Incorporation model for the spreading of extrinsic grain boundary dislocations

Scripta METALLURGICA et MATERIALIA

Vol.

24, pp. 1929-1934, 1990 Printed in the U.S.A.

Pergamon Press plc All rights reserved

INCORPORATION MODE[. FOR THE SPRFIDING OF EXTRINSIC GRAIN BOUNDARY DISLOCATIONS A.A.Nazarov ~

•

A.E.Romanov**

f

R Z.Valiev ~ •

• Institute of Metals Superplasticity Problems, USSR Academy of Sciences, Ufa 450001 USSR • *A.F.Ioffe Physico-Technlcal Institute, USSR Academy of Sciences, Leningrad, 1 9 4 0 2 1 U S S R (Received April 5, 1990) (Revised July 30, 1990) Introduction In a number of experimental works concerned with in situ annealing ot thin m~tal foils• spreading of trapped lattice dislocations (TLDs) in highangle grain boundarles has been observed: when heated above (0.2-0.5)T m (T m is the melting temperature), the e]ectrnn microscope images of TLDs are widened and weakened until the d~ffractlon contrast completely disappears (see, for example• 1-3). In order to give a quantltatlv~ description of this phen~menotl, two models have been pruposed: a model of continuous delocalizatinn (~F T[,D cores in grain boundary plane (2,4) and a model of dissociation nF T[.Ds ~[,t(~ extrinsic grain boundary dislocations (EGBDs) wlth small values of Burgers vectors and subsequent scattering of dissociation products (5-7). Both models give similar expresslons for the spreading time, an d neither of them can be preferred when compared with experimental data (8) . Thmse models have a significant disadvantage: the reconstruction of the graln boundary structure during the absorption process is taken into account in them only through the local change of mlsorlentation angle caused by the TLD. At the same time, Horton, Silcock and Kegg (9) and Pond and Smith (6) supposed that F~,BDs can be incorporated by Intrinsic (secondary) grain boundary dislocation (SGBD) networks. Such an approach would provide a more detailed consideration of the grain boundary structure changes, but the idea was referred only to low-angle and special boundaries and was used only for qualitative discussions. The recent results of computer simulation of the atomic structure of grain boundaries allow one to extend this idea to the case of random boundaries and develop a new quantitative model of the spreading of TLDs in grain boundaries, which is the elm of this paper. Description

of the Model and Main Klnet~c Formulae

As has been concluded from computer simulation results (i0), there exist two fundamental classes of grain boundaries: favoured boundaries consisting of a single type structural unit, and random boundaries whose structure can be represented as a mixture of structural units of two favoured boundaries taken in a certain proportion and arrangement. It has been shown (i0) that minority structural units in a random boundary correspond to the cores of SGBDs possessing the Burgers vector equal to a DSC-lattice vector of the favoured boundary composed of majority structural units. Recently, Sutton (Ii) shown the structural unlt/SGBD model to be of limited predictive power for boundaries with high-lndex misorlentatlon axes and/or mixed tilt and twist character. However, rather a large class of pure tilt or twist boundaries do contain SGBD networks. Proceeding from this notion, the process of TLD absorption which takes the shape of TLD spreading can be explained as follows. Lattice dislocations

1929 0036-9748/90 $3.00 + .00 Copyright (c) 1990 Pergamon Press plc

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GRAIN BOUNDARY DISLOCATIONS

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trapped in a grain boundary dissociate into EGBDs whose Burgers vectors are DSC-lattlce vectors of a favoured boundary and, in general, can have components both normal and parallel to the grain boundary plane. It is important that the distances between the EGBDs formed by each TLD do not exceed the SGBD spacings (which can be less than I-2 nm) and they cannot be observed separately in the electron microscope. The TLD-dlssoclatlon products interact between themselves as well as wlth secondary dislocations so that the relaxation of long-range stress fields created by T L D includes the motion of all these d~slocatlons towards new equilibrium positions. Canceling of the long-range stress f~elds leads to the spreading of TLD-Image. Since the relative displacements of EGBDs, are on the order of one grain boundary period, the kinetics of the process are completely limited only by the climb of secondary dislocations. Therefore, without the loss of generality, we can investigate separately the absorption of EGBDs with the Burgers vector equal to a single DSC-lattlce vector, normal or parallel to the boundary plane. In this paper we confine ourselves to cons~deratlon of the spreading of sessile EGBDs with a normal Burgers vector. Note that this case represents the real process of the TLD spreading if one of favoured ~oundarles whose structural elements compose the boundary is a perfect lattice plane (for boundaries which are known as medlum-angle boundaries). Let us suppose that in the initial state a tilt boundary contains a network of SGBDs with Burgers vector b = ( b , 0 , 0 ) and spacing h, and the EGBDs with the same Burgers vector be located symmetrically with respect to SGBDs, as shown in Fig.l (their spacing being H=2Nh). The EGBDs are assumed to be arranged periodically, because this slmpli[ies the analysis. If N becomes large, then we actually deal with the absorption of a single EGBD. The starting coordinates of secondary dislocations are yi = (i-I/2)h (i=I,2 ..... N). Under the influence of ~a component climb to new equilibrium positions period of a new the climb rate of the vacancy

