- Email: [email protected]
The migration of tilt boundaries and facet development in thin gold bicrystals
r(cta Mrtuhwga.
Vol. 25. pp. lC9-110‘.
Perpmon
Press
LY7: Printed
in Great
Bntam
THE MIGRATION OF TILT BOUNDARIES AND FACET DEVELOPMENT IN THIN GOLD BICRYSTALS* R. M. ALLEK Department of Materials Science and Engineering. Cornell Univ.ersity. Ithaca. U.S.A. and
P. J. GOODHEW Department
of Metallurgy and Materials Technology, University of Surrey. Guildford. England (Receirerl 20 &grrsr 1976; in rrrise[ifbrrn
14 Fehntar.r 1977)
Abstract-Thin films of gold were welded together to create bicrystals containing [OOl] twist grain boundaries. The bicrystals were heated in the transmission electron microscope and the migration of first twist and subsequently tilt regions of the boundary was observed directly. The tilt boundaries showed a tendency to facet which grew less pronounced as the twist angle was increased. The preferred boundary planes appeared to be controlled either by inhomogeneities m the thickness of the original single crystal films or by the lowest energy section of the O-lattice, w-hich in this case is the plane with the highest density of O-points. The interaction of a migrating boundary with bubbles of a range of sizes was also observed and the pinning effects are reported. R&urn&-On a soudi des films minces d’or aiin de fabriquer des bicristaux contenant un joint de torsion [OOl]. On a chauffe les bicristaux dans le microscope et Ton a observe directement la migration des regions du joint en torsion, puis des regions en flexion. Les joints de flexion avaient tendance a presenter des facettes dont l’importance diminuait lorsqu’on augmentait Tangle de torsion. Les plans preferentiels du joint semblaient control& par le hetirogtneites dans l’epaisseur des films monocristallins otiginaux, ou bien par la section du reseau 0 de moindre tnergie, qui est ici le plan dont la densiti de points’ 0 est la plus grande. On a egalement observe l’interaction dun joint en tours de migration avec des bulles de tailles diverses et l’on decrit les effets d’ancrage. Zusammenfassung-Diinne Goldfilme wurden miteinander verschweiBt. urn Bikristalle mit [OOl]-DriUkorngrenzen herzustellen. Die Bikristalle wurden im Durchstrahlungselcktronenmikroskop aufgeheizt; die Wanderung erst der Drillbereiche und darauffolgend der Knickbereiche der Komgrenzen wurde direkt beobachtet. Die Knickgrenzen neigten dazu, Facetten zu bilden, die mit zunehmendem Drillwinkel weniger ausgepragt wuchsen. Es schien, daB die bevorzugten Komgrenzebenen bestimmt sind entweder durch Inhomogenitlten in der Dicke der urspriinglichen Einkristallfilme oder durch den Abschnitt des 0-Gitters geringster Energie, welcher in diesem Fall die Ebene mit der griil3ten Koinzidenzpunktdichte ist. Die Wechselwirkung einer wandanden Korngrenze mit Blasen in einem be+ timmten GrBDenbereich wurde ebenfalls beobachtet; Verankerungseffekte werden mitgeteilt.
1. INTRODUCI’IOX
The simple thin film welding technique for the pteparation of bicrystal metal samples suitable for direct examination in the transmission electron microscope has proved useful in many studies of grain bounderies. [l-9] A sample of this type contains an interface of known inclination between two identically constituted grains at a known misorientation. Boundary plane, inclination and intergranular misorientation may thus be chosen as experimental variables. As-welded samples present large areas of grain boundary normal to the electron beam, making them well suited to the study of interfacial microstructure *This work was carried out while P. J. Goodhew was a visiting fellow in the Department of Materials Science and Engineering, Cornell University. R. M. Allen is now at the Department of Materials Science, M.I.T., Cambridge, Mass.
by electron microscopy. In particular, (001) twist boundaries in gold bicrystals have been extensively examined, and shown to have structures depending both on the planar density of coincidence sites (PCSD) within the interface [2] and the O-lattice geometry of the system. [S, 61 Thin film specimens containing extended grain boundaries parallel to their free surfaces are clearly metastable. It has been found [I, 7,9] that, during low-temperature annealing. gold specimens originally containing twist boundaries of this type will reduce their interfacial area by boundary migration. Figure 1 shows this process schematically. If the destruction of twist boundary area is carried to its conclusion, the sample will reach a new metastable state; this consists of a mosaic of columnar grains of two types, corresponding to the two original orientations of the starting bicrystal. Each grain spans the total specimen thickness and all are separated by tilt boundaries nor1095
1096 ALLEN
AND
GOODHEW:
MIGRATION
AND FACET DEVELOPMENT
OF TILT BOUNDARIES
$3011 a.
1 2
I,
I
b.
Fig. 1. The temperature range over which each of the eight bicrystals was heated in the microscope. The lowest temperature in each range indicates the onset of stow migration. At the tfighest temperature, fast migration was clearly occurnng in the small region being examined.
d.
stage of a JEOL JEM 200 microscope. Micrographs were taken during the heat-treatment with the specimen stabilized at a selected furnace setting. Figure 2 shows the range of temperature over which each of Fig. 1. A schematic illustration of the boundary migration the specimens was photographed. occurring in our experiments. The or;iginal twist boundary For the 2’ twist bicrystals, observations of the (a) migrates a smail distance until it Intersects the free surgrowing single crystal patches could be made directly face (c) forming segments of tilt boundary ~~nd~cu~ar to the film. The tilt boundaries migrate laterally until a11 with bright-field imaging since the tilt boundary the twist boundary is removed. leaving a columnar polymigration corresponds to the disappearance of the bicrystal. Grooves (shown dotted) may form at the bounvisible twist boundary dislocation structure. This was dary-surface intersectlon. not possible for hot stage observations from higher angle bicrystals. Instead, these samples were examined ma1 to the specimen surface. but having a variety of in a dark-field image formed from a beam singly difinclinations with respect to a reference direction in fracted from one of the crystals. A growing single the plane of the film. crystal patch was thus seen as an expanding dark It has been shown that. for bicrystals of fixed mis- region during the annea1. After the initial dynamic observations had been orientation, grain boundary energy is a function of interfacial plane inclination. [IO] Accordingly, it has recorded, the annealed specimens were transferred to been suggested [P] that the con~gurations adopted by a Siemens Elmiskop 102 microscope for a static SW the columnar grains of these annealed bicrystal films vey of the single crystal patch shapes which had been produced during the hot stage anneal. Micrographs reflect some type of Wuiff-Herring faceting. [ill Analysis of micrographs to find preferred orientations were taken from at least six different regions of each of the column walls would therefore be an extremely bicrystal. Each picture was marked with a standard useful method for determining the relative energies reference line and then computer analysed to obtain of various inclinations of boundary in a fixed-misorthe distribution of crystallographic directions which ientation system. The feasibility of such a study has made up the borders of the single crystal patches. The individual distributions from each specimen were already been established. [7.9] The purpose of this research was primarily to then processed to arrive at a final average distribution examine the role played by boundary migration kin- of orientations for each bicrystal misorientation. A etics in producing the final columnar configuration description of the computer program will be given in annealed gold bicrystals. Additionally, we hoped elsewhere. [9] to compare observed boundary migration behaviour 3. RESULTS AND DfSCLSS1ON with existing theories of pinning and migration kinetics. 3. I itritial sitrgle crystal patch fortnatiotz 2. EXPERMENTAL A series of bicrystal specimens containing (001) twist boundaries of various misorientation angles between 2 and 4.5’ was prepared from pairs of nominally 25nm thick gold single crystal films in the usual way. Cl] Each of these samples was then annealed at a low temperature (c270”C) in the hot
An idealized mechanism of the type depicted in Fig. 1 presents a substantial free energy barrier to the homogeneous nucleation of single crystal patches. A relatively large boundary area increase must occur before any part of the boundary reaches a free surface and lowers the free energy of the system by being eliminated. In practice, initial breakthrough in the 2” bicrystals takes place preferentially at the edges of
,ALLES
hSD
GOODHEW:
51IGR.4TION AND FACET DEVELOPMEXT
OF TILT BOUSDARIES
IW
Fig. 3. A region of 2. twist boundary showing many small pinholes and bubbles. Breakthrough ot single crystal patches has occurred at a pre-existing twin (T) and at one of the pinholes (P).
either weld bubbles or pinholes, and also at twins in either single crystal (Fig. 3). Presumably, these inhomogeneities act to reduce the energy barrier for breakthrough. In particular, bubbles and pinholes probably decrease the effective distance through which the boundary has to expand to reach a free surface. A thermal groove at the twin-surface intersection [I?] may perform the same function for breakthrough at a twin. Additionally, part of the energy cost involved in extending the grain boundary to the surface would then be met by the destruction of an area of twin interface at such locations. We were readily able to obtain as-welded 2’ specimens with large areas (- 1 x I pm) free from grownthrough single crystal patches. This was never the case for the higher-angle specimens. which suggests that they are structurally less stable or more mobile. There have been very few calculations of twist boundary energies. but the espectation that high-angle boundaries should have energies much greater than low-angle boundaries is reinforced by the recent calculations of Lodge and Fletcher. [ 131 These uorkers found that the energy of twist boundaries in aluminium should rise to a relatively constant high value ( _ 600 erg cm-‘) for twist angles aboLe IS-. .AGuming that this result is qualitatively applicable to other
f.c.c. metals, the observed relative stability of our boundaries is easily understood. Although preferred breakthrough sites were also active in the higher angle boundaries, most of the initial single crystal patches did not seem to be associated with such defects. Rather. the majoriF appeared as square regions. randomly distributed. exh several tens of nm across. with edges aligned with [lOO] directions in the crystal being grown through (Fig. 4). As we have mentioned above. these patches were generally formed during bicrystal welding: however. on the rare occasions when a breakthrough was observed during the hot stage anneal. the freshly created single crystal region quickly assumed the same general shape, size and directionalit! as the prsexisting patches. It was very common to find a grownthrough region associated with one of the single cry+ tals. in very close proximity to a patch produced by breakthrough of the other single crystal. In some cases, the complementary squares impinged on each other. However. the edge directions of an) particular square still matched the [ IOO]directions of the crystal being gronn through. This suggests that the mechanism responsible for the square orientation is independent of the local bounder> structure. Because it matches a IOU-index direction in one cr!sta.l. it must
1WS ALLEN XNIJ GOODHEW:
XlfGRATIOS
AND FACET DEVELOPMEW
OF TILT BOL’NDARIES
Fig. 4. A pair of complementary 200 dark-field images of a 37” twist boundary. The initial patches of growth-throu~b single crytaI are aligned parallel to [ IOO]in each crystal.
