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Dislocation structures in large-angle grain boundaries in hexagonal close-packed materials
Materials Science and Engineering, A 113 (1989) 121-127
121
Dislocation Structures in Large-angle Grain Boundaries in Hexagonal Close-packed Materials* KISOO SHIN and A. H. KING Departrnem of Materials Science and Engineering, State University of New York at Stony Brook, Stony Brook, NY 11794-2275 (U.S.A.) (Received September 26, 1988)
Abstract In materials with cubic crystal lattices, the dislocation structures of grain boundaries are believed to change in a continuous manner as misorientation changes, according to the well-established O-lattice theory developed by Bollmann. They have been shown to change discontinuously as grain boundary phase transformations occur with changes of composition or temperature. The possibilities of continuous and discontinuous structure changes with misorientation and temperature are explored for materials with hexagonal lattices. It is shown that the irrational c/a ratio of most hexagonal metals gives rise to the fikefihood of such transformations. A method for predicting the most likely structure of a boundary is given and shown to provide the correct structure in at least one case.
1. Introduction Bollman's O-lattice theory has been applied widely to the study of the structures of grain boundaries, particularly in the case of cubic materials [1]. A singular advantage of the Olattice theory over other formulations such as the coincidence site lattice theory is that the O-lattice changes continuously with misorientation. In essence, the O-lattice defines the points of best match between the lattices of the adjoining crystals, between which are the regions of worst match, where the lattice mismatch becomes discretized into dislocations. For large-angle grain boundaries it is common to use a similar formulation based upon the O2-1attice or secondary O-lattice which defines *Paper presented at the 2nd International Conference on Low-Energy Dislocation Structures, Charlottesville, VA, August 13-17, 1989. (1921-5093/89/$3.50
the points of best match between the so-called displacement-shift-complete (DSC) lattices defined at some suitable coincidence misorientation. As the deviation from the coincidence orientation increases, the O2-1attice continuously shrinks and rotates, defining the changes in the secondary dislocation array in the boundary. Until recently, this model for the structure of large-angle grain boundaries was a little problematical because the dislocations that it predicts would often be so close to each other as to lose their individual identities at most misorientations. A more recent advance in the understanding of high-angle grain boundary structure was the development of the structural unit model [2] which demonstrates that easily identifiable clusters of small numbers of atoms can be found in most symmetrical grain boundaries, and that certain "favored" grain boundaries .are made up exclusively of a single cluster type. This seems to occur at certain of the well known coincidence orientations, though not at all of them. As misorientation changes, moving away from a favored misorientation, the boundary structure begins to incorporate other structural units, and in particular it seems to take in units from a favored boundary structure which will eventually be reached if the misorientation change is continually increased. At any point in misorientation space between two favored boundary structures, the boundary structure can be described as a simple lever-rule mixture of the structural units that define the favored boundaries. At any point close in misorientation to one of the favored boundaries, the minority structural units (belonging to the other boundary structure) can also be identified as the cores of DSC dislocations, and this provides a bridge to the older theory. It also demonstrates that there is a physical reality underlying the mathematical formality of the © Elsevier Sequoia/Printed in The Netherlands
122
O-lattice even when the O-elements are very close together. In the following sections we investigate the possible dislocation structures of large-angle grain boundaries in h.c.p, materials and note that there are significant new degrees of freedom that do not exist in the cubic case.
2. Misorientation effects in h.c.p, materials Three dimensional coincidence site lattices (CSLs) are only possible in h.c.p, materials for rational values of (c/a) 2, except for rotations about the [0001] axis [3]. It is therefore necessary to introduce the concept of constrained coincidence to accommodate the irrational axial ratios of real materials. The fundamental idea is that any small deviation from an ideal c/a ratio can be accommodated by an array of dislocations in the interface, in the same way that a deviation from the perfect CSL misorientation is accommodated by an array of dislocations. This is illustrated in Fig. 1, where a grain boundary is schematically dissociated into three interfaces. The first interface separates the natural lattice of crystal 1 from a lattice with the ideal c/a ratio necessary to form a CSL: this may be considered to be a semicoherent interface. The second interface is a grain boundary between two ideal-ratio lattices and is in all respects similar to a coincidence-related boundary of the type familiar in cubic lattices. The third interface is again a semicoherent boundary between ideal-ratio and natural material. It is apparent that even at the exact misorientation required to form a constrained CSL, or CCSL, a grain boundary may contain an array of dislocations which exist in order to accommodate the deviation from the appropriate ideal c/a ratio. Chen and King [7] have shown that the experimentally observed dislocation structures of highangle grain boundaries in zinc can be rationalized
in this scheme and that the O-lattice formalisms can be modified to accommodate it. Various tables of CCSLs and their corresponding DSC lattices have been published for ideal cases with rational values of (c/a) 2 [4-6]. As can be seen in Fig. 2, the distribution of these CCSLs in misorientation space is very highly inhomogeneous, with dense clusters often being found, which is in marked contrast to the case of the cubic crystals. The mixing of structures continuously between the various CCSLs of a single cluster would give rise to a very rapid change of structures as a function of misorientation which could appear as a form of structural transformation. Such a transformation as a function of misorientation would occur only if there were more than a single favored structure in the cluster, and this cannot yet be ruled out. It is also conceivable that the selection of the preserved CCSL in such structures is strongly affected by the exact form of the dislocation array in the interface, and this changes rapidly with misorientation, as we have described above. The boundary structure is set by the requirement to minimize its energy, and the energy can be considered, to a fair approximation, as having two components: the energy of the ideal structure of the coincidence interface, and the energy of the superimposed dislocation array. In a CCSL cluster, there may be several choices of ideal structure, each with its associated dislocation array, and the energy balances between the various choices may change rapidly with misorientation, as do the densities and the dislocation arrays. (In fact, in the work of Chen and King [7], the large
0
Natural
Ideal
20
30
40
50
60
70
80
90
material
c/a
material
~....Sem i - c o h e r e n t interface
_~__~__~__~__~__~_Cotnctdence-related boundary Ideal
10
c/a
material
~ll~icob,+o++ +
+++
+
+
L
I
r
J
I
]
i
]
I
]
0
10
20
30
40
50
60
70
80
90
Angle
In
Rotation
de~rees
~. ~ ~....~. S e m l - c o h e r e n t [Natural
zzmaterial
l-
IntErface
Fig. 1. Schematic diagram illustrating the notional dissociation of a grain boundary in an h.c.p, material into three simple interfaces whose dislocation content are readily calculable using current theory.
