INTERNAL FRICTION ASSOCIATED WITH THE RE-SOLUTION AND PRECIPITATION OF THE θ - PHASE ALONG THE GRAIN BOUNDARIES AND IN THE MATRIX OF ALUMINIUM-COPPER ALLOYS

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INTERNAL FRICTION ASSOCIATED WITH THE RE-SOLUTION AND PRECIPITATION OF THE θ - PHASE ALONG THE GRAIN BOUNDARIES AND IN

THE MATRIX OF ALUMINIUM-COPPER ALLOYS

P. Cui, T. Kê

To cite this version:

P. Cui, T. Kê. INTERNAL FRICTION ASSOCIATED WITH THE RE-SOLUTION AND PRECIP- ITATION OF THE θ - PHASE ALONG THE GRAIN BOUNDARIES AND IN THE MATRIX OF ALUMINIUM-COPPER ALLOYS. Journal de Physique Colloques, 1987, 48 (C8), pp.C8-417-C8-422.

�10.1051/jphyscol:1987863�. �jpa-00227167�

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JOURNAL DE

PHYSIQUE

Colloque C8, supplbment au n012, Tome 48, dbcembre 1987

INTERNAL FRICTION ASSOCIATED WITH THE RE-SOLUTION AND PRECIPITATION OF THE

8

- PHASE ALONG THE GRAIN BOUNDARIES

AND

I N THE MATRIX OF ALUMINIUM-COPPER ALLOYS

A

P.

C U I

and T. S. KE

Institute of Solid State Physics, Academia Sinica, Hefei, China

A b s c r a c c Low f r e q u e n c y i n ; e r n a l frLc::on a n d e 1 a s : i c m o d u l u s measure men;^

w e r c made wi:h a l u m i n i u m c o n : a i n i n g 0 . 0 1 5 L O 1 . 2 w: % Cu. ( 1 ) Wnen Cu con:enn i s l o w e r :;Ian 0 . 5 wt %, a n i n z e r n a l f r i c ~ i o n p e a k P l a p p e a r s a i a l o w e r L e m p e r a t u i e ::Ian t h a t o f :he g i - a l n b o u n d a r y p e a k o b s e r v e d i n p u r e a l u m i n i u m . ( 2 ) When c o p p e r conZen: r e a c h e s 0.5 wc "L, P l a p p e a r s i n m e a s u r e m e n t s wi:n d e s c e n d i n g ;empera:ures.

However, t n i s p e a k i s l o w e r e d c o n s i d e r a b l y when m e a s u r e d w::h a s c e n d i n g cempera:ures a n d a n o c h e r p e a k P 2 a p p e a r s a r o u n d 3 6 0 % a n d i s i n d e p e n d e n : o t f r e q u e n c y b u ~ v a r i e s w i t h c o p p e r c o n p e n t , t h e way o f c h a n g i n g c e n p e r a t u r e s , a n d a g e i n g L e m 2 e r a t u r e a n d c i m e . Tne e l a s c i c m o d u l u s d r o p s a b r u p t l y c o r r e s p o n d i n g wi:h :he a p p e a r a n c e o f

%.

^{( 3 )}

When c o p p e r c o n t e n : r e a c h e s 1 . 2 wt %, a p e a k P 3 a p p e a r s a r o u n d 2 9 0 ' ~ when m e a s u r e d w l t n d e s c e n d i n g :empcra:ures. The b e h a v l o r o f P3 i s s i m i l a r t o t n a : o f

9,

e x c e p t t h e e l a s t i c m o d u l u s r i s e s instead of d r o p p i n g c o r r e s p o n d : n g w i t h t h e a p p e a r a n c e of P 3 '

Expe.riments wi:h s p e c i m e n s c o n : a i n i n g o n l y a few bamboo b o u n d a r i e s a n d X-ray a n a l y s i s show :ha: t h e a p p e a r a n c e o f P, a n d P 2 L s o r i g i n a c e d respectively f r o m t h e r e - _L s o l u t i o n a n d ;he p r e c i p I : a t i o n of 0 - p h a s e a r o n g t h e g r a i n b o u n d a r i e s .

