DEPINNING OF DISLOCATIONS AS REFLECTED IN ANOMALOUSLY SHARP DISLOCATION DAMPING PEAKS

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DEPINNING OF DISLOCATIONS AS REFLECTED IN ANOMALOUSLY SHARP DISLOCATION

DAMPING PEAKS

Y. Estrin, K. Lücke

To cite this version:

Y. Estrin, K. Lücke. DEPINNING OF DISLOCATIONS AS REFLECTED IN ANOMALOUSLY

SHARP DISLOCATION DAMPING PEAKS. Journal de Physique Colloques, 1983, 44 (C9), pp.C9-

627-C9-631. �10.1051/jphyscol:1983994�. �jpa-00223444�

JOURNAL _DE

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Colloque C9, suppl6ment au n012, Tome 44, dhcembre 1983 page C9-627

D E P I N N I N G OF D I S L O C A T I O N S AS REFLECTED I N ANOMALOUSLY SHARP D I S L O C A T I O N DAMPING PEAKS

Y. E s t r i n and K. ~ i i c k e ~

TechnicaZ University Hamburg-Harburg, 21 Hamburg 90, F .R. G . 'RWTH Aachen, 5 2 Aachen, F.R. G .

Resume

-

L " ' ~ v a p o r a t i o n " des d g f a u t s ponctuels e s t 2 l l o r i g i n e d ' u n f r e i n a g e (visqueux) des d i s l o c a t i o n s . On 6 t u d i e son i n f l u e n c e s u r l a forme e t s u r l a p o s i t i o n du maximum d'amortissement. Une methode e s t propos@e pour d6coupler l e s 6nergies d ' a c t i v a t i o n des processus mis en jeu.

A b s t r a c t

-

The e f f e c t o f "evaporation" of p o i n t defects, which g i v e r i s e t o p o i n t d e f e c t drag o f d i s l o c a t i o n s , on the p o s i t i o n and t h e shape o f t h e damping maximum i s considered. A procedure f o r "decoupl i n g " the a c t i v a t i o n energies o f t h e processes i n v o l v e d i s suggested.

I

-

INTRODUCTION

I n t h i s communication we consider t h e d i s l o c a t i o n damping associated w i t h t h e p o i n t d e f e c t drag o f d i s l o c a t > o n s /1,2/. Due t o a t t r a c t i v e d i s l o c a t i o n - p o i n t d e f e c t i n t e r a c t i o n , a c l o u d o f p o i n t d e f e c t s i s formed around a d i s l o c a t i o n , a c t i n g t h e r e as " s o f t " p i n n i n g p o i n t s capable o f d i f f u s i n g along w i t h the o s c i l l a t i n g d i s l o c a - t i o n . The c o n t r i b u t i o n t o t h e drag c o n s t a n t stemming from t h e p o i n t d e f e c t drag (and h e r e a f t e r considered predominant) i s g i v e n by /2/

B = nkT/DL, ( 1 )

where L i s t h e average d i s l o c a t i o n segment l e n g t h , n t h e number o f s o f t p i n n i n g p o i n t s per segment (we assume n > > l ) , D t h e d i f f u s i o n c o e f f i c i e n t f o r p o i n t d e f e c t d i f f u s i o n along w i t h t h e d i s l o c a t i o n , T t h e absolute temperature, and k t h e B o l t z - mann constant.

This drag g i v e s r i s e t o a damping maximum

where 6 i s t h e l o g a r i t h m i c decrement over a, w i s t h e frequency o f t h e e x t e r n a l s t r e s s f i e l d ; t h e r e l a x a t i o n s t r e n g t h A, and t h e r e l a x a t i o n time rr are r e s p e c t i v e - 1 y g i v e n by /2/

1 ~ b ~ 2

Ar = ~ ~9 f i ( 3 )

Here G i s t h e shear modulus, b the Burgers vector, A t h e d i s l o c a t i o n d e n s i t y , C,the d i s l o c a t i o n l i n e t e n s i o n ( C - Gb2/2), and B the drag constant. The numerical f a c t o r s

K and y a r e o f t h e order o f u n i t y and can be considered constant /2/.

