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Diffusion induced grain boundary migration in the zinc-cadmium system
Acta mater. Vol. 44, No. 7, pp. 2983-2998, 1996 Copyright 0 1996 Acta Metallurgica Inc. Published by Elsevier Science Ltd
Pergamon
Printed in Great Britain. All rights reserved 1359-6454/96 $15.00 + 0.00
DIFFUSION
INDUCED GRAIN BOUNDARY MIGRATION IN THE ZINC-CADMIUM SYSTEM L. LIANG and A. H. KING
Department
of Materials
Science and Engineering, State University NY 11794-2275, U.S.A.
of New York
at Stony
Brook,
(Received 16 March 199.5; in revised form 27 July 1995) Abstract-We present a series of experiments on the motion of grain boundaries in zinc during inward cadmium diffusion. Diffusion induced grain boundary migration is shown to occur in a hexagonal metal, for the first time. Using the cylindrical symmetry of the elastic constants in hexagonal crystals, we are able to investigate the applicability of the coherency-strain model and the influence of grain boundary structure, independently. Copyright 0 1996 Acta Metallurgica Inc.
1. INTRODUCTION
Diffusion induced grain boundary migration (DIGM) is characterized by the sideways motion of grain boundaries (GBs) in response to solute diffusion along them and it has a fairly complex phenomenology. It has been studied in a wide variety of metal alloy systems and a few ceramics, and the observations are very similar even in widely differing alloy systems [l]. However, almost all observations to date have been made in cubic crystals. Understanding a complex phenomenon such as DIGM is difficult. Many components of the phenomenon are not thoroughly understood individually, and their interactions are, therefore, doubly difficult to unravel. Much attention has been paid to the driving force for DIGM, with attention focussing primarily upon two contributions: the coherency strain energy [2] and the free energy of mixing which is made available by the formation of an alloyed (or de-alloyed) region behind the moving GB [3]. Early work suggested that the free energy of mixing was insufficient to explain some of the observed features of DIGM, including migration against very small radii of curvature [4]. Systematic studies carried out as a function of misfit strain, via alloying [5], and elastic modulus effects, via elastic anistropy [6], have tended to confirm that the coherency strain effect has an important influence upon DIGM. Grain boundary structure also exerts some influence upon the progress of DIGM, as illustrated in a detailed study as a function of misorientation using symmetrical tilt GBs [7]. It is not always clear how such an effect arises: it may relate to the variation of GB diffusivity with misorientation [8], or the variation of GB mobility with misorientation [9]. The
mechanism of GB diffusion may also vary with misorientation, although it is thought to occur by a vacancy mechanism at least in well-ordered GBs [lo]. Finally, the mechanism of GB migration varies with GB parameters and may occur by shear (in some cases, such as twins), uncorrelated atomic jumps or correlated atomic jumps [l 11.Any linkage between the mechanisms of GB diffusion and migration is obviously affected if either of the mechanisms changes. In order to distinguish the effects of GB structure from non-structure-related effects, it is desirable to perform experiments in which the driving force can be held constant, while the structure varies. Hexagonal crystals provide an opportunity for such experiments and we report a study of DIGM in the zinc-cadmium system in this paper.
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2. COHERENCY
STRAIN
THEORY
The coherency strain model, which was originally proposed for liquid film migration, suggests that the high solute concentration established in the GB leaks into the adjacent crystals [2]. The volume of the alloyed layer is altered because there is misfit between the solute and solvent, but it remains constrained by the underlying purer material with which it remains coherent. The strain in the alloyed layer then provides a local increase in free energy. This energy is proportional to an appropriate modulus, the square of the atomic misfit, ‘I, and the square of the solute concentration. Under conditions of asymmetrical strain, the energy of the system can be reduced if the GB migrates through the more highly strained layer. The velocity of the boundary for DIGM driven by
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coherency strain asymmetry et nl. [12]
DIFFUSION
is given by Handwerker
where D, is the diffusivity across the boundary, Q is the mean atomic volume, 6 is the width of the GB, Y(n,) is an appropriate, orientation dependent, elastic modulus in grain i, v]is the atomic misfit of the solute in the solvent lattice, C,, is the concentration in the “leakage layer”, C, is the original concentration of the material, and C, is the concentration in the boundary, y is the GB energy and K, is the curvature of the surface of grain i abutting the GB. The orientation dependence of Y for cubic crystals has been derived by Cahn [13] in terms of elastic constants and the direction cosines between the boundary normal and the standard crystallographic axes. Due to the anisotropy of the elastic modulus, the coherency strains are different in the surfaces of the two adjoining grains. The strain energy gradient sets up a diffusive flux of the solute toward the less stressed solid. From equation (l), we are able to calculate the variation of the expected velocity as the GB plane changes. Facet formation can be predicted and the facet planes should be planes of low indices. The model is testable using ternary solid solution materials in which the lattice parameter is continuously variable by means of composition control: this provides the ability to control the value of the misfit parameter, ‘I. Yoon and co-workers [5] demonstrated the predicted dependence of migration velocity on the square of r~. Faceting consistent with the variation of Y with orientation was also observed. These results strongly support the coherency strain model. DIGM observations in a series of (lOO)(OOl) asymmetrical boundaries in the Cu-Zn system show that the direction of migration is always as predicted by the coherency strain model, but the migration distance remains proportional to the initial driving force only for misorientations below approximately 25” [14]. There are also significant departures from the regular behavior for C17, I35 and C29 GBs. Two notable deviations from the coherency strain predictions were that the maximum migration distances were obtained for the boundaries close to 211 (51.41”) in spite of the very low driving force at
Y = f(C,, + 2C,,)
3-
C,, + 2(2C,
this misorientation, and that small or zero migration distances were measured for boundaries close to X3 (the primary twin) in spite of large driving forces.
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These observations indicate that the GB structure does affect the process of DIGM, while the coherency strain energy model does not take any account of the GB structure. There are also some less explicit, but still troubling experimental results that indicate problems with the model. For example, the Au-Ag system, with extremely small misfits and small elastic modulus, appears to undergo DIGM just as rapidly as systems with larger misfits and higher modulus [ 151. A similar study has been carried out on a Ni-19.4%Pd alloy in which little coherent strain energy is expected because of the vanishingly small lattice parameter misfit. However, DIGM occurs in diffusion couples of this alloy electroplated with copper at temperatures ranging from 773 to 1073 K for 48 h [16]. These results may indicate large contributions from the free energy of mixing, or from GB structure effects. 2.1. Elastic anisotropy efsects For most crystalline solid solutions, the lattice parameters vary with composition. If the lattice is to remain coherent in the presence of a composition variation, work has to be performed in straining the lattice. In the case of DIGM, coherency strains are developed as a result of lattice diffusion adjacent to the GB. The coherent diffusion zones are assumed to have the form of thin slices parallel to the GB plane. Hilliard [17] gave a procedure for calculation of the coherency strain energy in a thin slice by considering a process to form the strained layer in two steps. First, consider a slice of solid maintaining coherence with the substrate crystal during diffusion. When solute atoms diffuse into the slice, forming a solid solution, a hydrostatic strain, t, builds up in order to maintain coherence with the substrate materials. The stresses required to produce the hydrostatic strain, t, and thus the elastic strain energy per unit volume, W(l), can be calculated. In the second step, the sides of the slice parallel to the plane normal are clamped and the stress in this direction is relaxed reversibly. The work performed in this step, W(2), depends on orientation of the crystal. The net work required in order to achieve coherency for a layer parallel to (hk/) is w = W(1) - W(2) = Yc2. where Y is an orientation dependent given for cubic crystals by
C,, + 2G2 - C,, + C,,)(h2k2
+ k212 + I%‘)
1
(2)
elastic modulus
(3)
and C, are the usual elastic constants. However, this approach is questionable for noncubic systems. First, the formation of a solid solution
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does not cause isotropic strain along all of the crystal axes. The rate of increase of the c-axis with increasing Cd content in Zn is remarkably large; approximately 2.6 times the rate of increase of the u-axis [18]. Clearly, one of the basic assumptions in the Hilliard calculation, the inducement of hydrostatic strain by alloying, is invalid for this alloy system. Secondly, shear strain in the coherent layer was assumed by Hilliard to be very small and thus ignored in the calculation. This may not be true for a highly anisotopic material. An alternative method to calculate coherency strain energy is more straightforward. We resolve all strains and stresses present in the coherent layer. which is possible if the elastic modulus, the initial and final lattice parameters and the orientation of the crystal are known. When solute atoms diffuse into the slice, strains build up in the slice depending on the crystal orientation and lattice misfit parameters in the a and c directions. From Pearson [18] we find the lattice misfit parameters in the c and a directions to be 2.6 x 10m3 and 1.01 x 10m3, respectively. For a [OOI] orientated crystal, the stress-free strain, t”,, in the plane of the coherent layer can be obtained from the lattice parameter change with composition as
while the strain
perpendicular
6, =
to the plane
will be
cm - CWLl)= dj, cm
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can be generated by two rotations, firstly by an angle 4 around the c-axis, and secondly by an angle 0 around the u-axis. Thus, the rotation matrix can be written as 1 uli= agu4
=
0
0 -sine [ 0 cod
0 costI sin0
1
[-+$?__nab;]
[ aii =
sincos 0 sin 4 4 -cosQ
sin4
(7)
- sinsin0 4cos 4
cos0 0
cos e cos 4
sin 0
1
1. (8)
The transformations of Q and S,, then proceed according to the usual rules for second- and fourthrank tensors, respectively. Since stresses and strains are concentrated in the very thin coherent layer, a state of plane stress exists and there are six knowns and six unknowns among the 12 strain and stress components. Using the conventional shorthand notation e3 = Q~= e6 = 0; c,, cl, t6 are known from the above analysis; c,, CT?, 06, u3, 04, o5 are unknown. The unknown stresses and strains may be found by solving Hooke’s law. However, further simplification is possible if we assume that there are no shear strains in the plane perpendicular to the boundary plane, and thus the fourth and fifth equations (i = 4,s) can be ignored. Finally, four equations remain
where u(X) and 0(X,), c(X) and c(&) are the lattice parameters of the matrix lattice and the coherent layer, respectively. The associated stresses can be found from Hooke’s law
When the crystal orientation changes, both t,, and S,,,, must be transformed to the new coordinate system. This can be done by introducing a rotation matrix, aq. Generally speaking, the rotation matrix
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61 =
$,a, + S1?UZ +
s,,o,
(9)
62 =
&,a, + &a,
+
s*,o,
(10)
63 =
S,,a,
+
S,,a,
(11)
s,,o,
(12)
+
S320?
tfj = &,a, + S6?U2 + from which are
uI ,
CT?, o6 and c3 can be determined.