be assumed rather that the orientation stems from some property of the crystal being grown through. We base carefully examined the original single crystal thin films of old in the eiectron microscope. We frequently obserse. in very low contrast, square
and rectangular defects of approximately the same dimensions as the square breakthrough patches in the bicrystals. It has not proved possibte to determine whether these arise from the thin film growth mechanism or are a replication of a structure on the salt
ALLEN
.a\D
GOODHEW:
MIGRATIOS
.tND
substrsts. In either case. ho\vever. w can expect the welding process to introduce a locall>- higher strain around the square regions and thereb! provide a locall> higher driving force for boundary migration.
Several diRerent aspects of the dJnamica1 behaviour of the bicrystal were characterized. For instance, after a limited number of initial breakthrouyh patches had formed within a given specimen region (as described in the previous section). further single crystal grolvth took place primaril) by expansion of these esisting patches rather than bl; further isolated breakthroughs. X second common effect was that bubbles lying in the initial t\\-isi boundary were seen to interfere with patch growth by a pinning process. These bubbles (generally ranging in diameter from one to
FACET
DEVELOPblEST
OF TILT
BOCSDARIES
1099
tens of nm) are quite common in bicrystal >Tecimens of this t)v, and are believed to be produced by a sintering process during sample welding. [I] Such defects are distributed randomly in the specimen midplane and often exhibit Moire fringes [j] arising from double diffraction. [I41 Good examples can be seen in Fig. 5. Growth, within the areas examined in the hot stage, generally occurred in one of two distinctive modes. At lower annealing temperatures. single crystal patch expansion was extremely slow and the continuous progress of the advancing patch borders was visible. For instance. in the 1’ bicrystal the border appeared to move by the co-operative slip of several of @e real lattice dislocations present at the patch edge, concurring with the results of an earlier study. [I] At the higher temperatures used. growth became very
(b)
Fig. 5. lab .A large single crystal patch in a low-angle specimen. In two places. dislocations have &n removed from the migrating tilt boundary and remain attached to a bubble. The geometry is shoan schematically in (b).
hst xxi jerky. so that whole sections of the specimen seemed to become singie cr>rtal ail at once. in this case. the edges of the patch were only visible when its motion had been tzmpomrily arrested. as at a group of bubble obstacles. The migration behabiour was rstrcmely inhomogeneous on the scale of ths observations: some sections of the bicrystal became columnar at temperatures which produced no grotvth at all in other regions of the same specimen only tens of pm away. The hot stage used was a furnace type. heating by radiation. It is not apparent why this irregular specimen behaviour should arise. since there should be no gross temperature gradient within the furnace. but similar inhomogeneous structural response to annealing on roughly the same scale has been reported before in recrystallization experiments. [ 141 The fact that the columnization process occurred mainly by patch growth rather than repeated isolated breakthrough suggest that the patches are heterogenously nucleated. Their subsequent growth by tilt boundary migration is accompanied by the continuous reduction of specimen free energy, since twist boundary area is destroyed continuously. while fess tilt boundary is created. For the simplest realistic model. of a square breakthrough patch of side h in a bicrystal of half-thickness t. the driving force for boundary migration per unit area of the tilt boundary is
F = & r
_
brILr --, h
(1)
where ;‘rW and j’~,t7 are the grain boundary energies associated with the twist and tilt boundaries. respectively. Unless yT,\.and yIlLI arc very diRerent this driv-
dx
Section
Fig. 6. The geometry of thermal grooving and tilt boundary migration at a square patch of ~idr IL
ing force rises rapidly to a constant flue of ;‘r\t I as h increases. The two distinct growth rates obserl-ed may mercl! represent the behaviour at two different rsmperatures or may reflect two different migration modes for the tilt boundaries. This could be understmd in ternts of the ‘drag and breakaway’ concepts in the kinetic
(br
Fig. 7. A sequence of micrographs showing the interaction between a bubble 1A1and the migrating boundary. The boundarv segments at each side of the bubbb bow out [a) and (bt until the boun&r: puIIs free and re-establishes a straight edge (c). In this case, the Moire frinps which remain ins& the bubble indicate that a Ioop of tilt boundary has been pinched 0%. theories of interfacial migration. [ 16.171 In addition to tkeir intrinsic s~ruct~~re-de~e~deot mobility, at lower temperatures. the tilt boundaries may be subject to a drag force stemming either from the traditional solute cloud formation [Ia 173 or from the production of a thermal groove at tkeir intersection witk a free surface. [I& 73 It is dit?icult to gauge the mqnitude of the solute effect, but we can estimate tkc effect of grooving in
terms of its effect on equation I 1k The introduction of a groove of depth r, and root angle 2P is show;n in Fig. 6. The energy reduction AE on migration of the tilt boundary by an increment dx is thy AE = yT‘vk ds _ ;rILT[z &,.jt _ t, + dtt I ir dr]. The driving force for patch growth dE dx is thus dE - = y*\, It - j(T,,.T[zlr - t,i + h coz $1. dx
110’ .\LLEN
ASD
GOODHEW:
MIGRATIOlr; AND FACET DEVELOPMENT
since dr &c = cot 6. The force per unit are3 of tilt
boundw F=-
is then given by dE h(r - tJ ds ’ ’ a-- Ynv t--L
h,‘TEW
- &I f h cot @J,
[$
h(r - tJ
which reduces to e&ation (1) in the absence of grooving (r, = 0. cot 8 = 0). Typical values of 6 for the grooves at the intersection of high-angle grain boundaries with f.c.c. metal surfaces are in the region of 80”. [I21 For Iow-angle or speciai boundaries having lower energies, the value will be higher. We are therefore concerned with values of cot 0 in equation (2) of 0.2 or less. In order to make clear the variation of driving force with patch size we can replace (f - t& with t’, cot 8 with 0.2 and both yTw and Y~,,,~with 7 in equation (2). This gives the approximate expression .. I721
F+Cl&xJ.
while making the equivalent equation (1) produces
approximations
1
in
2t F =- ;l-7;. [ The maximum effect of groove pinning is therefore to change only slightly the critical size of patch (i.e. values of jr) above which a positive driving force exists for patch growth. The ultimate driving force at large It is scarei)’ affected, since 0.8/r’ and l/t are unlikely to differ significantIy. From the above analysis. we can deduce that pinning by thermal grooves should not be a serious effect in these experiments, although it has previously been reported in other thin-film experiments. [ISJ What is clear from the analysis is that the simple model described here gives rise to a negative driving force for patch growth until the patches are about 2t in diameter. The nucleation of the patches must therefore be heterogeneous, as postulated in the previous section. At high annealing temperatures, the tilt boundaries exhibit rapid. jerky motion which is characteristic of breakaway from the drag-producing entities, whether they be grooves or solute clouds. Although the temperature regimes in which migration was observed differed widely among the boundaries studied (Fig. 2), it is possible, making some assumptions, to deduce a very approximate activation energy for migration for each boundary. Our assumptions were (a) that the minimum and maximum migration velocities observed, corresponding to the lower and upper temperature limits in Fig. 2. were one fifth of a field of view (at 100,000 x ) in i h and a whole field of view in lOOs, and (b) that at both extremes of temperature the migration velocity o is git;en by c x DT?; where D is the appropriate diffusion coefficient.
OF TILT BOCSDARIES
For most of the boundaries studied, the activation energy, Q. was calculated to lie between 5 and 8 kJ mol- I. which is more than half of the activation energy for self di&sion in gotd (_ 10 kJ mol-‘1. Activation energies of this magnitude have been reported previously. Cl93 but it is probably unwise to place much reliance on the absolute values of Q from our experiments. since the data are so imprecise. For the remaining two boundary specimens. the estimated activation energies were much higher (10 and 30 kJ mot - ‘), and this is probably accounted for by the special nature of these boundaries. which were both low-angle boundaries with misorientations of approximately 2’. Our data are therefore in agreement with the evidence that activation energies for boundary migration increase greatly as the boundary misorientation decreases (e.g. refs. 22. 2.3;).but do not permit us to distinguish between the conclusions of Rutter and Aust [20] and Viswanathan and Bauer[X] as to the relative mobilitics of E:j and “random” boundaries. As mentioned above, the bubbles in the twist boundary interfered with the growth of single crystal patches. It was observed that whenever the perimeter of a patch struck a bubble. that section became pinned. The boundary segments on each side of the bubble were still free to move, however, and as a result the perimeter would bow out around the bubble. Eventualty, a limit was reached whereupon the pinned section would snap past the bubble catastrophically and catch up with the advancing boundary. Occasionally. the bypassed bubble region would remain bicrystalline, as could be seen from its Moirl fringes (e.g. at A in Fig. 7). A bubble in the interface is an area devoid of twist boundary. As a consequence of this, an advancing tilt boundary striking the bubble perimeter is suddenly bereft of the major driving force for further motion. This is probably sufficient to pin the boundary, but it is also possibIe that sintered bubbles may actually reduce each single crystal film thickness in their Iocality. Thus an adl-ancing tilt interface is able to penetrate into the bubble region, but becomes pinned because further motion would require the tilt boundary to increase its own area. to reassume the fufi film thickness awa? from the bubble. With no other driving force. this ensures that further motion in any direction requires a local free energy increase. As the tilt interface moves past the bubbIe during the bowing process, new tilt boundary area is continuously created. This stores more and more energy in the system until, finally, it is favourable for the pinned section to slip past the bubble and straighten the bow in one rapid release of energ. In some instances, passage across the bubble is so difficult that the bowing process pinches the bubble off instead and a smaI1 region of bicrystal remains behind the migrating interface. Atthough such a small loop of boundary should be rather unstable and would be expected to shrink, it seems probable that whatever effect
ALLES
.ASD
GOODHEW:
MIGRATION
AND FACET DEVELOPMEXT
OF TILT BOUSDARIES
1103
pinned it sufficiently to prevent it being dragged across the bubble. also stabilizes it after it is pinched Of?.
