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INTERNAL FRICTION AND VISCOSITY
ASSOCIATED WITH MOBILE INTERSTITIALS IN THE PRESENCE OF A KINK HARMONICALLY OR
UNIFORMLY MOVING IN ANISOTROPIC BODY-CENTERED CUBIC METALS
T. Ogurtani, A. Seeger
To cite this version:
T. Ogurtani, A. Seeger. INTERNAL FRICTION AND VISCOSITY ASSOCIATED WITH MOBILE
INTERSTITIALS IN THE PRESENCE OF A KINK HARMONICALLY OR UNIFORMLY MOV-
ING IN ANISOTROPIC BODY-CENTERED CUBIC METALS. Journal de Physique Colloques,
1983, 44 (C9), pp.C9-639-C9-644. �10.1051/jphyscol:1983996�. �jpa-00223446�
INTERNAL FRICTION AND VISCOSITY ASSOCIATED WITH MOBILE INTERSTITIALS IN THE PRESENCE OF A KINK HARMONICALLY OR UNIFORMLY MOVING IN
ANISOTROPIC BODY-CENTERED CUBIC METALS
T . O . O g u r t a n i and A . Seeger
Max-PZanck-Institut ftLr MetaZlforschung, Institut far Physik, Stuttgart, F.R.G.
and Middle East TechnicaZ University, Ankara, mrkey
Abstract.- The power dissipation due to mobile octahedral interstitial s in the stress field of a harmonically oscillating or uniformly moving kink of a dislocation line is derived for anisotropic BCC metals using the discrete Fourier transformation of the elastic Green's function. The viscosity and the drag force acting on the kink moving uniformly along the dislocation line are formulated which shows strong velocity and the temperature dependent behaviour.
1. Introduction.- In a previous paper
/ I /the present authors have successfully developed a unified linear-response theory of internal friction due to movements of interstitial impurities in the presence of periodically time dependent but spatially inhomogeneous stress field excitations in terms of discrete k-space Fourier trans- formation for a BCC lattice. This specific problem of energy dissipation is closely related to the atomic migration of point defects in time dependent, spatially fluc- tuating fields which has been studied extensively by Seeger and Hornung /2/ and Hornung /3/ using the conventional Fourier transformation in time and space. In a more recent theoretical investigation by Ogurtani and Seeger
/4/,this problem is re- formulated using the discrete Fourier k-space transformation with respect to space and supplemented by a Laplace transformation with respect to time. Also, the physi- cal scope of the problem is enlarged by considering the hopping motion with defect reactions between two or more non-equivalent sites for an arbitrary lattice structure.
The desired explicit expression for the Fourier transform of the "partial concentra- tion" of the interstitial impurities at each energetically distinquishable sites is obtained, and later is employed in the energy dissipation calculation.
2. Power Dissipation in Inhomoqeneous Fields.- The power input I(t) for the selec- ted super-cell sample may be written in the following form /1,4/ which is valid even for a nonlinear response system:
where N1, N2, and N3 are even integer numbers representing the size of the sampling domain, and Vk is the corresponding vector set in the rectprocal lattice space /I/.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1983996
C9-640 JOURNAL DE PHYSIQUE
Here, llk(t) are the discrete Fourier &-space transforms of the kink-interstitial in- - -
teraction energy fields, and Ck(t) are the partial concentrations of octahedral interstitial atoms at the three types of interstitial sites in a BCC lattice in the discrete Fourier &-space representation. The partial concentrations denoted by a vectorial quantity which has three components according to the three different tetra- gonality axes. We denote the octahedral sites with axes parallel to
[1001 ,[0101 , or
[OOI] by the subscript 1,2 or 3, respectively. As far as the time dependence of the interaction energy field is concerned, we deal with two different but similar cases in this paper:
and
where 2 corresponds to the static part (bias) of the field, and 2; is related to the amplTtude of the dynamic component of the excitation (induced or external 1. Ex- pression
(2)indicates that the individual Fourier components have their own oscil- lation frequency denoted by w k (dispersion), which will be the case for a kink moving rigidly and uniformly along tKe dislocation line. Relationship
(3),on the other hand, corresponds to an inhomogeneous field which has simple harmonic oscillations in time, with an arbitrary frequency, w, about a static field or bibs, 3;. The partial concen- trations related to these fields can be written down using the general theory advoca- ted by the present authors /1,4/ in the following formats, respectively:
and
.- 3
-
Pi(k)
+iw
-
where Ck(m) is the thermodynamic equil ibrium concentration in &-space which is closely related to the static part of the interaction field. In the above expressions (4) and (5), Pi(&) are all real positive numbers in the closed interval (0,21, zero occurs only when &=O and i=l, (acoustic branch /I/), the Pi(&) correspond to the inverse relaxation time spectrum which is normalized wuth respect to 4r, where r is
the atomic jump frequency of interstitials. C ' is the initial, uniform, total con- centration of the interstitial species in the matrix.