They
(13)
(14)
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(15)
(16) The coherency
strain energy per unit volume is given
by E =&CT;
+ S& i- W,*~,fJ,
+ &CT; + S,,~,~,
+ &,%%)I.
(17)
Figure 1 shows the three-dimensional representative surface of variation of the coherency strain energy with boundary plane normal for Zn1.lwt%Cd, which shows the coherency strain energy in a thin layer can be represented by a surface. As we can see, the coherency strain energy in this alloy is cylindrically symmetrical about the c-axis and only a function of 6. It does not change with (p. Figure 2 shows a central section of a representive surface for coherency strain energies in the coherently strained layer in zinc for e&/c y1= 2.6(Zn-1.1 wt%Cd). The length of the radius vectors represents the coherency strain energy in the corresponding direction. As shown in Fig. 2, the coherency strain energy increases with increase of 8, and reaches a maximum when 0 = 90”. For a GB with known plane normal for both grains, the coherency strain energies on both sides of the boundary can be calculated, and the difference of those energies should be the driving force for DIGM.
3. EXPERIMENTAL
TECHNIQUES
Polycrystalline zinc was used in addition to three kinds of zinc bicrystals grown in a vacuum Bridgman furnace under conditions described in detail elsewhere [19]. The first bicrystal specimens were [OOl] tilt bicrystals. In order to make symmetric tilt bicrystals, 20 mm long seeds were cut from single crystals. The cylindrical single crystals were accurately oriented by the X-ray Laue technique, using a three-axis goniometer. The seed crystals were then cut into two halves along chosen planes, parallel to their axes and one of the halves was rotated 180” about the normal to the cut surface, to provide symmetrical seeds with respect to the chosen boundary plane. The specimens
Fig. 2. A central section through the strain energy surface for a coherently strained 1.1 wt% Cd layer on a zinc surface, using E’$/L~, = 2.6. The length of the radius vector in any orientation is proportional to the coherency strain energy for that surface normal and the axes are marked in units of MN/m’.
grown with this method were symmetric tilt bicrystals with their tilt axes parallel to [OOl]. The second kind of specimens were grown in such a way that the bicrystals have one grain with its [OOl] direction parallel to the GB plane, i.e. its crystallographic c-axis perpendicular to the GB plane normal, and the other has an angle between the boundary plane normal and its crystallographic c-axis. The seed pair comprised one as described above and the other cut from a randomly oriented single crystal. The third kind of specimen was grown from two randomly oriented seeds, forming bicrystals with general GBs. All seeds were chemically polished in a solution of 10% hydrochloric acid and ethyl alcohol to remove any damage introduced during cutting. The graphite mold used to grow these bicrystals was identical to that used by Chen and King [20]. The misorientations of the bicrystals were checked after growth by the Laue method. Slices about 3 mm thick, with their flat surface perpendicular to the tilt axis, were cut from the bicrystals using a low speed diamond saw. The specimens were polished on silicon carbide grinding paper (240-600 grit) first. At least 1 mm of the surface of the section was removed in these operations in order to remove the deep deformation twinning band resulting from the sawing operation which can be seen after etching. The specimens were then polished using c( alumina powder (5-0.05 pm). Finally, the specimens were electropolished in a solution of 15% nitric acid in ethanol giving a clean, stress-free surface. Prepared specimens were encapsulated under vacuum (10m6 T) along with about 12 g of fine filings of a Cd source alloy, which were made by vacuum casting high purity zinc and cadmium to produce Zn-30 wt% Cd and Zn-10 wt% Cd alloys. The specimens then were annealed for the chosen time and temperature, depending on the purpose of the experiments. 4. RESULTS
4.1. General phenomenology Fig. 1. Three-dimensional representation of the variation of coherency strain energy with surface orientation, for t&/t:, = 2.6.
DIGM specimens
was found to occur for polycrystalline exposed to cadmium vapor in the tempera-
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DIFFUSION
ture regime from 160 to 260°C. The extent of DIGM varies in different conditions. At lower temperatures, boundary migration only occurs in a few boundaries, which have short migration distances. Figure 3 is an optical micrograph of a polycrystalline zinc specimen after annealing at 250°C for 4 days. The specimen is as-annealed, without further polishing. In this condition, the original and final positions of GBs can be observed. The initial positions of the boundaries were revealed by traces of polishing edges caused by differential polishing of the adjacent grains, while the final positions of the GBs were decorated by thermal grooves which appear as black lines in optical micrographs. Grain boundary migration is not observed in similar anneals without Cd sources, indicating that the observations truly correspond to DIGM and are not related to strains or other effects of anisotropic thermal expansion. Shin and King have shown that grain boundaries in zinc readily accommodate strains deriving from anisotropic expansion, even at very low temperatures [21]. The rate of DIGM was not uniform from boundary to boundary. Even for single boundaries, some segments might migrate by DIGM while others showed no migration. The boundaries usually migrated in a way that increased their areas, independently of the local boundary curvature, as for DIGM in other metal and ceramic systems. There were also some very thin lines, so-called “ghost lines” in the migrated regions, which may have resulted from the discontinuous nature of GB migration. Some
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boundaries migrated unevenly, resulting in a wavy boundary morphology. GB migration reversal after long anneals was frequently observed. Surface pits were also sometimes observed. The Cd content in several DIGM regions was measured by wavelength dispersive spectrometry (WDS) in a specimen after a diffusion anneal at 18O’C for 4 days. Figure 4(a) shows the variation of Cd content across the region where DIGM had taken place, while Fig. 4(b) shows the image of the migrated boundary where the spectra were taken. A compositional discontinuity was observed at the initial boundary position, where a relatively high Cd content was found. Although the Cd concentration was observed to vary with position in the alloyed zone, it tended to decrease as DIGM proceeded, which is consistent with observations by Cahn et al. in the Au-Cu system [ 151. The background level of Cd detected outside the migrated zone may be due to the presence of Cd atoms which diffused into the surface of zinc by lattice diffusion during the DIGM anneal. The measured composition profiles again confirm that we have observed DIGM, and not any other form of GB migration. 4.2. DIGM studies in bicrystal specimens Our primary concern in this study was the extent and direction of GB migration induced by Cd diffusion. We have used two measures of the extent of migration, as illustrated in Fig. 5. The first is the fraction of the total length of the boundary which
Fig. 3. Optical micrograph showing DIGM in polycrystalline zinc after annealing at 250°C for 4 days.
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0.0
I
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B ,
I
b
I
I
20 10 Distance (pm)
Fig. 4. (a) Composition profile across an alloyed zone formed by DIGM after a diffusion anneal at 180°C for 4 days; (b) micrograph showing the region from which the profile was obtained.
