- Email: [email protected]
Regular defect structures in high angle grain boundaries
REGULAR
DEFECT
STRUCTURES B.
LOBERG?
IN HIGH
ANGLE
GRAIN
BOUNDARIES*
and H. NORDENt
The structure of high angle grain boundaries in tungsten has been studied using field ion microscopy and electron microscopy. The observed linear defect structure is explained in terms of plane matching across the grain boundary; this model of relaxation across the interface is assumed to apply more generally. STRUCTURES
REGULIERES
DES
DEFAUTS DANS LES JOINTS DESORIENTATION
DE
GRAINS
A FORTE
La structure des joints de grains B forte d&orientation dans le tungstl?ne a Bti Btudibe par microsoopie a Qmission d’ions et par microscopic Blectronique. La structure linbaire des d6fauts qui a Bt6 observk est expliqube par la correspondance des plans B travers le joint de grains; on suppose que ce modele de relaxation ZLtravers l’interface peut s’appliquer d’une fapon plus g&&ale. REGELMdPIGE
DEFEKTSTRUKTUREN
IN GROBWINKELKORNGRENZEN
Die Struktur von GroBwinkelkorngrenzen in Wolfram wurde mit Hilfe der Feldionenmikroskopie und Elektronenmikroskopie unteraucht. Die beobachtete lineare Defektstruktur wird mit einem Model1 ebener Anpassung an der Korngrenze erkliirt. Es wird angenommen, daB dieses Model1 der Relaxation an der Grenzfl&che noch allgemeiner anwendbar ist.
INTRODUCTION Recently there have appeared in the literature numerous reports of electron microscope observations of periodic defect structures in high angle grain boundaries. In some cases networks have been seen,(ls2s3) but mostly the observed contrast has consisted of roughly parallel lines. (4*5*6)It seems clear that phenomena such as large steps in the interface and absorbed lattice dislocations although giving contrast in the electron microscope in general cannot account for all these regular structures. It has therefore been proposed that dislocation-like defects with Burgers vectors normally not present in a crystal can appear in grain boundaries.(4s7) Such dislocations have in particular been predicted to exist when the grains are near a coincidence relationship.@) In such cases a highly regular structure with small repeat distances could be achieved in the interface when t,his lies in a symmetric position, i.e. is along equivalent, lattice planes in the two grains. Deviations from this ideal configuration could then be localized in misfit dislocations with certain Burgers vectors which would preserve the regularity of the boundary structure. Detailed pictures of the atomic configurations in coincidence boundaries has been produced by computer simulations,‘B,lO) and observed linear defects in interfaces between crystals near coincidence relationships have been interpreted as specific misfit dislocat,ions.(l*2*11)
The coincidence model however explains the defects observed in the electron microscope only when the misorientation lies within a narrow range around specific axes ; it also presupposes that the interface * Received May 30 1972. t Department of Physics, Chalmers University of Technology, S-402 20 Gothcnburg 5, Sweden. ACTA 3
METALLURGICA,
VOL.
21, MARCH
1973
213
is ordered and lies along specific planes. It seems clear that a more general model is needed to account for defects observed when these conditions arc not fulfilled. Combined electron and field ion microscope observations in drawn tungsten wire suggest that the existing periodic defects are due to relaxation in the boundary.(12) This idea is further developed in the present paper. EXPERIMENTAL
Most of the field-ion specimens were prepared from as received tungsten wire with a diameter of 0.1 mm and of commercial purity. The wire had the usual pronounced fibre texture with a [llO] axis. The high angle grain boundaries to be discussed here are the [ 1lo] predominantly tilt boundaries abundantly observed in such a material. Generally, the [llO] directions of the two grains do not coincide but lie at a small angle to each other. For purposes of making this clear the relative orientation of the two grains can be described by adding two small rotations to the main tilt around [IlO] : One about an axis perpendicular to [llO] in the grain boundary, or a “tilt misorientation”, and one about the normal to the boundary or a “twist misorientation”. When an additional small twist misorientation is present linear defects have regularly been observed in high angle boundaries of as-polished specimens when examined in the electron microscope. The defect contrast consists of a set of lines, roughly parallel to the [llO] direction. The lines are visible in dark field micrographs, using diffracted beams from either grain and they must therefore represent defects which have strain fields in both grains. At larger twist angles the lines become more closely spaced and begin to look like moire fringes.(13) The closest observed
214
ACTA
METALLURGICA,
VOL.
21,
1973
spacing of the defects is about 50 A whereas at larger twist angles it appears that the localized line structure is replaced by ordinary moire fringes. Examples of the described line contrast are given in Figs. 1-2; others have been published elsewhere.(5*12)
Fm. 2. Dark field micrograph using the common [110] reflexion of a specimen with two grains with a relative rotation of 10”. The boundary defects appear to be discontinuous at inclusions in the boundary; the specimen wire had been doped with alumina. .
Fro. 1. Bright field electron micrograph of a field evapTwo grains with the interface orated specimen. approximately perpendicular to the electron beam extend to the tip and a third grain terminates 800 A up the shank. The relative rotation between the first two grains is 13” around an axis close to [llOl and the boundary plane is approximately (118) and (113). The relative rotation across the second visible interface with linear defects is 63”.