!'(&)is a dyadic uniquely de- termined by the crystal structure in the present case (no chemical reaction).
If one uses the steady-state solutions given by, eqs. (4) and
( 5 ) ,which clearly indi-
cate that the system has a linear response, together with the fundamental power rela-
tionship (I), the following can be obtained for the loss or dissipation part /1,5/:
Pi(k) + Wk
-
where we have assumed t h a t wk =
-
w-k which i s t h e case f o r a k i n k moving w i t h a cons- t a n t v e l o c i t y along t h e d i s l o c a t i o n l i n e . For a harmonic o s c i l l a t i o n i n time, a s i m i l a r expression may be o b t a i n e d from eqs. ( 1 ) and ( 5 ) where one should r e p l a c e wk by w i n eq. (6). The energy d i s s i p a t i o n p e r c y c l e f o r a harmonic system can be c a l c u l a t e d from t h e f o l l o w i n g expression:which y i e l d s immediately:
where -ri(k) i s t h e r e l a x a t i o n t i m e associated w i t h t h e i ' t h branch o f t h e spectrum,
0 0
given by ri(&) = l/Pi(k). Also, CT = C II Ni. As shown i n r e f e r e n c e /4/, t h e r e l a x - a t i o n t i m e spectrum i s affected by t h e presence o f t h e s t a t i c f i e l d , and i t can be c a l - c u l a t e d u s i n g t h e f i r s t o r d e r p e r t u r b a t i o n theory.
3. The E l a s t i c D i p o l e I n t e r a c t i o n . - A c t u a l l y t h e general d i s s i p a t i o n and storage r e l a t i o n s h i p deduced i n t h i s paper (see / 5 / ) i s v a l i d regardless o f t h e n a t u r e o f t h e i n t e r a c t i o n energy f i e l d , e.g. d i p o l e , quadrupole, s t r e s s , .magnetic, e l e c t r i c , e t c . I n t h e case o f s t r e s s f i e l d i n t e r a c t i o n s , one can w r i t e t h e f o l l o w i n g r e l a t i o n s h i p u s i n g t h e e l a s t i c d i p o l e approximation /5,6/
where n=(X1
-
XZ)/h2, which i s c l o s e l y r e l a t e d t o t h e shape f a c t o r o f t h e p o i n t de- f e c t . Iq i s t h e summation o p e r a t o r w i t h r e s p e c t t o dummy i n d i c e s q. I n above equa- t i o n , vA i s t h e atomic volume, A , and X2 a r e t h e p r i n c i p a l values o f t h e t e t r a g o - n a l e l a s t i c d i p o l e tensor, and 3(&,t) i s t h e d i s c r e t e F o u r i e r t r a n s f o r m o f t h e i n t e r - a c t i n g s t r e s s tensor. Eq. ( 9 ) c l e a r l y i n d i c a t e s t h a t i n t h e c a l c u l a t i o n o f t h e energy d i s s i p a t i o n and t h e power consumption, o n l y t h e diagonal m a t r i x elements o f t h e s t r e s s t e n s o r i n t h e p r i n c i p a l - a x i s r e p r e s e n t a t i o n a r e important. T h i s i s due t o t h e f a c t t h a t t h e e l a s t i c d i p o l e t e n s o r associated w i t h t h e octahedral o r t e t r a h e d r a l i n t e r s t i - t i a l s i t e s i n a BCC s t r u c t u r e i s completely d i a g o n a l i z e d i n t h e p r i n c i p a l - a x i s repre- s e n t a t i o n .C9-642 JOURNAL
DE
PHYSIQUE4. The Stress F i e l d o f a Kinked D i s l o c a t i o n Line.- The d i s c r e t e F o u r i e r t r a n s f o r m o f t h e s t r e s s f i e l d associated w i t h a d i s l o c a t i o n l i n e i n an i n f i n i t e , homogeneous and a n i s t r o p i c l i n e a r e l a s t i c continuum may be w r i t t e n down as /5/
-
- i k . r lajk(k,t) = Cjkmn CRpts bR
(&. k k . 1,) G ( k ) j e -as,!, ,
(10)
t m
where Gtm(k) be c a l
&