took part in migration, known as the “migrating fraction”. The second measure of DIGM is the average displacement of the boundary within the segments that actually moved: this is called the “migration distance”. Both measures are made on the surfaces of specimens after diffusion anneals, without any further polishing or etching. A typical observation of DIGM in a bicrystal specimen is shown in Fig. 6. Figure 7 shows the average migration versus DIGM time for a [110]/41.9” bicrystal at different temperatures. The displacement of migrating seg-
ments is reduced at lower diffusion temperatures. For example, the longest migration distance is only 3 pm at 150°C after 8 days of diffusion treatment. The displacement of the migrating segments increased with annealing time and the highest velocity occurred at the early stage of DIGM, gradually leveling off after 5 days of annealing. The higher temperatures also give higher initial migration rates, consistent with a thermally activated process. 4.2.1. [OOI] Rotation axes. DIGM experiments were attempted using 17 different symmetric [OOl] tilt bicrystals and eight different asymmetric tilt
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L
c
Final boundary position
\
Ongmal boundary position
Fig. 5. Schematic illustration of the measures of DIGM used in this work. The “migrating fraction” of the boundary is (ZI,)/L and the “average migration distance” is W,Y(W Annealing Time (days)
specimens (see Tables 1 and 2). It is noteworthy that both “general” and “coincidence” boundaries were included in the study. The range of misorientation angles is from 6” to 57”. Figure S(a) is an optical micrograph of a 42”/[001] symmetric tilt GB after annealing at 230°C for 4 days with a 30 wt% Cd source. No GB migration is detectable and the result is typical of the entire series of [OOl] tilt CBS, which were apparently immobile, or resistant to nucleation Table 1. Details of the symmetric tilt bicrystals used for the experiments Misorientation (degrees) 6 8 11.5 14 18.5 24.5 36.5 41.5 51 20 21 33 33 17.5 9.2 15 15.5
Fig. 6. Typical
Boundary type
General boundaries
Near Near Near Near Near Near Near Near
case of DIGM
X7 27 El3 213 231 231 243 X43
Fig. 7. Average migration distance vs time for a [I 10]/41.9” bicrystal, at different temperatures. Table 2. Details of the asymmetric tilt bicrystals used for the experiments
8,
02
(degrees)
(degrees)
5 5 18.5 9 11.2 20
14 14 21.5 14 17.2 30
Misorientation (degrees) 19 19 40 23 28.4 50
Boundary type General boundaries Near 27 Near 27 Near 213 Near 237
under the experimental conditions. Identical results were obtained from [OOl] asymmetric tilt boundaries at the same DIGM annealing conditions, as shown in Fig. 8(b). These results are consistent with the prediction of the coherency strain model which suggests that for [OOl] tilt boundaries between hexagonal structure crystals, the elastic isotropy of the basal plane results in the absence of driving force (coherency strain energy) and thus no boundary migration is expected. This result in the Zn-Cd system makes a very interesting and instructive comparison with observations of DIGM in symmetric tilt boundaries in the case of Cu-Zn. Chen and King have shown that DIGM occurs extensively in many of
in a bicrystal specimen. Note the traces of the basal planes, boundary plane in this optical micrograph.
parallel
to the
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Fig. 8. Optical micrographs of 8” (a) and 42” (b) symmetric tilt [OOl] grain boundaries after diffusion anneals at 230°C for 4 days. No DIGM is detectable.
these boundaries, in spite of the apparent lack of a coherency strain driving force, which is negated by the boundary symmetry [20]. In the case of the cubic Cu-Zn alloy, however, any deviation of the boundary plane from its symmetric orientation creates a difference of coherency strain, so local fluctuations of the boundary position can allow the coherency strain mechanism to operate. In the case of the hexagonal crystals, however, changes of boundary plane do not result in differences of coherency strain energy, because of the cylindrical symmetry of the elastic constants. Local fluctuations of the boundary plane do not, therefore, set up coherency strain driving forces, and the boundaries are immune to DIGM deriving from this mechanism. Although no migration was observed for coincidence boundaries in this series of experiments, they did exhibit one feature not observed in the general boundaries. Features with faceted edges on both sides emerge along these boundaries. As the anneal time increases, these features nucleate at other positions along the boundary, and eventually grow together. Figure 9 shows scanning electron microscope @EM) images of a C7[001] symmetric tilt boundary after 4 and 8 days DIGM anneal at 230°C respectively. Detailed analysis and cross-section view of such regions indicate that they were not due to boundary migration, but are surface pits (Fig. 10). A similar phenomenon has been reported in
annealed aluminum 1221, and has been attributed to vacancy diffusion along the GB toward the free surface. 4.2.2. Other rotation axes. The theory predicts that differences in coherency strain energy arise if the orientation relation between the two grains differs from [OOl] tilt. Significant DIGM is then expected for such boundaries. The theoretical result illustrated in Fig. 11 also indicates that for the hexagonal close packed (h.c.p.) crystal structure, the coherency strain energy in each grain depends only on the angle between the crystallographic c-axis and the boundary normal. Thus, a systematic study of GB behaviors was performed by using a series of bicrystals with fixed orientation of grain 1 and varying orientation of grain 2. The occurrence of DIGM after Cd diffusion can be identified by optical microscopy in these specimens. Figure 12 shows the DIGM morphology of a 77.6” [210] bicrystal after different DIGM treatments. The diffuse dark line shows the position of the original boundary, while the final boundary is the thin line on the upper side of the original boundary. The general features of the GB after the different DIGM processes are similar, except that faceting occurred after the 260°C 10 day DIGM anneal. The specimen surface of the upper grain in the pictures has the basal plane perpendicular to the GB plane. In all three cases, the GB segments moved into the upper grain.
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Fig. 9.
Set
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BOUNDARY
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:ondary electron SEM images showing the faceted pits on a X7 coincidence bound >ary atfter a diffusion anneal at 230°C for 4 days (a) and 8 days (b).
In other words, the grain with [OOl] perpendicular to the GB plane normal shrank and the other grew. This is in agreement with the coherency strain model, which predicts that the boundaries will migrate into the grain with the higher modulus. A cross-sectional view of the DIGM sample (Fig. 12) confirms the above observation. The migration distance decreases with increasing penetrating depth and the boundary exhibits a convex curvature toward the grain into which the boundary was migrating. A series of experiments was performed using bicrystals with one grain, named grain 1, always having its [OOl] axis lying in the GB plane (0, = 90@) and normal to the bicrystal surface. The other crystal, grain 2, had varying orientations (0, varies) and thus has varying strain energy in the coherent layer. Figure 13 shows the average migration distance for migrating segments as a function of &, the angle
between the boundary normal and the c-axis of grain 2, after various annealing times at 230°C. The theoretical coherency strain energy difference between grains 1 and 2, corresponding to the driving force, is superimposed on the graph. The migration distance data generally match the theoretical driving force. For bicrystals having large OX, and thus only small driving force, migration in the direction opposite to the driving force was also observed. For bicrystals with smaller Q2 ( < 60”), the migration distance kept increasing as annealing time increased, while for the bicrystals with larger f3?, the migration tends to cease after rapid motion in the early stage. Figure 14 shows the percentage of the original boundary length that underwent migration, as a function of &. The calculated curve of coherency strain energy difference is also superimposed. It is noticed that for specimens with smaller 8, ( < 55”)
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Fig. 10. Optical micrograph taken in cross section, showing a pit formed at the intersection of a 27 boundary with the surface. The specimen was annealed for 8 days at 230°C.
large portions of the boundaries migrated in the early stages of DIGM, and no significant increase in the portion occurred even after long annealing times. For bicrystals with larger 0, (> 60”), however, DIGM started from relatively small portions of the boundaries at the early stage of DIGM, and there was a marked increase in the migrating portion as DIGM continued. Generally speaking, non-uniform nucleation of DIGM was found in the Zn-Cd system. Those segments undergoing initial-stage DIGM extended laterally and migrated forward after long period anneals. It can be seen in Fig. 15, for example, that “ghost lines” show intervals of migration. Migration direction reversal was observed, as shown in Fig. 16(a). Occasionally, twinning
occurred as a result of A second series of pleted, using the same faces cut perpendicular instead of parallel to it differ in no significant here.
DIGM, as seen in Fig. 16(b). experiments has been combicrysals but with their surto the c-axis of crystal 1 [19]. The results of this series way from the series reported
5. DISCUSSION
We have identified DIGM in both polycrystalline and bicrystal zinc, and its characteristics are similar to those in other metal and ceramic systems. Only in a few cases, when the coherency strain energy difference
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was small or absent, did DIGM not follow coherency strain model predictions. 5.1. Driving forces It is instructive to consider the balance of driving forces at the point of nucleation of DIGM. Assuming that the original GB is planar, GB energy is constant and bulging may occur in either side of the boundary in the form of a semicylinder with radius r, as shown in Fig. 17, then the total energy change associated with the DIGM initiation will have the following three major contributions. (1) The free energy of mixing, AG,, arising from the formation of the alloyed (or dealloyed) regions.
W Fig. 12. Cross-sectional optical micrograph of a 77.6”/[210] asymmetric tilt boundary after annealing at 230°C for 4 days. The line diagram illuminates the initial and final boundary positions; the final position being the darker line in this schematic.
J Inclination of Boundary Plane from Basal Plane of Grain 2, in degrees Fig. 11. Plan view optical micrographs of a 77.6/[210] asymmetric tilt boundary after various annealing treatments: (a) 260°C (10 days); (b) 230°C (4 days); and (c) 200°C (8 days).
Fig. 13. Migration distance vs inclination of grain 2 (0,) for a series of bicrystal specimens annealed at 230°C for different annealing times. The heavy continuous curve is the functional form of the driving force derived from coherency strain.
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E”100 , 3 ‘2
80
&? h 60 2 -u g 40 m” M 20 .g E .9 I 0
2
20
40
60
80
Inclination of Boundary Plane from Basal Plane of Grain 2, in degrees
Fig. 14. Migrating boundary fraction vs inclination of grain 2 (0,) for a series of bicrystal specimens annealed at 230°C for different annealing times. The heavy continuous curve is the functional form of the driving force derived from coherency strain.
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shown in the lower case of Fig. 17(a)]. In the second case, the bulge may have the strain corresponding to the locally growing grain, as shown in Fig. 17(b). AG, is then negative for bulges formed in the “forward” direction defined by the coherency strain model, and positive in the “backward” direction. Under the latter set of circumstances, the difference between “forward” and “backward” nucleation is twice as large as in the former case, but the absolute effect on the forward nucleation rate will be smaller. (3) Assuming the boundary free energy is isotropic and constant, the difference of GB area for original and final boundary will cause an energy increase of AA y, where y is the free energy per unit area of GB. Hence, the total energy change is given, in general, as AG = AG, + AG, + AA,,.