Linear defects have been observed both in specimens with symmetric and asymmetric boundaries containing [llO] as well as in boundaries where the interface is inclined to [IlO] (Fig. 3). This suggests that the presence of the defects is insensitive to the orientation of the boundary plane. That the defects themselves are indeed related to the misalignment of the [IlO] directions of the two grains has been shown by subsequent field-ion microscopy of the specimens.
Fro. 3. Grain boundary defects in tungsten, annealed at 1200°C for 8 hours. The relative rotation is 20” around an axis near [110] and the grain boundary plane is approximately (lOI) of the top grain and (5I3) of the other and the defects lie in the [212] direction of the top grain. (c) Dark field using [ST11 of the other grain. (a) Bright field. (b) Dark field using [200] of top grain. The specimen has been rotated 8’ between the three miorographs.
LOBERG
AND
NORDEN:
REGULAR
DEFECT
In a field-ion micrograph of a single crystal, the [llO] pole is surrounded by a prominent set of rings which are the images of the edges of consecutive (110) planes. In those cases where the misorientation between two grains is a pure rotation around [llO] and the grain boundary runs across the [llO] pole of the micrograph, the (110) planes will always match up at the interface and thus the rings continue unbroken across the boundary. An example of this is shown in Fig. 4. When the [llO] directions are slightly misaligned, the expected image of a boundary in the [ 1lo] region would consist of two sets of offset part rings, i.e. t’he edges of the (110) planes of the two crystals would in general not meet across the boundary. Such an image is not seen; the (110) planes appear to relax so that good matching is achieved across the interface except at the few places where spirals begin.(14*15)An example of this is shown in Fig. 5. In our studies of grain boundaries in tungsten a oneto-one correspondence has always been observed betueen the lines in the electron micrographs and the spirals in the field-ion ones. It is thus possible to predict with accuracy the number of spirals that will appear in the field-ion image from the number of lines present in the electron microscope image of an aspolished specimen. It is also possible to show that the positions of the lines correspond to those of the spirals. The positions of the spirals seen during an evaporation sequence can be plotted on an electron micrograph of the specimen if the evaporation end form, the distance
STRUCTURES
IN
GRAIN
BOUNDARIES
215
Fm. 5. The relative rotation of the grains is approximately 69’ around an axis near the central [llO]. The [ 1lo] poles of the two grains are marked with dots and spirals beginning in the interface with arrows. The corresponding lines observed in the electron microscope were parallel to [llO] within 3” although a tilt misorientation is present as seen in the micrograph above.
removed by evaporation and the magnification and orientation of the electron micrograph are known. The positions of the spirals coincide with the lines, as has been shown elsewhere.(5) It is therefore clear that the linear contrast and the spirals have the same origin and can be related to a slight misorientation of the [l lo] directions of the grains. It also implies that the observed good matching of (110) planes across the boundary in the field-ion image is not due to a relaxation at the specimen surface. INTERPRETATION
seems improbable that the linear defects should be interpreted as absorbed lattice dislocations. If they were, the correspondence between lines and spirals would mean that in all cases there had been a perfect alignment of [llO] directions of the two grains before the absorption of dislocations. Furthermore, the absorbed dislocations must all either have moved in slip planes containing [IlO] or realigned after absorption since the lines are approximately parallel to this direction. They must all have had a Burgers vector with a component of a/2 [llO] as they give spirals in the field-ion microscope and all be of the same sign as all spirals go either clockwise or anti-clockwise. It is unlikely that such dislocations should be the only ones absorbed. An interpretation of the observed contrast as due to ledges can be ruled out for two reasons: firstly, no It
Fxa. 4. Field-ion micrograph of a specimen containing two grains with a relative rotation of 18’ around the central [ 1101. The rings around the centre are formed by the edges of consecutive (110) planes and are continuous across the interface.
216
ACTA
METALLURGICA,
steps are in general seen in field-ion micrographs near the positions in the boundaries where the lines appear ; secondly, the observed spirals cannot be explained by a ledge structure. An interpretation of the defects as the dislocations postulated by the coincident site lattice theory is impossible as the lines are observed even when the orientation between the two crystals is far from a high density coincidence relationship. For example, lines spaced more than 50 A apart have been observed for relative rotations of 13”, 34’ and 59’ around axes near [I IO]. In each case the angle could be measured to within 2” from field-ion micrographs. These rotations and large line spacings would lead to unrealistically large Burgers vectors while those proposed by the coincident lattice theory are smaller than for lattice dislocations. For example, the Burgers vector of the defects in Fig. 1 must be about 15 A if the usual maximum va.lue of 19 is used for the inverse density of coincident sites, C. Disregarding this limit, a value of C = 73 is needed to give the defects a Burgers vector less than 3 8. Furthermore, the coincident site lattice theory does not explain the presence of the screw component along [llO]; that the lines corresponded to spirals could be checked for the rotations mentioned above as the boundary appeared in the [llO] pole of the field-ion micrographs. In view of the combined observations in the two types of microscopes it is suggested that the linear contrast is the result of a relaxation process in the boundary. The relaxation is such that the (110) planes of the two grains are in register over most of the interface and the misfit in the [llO] direction has been localized along the lines. Figure 6 is a schematic drawing of the proposed configuration. A similar interpretation of observed lines in bicrystals of aluminium, essentially based on geometrical considerations, has been given by Levy.c4) DISCUSSION
Generally, a regular defect structure in a planar high angle boundary must indicate that there exists some periodicity in the boundary. This periodicity should be at least one-dimensional and have a crystallographic meaning, i.e. it must be related either to lattice points or to lattice planes in both grains. In most boundaries there are no such repeat distances related to lattice points, as this requires that a vector in the interface is a lattice vector in both grains. The most general boundary where this condition is fulfilled is an asymmetric pure tilt boundary with the two grains rotated relative to each other around an axis [hkl] in the boundary plane. In this case a vector a - (h, k, 2)
VOL.