Fig. 2. Examples of misorientations producing three-dimensional CSLs with o less than 50. Note the dense clusters of CSLs in the h.c.p, case, as compared with the relatively widely spaced occurrences in the cubic system. The pecked horizontal lines represent the range of c/a ratios that zinc can take at various temperatures.
123
variation of structure with differing CCSLs in a single cluster was used to identify the actual CCSL that was being preserved by the dislocation array.) This leads to the suggestion that there may exist true first-order structural transformations as a function of misorientation, as one structure is replaced by another.
2.1. Temperature effects A second consideration with respect to the structure of high-angle grain boundaries in hexagonal materials is that the c/a ratio can be quite strongly temperature dependent, as shown in Fig. 3. This means that the component of the dislocation array that exists to accommodate the deviation of the actual ratio from the ideal value must change with temperature, so the total dislocation content will vary as a function of temperature. The problem is further complicated by the fact that the rotation axis for a grain boundary will also change with temperature if it lies between the c axis and the basal plane, and assuming that the crystals are not free to rotate to accommodate the change. When the value of c/a changes to k(c/a), a rotation axis, say [u, v, w], will change to [u, v, w/k], while the angle of rotation is unchanged [8]. Just as we have postulated a first-order phase transformation as a ruction of misorientation, driven by the energy of the interfacial dislocation array, we can also postulate the existence of similar transformations as a function of temperature, because the dislocation arrays also change with that parameter. This is different from the kind of phase transformation which has been demonstrated by Sickafus and Sass [9], since it derives from a change in the crystallography of the interface rather than from a change in the composi-
tion. The types of transformation that we are describing here might be expected in any noncubic crystal system where coincidence is predicated upon the existence of special axial ratios, and to this extent the cubic crystals must be regarded as a somewhat special case, where such transformations are forbidden.
3. O2-Lattice equation in the h.c.p, structure The purpose of this section is to describe how the vectors X(02) are determined in the case of large-angle grain boundaries in the h.c.p, structure. These vectors define the O2-1attice and therefore the separation and orientation of dislocations in the interface. Our development is intended to illustrate the parallels with the familiar form of the O2-1attice equation X(OZ)=(I-RcR
,)-, ba~
where R c is an exact coincidence rotation and b d~c is a possible Burgers vector for a perfect dislocation in the interface. The matrix RR~.- ~ is usually known as R (02). The O2-1attice equation describing dislocation structures to accommodate deviations from the ideal c/a ratio and the exact misorientation in the h.c.p, structure can be formulated in an orthonormal coordinate system, which is the most convenient for numerical calculations. In the h.c.p. materials, coincidence rotation matrices, CSLs and DSC lattices have all been given in hexagonal coordinates. Experimental rotation matrices Rexp are also determined in hexagonal coordinates. These are all transformed into orthonormal coordinates by means of the structure matrix S of the hexagonal system as follows: rotation matrices R by R,,rth = S R h ~ x S
1
1.9
and vectors by 1.aa
~ , ~ ~
borth = Sbh~
Cd
c/a
where
1.86 )
~
-
. ~ / z ~Tf
1,a4
1
1
2
31/2 1,82
---~
I 50
100
..
i
i
150
200
I 250
i
i
300
350
L 400
S=
i _ _ ~ 450
500
550
0
800
2
T E ~ E ~ T U R E (X)
Fig. 3. The variation of c/a with temperature for zinc and cadmium.
