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INTERPRETATION OF SIGNATURES MEASURED IN THE BORDONI RELAXATION TEMPERATURE RANGE IN 5N AND 6N ALUMINUM USING A NEW
MODEL OF DISLOCATION MOTION
B. Quenet, G. Gremaud, M. Bujard, W. Benoit
To cite this version:
B. Quenet, G. Gremaud, M. Bujard, W. Benoit. INTERPRETATION OF SIGNATURES MEA-
SURED IN THE BORDONI RELAXATION TEMPERATURE RANGE IN 5N AND 6N ALU-
MINUM USING A NEW MODEL OF DISLOCATION MOTION. Journal de Physique Colloques,
1987, 48 (C8), pp.C8-125-C8-130. �10.1051/jphyscol:1987815�. �jpa-00227119�
INTERPRETATION OF SIGNATURES MEASURED I N THE BORDONI RELAXATION TEMPERATURE RANGE I N 5N AND 6N ALUMINUM USING A NEW MODEL OF DISLOCATION MOTION
B. QUENET, G. GREMAUD, M. BUJARD and W. B E N O I T
I n s t i t u t de Genie Atomique, Ecole Polytechnique F e d e r a l e de Lausanne, CH-1015 Lausanne, S w i t z e r l a n d
Rdsumd : U n e ktude de l a relaxation de Bordoni dans l'aluminium 5N et 6N est e f f e c t u k au myen de l a technique des ondes coupl&sl,2,4. Les signatures obtenues pour l'une e t l'autre puretks sont t & s differentes. I1 est toutefois possible de l e s interpreter avec un nouveau d & l e de muvement des dislocations, bas6 sur l a formation de paires de d6crochawnts. Dans cet article, on trouvera l e s bases du d e l e e t une c ~ r a i s o n qualitative avec les e e r i e n c e s . On sera conduit conclure ?I une relation entre l a relaxation de Bordoni observee dans les k t a u x cfc par frotterrwt interieur e t l e &canisme de creation a c t i d thermiquawnt de paires de decrochements.
A b s t r a c t : The two waves acoustic coupling mthodl,2,4 i s employed for the study of the Bordoni relaxation in 5 N and 6N aluminum. The masured signatures are completely different, when the purity changes. But it i s possible to interpret them with a new model of dislocation mtion based on the Kink Pair Fomtion (KPF)
.
In this paper,the basis of the model and a qualitative comparison with the expriments are presented. It w i l l be concluded t h a t the Bordoni relaxation observed in FCC metal3 by internal friction is related with the thermally activated KPF mchanism.A new model of dislocation mtion, only based on k i n k interactions, i s introduced here.It i s strongly supported by the results of experiments using the two waves coupling method.
Before entering the hypothesis of t h i s model, l e t us recall what "kinks" are. The crystalline materials present a periodicity whose consequence i s a periodic potential f e l t by the dislocation in i t s glide plane. In order t o minimize i t s shape energy, kinks are formed along the dislocation l i n e . ( f i g . l a )
The mxiel i s based on two simple hypotheses
-In an i n f i n i t e crystal, the formation energy of one k i n k on an infinite rectilinear dislocation is Ek,where Ek includes an elastic contribution and core effects.
-Kinks have e l a s t i c interactions.The interaction strength between kinks i s attractive, when kinks have opposite orientations (fig. 1 b ) ; i t is repulsive, when they have the same orientation (fig. lc)
.
The expression for this strength can be written as
where p is the shear m>dulus and b the Biirger's vector
For the calculation, it w i l l be considered that kinks are abrupt.
*
A detailed calculation w i l l be soon presented elsewhereArticle published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987815
C8-126 JOURNAL DE PHYSIQUE
Consickring the mxhanical equilibrium under stress of a dislocation w i t h kink pairs, the condition about the length L of the dislocation segwnt can be written
&+
2Z N-1tij
= L (2) (fig. ld) j=1(a)
This relation leads t o the following
F F , equation, sinply by considering the
(b)
'1
/ = a t t r a c t i v e and repulsive strength of the6 nearest neighkern kinks :
Fig.1 a) D i s l a 3 t h with kinks; b)l\ttracrion t e r m two kinks of -site s i q . 5 ; ~ ) Rqdsion b33een two kinks with the s a r e sign; d) W of dislocaticm with abnat kink
in which U i s the stress, N the number of kink pairs and Fo the attractive strength between the kinks of the f i r s t pair.