(2) The coherency strain energy, AG,, which arises from the difference of coherency strain adjacent to the GBs, as solute diffuses into the adjacent grains. At least two different situations can be envisaged with respect to this component, as illustrated very schematically in Fig. 17. In the first case, a strain-free (or lower-strain) bulge nucleates in the coherently strained region, and AG, is negative. The magnitude of AG, depends on the square of the strain that is relieved, so it is greater for a bulge forming in the more highly strained grain [as shown in the upper case of Fig. 17(a)] than in the less-strained grain [as
(18)
Now we assume that a critical-sized
bulge is of the same order of magnitude as the thickness of the coherently strained layer, and we can write AG = Y(Ag,,,) + I’(Ag,) + AAy
$
[(Ag, + Ag,) + (rr - 2)Zry,
(19)
where V, 1 and r are the volume, length and radius of the bulge, respectively; and Ag,,, and Ag, are the
Fig. 15. The same area after two successive diffusion anneals, showing the lateral and forward of the alloyed region, and the appearance of “ghost lines” marking the intermediate boundary
extension position.
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Fig. 16. Optical micrographs showing (a) reversal of the GB migration direction (the initial migration direction was into the upper grain in this image), and (b) the training of twins by a migrating boundary.
changes in free energy of mixing per unit volume and coherency strain energy per unit volume. The critical radius of an embryonic bulge is then found to be r*=
(n-37
Wgm+W’ The bulge may grow only when its radius is larger than r*, and this defines the point at which an embryonic bulge can be considered as a nucleus for DIGM. Clearly, a large strain energy density, Ag,, favors nucleation. Cahn and Balluffi [4] have argued that the nucleation of DIGM by the formation of small bulges cannot be achieved by the free energy of mixing alone because the increase of free energy associated with the increased GB area is much larger than the reduction achieved through alloying when the bulges are of the order of 0.1 pm in diameter. Data are not available
for the Zn-Cd system, however, using data from the Cu-Zn system, which is very commonly investigated, the GB energy is 0.625 J/m2 [23] and the free energy of mixing has been taken as 78 MJ/m3 at 773 K [24]. We find that the energy changes for both free energy of mixing and increase of GB area are of the same order when r = 0.1 pm, which means once bulges have been formed at this size, the free energy of mixing has to be taken into account. On the other hand, from the nucleation point of view, if we ignore the coherency strain term in equation (20), the critical radius is found to be 2.9 nm, which is still too large to be reached by random thermal fluctuations in the position of GB at common DIGM temperatures. Our expermental results in the Zn-Cu system show that there is no DIGM in [OOl] tilt boundaries no matter whether the boundaries are of symmetric or asymmetric type, though high Cd concentration was
2996
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(low strain)
(a> Fig. 17. Schematic illustration nucleation by the formation of embryonic bulges. In case (a) the bulges are formed strain-free, while in case (b) they embody the strain of the underlying, or growing, crystal.
built up in the GB regions [19]. Meanwhile, DIGM was identified in polycrystalline specimens and bicrystals with large angle GBs where coherency strain energy differences were present. These results suggest that the reduction of free energy caused by alloying is not a sufficient condition for causing motion of the boundary and the coherency strain energy is thus a critical component of the driving force for the initiation of DIGM. The coherency strain energy increases or decreases the effective driving force, depending on its sign in equation (20). If the bulge moves toward a more highly strained layer, the effective driving force is increased. However, if the bulge moves toward the lower strained layer a reduction of the effective driving force of the system is expected, in cases where coherency loss does not occur. Then, r * (forward) is smaller than r* (backward), meaning that the “forward” process is
more highly favored than the “backward” one because the higher coherency strain energy reduces the size of the critical nuclei and increases the probability of boundary migration into the predicted directions. Other mechanisms can also complicate the nucleation process. As discussed above, coherency loss leading to strain-free bulges can increase the nucleation rate and also reduce the directional selectivity of the coherency strain. Figure 18 shows a random large angle GB with a misorientation yielding almost the same coherency strain energies on both sides. The misorientation of this boundary was 39.4” [?83] and the angles, 0, and &, between the c-axes and the boundary normal for the two grains were 72.8 and 73.4”, respectively. The boundary still migrated and a dislocation wall was left at the original position of the boundary, which implies that dislocation
Fig. 18. Transmission electron micrograph showing a moving boundary with very small coherency strain energy difference, after a 24 h diffusion anneal at 200°C. A dislocation array is visible at the original position of the boundary indicating a coherency loss mechanism which is probably responsible for the migration.
LIANG
and KING:
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mechanisms may play a role by allowing coherency loss on one side leading to asymmetry of the strain energy. 5.2. Inhomogeneity of DIGM In some of the previous DIGM studies, especially in polycrystalline specimens, it has often been reported that boundary migration is not uniform [I]. Even for a single boundary, some segments migrate and some do not. It can be argued that non-uniform migration is caused by different curvature of different parts of the boundary or small orientation differences of the GB normal along the boundary, according to the coherency strain model. On the other hand, if a boundary is perfectly planar and the GB structure is homogenous everywhere along the boundary, the driving force is expected to be the same along the entire boundary. DIGM should then be initiated uniformly along the boundary, which has been the case for DIGM experiments in symmetric and asymmetric bicrystals of Cu-Zn [6]. However, this is not the case in Zn-Cd, as shown in Fig. 14. Boundaries migrated in certain places and the percentage of migrating segments increased with increasing driving force. Migration progressed by the migrating segments moving further and extending laterally after long treatment times. This difference between the Cu-Zn and Zn-Cd systems may arise from several causes. Firstly, the microscopic regularity of these synthetic boundaries is questionable although there is no clear reason why a copper bicrystal should be more regular or homogeneous than a zinc one. Secondly, the solubility is about 30 wt% for the Cu-Zn system compared with 1.5 wt% for ZnCd at the DIGM temperature, so it is easier to build up a relatively high solute content in the Cu-Zn case. This may minimize any effects caused by irregularity of GB structure. Thirdly, in cubic systems, there is no misfit anisotropy while the hexagonal ZnCd system has a large misfit anisotropy between the a- and c-axes. This allows for small structural irregularities to generate non-uniform driving force along the boundaries. 5.3. Grain boundary d#iision mechanism The GB self-diffusivity of zinc at the DIGM temperature is between 1.1 x lo-l3 and 1.9 x lo-l3 m’/s, while the diffusivity of Cd is 1.9 x lo-‘* m*/s [25]. The diffusivity of Cd is thus one order of magnitude larger than that of Zn. These unequal fluxes of solvent and solute should, therefore, entail a balancing point defect flux if the diffusion occurs by a defect mechanism. The surface pits observed at [OOl] tilt, nearcoincidence boundaries probably result from the vacancy flux induced by such a GB Kirkendall effect, although it is also possible that they are formed as a result of an interstitialcy mechanism of Cd diffusion. Computer simulation experiments [lo] indicate the likelihood of a vacancy mechanism of diffusion in coincidence boundaries. General (or non-
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BOUNDARY
2991
coincidence) boundaries were not observed to form these pits in our experiments and we surmise that diffusion in these interfaces may proceed by a mechanism that differs in some detail from that in the coincidence boundaries, that does not result in a vacancy flux to the specimen surface. The difference between the two cases may be as drastic as a completely different mechanism of atomic motion, not involving point defects in the general boundaries, or it may derive from the poorer ability of coincidence boundaries to absorb vacancies, which must then be conducted to the surface. Aperiodic boundaries are unfortunately not amenable to study by computer simulation to clarify this issue. We believe that our observation is the first direct indication of a difference of diffusion mechanism between coincidence and general boundaries. 5.4. Some comments about DIGM tems
in hexagonal sys-
The present study is based on the elastic properties of the hexagonal system, which presents special advantages to the experimenter. An elastic isotropy condition obtained within the basal plane and many elastic properties depend only on the angle between an arbitrary reference direction and the crystallographic c-axis. These properties allow us to determine the coherency strains for any GB plane and, potentially, to vary the misorientation while maintaining a constant coherency strain. On the other hand, there are disadvantages to the use of hexagonal systems for DIGM studies because of the low solubilities in hexagonal solvents. In most cases, the hexagonal phase remains stable only up to one or one-and-a-half weight percent of solute. This means less free energy of mixing is available as the driving force for DIGM. From a technical point of view, quantitative compositional analysis of the alloyed zone is more difficult because of the very small solute concentrations involved.
6. CONCLUSIONS
DIGM is found to occur in the ZnCd system, with phenomenology similar to that observed in cubic metal systems, although there is less homogeneity of migration in the hexagonal bicrystal specimens. The progress of GB migration appears to scale with the driving force derived from coherency strain energy. There is evidence in support of a point defect mechanism of GB diffusion in coincidence boundaries, but not more general cases. Acknowledgements-This work was supported by the U.S. National Science Foundation, grant number DMR9404589. Professor R. W. Balluffi provided insightful comments upon the question of the GB diffusion mechanism.