I t i s recognized t h a t t h e damping maximum has a Debye form when considered as a f u n c t i o n o f frequency. Furthermore, when L and n remain constant throughout a damping measurement, b o t h the r e l a x a t i o n s t r e n g t h and the r e l a x a t i o n t i m e rr

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1983994

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a r e constant, and t h e maximum e x h i b i t s a "standard" h a l f - w i d t h AT (which c o r r e s - ponds t o a pure r e l a x a t i o n maximum associated w i t h a s i n g l e t h e r m a l l y a c t i v a t e d process). This standard h a l f - w i d t h i s r e l a t e d t o t h e maximum temperature Tm (de- termined by t h e c o n d i t i o n wTr=l) and t o t h e a c t i v a t i o n energy Hr f o r t h e relaxa- t i o n process ( i .e., i n t h e case under c o n s i d e r a t i o n , t o t h e a c t i v a t i o n energy f o r p o i n t d e f e c t m i g r a t i o n along w i t h t h e d i s l o c a t i o n ) through /3/

However, measuring t h e damping as a f u n c t i o n o f temperature i s commonly c a r r i e d o u t by c o n t i n u o u s l y h e a t i n g t h e specimen. The p o p u l a t i o n o f t h e c l o u d o f s o f t , movable p i n n i n g p o i n t s w i l l then vary, whereas t h e number o f f i r m , immobile p i n - n i n g p o i n t s determining L (such as d i s l o c a t i o n nodes, i n t e r s e c t i o n s , e t c . ) w i l l t y p i c a l l y remain unchanged. That i s t o say, t h e r e l a x a t i o n s t r e n g t h w i l l be con- s t a n t w h i l e t h e r e l a x a t i o n time w i l l a c q u i r e an a d d i t i o n a l temperature dependence owing t o t h e temperature dependent n. E v i d e n t l y , t h i s w i l l l e a d t o a d i s t o r t i o n o f t h e shape o f t h e maximum i n t h e 6 vs. T curve. P a r t i c u l a r l y , t h e r e l a t i o n s h i p ( 5 ) w i l l no l o n g e r be v a l i d . We s h a l l c o n s i d e r t h i s d i s t o r t i o n , i n o r d e r t o l e a r n what i n f o r m a t i o n can be gained from i t as regards t h e concurrent processes o f p o i n t de- f e c t drag and p o i n t d e f e c t "evaporation" o u t o f t h e c l o u d round t h e d i s l o c a t i o n ( h e r e a f t e r r e f e r r e d t o as depinning). Below we summarize t h e r e s u l t s which p r o v i d e a b a s i s f o r determining t h e a c t i v a t i o n energies f o r these two processes. A more d e t a i l e d treatment o f t h e problem w i l l be given elsewhere (Y. ESTRIN and K. LUCKE, J. Phys. F: Metal Physics, t o be published).

I 1

-

THE EFFECT OF DEPINNING ON THE DAMPING MAXIMUM

The e v o l u t i o n o f t h e number o f p o i n t d e f e c t s w i t h t h e h e a t i n g t i m e can be described

~ d i s a t y p i c a l depinning time, o r t h e l i f e - t i m e o f a p o i n t d e f e c t i n t h e c l o u d round t h e d i s l o c a t i o n : a f t e r t h e time ~ d a p o i n t d e f e c t i n i t i a l l y trapped w i t h i n t h e c1 oud w i l l escape i t b y way o f thermal a c t i v a t i o n . The form o f Eq. ( 6 ) i m p l i e s t h a t

( i ) a l l p o i n t d e f e c t s w i t h i n t h e cloud, regardless o f t h e i r d i s t a n c e from t h e d i s l o c a t i o n l i n e , a r e counted as movable pinners;

( i i ) n i s l a r g e compared t o i t s thermal e q u i l i b r i u m v a l u e a t temperatures near t h e maximum, i . e . t h e i n f l u x o f p o i n t d e f e c t s from t h e b u l k i n t o t h e c l o u d i s negl i g i b l e .

Assumption ( i ) i s w e l l grounded p r o v i d e d t h a t t h e c h a r a c t e r i s t i c i n t e r a c t i o n r a d i u s between t h e d i s l o c a t i o n s and t h e p o i n t d e f e c t s ( d e f i n e d as t h e r a d i u s o f t h e r e g i o n where 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 i n t e r a c t i o n energy exceeds t h e thermal energy kT) i s o f t h e o r d e r o f b. Then t h e depinning time i s g i v e n by

where t h e a c t i v a t i o n energy f o r depinning, Hd, i s made up o f t h e b i n d i n g energy Vo and t h e a c t i v a t i o n energy f o r p o i n t d e f e c t m i g r a t i o n i n p r o x i m i t y t o d i s l o c a t i o n ( t h e l a t t e r q u a n t i t y being r o u g h l y equal t o Hr); ^T, can be evaluated as t h e i n v e r s e of t h e Debye frequency. For a spread c l o u d extending o v e r several i n t e r a t o m i c d i s - tances t h e parameters Hd and T~ would have a meaning o f some averages over v a r i o u s d i f f u s i o n paths w i t h i n t h e c l o u d l e a d i n g t o escape. The preexponential f a c t o r ^{T O} then acquires a temperature dependence ( Y . ESTRIN and K. LUCKE