In figure 2, the theoretical behaviour of K(N) (Fo) i s presented. The l e f t member G of equation (3) does not depend on Fo ; in a given material, for a given s t r e s s Q and for a fixed value of L, G takes a unique value.When K(N1 (Fo) has intersections with G, a configuration with N kink pairs is possible. Because G i s an increasing function of 0, there is a minimum value of the s t r e s s under which the existence of a dislocation with N kink pairs is no longer possible. Then, i f (3 i s increased, the equation (3) is verified only for two values of Fo :F0(1) and Fo(2)
.
By considering what happens when there 1s a perturbation on the system, it is easy t o deduce that the smaller value Fo(1) appears for a stable conf iguration and the greater value Fo (2) for an unstable one. ( f i g . 3a)Fig..? Variation of K(n) (fO) for different values of N a given N f the area swept the dislocation i s given by ( f i g . 3b) :
Fo(1) and F0(2), solutions of the equation ( 3 ) , can be expressed in terms of p,a,b,L and 0, so t h a t the expressions for the area become :
The corresponding configuration enthalpy of the dislocation segroent i s given by H (N) = kink formation energy RNEkl
-
attractive potential energy of the first kink pair+
repulsive potential energy betheen the nearest neigfibour kinks-
work of the resolved stressrespectively
Fig.3 a) Stable and unstable configurations with N kink plirs;b) Area s e by the d i s l a z i c n ; ~ ) Craation of the Nth ldnk pair;d) m a t i o n of the W h kink PdLI
The creation enthalpy and the annihilation enthalpy of the Nth kink pair can then be calculated.
In order t o create the NUI pair, the dislocation has t o pass f m the stable situation with (N-1) pairs to the unstable situation with N pairs
.
(fig. 3c)The creation enthalpy can thus be written a s :
In order t o annihilate the N t h kink pair, the dislocation has t o pass f m the stable situation with N pairs to the unstable situation with N pairs. (fig. 3d)
The annihilation enthalpy of the Nth pair becomes :
For a given material, p and b are fixed; by considering only the kinks in compact planes, a is fixed too. To see the variation of
AH
..(N) and AH ,(N) with G, only two parameters can be changed : Ek and L.The behaviours of AH ..(N) and AH ,(N) a s a function of the s t r e s s are plotted in the upper part of figure 4 .
Now it i s possible t o consider the effect of the thermal activation on the creation and the annihilation of kink pairs.
The jump frequency
r
of creation o r annihilation depends on the temperature T, and for a given value ofA%
the enthalpy of one of these two mechanisms, it can be written a s :I t has then a sense t o assume that, i f an internal friction experiment is p e r f o d a t a measurement frequency
o,
the mechanism of creation or annihilation w i l l beccme e f f i c i e n t when the jump frequencyr
is a h s t equal t o the measurement frequency O:In other terms, one can say that, a t a given frequency o and a given temperature T, there e x i s t s one value of AH which s a t i s f i e s the preceding equation. This value can be introduced a s a normalized temperature r, depending on T, O and Ek:
C8-128 JOURNAL DE PHYSIQUE
When z equals )N(.. o r
A H
,(N),
one obtains respectively t h e creation o r t h e a n n i h i l a t i o n stress f o r t h e Nth pair. ( f i g . 4).
