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GRAIN BOUNDARY STRUCTURAL
TRANSFORMATIONS IN HEXAGONAL CLOSE PACKED METALS
A. King, Kisoo Shin
To cite this version:
A. King, Kisoo Shin. GRAIN BOUNDARY STRUCTURAL TRANSFORMATIONS IN HEXAGO- NAL CLOSE PACKED METALS. Journal de Physique Colloques, 1990, 51 (C1), pp.C1-203-C1-208.
�10.1051/jphyscol:1990131�. �jpa-00230289�
COLLOQUE DE PHYSIQUE
C o l l o q u e Cl, s u p p l k m e n t a u n O l , Tome 5 1 , j a n v i e r 1990
GRAIN BOUNDARY STRUCTURAL TRANSFORMATIONS IN HEXAGONAL CLOSE PACKED METALS
A.H. KING and KISOO SHIN
D e p a r t m e n t of M a t e r i a l s S c i e n c e and E n g i n e e r i n g , S t a t e U n i v e r s i t y o f New Y o r k a t S t o n y B r o o k s , S t o n y Brook NY 1 1 7 9 4 - 2 2 7 5 , U.S.A.
Abstract - Hexagonal metals differ in three important ways from cubic materials. First, the irrational c/a ratio precludes the formation of 3-dimensional CSLs, in general, so lattice coincidence is achieved on the basis of "constrained CSLs" or CCSLs. Second, t h e c/a ratio is temperature dependent, meaning that grain boundary geometry also changes with temperature. Third, the distribution of coincidence systems is highly inhomogeneous in misorientation space, meaning that clusters of CCSLs exist and it is possible for a boundary structure to transform from one CCSL-related form t o a form related to another CCSL.
1 - INTRODUCTION
Grain boundary structure has been studied extensively in cubic materials, through the application of electron microscopy, x-ray diffraction, and various other techniques. In most cases, results are interpreted in terms of the coincidence site lattice (CSL) or 0-lattice models which stress the periodic nature of the grain boundary structure. In non-cubic materials, however, entirely new problems arise when we try t o interpret grain boundary structure i n terms of some kind of periodicity. Most importantly, it is not generally possible to form threedimensional CSLs because of irrational axial ratios. I t is necessary that the squares of the lattice parameters be simple fractions of each other for the formation of CSLs i n hexagonal, tetragonal and orthorhombic crystal systems. Alternatively, the crystals must be rotated about a common basal plane normal. These conditions are not generally fulfilled for real crystals, so new forms of coincidence must be sought. The crystals which abut a t a grain boundary i n a hexagonal metal, for example, can be thought of as being constrained t o a suitable c/a ratio for which CSLs can be formed. The necessary constraint may be quite small - corresponding to a strain well within the elastic limit, and indeed there may be several available choices of suitable c/a ratios. We refer t o CSLs formed by this means as "Constrained CSLsl' or CCSLs.
The structure of a large angle grain boundary in a cubic material is generally thought of as consistin of a "core" structure essentially that of a boundary a t a n exact coincidence misorientation of a favored type!
superimposed upon W
\
ich are arrays of interfacial dislocations that accommodate the deviation of the actual misorientation from that necessary t o form the CSL. Provided that the CSL in question corresponds to a favored boundary as described by Sutton and Vitek [l], and the deviation of the misorientation is not too great, the dislocations are well separated in the boundary andareamenable t o analysis using transmission electron microscopy. For cases of constrained coincidence, the situation is a little more complicated, and it is illustrated in Fig.1, where we schematically dissociate a large angle grain boundary into three interfaces. The first interface separates t h e natural material from material with the ideal c f a ratio necessary t o form a CCSL:this may be regarded as a semi-coherent interface and should be made up of arrays of lattice dislocations.
The second interface is a conventional coincidence-related grain boundary made up of a core structure and a superimposed array of dislocations accommodating the deviation from the ideal misorientation. Finally, we have a third interface, like the first, between constrained and natural material. In reality, of course, all three interfaces are combined into a single one which comprises a periodic core structure with superimposed arrays of dislocations. I n this case, however, the arrays of dislocations accommodate the deviation from the ideal c/a ratio necessary t o form the CCSL, i n addition t o the deviation from the ideal misorientation. The dislocation arrays may take the form of D S c dislocations or lattice dislocations.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1990131
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N a t u r a l m a t e r i a l
.)".,.,v ....$*....v . . . . > . . .
S e m i - c o h e r en t i n t e r f a c e
Ideal c / a m e t e r l e l
C o i n c i d e n c e - r e l a t e d -+-+-+-+-+--l- b " d a
Ideal c / e m a t e r i a l
. . . F ....
4...)LSem
i- c o h e r en t i n t e r f a c e
N a t u r a l m a t e r i a l
Figure l. Schematic dissociation of a large angle grain boundary in a hexagonal crystal system into three simpler interfaces.
We create material with the necessary c/a ratio to form a CSL by inserting semi--coherent interfaces between the natural and ideal crystals.