0
0
c
-
a
124
Now we need to determine how an experimentally determined rotation matrix Rex p c a n be expanded into the transformation matrices corresponding to the three schematic interfaces described above. First, vectors XI~ in ideal-ratio lattice 1 are transformed into vectors X~ in the equivalent natural lattice:
We multiply this equation by E - 1 to yield (E -1 - RcE- IRexp- 1)X(O2) = basc and if we define T = E- 1
_
_
Rc E - mRexp- 1
then
X l = EXxc
X(O2) = T - lbdsc
where
Here, a subscript "n" refers to the natural material and "c" refers to material with the ideal c/a ratio to form a particular coincidence site lattice. The vector X~ can now be rotated by the crystal misorientation to give a vector X2 in lattice 2, which deviates slightly from a CCSL misorientation:
X(O2) is now given in orthonormal coordinates related to the natural lattice of crystal 1. If the natural material has the required ideal c/a ratio, this equation degenerates to the well known O2-1attice equation in the cubic system. The equation that we have derived is the same as that given by Bollmann and Warrington [10] except that their R d is substituted by RexpRc- I. Now the spacings and orientations of the dislocations can be calculated for any grain boundary plane:
X2
dislocation line direction
E=
=
[1 0 0
00
]
1
0
0
(c/a)n/(C/a) c
RexpX 1
= RexpEXl~
D = n x X(O2)
We can now introduce the exact transformation R~ relating Xlc and X2c at a coincidence misorienration. It is the difference between the actual (or experimental) rotation and the coincidence misorientation that gives rise to the dislocation content of the second of our three conceptual interfaces: XI c = R c - JX2c =Rc-IE-IX~ Our frame of reference is still the orthogonal frame derived from the natural lattice of crystal 1, and R c relates the ideal-ratio lattices; it is therefore necessary to transform R c into the natural lattice coordinates. Thus X 2 -~ R e x p E R c -
1E -
1X2¢
So R(O2) = RexpERc- 1E - 1 and R(O2)- 1 = ERcE - 1Rexp- 1 Since the dsc lattice is also defined in the lattice of the constrained material, we must use E b ds~ instead of b ds~ when we work in natural crystal coordinates. Our O2-1attice equation therefore becomes (I - ERcE -
1Rex
p -
1)X(O2) = g b d s c
dislocation spacing
S = {[ X(O2)I/(X(O2)}[n x {n x X(O2)}] where n is the grain boundary normal. 4. C a l c u l a t i o n s o f d i s l o c a t i o n s t r u c t u r e s
For a given rotation axis and grain boundary normal, we calculate the spacings and orientations of dislocations in zinc as a function of misorientation and temperature. Two groups of clustered CCSLs are considered in this paper, and the details are given in Tables 1 and 2. In all of these cases, det(T ) is zero, so rank( T ) is 2. T was inverted by the procedure which has been developed by Bollmann [11]. Plots of dislocation spacing as a function of both misorientation and temperature were made. In each case, the dislocation spacing tends to infinity when the ideal c/a ratio and the exact coincidence misorientation are achieved, as expected. It is also found that there exists either one line of maximum spacings or a pair of crossed lines, as shown in Figs. 4 and 5. These lines correspond to loci of counteraction between the dislocations required to accommodate the misorientation and the lattice constraint. The dislocation spacing changes continuously with changes in either temperature or misorientation for structures referred to any single CCSL.
125
TABLE 1 The single cluster of six different coincidence systems close to 86°/[100]
Z/[uvw]/O (deg)
(c/a):
Y DSC 1
Y DSC2
13/[100]/85.59 15/[100]/86.18 17/[100]/86.62 24/[100]/85.21 28/[100]/85.9 32/[100]/86.42
7/2 24/7
[1 [1 [1 [I [1 [1
[3 - 7 [3 - 9 [4 - 9 [5 - 1 4 [5 - 1 8 [5 - 2 2
TABLE 2
27/8 39/11 45/13 17/5
2 2 2 2 2 2
-1] -1] - 1] -1] -1] -1]
Z DSC3 -31 -31 -4] -5] -5] -5]
[ - lo
-7 I - 12 - 9 [ - 13 - 9 [-19 -14
-3] -3] -41 -5]
[-23 -18 -5] [-27 -22 -5]
The single cluster of six different coincidence systems close to 57°/[210]
X/[uvw]/O (deg)
(c/a) ~-
Z DSC 1
Z DSC2
9/[210[/56.25 22/[210]/56.944 31/[210]/56.74 35/[210]/57.12 41t/[210]/56.63 50/[210]/55.944
7/2 17/5 24/7 27/8 31/9 39/11
[0 [0 [0 [(/ [0 [0
[0 [0 [0 [0 [() [0
l l 1 I 1 1
- l] -1] -1] -1] - 1] -1[
-7 -17 -26 -3(1 -35 -45
Z DSC3 -2l 5] -5] -51 -5] -5]
9
8
l]
-22 -20 -2] -31 -28 -3] -35 -31 -4] 40 4 - 4] 50 5 -5]
I
k, • .~-
a.a..
Fig. 4. Calculated dislocation spacings for DSC3 (11/39) [ - 13, - 4, 17, - 9]) as a function of misorientation and temperature. Grain boundary structures are constrained to Y 13 (85.59°/[ 100]). Spacings are artificially truncated at 50 nm in this figure.
Investigations of the dislocation orientations show that these also change continuously with misorientation and temperature. The magnitude o f t h e s e v a r i a t i o n s , h o w e v e r , is v e r y s m a l l if o n l y a s i n g l e line of m a x i m u m s p a c i n g s exists, b u t q u i t e l a r g e w h e n a p a i r o f c r o s s e d lines is o b t a i n e d . T y p i c a l v a r i a t i o n s o f line d i r e c t i o n f o r t h e l a t t e r c a s e a r e i l l u s t r a t e d in F i g . 6.
Fig. 5. Calculated dislocation spacings for DSC3 (11/96) [ - 3 2 , - 17, 49, - 15]) as a function of misorientation and temperature. Grain boundary structures are constrained to Z32 186.42°/[100]). Spacings are artificially truncated at 50 nm in this figure.
5. Choice of lowest energy dislocation structure In the h.c.p, metals it is typical, as w e have shown, that several C C S L s are clustered in a very narrow region of misorientation space, and the ideal c/a ratio for each C C S L is different. Dislocation structure is frequently f o u n d in the grain boundaries close in misorientation to a cluster of
126
LINES OF PREFERRED STRTES g ':R. g~,.q.
La~
ZOO F-.., u~. aE co
CDu~
Fig. 6. Dislocation line orientations, relative to an arbitrarily chosen reference direction, as a function of misorientation and temperature. The boundary structure is the same as that used for Fig. 5.