t h e function 2 ( T,O,Ek) is increasing
I
with t h e temperature, t h e following behaviours appear :-for a temperature TI, 'Q i s t o l o w t o i n t e r s e c t t h e AM,,(N) curves, in t h e ranqe of stress represented on f i g u r e 4, and no kink p a i r can be created; t h e d i s l o c a t i o n seqnmt is not mile
-for a temperature T2, 'Q i n t e r s e c t s s e v e r a l AH C r ( ~ ) and AH ,(N), but t h e c r e a t i o n and t h e a n n i h i l a t i o n f o r a given value N of k i n k p a i r do not occur a t the same stress; t h a t leads t o an h y s t e r e t i c m t i o n of t h e d i s l o c a t i o n segment
-for a temperature T3, Q does not i n t e r s e c t any curve; t h e kink p a i r s are created and a n n i h i l a t e d a t t h e t h e d y n a m i c equilibrium, which takes place f o r t h e s t r e s s given by t h e i n t e r s e c t i o n of t h e curves .,(N) and d B
an(^). In t h i s case, t h e motion of t h e d i s l o c a t i o n segment is purely r e v e r s i b l e . As t h e i n t e r n a l f r i c t i o n I F is associated
Fig.4 -K graphic: ewlmion of &(n) ard Wn with t h e a n e l a s t i c s t r a i n due t o t h e
(n) E O ~ &=o.i ev and L r 5 0 C b . m g r d c : n m of d i s l o c a t i o n motion, it i s given by the
rink pairs vcrsus the a r e s s . for r e l a t i o n :
t-ratures. IF
=I
Q.&= area inclosed i n t h e curve N ( 0 ) ( f i g . 4 )
It is c l e a r that ar TF relaxation, due t o t h i s ~ c h a n i s m of KPF, has t o appear as a function of the t e n ~ r a t u r e T, or the frequency w.
Aluminum2,3
,
a s t h e o t h e r FCC m t a l s ,2 presents a t low temperature an i n t e r n a l
f r i c t i o n peak which i s c a l l e d t h e Bordoni r e l a x a t i o n . ( f i g . 5).The a c t i v a t i o n energy
1 EB of t h i s relaxation can be measured.
From t h e model described before, one can deduce t h a t t h e peak which corresponds t o
0 t h e KPF mechanism has an a c t i v a t i o n energy
o 100 T[K] Ead depending on t h e kink formation energy
(a) Ek
F i g . 5 mldoni relaxation m a s u r e d m a swle w16 E a e =
f
( Ek) (13)uorkcd 0.5% a t 1Ok:a-1.7 H Z .
If it is assumxi t h a t the Bordoni relaxation observed in aluminum is really the KPF mechanisn relaxation, then t h e r e is a unique r e l a t i o n between Ek and E g :
E k = f - l ( E ~ ) ( 1 4 )
A s Ek t a k e s a f i x e d value, t h e &el has f i n a l l y only one parameter t h a t can be changed: the dislocation length L.
w i l l be now c a p r e d q u a l i t a t i v e l y w i t h those which
can
be expcted f r a n t h e W F model described in the formalism of the abrupt kinks.The
two
waves coupling methodl,2,4 amsists~ a [ d ~ l p s ] of applying simultaneously on a sample a
l o w frequency hanmnic stress qE and T E 84.7K
o.mL-.
5N A' * ultrasonic longitudinal waves. The attenuationa
of these waves is nrasured and t h e closed curves A a ( *) which area [kglcm21 Obtained are called "signaturesn.
0 Before applying
*,
the sample i s sutrnitted0 10 20 t o a cold-working stress %. When t h i s
(a) stress is m v e d t h e r e is a remaining
i n t e r n a l stress
oil
which has t o be in the same magnitude range a s %, but with an opposite sign.Since and have the same sign, oi i s n e g a t i w when positive values of-
are considered.A t temperatures which correspond t o the rnaxinrm of t h e Bordoni peak in aluminum, u[kg1cm2]
-
f o r t h e frequency 6l of t h e applied s t r e s s 0 10 20-,
t h e e x p e r h t a l r e s u l t s which a r e(b) obtained in SN and 6N aluminum samples are
presented in figure 6
Ng.6 a1 Signature observed on a 5N A1 -1e subnitfed to QflW kgf/unki.EZ-~~; bl Signature
&served on a ffl A1 smple subnitted to cbpMkg€/* .3E-4&
the
p
One a l s o deduces t h e expression of Aa
,
t h e variation of t h e attenuation of the u l t r a s o n i c waves, from t h e model previously presented; it is proportional t o :In which % = oi