Our general observation that bubbles do not totally
stop boundary motion is in accord with a simple consideration of boundary pinning by spherical obstacles. As Zener first pointed out (241 the maximum pinning force exerted by an obstacle of radius r is my;. Since the driving force per unit area is approximately ;‘,t, in our experiments this will exceed the pinning force unless the bubbles are closer than about 3r apart. Since we have almost never seen this density of bubbles. complete pinning should indeed not occur in our specimens. In the 2’ mist boundaries, the smallest twist interface bubbles had a diameter of the order of the dislocation spacing ( _ 10 nm). It was very common to find such a bubble ‘sitting astride’ a single dislocation line of the orthogonal grid. In that case, during annealing, the bubble pinned only that single line in place as the single crystal patch grew past. This produced a charateristic feature: a small bubble in a single crystal region frequently appears to be connected to the remnant twist grid by a single dislocation (Fig. 5). An equivalent description of the situation is that interaction of the migrating low-angle tilt boundary with the bubble serves to remove a single dislocation line from the tilt boundary. This directly confirms the hypothesized mechanism for low-angle boundary; small obstacle interaction advanced as a result of earlier macroscopic studies of boundary migration in metals by Bainbridge et al. [25].
1,
mJl
m ccw
Fig. 8. A polar plot produced by computer analysis of several micrographs of an 11’ twist bicrystal. The plot shows the percentage of the total tilt boundary length vrhich lies within each 5’ inclination range. The [OlO] and [llO] directions for each gram are indicated. as is the quadrant which would be expected if the boundary inchnations acre randomly distributed. The [110-j, peak clearI?: extends far beyond the kd limit attached to the random value.
A typical computer result from the static survey of the directions adopted by the borders of the patches is shown in Fig. Y. Such polar plots demonstrate the distribution of inclinations of the tilt boundaries forming the borders of single crystal patches. Only one point has been plotted per 5’ interval in the quadrant. The connecting curves are purely for visual convenience. A subscript cw or ccw on a crystallographic direction indicates that the direction belongs to either the clockwise or counterclockwise single crystal of the bicrystal as it was positioned during the entire static survey. A subscript m indicates a ‘mean’ direction: i.e. the [IOO], bisects the smaller angle between the [IOO],,, and [IOO],,, of a system. The fact that the ccw directions appear to be rotated clockwise from the cw directions is an artifact of the data processing. Each sample was allowed to undergo fast growth in a hot stage anneal before being trdnsferred for the static survey. Thus growth was stopped before the specimen became wholly columnar, in contrast to previously reported experiments. [7,9] During the survey, only those single crystals patches clearly very much larger (> 1 nm’) than the initial breakthrough patches were photographed. The major part of the border of such regions must have been created during growth fast enough for the boundary to have broken away from any estrinsic drag structures. The error in positioning the distributions with respect to the known sample directions (measured from diffraction patterns) was estimated from the I I specimen results. The tilt boundaries in this specimen showed a strong tendency to line up along the mean (110) directions in the (001) plane; that is. along tine directions in the dislocation grid. This is reflected in the large peak of the corresponding distribution, centered on [llo],. .Angular error for all the distributions was taken as the half-width of this peak. or roughly one interval. To give an indication of peak significance. each distribution was marked with an arc representing the orientation distribution of a perfectly random system (i.e. with circular patches). Only two major features of the esperimental plots are found consistently. within the angular error of the experiment: a fairly strong [lOO],, = [OIOICN peak (in all but the 2’ boundary) and a [1 lo], peak. strong for low-angle boundaries but decreasing and eventually disappearing with increasing twist angle. For those specimens surveyed by dark-field microscopy. by chance. nearly twice as many photographs were taken of regions where the ccw component crystal was growing through the cw crystal. rather than vice versa. Since asymmetric behaviour seems unreasonable. we assume that the [100-J,,, peaks merely represent a patch shaping dependence on some property of the crystal being grown through during the anneal. Apparently, the same mechanism which governs the
lit34 ALLEN .+ED GOODHEW:
MIGR-\TION AND FACET DEVELOPMENT
OF TILT BOUSDARIES
Table I. Gtxm~tricd data on the O-lattice and coincidence site lattice at the three misoriea~a~ions of ~~FI.SS~ E (highest density of coincident sites) for the @IIt twist boundaries O-lattice
Parameter 0
(nrn!
Coincidence site lattice (CSL)
Densest direction
E:
Parameter (nm)
initial breakthrough shapes is also active to some estznt during fast growth. Once again, the 1’ bound-
ary was the only exception to this mechanism. although since bright-field imaging was used we do not know whether or not the data was prejudiced in favour of one component crystal. The [ 1IO],,, directions do not generally correspond to low index directions in either of the two single crystals of a bicrystai. However. for this twist system of boundaries (i.e. with the two crystals misoriented by a rotation about a mutual [OOI] axis) the [I lOJ,,, direction is a low index direction in both the coincident site and O-lattices (see Table 1). For a review of the concepts and geometrical significance of these ConStructions the reader is referred to the book by Bollmann. [263 The orientation and lattice parameters of the coincident site lattice (CSL) are discontinuous functions of the twist angle in these systems. The [ 1IO], peaks might be expected to be produced by a mechanism dependent on some property of the CSL, such as a preferential seiection of tilt planes with the highest pianar density of coincident sites (PCSD). or preferred tilt boundary orientations along secondary grain boundary dislocation lines. However, the observed peak behaviour as a function of twist angle is not well correlated with the CSL behaviour. Certainly, no peak is produced at the 37’ twist angle, which has the lowest non-unity Z (ratio of real lattice site density to CSL density) of any (001) bicrystal angle. Yet it is precisely at this angle that a CSL-determined effect should be strongest. Hence we conclude that the [l lo],, peaks do not arise from a CSL-related mechanism. On the other hand. the orientation and lattice parameters of the O-lattice are continuous functions of twist angle. The most dense O-lattice plane is the tilt boundary containing a [llO],,, direction. The O-lattice parameter is a monotonically decreasing function of twist angle. The hypothesis that the preferred tilt boundary orientation is governed by the planar density of O-points. while the peak height varies with O-lattice parameter. yields a good correlation. with observed peak behaviour. Another possibiIity is that the tilt boundary prefer-
Orientation of cell edge
Highest PCSD direction
Direction of secondary gbds
ence is governed by kinetic effects. In that case, either the interaction with obstacles or one of the two passible drag mechanisms must operate to select the preferred tilt boundaries during growth. \Ve will now consider these possibilities. On the scale of the micrographs processed in the static survey. the bubbIes were essentia& a randomlydistributed array of point obstacles in the (001) plane. As such, it is diRicult to see how the co-operative effort of several pinning bubbles could consistently select tilt boundaries aligned along El lo], directions. Also. as can be seen in Fig. 9, the bubble density was low enough to allow manifestation of the [I IO],,, preference between pinning locations (i.e. in spite of the bubbles). We have shown that most of the suneyed patch perimeter length must have been formed during fast growth. that is. after the tilt boundaries have broken away from their drag structures. Theoret& calculations have suggested that in this regime the interfacial migration rate is determined by how rapidly the boundary moves through the bulk, rather than by any surface thermal grooving erect. [ig) This implies that the determining kinetic mechanism should be the interaction of the moving interface with solute impurity atoms in the bulk. Models of such behaviour thus far point to the ability of a grain bout&r)- to entrap solute atoms as an important factor in migration. [17] It has been shown experimentally that gain boundaries with high PCSD are not good solute traps. [17] On this basis. such boundaries in Cc.c. metals in general might be espected to move faster. Evidence for this effect has been found in studies of migration rates.[Zg] The [llOJ, tilt boundaries in these systems are general@ among the hrgest PCSD tilt planes posssible. The considerations just outlined would seem to indicate that. if the kinetics governed the growth shaping. these planes would move faster than others and hence would ‘grow out’ of the perimeter rather than remaining behind preferentially. In addition. the kinetic shaping process has been linked to the CSL, and so should have a strong effect in the X5 bicrystal. No such effect is in evidence. For these reasons, we argue that the growth kinetics do not produce the [I IO],,, peaks observed.