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REFERENCES 1. A. H. King, Znt. Mater. Rev. 32, 173 (1987). 2. M. Hillert, Scrinta metall. 17, 237 (1983). 3. C. Y. Ma, W. Gust, R. A. Fournklle and B. Predel, Mater. Sci. Forum 126-128, 317 (1993). 4. J. W. Cahn and R. W. Balluffi, ‘Acta’metall. 29, 493 (1981). 5. W. H. Rhee, Y. D. Song and D. N. Yoon, Acta metall. 35, 57 (1987). 6. F.-S. Chen, G. Dixit, A. J. Aldykiewicz and A. H. King, Metall. Trans. A 21, 2363 (1990). 7. F.-S. Chen and A. H. King, Acta metall. 36,2827 (1988). of Grain and 8. I. Kaur and W. Gust, Fundamentals Interphase Boundary Diffusion. Ziegler Press, Stuttgart (1989). 9. V. Y. Aristov, V. L. Mirochnik and L. S. Shvindlennan, Fiz. Tverd. Tela. 18, 137 (1976). 10. R. W. Balluffi, T. Kwok, P. D. Bristowe, A. Brokman, P. S. Ho and S. Yip, Scripta metall. 15, 951 (1981). 11. D. A. Smith, C. M. F. Rae and C. R. M. Grovenor, in Grain Boundary Structure and Kinetics (edited by R. W. Balluffi). ASM, Materials Park, OH (1980). 12. C. A. Handwerker, J. W. Cahn, D. N. Yoon and J. E. Blendell, in Atomic Transport in Alloys: Recent Develop-
13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.
25.
GRAIN
BOUNDARY
ments (edited by G. E. Murch and M. A. Dayananda). TMS-AIME, Warrendale, PA (1985). _ J. W. Cahn. Acta metall. 7. 18 (1959). A. H. King and G. Dixit, d. Physique COB. Cl 51, 545 (1990). J. W. Cahn, J. D. Pan and R. W. Balluffi, Scripta metall. 13, 503 (1979). C. Y. Ma, W. Laub, R. A. Fournelle, W. Gust and B. Predel, Z. MetaUk. 83, 633 (1992). J. E. Hilliard, Znd. Engng Chem. 58, 19 (1966). W. B. Pearson, Handbook of Lattice Spacings and Structures of Metals. Pergamon, New York (1964). L. Liang, Ph.D. dissertation, State University of New York at Stony Brook (1994). F.-S. Chen and A. H. King, Acta metall. 36, 2827 (1988). K. Shin and A. H. King, Phil. Mug. A 63, 1023 (1991). G. A. Chadwick, Phil. Mug. 14, 1295 (1966). L. E. Murr, Interfacial Phenomena in Metals and Alloys. Addison-Wesley, Reading, MA (1975). R. Hultgren, P.-D. Desai;D. T. Hawkins, M. Gleiser and K. K. Kellv, Selected Values of the Thermodynamic Properties of Binary Alloys. ASM, Metals Park, OH (1973). N. L. Peterson, Znt. MetaN. Rev. 28, 65 (1983).
Pergamon
Printed in Great Britain. All rights reserved 1359-6454/96 $15.00 + 0.00
DIFFUSION
INDUCED GRAIN BOUNDARY MIGRATION IN THE ZINC-CADMIUM SYSTEM L. LIANG and A. H. KING
Department
of Materials
Science and Engineering, State University NY 11794-2275, U.S.A.
of New York
at Stony
Brook,
(Received 16 March 199.5; in revised form 27 July 1995) Abstract-We present a series of experiments on the motion of grain boundaries in zinc during inward cadmium diffusion. Diffusion induced grain boundary migration is shown to occur in a hexagonal metal, for the first time. Using the cylindrical symmetry of the elastic constants in hexagonal crystals, we are able to investigate the applicability of the coherency-strain model and the influence of grain boundary structure, independently. Copyright 0 1996 Acta Metallurgica Inc.
1. INTRODUCTION
Diffusion induced grain boundary migration (DIGM) is characterized by the sideways motion of grain boundaries (GBs) in response to solute diffusion along them and it has a fairly complex phenomenology. It has been studied in a wide variety of metal alloy systems and a few ceramics, and the observations are very similar even in widely differing alloy systems [l]. However, almost all observations to date have been made in cubic crystals. Understanding a complex phenomenon such as DIGM is difficult. Many components of the phenomenon are not thoroughly understood individually, and their interactions are, therefore, doubly difficult to unravel. Much attention has been paid to the driving force for DIGM, with attention focussing primarily upon two contributions: the coherency strain energy [2] and the free energy of mixing which is made available by the formation of an alloyed (or de-alloyed) region behind the moving GB [3]. Early work suggested that the free energy of mixing was insufficient to explain some of the observed features of DIGM, including migration against very small radii of curvature [4]. Systematic studies carried out as a function of misfit strain, via alloying [5], and elastic modulus effects, via elastic anistropy [6], have tended to confirm that the coherency strain effect has an important influence upon DIGM. Grain boundary structure also exerts some influence upon the progress of DIGM, as illustrated in a detailed study as a function of misorientation using symmetrical tilt GBs [7]. It is not always clear how such an effect arises: it may relate to the variation of GB diffusivity with misorientation [8], or the variation of GB mobility with misorientation [9]. The
mechanism of GB diffusion may also vary with misorientation, although it is thought to occur by a vacancy mechanism at least in well-ordered GBs [lo]. Finally, the mechanism of GB migration varies with GB parameters and may occur by shear (in some cases, such as twins), uncorrelated atomic jumps or correlated atomic jumps [l 11.Any linkage between the mechanisms of GB diffusion and migration is obviously affected if either of the mechanisms changes. In order to distinguish the effects of GB structure from non-structure-related effects, it is desirable to perform experiments in which the driving force can be held constant, while the structure varies. Hexagonal crystals provide an opportunity for such experiments and we report a study of DIGM in the zinc-cadmium system in this paper.
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2. COHERENCY
STRAIN
THEORY
The coherency strain model, which was originally proposed for liquid film migration, suggests that the high solute concentration established in the GB leaks into the adjacent crystals [2]. The volume of the alloyed layer is altered because there is misfit between the solute and solvent, but it remains constrained by the underlying purer material with which it remains coherent. The strain in the alloyed layer then provides a local increase in free energy. This energy is proportional to an appropriate modulus, the square of the atomic misfit, ‘I, and the square of the solute concentration. Under conditions of asymmetrical strain, the energy of the system can be reduced if the GB migrates through the more highly strained layer. The velocity of the boundary for DIGM driven by
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coherency strain asymmetry et nl. [12]
DIFFUSION
is given by Handwerker
where D, is the diffusivity across the boundary, Q is the mean atomic volume, 6 is the width of the GB, Y(n,) is an appropriate, orientation dependent, elastic modulus in grain i, v]is the atomic misfit of the solute in the solvent lattice, C,, is the concentration in the “leakage layer”, C, is the original concentration of the material, and C, is the concentration in the boundary, y is the GB energy and K, is the curvature of the surface of grain i abutting the GB. The orientation dependence of Y for cubic crystals has been derived by Cahn [13] in terms of elastic constants and the direction cosines between the boundary normal and the standard crystallographic axes. Due to the anisotropy of the elastic modulus, the coherency strains are different in the surfaces of the two adjoining grains. The strain energy gradient sets up a diffusive flux of the solute toward the less stressed solid. From equation (l), we are able to calculate the variation of the expected velocity as the GB plane changes. Facet formation can be predicted and the facet planes should be planes of low indices. The model is testable using ternary solid solution materials in which the lattice parameter is continuously variable by means of composition control: this provides the ability to control the value of the misfit parameter, ‘I. Yoon and co-workers [5] demonstrated the predicted dependence of migration velocity on the square of r~. Faceting consistent with the variation of Y with orientation was also observed. These results strongly support the coherency strain model. DIGM observations in a series of (lOO)(OOl) asymmetrical boundaries in the Cu-Zn system show that the direction of migration is always as predicted by the coherency strain model, but the migration distance remains proportional to the initial driving force only for misorientations below approximately 25” [14]. There are also significant departures from the regular behavior for C17, I35 and C29 GBs. Two notable deviations from the coherency strain predictions were that the maximum migration distances were obtained for the boundaries close to 211 (51.41”) in spite of the very low driving force at
Y = f(C,, + 2C,,)
3-
C,, + 2(2C,
this misorientation, and that small or zero migration distances were measured for boundaries close to X3 (the primary twin) in spite of large driving forces.