E x p e r i m e n c s wi:h A1-4 wt % show t h a : P' ( 4 3 0 ' ~ ) a n d P ' ( 4 1 0 ' ~ ) p e a k s a s s o c i a t e d

2 .

w i t h t h e r e - s o l u t i o n a n d p r e c i p i ~ a t i o n o f O-pilase i n s i d e

2,e

g r a i n s p r o p e r a p p e a r L o g e t h e r w i t n p e a k s P a n d P

2 3 '

I. I n t r o d u c t i o n

S i n c e t h e d i s c o v e r y of t h e g r a i n b o u n d a r y i n t e r n a l f r i c t i o n p e a k o f 9 9 . 9 9 a l u - minium by KG I n 1947 [ I ] , n u m e r o u s r e s e a r c h w o r k s h a v e b e e n d o n e on t n e e f f e c : of a l l o y i n g e l e m e n t s on : i ~ L s ; n : e r n a l f r i c t i o n p e a k [ 2 ] . T h i s p e a k was f o u n d 2 0 s n i f : i o l o w e r c e n p e r a t u r e s t o r l e s s p u r e 9 9 . 6 % alum:n<um [ 3 ] . However, r e c e n : r e s u l t s show ;ha: :he op:imum :enpera:ures f o r :he g r a i n b o u n d a r y p e a k o b s e r v e d I n 9 9 . 9 9 , 9 9 . 9 9 9 a n d 9 9 . 9 9 9 9 h i g h - p u r i : y a l u m i n i u m a r e r e s p e c : i v e l y 2 9 0 , 270 a n d 210UC r e f e r r i n g

;o f r e q u e n c y = 1 Hz [ 4 ] . T h i s i n d i c a : e s :hat :he pea^ s ; ~ i f t s t o n i g n e r L e m p e r a i u r e s f o r l e s s p u r e s p e c i m e n s . Sucn a contradiction o f r e s u l t s may r e f l e c t t h e f a c : :na:

:he s:a:e of t h e a l l o y i n g e l e n e n t s a: :ne q r a l n b o u n d a r i e s , i n s o l i d so1u:Ion o r i n p r e c i p i t a ~ i o n s t a t e , may be c r u c i a l i n a f f e c t i n g :lie g r a i n b o u n d a r y : e l a x a t i o n p r o - c e s s g i v i n g r i s e i o :he i n t e r n a l f r i c c l o n p e a k . A f o r w a r d s ~ e p was made by Q u a d e r [ 5 ] ;n 11;s i n ~ e r n a l f r i c i i o n s:udy on a l u n l n L u m a l l o y s wi::~ 2 a n d 2 . 9 5 w: % Cu. Two t r a n s i c n : I n c e r n a l f r i c t i o n p e a k s w e r e o b s e r v e d : o n e p e a k i s c o n n e c c c d wi:n p r o c e s s of p r e c i p i t a t i o n from s o l i d so1u:ion a n d :he o t h e r Ls c o n n e c z e d wi:n i h e p r o c e s s of r e - s o l u t i o n of :he p r e c l p i : a t l o n s : a i e . I n :he p r e s c n : r e s e a r c h , a s y s t c n a ; i c s t u d y a l o n g : h i s l ; n e was made w i c k l o w e r ( 0 . 0 1 5 :o 1 . 2 wi %! a n d h i g n e r ( 4 w: %) c o p p e r con:cn;s i n a l u m i n i u n wi:n :he p u r p o s e o f c 1 a r ; f y i n g :he c o n t r a d i c t o r y r e s u l ~ s r e - p o r t e d I n :he l i c e r a t u r e . X-ray s ~ u d i e s w e r e made I n c o r r e 1 a : i o n w i t h i n t e r n a l f r : c - t i o n a n d e l a s ~ i c m e d u l u s m e a s u r e m e n t s .

TI. The S p e c i m e n s

The s p e c i m e n s u s e d w e r e s u p p l i e d by Fusnun Aluminium P l a n : o f C h i n a wiLk compo- s;:ion shown i n T a b l e 1 . The 3-mn r o d s r e c e i v e d w e r c a n n e a l e d a t 550°C f o r 8 5 ,

T a b l e 1 . C o m p o s i t i o n o f t h e A1-Cu a l l o y s

S p e c i m e n s No. 1 2 3 4 5 6 7 8

Cu con:en:(wt%! 0 . 0 1 5 0 . 0 4 5 0 . 1 3 0 . 2 9 0 . 5 1 0 . 8 0 1 . 2 4 . 0 w a t e r q u e n c h e d a n d t h e n c o l d - d r a w n t o 1 mm wLre a: room t e m p e r a c u r e . 'The 1eng:h of

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987863

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:he specimens used in lnternal fric:lon and elastic modulus measurements is 100 mm.