L1’ bicrptnl in which two singk crystal patches (SIX) have bqun tr, meet funrAg tilt boundary at T. .A iargr itre~l of bicrystnl containing twist boundary remak fBI?i! 2nd the tilt boundaries forming the edges of this area can be seen to bz aligned along the f I ii& dirrcrbn Fig 9. A region of
a pure
despite being pinned by bubbles at several points {Bg
Based on this anafysis, the following exptanation of the or&in of the [f tO],,l peak is postulated. The patch gowth, during annealing, satisfies the theoretical requirements for WuK-Herring faceting to MXI.K. [l 13 The patch shaping. and hence the preference for [IlO];,, boundaries, is therefore determined by such faceting. The [l lo],,, tilt boundaries can be concluded to represent distinguishably lower sner_ey orientations of tilt boundary planes. The presence of a significant number of grain boundary; ledges co&d possibly lead to misleading apparent facets, but we have no evidence for their presence: no iedges were resolvable either during or after patch growth in these experiments. Also, if boundary migration had occutred by a tedge mechanism, we would expect the ledges to join regions of close-packed plane. [29] Woweser. in other experiments with thin gold bicrytaij, aI! the small steps and facets which we have resolved proved not to consist of close-packed planes in either grain. [9] so there is not even inciirrct evidence for such led_ees in these specimens.
faces consist
of nets of real fatticr 3x5 distoeritions. this resutt is not surprising. The [I 10]j, boundaries are symmetrical tilt boundaries and has: y&e lowest dislocation density of any possible plane orientation. As such. they would bc expected to be of lower energy for a fixed tivist angle. [Wj The question remains wh\’ the facet&g process takes place in those sytems wvjrh t\*is: angles too large for the grain boundary to be rea!istisaDy considered as a network of discrete fartics dislocations. We feel that the correlation between &c obsersed faceting and the behafiour of the U-tarii=~ suggests m ansbvrr: the primary grGn boundan- &locations igbdsl on the [1 IfQ,~ pI:nx:s bar-e the priodicity of the O-lattice and so their spacing. althouph the widest of all possible tilt gbd arrays at a &en r.vist angle. decreases continuousl\- and monotonic:& as a function of ttvist angle. At low twist an$es. the primary r&ss!ion structure results directly in the familiar nctuorks of lattice dislocations. Xt hi@er an&r. although ir is unlikely that primdry pbd structure esists in the i;?rm of discrete line defects. electron diffkaaion cGdence indi-
1106 ALLEN ASL) GOODHEW:
~MtGR?ITION AND FACET DEVELOPMENT
fates
at
that
the
atomic
reiaxation
grain
boundaries
welded gold bicrystat specimens) still exhibits the periodicit)- of the O-lattice. [6] Therefore we suggest that [IlO], tilt planes appear to exhibit lower energies at twist angles ~30’ due to the relatively large periodicit! of their atomic relaxation structure lbvhether manifested in discrete line deftits or not) compared to that found on other planes. This allows the system to preferentially select these planes by a Wuff-Herring faceting mechanism. producing the observed [I IO], peaks in the orien~tion distribution. We then suggest that for hicrystals with angles >30’ the O-httice periodicity has been reduced to too fine a scale (GO.5 nmf for a significant energy difference to exist between various tilt orientations. This could well correspond to the merging of discrete dislocation cores into a continuous plate of core material in the manner suggested by Li. L-311 Although any inclination of such a boundary will result in an energetically equivalent plate of core material. the atoms still relax (albeit in a non-Hookean manner) periodically. Therefore no preferential selection of tilt planes occurs on this basis and the [ 1IO],,, peaks disappear from the high-angle distributions. Similar results have been found when tilt boundary distributions were plotted for completely columnar annealed gold bicrystals. [9] However, it should be mentioned that the misorientation system used in these samples leads to a particularly simple CSL and O-lattice structure. There is no reason to assume that for boundary systems with more complex geometries (such as rotation about a mutual [I IO] axis [S]) the same sort of relaxation structure criterion could be applied to determine ~7priori the lowest energy tilt boundary plane. In addition_ it must be reaIized that geometrical models of grain boundaries are not always sufficient in themselves to predict boundary structure or behaviour. [9,3t] Although such models may prove useful. a complete understanding of interfacial structure and behaviour awaits the day when reliable computations can be made to determine atomic relaxations at grain boundaries directly from the interatomic potential. Finally, it should be noted that an earlier experiment performed by Gleiter [lo] on a macroscopic polycrystalline sample (of the same f.c.c. [OOl] twist system) at the X5 f37-) special miso~entation showed that the symmet~c[llO]~) tilt boundaries also appeared to have a slightly tower energy than other boundary orientations. We take this as evidence in favour of our hypothesis. It implies that there is still a very small energy minimum at such high misorientations (and correspondingly small O-spacings). We would expect a bulk true-equilibrium technique such as Gleiter employed to be more sensitive to a small energy minimum than our quasi-equilibrium experiments. We would therefore expect that the type of experiment conducted by Gleiter would show a larger energy minimum for tilt boundaries of lower mis(at
least
in
OF TILT BOCNDARIES
orientatioa even for misorientations special low-T case.
far from
a
4. coscLL~sIoss (I) For all the bicrpstal twist an&s which we observed. the initial single crystal breakthrough on annealing occurred most frequently at inhomogeneities in the films, which served to reduce the free energy cost of tilt boundary formation. (2) The initial breakthroL[gh in the higher angle ~LIn~ries usually had a characteristic size and shape governed by the single crystal being grown through. (3) Observations of tilt boundary motion made during annealing showed the migration dlnamics to be governed by the familiar concept of drag and breakaway. The two possible origins of extrinsic drag forces on a moving interface in these specimens are either thermal grooves which form where the rilt boundary intersects the surface or solute clouds sgregated to the boundary within the film. (1) Bubbles within the twist interface kit over from the bicrystal manu~dctLIring process proved to be strong obstacles to migration. In the ? ru.ist b&stafs. it was observed that indi~~dual Istrice dislocations within the migrating tilt boundaries coutd interact strongly with small bubbles and be removed from the advancing interface. This provides evidence for a mechanism suggested in an earlier macroscopic study of low angle boundary migration ir, metals. (251 (5) Computer analysis of the distriba:ion of tilt boundary orientations in the samples after some migration had occurred showed that two r)pes of directions were favoured signficantly. One such preference probably resulted from inhomogcccities in the initial single crystal films. The sec0r.d preferred boundary direction arises from a WL~l~-Herring faceting mechanism. The resuftanr orientations correspond to the lowest energy tilt boundary orientation in each bicrystal. A preferrred boundary orientation is only evident for twist angles of 30’ or less, in which case the boundary plane follows the plane cf maximum O-point density rather than the plane containing the maximum density of coincident sites. The results SUMgest the presence of a physically significant periodic relaxation structure in boundaries with a$-ist angles as high as 30’. Beyond this limit. the dominant effect of the O-lattice disappears. probably b=eause of the very small period of the atomic relaxation. Ack~lowl~~go)~e~z~~-This work UBE supprrtd by the Energy Research and Development Adminis:r;lrion under contract E(1 l-1 t2679. Additional support w.5 received from the NSF through the Materials Scienx Cornell University.
Center at
REFERENCES 1. T. Schober and R. W. BalMi.
PM
Jlq.
20. SI 1
(1969). 2. T. Schober and R. W. BalMi, t 1970).
PM
Jig.
21, 109
ALLEN
AND
GOODHEW:
MIGRATION
AND FACET DEVELOPMENT
3. T. Schober and R. W. Balluffi, Phps. Stat. Sol. (b) 44, 103 (1971). 4. Y. Komem, P. Petroff and R. W. Balluffi, Phil. lciag. 26. 239 (1972). 5. T. Y. Tan, S. L. Sass and R. W. Balluffi, Phil. Msg. 31. 559 (1975). 6. S. L. Sass, T. Y. Tan and R. W. Ballufli, Phil. Mug. 31. 575 (1975). 7. T. Y. Tan, J. C. M. Hwang, P. J. Goodhew and R. W. Batluffi. Tkin S&d Films 33. 1 (1976). 8. P. J. Goodhew and R. W. BaRufB: to be pub~sh~. 9. P. J. Goodhew, T. Y. Tan and R. W. BailuR, to be published. 10. H. Gleiter, Aera Met. IS. 23 (1970). 11. C. Herring, Phys. Ret 82. 87 (1951). 12. J. M. Blakely, Introduction ro the Properries of Crystal Surfaces. Pernamon Press, Oxford (19731 13. K.-W. Lodge-and N. H. Fletcher, $hif..Mag. 31. 529 (1975). 14. P. B. Hirsch et al.. Electron Microscopy of Thin Cry> tnls, p. 358. Butterworths, London (1965). 15. J. E. Bailey, Phil. Msg. 5. 833 (1960). 16. K. Lucke and K. Detert, Acta Met. 5. 628 (19.57). 17. J. W. Cahn. Acca Met. IO. 789 (1968). 1X. W. W. Mullins. Acra :Mer. 6. 414 (1958).
OF TILT BOUNDARIES
1107
19. K. Lucke. R. Rixen and F. W. Rosenbaum. in .Vature and Behau’our of Grain Boundaries. edited bv H. Hu. p. 245. Plenum Press. New York (1972). . 20. J. W. Rutter and K. T. Aust. rlcta Me?. 13, 181 (1965). 21. R. Viswanathan and C. L. Bauer, Merall. Trans. 5, 1690 (1974). 22. R. C. Sun and C. L. Bauer. Acta Jlet. 18. 639 (1970). 23. A. V. Antonov. C. V. Kopetsky, L. S. Shvindlerman and V. G. Surpeeva, Do&f. Akad. &at& SSSR 213. 318 (1973). 24. C. Zener quoted in C. S. Smith. Tram A.I.M.E. 175. 15 (1948). 25. D. W. Bainbridge. C. H. Li and E. H. Edwards. Acta Met. 2. 322 (19%). 26. W. Bollmann. ~~staf Dejkrs and Crysfufline Interfhces. Sprmger. New York (1970). 27. H. Gleiter. Acta Met. 18. 117 (1970). 28. K. T. Aust and J. W. Rutter, Trans. A.I.M.E. 215. 119 (1959). 29. H. Gleiter, Am Met. If. 565 (1969). 30. W. T. Read and W. Shockley, Phq’s. Rev. 78.275 (1950). 31. J. C. M. Li, J. appl. Phys. 32. 525 (1961). 32. G. Herrmann. H. Sautter, G. B&o and H. Gleiter, froc. 4th Bolton hiding Con& Grain Boundaries in ~~gi~eeri~g ~~~uferju~s,1974.