CCSLs, but it is a difficult task to determine to which member of the cluster the structure is related. In the case of cubic crystals, it is sufficient to determine merely which CSL is closest in misorientation to an experimental boundary, but in the hexagonal materials the c/a ratio must also be considered. The type of calculation that we have described above may be performed for each of the several CCSLs in a cluster close to the misorientation of a real grain boundary. Each calculation yields different results because the coincidence structures and ideal c/a ratios are all different. As noted by Chen and King [7], however, the Burgers vectors of the DSC dislocations are very nearly the same. Because of the similarity of the Burgers vectors, a simple criterion for the lowest-energy structure is just the one that provides the greatest dislocation spacings (or lowest dislocation density). We can also avoid the graphical complexity of overlaying several three-dimensional plots by projecting the lines of maximum dislocation spacing for each possible Burgers' vector onto the plane containing the c/a ratio and misorientation axes. On such a two-dimensional plot, proximity to the projected line denotes large dislocation spacing and therefore low energy. An example of the type of map that we propose is given in Fig. 7. This case corresponds to a grain boundary with a misorientation of 85.21°/ [100], which is precisely the correct misorientation to form Y24. In this case, however, the
1.8352
1.B~62
1.~s62
~.~5~2
1.8~s2
1.B862
Fig. 7. Preferred states as a function of misorientation and c/a ratio. The six lines represent maximum dislocation spacings for each DSC1 vector, based on the members of a cluster of six different coincidence systems close to 86°/[100].
LINES OF PREFERRED STFITES g
--m
',
~. z~.
,, .
1.8362
1.B462
.
.
.
1.8562
I .B662
1 .B762
1.8862
C/R
Fig. 8. Experimentally determined misorientation from ref. 7 (85.38°/[98, - 3 , 0]) plotted at the room-temperature' c/a value on the corresponding dislocation spacing map for the DSC3 dislocations, which embody the largest grain boundary Burgers vectors.
choice of Z13 would provide lower energy because of the better match with the c/a ratio of zinc at room temperature (1.8562), leading to a greater overall dislocation spacing. Figure 8
127
shows a further example of a dislocation spacing map, corresponding to an experimental case reported in ref. 7. The experimentally determined misorientation is plotted, for the room temperature c/a ratio. This point is closest to the line corresponding to 5'13, which was identified experimentally as the structure of the boundary. It is clearly possible to move about within a dislocation spacing map either by changing the misorientation of the grain boundary, or by changing the temperature. Either variable may cause the minimum energy structure to change, as the relative dislocation spacings of different structures are altered. It is thus reasonable to postulate that structural transformations may occur as a function of either of these variables. We note further that the structures of grain boundaries in non-cubic crystals are very sensitive to small changes of misorientation or crystallography, and this statement might be extended to phase boundaries as well. For this reason, it is of the utmost importance to measure all of these parameters with the greatest possible accuracy in any experiment aiming to compare experimental with theoretical structures.
modate the misorientation is required, and this only occurs at a particular displacemen! from the coincidence misorientation. The existence of clusters of CCSLs in misorientation space also means that several cusps may merge into a broad relatively smooth minimum in the free energy surface. It would then require a very sensitive technique to distinguish single cusps from multiple ones. Present techniques for investigating the free energy surface would not appear to be sufficient for this task.
7. Summarizing remarks Interfacial dislocations exist in crystals with irrational axial ratios to accommodate deviations from ideal ratios as well as from ideal misorientations. Since constrained CSLs may occur in clusters, transformations from one structure to another are likely as functions of either misorientation or temperature.
Acknowledgments This work was supported by the National Science Foundation, Grant DMR 8601433.
6. Discussion An interesting aspect of the features that we have presented in this paper is the fact that the variation of grain boundary energy with misorientation in the h.c.p, metals may be very different from the form that has become familiar in cubic cases. It is generally accepted that in the largeangle regime for the cubic metals, the grain boundary energy is sharply cusped at points corresponding to coincidence misorientations. The variation of the energy in the vicinity of these cusps is expected to follow a Read-Shockley type of curve, since it is principally determined by the elastic energy of the dislocation arrays in the grain boundary. For the h.c.p, cases, where there is rarely a precise match of the natural and ideal c/a ratios, the positions of cusps in the energy surface will be displaced from the coincidence misorientation. This is because some counteraction of the dislocations that accommodate the constraint of the c/a ratio by the dislocations that accom-
References 1 W. Bollmann, C~stal l)efects and ('rvstalline lnterJ~wes, Springer, Berlin, 1970. 2 A. P. Sutton and V. Vitek, l'hilos. Trans. R. Soc. London, Ser. A, 309 (1983) 37. 3 G. A. Bruggemann, G. H. Bishop and W. H. Hartt, in H. Hu (ed.), The Nature and Behavior o[ Grain Boundaries, Plenum, New York, 1972, p. 83. 4 D. H. Warrington, J. Phys. Paris, Colloq. (4, 36 (1975) 87. 5 R. Bonnet, E. Cousineau and D. H. Warrington, Acta Crystallogr. A, 37(1981) 184. 6 F. R. Chen, Ph.D. Dissertation, State University of New York at Stony Brook, 1986. 7 F. R. Chen and A. H. King, Philos. Mag. A. 57(1988) 431. 8 M. A. Fortes and D. A. Smith. Scr. Memll.. 10 (1976) 575. 9 K. E. Sickafus and S. L. Sass, Acta Metall., 35 (1987) 69. 10 W. Bollmann and D. H. Warrington, in W. Bollmann (ed.), C~vstal Lattices, Interlaces, Matrices, 1982, p. 214. 11 W. Bollmann (ed.), Crystal Lattices, Interfaces, Matrices, 1982.
