

-

^{~ ~ - 3}

i i ) The e v o l u t i o n s o f modulus d e f e c t and i n t e r n a l f r i c t i o n depend on t h e e x i s - tance of the PE. Indeed, l e t ' s consider t h e expressions o f M and F g i v e n by r e l a t i o n s (15) f o r two cases :

.

.

F = x t m ( a ~ t ) - ~ ~ M = Thus, F a M( 5-m/n)/2

and thus v a r i e s The slope o f t h e l i n e l o g M = f ( l o g F ) i s equal t o p = 3-m/n

from specimen t o specimen. I n F i g . 7 (b, c ) , we have n = 0.3 and m = 1/3 Qr 2/3 and p 5 0,51 o r 0.72 r e s p e c t i v e l y .

i i i ) The p o s i t i o n of t h e PE maximum 4 i s l i n k e d t o the value o f x = ko E Ko/Bo ( f i g . 8) : i t v a r i e s as l o g x (excepted f o r

?

4 i ) The e f f e c t o f p r i n c i p a l parameters on the PE i s e l u c i d a t e d by the deriva- t i v e d Q - l / d ~ .

a)The PE h e i g h t depends on the temperature. I n the range T. < Tp, Tp : tem- p e r a t u r e o f a r e l a x a t i o n peak (due t o t h e term T,) , t h e r e i s no PE ( t h e d e r i v a t i v e i s negative). I n the range T>Tp, appears b u t becomes decreasingly as Tnears Tp.

Indeed, t h e i n i t i a l background increases and so tne rt decreases as i n d i c a t e d i n i ) . b ) The PE v a r i a t i o n w i t h t h e s t r a i n amplitude ( o r Qo) shows a s a t u r a t i o n because t h e d e r i v a t i v e d Q - l / d ~ decreases when

or

5 i ) When t h e term Eo i s independent o f frequency, the PE i s a l s o independent on frequency as shown by t h e x parameter ( r e l a t i o n (16)) :

x = k 0 - - - k o ~ w A0

Q-l, _"

This hypothesis r e q u i r e s t h a t the background Q-l0 i s p r o p o r t i o n a l t o w, A, being constant. This v a r i a t i o n has been observed o f t e n i n t h e khz and !4hz range b u t i s n o t observed a t low frequency.

I 1 1 - COMPARISON WITH EXPERIMENTAL RESULTS

- i ) a) Tile general form of t h e curves ~ - l ( t ) / Q-1 and t h e i r e v o l u t i o n w i t h ^E ( f i g . 5) a r e v e r y s i m i l a r t o t h e experimental r e s u l f s . The c o e f f i c i e n t s a,

n and m must be f i t t e d t o o b t a i n t h e b e s t agreement. a and n a r e obtained by measu- r i n g simultaneously i n t e r n a l f r i c t i o n and modulus d e f e c t ; indeed, we have :

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P i g . 5 - PE c a l c u l a t e d from e q u a t i o n ( 1 5 ) f o r d i f f e r e n t v a l u e s o f X a = 0 . 7 x 10-15 cm2 , n = 0 . 3 , m = 1 1 3 ( a ) , ^{m =} 2 / 3 ( b ) .

F i g . 6 - V a r i a t i o n o f t h e PE h e i g h t ( a ) w i t h ^E : ( 1 ) ~ : ' = 2 . 1 0 - ~ ( 2 ) ( 3 ) 6 . 1 0 - ~

((43

4 . 1 0 - 4 ( 4 ) 4 . 1 0 - ~ ; ( b ) w i t h Q ; ~ : ( 1 ) x = 0 . 8 3 ( 2 ) x = 1 . 6 7 ( 3 ) x = 4 . 1 7 ( a = 0 . 7 x 10- , m = 1 / 2 , n = 0 . 3 ) .

F i g . 7 - V a r i a t i o n o f l o g AM/M a s a f u n c t i o n o f l o g Q-1 : a = 0 . 7 10-l5 n = 0 . 3 , ( a ) m = 1 / 3 , x = 0.167 ( a ' ) m = 2 / 3 x =,0.167 ( b ) = m = 1 / 3 x = 3.17 ( c ) m = 2 / 3 x = 3.17.

i n Cu ( 4 ) . The maximum p o s i t i o n @Ij,,.gives m b u t a more p r e c i s e value can be o b t a i - ned by u s i n g t h e v a r i a t i o n o f @tr,, w ~ t h ^E.

b ) The l i n e a r dependence of t h e PE h e i g h t w i t h Qo observed by Caro (5) i s w e l l described by our model ( f i g . 6b 1, except f o r t h e h i g h values o f Q - ~

low PE). The slopes o f t h e l i n e s 9-1 px/Q-lo = f ( Q o ) are a p p m x i n a t e l y ~o;'~:~

(4.5 x f o r wko ^{E =} f i g . 6 by1 .e. near 10-4 w i t h A '0.1. T h i s v a l u e i s near t h e v a l u e obtained by Caro (5) i n t h e case o f Cu ( 5 x 18-4).

c ) The Grenoble Group has shown t h a t the PE decreases WWn the amount o f p l a s t i c deformation increases i n agreement w i t h o u r model

.

i i ) The slope p o f t h e experimental l i n e l o g M versus l o g f i s always one h a l f when t h e r e i s no PE and E i s low ( 4 ) . !:'hen t h e PE appears, t h i s s l o p i s about 1. According t o our model, p i s equal t o 1/2 w i t h o u t PE and p = 2 / ( 5 - m/n) f o r long time when the PE i s present. This v a l u e o f p depends on t h e experimental con- d i ti o n s , m i c r o s t r u c t u r a l s t a t e af the specimen and i t s temperature ( 4 ) .

i i i ) The l i n e a r dependence of t h e PE maximum p o s i t i o n

$,,

versus l o g E we can o b t a i n a p r e c i s e v a l u e o f exponent m (m = 0.45 i n Rg). One can note t h a t a r e d u c t i o n o f l o o p l e n g t h by p l a s t i c deformation (4) produces a s i m i l a r e f f e c t t o a decrease o f E.