2 - CALCULATIONS OF GRAIN BOUNDARY STRUCTURE
Previous work [2] has shown that CCSLs are densely clustered in misorientation space for the HCP metals, and that observed misorientations often fall within these clusters. It is therefqre of considerable interest to investigate what determines the actual structure adopted by a given grain boundary. Although it has been observed that the densest available CCSL was adopted in the few cases that have been analyzed in full, there is no particular energetic reason for this to be the case. The approach that we have adopted is to calculate all of the available structures, and then to try to determine which of them embodies the lowest energy. We can perform these calculationsas function of temperature in addition to misorientation, since the variation of c/a for zinc (which is a convenient experimental material) is known.
We restrict ourselves to studying grain boundaries that have misorientations close to 85.50 about [loo], since these are observed frequently in experiments, and there is a variety of CCSL systems close to this misorientation, including X13, 15, 17, 24, 28 and 32. The DSc vectors for all of these systems are ~ery~similar in magnitude and orientation, but the constraints vary significantly. We have shown elsewhere [3] that the 02-lattice equation can be written as follows, for the case of hexagonal materials when we must consider constraint of the c/a ratio in addition to deviation from the ideal misorientation:
Here
$
is the rotation matrix describing the exact coincidence rotation for the material with an ideal c/a ratio,Bexp
is the actual (or experimentally determined) rotation matrix, andE
is the matrix relating the ideal (c) and natural (n) materials:It should also be noted for more general cases that as the c/a ratio changes, say to k(c/a), the axis of rotation changes such that [uvw] becomes [uv(w/k)].
Using equation 1, we can calculate the spacing of a given set of dislocations in a grain boundary plane as a function of misorientation and temperature, and two examples of the results that are obtained are given in Fig.2. It is interesting to note that the spacing is maximized not just at a single point where exact coincidence is obtained, but along at least one line in the temperature/misorientation plane, where the dislocation content required to accommodate the misorientation deviation is canceled by that required to accommodate the axial ratio constraint. In some cases, as shown in Fig.2b, there are two lines of maximum spacing, crossing at the exact coincidence point. For most temperatures, the maximum dislocation spacing does not occur at the misorientation corresponding to the exact CCSL. We are also able to calculate the orientation of the dislocation lines, as shown in Fig.3 where we demonstrate that they also rotate as a function of both temperature and misorientation.
Figure 2. Dislocation spacing as a function of misorientation and temperature for single dislocation arrays in single coincidence systems. The graphs are truncated at 50nm.
Figure 3. Dislocation orientation as a function of misorientation and temperature for a single dislocation array in a single coincidence system. The orientation is plotted as the angle between the dislocations and an arbitrary reference direction lying in the grain boundary plane.
In order t o give a complete description of the structure of a grain boundary we must specify all of the dislocation arrays, in terms of their Burgers vectors, spacings and orientations. For the cases a t hand, we must also specify the coincidence system, since that becomes a degree of freedom when multiple choices exist.
For any single coincidence system, we find that there are two choices of dislocation structure for the boundary plane i n question. These are (a) an array of lattice dislocations, +[1210], plus a n array of DSc dislocations, termed D S c 1 in the notation of Chen and King [2]; and (b) two arrays of D S c dislocations, DSc1 and DSC3. We compare the elastic energies of the various possible structures by applying the Read-Shockley formula t o each array of dislocations, then summing the results t o give the total elastic energy of the grain boundary. This, of course, presumes that there are no elastic interactions between the various dislocation arrays, and it specifically excludes from consideration any effects arising from core enggy differences between the different CCSLs. The latter problem vanishes for a single boundary plane, (0112), which has a structure that is invariant with changes of CCSL.
Calculating the elastic energy of the interface as described above, and always selecting the lowest calculated value, we obtain an energy surface that is represented i n Fig.4. Multiple minima are apparent in the region of the cluster of CCSLs. Fig.5 shows which of the calculations provided the minimum energy at each point i n the misorientation/temperature plane, and this corresponds t o a form of phase diagram for the grain boundary. As we change temperature andtor rnisorientation we expect the grain boundary structure to transform from one CCSL structure t o another as we cross the boundaries shown in Fig.5. We have set out to demonstrate this effect experimentally.
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Figure 4. The energy surface in misorientation-temperature space for grain boundaries in a cluster of CCSLs. The lowest energy structure (based upon elastic energy only) is selected a t every point.
d
l /0.0 10.0 2 . 0 30.0 40.0 500.0 60'0.0 TEMPERRTURE I K I
Figure 5. Diagram illustrating the favored grain boundary structures as a function of temperature and misorientation, corresponding to Fig.4. The Q structures contain lattice dislocations, while the
P
structures are made up of primitive dsc dislocations. The points A and B refer to the grain boundary illustrated in Fig.6, and the points C and D refer to the grain boundary illustrated in Fig.7.3
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EXPERIMENTALPolycrystalline zinc sheet was cold rolled t o a thickness of 0.23mm and 3mm discs were punched from it. These discs were annealed a t 850C in vacuo for 30 minutes t o produce an experimentally convenient grain size and well equilibrated grain boundary structures. The specimens were then double-jet electropolished in a 10% nitric acid solution in methanol, a t -10oC. Electron microscopy was performed a t 120kV using a Philips CM12 equipped with a Gatan liquid nitrogen cooled double tilt stage. The crystallographic details of the boundaries were determined with high accuracy using the techniques developed by Chen and King [4].