121
Dislocation Structures in Large-angle Grain Boundaries in Hexagonal Close-packed Materials* KISOO SHIN and A. H. KING Departrnem of Materials Science and Engineering, State University of New York at Stony Brook, Stony Brook, NY 11794-2275 (U.S.A.) (Received September 26, 1988)
Abstract In materials with cubic crystal lattices, the dislocation structures of grain boundaries are believed to change in a continuous manner as misorientation changes, according to the well-established O-lattice theory developed by Bollmann. They have been shown to change discontinuously as grain boundary phase transformations occur with changes of composition or temperature. The possibilities of continuous and discontinuous structure changes with misorientation and temperature are explored for materials with hexagonal lattices. It is shown that the irrational c/a ratio of most hexagonal metals gives rise to the fikefihood of such transformations. A method for predicting the most likely structure of a boundary is given and shown to provide the correct structure in at least one case.
1. Introduction Bollman's O-lattice theory has been applied widely to the study of the structures of grain boundaries, particularly in the case of cubic materials [1]. A singular advantage of the Olattice theory over other formulations such as the coincidence site lattice theory is that the O-lattice changes continuously with misorientation. In essence, the O-lattice defines the points of best match between the lattices of the adjoining crystals, between which are the regions of worst match, where the lattice mismatch becomes discretized into dislocations. For large-angle grain boundaries it is common to use a similar formulation based upon the O2-1attice or secondary O-lattice which defines *Paper presented at the 2nd International Conference on Low-Energy Dislocation Structures, Charlottesville, VA, August 13-17, 1989. (1921-5093/89/$3.50
the points of best match between the so-called displacement-shift-complete (DSC) lattices defined at some suitable coincidence misorientation. As the deviation from the coincidence orientation increases, the O2-1attice continuously shrinks and rotates, defining the changes in the secondary dislocation array in the boundary. Until recently, this model for the structure of large-angle grain boundaries was a little problematical because the dislocations that it predicts would often be so close to each other as to lose their individual identities at most misorientations. A more recent advance in the understanding of high-angle grain boundary structure was the development of the structural unit model [2] which demonstrates that easily identifiable clusters of small numbers of atoms can be found in most symmetrical grain boundaries, and that certain "favored" grain boundaries .are made up exclusively of a single cluster type. This seems to occur at certain of the well known coincidence orientations, though not at all of them. As misorientation changes, moving away from a favored misorientation, the boundary structure begins to incorporate other structural units, and in particular it seems to take in units from a favored boundary structure which will eventually be reached if the misorientation change is continually increased. At any point in misorientation space between two favored boundary structures, the boundary structure can be described as a simple lever-rule mixture of the structural units that define the favored boundaries. At any point close in misorientation to one of the favored boundaries, the minority structural units (belonging to the other boundary structure) can also be identified as the cores of DSC dislocations, and this provides a bridge to the older theory. It also demonstrates that there is a physical reality underlying the mathematical formality of the © Elsevier Sequoia/Printed in The Netherlands
122
O-lattice even when the O-elements are very close together. In the following sections we investigate the possible dislocation structures of large-angle grain boundaries in h.c.p, materials and note that there are significant new degrees of freedom that do not exist in the cubic case.
2. Misorientation effects in h.c.p, materials Three dimensional coincidence site lattices (CSLs) are only possible in h.c.p, materials for rational values of (c/a) 2, except for rotations about the [0001] axis [3]. It is therefore necessary to introduce the concept of constrained coincidence to accommodate the irrational axial ratios of real materials. The fundamental idea is that any small deviation from an ideal c/a ratio can be accommodated by an array of dislocations in the interface, in the same way that a deviation from the perfect CSL misorientation is accommodated by an array of dislocations. This is illustrated in Fig. 1, where a grain boundary is schematically dissociated into three interfaces. The first interface separates the natural lattice of crystal 1 from a lattice with the ideal c/a ratio necessary to form a CSL: this may be considered to be a semicoherent interface. The second interface is a grain boundary between two ideal-ratio lattices and is in all respects similar to a coincidence-related boundary of the type familiar in cubic lattices. The third interface is again a semicoherent boundary between ideal-ratio and natural material. It is apparent that even at the exact misorientation required to form a constrained CSL, or CCSL, a grain boundary may contain an array of dislocations which exist in order to accommodate the deviation from the appropriate ideal c/a ratio. Chen and King [7] have shown that the experimentally observed dislocation structures of highangle grain boundaries in zinc can be rationalized
in this scheme and that the O-lattice formalisms can be modified to accommodate it. Various tables of CCSLs and their corresponding DSC lattices have been published for ideal cases with rational values of (c/a) 2 [4-6]. As can be seen in Fig. 2, the distribution of these CCSLs in misorientation space is very highly inhomogeneous, with dense clusters often being found, which is in marked contrast to the case of the cubic crystals. The mixing of structures continuously between the various CCSLs of a single cluster would give rise to a very rapid change of structures as a function of misorientation which could appear as a form of structural transformation. Such a transformation as a function of misorientation would occur only if there were more than a single favored structure in the cluster, and this cannot yet be ruled out. It is also conceivable that the selection of the preserved CCSL in such structures is strongly affected by the exact form of the dislocation array in the interface, and this changes rapidly with misorientation, as we have described above. The boundary structure is set by the requirement to minimize its energy, and the energy can be considered, to a fair approximation, as having two components: the energy of the ideal structure of the coincidence interface, and the energy of the superimposed dislocation array. In a CCSL cluster, there may be several choices of ideal structure, each with its associated dislocation array, and the energy balances between the various choices may change rapidly with misorientation, as do the densities and the dislocation arrays. (In fact, in the work of Chen and King [7], the large
0
Natural
Ideal
20
30
40
50
60
70
80
90
material
c/a
material
~....Sem i - c o h e r e n t interface
_~__~__~__~__~__~_Cotnctdence-related boundary Ideal
10
c/a
material
~ll~icob,+o++ +
+++
+
+
L
I
r
J
I
]
i
]
I
]
0
10
20
30
40
50
60
70
80
90
Angle
In
Rotation
de~rees
~. ~ ~....~. S e m l - c o h e r e n t [Natural
zzmaterial
l-
IntErface
Fig. 1. Schematic diagram illustrating the notional dissociation of a grain boundary in an h.c.p, material into three simple interfaces whose dislocation content are readily calculable using current theory.