i s t h e u n i t Cartesian v e c t o r s e t and Cjkmn a r e t h e e l a s t i c s t i f f n e s s e s . i s the d i s c r e t e F o u r i e r t r a n s f o r m o f t h e e l a s t i c Green's f u n c t i o n , which can c u l a t e d from t h e conventional F o u r i e r i n t e g r a l t r a n s f o r m o f t h e e l a s t i c Green's f u n c t i o n as d e f i n e d by Dederichs and L e i b f r i e d /7/. We want t o u n d e r l i n e t h e f a c t t h a t t h e i n t e g r a l i n eq.(lO) i s t h e o n l y term which depends upon t h e c o n f i g u r a t i o n of d i s l o c a t i o n l i n e i n i t s s l i p plane because t h e surface i n t e g r a l i s bounded by t h e d i s - l o c a t i o n l i n e i t s e l f . Any t i m e dependent p e r t u r b a t i o n s o f t h e d i s l o c a t i o n l i n e w i l l be r e f l e c t e d i n t h e F o u r i e r t r a n s f o r m o f t h e s t r e s s t e n s o r by the surface i n t e g r a l term, S(k,t). Therefore, i t i s v a l i d t o c a l l S(k,t) i s t h e dynamic form f a c t o r o f a p l a n a r c u r v i l i n e a r d i s l o c a t i o n 1 in e ( q u a s i - s t a t i c approximation).5. The Dynamic Form F a c t o r Associated H i t h t h e Kinked D i s l o c a t i o n Performing Uniform o r Harmonic Motions.- The d e t a i l e d c a l c u l a t i o n s o f t h e dynamic form f a c t o r f o r v a r i o u s d i s l o c a t i o n c o n f i g u r a t i o n s , s t a t i c , o s c i l l a t i n g harmonically w i t h t i m e o r moving u n i f o r m l y and r i g i d l y , a r e presented elsewhere /5,8/. The exact expression f o r t h e dynamic form f a c t o r f o r a k i n k moving harmonically along t h e d i s l o c a t i o n l i n e as
where ~ ( + ) ( k ) - i s t h e form f a c t o r f o r a s t r a i g h t d i s l o c a t i o n l y i n g along t h e +a/2 P e i e r l y v a l l e y /5/. The expression (11) shows t h e generation o f t h e h i g h e r harmonics i n t h e F o u r i e r t r a n s f o r m o f t K e s t r e s s f i e l d . However, when t h e k i n k o s c i l l a t i o n am- p l i t u d e , AK, i s small compared t o t h e p r o j e c t i o n o f t h e wave l e n g t h along t h e d i s l o c a - t i o n l i n e , t h e f o l l o w i n g l i n e a r approximation can be performed:
S (k,t) h
=
Ca AK s i n ( k ./i. r,
l s i n w t + S(k),
KiEk
-
STatic Kinkwhere "a" i s t h e distance between two nearest P e i e r l s v a l l e y , and
3
i s a u n i t tan- gent v e c t o r along t h e s t r a i g h t d i s l o c a t i o n (screw o r edge). S i m i l a r l y , r+.,=(5+WK>)/2, where WK i s t h e k i n k w i d t h along t h e d i s l o c a t i o n l i n e .The dynamical form f a c t o r f o r a k i n k moving w i t h a constant v e l o c i t y
xK
along t h e d i s - l o c a t i o n can be c a l c u l a t e d /5/ which r e s u l t yi a s i n ( k . L) K
( + I
s"(&,t) = - e x p ( - i w k t ) + S ( k ) ,
-
(13)Kink
-
k.3
(k.K
-
where
wk =
k. yK,
which shows t h a t o s c i l l a t i o n frequencies o f t h e i n d i v i d u a l F o u r i e r -6. The V i s c o s i t y and t h e Drag Force A c t i n g on a Kink Moving Uniformly Along t h e D i s l o c a t i o n Line.- The k i n k v i s c o s i t y due t o i n t e r s t i t i a l d e f e c t s can be obtained as a s c a l a r q u a n t i t y from t h e f o l l o w i n g o p e r a t i o n a l r e l a t i o n s h i p
U 2
I D ( t ) =
- v K
V K 3which according t o eq.(6), ( 9 ) , (10) and (13) r e s u l t s :
2
...