,

J. Phys. F: Metal Physics, t o be published).

Assumption ( i i ) i s r e l e v a n t f o r a t y p i c a l experimental s i t u a t i o n where a p r e - t r e a t - ment o f t h e specimen (e.g. i r r a d i a t i o n o r p l a s t i c deformation) takes p l a c e i n t h e temperature range where p i n n i n g i s e f f e c t i v e . This leads t o accumulating p o i n t d e f e c t s on the d i s l o c a t i o n s and t o d e p l e t i n g t h e b u l k . Since t h e depinning stage l i e s above t h e corresponding p i n n i n g stage (owing t o t h e b i n d i n g 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 ) , i n c r e a s i n g t h e temperature and thus decreasing t h e thermal e q u i l i b r i u m c o n c e n t r a t i o n w i l l l e a d t o s u p e r s a t u r a t i o n i n t h e p o i n t d e f e c t cloud. The k i n e t i c s o f decay o f t h i s supersaturation, i . e . o f depinning, i s des- c r i b e d by Eq. (6).

Most common1 y, the i n t e r n a l f r i c t i o n i s measured when c o n t i n u o u s l y h e a t i n g t h e specimen w i t h a constant h e a t i n g r a t e h:

T = h = const. ( 8 )

Combining Eqs. ( 6 ) and ( 8 ) and i n t e g r a t i n g w i t h respect t o temperature y i e l d s

n / n i

-

exp {-x(kT/Hd)2exp(-Hd/kT)

3 ,

( 9 )

where

x =Hd/khr0, (10)

and n i denotes an i n i t i a l v a l u e o f n. Eq. (9) shows t h a t a t r a n s i t i o n from t h e pinned s t a t e , n / n i = l , t o t h e unpinned s t a t e , n / n i < c l , occurs i n a narrow tempera- t u r e i n t e r v a l around t h e depinning temperature Td, which can be d e f i n e d as t h e temperature whereldn/dTI e x h i b i t s a maximum. Both t h i s temperature and t h e c o r r e s - ponding depinning i n t e r v a l depend, through t h e dimension1 ess parameter x, on t h e h e a t i n g r a t e h. P a r t i c u l a r l y , w i t h i n c r e a s i n g h, t h e depinning temperature i n c r e a - ses.

One can e a s i l y recognize ( c f . Ref. 4) t h a t t h e shape o f t h e damping vs. tempe- r a t u r e curve e s s e n t i a l l y depends on t h e r e l a t i v e p o s i t i o n o f t h e unperturbed r e l a x a t i o n peak, associated w i t h p o i n t d e f e c t drag a t constant n (maximum a t Tr), and t h e depinning temperature Td. No damping due t o t h e mechanism under conside- r a t i o n w i l l be recorded i f Tr i s a p p r e c i a l l y l a r g e r than Td, w h i l e t h e unperturbed damping peak w i l l be observed when Tr<cTd. A n o n - t r i v i a l r e s u l t i s obtained i f t h e two temperatures l i e c l o s e t o each other. The damping maximum then s h i f t s towards lower temperatures and becomes sharper than t h e standard one. ( I t s h e i g h t remains unchanged, however). The e f f e c t o f peak sharpening can be v i s u a l i z e d as f o l l o w s . The peak temperature, Tm, i s again determined by wrr=l which y i e l d s

Here the dimensionless parameter

has been introduced, w i t h D denoting t h e preexponential f a c t o r i n t h e expression f o r t h e d i f f u s i o n constant

8:

It i s e a s i l y recognized from Eq. (1 1) t h a t Tm, considered as an imp7 i c i t f u n c t i o n o f w (through p ) and h (through x ) , increases w i t h b o t h w and h. Furthermore, t h e peak s h i f t s towards 1 ower temperatures as the r a t i o p=Hr/Hd o f t h e a c t i v a t i o n energies f o r t h e two processes i n v o l v e d increases. (Note, however, t h a t p cannot exceed u n i t y , as seen from t h e d e f i n i t i o n o f t h e two a c t i v a t i o n energies).

The h a l f - w i d t h o f t h e peak i s now g i v e n by

where t h e sharpening parameter

m o d i f i e s t h e expression ( 5 ) f o r t h e unperturbed r e l a x a t i o n peak. T h i s parameter decreases w i t h i n c r e a s i n g p ( a t f i x e d w and h ) i n d i c a t i n g t h a t t h e normalized peak h a l f - w i d t h AT/T, decreases w i t h i n c r e a s i n g p more r a p i d l y than the maximum

JOURNAL DE PHYSIQUE

I11 - DETERMINING

THE

ACTIVATION ENERGIES

I t should be emphasized t h a t the sharpening e f f e c t i s predicted in terms of t h e

t r u e a c t i v a t i o n energy

H r f o r point defect diffusion. Commonly, an

apparent a c t i - v a t i o n enerau

HE i s determined in experiment. The usual procedure of determinins

H;

consistsu;n heasuring the s h i f t of t h e maximum temperature on varying the f r e - quency a t a constant heating r a t e :

The apparent a c t i v a t i o n energy i s related t o the t r u e one by

and , f o r p#O, i s always an overestimate of t h e l a t t e r ( c f .