Fig. 2. Examples of misorientations producing three-dimensional CSLs with o less than 50. Note the dense clusters of CSLs in the h.c.p, case, as compared with the relatively widely spaced occurrences in the cubic system. The pecked horizontal lines represent the range of c/a ratios that zinc can take at various temperatures.
123
variation of structure with differing CCSLs in a single cluster was used to identify the actual CCSL that was being preserved by the dislocation array.) This leads to the suggestion that there may exist true first-order structural transformations as a function of misorientation, as one structure is replaced by another.
2.1. Temperature effects A second consideration with respect to the structure of high-angle grain boundaries in hexagonal materials is that the c/a ratio can be quite strongly temperature dependent, as shown in Fig. 3. This means that the component of the dislocation array that exists to accommodate the deviation of the actual ratio from the ideal value must change with temperature, so the total dislocation content will vary as a function of temperature. The problem is further complicated by the fact that the rotation axis for a grain boundary will also change with temperature if it lies between the c axis and the basal plane, and assuming that the crystals are not free to rotate to accommodate the change. When the value of c/a changes to k(c/a), a rotation axis, say [u, v, w], will change to [u, v, w/k], while the angle of rotation is unchanged [8]. Just as we have postulated a first-order phase transformation as a ruction of misorientation, driven by the energy of the interfacial dislocation array, we can also postulate the existence of similar transformations as a function of temperature, because the dislocation arrays also change with that parameter. This is different from the kind of phase transformation which has been demonstrated by Sickafus and Sass [9], since it derives from a change in the crystallography of the interface rather than from a change in the composi-
tion. The types of transformation that we are describing here might be expected in any noncubic crystal system where coincidence is predicated upon the existence of special axial ratios, and to this extent the cubic crystals must be regarded as a somewhat special case, where such transformations are forbidden.
3. O2-Lattice equation in the h.c.p, structure The purpose of this section is to describe how the vectors X(02) are determined in the case of large-angle grain boundaries in the h.c.p, structure. These vectors define the O2-1attice and therefore the separation and orientation of dislocations in the interface. Our development is intended to illustrate the parallels with the familiar form of the O2-1attice equation X(OZ)=(I-RcR
,)-, ba~
where R c is an exact coincidence rotation and b d~c is a possible Burgers vector for a perfect dislocation in the interface. The matrix RR~.- ~ is usually known as R (02). The O2-1attice equation describing dislocation structures to accommodate deviations from the ideal c/a ratio and the exact misorientation in the h.c.p, structure can be formulated in an orthonormal coordinate system, which is the most convenient for numerical calculations. In the h.c.p. materials, coincidence rotation matrices, CSLs and DSC lattices have all been given in hexagonal coordinates. Experimental rotation matrices Rexp are also determined in hexagonal coordinates. These are all transformed into orthonormal coordinates by means of the structure matrix S of the hexagonal system as follows: rotation matrices R by R,,rth = S R h ~ x S
1
1.9
and vectors by 1.aa
~ , ~ ~
borth = Sbh~
Cd
c/a
where
1.86 )
~
-
. ~ / z ~Tf
1,a4
1
1
2
31/2 1,82
---~
I 50
100
..
i
i
150
200
I 250
i
i
300
350
L 400
S=
i _ _ ~ 450
500
550
0
800
2
T E ~ E ~ T U R E (X)
Fig. 3. The variation of c/a with temperature for zinc and cadmium.