4 i ) a) Often, the PE has been s t u d i e d a t temperature h i g h e r than the tempe- r a t u r e Tp o f Bordoni peaks. When the temperature i s near T , the PE becomes low and disappears f o r T = Tp (4, 11). This r e s u l t agrees w e l l w i t 1 o u r moael. However, some authors nave observed a PE el ow t n e Bordoni peaK (4K), which can be due t o the secon k i n d o f P e i e r i s v a l l e y s ( 1 1 ) .

b ) The e x p e r i m e n t a l l y observed s a t u r a t i o n o f t h e PE h e i g h t w i t h t h e s t r a i n amplitude ^E ( f i g . 3 ) i s due t o t h e f a c t t h a t t h e i n t e r n a l f r i c t i o n cannot exceed t h e t e l a x a t i o n s t r e n g t h . Thus t h e PE dependence on ^E v a r i e s w i t h temperature : the s a t u r a t i o n i s observed e a r l i e r when t h e temperature i s nearer t h e Bardoni peak as seen on F i g . 3.

5 i ) The PE independence o f t h e a p p l i e d frequency shows t h a t tne r a t i o K /BO must s t a y constant ( r e l a t i o n (15)). But, the i n i t i a l background Q-1,

xo

1V - CONCLUSION

The comparison o f experimental r e s u l t s w i t h the preceding t h e o r e t i c a l ana- l y s i s shows t h a t our model describes t h e p r i n c i p a l f e a t u r e s of the PE. Our mode l can be improved by t a k i n g i n t o account some secondary e f f e c t s such as t h e depinning o f l o o p l e n g t h s w i t h e, r e s p o n s i b l e f o r t h e v a r i a t i o n o f Q-1, w i t h ^E. Thus, a b e t t e r ag'ree- ment w i t h experimental r e s u l t s can be obtained. Eut, such an e f f e c t i s n o t neces-

sary t o o b t a i n a PE as i n the Caro and llondino model. A b e t t e r aqreement w i t h ex- periments can be a l s o obtained by ciioosing a b e t t e r law f o r t h e v a r i a t i o n o f l o o o lengths : the law L = Lo ( 1

+

C9-568 JOURNAL DE PHYSIQUE

L a s t l y , we have shown t h a t t h e e x i s t e n c e and t h e n o n - l i n e a r e f f e c t s o f r h e PE a r e n o t l i n k e d t o t h e i n i t i a l background dependence on t h e s t r a i n a m p l i t u d e t h i s dependence i s o n l y r e s p o n s i o l e f o r some secondary e f f e c t s on1 y.

XEFERE?lCES

( 1 ) SIWPSOll H.M., SOSIR A. and JOKWS0I.I D.F., Phys. Rev. B, 5, 4, (1972) 1393 ( 2 ) SEIFFERT S.L., SITlPSOPi i:.M. and SOSI;: A, J. Appl. Phys. 44, 8, (1973) 3404 ( 3 ) GIRAR3 ?.andMINIER C., J. Phys. 39, (1978) 981

( 4 ) LAUZIER J . , These, U n i v e r s i t P de Grenoble, 1981 ( 5 ) CARO J., ThPse, E.P.F. Lausanne, 1981

( 6 ) CARO J., !;lOADINO I!., J. Appl. Phys. 52, (1981) 7147

( 7 ) LAUZIER J., :lINIER C.andSEIFFERT S.L., P h i l . Mag. 31, (1975) 893 ( 8 ) SIPIPSON H.N., SOSIN A.andSEIFFERT S.L., J. Appl. Phys. 42, (1971) 3977 ( 9 ) 8011JOUR C., ThPse, E.P.F. Lausanne, 1978

( 1 0 ) F4INIER C., LAUZIER J., ESNOUF C. FANTOZZI G., Phys. S t a t . S o l . ( a ) 7 1 (1982)381 ( 1 1 ) MINIER C . , LAUZIER J., ESNOUF C., FANTOZZI G., t h i s c o n f e r e n c e

(12) ESNOUF C. and FAFITOZZI G., 3 ECIFUP.S, I l a n c h e s t e r , ed. C.C. SMITH, Pergamon Press, (1980) p. 109

(13) CARO J. , GLP,SS 8.E. and !10l~IDI110 Yl., J. P.ppl. Phys. 53, 7, (1982) 4854 (14) GRAMTO A.V. and LUCKE K., J . Appl

.

(15) RITCtlIE I .G., ATREIIS A., SO C.B. and SPRUNGHAIX: K.W., J. Phys. C5, 10,42(1981)310 (16) FP,NTOZZI G., ESrIOUF C., SEPIOIT W., RITCI!IE I., Prog. N a t . S c i . 27, 3-4(1982) 311

0.027 2.77

Acknowledgments : t h e a u t h o r s a r e g r a t e f u l t o D r . I. RITCHIE f o r t r a n s l ' a t i o n . -

2 --

F i g . 8 - V a r i a t i o n o f t h e i n t e g r a t e d

i

: ( a ) m = 1 / 3 ( b ) m = 213 (c) m = 1 ( a = 0 . 7 x 10-l5 , n = 0 . 3 , Q G ~ =

1 -- A ~ ~ 2 ) .

X

0

-