Eq.

( 1 4 ) ) . The degree of overe@imate can be q u i t e large. Take a typical numerical example of

H

/kTm=30 and x=10 (corresponding t o the heating r a t e of order of tenths of a degqee Per second). Then, with a reasonable value of p, e.g. from the interval (0.5,1), t h e r a t i o H P / H r will have the order of

10.

In terms of the apparent a c t i v a t i o n energy, Eq. (13) assumes the form

i .e. formally recovers the relationship ( 5 ) f o r t h e standard relaxation peak. This means t h a t the sharpening e f f e c t , however l a r g e i t may be, i s a

hidden

one: i t will never be detected i f one remains i n t h e framework of conventional analysis based on measuring the frequency dependence of the maximum temperature Tm only.

To reveal the peak sharpening e f f e c t , one has t o consider the dependence of Tm on t h e heating r a t e h a l s o . Calculating the derivative of T, with respect t o h from

EQ. ( 1 1 )

yields the followinq r e l a t i o n

where

a = 1.lkTm/Hr =

m(~b2/Do)(kT,/Cb) (L/b)ni.

1

In view of the f a c t t h a t t h i s quantity enters Eq. (18) logarithmically, and since

Hr/kTm

i s t y p i c a l l y l a r g e , an

estimated

value of

a

can be used in Eq. (18).

Even an order-of-magnitude e r r o r i n t h i s estimate would not give r i s e t o an e r r o r in the t r u e activation energy exceeding experimental e r r o r s . For the case of h-independent damping maximum, (alnTm/alnh),=O, Eq. (18) reduces t o t h e

relation

t h a t follows from t h e condition W'Cr=l f o r constant n .

Eq. (18) provides an expression f o r the sharpening parameter f

i n

terms of t h e derivatives of

Tm

with respect t o the frequency and t h e heating r a t e : comparing t h i s equation with Eqs. ( 1 5 ) , (16) y i e l d s

For the depinning energy

H

the following expression i s obtained by combining Eqs. ( I I ) , (14)-(16), and f18):

kTm/Hr-(alogTm/alosw)h Hd

=

H r

(

alogTm/alog

h ~ , ~

(19)

Together with Eq. ( 1 8 ) , t h i s equation provides a means of determining t h e activa-

t i o n energies f o r both processes involved from experimental data on the de~endence

o f Tm on IA and h. With t h e t r u e a c t i v a t i o n energy thus determined, t h e anomalous sharpness o f t h e damping maximum can be revealed,

The above c o n s i d e r a t i o n shows t h a t t h e s a l i e n t f e a t u r e of the damping maximum a f f e c - t e d by t h e depinning process i s t h e dependence o f i t s p o s i t i o n and shape on t h e h e a t i n g r a t e . Therefore checking whether t h e damping maximwn i s heating r a t e depen- dent i s mandatory f o r j u d g i n g whether t h e a c t i v a t i o n energy determined by means o f t h e conventional procedure i s t h e t r u e a c t i v a t i o n energy f o r t h e u n d e r l y i n g r e l a x a - t i o n process. This dependence being known, one can f i n d , w i t h t h e h e l p o f Eqs. ( 1 8 ) and ( 1 9 1 , b o t h t h e a c t i v a t i o n energy f o r p o i n t d e f e c t d i f f u s i o n and t h e a c t i v a t i o n energy f o r t h e i n t e r f e r i n g depinning process.

REFERENCES

/ I / SIMPSON H.M. and SOSIN A., Phys. Rev. 616 ( 1 9 7 7 ) 1489.

/ 2 / GRANATO A.V. and LOCKE K., Phys. Rev.

B24

( 1 9 8 2 ) 7007.

/3/ NOWICK A.S. and BERRY B.S., Anelastic =axation i n CrystalZine S o l i d s , Academic Press, N.Y. (1972).

/ 4 / LOCKE K., SCHNELL G., and SOKOLOWSKI G., i n : Internal F r i c t i o n and Ultrasonic Attenuation i n S o l i d s , Eds. R.R.Hasiguti and N.Mikoshiba, Tokyo U n i v e r s i t y Press ( 1 9 7 7 ) p . 99.

ACKNOWLEDGEMENTS

F i n a n c i a l support from t h e DFG i s g r a t e f u l l y acknowledged.