Grain boundaries were observed a t temperatures ranging between approximately 140 and 300K.
Figure 6. Transmission electron micrographs of a grain boundary in zinc. The misorientation is 86.7°/[1.011, 0.023, 0.0041.
The first image is taken a t room temperature, the second a t liquid nitrogen temperature, and the third after returning to room temperature.
'Figure 7. Transmission electron micrographs of a grain boundary in zinc. The misorientation is 86.~~/[1.007, 0.0114, 01. The first image is taken a t room temperature, the second a t liquid nitrogen temperature, and the third after returning to room temperature.
4
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RESULTSFigure 6 shows the structure of a large angle grain boundary in zinc at room temperature (a) after cooling to liquid nitrogen temperature (b) and after returning to room temperature (c). It can be seen that the grain boundary dislocations are parallel and uniformly spaced in Fig.6a, but that the structure is disrupted in Fig.Gb, as certain dislocation segments have rotated into a new orientation. Upon returning to room temperature, as shown in Fig.Gb, the dislocations adopt a uniform structure again. The points marked B and A in Fig.5 correspond to this grain boundary at room temperature and liquid nitrogen temperature, respectively. It can be seen that changing the temperature should cause a transformation from 232 to 217, provided that there are no effects of the grain boundary core on the interfacial energy. Analysis of the diffraction contrast obtained from this boundary indicates that the visible set of dislocations had Burgers vectors of the lattice type discussed in section 2.
Figure 7 shows a similar example of a phase transformation in a grain boundary which is represented by the pojnts D and C in Fig.5. This interface is believed to transform from 2'15 to 232 on coooling from room to liquid nitrogen temperature.
COLLOQUE DE PHYSIQUE
5 - DISCUSSION
Figures 6 and 7 gives a clear indication of a change of the grain boundary structure occurring a t low temperature. It was unfortunately not possible t o see t h e transformation go to completeness in either case because of the slow kinetics at 140K: the rotation of the dislocations into their new orientation would require a combination of glide and climb, so the motion of point defects in the grain boundary plane determines the rate of the transformation.
It is also important to note that the structure of the boundary as shown in Fig.6a and c corresponds to that predicted in Fig.5. The dislocations have lattice Burgers vectors rather than primitive D S c vectors, and the orientation of the dislocations is as predicted for the case of 232. Because of the incomplete nature of the transformation t o t h e low-temperature structure, it is not possible t o confirm that E17 is stable a t the temperature of liquid nitrogen.
Previous observations of grain boundary structural transformations [5,6,7,8] have all been made in cubic crystal systems, in cases where the transformation occurred as a result of a compositional change in the boundary, a change in temperature, or an approach t o the bulk melting point. The kind of transformation that we report here is somewhat different since i t occurs in a pure material at low temperature and is driven primarily by the minimization of elastic energy. This type of transformation will not occur in cubic crystals, since they exhibit isotropic thermal expansion, but it is possible in all crystal classes other than cubic. I t is more likely t o occur where there is a large variation of axial ratio with temperature.
We have previously inferred that the selection of the CCSL system with the smallest value of E was favored, for boundaries falling within a cluster of CCSLs [2]. We have later shown, however, that the best documented case i n reference 2 would also have been predicted the same way on the basis of the elastic calculations described here [3]. The favoring of the smallest coincidence cell would indicate the predominance of grain boundary core energy in determining the structure, whereas the calculations described here are concerned only with the elastic energy of the boundary. The previous results provide no basis for determining that core energy or elastic energy plays a predominant role, since the inferences are t h e same i n both cases.
In the present experiment, however, the phase transformation of the boundary structure appears to be driven by the elastic energy of the grain boundary dislocation arrays.
6
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CONCLUSIONSWe have shown that i t is possible to estimate the elastic energy of a grain boundary situated in a cluster of CCSLs, and predict t h e occurrence of transformations of the grain boundary structure. These transformations have been observed to occur below room temperature.
ACKNOWLEDGEMENT
This work is supported by the National Science Foundation, under grant number DMR-8901994.
REFERENCES
1. A.P. Sutton and V. Vitek: Phil. Trans. R. Soc. Lond. (1983) 37.
2. Fu-Rong Chen and A.H. King: Phil. Mag.
A57
(1988) 431.3. Kisoo Shin and A.H. King: Mater. Sci. Eng. (1989) 121.
4. Fu-Rong Chen and A.H. King: J. Electr. Micr. Tech. 6 (1987) 55.
5. A.N. Aleshin, S.I. Prokofjev and L.S. Shvindlerman: Scripta Met. 19 (1985) 1135.
6. K.E. Sickafus and S.L. Sass: Acta Met. 35 (1987) 69.
7. E.L. Maksimova, L.S. Shvindlerman and B.B. Straumal: Acta Met. 36 (1988) 1573.
8. T.E. Hsieh and R.W. Balluffi: Acta Met. 2 (1989) 2133.