21,
1973
Grain A GrainB I
(b)
Groin A Grain
B
Fm. 6. Orientation of the (110) lattice planes seen edge on (a) near and (b) far away from the interface. The boundary plane lies in the plane of the paper. The angle between [I lo] of crystals A and B is exaggerated relative to the observed cases.
in the interface connects points surrounded by exactly the same configurations of atoms. In the particular case where [hkZ] is equal to [llO] the periodicity vector is a * (1, 1,O) where a is the lattice parameter. The observed linear defects are however connected with a relative displacement of the grains of only half this size and equal to the spacing of the (110) lattice planes. The configuration achieved by introducing the defects is one where the (110) planes in the two crystals are aligned without the introduction of the same periodic atom arrangement that would be present in the interface with perfect alignment of [llO]. The periodicity which facilitates the creation of the defects is therefore due to lattice planes and not to lattice points. This would explain why the lines are observed not only on asymmetric tilt boundaries but also in boundaries inclined to the [llO] direction. The proposed relaxation is only possible if the energy gained by rearranging the atoms in the boundary region to align the (110) planes across the boundary is greater than the energy necessary to build up the strain field extending into the two crystals around each line. That atoms near an interface indeed adjust their positions from those of two undistorted crystals is clear from the cohesion of grain boundaries and modern theories emphasize the possibility of boundary rearrangement at the expense of some lattice strain.@) The fact that field-ion micrographs of bicrystals with a perfect alignment of a pole with low indices with few exceptions always show a ring pattern continuing across the interface as in Fig. 4 is a strong indication that the matching of planes across the boundary is a favored configuration as compared to one where the planes are staggered across the boundary. Computer
LOBERG
x4s-i~ SORDEN:
REGULAR
DEFECT
simulations of atom arrangements in coincidence tilt boundaries suggest the same.{16) Although the con~gurations with lowest energies in this case were the ones where actual coincidence positions in the interface were lost, the planes perpendicular to the tilt axis with one exception were in register and not staggered. It must be considered how a small twist misorient,ation from a pure tilt posit,ion could give rise to only one set of visible linear defects in the elect,ron microscope. In an ordinary low angle twist boundary at least two sets of screw dislocations must be present to remove long range stresses. If the lines are due to an interaction of nearly aligned planes across the boundary, ILsecond set of defects would presumably be due to a similar interaction between other planes. That other nearly aligned planes exist is evident, as any orientation between two grains can be described by rotations around twenty-four different axes.*(“) However, for all rota,tions around [llO] except those distinguishing coincidence relationships the Millerindices of directions near t,he other axes of rotations are high. This means that the atomic densities in other nearly aligned planes are low and therefore their interaction across the interface weak. Even if a relaxation t,akes place in a second set of planes,? their interplanar spacing is small and the strain from the introduced defects equivalent to that of a set of closely spaced screw dislocations with a small Burgers vector. The contrast in the electron microscope from such defects would be very weak and in most cases invisible. Kor can their existence be easily checked in the field-ion microscope, as rings do not develop around poles with high indices. However, if such a network of two sets of defects is achieved, all planes perpendicular to the twenby-four different axes of rotation are then aligned over each mesh in the netrvork, So far the discussion has been concerned with smalI twist components added to the main tilt rotation. If a small tilt around an axis perpendicular to [llO] is included as well. this can be accommodated by inclining the defects slightly to the [llO] direction(l*~ The spacing of the lines for a superposed twist component of angle /l is d/b, where d is the spacing of the relaxing planes. The spacing for a small superposed tilt component of angle a is 2d/a2, if [ 1lo] for one grain lies in the boundary plane. The angle between the defects and [IlO] would therefore be approximately arctan a2,12j3,which always is small unless #? approaches zero (Fig. 5). Only in asymmetric boundaries * 23 in this case, as (110) has a twofold symmetry. 7 Tentatively in planes approximately perpendicular boundary plane and to (110).