Vol. 25. pp. lC9-110‘.
Perpmon
Press
LY7: Printed
in Great
Bntam
THE MIGRATION OF TILT BOUNDARIES AND FACET DEVELOPMENT IN THIN GOLD BICRYSTALS* R. M. ALLEK Department of Materials Science and Engineering. Cornell Univ.ersity. Ithaca. U.S.A. and
P. J. GOODHEW Department
of Metallurgy and Materials Technology, University of Surrey. Guildford. England (Receirerl 20 &grrsr 1976; in rrrise[ifbrrn
14 Fehntar.r 1977)
Abstract-Thin films of gold were welded together to create bicrystals containing [OOl] twist grain boundaries. The bicrystals were heated in the transmission electron microscope and the migration of first twist and subsequently tilt regions of the boundary was observed directly. The tilt boundaries showed a tendency to facet which grew less pronounced as the twist angle was increased. The preferred boundary planes appeared to be controlled either by inhomogeneities m the thickness of the original single crystal films or by the lowest energy section of the O-lattice, w-hich in this case is the plane with the highest density of O-points. The interaction of a migrating boundary with bubbles of a range of sizes was also observed and the pinning effects are reported. R&urn&-On a soudi des films minces d’or aiin de fabriquer des bicristaux contenant un joint de torsion [OOl]. On a chauffe les bicristaux dans le microscope et Ton a observe directement la migration des regions du joint en torsion, puis des regions en flexion. Les joints de flexion avaient tendance a presenter des facettes dont l’importance diminuait lorsqu’on augmentait Tangle de torsion. Les plans preferentiels du joint semblaient control& par le hetirogtneites dans l’epaisseur des films monocristallins otiginaux, ou bien par la section du reseau 0 de moindre tnergie, qui est ici le plan dont la densiti de points’ 0 est la plus grande. On a egalement observe l’interaction dun joint en tours de migration avec des bulles de tailles diverses et l’on decrit les effets d’ancrage. Zusammenfassung-Diinne Goldfilme wurden miteinander verschweiBt. urn Bikristalle mit [OOl]-DriUkorngrenzen herzustellen. Die Bikristalle wurden im Durchstrahlungselcktronenmikroskop aufgeheizt; die Wanderung erst der Drillbereiche und darauffolgend der Knickbereiche der Komgrenzen wurde direkt beobachtet. Die Knickgrenzen neigten dazu, Facetten zu bilden, die mit zunehmendem Drillwinkel weniger ausgepragt wuchsen. Es schien, daB die bevorzugten Komgrenzebenen bestimmt sind entweder durch Inhomogenitlten in der Dicke der urspriinglichen Einkristallfilme oder durch den Abschnitt des 0-Gitters geringster Energie, welcher in diesem Fall die Ebene mit der griil3ten Koinzidenzpunktdichte ist. Die Wechselwirkung einer wandanden Korngrenze mit Blasen in einem be+ timmten GrBDenbereich wurde ebenfalls beobachtet; Verankerungseffekte werden mitgeteilt.
1. INTRODUCI’IOX
The simple thin film welding technique for the pteparation of bicrystal metal samples suitable for direct examination in the transmission electron microscope has proved useful in many studies of grain bounderies. [l-9] A sample of this type contains an interface of known inclination between two identically constituted grains at a known misorientation. Boundary plane, inclination and intergranular misorientation may thus be chosen as experimental variables. As-welded samples present large areas of grain boundary normal to the electron beam, making them well suited to the study of interfacial microstructure *This work was carried out while P. J. Goodhew was a visiting fellow in the Department of Materials Science and Engineering, Cornell University. R. M. Allen is now at the Department of Materials Science, M.I.T., Cambridge, Mass.
by electron microscopy. In particular, (001) twist boundaries in gold bicrystals have been extensively examined, and shown to have structures depending both on the planar density of coincidence sites (PCSD) within the interface [2] and the O-lattice geometry of the system. [S, 61 Thin film specimens containing extended grain boundaries parallel to their free surfaces are clearly metastable. It has been found [I, 7,9] that, during low-temperature annealing. gold specimens originally containing twist boundaries of this type will reduce their interfacial area by boundary migration. Figure 1 shows this process schematically. If the destruction of twist boundary area is carried to its conclusion, the sample will reach a new metastable state; this consists of a mosaic of columnar grains of two types, corresponding to the two original orientations of the starting bicrystal. Each grain spans the total specimen thickness and all are separated by tilt boundaries nor1095
1096 ALLEN
AND
GOODHEW:
MIGRATION
AND FACET DEVELOPMENT
OF TILT BOUNDARIES
$3011 a.
1 2
I,
I
b.
Fig. 1. The temperature range over which each of the eight bicrystals was heated in the microscope. The lowest temperature in each range indicates the onset of stow migration. At the tfighest temperature, fast migration was clearly occurnng in the small region being examined.
d.
stage of a JEOL JEM 200 microscope. Micrographs were taken during the heat-treatment with the specimen stabilized at a selected furnace setting. Figure 2 shows the range of temperature over which each of Fig. 1. A schematic illustration of the boundary migration the specimens was photographed. occurring in our experiments. The or;iginal twist boundary For the 2’ twist bicrystals, observations of the (a) migrates a smail distance until it Intersects the free surgrowing single crystal patches could be made directly face (c) forming segments of tilt boundary ~~nd~cu~ar to the film. The tilt boundaries migrate laterally until a11 with bright-field imaging since the tilt boundary the twist boundary is removed. leaving a columnar polymigration corresponds to the disappearance of the bicrystal. Grooves (shown dotted) may form at the bounvisible twist boundary dislocation structure. This was dary-surface intersectlon. not possible for hot stage observations from higher angle bicrystals. Instead, these samples were examined ma1 to the specimen surface. but having a variety of in a dark-field image formed from a beam singly difinclinations with respect to a reference direction in fracted from one of the crystals. A growing single the plane of the film. crystal patch was thus seen as an expanding dark It has been shown that. for bicrystals of fixed mis- region during the annea1. After the initial dynamic observations had been orientation, grain boundary energy is a function of interfacial plane inclination. [IO] Accordingly, it has recorded, the annealed specimens were transferred to been suggested [P] that the con~gurations adopted by a Siemens Elmiskop 102 microscope for a static SW the columnar grains of these annealed bicrystal films vey of the single crystal patch shapes which had been produced during the hot stage anneal. Micrographs reflect some type of Wuiff-Herring faceting. [ill Analysis of micrographs to find preferred orientations were taken from at least six different regions of each of the column walls would therefore be an extremely bicrystal. Each picture was marked with a standard useful method for determining the relative energies reference line and then computer analysed to obtain of various inclinations of boundary in a fixed-misorthe distribution of crystallographic directions which ientation system. The feasibility of such a study has made up the borders of the single crystal patches. The individual distributions from each specimen were already been established. [7.9] The purpose of this research was primarily to then processed to arrive at a final average distribution examine the role played by boundary migration kin- of orientations for each bicrystal misorientation. A etics in producing the final columnar configuration description of the computer program will be given in annealed gold bicrystals. Additionally, we hoped elsewhere. [9] to compare observed boundary migration behaviour 3. RESULTS AND DfSCLSS1ON with existing theories of pinning and migration kinetics. 3. I itritial sitrgle crystal patch fortnatiotz 2. EXPERMENTAL A series of bicrystal specimens containing (001) twist boundaries of various misorientation angles between 2 and 4.5’ was prepared from pairs of nominally 25nm thick gold single crystal films in the usual way. Cl] Each of these samples was then annealed at a low temperature (c270”C) in the hot
An idealized mechanism of the type depicted in Fig. 1 presents a substantial free energy barrier to the homogeneous nucleation of single crystal patches. A relatively large boundary area increase must occur before any part of the boundary reaches a free surface and lowers the free energy of the system by being eliminated. In practice, initial breakthrough in the 2” bicrystals takes place preferentially at the edges of
,ALLES
hSD
GOODHEW:
51IGR.4TION AND FACET DEVELOPMEXT
OF TILT BOUSDARIES
IW
Fig. 3. A region of 2. twist boundary showing many small pinholes and bubbles. Breakthrough ot single crystal patches has occurred at a pre-existing twin (T) and at one of the pinholes (P).
either weld bubbles or pinholes, and also at twins in either single crystal (Fig. 3). Presumably, these inhomogeneities act to reduce the energy barrier for breakthrough. In particular, bubbles and pinholes probably decrease the effective distance through which the boundary has to expand to reach a free surface. A thermal groove at the twin-surface intersection [I?] may perform the same function for breakthrough at a twin. Additionally, part of the energy cost involved in extending the grain boundary to the surface would then be met by the destruction of an area of twin interface at such locations. We were readily able to obtain as-welded 2’ specimens with large areas (- 1 x I pm) free from grownthrough single crystal patches. This was never the case for the higher-angle specimens. which suggests that they are structurally less stable or more mobile. There have been very few calculations of twist boundary energies. but the espectation that high-angle boundaries should have energies much greater than low-angle boundaries is reinforced by the recent calculations of Lodge and Fletcher. [ 131 These uorkers found that the energy of twist boundaries in aluminium should rise to a relatively constant high value ( _ 600 erg cm-‘) for twist angles aboLe IS-. .AGuming that this result is qualitatively applicable to other
f.c.c. metals, the observed relative stability of our boundaries is easily understood. Although preferred breakthrough sites were also active in the higher angle boundaries, most of the initial single crystal patches did not seem to be associated with such defects. Rather. the majoriF appeared as square regions. randomly distributed. exh several tens of nm across. with edges aligned with [lOO] directions in the crystal being grown through (Fig. 4). As we have mentioned above. these patches were generally formed during bicrystal welding: however. on the rare occasions when a breakthrough was observed during the hot stage anneal. the freshly created single crystal region quickly assumed the same general shape, size and directionalit! as the prsexisting patches. It was very common to find a grownthrough region associated with one of the single cry+ tals. in very close proximity to a patch produced by breakthrough of the other single crystal. In some cases, the complementary squares impinged on each other. However. the edge directions of an) particular square still matched the [ IOO]directions of the crystal being gronn through. This suggests that the mechanism responsible for the square orientation is independent of the local bounder> structure. Because it matches a IOU-index direction in one cr!sta.l. it must
1WS ALLEN XNIJ GOODHEW:
XlfGRATIOS
AND FACET DEVELOPMEW
OF TILT BOL’NDARIES
Fig. 4. A pair of complementary 200 dark-field images of a 37” twist boundary. The initial patches of growth-throu~b single crytaI are aligned parallel to [ IOO]in each crystal.
be assumed rather that the orientation stems from some property of the crystal being grown through. We base carefully examined the original single crystal thin films of old in the eiectron microscope. We frequently obserse. in very low contrast, square
and rectangular defects of approximately the same dimensions as the square breakthrough patches in the bicrystals. It has not proved possibte to determine whether these arise from the thin film growth mechanism or are a replication of a structure on the salt
ALLEN
.a\D
GOODHEW:
MIGRATIOS
.tND
substrsts. In either case. ho\vever. w can expect the welding process to introduce a locall>- higher strain around the square regions and thereb! provide a locall> higher driving force for boundary migration.