INDUCED
GRAIN BOUNDARY
These observations indicate that the GB structure does affect the process of DIGM, while the coherency strain energy model does not take any account of the GB structure. There are also some less explicit, but still troubling experimental results that indicate problems with the model. For example, the Au-Ag system, with extremely small misfits and small elastic modulus, appears to undergo DIGM just as rapidly as systems with larger misfits and higher modulus [ 151. A similar study has been carried out on a Ni-19.4%Pd alloy in which little coherent strain energy is expected because of the vanishingly small lattice parameter misfit. However, DIGM occurs in diffusion couples of this alloy electroplated with copper at temperatures ranging from 773 to 1073 K for 48 h [16]. These results may indicate large contributions from the free energy of mixing, or from GB structure effects. 2.1. Elastic anisotropy efsects For most crystalline solid solutions, the lattice parameters vary with composition. If the lattice is to remain coherent in the presence of a composition variation, work has to be performed in straining the lattice. In the case of DIGM, coherency strains are developed as a result of lattice diffusion adjacent to the GB. The coherent diffusion zones are assumed to have the form of thin slices parallel to the GB plane. Hilliard [17] gave a procedure for calculation of the coherency strain energy in a thin slice by considering a process to form the strained layer in two steps. First, consider a slice of solid maintaining coherence with the substrate crystal during diffusion. When solute atoms diffuse into the slice, forming a solid solution, a hydrostatic strain, t, builds up in order to maintain coherence with the substrate materials. The stresses required to produce the hydrostatic strain, t, and thus the elastic strain energy per unit volume, W(l), can be calculated. In the second step, the sides of the slice parallel to the plane normal are clamped and the stress in this direction is relaxed reversibly. The work performed in this step, W(2), depends on orientation of the crystal. The net work required in order to achieve coherency for a layer parallel to (hk/) is w = W(1) - W(2) = Yc2. where Y is an orientation dependent given for cubic crystals by
C,, + 2G2 - C,, + C,,)(h2k2
+ k212 + I%‘)
1
(2)
elastic modulus
(3)
and C, are the usual elastic constants. However, this approach is questionable for noncubic systems. First, the formation of a solid solution
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does not cause isotropic strain along all of the crystal axes. The rate of increase of the c-axis with increasing Cd content in Zn is remarkably large; approximately 2.6 times the rate of increase of the u-axis [18]. Clearly, one of the basic assumptions in the Hilliard calculation, the inducement of hydrostatic strain by alloying, is invalid for this alloy system. Secondly, shear strain in the coherent layer was assumed by Hilliard to be very small and thus ignored in the calculation. This may not be true for a highly anisotopic material. An alternative method to calculate coherency strain energy is more straightforward. We resolve all strains and stresses present in the coherent layer. which is possible if the elastic modulus, the initial and final lattice parameters and the orientation of the crystal are known. When solute atoms diffuse into the slice, strains build up in the slice depending on the crystal orientation and lattice misfit parameters in the a and c directions. From Pearson [18] we find the lattice misfit parameters in the c and a directions to be 2.6 x 10m3 and 1.01 x 10m3, respectively. For a [OOI] orientated crystal, the stress-free strain, t”,, in the plane of the coherent layer can be obtained from the lattice parameter change with composition as
while the strain
perpendicular
6, =
to the plane
will be
cm - CWLl)= dj, cm
INDUCED
can be generated by two rotations, firstly by an angle 4 around the c-axis, and secondly by an angle 0 around the u-axis. Thus, the rotation matrix can be written as 1 uli= agu4
=
0
0 -sine [ 0 cod
0 costI sin0
1
[-+$?__nab;]
[ aii =
sincos 0 sin 4 4 -cosQ
sin4
(7)
- sinsin0 4cos 4
cos0 0
cos e cos 4
sin 0
1
1. (8)
The transformations of Q and S,, then proceed according to the usual rules for second- and fourthrank tensors, respectively. Since stresses and strains are concentrated in the very thin coherent layer, a state of plane stress exists and there are six knowns and six unknowns among the 12 strain and stress components. Using the conventional shorthand notation e3 = Q~= e6 = 0; c,, cl, t6 are known from the above analysis; c,, CT?, 06, u3, 04, o5 are unknown. The unknown stresses and strains may be found by solving Hooke’s law. However, further simplification is possible if we assume that there are no shear strains in the plane perpendicular to the boundary plane, and thus the fourth and fifth equations (i = 4,s) can be ignored. Finally, four equations remain
where u(X) and 0(X,), c(X) and c(&) are the lattice parameters of the matrix lattice and the coherent layer, respectively. The associated stresses can be found from Hooke’s law
When the crystal orientation changes, both t,, and S,,,, must be transformed to the new coordinate system. This can be done by introducing a rotation matrix, aq. Generally speaking, the rotation matrix
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GRAIN BOUNDARY
61 =
$,a, + S1?UZ +
s,,o,
(9)
62 =
&,a, + &a,
+
s*,o,
(10)
63 =
S,,a,
+
S,,a,
(11)
s,,o,
(12)
+
S320?
tfj = &,a, + S6?U2 + from which are
uI ,
CT?, o6 and c3 can be determined.
They
(13)
(14)
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(15)
(16) The coherency
strain energy per unit volume is given
by E =&CT;
+ S& i- W,*~,fJ,
+ &CT; + S,,~,~,
+ &,%%)I.
(17)
Figure 1 shows the three-dimensional representative surface of variation of the coherency strain energy with boundary plane normal for Zn1.lwt%Cd, which shows the coherency strain energy in a thin layer can be represented by a surface. As we can see, the coherency strain energy in this alloy is cylindrically symmetrical about the c-axis and only a function of 6. It does not change with (p. Figure 2 shows a central section of a representive surface for coherency strain energies in the coherently strained layer in zinc for e&/c y1= 2.6(Zn-1.1 wt%Cd). The length of the radius vectors represents the coherency strain energy in the corresponding direction. As shown in Fig. 2, the coherency strain energy increases with increase of 8, and reaches a maximum when 0 = 90”. For a GB with known plane normal for both grains, the coherency strain energies on both sides of the boundary can be calculated, and the difference of those energies should be the driving force for DIGM.
3. EXPERIMENTAL
TECHNIQUES
Polycrystalline zinc was used in addition to three kinds of zinc bicrystals grown in a vacuum Bridgman furnace under conditions described in detail elsewhere [19]. The first bicrystal specimens were [OOl] tilt bicrystals. In order to make symmetric tilt bicrystals, 20 mm long seeds were cut from single crystals. The cylindrical single crystals were accurately oriented by the X-ray Laue technique, using a three-axis goniometer. The seed crystals were then cut into two halves along chosen planes, parallel to their axes and one of the halves was rotated 180” about the normal to the cut surface, to provide symmetrical seeds with respect to the chosen boundary plane. The specimens
Fig. 2. A central section through the strain energy surface for a coherently strained 1.1 wt% Cd layer on a zinc surface, using E’$/L~, = 2.6. The length of the radius vector in any orientation is proportional to the coherency strain energy for that surface normal and the axes are marked in units of MN/m’.
grown with this method were symmetric tilt bicrystals with their tilt axes parallel to [OOl]. The second kind of specimens were grown in such a way that the bicrystals have one grain with its [OOl] direction parallel to the GB plane, i.e. its crystallographic c-axis perpendicular to the GB plane normal, and the other has an angle between the boundary plane normal and its crystallographic c-axis. The seed pair comprised one as described above and the other cut from a randomly oriented single crystal. The third kind of specimen was grown from two randomly oriented seeds, forming bicrystals with general GBs. All seeds were chemically polished in a solution of 10% hydrochloric acid and ethyl alcohol to remove any damage introduced during cutting. The graphite mold used to grow these bicrystals was identical to that used by Chen and King [20]. The misorientations of the bicrystals were checked after growth by the Laue method. Slices about 3 mm thick, with their flat surface perpendicular to the tilt axis, were cut from the bicrystals using a low speed diamond saw. The specimens were polished on silicon carbide grinding paper (240-600 grit) first. At least 1 mm of the surface of the section was removed in these operations in order to remove the deep deformation twinning band resulting from the sawing operation which can be seen after etching. The specimens were then polished using c( alumina powder (5-0.05 pm). Finally, the specimens were electropolished in a solution of 15% nitric acid in ethanol giving a clean, stress-free surface. Prepared specimens were encapsulated under vacuum (10m6 T) along with about 12 g of fine filings of a Cd source alloy, which were made by vacuum casting high purity zinc and cadmium to produce Zn-30 wt% Cd and Zn-10 wt% Cd alloys. The specimens then were annealed for the chosen time and temperature, depending on the purpose of the experiments. 4. RESULTS
4.1. General phenomenology Fig. 1. Three-dimensional representation of the variation of coherency strain energy with surface orientation, for t&/t:, = 2.6.
DIGM specimens
was found to occur for polycrystalline exposed to cadmium vapor in the tempera-
LIANG and KING:
DIFFUSION
ture regime from 160 to 260°C. The extent of DIGM varies in different conditions. At lower temperatures, boundary migration only occurs in a few boundaries, which have short migration distances. Figure 3 is an optical micrograph of a polycrystalline zinc specimen after annealing at 250°C for 4 days. The specimen is as-annealed, without further polishing. In this condition, the original and final positions of GBs can be observed. The initial positions of the boundaries were revealed by traces of polishing edges caused by differential polishing of the adjacent grains, while the final positions of the GBs were decorated by thermal grooves which appear as black lines in optical micrographs. Grain boundary migration is not observed in similar anneals without Cd sources, indicating that the observations truly correspond to DIGM and are not related to strains or other effects of anisotropic thermal expansion. Shin and King have shown that grain boundaries in zinc readily accommodate strains deriving from anisotropic expansion, even at very low temperatures [21]. The rate of DIGM was not uniform from boundary to boundary. Even for single boundaries, some segments might migrate by DIGM while others showed no migration. The boundaries usually migrated in a way that increased their areas, independently of the local boundary curvature, as for DIGM in other metal and ceramic systems. There were also some very thin lines, so-called “ghost lines” in the migrated regions, which may have resulted from the discontinuous nature of GB migration. Some
INDUCED
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2987
boundaries migrated unevenly, resulting in a wavy boundary morphology. GB migration reversal after long anneals was frequently observed. Surface pits were also sometimes observed. The Cd content in several DIGM regions was measured by wavelength dispersive spectrometry (WDS) in a specimen after a diffusion anneal at 18O’C for 4 days. Figure 4(a) shows the variation of Cd content across the region where DIGM had taken place, while Fig. 4(b) shows the image of the migrated boundary where the spectra were taken. A compositional discontinuity was observed at the initial boundary position, where a relatively high Cd content was found. Although the Cd concentration was observed to vary with position in the alloyed zone, it tended to decrease as DIGM proceeded, which is consistent with observations by Cahn et al. in the Au-Cu system [ 151. The background level of Cd detected outside the migrated zone may be due to the presence of Cd atoms which diffused into the surface of zinc by lattice diffusion during the DIGM anneal. The measured composition profiles again confirm that we have observed DIGM, and not any other form of GB migration. 4.2. DIGM studies in bicrystal specimens Our primary concern in this study was the extent and direction of GB migration induced by Cd diffusion. We have used two measures of the extent of migration, as illustrated in Fig. 5. The first is the fraction of the total length of the boundary which
Fig. 3. Optical micrograph showing DIGM in polycrystalline zinc after annealing at 250°C for 4 days.