to the
STRGCTI-RES
IX
GRAIR
BOUSDARIES
“17
with J? equal to zero can lines perpendicular to [ 1lo] and behaving like edge diclocations be present. Equally, the spacing and to a smaller extent, the direction of these defects would change with boundary curveture. Examples of this behaviour of lines in nonplanar boundaries have been shown in the literature.c6) Bicrystals in a coincidence position have several directions with relatively low indices in common. Relaxation in several sets of planes is possible at small deviations in the orientation and networks might be visible in the electron microscope.‘1~2~3) With the exception of the first order twin position there is however only one common direction of the (loo), (llO\ or (111) types. If this direction is not too much inclined to the boundary plane, the contrast. due to relaxation in the corresponding set of planes would presumably dominate. For twist. boundaries the planes with the lowest indices are parallel to the interface and the strains from relaxation of other planes would be the most, pronounced. The question of dislocations in coincidence boundaries has been investigated in detail by Bollman. The predicted Burgers vectors of such dislocations are displacements of minimum length preserving the atomic regularity in the boundary plane instead of displacements preserving the matching of planes across the boundary as proposed here. In many cases the displacements predicted by the two models would be identical, but relaxation due to matching of planes is less sensitive to the posit.ion of the boundary plane and the actual atomic regularity of the very interface In the model proposed here linear defects would therefore appear also in those cases where a high degree of order is not achieved. For twins, there are three sets of planes with low indices in common. (111) for f.e.e. and {llOj for b.e.c. Ot.her planes of relatively low indices are however common and the actual Burgers vectors could presumably depend on the indices of the boundary plane, If the term grain boundary dislocations is to be used for these defects a Burgers vector with a crystallographic IneaIling should be asigned to them. As the defects are associa,ted with the alignment of a set of planes and the relaxation is expected to take place parallel to the interface, the Burgers vector lies in the boundary and is the distance between the planes as measured in the boundary plane. Thus the Burgers vector does not need to be a lattice vector in either grain. It has been shown above that the direction of the dislocations is such that they in most cases are predominantly screw dislocations. Their strain fields might however be modified by the presence of the
ACTX
218
METALLURGIC.%,
interface. The relaxation process must also depend on both the direction and the actual atomic configuration of the boundary and these could vary from defect to defect in one boundary. While the geometrical significance of the defects is clear the strain field around each of them might therefore be complex and different from its neighbours. VALIDITY
AND
VOL.
1973
into a grain boundary with plane matching should be essentially similar to its strain field in the grain since t,he matching resists the introduction of dislocations. When grain boundary dislocations are present in the boundary, lattice dislocations which decrease the average m&orientation between the matching planes might be preferentially absorbed.
IMPLICATIONS
Levy observed lines in t,ilt boundaries around [loo], [110] and [ill] in grown bicrystals of aluminium and explained them as due to a twist misorientation between the grains. The same type of defects have been found in cold worked tungsten and it has been shown that they are due to a relaxation whereby the (110) planes align across the boundary and that the total misorientation of [l lo] in the interface can be localized along the defects. It therefore seems reasonable that the same type of relaxation should be a general phenomenon and possibly exist even for planes with higher indices, although in those cases the strain fields might not be visible in the electron microscope. Lines should be commonly observed in coldworked material where a texture with approximate alignment of a direction with low Miller indices is present. They should be rare in materials with a more random orientation between the grains. The presence of the defects must imply that an L‘amorphous” layer wit,h a random orientation does not exist between the grains and the plane matching suggests that more than nearest neighbour interactions should be taken into account for calculation of boundary energies, The strain field of a lattice dislocation after moving
21,
ACKNOWLEDGEMENTS
The authors are grateful to Professor G. Brogren for his encouragement and support. The work was financially supported by The Swedish Natural Research Council and The Swedish Board for Technical Development. REFERENCES 1. T. SCHOBER and R. W. B~LLUFFI, Phil. Mag.
21, 109
(1970). 2. T. SCHOBER, Phil. Msg. 22, 1063 (1970). 3. B. LOBERO, H. NORDEN and D. A. SYITH, Ark. Fys. 40, 513 (1970). 4. J. LEVY, Phys. Status. Solidi 31, 193 (1969). 0. B. LOBERQ and H. NORDEX, Ark. Fys. 40, 413 (1970). Phil. Mog. 24,497 6. G. BUZZICEELLI and A. MASCANZONI, (1971). 7. H. GLEITER, E. HOR~BOGEN and G. BIRO, Acta Met. 16, 1053 (1968). W. BOLLMAN, Phil. Nag. 16,363 (1967); 16,383 (1967). :: G. H. BISHOP and B. CHALMERS, Scripta Met. 2, 133 (1968). 10. M. WEINS, B. CHALMERS, H. GLEITER and M. ASHBY, Scri#a Illet. 3, 601 (1969). 11. T. SCHOBERand R. W. B~LL~FFI. Phvs. Statwr Solidi fB) 44,115 (1971). 12. B. LOBERO, H. NORD$N and D. A. S~IITFI,Phil. Mug. 24, 897 (1971). 13. A. R. THBLEN, Phvs. Status Solidi (A) 2. 53i (1970). 14. H. F. Rr.4~ and J: SUITER, Phil. Cilia;. i0, 71’( 1964). 15. S. RANOANATHAN, Acta Cryet. 21, 197 (1966). 16. M. J. WEINS, H. GLEITER and B. CHALMERS, J. appl. Phys. 42, 2639 (19il). IT. C. Goes, Bull. Cercle d’&tudes Nef. 8, 185 (1961). 18. P. H. PUMPHRET, Scripta Met. 6, 107 (1972).