Measurements were made in an inver:cd cors:on pendulum. X-ray stud:cs were madc witn Acomat<.c Power 1700 Diffrac:ometer made in Philips, Holland.

111. E x ~ e r i m e n t i a l results

1. t1easuremen;s on specimens witn low Cu conten:

In Fig. 1 are shown the variation of internal friction and elastic modulus with temperature for specimens with copper contents o f 0.015, 0.045, 0.13 and 0.29 w: % (corresponding :o curve 1 to 4). The specimens were annealed in-s;:u in the torsion apparatus at 450°C for 2 h. An in:ernal fric:ion pea^ P appears for all specimens whether :ne measurements were taken witn descending or ascending tempera;ures. Tne 1 elasYcmodulus drops considerably wi:h the temperature range of ;ne internal frictior.

pea^. The ac'ivation energy a s s o c i a ~ e d with this pea^ a s determined by changing the frequency of measurement is 3 2 kcal/mol, which is stmilar to tha: for 99.99 aluminium. Tne height of the peak is somewflat lower(0.07 vs 0.09) and does no: change by age:ng. The optimum temperazure of :he peak which is abouc 275'C with

E

= 2 Hz seems no: to change with the copper content, nu; i: is d e f i n i ~ e l y lower than that of 99.99 aluminium. This agrees with the resul: by Kg [3] for 99.6 % aluminium. The apparen:

independency of the optimum temperature with Cu concent may be due to the superposi-

- :

L ~ o n of two different effects: ( I ) For lower Cu contents which can only form solid solucion at the grain boundaries, :he peak will shift to higher temperatures wi;n an increase of Cu conten:. (2) For higlier Cu contents which may cause precipitation at the grain boundaries, the peak will shift :o lower temperatures with an increase of the Cu content. The Cu content ranging from 0.015 to 0.51 wt % may well be ihe demr- cation of :he lower and :he higher Cu contents giving opposite effects.

T('C)

450 350 250 I 50

Fig. 1. Internal friction and elastic modulus curves of specimens wi:h various C u contents. Curve 1 to 4: 0.015, 0.045, 0.13 and 0.29 w: %. f = 2 H z .

Fig. 2. Effec: of various kinds of heat Zreatments on the internal fric::on a n d elastic modulus of the specimen con:a:ning 0.8 w: % of Cu annealed at 450°C f o r 2 h. Curve 1: Measured wi:n descending :empera:ures. Curve 2: Measured witn ascendLng temperatures and ageing at 270'~. Curve 2': Measured w;:n ascending :em- peratures after an ageing at 270% for 3 h. Curvc 1 , 2, 2': WL:b f = 2 Hz. Curve 3 and 3': Heat :reatmenis :he same as :ha: for curve 2 and 2 ' , ou: measuredwithE=O.Sllz.

2. Eeasuremen:~ on specimens wl:k medlum Cu conZen:

Tnc internal friction and modulus curves for :he specimen con:a;ning 0.80 wt % of C u a r e shown in Fig. 2. Af:c; :he specimen was annealed a: 450°C for 2 ;i and aea-

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surements taken wi:h descending :cmpera:ures, only peak Pl appears a s shown by curve 1 (f = 2 8 2 ) . The height of :he peak is only a 1i:tle h i : lower than that shownin Fig.l, Indicating :hat prec5pI:acion has no; taken place very much when the specimeniscoolec' down from 4 5 0 ' ~ co P l p c a k temperature. Af:er :he spectmen w a s cooled down :o room :empers:ure and measurrnentc:akenwith ascending tcmpcraEurss, rhe heigh: of P peak is considerably reduced a s shown b y curve 2 . This is evidently due to che progress of 1 precipi:a:ion at grain boundarieswhen the specimen w a s cooled down to room :em~era- :!!re and heated to peak :empera:ure. The con:inuous lowerlng of P I d u r i n g an ageIng at 2 7 0 ' ~ is also shown b y curve 2 . This indicates :ha: 0.8 w: % Cu is much higher :ban ihe solubili2y of Cu in A1 at 2 7 0 ~ ~ . A saturacion value of the heigh: of is reached after an ageing at 2 7 0 ' ~ for 3 h. When meas:uemen;s are :aken again f r o m r o o m cemperacure, curve 2 ' shows tha: :hc hcign: of Pl is reduced very much and L:s 1oca.-

<:on sn;f:s to a lower :empe:a<urc, and a peak P, appeals around 390°c.. The modulus drops abruptly corresponding wi:h the appearance of L Q

.