Several diRerent aspects of the dJnamica1 behaviour of the bicrystal were characterized. For instance, after a limited number of initial breakthrouyh patches had formed within a given specimen region (as described in the previous section). further single crystal grolvth took place primaril) by expansion of these esisting patches rather than bl; further isolated breakthroughs. X second common effect was that bubbles lying in the initial t\\-isi boundary were seen to interfere with patch growth by a pinning process. These bubbles (generally ranging in diameter from one to
FACET
DEVELOPblEST
OF TILT
BOCSDARIES
1099
tens of nm) are quite common in bicrystal >Tecimens of this t)v, and are believed to be produced by a sintering process during sample welding. [I] Such defects are distributed randomly in the specimen midplane and often exhibit Moire fringes [j] arising from double diffraction. [I41 Good examples can be seen in Fig. 5. Growth, within the areas examined in the hot stage, generally occurred in one of two distinctive modes. At lower annealing temperatures. single crystal patch expansion was extremely slow and the continuous progress of the advancing patch borders was visible. For instance. in the 1’ bicrystal the border appeared to move by the co-operative slip of several of @e real lattice dislocations present at the patch edge, concurring with the results of an earlier study. [I] At the higher temperatures used. growth became very
(b)
Fig. 5. lab .A large single crystal patch in a low-angle specimen. In two places. dislocations have &n removed from the migrating tilt boundary and remain attached to a bubble. The geometry is shoan schematically in (b).
hst xxi jerky. so that whole sections of the specimen seemed to become singie cr>rtal ail at once. in this case. the edges of the patch were only visible when its motion had been tzmpomrily arrested. as at a group of bubble obstacles. The migration behabiour was rstrcmely inhomogeneous on the scale of ths observations: some sections of the bicrystal became columnar at temperatures which produced no grotvth at all in other regions of the same specimen only tens of pm away. The hot stage used was a furnace type. heating by radiation. It is not apparent why this irregular specimen behaviour should arise. since there should be no gross temperature gradient within the furnace. but similar inhomogeneous structural response to annealing on roughly the same scale has been reported before in recrystallization experiments. [ 141 The fact that the columnization process occurred mainly by patch growth rather than repeated isolated breakthrough suggest that the patches are heterogenously nucleated. Their subsequent growth by tilt boundary migration is accompanied by the continuous reduction of specimen free energy, since twist boundary area is destroyed continuously. while fess tilt boundary is created. For the simplest realistic model. of a square breakthrough patch of side h in a bicrystal of half-thickness t. the driving force for boundary migration per unit area of the tilt boundary is
F = & r
_
brILr --, h
(1)
where ;‘rW and j’~,t7 are the grain boundary energies associated with the twist and tilt boundaries. respectively. Unless yT,\.and yIlLI arc very diRerent this driv-
dx
Section
Fig. 6. The geometry of thermal grooving and tilt boundary migration at a square patch of ~idr IL
ing force rises rapidly to a constant flue of ;‘r\t I as h increases. The two distinct growth rates obserl-ed may mercl! represent the behaviour at two different rsmperatures or may reflect two different migration modes for the tilt boundaries. This could be understmd in ternts of the ‘drag and breakaway’ concepts in the kinetic
(br
Fig. 7. A sequence of micrographs showing the interaction between a bubble 1A1and the migrating boundary. The boundarv segments at each side of the bubbb bow out [a) and (bt until the boun&r: puIIs free and re-establishes a straight edge (c). In this case, the Moire frinps which remain ins& the bubble indicate that a Ioop of tilt boundary has been pinched 0%. theories of interfacial migration. [ 16.171 In addition to tkeir intrinsic s~ruct~~re-de~e~deot mobility, at lower temperatures. the tilt boundaries may be subject to a drag force stemming either from the traditional solute cloud formation [Ia 173 or from the production of a thermal groove at tkeir intersection witk a free surface. [I& 73 It is dit?icult to gauge the mqnitude of the solute effect, but we can estimate tkc effect of grooving in
terms of its effect on equation I 1k The introduction of a groove of depth r, and root angle 2P is show;n in Fig. 6. The energy reduction AE on migration of the tilt boundary by an increment dx is thy AE = yT‘vk ds _ ;rILT[z &,.jt _ t, + dtt I ir dr]. The driving force for patch growth dE dx is thus dE - = y*\, It - j(T,,.T[zlr - t,i + h coz $1. dx
110’ .\LLEN
ASD
GOODHEW:
MIGRATIOlr; AND FACET DEVELOPMENT
since dr &c = cot 6. The force per unit are3 of tilt
boundw F=-
is then given by dE h(r - tJ ds ’ ’ a-- Ynv t--L
h,‘TEW
- &I f h cot @J,
[$
h(r - tJ
which reduces to e&ation (1) in the absence of grooving (r, = 0. cot 8 = 0). Typical values of 6 for the grooves at the intersection of high-angle grain boundaries with f.c.c. metal surfaces are in the region of 80”. [I21 For Iow-angle or speciai boundaries having lower energies, the value will be higher. We are therefore concerned with values of cot 0 in equation (2) of 0.2 or less. In order to make clear the variation of driving force with patch size we can replace (f - t& with t’, cot 8 with 0.2 and both yTw and Y~,,,~with 7 in equation (2). This gives the approximate expression .. I721
F+Cl&xJ.
while making the equivalent equation (1) produces
approximations
1
in
2t F =- ;l-7;. [ The maximum effect of groove pinning is therefore to change only slightly the critical size of patch (i.e. values of jr) above which a positive driving force exists for patch growth. The ultimate driving force at large It is scarei)’ affected, since 0.8/r’ and l/t are unlikely to differ significantIy. From the above analysis. we can deduce that pinning by thermal grooves should not be a serious effect in these experiments, although it has previously been reported in other thin-film experiments. [ISJ What is clear from the analysis is that the simple model described here gives rise to a negative driving force for patch growth until the patches are about 2t in diameter. The nucleation of the patches must therefore be heterogeneous, as postulated in the previous section. At high annealing temperatures, the tilt boundaries exhibit rapid. jerky motion which is characteristic of breakaway from the drag-producing entities, whether they be grooves or solute clouds. Although the temperature regimes in which migration was observed differed widely among the boundaries studied (Fig. 2), it is possible, making some assumptions, to deduce a very approximate activation energy for migration for each boundary. Our assumptions were (a) that the minimum and maximum migration velocities observed, corresponding to the lower and upper temperature limits in Fig. 2. were one fifth of a field of view (at 100,000 x ) in i h and a whole field of view in lOOs, and (b) that at both extremes of temperature the migration velocity o is git;en by c x DT?; where D is the appropriate diffusion coefficient.
OF TILT BOCSDARIES
For most of the boundaries studied, the activation energy, Q. was calculated to lie between 5 and 8 kJ mol- I. which is more than half of the activation energy for self di&sion in gotd (_ 10 kJ mol-‘1. Activation energies of this magnitude have been reported previously. Cl93 but it is probably unwise to place much reliance on the absolute values of Q from our experiments. since the data are so imprecise. For the remaining two boundary specimens. the estimated activation energies were much higher (10 and 30 kJ mot - ‘), and this is probably accounted for by the special nature of these boundaries. which were both low-angle boundaries with misorientations of approximately 2’. Our data are therefore in agreement with the evidence that activation energies for boundary migration increase greatly as the boundary misorientation decreases (e.g. refs. 22. 2.3;).but do not permit us to distinguish between the conclusions of Rutter and Aust [20] and Viswanathan and Bauer[X] as to the relative mobilitics of E:j and “random” boundaries. As mentioned above, the bubbles in the twist boundary interfered with the growth of single crystal patches. It was observed that whenever the perimeter of a patch struck a bubble. that section became pinned. The boundary segments on each side of the bubble were still free to move, however, and as a result the perimeter would bow out around the bubble. Eventualty, a limit was reached whereupon the pinned section would snap past the bubble catastrophically and catch up with the advancing boundary. Occasionally. the bypassed bubble region would remain bicrystalline, as could be seen from its Moirl fringes (e.g. at A in Fig. 7). A bubble in the interface is an area devoid of twist boundary. As a consequence of this, an advancing tilt boundary striking the bubble perimeter is suddenly bereft of the major driving force for further motion. This is probably sufficient to pin the boundary, but it is also possibIe that sintered bubbles may actually reduce each single crystal film thickness in their Iocality. Thus an adl-ancing tilt interface is able to penetrate into the bubble region, but becomes pinned because further motion would require the tilt boundary to increase its own area. to reassume the fufi film thickness awa? from the bubble. With no other driving force. this ensures that further motion in any direction requires a local free energy increase. As the tilt interface moves past the bubbIe during the bowing process, new tilt boundary area is continuously created. This stores more and more energy in the system until, finally, it is favourable for the pinned section to slip past the bubble and straighten the bow in one rapid release of energ. In some instances, passage across the bubble is so difficult that the bowing process pinches the bubble off instead and a smaI1 region of bicrystal remains behind the migrating interface. Atthough such a small loop of boundary should be rather unstable and would be expected to shrink, it seems probable that whatever effect
ALLES
.ASD
GOODHEW:
MIGRATION
AND FACET DEVELOPMEXT
OF TILT BOUSDARIES
1103
pinned it sufficiently to prevent it being dragged across the bubble. also stabilizes it after it is pinched Of?.