0
0
c
-
a
124
Now we need to determine how an experimentally determined rotation matrix Rex p c a n be expanded into the transformation matrices corresponding to the three schematic interfaces described above. First, vectors XI~ in ideal-ratio lattice 1 are transformed into vectors X~ in the equivalent natural lattice:
We multiply this equation by E - 1 to yield (E -1 - RcE- IRexp- 1)X(O2) = basc and if we define T = E- 1
_
_
Rc E - mRexp- 1
then
X l = EXxc
X(O2) = T - lbdsc
where
Here, a subscript "n" refers to the natural material and "c" refers to material with the ideal c/a ratio to form a particular coincidence site lattice. The vector X~ can now be rotated by the crystal misorientation to give a vector X2 in lattice 2, which deviates slightly from a CCSL misorientation:
X(O2) is now given in orthonormal coordinates related to the natural lattice of crystal 1. If the natural material has the required ideal c/a ratio, this equation degenerates to the well known O2-1attice equation in the cubic system. The equation that we have derived is the same as that given by Bollmann and Warrington [10] except that their R d is substituted by RexpRc- I. Now the spacings and orientations of the dislocations can be calculated for any grain boundary plane:
X2
dislocation line direction
E=
=
[1 0 0
00
]
1
0
0
(c/a)n/(C/a) c
RexpX 1
= RexpEXl~
D = n x X(O2)
We can now introduce the exact transformation R~ relating Xlc and X2c at a coincidence misorienration. It is the difference between the actual (or experimental) rotation and the coincidence misorientation that gives rise to the dislocation content of the second of our three conceptual interfaces: XI c = R c - JX2c =Rc-IE-IX~ Our frame of reference is still the orthogonal frame derived from the natural lattice of crystal 1, and R c relates the ideal-ratio lattices; it is therefore necessary to transform R c into the natural lattice coordinates. Thus X 2 -~ R e x p E R c -
1E -
1X2¢
So R(O2) = RexpERc- 1E - 1 and R(O2)- 1 = ERcE - 1Rexp- 1 Since the dsc lattice is also defined in the lattice of the constrained material, we must use E b ds~ instead of b ds~ when we work in natural crystal coordinates. Our O2-1attice equation therefore becomes (I - ERcE -
1Rex
p -
1)X(O2) = g b d s c
dislocation spacing
S = {[ X(O2)I/(X(O2)}[n x {n x X(O2)}] where n is the grain boundary normal. 4. C a l c u l a t i o n s o f d i s l o c a t i o n s t r u c t u r e s
For a given rotation axis and grain boundary normal, we calculate the spacings and orientations of dislocations in zinc as a function of misorientation and temperature. Two groups of clustered CCSLs are considered in this paper, and the details are given in Tables 1 and 2. In all of these cases, det(T ) is zero, so rank( T ) is 2. T was inverted by the procedure which has been developed by Bollmann [11]. Plots of dislocation spacing as a function of both misorientation and temperature were made. In each case, the dislocation spacing tends to infinity when the ideal c/a ratio and the exact coincidence misorientation are achieved, as expected. It is also found that there exists either one line of maximum spacings or a pair of crossed lines, as shown in Figs. 4 and 5. These lines correspond to loci of counteraction between the dislocations required to accommodate the misorientation and the lattice constraint. The dislocation spacing changes continuously with changes in either temperature or misorientation for structures referred to any single CCSL.
125
TABLE 1 The single cluster of six different coincidence systems close to 86°/[100]
Z/[uvw]/O (deg)
(c/a):
Y DSC 1
Y DSC2
13/[100]/85.59 15/[100]/86.18 17/[100]/86.62 24/[100]/85.21 28/[100]/85.9 32/[100]/86.42
7/2 24/7
[1 [1 [1 [I [1 [1
[3 - 7 [3 - 9 [4 - 9 [5 - 1 4 [5 - 1 8 [5 - 2 2
TABLE 2
27/8 39/11 45/13 17/5
2 2 2 2 2 2
-1] -1] - 1] -1] -1] -1]
Z DSC3 -31 -31 -4] -5] -5] -5]
[ - lo
-7 I - 12 - 9 [ - 13 - 9 [-19 -14
-3] -3] -41 -5]
[-23 -18 -5] [-27 -22 -5]
The single cluster of six different coincidence systems close to 57°/[210]
X/[uvw]/O (deg)
(c/a) ~-
Z DSC 1
Z DSC2
9/[210[/56.25 22/[210]/56.944 31/[210]/56.74 35/[210]/57.12 41t/[210]/56.63 50/[210]/55.944
7/2 17/5 24/7 27/8 31/9 39/11
[0 [0 [0 [(/ [0 [0
[0 [0 [0 [0 [() [0
l l 1 I 1 1
- l] -1] -1] -1] - 1] -1[
-7 -17 -26 -3(1 -35 -45
Z DSC3 -2l 5] -5] -51 -5] -5]
9
8
l]
-22 -20 -2] -31 -28 -3] -35 -31 -4] 40 4 - 4] 50 5 -5]
I
k, • .~-
a.a..
Fig. 4. Calculated dislocation spacings for DSC3 (11/39) [ - 13, - 4, 17, - 9]) as a function of misorientation and temperature. Grain boundary structures are constrained to Y 13 (85.59°/[ 100]). Spacings are artificially truncated at 50 nm in this figure.
Investigations of the dislocation orientations show that these also change continuously with misorientation and temperature. The magnitude o f t h e s e v a r i a t i o n s , h o w e v e r , is v e r y s m a l l if o n l y a s i n g l e line of m a x i m u m s p a c i n g s exists, b u t q u i t e l a r g e w h e n a p a i r o f c r o s s e d lines is o b t a i n e d . T y p i c a l v a r i a t i o n s o f line d i r e c t i o n f o r t h e l a t t e r c a s e a r e i l l u s t r a t e d in F i g . 6.
Fig. 5. Calculated dislocation spacings for DSC3 (11/96) [ - 3 2 , - 17, 49, - 15]) as a function of misorientation and temperature. Grain boundary structures are constrained to Z32 186.42°/[100]). Spacings are artificially truncated at 50 nm in this figure.
5. Choice of lowest energy dislocation structure In the h.c.p, metals it is typical, as w e have shown, that several C C S L s are clustered in a very narrow region of misorientation space, and the ideal c/a ratio for each C C S L is different. Dislocation structure is frequently f o u n d in the grain boundaries close in misorientation to a cluster of
126
LINES OF PREFERRED STRTES g ':R. g~,.q.
La~
ZOO F-.., u~. aE co
CDu~
Fig. 6. Dislocation line orientations, relative to an arbitrarily chosen reference direction, as a function of misorientation and temperature. The boundary structure is the same as that used for Fig. 5.