When measured wi:h a frequency of 0 . 4 H z , :he locarion of P2does no: cnange nu: :hat of P.shif:s to a lower temperature. Thts i n d i c a ~ e s :ha: P2is no: a zhermal-ac:iva:ed relaxation process bu:

P,is. Similar behavior of PI is shown by comparing curve 3 w;:n curve 2 which are caken rcspec::vely wirh f = 0 . 4 and 2 Hz. I: is seen that :he op:imum t e m p e r a ~ u r e of PI shown by curve 3 ' Is abou; 227 OC which is ',den",cal wich :ha: previously observed w i ~ h 9 9 . 2 :o 99.6 % aluminium[3]. The response of peak

q,

to :he described heat treat- men: indicates :hat 1: is :he s:ablc grain boundary precipitation peak, of whlch :he

lower :he helgh: is, the more the amount of precipi:ation at :he grain boundaries.

The P l a n d

q

peaks also appear in the specimen con:ainLng 0.51 w: % C u , and :heir behavior is slmilar :o that in :ne specimen concalning 0 . 8 w: % Cu. The reduction of

;he heigh: of P l a n d &?if; of ;:s 1oca:ion to lower cempera:ure is less. And

5

^appears

around 360°c which is lower than :he 3 9 0 ' ~ for che 0 . 8 w: % Cu specimen.

3. measure men:^ o n specimen con:a:nlng 1.2 w: %

For :11e A1 - 1 . 2 w: % C u specimen annealed a: 450°C for 2 h and measurements taken wi:h descending :empera:ures(f = 2 H z ) , a peak

8

appears a t abouc 290°C in add- tion to peak P,(curve I of Pig. 3 ! . During :he ageing at ::?e o p ~ i m u m temperature of $, the Internal friction decreases and modulus increases gradually a s shown by curve 2.

The increase of modulus during ageing indica:es that a hardening process cakes place dur;ng agelng. This is evidenzly a process of precipitat:on hardening. Afzer an ageing of 2 h and xeasurements :aken with ascending temperazures,

%

peak disappears and P,, P2pcaks appear a s shown by curve 3. When measured wlth a frequency of 0 . 4 i l z , :he 1oca:lon of P2and

9

does no: change bu: :hat of PI shif:s :3a lower :emperaturc a s

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Pig. 3. Effect of various kinds of heat treatments on the internal friction and elastlc modulus oL :he specimen containing 1.2 wt % Cu annealed a: 450'~ for 2 h. Curve 1: Measured with descending temperatures. Curve 2: Specimen aged again at 4 5 0 ' ~ for 0.5 h and measured with descending temperaiures and ageing at 3 0 0 ' ~ for 2 h.Curve 3: Specimen furnace-cooled from 450'~. measured with ascendicg zernpera- tures. Curve 1-3: With f = 2 Hz. Curve 4, 5: Heat treatment :he same as that for curve 1 , 3, but measured with f = 0.4 Hz.

Fig. 4. Internal friction and elastic modulus curves of "single crystal" Al- 1.2 wt % Cu specimen. Curve l: Annealed at 4 5 0 % for 2 h and measurements taken with descending temperatures. Curve 2: Measured wieh ascending cempera:ures,f = 2Hz.

shown by curve 4 and 5. This indicates cha: both peak P2and

8

are no: relaxation peaks.

Auxiliary expertments show tna: before the grain boundary precipi2atlon reaches its saturated state, Pland P2are complementary in that one raises while o:her drops by differen: age;ngs.So P2is connected w::h :he process of re-solution of :he gra;n bouw- dary precipitates. On ihe contrary, Pjis connected wi:h the process of precipi:a:;on to the grain boundaries and the height of PI is reduced accompanying wiih the appearance of PS

"Single crystals" of s p e c i m ~ n s containing 0.8 and 1.2 wt % Cu were prepared by static and dynamic strain-annealing me:hod. The "single crys:all' grown contains only 4 bamboo boundar;es in a l e n g ~ h of 100 mm. Measurements were caken under idencralcon- d;tions as ihosc for t h e fine grain-sized specimens. No internal friction peaks(q,P2, P3) were observed for such specimens. The internal friciion and e l a s ~ i c modulus curve of A1-1.2 wt % Cu specimen are shown in Fig. 4. These resulcs demonsirate conclusively tnat peaks P,, P,and P,are all connected with the grain boundaries in the sDecimens.

a L

-

4. Measurements on specimens wtth 4 wc % Cu conten;

For an A1-4 w: % Cu specimen annealed a? 550'~ for 2 h. onlv ~ e a k _,

.