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LIANG and KING:
0.0
I
DIFFUSION
INDUCED GRAIN BOUNDARY
B ,
I
b
I
I
20 10 Distance (pm)
Fig. 4. (a) Composition profile across an alloyed zone formed by DIGM after a diffusion anneal at 180°C for 4 days; (b) micrograph showing the region from which the profile was obtained.
took part in migration, known as the “migrating fraction”. The second measure of DIGM is the average displacement of the boundary within the segments that actually moved: this is called the “migration distance”. Both measures are made on the surfaces of specimens after diffusion anneals, without any further polishing or etching. A typical observation of DIGM in a bicrystal specimen is shown in Fig. 6. Figure 7 shows the average migration versus DIGM time for a [110]/41.9” bicrystal at different temperatures. The displacement of migrating seg-
ments is reduced at lower diffusion temperatures. For example, the longest migration distance is only 3 pm at 150°C after 8 days of diffusion treatment. The displacement of the migrating segments increased with annealing time and the highest velocity occurred at the early stage of DIGM, gradually leveling off after 5 days of annealing. The higher temperatures also give higher initial migration rates, consistent with a thermally activated process. 4.2.1. [OOI] Rotation axes. DIGM experiments were attempted using 17 different symmetric [OOl] tilt bicrystals and eight different asymmetric tilt
LIANG
and KING:
DIFFUSION
INDUCED
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BOUNDARY
L
c
Final boundary position
\
Ongmal boundary position
Fig. 5. Schematic illustration of the measures of DIGM used in this work. The “migrating fraction” of the boundary is (ZI,)/L and the “average migration distance” is W,Y(W Annealing Time (days)
specimens (see Tables 1 and 2). It is noteworthy that both “general” and “coincidence” boundaries were included in the study. The range of misorientation angles is from 6” to 57”. Figure S(a) is an optical micrograph of a 42”/[001] symmetric tilt GB after annealing at 230°C for 4 days with a 30 wt% Cd source. No GB migration is detectable and the result is typical of the entire series of [OOl] tilt CBS, which were apparently immobile, or resistant to nucleation Table 1. Details of the symmetric tilt bicrystals used for the experiments Misorientation (degrees) 6 8 11.5 14 18.5 24.5 36.5 41.5 51 20 21 33 33 17.5 9.2 15 15.5
Fig. 6. Typical
Boundary type
General boundaries
Near Near Near Near Near Near Near Near
case of DIGM
X7 27 El3 213 231 231 243 X43
Fig. 7. Average migration distance vs time for a [I 10]/41.9” bicrystal, at different temperatures. Table 2. Details of the asymmetric tilt bicrystals used for the experiments
8,
02
(degrees)
(degrees)
5 5 18.5 9 11.2 20
14 14 21.5 14 17.2 30
Misorientation (degrees) 19 19 40 23 28.4 50
Boundary type General boundaries Near 27 Near 27 Near 213 Near 237
under the experimental conditions. Identical results were obtained from [OOl] asymmetric tilt boundaries at the same DIGM annealing conditions, as shown in Fig. 8(b). These results are consistent with the prediction of the coherency strain model which suggests that for [OOl] tilt boundaries between hexagonal structure crystals, the elastic isotropy of the basal plane results in the absence of driving force (coherency strain energy) and thus no boundary migration is expected. This result in the Zn-Cd system makes a very interesting and instructive comparison with observations of DIGM in symmetric tilt boundaries in the case of Cu-Zn. Chen and King have shown that DIGM occurs extensively in many of
in a bicrystal specimen. Note the traces of the basal planes, boundary plane in this optical micrograph.
parallel
to the
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Fig. 8. Optical micrographs of 8” (a) and 42” (b) symmetric tilt [OOl] grain boundaries after diffusion anneals at 230°C for 4 days. No DIGM is detectable.
these boundaries, in spite of the apparent lack of a coherency strain driving force, which is negated by the boundary symmetry [20]. In the case of the cubic Cu-Zn alloy, however, any deviation of the boundary plane from its symmetric orientation creates a difference of coherency strain, so local fluctuations of the boundary position can allow the coherency strain mechanism to operate. In the case of the hexagonal crystals, however, changes of boundary plane do not result in differences of coherency strain energy, because of the cylindrical symmetry of the elastic constants. Local fluctuations of the boundary plane do not, therefore, set up coherency strain driving forces, and the boundaries are immune to DIGM deriving from this mechanism. Although no migration was observed for coincidence boundaries in this series of experiments, they did exhibit one feature not observed in the general boundaries. Features with faceted edges on both sides emerge along these boundaries. As the anneal time increases, these features nucleate at other positions along the boundary, and eventually grow together. Figure 9 shows scanning electron microscope @EM) images of a C7[001] symmetric tilt boundary after 4 and 8 days DIGM anneal at 230°C respectively. Detailed analysis and cross-section view of such regions indicate that they were not due to boundary migration, but are surface pits (Fig. 10). A similar phenomenon has been reported in
annealed aluminum 1221, and has been attributed to vacancy diffusion along the GB toward the free surface. 4.2.2. Other rotation axes. The theory predicts that differences in coherency strain energy arise if the orientation relation between the two grains differs from [OOl] tilt. Significant DIGM is then expected for such boundaries. The theoretical result illustrated in Fig. 11 also indicates that for the hexagonal close packed (h.c.p.) crystal structure, the coherency strain energy in each grain depends only on the angle between the crystallographic c-axis and the boundary normal. Thus, a systematic study of GB behaviors was performed by using a series of bicrystals with fixed orientation of grain 1 and varying orientation of grain 2. The occurrence of DIGM after Cd diffusion can be identified by optical microscopy in these specimens. Figure 12 shows the DIGM morphology of a 77.6” [210] bicrystal after different DIGM treatments. The diffuse dark line shows the position of the original boundary, while the final boundary is the thin line on the upper side of the original boundary. The general features of the GB after the different DIGM processes are similar, except that faceting occurred after the 260°C 10 day DIGM anneal. The specimen surface of the upper grain in the pictures has the basal plane perpendicular to the GB plane. In all three cases, the GB segments moved into the upper grain.
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:ondary electron SEM images showing the faceted pits on a X7 coincidence bound >ary atfter a diffusion anneal at 230°C for 4 days (a) and 8 days (b).
In other words, the grain with [OOl] perpendicular to the GB plane normal shrank and the other grew. This is in agreement with the coherency strain model, which predicts that the boundaries will migrate into the grain with the higher modulus. A cross-sectional view of the DIGM sample (Fig. 12) confirms the above observation. The migration distance decreases with increasing penetrating depth and the boundary exhibits a convex curvature toward the grain into which the boundary was migrating. A series of experiments was performed using bicrystals with one grain, named grain 1, always having its [OOl] axis lying in the GB plane (0, = 90@) and normal to the bicrystal surface. The other crystal, grain 2, had varying orientations (0, varies) and thus has varying strain energy in the coherent layer. Figure 13 shows the average migration distance for migrating segments as a function of &, the angle
between the boundary normal and the c-axis of grain 2, after various annealing times at 230°C. The theoretical coherency strain energy difference between grains 1 and 2, corresponding to the driving force, is superimposed on the graph. The migration distance data generally match the theoretical driving force. For bicrystals having large OX, and thus only small driving force, migration in the direction opposite to the driving force was also observed. For bicrystals with smaller Q2 ( < 60”), the migration distance kept increasing as annealing time increased, while for the bicrystals with larger f3?, the migration tends to cease after rapid motion in the early stage. Figure 14 shows the percentage of the original boundary length that underwent migration, as a function of &. The calculated curve of coherency strain energy difference is also superimposed. It is noticed that for specimens with smaller 8, ( < 55”)
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Fig. 10. Optical micrograph taken in cross section, showing a pit formed at the intersection of a 27 boundary with the surface. The specimen was annealed for 8 days at 230°C.
large portions of the boundaries migrated in the early stages of DIGM, and no significant increase in the portion occurred even after long annealing times. For bicrystals with larger 0, (> 60”), however, DIGM started from relatively small portions of the boundaries at the early stage of DIGM, and there was a marked increase in the migrating portion as DIGM continued. Generally speaking, non-uniform nucleation of DIGM was found in the Zn-Cd system. Those segments undergoing initial-stage DIGM extended laterally and migrated forward after long period anneals. It can be seen in Fig. 15, for example, that “ghost lines” show intervals of migration. Migration direction reversal was observed, as shown in Fig. 16(a). Occasionally, twinning
occurred as a result of A second series of pleted, using the same faces cut perpendicular instead of parallel to it differ in no significant here.