flux

of the stress field of EGBDs yl=zn ' , where h'=2Nh/(2N+l)

equillbrlated dislocation i-th dislocation will be

from

from i-th to "upper" Following the first J=-Db~/ilkT, where

"lower"

((i-l)-st)

network. According to (12), v i = ( I I - l u ) ~ / b , where I 1 is

to i-th dislocation,

I u is the same

((i+l)-st) dislocation, and /~ is the atomic volume. Fick's law, the vacancy flux density Is determined by D b is the grain boundary self-dlffuslon coefficient and

~ is the gradient of chemical potential for vacancies. Since vacancy flux along the boundary equals to I=J~= -Db~ v ~ x / k T diffusion width of the boundary) (12) . Substituting expression for the dislocations rate we obtain

where

~-~ i s

the stress

value

H-y N. Calculatlng__ 2 t ,stresses h e__ and time T=£~,~J~/~-9)~ ~ cations in the following form:

"

,here

near the i-th

dislocation

, we can write

k;,k N

-

; numerlcally,

V~=v~'[, (~

is

this formula

the the

into the

c o r e and yo=O and YN+] =

(13) and using dimensionless

NO =O, Solving Eqs.2

the SGBDs is the

coordinates ~ i = Y l / h

the motion equations

for dislo-

N

+ z we

can

Investigate

the

kinetics

of the

Vol.

_~, No

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GRAIN BOUNDARY DISLOCATIONS

1931

relaxation of the boundary with EGBDs into a new equilibrium state. The degree of the boundary nonequillbrium can be determined by the parameter ~=£~,~a)]2,1/2 {=I tYl-Yi 2 having maximum value ~ 0 = 0 . 2 5 at t=0 and reaching zero at the new equilibrium state. In the delocallzation and dissociation models it was assume,i that thu EGBD image disappears when the dislocation core width (or the width of its dissociation products complex) reaches about two extinction distances (2,4,8). In the proposed model such a simple criterion which includes a structural parameter is absent because all the SGBDs undergo slmultane~JS d ~ p l a c e m e n t s . A more detailed investigation of the variations of long-range stress fiulds of the "grain boundary+EGBDs" complex with time is needed. One can expect (and this will be confirmed by numerical calculations) that the stress field of the complex is analogous to that of some finite dislocation wall with increasing length. The shear stress field of the finite dislocation wall wlth a length changes along the normal to it passing through its middle as the function

xl[x2+( [12 )2] . This function has its maximum at the point we can introduce an "effective width of the dislocation"

Xmax=~/2 equal to

(14). Then ~=2Xma x.

Results Fig.2 represents the abating curves of shear stress fields ~ created by the boundary on llne y=0 at different moments of time. If N is large enough (Eqs.2 are solved for N=25,50, and i00), at the same values of dimensionless time T = £D$~A/4~-~)~T~. ~ these curves are invarlant with respect to N. They are very similar to ~x, (X) -curves for finite dislocation walls: at small values of x there is an ~ p o n e n t i a l dependence of ~xS on x, then the curves pass through the maximum and as x increases approach the stress field curve of a single dislocation (curve i). In complete accordance with the "widening" of EGBDs, the maximum point on the curves moves to the right as tlme ~ncreases. The time dependence of the ef[ectlve width of the EGBDs is presented in Fig.3. This dependence is linear in logarithmic coordinates, and we obtain the following relationship "dislocation width" at spreading time

between ~ and [ : T ~ 0 . 0 1 2 ( [ / h ) 3 • Designating the which its image disappears as S, we can write for the

}s~ ~ ~°3ETS3gA The s p r e a d i n g o f t h e images o f EGBDs absorption looks from the outside, and tsp r the

relaxation

time

of the

grain boundary

o n l y shows how t h e p r o c e s s o f EGBD does not necessarily coincide with structure.