+
om represents t h e resolved applied s t r e s s f e l t by the dislocation segment andA
is t h e area swept by t h e dislocation.With E k fixed by t h e r e l a t i o n (14), and by choosing L, t h e curves
A M c r ( ~ ) (a),&# ,(N)
(o)
and[d
st (N) /do12 can be simulatedBecause p o s i t i v e and negative s t r e s s e s have t o be taken i n t o account, t h e graphics a r e presented symnetrically in figures 7 and 8.By considering t h e temperature T of the expriments, the value of T can be calculated and shown in figures 7 and 8
C8-130 JOURNAL DE PHYSIQUE
Fig.7 -0.1 eV fig.8 4.1 e~
L -1- L - 2 m
W i t h L =L1 = 1000 b ( f i g . 7 ) , t h e i n t e r n a l stress seen by the d i s l o c a t i o n when qE equals zero is chosen here s t r o n g enough t o form kink p a i r s . By increasing the c y c l i c stress amplitude, one begins t o a n n i h i l a t e t h e previous kink p a i r s , and nothing mre! Kink p a i r s can not be c r e a t e d in t h e d i r e c t i o n of applied stress
-
p o s i t i v e d i r e c t i o n -because t h e gap bet- the c r e a t i o n s of t h e f i r s t kink pair, r e s p e c t i v e l y in t h e negative d i r e c t i o n and t h e p o s i t i v e one, i s t o o b i g anpared with t h e amplitude o f qE. By decreasing t h e applied s t r e s s , kink p a i r s in t h e negative d i r e c t i o n can again be created. The corresponding v a r i a t i o n of
[d
/*, then ofAa, can be followed on t h e lower graphic of f i g u r e 7, and presents e x a c t l y t h e f e a t u r e of t h e s i g n a t u r e obtained i n t h e 5 N aluminum sample ( f i g . 6.3)With L =Lg =2000 b = U1 ( f i g . 8), one can see that t h e n h r of kink p a i r s t h a t can be c r e a t e d and a n n i h i l a t e d a t given temperature and stress range i s much g r e a t e r than ,in t h e previous case; and t h e "gap", a s defined before, is s m l l e r . T h e r e l a t i o n L2> L1 has been chosen because it i s e x p c t e d that t h e d i s l o c a t i o n segments i n an aluminum sanple with a higher p u r i t y a r e longer. Then in order t o do a c a p a r i s o n with t h e r e s u l t of t h e experiment done on a 6N aluminum sample, one has t o consider now an i n t e r n a l stress less negative than previously, because t h e cold-working stress % e x p e r h e n t a l l y used i n t h i s case was less important. By increasing t h e applied stress t o t h e same value a s in t h e former case, it is now p o s s i b l e t o c r e a t e kink p a i r s in t h e p o s i t i v e d i r e c t i o n , a f t e r t h e a n n i h i l a t i o n of those which w e r e perhaps formed in t h e negative one. The v a r i a t i o n of [& /do12 i s t h e r e f o r e very d i f f e r e n t , but t h e f e a t u r e of t h e s i g n a t u r e experinrentally obtained with t h e 6N aluminum sample can be recognized ( f i g . 6b)
.
The two waves coupling experiments g i v e a c o n f i m t i o n of t h e hypothesis t h a t t h e Bordoni r e l a x a t i o n is r e l a t e d t o t h e KPF mechanism5, since t h e s i g n a t u r e s obtained a t corresponding temperatures, with t h e i r s o d i f f e r e n t f e a t u r e s when t h e p u r i t y of t h e sample changes, a r e very w e l l explained by t h i s mechanism,when described i n t h e f o m l i s m of t h e abrupt kinks.
Acknowledgment
This work was p a r t i a l l y supported by the Swiss National Science Fondation.
References
(1) G.Gremaud, M.Wljard, W. Benoit;J.Appl.Physics,fil (1987) 5.
(2) G . G m u d , W i s (Ecole p l y t e c h n i q u e F&i&rale de Lausanne, Switzerland, 1981)
(3) C-Esnouf, thesis (UniversitB Claude B e r n a r d Lyon I, France, 1978) (4) M.Eujard, thesis (Ecole Polytechnique F e a l e de Lausame,
Switzerland, 1985)
(5) G.Fantozzi, C.Esnouf, W.Benoit, I .G.Ritchie; Progress in Material Science