P-aooeared J

..

when measured with descending temperatures (curve 1 of Fig. 5), and P;,P2 appear when

Pig. 5. Internal friction ^{T ('C}⁾

5% 4% 350 2 5 0 150

and elascic modulus curves of I I

polycryscalline and "single crystal" A1-4 wc % Cu specimens.

Curve 1, 2: Polycrystalline specimen annealed ai 550'~ for 2 h and measurements :aken respect:vely with decend:ng and with ascending

:empera:ures. Curve 3, 4: Same as curve 1 , 2 , but for "single crys - tal" specimen. f = 2 Hz.

5 measured with ascending tempera- -

tures again (curve 21. For "single '4 crystal" specimen wich a few bamboo boundaries, a new peak Pf 3 appears when measured wl:n dcscen- ding temperatu,:es (curve 3:, and 2 another peau Pd appears whenmea- sured with ascend;ng temperacures I (curve 4:.

It is to be noced :ha:: (1) _I2

Correspondfng to :he appearance lA ¹⁶ ^I&_IO?T(%)^2.0 ^2.2 ^2.4 of P2(:he re-solu~ion peak asso-

ciated with the re-solution process caking place at grain boundaries!, che elastic modulus decreases as shown in Fig. 2, 3, and 5. The elasclc modulus decreases cori-es- ponding to the beginning of the emergence of Pi foi :'single crystal:' spectmen when measurements are taken in ascending cemperacures (curve 4 of Fig. 5). Consequently, we believe tnat Pi is associated w i t i t che re-solution process taking place in ihe

interior of :he grains. 1: is seen chac both T a n d Pi appear in :he case of polycrys- tals and Pi appears at a lower temperature :han thai o f

9.

This i n d i c a ~ e s that the re-soluiion process scarts earlier at a lower :emperature. (2) Corresponding to the appearance of Pj(the preclpitacion peak associated wizn :he precipi:ation process

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Faking place at grain boundarles),:he elas:ic modulus increases corresponding :o the appearance of Pj for "single crystal" specimen whcn measuremen:s are taken in descending tcmpera~ures (curve 3 of Fig. 5). Consequenrly, we believe that

3'

^{is asso-}

ciated with the precipitation process taking place in the inzerior of the grains. It is seen tha: P appears ac somewna: higi1e.r ,:ern?erarure than tha: of P;'. This indi.ca:es that the precigi:ation 3 process starts earlier at the grain boundaries at a higher rempera:ure.

5 . Results of X-ray diffraction analysis

The A1-1.2 w: % specimen was annealed at 450°C for 2 h, furnace-cooled down to room cemperacurc and heated up :o 290°C, and then aged a: 290°C for 7 h. This :emperature corresponds to :he peak remperature of P3 (:he precipi;a:ion peak) as shown by curve (a) of F<g. 3. X-ray diffraction analysis of this specimen at 200C shows

Fig. 6, X-ray dlffrac:ion graph of A 1 - 1.2 wt '% Cu specimtn.(a) Specimen a g e d at 290°C for 7 h and analysis made a c 20 OC. (b! Specimen heated in-situ :o 300% from 20°C. ( c ) Specimen heated in- sicu to 4 0 0 % from 300°C. The six srrong- es: diffrac:;on peaks of 0-phase are mark- ed wi:h 1-6. The two strong peaks are K, peaks of aluminium.

Fig. 7. Variat<on of che 0 diffrac- :ion peak (d-4.30298,(110! ) of A1-1.2 w:

X C u specimen w;:h ageing a: dlfferenr temperacures for various rimes. (a):Grow- :h of :he Q peak during ageing a: 3 0 0 ' ~ (20 -c 300°c?. (b)Disappearance of the 8 peak during ageing at 400'C(300 4 400'~).