Our general observation that bubbles do not totally
stop boundary motion is in accord with a simple consideration of boundary pinning by spherical obstacles. As Zener first pointed out (241 the maximum pinning force exerted by an obstacle of radius r is my;. Since the driving force per unit area is approximately ;‘,t, in our experiments this will exceed the pinning force unless the bubbles are closer than about 3r apart. Since we have almost never seen this density of bubbles. complete pinning should indeed not occur in our specimens. In the 2’ mist boundaries, the smallest twist interface bubbles had a diameter of the order of the dislocation spacing ( _ 10 nm). It was very common to find such a bubble ‘sitting astride’ a single dislocation line of the orthogonal grid. In that case, during annealing, the bubble pinned only that single line in place as the single crystal patch grew past. This produced a charateristic feature: a small bubble in a single crystal region frequently appears to be connected to the remnant twist grid by a single dislocation (Fig. 5). An equivalent description of the situation is that interaction of the migrating low-angle tilt boundary with the bubble serves to remove a single dislocation line from the tilt boundary. This directly confirms the hypothesized mechanism for low-angle boundary; small obstacle interaction advanced as a result of earlier macroscopic studies of boundary migration in metals by Bainbridge et al. [25].
1,
mJl
m ccw
Fig. 8. A polar plot produced by computer analysis of several micrographs of an 11’ twist bicrystal. The plot shows the percentage of the total tilt boundary length vrhich lies within each 5’ inclination range. The [OlO] and [llO] directions for each gram are indicated. as is the quadrant which would be expected if the boundary inchnations acre randomly distributed. The [110-j, peak clearI?: extends far beyond the kd limit attached to the random value.
A typical computer result from the static survey of the directions adopted by the borders of the patches is shown in Fig. Y. Such polar plots demonstrate the distribution of inclinations of the tilt boundaries forming the borders of single crystal patches. Only one point has been plotted per 5’ interval in the quadrant. The connecting curves are purely for visual convenience. A subscript cw or ccw on a crystallographic direction indicates that the direction belongs to either the clockwise or counterclockwise single crystal of the bicrystal as it was positioned during the entire static survey. A subscript m indicates a ‘mean’ direction: i.e. the [IOO], bisects the smaller angle between the [IOO],,, and [IOO],,, of a system. The fact that the ccw directions appear to be rotated clockwise from the cw directions is an artifact of the data processing. Each sample was allowed to undergo fast growth in a hot stage anneal before being trdnsferred for the static survey. Thus growth was stopped before the specimen became wholly columnar, in contrast to previously reported experiments. [7,9] During the survey, only those single crystals patches clearly very much larger (> 1 nm’) than the initial breakthrough patches were photographed. The major part of the border of such regions must have been created during growth fast enough for the boundary to have broken away from any estrinsic drag structures. The error in positioning the distributions with respect to the known sample directions (measured from diffraction patterns) was estimated from the I I specimen results. The tilt boundaries in this specimen showed a strong tendency to line up along the mean (110) directions in the (001) plane; that is. along tine directions in the dislocation grid. This is reflected in the large peak of the corresponding distribution, centered on [llo],. .Angular error for all the distributions was taken as the half-width of this peak. or roughly one interval. To give an indication of peak significance. each distribution was marked with an arc representing the orientation distribution of a perfectly random system (i.e. with circular patches). Only two major features of the esperimental plots are found consistently. within the angular error of the experiment: a fairly strong [lOO],, = [OIOICN peak (in all but the 2’ boundary) and a [1 lo], peak. strong for low-angle boundaries but decreasing and eventually disappearing with increasing twist angle. For those specimens surveyed by dark-field microscopy. by chance. nearly twice as many photographs were taken of regions where the ccw component crystal was growing through the cw crystal. rather than vice versa. Since asymmetric behaviour seems unreasonable. we assume that the [100-J,,, peaks merely represent a patch shaping dependence on some property of the crystal being grown through during the anneal. Apparently, the same mechanism which governs the
lit34 ALLEN .+ED GOODHEW:
MIGR-\TION AND FACET DEVELOPMENT
OF TILT BOUSDARIES
Table I. Gtxm~tricd data on the O-lattice and coincidence site lattice at the three misoriea~a~ions of ~~FI.SS~ E (highest density of coincident sites) for the @IIt twist boundaries O-lattice
Parameter 0
(nrn!
Coincidence site lattice (CSL)
Densest direction
E:
Parameter (nm)
initial breakthrough shapes is also active to some estznt during fast growth. Once again, the 1’ bound-
ary was the only exception to this mechanism. although since bright-field imaging was used we do not know whether or not the data was prejudiced in favour of one component crystal. The [ 1IO],,, directions do not generally correspond to low index directions in either of the two single crystals of a bicrystai. However. for this twist system of boundaries (i.e. with the two crystals misoriented by a rotation about a mutual [OOI] axis) the [I lOJ,,, direction is a low index direction in both the coincident site and O-lattices (see Table 1). For a review of the concepts and geometrical significance of these ConStructions the reader is referred to the book by Bollmann. [263 The orientation and lattice parameters of the coincident site lattice (CSL) are discontinuous functions of the twist angle in these systems. The [ 1IO], peaks might be expected to be produced by a mechanism dependent on some property of the CSL, such as a preferential seiection of tilt planes with the highest pianar density of coincident sites (PCSD). or preferred tilt boundary orientations along secondary grain boundary dislocation lines. However, the observed peak behaviour as a function of twist angle is not well correlated with the CSL behaviour. Certainly, no peak is produced at the 37’ twist angle, which has the lowest non-unity Z (ratio of real lattice site density to CSL density) of any (001) bicrystal angle. Yet it is precisely at this angle that a CSL-determined effect should be strongest. Hence we conclude that the [l lo],, peaks do not arise from a CSL-related mechanism. On the other hand. the orientation and lattice parameters of the O-lattice are continuous functions of twist angle. The most dense O-lattice plane is the tilt boundary containing a [llO],,, direction. The O-lattice parameter is a monotonically decreasing function of twist angle. The hypothesis that the preferred tilt boundary orientation is governed by the planar density of O-points. while the peak height varies with O-lattice parameter. yields a good correlation. with observed peak behaviour. Another possibiIity is that the tilt boundary prefer-
Orientation of cell edge
Highest PCSD direction
Direction of secondary gbds
ence is governed by kinetic effects. In that case, either the interaction with obstacles or one of the two passible drag mechanisms must operate to select the preferred tilt boundaries during growth. \Ve will now consider these possibilities. On the scale of the micrographs processed in the static survey. the bubbIes were essentia& a randomlydistributed array of point obstacles in the (001) plane. As such, it is diRicult to see how the co-operative effort of several pinning bubbles could consistently select tilt boundaries aligned along El lo], directions. Also. as can be seen in Fig. 9, the bubble density was low enough to allow manifestation of the [I IO],,, preference between pinning locations (i.e. in spite of the bubbles). We have shown that most of the suneyed patch perimeter length must have been formed during fast growth. that is. after the tilt boundaries have broken away from their drag structures. Theoret& calculations have suggested that in this regime the interfacial migration rate is determined by how rapidly the boundary moves through the bulk, rather than by any surface thermal grooving erect. [ig) This implies that the determining kinetic mechanism should be the interaction of the moving interface with solute impurity atoms in the bulk. Models of such behaviour thus far point to the ability of a grain bout&r)- to entrap solute atoms as an important factor in migration. [17] It has been shown experimentally that gain boundaries with high PCSD are not good solute traps. [17] On this basis. such boundaries in Cc.c. metals in general might be espected to move faster. Evidence for this effect has been found in studies of migration rates.[Zg] The [llOJ, tilt boundaries in these systems are general@ among the hrgest PCSD tilt planes posssible. The considerations just outlined would seem to indicate that. if the kinetics governed the growth shaping. these planes would move faster than others and hence would ‘grow out’ of the perimeter rather than remaining behind preferentially. In addition. the kinetic shaping process has been linked to the CSL, and so should have a strong effect in the X5 bicrystal. No such effect is in evidence. For these reasons, we argue that the growth kinetics do not produce the [I IO],,, peaks observed.