DEFECT
STRUCTURES B.
LOBERG?
IN HIGH
ANGLE
GRAIN
BOUNDARIES*
and H. NORDENt
The structure of high angle grain boundaries in tungsten has been studied using field ion microscopy and electron microscopy. The observed linear defect structure is explained in terms of plane matching across the grain boundary; this model of relaxation across the interface is assumed to apply more generally. STRUCTURES
REGULIERES
DES
DEFAUTS DANS LES JOINTS DESORIENTATION
DE
GRAINS
A FORTE
La structure des joints de grains B forte d&orientation dans le tungstl?ne a Bti Btudibe par microsoopie a Qmission d’ions et par microscopic Blectronique. La structure linbaire des d6fauts qui a Bt6 observk est expliqube par la correspondance des plans B travers le joint de grains; on suppose que ce modele de relaxation ZLtravers l’interface peut s’appliquer d’une fapon plus g&&ale. REGELMdPIGE
DEFEKTSTRUKTUREN
IN GROBWINKELKORNGRENZEN
Die Struktur von GroBwinkelkorngrenzen in Wolfram wurde mit Hilfe der Feldionenmikroskopie und Elektronenmikroskopie unteraucht. Die beobachtete lineare Defektstruktur wird mit einem Model1 ebener Anpassung an der Korngrenze erkliirt. Es wird angenommen, daB dieses Model1 der Relaxation an der Grenzfl&che noch allgemeiner anwendbar ist.
INTRODUCTION Recently there have appeared in the literature numerous reports of electron microscope observations of periodic defect structures in high angle grain boundaries. In some cases networks have been seen,(ls2s3) but mostly the observed contrast has consisted of roughly parallel lines. (4*5*6)It seems clear that phenomena such as large steps in the interface and absorbed lattice dislocations although giving contrast in the electron microscope in general cannot account for all these regular structures. It has therefore been proposed that dislocation-like defects with Burgers vectors normally not present in a crystal can appear in grain boundaries.(4s7) Such dislocations have in particular been predicted to exist when the grains are near a coincidence relationship.@) In such cases a highly regular structure with small repeat distances could be achieved in the interface when t,his lies in a symmetric position, i.e. is along equivalent, lattice planes in the two grains. Deviations from this ideal configuration could then be localized in misfit dislocations with certain Burgers vectors which would preserve the regularity of the boundary structure. Detailed pictures of the atomic configurations in coincidence boundaries has been produced by computer simulations,‘B,lO) and observed linear defects in interfaces between crystals near coincidence relationships have been interpreted as specific misfit dislocat,ions.(l*2*11)
The coincidence model however explains the defects observed in the electron microscope only when the misorientation lies within a narrow range around specific axes ; it also presupposes that the interface * Received May 30 1972. t Department of Physics, Chalmers University of Technology, S-402 20 Gothcnburg 5, Sweden. ACTA 3
METALLURGICA,
VOL.
21, MARCH
1973
213
is ordered and lies along specific planes. It seems clear that a more general model is needed to account for defects observed when these conditions arc not fulfilled. Combined electron and field ion microscope observations in drawn tungsten wire suggest that the existing periodic defects are due to relaxation in the boundary.(12) This idea is further developed in the present paper. EXPERIMENTAL
Most of the field-ion specimens were prepared from as received tungsten wire with a diameter of 0.1 mm and of commercial purity. The wire had the usual pronounced fibre texture with a [llO] axis. The high angle grain boundaries to be discussed here are the [ 1lo] predominantly tilt boundaries abundantly observed in such a material. Generally, the [llO] directions of the two grains do not coincide but lie at a small angle to each other. For purposes of making this clear the relative orientation of the two grains can be described by adding two small rotations to the main tilt around [IlO] : One about an axis perpendicular to [llO] in the grain boundary, or a “tilt misorientation”, and one about the normal to the boundary or a “twist misorientation”. When an additional small twist misorientation is present linear defects have regularly been observed in high angle boundaries of as-polished specimens when examined in the electron microscope. The defect contrast consists of a set of lines, roughly parallel to the [llO] direction. The lines are visible in dark field micrographs, using diffracted beams from either grain and they must therefore represent defects which have strain fields in both grains. At larger twist angles the lines become more closely spaced and begin to look like moire fringes.(13) The closest observed
214
ACTA
METALLURGICA,
VOL.
21,
1973
spacing of the defects is about 50 A whereas at larger twist angles it appears that the localized line structure is replaced by ordinary moire fringes. Examples of the described line contrast are given in Figs. 1-2; others have been published elsewhere.(5*12)
Fm. 2. Dark field micrograph using the common [110] reflexion of a specimen with two grains with a relative rotation of 10”. The boundary defects appear to be discontinuous at inclusions in the boundary; the specimen wire had been doped with alumina. .
Fro. 1. Bright field electron micrograph of a field evapTwo grains with the interface orated specimen. approximately perpendicular to the electron beam extend to the tip and a third grain terminates 800 A up the shank. The relative rotation between the first two grains is 13” around an axis close to [llOl and the boundary plane is approximately (118) and (113). The relative rotation across the second visible interface with linear defects is 63”.