Complete

achieved when EGBDs take up positions determined by termlna]

re]awat~on ~s coordinates Yi (t)

In order to estimate the characteristic time of th~s process and determine its dependence on structural parameters, the curve ~({) has been calculated and plotted in Fig.4. In tcrms of dimensionless time ~ = ~ G ~ / ~ ( ~ - ~ ) ~ T N 3 ~ 3 this curve is Invarlant with respect t.o N. As can be seen fr(im the f l g u r e , ~ becomes less than 10 -3 at TI>0.1, Therefore, we can say that the complete recovery of the grain boundary structure occurs in a time interval of the order

&O3k T H~

(4)

Discussion To begin with, let us determine the range of validity of the formula /3/. As a rule, the TLD spreading is observed in random boundaries which cof,taln no SGBD networks visible in an electron microscope (1-3). For such boundaries h<5 nm (15). Then, at the minimum value of N which was used in the investigation

1932

(N=25),

GRAIN BOUNDARY DISLOCATIONS

the EGBD density

~=ll2Nh)4xl04

Vol.

cm/cm 2. Therefore,

24, No.

the formula

I0

131 is

applicable at least for EGBD densities less than 4x104 cm/cm 2. When N(25, the results" are considerably affected by the periodicity of EGBDs arrangement and the spreading crlterlon used in th~s work becomes invalid. Hence, the use of more precise criterion is required to estimate tsp r at TLD densities greater than 4x104 cm/cm 2. Besides this limitation, which is concerned with the calculation technique, there is also a physical restriction for the formula /3/. As can be seen from the comparison of formulae /3/ and 24/, the relaxation time of grain boundary structure and the spreading time of EGBDs are as EGBD density

~=I/H<
then

non-equillbrlum

after complete

I/S. This

that not

means

large EGBD densities

trel)>tspr, spreading

trel/tspr~(H/S)3.

and the boundary

of EGBDs.

However,

structure trel<(tspr

tsp r but tre I shol,]d be taken as a spreading

and a sharp decrease

of the spreading

If the remains

at

~)) --

time at

time with increasing

TLD density may be observed. The transition region ~ I / S ~ I 0 5 cm/cm 2 is approximately the same as the max|mum density of TLDs ob6erved during in s~tu deformations (16) and slightly higher than the region where our spreading cr~teriun stops being valid. It should be pointed out that the same relationships between tre I and tsp r are given by previous models (4,5,7), so this limitation is common for all models of the spreading. A special investigation should be carried old to learn how the spreading process occurs in the transition region ~I/S. As follows from formula /3/, In the range of its appllcabilit~ the spreading time of EGBD images is Independent of their density. This conclusion seems to be supported by experimental data (16). Formula /3/ shows also the absence of any direct dependence of spreading time on SGBD spacinq. Th~s has been qualitatively explained by Pond and Smith (6): at low densities of SGBDs relatively large climb of few dislocations occltrs, whilst at h~gh denslt|es many d~slocat~ons climb lesser distances; however, the total vacancy flux required for the process is constant. Therefore, provided all the other conditions are the same, the spreading of EGBD occurs identically in all the random boundaries. However, the grain-boundary self-dlffuslorl coefficient D b strongly depends un the boundary :~t~ucture (17): it car, be by an order of magnitude lower in special boundaries than in random boundaries. Accordingly, at the same temperature, the spreading time In special boundaries will be by an order of magnitude longer than in random boundaries. Formula 131 ~s s~milar to those ol,t.alned In models of r.oJkowski-Grabskl (4) and Johannesson-Tholen (5) and differs from them only by numeric factor. Let us compare all formulae both among themselves and with experimental data. As has been established by Pumphrey and Glelter (2), the spreading time of TLDs in random boundaries in nickel is about 30s at T=493 K. The LoJkowski-GrabskI and Johannesson-Tholen models give for this case 4.9xi03 s and 1.6x103 s respectlve]y (8) . The dissociation model presented in (7) provides approximately the same estimation. Under the same conditions, it follows from formula /3/ that tspr~15 s, which is closer to experimental data and much lower than in delocallzatlon and dissociation models. This large difference can be explained qualitatively as follows. One can show that for a given density of TLDs the same number of vacancies is necessary for both the widening of the complex of dissociation products and incorporation of EGBDs; the diffusion paths are also approximately equal. But in the first case due, to small values of Burgers vectors of dissociation products, the driving force for the process is less than in the second case. Besides this, the motion of SGBDs surrounding the EGBD begins simultaneously along the whole grain boundary plane, while the dissociation products complex covers it up gradually. Hence, the spreading time in the incorporation model Is much less than in the dissociation model. The same comparison is valid also for the delocallzation model because it can be considered as a limit of the dissociation model.