L1’ bicrptnl in which two singk crystal patches (SIX) have bqun tr, meet funrAg tilt boundary at T. .A iargr itre~l of bicrystnl containing twist boundary remak fBI?i! 2nd the tilt boundaries forming the edges of this area can be seen to bz aligned along the f I ii& dirrcrbn Fig 9. A region of
a pure
despite being pinned by bubbles at several points {Bg
Based on this anafysis, the following exptanation of the or&in of the [f tO],,l peak is postulated. The patch gowth, during annealing, satisfies the theoretical requirements for WuK-Herring faceting to MXI.K. [l 13 The patch shaping. and hence the preference for [IlO];,, boundaries, is therefore determined by such faceting. The [l lo],,, tilt boundaries can be concluded to represent distinguishably lower sner_ey orientations of tilt boundary planes. The presence of a significant number of grain boundary; ledges co&d possibly lead to misleading apparent facets, but we have no evidence for their presence: no iedges were resolvable either during or after patch growth in these experiments. Also, if boundary migration had occutred by a tedge mechanism, we would expect the ledges to join regions of close-packed plane. [29] Woweser. in other experiments with thin gold bicrytaij, aI! the small steps and facets which we have resolved proved not to consist of close-packed planes in either grain. [9] so there is not even inciirrct evidence for such led_ees in these specimens.
faces consist
of nets of real fatticr 3x5 distoeritions. this resutt is not surprising. The [I 10]j, boundaries are symmetrical tilt boundaries and has: y&e lowest dislocation density of any possible plane orientation. As such. they would bc expected to be of lower energy for a fixed tivist angle. [Wj The question remains wh\’ the facet&g process takes place in those sytems wvjrh t\*is: angles too large for the grain boundary to be rea!istisaDy considered as a network of discrete fartics dislocations. We feel that the correlation between &c obsersed faceting and the behafiour of the U-tarii=~ suggests m ansbvrr: the primary grGn boundan- &locations igbdsl on the [1 IfQ,~ pI:nx:s bar-e the priodicity of the O-lattice and so their spacing. althouph the widest of all possible tilt gbd arrays at a &en r.vist angle. decreases continuousl\- and monotonic:& as a function of ttvist angle. At low twist an$es. the primary r&ss!ion structure results directly in the familiar nctuorks of lattice dislocations. Xt hi@er an&r. although ir is unlikely that primdry pbd structure esists in the i;?rm of discrete line defects. electron diffkaaion cGdence indi-
1106 ALLEN ASL) GOODHEW:
~MtGR?ITION AND FACET DEVELOPMENT
fates
at
that
the
atomic
reiaxation
grain
boundaries
welded gold bicrystat specimens) still exhibits the periodicit)- of the O-lattice. [6] Therefore we suggest that [IlO], tilt planes appear to exhibit lower energies at twist angles ~30’ due to the relatively large periodicit! of their atomic relaxation structure lbvhether manifested in discrete line deftits or not) compared to that found on other planes. This allows the system to preferentially select these planes by a Wuff-Herring faceting mechanism. producing the observed [I IO], peaks in the orien~tion distribution. We then suggest that for hicrystals with angles >30’ the O-httice periodicity has been reduced to too fine a scale (GO.5 nmf for a significant energy difference to exist between various tilt orientations. This could well correspond to the merging of discrete dislocation cores into a continuous plate of core material in the manner suggested by Li. L-311 Although any inclination of such a boundary will result in an energetically equivalent plate of core material. the atoms still relax (albeit in a non-Hookean manner) periodically. Therefore no preferential selection of tilt planes occurs on this basis and the [ 1IO],,, peaks disappear from the high-angle distributions. Similar results have been found when tilt boundary distributions were plotted for completely columnar annealed gold bicrystals. [9] However, it should be mentioned that the misorientation system used in these samples leads to a particularly simple CSL and O-lattice structure. There is no reason to assume that for boundary systems with more complex geometries (such as rotation about a mutual [I IO] axis [S]) the same sort of relaxation structure criterion could be applied to determine ~7priori the lowest energy tilt boundary plane. In addition_ it must be reaIized that geometrical models of grain boundaries are not always sufficient in themselves to predict boundary structure or behaviour. [9,3t] Although such models may prove useful. a complete understanding of interfacial structure and behaviour awaits the day when reliable computations can be made to determine atomic relaxations at grain boundaries directly from the interatomic potential. Finally, it should be noted that an earlier experiment performed by Gleiter [lo] on a macroscopic polycrystalline sample (of the same f.c.c. [OOl] twist system) at the X5 f37-) special miso~entation showed that the symmet~c[llO]~) tilt boundaries also appeared to have a slightly tower energy than other boundary orientations. We take this as evidence in favour of our hypothesis. It implies that there is still a very small energy minimum at such high misorientations (and correspondingly small O-spacings). We would expect a bulk true-equilibrium technique such as Gleiter employed to be more sensitive to a small energy minimum than our quasi-equilibrium experiments. We would therefore expect that the type of experiment conducted by Gleiter would show a larger energy minimum for tilt boundaries of lower mis(at
least
in
OF TILT BOCNDARIES
orientatioa even for misorientations special low-T case.
far from
a
4. coscLL~sIoss (I) For all the bicrpstal twist an&s which we observed. the initial single crystal breakthrough on annealing occurred most frequently at inhomogeneities in the films, which served to reduce the free energy cost of tilt boundary formation. (2) The initial breakthroL[gh in the higher angle ~LIn~ries usually had a characteristic size and shape governed by the single crystal being grown through. (3) Observations of tilt boundary motion made during annealing showed the migration dlnamics to be governed by the familiar concept of drag and breakaway. The two possible origins of extrinsic drag forces on a moving interface in these specimens are either thermal grooves which form where the rilt boundary intersects the surface or solute clouds sgregated to the boundary within the film. (1) Bubbles within the twist interface kit over from the bicrystal manu~dctLIring process proved to be strong obstacles to migration. In the ? ru.ist b&stafs. it was observed that indi~~dual Istrice dislocations within the migrating tilt boundaries coutd interact strongly with small bubbles and be removed from the advancing interface. This provides evidence for a mechanism suggested in an earlier macroscopic study of low angle boundary migration ir, metals. (251 (5) Computer analysis of the distriba:ion of tilt boundary orientations in the samples after some migration had occurred showed that two r)pes of directions were favoured signficantly. One such preference probably resulted from inhomogcccities in the initial single crystal films. The sec0r.d preferred boundary direction arises from a WL~l~-Herring faceting mechanism. The resuftanr orientations correspond to the lowest energy tilt boundary orientation in each bicrystal. A preferrred boundary orientation is only evident for twist angles of 30’ or less, in which case the boundary plane follows the plane cf maximum O-point density rather than the plane containing the maximum density of coincident sites. The results SUMgest the presence of a physically significant periodic relaxation structure in boundaries with a$-ist angles as high as 30’. Beyond this limit. the dominant effect of the O-lattice disappears. probably b=eause of the very small period of the atomic relaxation. Ack~lowl~~go)~e~z~~-This work UBE supprrtd by the Energy Research and Development Adminis:r;lrion under contract E(1 l-1 t2679. Additional support w.5 received from the NSF through the Materials Scienx Cornell University.
Center at
REFERENCES 1. T. Schober and R. W. BalMi.
PM
Jlq.
20. SI 1
(1969). 2. T. Schober and R. W. BalMi, t 1970).
PM
Jig.
21, 109
ALLEN
AND
GOODHEW:
MIGRATION
AND FACET DEVELOPMENT
3. T. Schober and R. W. Balluffi, Phps. Stat. Sol. (b) 44, 103 (1971). 4. Y. Komem, P. Petroff and R. W. Balluffi, Phil. lciag. 26. 239 (1972). 5. T. Y. Tan, S. L. Sass and R. W. Balluffi, Phil. Msg. 31. 559 (1975). 6. S. L. Sass, T. Y. Tan and R. W. Ballufli, Phil. Mug. 31. 575 (1975). 7. T. Y. Tan, J. C. M. Hwang, P. J. Goodhew and R. W. Batluffi. Tkin S&d Films 33. 1 (1976). 8. P. J. Goodhew and R. W. BaRufB: to be pub~sh~. 9. P. J. Goodhew, T. Y. Tan and R. W. BailuR, to be published. 10. H. Gleiter, Aera Met. IS. 23 (1970). 11. C. Herring, Phys. Ret 82. 87 (1951). 12. J. M. Blakely, Introduction ro the Properries of Crystal Surfaces. Pernamon Press, Oxford (19731 13. K.-W. Lodge-and N. H. Fletcher, $hif..Mag. 31. 529 (1975). 14. P. B. Hirsch et al.. Electron Microscopy of Thin Cry> tnls, p. 358. Butterworths, London (1965). 15. J. E. Bailey, Phil. Msg. 5. 833 (1960). 16. K. Lucke and K. Detert, Acta Met. 5. 628 (19.57). 17. J. W. Cahn. Acca Met. IO. 789 (1968). 1X. W. W. Mullins. Acra :Mer. 6. 414 (1958).
OF TILT BOUNDARIES
1107
19. K. Lucke. R. Rixen and F. W. Rosenbaum. in .Vature and Behau’our of Grain Boundaries. edited bv H. Hu. p. 245. Plenum Press. New York (1972). . 20. J. W. Rutter and K. T. Aust. rlcta Me?. 13, 181 (1965). 21. R. Viswanathan and C. L. Bauer, Merall. Trans. 5, 1690 (1974). 22. R. C. Sun and C. L. Bauer. Acta Jlet. 18. 639 (1970). 23. A. V. Antonov. C. V. Kopetsky, L. S. Shvindlerman and V. G. Surpeeva, Do&f. Akad. &at& SSSR 213. 318 (1973). 24. C. Zener quoted in C. S. Smith. Tram A.I.M.E. 175. 15 (1948). 25. D. W. Bainbridge. C. H. Li and E. H. Edwards. Acta Met. 2. 322 (19%). 26. W. Bollmann. ~~staf Dejkrs and Crysfufline Interfhces. Sprmger. New York (1970). 27. H. Gleiter. Acta Met. 18. 117 (1970). 28. K. T. Aust and J. W. Rutter, Trans. A.I.M.E. 215. 119 (1959). 29. H. Gleiter, Am Met. If. 565 (1969). 30. W. T. Read and W. Shockley, Phq’s. Rev. 78.275 (1950). 31. J. C. M. Li, J. appl. Phys. 32. 525 (1961). 32. G. Herrmann. H. Sautter, G. B&o and H. Gleiter, froc. 4th Bolton hiding Con& Grain Boundaries in ~~gi~eeri~g ~~~uferju~s,1974.