Linear defects have been observed both in specimens with symmetric and asymmetric boundaries containing [llO] as well as in boundaries where the interface is inclined to [IlO] (Fig. 3). This suggests that the presence of the defects is insensitive to the orientation of the boundary plane. That the defects themselves are indeed related to the misalignment of the [IlO] directions of the two grains has been shown by subsequent field-ion microscopy of the specimens.
Fro. 3. Grain boundary defects in tungsten, annealed at 1200°C for 8 hours. The relative rotation is 20” around an axis near [110] and the grain boundary plane is approximately (lOI) of the top grain and (5I3) of the other and the defects lie in the [212] direction of the top grain. (c) Dark field using [ST11 of the other grain. (a) Bright field. (b) Dark field using [200] of top grain. The specimen has been rotated 8’ between the three miorographs.
LOBERG
AND
NORDEN:
REGULAR
DEFECT
In a field-ion micrograph of a single crystal, the [llO] pole is surrounded by a prominent set of rings which are the images of the edges of consecutive (110) planes. In those cases where the misorientation between two grains is a pure rotation around [llO] and the grain boundary runs across the [llO] pole of the micrograph, the (110) planes will always match up at the interface and thus the rings continue unbroken across the boundary. An example of this is shown in Fig. 4. When the [llO] directions are slightly misaligned, the expected image of a boundary in the [ 1lo] region would consist of two sets of offset part rings, i.e. t’he edges of the (110) planes of the two crystals would in general not meet across the boundary. Such an image is not seen; the (110) planes appear to relax so that good matching is achieved across the interface except at the few places where spirals begin.(14*15)An example of this is shown in Fig. 5. In our studies of grain boundaries in tungsten a oneto-one correspondence has always been observed betueen the lines in the electron micrographs and the spirals in the field-ion ones. It is thus possible to predict with accuracy the number of spirals that will appear in the field-ion image from the number of lines present in the electron microscope image of an aspolished specimen. It is also possible to show that the positions of the lines correspond to those of the spirals. The positions of the spirals seen during an evaporation sequence can be plotted on an electron micrograph of the specimen if the evaporation end form, the distance
STRUCTURES
IN
GRAIN
BOUNDARIES
215
Fm. 5. The relative rotation of the grains is approximately 69’ around an axis near the central [llO]. The [ 1lo] poles of the two grains are marked with dots and spirals beginning in the interface with arrows. The corresponding lines observed in the electron microscope were parallel to [llO] within 3” although a tilt misorientation is present as seen in the micrograph above.
removed by evaporation and the magnification and orientation of the electron micrograph are known. The positions of the spirals coincide with the lines, as has been shown elsewhere.(5) It is therefore clear that the linear contrast and the spirals have the same origin and can be related to a slight misorientation of the [l lo] directions of the grains. It also implies that the observed good matching of (110) planes across the boundary in the field-ion image is not due to a relaxation at the specimen surface. INTERPRETATION
seems improbable that the linear defects should be interpreted as absorbed lattice dislocations. If they were, the correspondence between lines and spirals would mean that in all cases there had been a perfect alignment of [llO] directions of the two grains before the absorption of dislocations. Furthermore, the absorbed dislocations must all either have moved in slip planes containing [IlO] or realigned after absorption since the lines are approximately parallel to this direction. They must all have had a Burgers vector with a component of a/2 [llO] as they give spirals in the field-ion microscope and all be of the same sign as all spirals go either clockwise or anti-clockwise. It is unlikely that such dislocations should be the only ones absorbed. An interpretation of the observed contrast as due to ledges can be ruled out for two reasons: firstly, no It
Fxa. 4. Field-ion micrograph of a specimen containing two grains with a relative rotation of 18’ around the central [ 1101. The rings around the centre are formed by the edges of consecutive (110) planes and are continuous across the interface.
216
ACTA
METALLURGICA,
steps are in general seen in field-ion micrographs near the positions in the boundaries where the lines appear ; secondly, the observed spirals cannot be explained by a ledge structure. An interpretation of the defects as the dislocations postulated by the coincident site lattice theory is impossible as the lines are observed even when the orientation between the two crystals is far from a high density coincidence relationship. For example, lines spaced more than 50 A apart have been observed for relative rotations of 13”, 34’ and 59’ around axes near [I IO]. In each case the angle could be measured to within 2” from field-ion micrographs. These rotations and large line spacings would lead to unrealistically large Burgers vectors while those proposed by the coincident lattice theory are smaller than for lattice dislocations. For example, the Burgers vector of the defects in Fig. 1 must be about 15 A if the usual maximum va.lue of 19 is used for the inverse density of coincident sites, C. Disregarding this limit, a value of C = 73 is needed to give the defects a Burgers vector less than 3 8. Furthermore, the coincident site lattice theory does not explain the presence of the screw component along [llO]; that the lines corresponded to spirals could be checked for the rotations mentioned above as the boundary appeared in the [llO] pole of the field-ion micrographs. In view of the combined observations in the two types of microscopes it is suggested that the linear contrast is the result of a relaxation process in the boundary. The relaxation is such that the (110) planes of the two grains are in register over most of the interface and the misfit in the [llO] direction has been localized along the lines. Figure 6 is a schematic drawing of the proposed configuration. A similar interpretation of observed lines in bicrystals of aluminium, essentially based on geometrical considerations, has been given by Levy.c4) DISCUSSION
Generally, a regular defect structure in a planar high angle boundary must indicate that there exists some periodicity in the boundary. This periodicity should be at least one-dimensional and have a crystallographic meaning, i.e. it must be related either to lattice points or to lattice planes in both grains. In most boundaries there are no such repeat distances related to lattice points, as this requires that a vector in the interface is a lattice vector in both grains. The most general boundary where this condition is fulfilled is an asymmetric pure tilt boundary with the two grains rotated relative to each other around an axis [hkl] in the boundary plane. In this case a vector a - (h, k, 2)
VOL.