CCSLs, but it is a difficult task to determine to which member of the cluster the structure is related. In the case of cubic crystals, it is sufficient to determine merely which CSL is closest in misorientation to an experimental boundary, but in the hexagonal materials the c/a ratio must also be considered. The type of calculation that we have described above may be performed for each of the several CCSLs in a cluster close to the misorientation of a real grain boundary. Each calculation yields different results because the coincidence structures and ideal c/a ratios are all different. As noted by Chen and King [7], however, the Burgers vectors of the DSC dislocations are very nearly the same. Because of the similarity of the Burgers vectors, a simple criterion for the lowest-energy structure is just the one that provides the greatest dislocation spacings (or lowest dislocation density). We can also avoid the graphical complexity of overlaying several three-dimensional plots by projecting the lines of maximum dislocation spacing for each possible Burgers' vector onto the plane containing the c/a ratio and misorientation axes. On such a two-dimensional plot, proximity to the projected line denotes large dislocation spacing and therefore low energy. An example of the type of map that we propose is given in Fig. 7. This case corresponds to a grain boundary with a misorientation of 85.21°/ [100], which is precisely the correct misorientation to form Y24. In this case, however, the
1.8352
1.B~62
1.~s62
~.~5~2
1.8~s2
1.B862
Fig. 7. Preferred states as a function of misorientation and c/a ratio. The six lines represent maximum dislocation spacings for each DSC1 vector, based on the members of a cluster of six different coincidence systems close to 86°/[100].
LINES OF PREFERRED STFITES g
--m
',
~. z~.
,, .
1.8362
1.B462
.
.
.
1.8562
I .B662
1 .B762
1.8862
C/R
Fig. 8. Experimentally determined misorientation from ref. 7 (85.38°/[98, - 3 , 0]) plotted at the room-temperature' c/a value on the corresponding dislocation spacing map for the DSC3 dislocations, which embody the largest grain boundary Burgers vectors.
choice of Z13 would provide lower energy because of the better match with the c/a ratio of zinc at room temperature (1.8562), leading to a greater overall dislocation spacing. Figure 8
127
shows a further example of a dislocation spacing map, corresponding to an experimental case reported in ref. 7. The experimentally determined misorientation is plotted, for the room temperature c/a ratio. This point is closest to the line corresponding to 5'13, which was identified experimentally as the structure of the boundary. It is clearly possible to move about within a dislocation spacing map either by changing the misorientation of the grain boundary, or by changing the temperature. Either variable may cause the minimum energy structure to change, as the relative dislocation spacings of different structures are altered. It is thus reasonable to postulate that structural transformations may occur as a function of either of these variables. We note further that the structures of grain boundaries in non-cubic crystals are very sensitive to small changes of misorientation or crystallography, and this statement might be extended to phase boundaries as well. For this reason, it is of the utmost importance to measure all of these parameters with the greatest possible accuracy in any experiment aiming to compare experimental with theoretical structures.
modate the misorientation is required, and this only occurs at a particular displacemen! from the coincidence misorientation. The existence of clusters of CCSLs in misorientation space also means that several cusps may merge into a broad relatively smooth minimum in the free energy surface. It would then require a very sensitive technique to distinguish single cusps from multiple ones. Present techniques for investigating the free energy surface would not appear to be sufficient for this task.
7. Summarizing remarks Interfacial dislocations exist in crystals with irrational axial ratios to accommodate deviations from ideal ratios as well as from ideal misorientations. Since constrained CSLs may occur in clusters, transformations from one structure to another are likely as functions of either misorientation or temperature.
Acknowledgments This work was supported by the National Science Foundation, Grant DMR 8601433.
6. Discussion An interesting aspect of the features that we have presented in this paper is the fact that the variation of grain boundary energy with misorientation in the h.c.p, metals may be very different from the form that has become familiar in cubic cases. It is generally accepted that in the largeangle regime for the cubic metals, the grain boundary energy is sharply cusped at points corresponding to coincidence misorientations. The variation of the energy in the vicinity of these cusps is expected to follow a Read-Shockley type of curve, since it is principally determined by the elastic energy of the dislocation arrays in the grain boundary. For the h.c.p, cases, where there is rarely a precise match of the natural and ideal c/a ratios, the positions of cusps in the energy surface will be displaced from the coincidence misorientation. This is because some counteraction of the dislocations that accommodate the constraint of the c/a ratio by the dislocations that accom-
References 1 W. Bollmann, C~stal l)efects and ('rvstalline lnterJ~wes, Springer, Berlin, 1970. 2 A. P. Sutton and V. Vitek, l'hilos. Trans. R. Soc. London, Ser. A, 309 (1983) 37. 3 G. A. Bruggemann, G. H. Bishop and W. H. Hartt, in H. Hu (ed.), The Nature and Behavior o[ Grain Boundaries, Plenum, New York, 1972, p. 83. 4 D. H. Warrington, J. Phys. Paris, Colloq. (4, 36 (1975) 87. 5 R. Bonnet, E. Cousineau and D. H. Warrington, Acta Crystallogr. A, 37(1981) 184. 6 F. R. Chen, Ph.D. Dissertation, State University of New York at Stony Brook, 1986. 7 F. R. Chen and A. H. King, Philos. Mag. A. 57(1988) 431. 8 M. A. Fortes and D. A. Smith. Scr. Memll.. 10 (1976) 575. 9 K. E. Sickafus and S. L. Sass, Acta Metall., 35 (1987) 69. 10 W. Bollmann and D. H. Warrington, in W. Bollmann (ed.), C~vstal Lattices, Interlaces, Matrices, 1982, p. 214. 11 W. Bollmann (ed.), Crystal Lattices, Interfaces, Matrices, 1982.