As i n (k.
-
G) 2X Tr(Q(k)
I 1
lq eoq,i(k)1
22 -
(1.
- *
h*
0...
A+ 211 T r ( g ( k ) ) l q eOq,i(&) e n,i(*) Qnn(k)
-
-
where Qjk(_k) = Cjkmn Cipts 2
-
np bbQ kn ks ?Itm(k), and
?Itrn
=k
Gtm&). I n above expression k i s t h e u n i t v e c t o r i n t h e d i r e c t i o n o fk,
and g:i(k) corresponds t o t h e orthogonalized s e t o f eigen u n i t v e c t o r s associated w i t h t h e i n t e r s t i t i a l ' s hop- p i n g m a t r i x /I/, where - Bi(k) -1 1
eoi><Cil .
S i m i l a r l y , t h e viscous drag force, FK- Y
a c t i n g on t h e moving kink-with u n i f o r m v e l o c i t y and r i g i d l y along t h e d i s l o c a t i o n l i n e can be obtained from t h e f o l l o w i n g expression:
7. The I n t e r n a l F r i c t i o n Enhancement Due t o Small Amplitude Kink O s c i l l a t i o n . - The energy d i s s i p a t i o n p e r c y c l e can be obtained from eqs.(8), (9), (10) and (12) as
f o l l o w s :
...
A2 2 -w-c
nc0a2 h 2 s 3 ~
I
<~,,ilo
>1 '
nwkink
=- r -
TI k 2 I-
J--
}(f
N,) 18 kBT 1 + w2.r2 s -I
~ r ( < ~ i i ) >1
-.
A h - A+ 2n T r ( q ( k ) )
-
lq e"gyi(~) eonYi(X) Qnn(k) + 1-i2 eOm:i(~) Q,(~)I~B (17)-
def.-
Awhere bjYk(k,t) = S(k,t) QjYk(k), and < > denotes t h e angular averaging process.
C9-644 JOURNAL DE PHYSIQUE
8. Discussion.- Without making any extensive computer s i m u l a t i o n s t u d i e s which w i l l be presented elsewhere, t h e f o l l o w i n g conclusions can be e a s i l y reached: a). The energy d i s s i p a t i o n i s d i r e c t l y p r o p o r t i o n a l t o t h e mean c o n c e n t r a t i o n of i n t e r s t i t i a l s C', which i s assumed t o be u n i f o r m l y d i s t r i b u t e d i n i t i a l l y . b ) . The induced i n t e r n a l f r i c t i o n s t r e n g t h depends q u a d r a t i c a l l y on b o t h t h e amplitude o f t h e k i n k o s c i l l a t i o n and t h e i n t e n s i t y f a c t o r o f the e l a s t i c d i p o l e tensor. c ) . The i n t e r s t i t i a l s which can be represented by a simple i s o t r o p i c d i l a t i o n
n=
0, can s t i l l cause i n t e r n a l f r i c - t i o n and viscous drag due t o t h e inhomogeneity o f t h e s t r e s s f i e l d which i s character- i z e d by a non-vanishing t r a c e o f t h e F o u r i e r t r a n s f o r m o f t h e s t r e s s tensor. d). Also, one can show r i g o r o u s l y t h a t Tr(<Q(k)>) i s i d e n t i c a l l y zero f o r non-sessile d i s l o c a - t i o n c o n f i g u r a t i o n s /5/. Therefore, f o r an i s o t r o p i c BCC metal t h e induced Snoek peak ( u n i f o r m modes), represented by t h e f i r s t term i n eq.(17) i s completely missing.However, one can s t i l l g e t an a p p r e c i a b l e amount o f energy d i s s i p a t i o n i n t h e v i c i n i t y o f t h e pure Snoek r e l a x a t i o n ( T ~ ) frequency v i a o p t i c a l r e l a x a t i o n modes (i=2,3) f o r t h e non-vanishing
&
vectors. The l a t e r c o n t r i b u t i o n s , however, a r e v e r y s e n s i t i v e t o t h e amplitude modulation f a c t o r through t h e k i n k width, WK. As one might a n t i c i - pate, t h e g r e a t e r t h e k i n k width, t h e s m a l l e r t h e s p e c t r a l broadening of t h e i n t e r n a l f r i c t i o n peak (induced), again due t o t h e modulation f a c t o r , s i n(k. r+,)/(k. G) ,
which i s c a l c u l a t e d f o r v a r i o u s k i n k widths i n reference /5/.