21,
1973
Grain A GrainB I
(b)
Groin A Grain
B
Fm. 6. Orientation of the (110) lattice planes seen edge on (a) near and (b) far away from the interface. The boundary plane lies in the plane of the paper. The angle between [I lo] of crystals A and B is exaggerated relative to the observed cases.
in the interface connects points surrounded by exactly the same configurations of atoms. In the particular case where [hkZ] is equal to [llO] the periodicity vector is a * (1, 1,O) where a is the lattice parameter. The observed linear defects are however connected with a relative displacement of the grains of only half this size and equal to the spacing of the (110) lattice planes. The configuration achieved by introducing the defects is one where the (110) planes in the two crystals are aligned without the introduction of the same periodic atom arrangement that would be present in the interface with perfect alignment of [llO]. The periodicity which facilitates the creation of the defects is therefore due to lattice planes and not to lattice points. This would explain why the lines are observed not only on asymmetric tilt boundaries but also in boundaries inclined to the [llO] direction. The proposed relaxation is only possible if the energy gained by rearranging the atoms in the boundary region to align the (110) planes across the boundary is greater than the energy necessary to build up the strain field extending into the two crystals around each line. That atoms near an interface indeed adjust their positions from those of two undistorted crystals is clear from the cohesion of grain boundaries and modern theories emphasize the possibility of boundary rearrangement at the expense of some lattice strain.@) The fact that field-ion micrographs of bicrystals with a perfect alignment of a pole with low indices with few exceptions always show a ring pattern continuing across the interface as in Fig. 4 is a strong indication that the matching of planes across the boundary is a favored configuration as compared to one where the planes are staggered across the boundary. Computer
LOBERG
x4s-i~ SORDEN:
REGULAR
DEFECT
simulations of atom arrangements in coincidence tilt boundaries suggest the same.{16) Although the con~gurations with lowest energies in this case were the ones where actual coincidence positions in the interface were lost, the planes perpendicular to the tilt axis with one exception were in register and not staggered. It must be considered how a small twist misorient,ation from a pure tilt posit,ion could give rise to only one set of visible linear defects in the elect,ron microscope. In an ordinary low angle twist boundary at least two sets of screw dislocations must be present to remove long range stresses. If the lines are due to an interaction of nearly aligned planes across the boundary, ILsecond set of defects would presumably be due to a similar interaction between other planes. That other nearly aligned planes exist is evident, as any orientation between two grains can be described by rotations around twenty-four different axes.*(“) However, for all rota,tions around [llO] except those distinguishing coincidence relationships the Millerindices of directions near t,he other axes of rotations are high. This means that the atomic densities in other nearly aligned planes are low and therefore their interaction across the interface weak. Even if a relaxation t,akes place in a second set of planes,? their interplanar spacing is small and the strain from the introduced defects equivalent to that of a set of closely spaced screw dislocations with a small Burgers vector. The contrast in the electron microscope from such defects would be very weak and in most cases invisible. Kor can their existence be easily checked in the field-ion microscope, as rings do not develop around poles with high indices. However, if such a network of two sets of defects is achieved, all planes perpendicular to the twenby-four different axes of rotation are then aligned over each mesh in the netrvork, So far the discussion has been concerned with smalI twist components added to the main tilt rotation. If a small tilt around an axis perpendicular to [llO] is included as well. this can be accommodated by inclining the defects slightly to the [llO] direction(l*~ The spacing of the lines for a superposed twist component of angle /l is d/b, where d is the spacing of the relaxing planes. The spacing for a small superposed tilt component of angle a is 2d/a2, if [ 1lo] for one grain lies in the boundary plane. The angle between the defects and [IlO] would therefore be approximately arctan a2,12j3,which always is small unless #? approaches zero (Fig. 5). Only in asymmetric boundaries * 23 in this case, as (110) has a twofold symmetry. 7 Tentatively in planes approximately perpendicular boundary plane and to (110).