(c) Grow:h of chc 0 peak during ageing a: 300°C (400 -+ 300°C).

i:~ac ;he six strongest diffraciion peaks of the 0 pnase (CuA12\ all appear. After rhe speccaen was h e a ~ e d in-si:u of the X-ray diffrac:oae:er to 300°c, :he dlffrac- Lion peaks o f 8 phase do no: cnange. However, w;?en :he specimen was heated from300°C

L O 4009: which corresponds L O :he peak Cemperacure of

9

( z i ~ e re-soluiion peak) as shown (c) of Flg. 6, all :he d i f f r a c ~ i o n peaks of 0 phase alsappear.

In-si:u observation was made on one strong diffracrion pea^ of :he 0 pnase (d = 4.3029

1 ,

(110)) during ageing a: differen: temperatures for various :imes. I: is shown in Fig. 7(a) tnat tor a solut:on-:reared Al-1.2 w; % Cu specimen water - quenched :o room zemperacure and :hen hea:ed ;n-sicu :o 3 0 0 ' ~ (which rakes abou: 1 nin), the 0 peak appears af:er an agefng o f 9 min. The specimen was kept In 3 0 0 ' ~ for 2 i7 and :hen hearcd LO 4 0 0 ~ ~ (in 1 min), :he 0 peak drops rapidly and disappears afzer an ageing of 6 m;n as shown by Fig. 7 ( b ) . When the specimen was cooled down from 400 to 300°C (in I min) and aged :here for 12 min, :he 0 peak appears gradually and raises 20 its naximus neigh: afzer 2 h as shown by Fig. 7(c).

The resu1:s of X-ray analysis si~ow tna:: (1) The condltion for he appearance of Ppis tha: t;le 0 phase has exis:cd already in chc specimen. The 0 phase disappears gradually accompanying thc appearance of P 2 . A s such, P2Ls connected wicn che re-solution of :hc 0 phase. ( 2 : The condition tor the appearance of P3 is :ha: :nc specimen was solu::on - created ( without 0 phase ) . And 0 phase forms gradually accompanying :he appearance of P3. S o P J i s ;he reverse process of P, _L

.

^{I i}^is

conneczcd w i ~ h :he precip;;a:ion of 0 pnase

.

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VI. Concluding Remarks

I n con:rast wi:h the conventional grain boundary internal friction peak ( KG pcak 1, f;ve internal friction peaks were observed in A1 - Cu alloys under different conditions. Arranged in ascending order of :heir optimum tempera:ures, :ney are: PI, P;, pi, P3and P2.

( 1 ! P1 is the grain boundary peak ( a relaxation peak ) wi:h prec;pi:ates ac the grain boundaries.

( 2 ) Pi is :he re-su1u:ion peak associated w;th the re-so1u::on process :aking place in :he interior of the grains.

( 3 )

Pj

is the precipita:ion 2eak associa:ed :he precipitation process LakLng place in the interior o f the grains,

( 4 ) P3 is the precipi:ac;on peak associated wich the precipitation process taking place a: the grain boundaries.

( 5 ) PZ is :he re-solui.ion pcak associated w;ch the re-solucion process taklng place at the grain boundaries.

I n addition, there should be a grain boundary 'nrernal f-:c:'o,i pea4

;

a relaxation peak ) with solute atoms existing in the solid solution state at 'the grain boundaries. Such a peak ( the solu:c 2ra:n boundary inrernal fr;.c:-on peak ) should ap7ear in aluminium spec:mens with Cu content less :han that studied in :he present research (0.015 W L "L C < I > .

Acknowledgement

Thanks are due to B. P. Cheng, Y. M. Gao tor help in X-ray experiments, and to Q. Huang and other members of Internal Friction Laboratory of the Lns:itu:e for help in experimental work and discussions.

References

[I]

T. S. KC, Phys. Rev. 71. 533(1949).

[2] Cf. for example: H. Gleiier and Chalmers, High-Angle Grain Boundaries, Progress in MacerLals Science, Vol. 16(Pergamon Process, 0xford.New.Yor~.

Toronto. Sydney. Braunschweig, 1972!, Chap. 8, pp.219-254.

[3] T. S. K C , J. appl. Phys. 20 274(1949).

[4] T. S. KC, P. Cui, S. C. Yan, and Q. Huang, phys. stat. sol. (a) 86, 593(1984).

[ 5 ! M. A . Quader, J. appl. Pnys. 33, 1922(1962).