We can s t a t e t h a t one m i g h t g e t a considerable amount o f power d i s s i p a t i o n r e l a t e d t o t h e i n t e r n a l f r i c t i o n o r viscous drag from t h e mobile (even i s o t r o p i c ) i n t e r s t i t i a l s i f one has r e l a t i v e l y narrow k i n k o s c i l l a t i n g o r moving u n i f o r m l y along t h e pure screw d i s l o c a t i o n . The s i t u a t i o n f o r 71-0 mixed d i s l o c a t i o n s i s more f a v o r a b l e as f a r as these i s o t r o p i c p o i n t d e f e c t s a r e concerned f o r which power d i s s i p a t i o n s t i l l takes p l a c e even f o r very wide geometrical kinks. As a f i n a l remark t h e authors want t o s t a t e very c l e a r l y t h a t one can have an observable amount o f v i s c o s i t y o r i n t e r n a l f r i c t i o n enhancement due t o m o b i l e i s o t r o p i c d e f e c t s i n t h e presence o f kinked screw d i s l o c a t i o n s , even f o r l a r g e k i n k widths, as l o n g as t h e metal shows a h i g h degree o f e l a s t i c anisotropy, which i s t h e case f o r most metals and a l l o y s i n p r a c t i c e .
REFERENCES
1. T. 0. Ogurtani and A. Seeger, "The I n t e r n a l F r i c t i o n Associated w i t h M o b i l e Octa- hedral I n t e r s t i t i a l s i n t h e Presence o f Inhomogeneous F i e 1 ds i n BCC Metals", J. Appl
.
Phys. , ( t e n t a t i v e issue: 1 J u l y 1963j.2. A. Seeger and W. Hornung, I n t . F r i c t i o n and U l t r a s o n i c A t t e n u a t i o n i n C r y s t a l l i n e Sol i d s (D. Lenz and K. Lucke, ed.) Springer, B e r l i n , Vol .I, 222 (1975).
3.
W.
Hornung, Phys. S t a t . Sol.(b) 54, 341 (1972).4. T. 0. Ogurtani and A. Seeger, "The K i n e t i c s o f D i f f u s i o n o f I n t e r s t i t i a l s w i t h Chemical r e a c t i o n s i n A r b i t r a r y Time Dependent Inhomogeneous F i e l d s " , 3. Chem.
Phys. , ( t e n t a t i v e issue: 15 November 1983).
5. T. 0. Ogurtani and A. Seeger, " I n t e r n a l F r i c t i o n and Viscosi.ty Associated w i t h Mobile I n t e r s t i t i a l s i n t h e Presence o f a Kink Moving Harmonically o r Uniformly i n A n i s o t r o p i c BCC Metals", t o be p u b l i s h e d i n J. Appl. Phys.(1983).
6. E. Kroner, Kontinuumstheorie der Versetzungen und Eigenspannungen, Springer,Berlin, (1958).
7. P. Dederichs and G. L e i b f r i e d , Phys. Rev., Vol. 188, 1175 (1969).
8. A. D. B r a i l s f o r d , J. Appl. Phys., Vol