to the
STRGCTI-RES
IX
GRAIR
BOUSDARIES
“17
with J? equal to zero can lines perpendicular to [ 1lo] and behaving like edge diclocations be present. Equally, the spacing and to a smaller extent, the direction of these defects would change with boundary curveture. Examples of this behaviour of lines in nonplanar boundaries have been shown in the literature.c6) Bicrystals in a coincidence position have several directions with relatively low indices in common. Relaxation in several sets of planes is possible at small deviations in the orientation and networks might be visible in the electron microscope.‘1~2~3) With the exception of the first order twin position there is however only one common direction of the (loo), (llO\ or (111) types. If this direction is not too much inclined to the boundary plane, the contrast. due to relaxation in the corresponding set of planes would presumably dominate. For twist. boundaries the planes with the lowest indices are parallel to the interface and the strains from relaxation of other planes would be the most, pronounced. The question of dislocations in coincidence boundaries has been investigated in detail by Bollman. The predicted Burgers vectors of such dislocations are displacements of minimum length preserving the atomic regularity in the boundary plane instead of displacements preserving the matching of planes across the boundary as proposed here. In many cases the displacements predicted by the two models would be identical, but relaxation due to matching of planes is less sensitive to the posit.ion of the boundary plane and the actual atomic regularity of the very interface In the model proposed here linear defects would therefore appear also in those cases where a high degree of order is not achieved. For twins, there are three sets of planes with low indices in common. (111) for f.e.e. and {llOj for b.e.c. Ot.her planes of relatively low indices are however common and the actual Burgers vectors could presumably depend on the indices of the boundary plane, If the term grain boundary dislocations is to be used for these defects a Burgers vector with a crystallographic IneaIling should be asigned to them. As the defects are associa,ted with the alignment of a set of planes and the relaxation is expected to take place parallel to the interface, the Burgers vector lies in the boundary and is the distance between the planes as measured in the boundary plane. Thus the Burgers vector does not need to be a lattice vector in either grain. It has been shown above that the direction of the dislocations is such that they in most cases are predominantly screw dislocations. Their strain fields might however be modified by the presence of the
ACTX
218
METALLURGIC.%,
interface. The relaxation process must also depend on both the direction and the actual atomic configuration of the boundary and these could vary from defect to defect in one boundary. While the geometrical significance of the defects is clear the strain field around each of them might therefore be complex and different from its neighbours. VALIDITY
AND
VOL.
1973
into a grain boundary with plane matching should be essentially similar to its strain field in the grain since t,he matching resists the introduction of dislocations. When grain boundary dislocations are present in the boundary, lattice dislocations which decrease the average m&orientation between the matching planes might be preferentially absorbed.
IMPLICATIONS
Levy observed lines in t,ilt boundaries around [loo], [110] and [ill] in grown bicrystals of aluminium and explained them as due to a twist misorientation between the grains. The same type of defects have been found in cold worked tungsten and it has been shown that they are due to a relaxation whereby the (110) planes align across the boundary and that the total misorientation of [l lo] in the interface can be localized along the defects. It therefore seems reasonable that the same type of relaxation should be a general phenomenon and possibly exist even for planes with higher indices, although in those cases the strain fields might not be visible in the electron microscope. Lines should be commonly observed in coldworked material where a texture with approximate alignment of a direction with low Miller indices is present. They should be rare in materials with a more random orientation between the grains. The presence of the defects must imply that an L‘amorphous” layer wit,h a random orientation does not exist between the grains and the plane matching suggests that more than nearest neighbour interactions should be taken into account for calculation of boundary energies, The strain field of a lattice dislocation after moving
21,
ACKNOWLEDGEMENTS
The authors are grateful to Professor G. Brogren for his encouragement and support. The work was financially supported by The Swedish Natural Research Council and The Swedish Board for Technical Development. REFERENCES 1. T. SCHOBER and R. W. B~LLUFFI, Phil. Mag.
21, 109
(1970). 2. T. SCHOBER, Phil. Msg. 22, 1063 (1970). 3. B. LOBERO, H. NORDEN and D. A. SYITH, Ark. Fys. 40, 513 (1970). 4. J. LEVY, Phys. Status. Solidi 31, 193 (1969). 0. B. LOBERQ and H. NORDEX, Ark. Fys. 40, 413 (1970). Phil. Mog. 24,497 6. G. BUZZICEELLI and A. MASCANZONI, (1971). 7. H. GLEITER, E. HOR~BOGEN and G. BIRO, Acta Met. 16, 1053 (1968). W. BOLLMAN, Phil. Nag. 16,363 (1967); 16,383 (1967). :: G. H. BISHOP and B. CHALMERS, Scripta Met. 2, 133 (1968). 10. M. WEINS, B. CHALMERS, H. GLEITER and M. ASHBY, Scri#a Illet. 3, 601 (1969). 11. T. SCHOBERand R. W. B~LL~FFI. Phvs. Statwr Solidi fB) 44,115 (1971). 12. B. LOBERO, H. NORD$N and D. A. S~IITFI,Phil. Mug. 24, 897 (1971). 13. A. R. THBLEN, Phvs. Status Solidi (A) 2. 53i (1970). 14. H. F. Rr.4~ and J: SUITER, Phil. Cilia;. i0, 71’( 1964). 15. S. RANOANATHAN, Acta Cryet. 21, 197 (1966). 16. M. J. WEINS, H. GLEITER and B. CHALMERS, J. appl. Phys. 42, 2639 (19il). IT. C. Goes, Bull. Cercle d’&tudes Nef. 8, 185 (1961). 18. P. H. PUMPHRET, Scripta Met. 6, 107 (1972).