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1933

The proposed model allows us to predict some pecularlttes of the absorption of lattice dislocations by special grain boundaries. Since favoured boundaries have no SGBDs, TLD does not spread in them, but its d~ssoc~ation into discrete EGBDs can occur. In grain boundaries close to favoured ones there exist SGBD networks resolvable by electron microscope. Extrinsic dislocations incorporate into these networks without losing their images. As the self-d~ffusion coeff~clent for these boundaries is small, according to formula /4/ the relaxation time is long and no visible processes in such boundaries in usual observation time intervals can be detected. Special but non-favoured boundaries have a structure similar to that of long-period random boundaries but they share much lower self-diffusion coefficient. That is why the EGBD spreading in these boundaries occurs at a higher temperature or in a longer time interval. In grain boundaries with misorientatlon angles close to those of nonfavoured special boundaries, virtual dislocation networks induced by the superpos~t~on of SGBD stress fields a~e observed (i0). In such boundaries one of the following two things may occur: either the EGBD spreads only partially and the rest incorporates into the virtual dislocation network, or it spreads completely, causing some changes in the location of virtual dislocations. To know which of them occurs, a more detailed investigation of the processes taking place in these boundaries is needed. Conclusions I. The spreading of trapped lattice dislocations in high-angle grain boundaries can be well described in the model of incorporation of their dissociation products CEGBDs) into the network of secondary intrinsic dislocations. 2. The spreading tlme calculated from the incorporation model is by two ordet-s of magnitude shorter than the time given by the deloca]ization and dissociation models, which is in a good agreement with the experimental data. 3. To study the spreadinq kinetics at EGBD densities nearly equal to I/S a more precise criterion of EGBD image disappearing ~s required. This is common for all models of the spreading. Acknowledqm__ee[,t~ The Metal

authors are grateful

to Dr. V.Yu. Gertsman,

member of the Tnstltute of

Superplasticity Problems, for reading the manuscript and helpful discus-

ston.

References I. 2. 3. 4. 5. 6. 7. 8. 9. I0. II. 12. 13. 14. 15. 16. 17.

Y.l~hida, T.Hasegawa and F.Nagata, Trans.JIM g (suppl.), 504 (1968). P.H.Pumphrey and H.Gleiter, Phil. Mag. 3__0_0,593 (1974). R.A.Varln, Phys. Stat. Sol.(a) 52, 347 (1979). W.LoJkowskl and M.W.Grabskl, Deformation of Polycrysta]s: Mechanisms and Microstructures, p.329, Riso Nat. r.ah., Roskilde (1981). T.Johannesson and A.Tholen, Metal Sol. J. 6, 189 (1972). R.C.Pond and D.A.Smith, Phil. Mag. 9__6, 353 (Ig77). R.Z.Vallev, V.Yu.Gertsman and O.A.Kalbyshev, Phys. St at. Sol. Ca) 78, 177 (1983). K.Kurzydl~wskl, J.Wyrzykowsk~ and H.Garbacz, Fizlka Metallnw i Metallowed~nie 65, 385 (1988) (in Russian). C.A.P.Horton, J.M.Silcock and G.R.Kegg, Phys. Star. So]. Ca) 26, 215 (1974). A.P.Sutton and V.Vitek, Ph~l. Trans. R. Soc. Lond. A309, I, (1983). A.P.Sutton, Phil. Mag. Lett. 59, 53 (198g). ~.Arzt, M.F.Ashby and R.A.Verrall, Acta Met. 3_1 I, ]977 (1983). J.P.Hirth and J.Lothe, Theory of D~slocations, John-Wiley & Sons, New York (1982). A.M.Kosevlch, Dislocations in Solids, vol.l, p.33, North-Holland, Amsterdam (]g79). R.W.Balluffl, Y.Komem and T.Schober, Surf. ScJ. 31, 68 (1972). R.A.Varin and K.Tangri, Met. Trans. 12A, 1859 (1981). N.L.Peterson, Int. Met. Rev. 28, 65 (1983).

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GRAIN BOUNDARY DISLOCATIONS

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t: ~'=0. 2: 3: : 5:

-¢

6:

0 l0 0.~u 0..=0 tOO

,4=

7: g: 9

"aO.OO ~o.cc 600.00

OJo

L~

4.00

~.06~

6 d.

I

C,~2

I 9 1o

Fig.l. /nltlal structure of a tilt boundary containing SGBD network and EGBDs

2.6

Fig.~. Change of long-range stress f~elds creatud by the boundary on llne y:0 w i ! . h tlme

0"25i 0.20 0.f~

0.IC

i.o

0.0;

0.6 '

~[~~G~h(~'~)fTk 3] Fig.3. D e p e n d e n c e of the effective wldht of EGBDs ~n tlm~

i

0.05 ~.I)=~G~/~ (~-v}kTN'~~

Fig. 4. T h e c u r v e r e p r e = e n t l n g the relaxatlon of the boundary structure to a ~(;~ ~qu/llbrlum state

OAO