DIGM, as seen in Fig. 16(b). experiments has been combicrysals but with their surto the c-axis of crystal 1 [19]. The results of this series way from the series reported
5. DISCUSSION
We have identified DIGM in both polycrystalline and bicrystal zinc, and its characteristics are similar to those in other metal and ceramic systems. Only in a few cases, when the coherency strain energy difference
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was small or absent, did DIGM not follow coherency strain model predictions. 5.1. Driving forces It is instructive to consider the balance of driving forces at the point of nucleation of DIGM. Assuming that the original GB is planar, GB energy is constant and bulging may occur in either side of the boundary in the form of a semicylinder with radius r, as shown in Fig. 17, then the total energy change associated with the DIGM initiation will have the following three major contributions. (1) The free energy of mixing, AG,, arising from the formation of the alloyed (or dealloyed) regions.
W Fig. 12. Cross-sectional optical micrograph of a 77.6”/[210] asymmetric tilt boundary after annealing at 230°C for 4 days. The line diagram illuminates the initial and final boundary positions; the final position being the darker line in this schematic.
J Inclination of Boundary Plane from Basal Plane of Grain 2, in degrees Fig. 11. Plan view optical micrographs of a 77.6/[210] asymmetric tilt boundary after various annealing treatments: (a) 260°C (10 days); (b) 230°C (4 days); and (c) 200°C (8 days).
Fig. 13. Migration distance vs inclination of grain 2 (0,) for a series of bicrystal specimens annealed at 230°C for different annealing times. The heavy continuous curve is the functional form of the driving force derived from coherency strain.
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E”100 , 3 ‘2
80
&? h 60 2 -u g 40 m” M 20 .g E .9 I 0
2
20
40
60
80
Inclination of Boundary Plane from Basal Plane of Grain 2, in degrees
Fig. 14. Migrating boundary fraction vs inclination of grain 2 (0,) for a series of bicrystal specimens annealed at 230°C for different annealing times. The heavy continuous curve is the functional form of the driving force derived from coherency strain.
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shown in the lower case of Fig. 17(a)]. In the second case, the bulge may have the strain corresponding to the locally growing grain, as shown in Fig. 17(b). AG, is then negative for bulges formed in the “forward” direction defined by the coherency strain model, and positive in the “backward” direction. Under the latter set of circumstances, the difference between “forward” and “backward” nucleation is twice as large as in the former case, but the absolute effect on the forward nucleation rate will be smaller. (3) Assuming the boundary free energy is isotropic and constant, the difference of GB area for original and final boundary will cause an energy increase of AA y, where y is the free energy per unit area of GB. Hence, the total energy change is given, in general, as AG = AG, + AG, + AA,,.
(2) The coherency strain energy, AG,, which arises from the difference of coherency strain adjacent to the GBs, as solute diffuses into the adjacent grains. At least two different situations can be envisaged with respect to this component, as illustrated very schematically in Fig. 17. In the first case, a strain-free (or lower-strain) bulge nucleates in the coherently strained region, and AG, is negative. The magnitude of AG, depends on the square of the strain that is relieved, so it is greater for a bulge forming in the more highly strained grain [as shown in the upper case of Fig. 17(a)] than in the less-strained grain [as
(18)
Now we assume that a critical-sized
bulge is of the same order of magnitude as the thickness of the coherently strained layer, and we can write AG = Y(Ag,,,) + I’(Ag,) + AAy
$
[(Ag, + Ag,) + (rr - 2)Zry,
(19)
where V, 1 and r are the volume, length and radius of the bulge, respectively; and Ag,,, and Ag, are the
Fig. 15. The same area after two successive diffusion anneals, showing the lateral and forward of the alloyed region, and the appearance of “ghost lines” marking the intermediate boundary
extension position.
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Fig. 16. Optical micrographs showing (a) reversal of the GB migration direction (the initial migration direction was into the upper grain in this image), and (b) the training of twins by a migrating boundary.
changes in free energy of mixing per unit volume and coherency strain energy per unit volume. The critical radius of an embryonic bulge is then found to be r*=
(n-37
Wgm+W’ The bulge may grow only when its radius is larger than r*, and this defines the point at which an embryonic bulge can be considered as a nucleus for DIGM. Clearly, a large strain energy density, Ag,, favors nucleation. Cahn and Balluffi [4] have argued that the nucleation of DIGM by the formation of small bulges cannot be achieved by the free energy of mixing alone because the increase of free energy associated with the increased GB area is much larger than the reduction achieved through alloying when the bulges are of the order of 0.1 pm in diameter. Data are not available
for the Zn-Cd system, however, using data from the Cu-Zn system, which is very commonly investigated, the GB energy is 0.625 J/m2 [23] and the free energy of mixing has been taken as 78 MJ/m3 at 773 K [24]. We find that the energy changes for both free energy of mixing and increase of GB area are of the same order when r = 0.1 pm, which means once bulges have been formed at this size, the free energy of mixing has to be taken into account. On the other hand, from the nucleation point of view, if we ignore the coherency strain term in equation (20), the critical radius is found to be 2.9 nm, which is still too large to be reached by random thermal fluctuations in the position of GB at common DIGM temperatures. Our expermental results in the Zn-Cu system show that there is no DIGM in [OOl] tilt boundaries no matter whether the boundaries are of symmetric or asymmetric type, though high Cd concentration was
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(a> Fig. 17. Schematic illustration nucleation by the formation of embryonic bulges. In case (a) the bulges are formed strain-free, while in case (b) they embody the strain of the underlying, or growing, crystal.
built up in the GB regions [19]. Meanwhile, DIGM was identified in polycrystalline specimens and bicrystals with large angle GBs where coherency strain energy differences were present. These results suggest that the reduction of free energy caused by alloying is not a sufficient condition for causing motion of the boundary and the coherency strain energy is thus a critical component of the driving force for the initiation of DIGM. The coherency strain energy increases or decreases the effective driving force, depending on its sign in equation (20). If the bulge moves toward a more highly strained layer, the effective driving force is increased. However, if the bulge moves toward the lower strained layer a reduction of the effective driving force of the system is expected, in cases where coherency loss does not occur. Then, r * (forward) is smaller than r* (backward), meaning that the “forward” process is
more highly favored than the “backward” one because the higher coherency strain energy reduces the size of the critical nuclei and increases the probability of boundary migration into the predicted directions. Other mechanisms can also complicate the nucleation process. As discussed above, coherency loss leading to strain-free bulges can increase the nucleation rate and also reduce the directional selectivity of the coherency strain. Figure 18 shows a random large angle GB with a misorientation yielding almost the same coherency strain energies on both sides. The misorientation of this boundary was 39.4” [?83] and the angles, 0, and &, between the c-axes and the boundary normal for the two grains were 72.8 and 73.4”, respectively. The boundary still migrated and a dislocation wall was left at the original position of the boundary, which implies that dislocation
Fig. 18. Transmission electron micrograph showing a moving boundary with very small coherency strain energy difference, after a 24 h diffusion anneal at 200°C. A dislocation array is visible at the original position of the boundary indicating a coherency loss mechanism which is probably responsible for the migration.
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mechanisms may play a role by allowing coherency loss on one side leading to asymmetry of the strain energy. 5.2. Inhomogeneity of DIGM In some of the previous DIGM studies, especially in polycrystalline specimens, it has often been reported that boundary migration is not uniform [I]. Even for a single boundary, some segments migrate and some do not. It can be argued that non-uniform migration is caused by different curvature of different parts of the boundary or small orientation differences of the GB normal along the boundary, according to the coherency strain model. On the other hand, if a boundary is perfectly planar and the GB structure is homogenous everywhere along the boundary, the driving force is expected to be the same along the entire boundary. DIGM should then be initiated uniformly along the boundary, which has been the case for DIGM experiments in symmetric and asymmetric bicrystals of Cu-Zn [6]. However, this is not the case in Zn-Cd, as shown in Fig. 14. Boundaries migrated in certain places and the percentage of migrating segments increased with increasing driving force. Migration progressed by the migrating segments moving further and extending laterally after long treatment times. This difference between the Cu-Zn and Zn-Cd systems may arise from several causes. Firstly, the microscopic regularity of these synthetic boundaries is questionable although there is no clear reason why a copper bicrystal should be more regular or homogeneous than a zinc one. Secondly, the solubility is about 30 wt% for the Cu-Zn system compared with 1.5 wt% for ZnCd at the DIGM temperature, so it is easier to build up a relatively high solute content in the Cu-Zn case. This may minimize any effects caused by irregularity of GB structure. Thirdly, in cubic systems, there is no misfit anisotropy while the hexagonal ZnCd system has a large misfit anisotropy between the a- and c-axes. This allows for small structural irregularities to generate non-uniform driving force along the boundaries. 5.3. Grain boundary d#iision mechanism The GB self-diffusivity of zinc at the DIGM temperature is between 1.1 x lo-l3 and 1.9 x lo-l3 m’/s, while the diffusivity of Cd is 1.9 x lo-‘* m*/s [25]. The diffusivity of Cd is thus one order of magnitude larger than that of Zn. These unequal fluxes of solvent and solute should, therefore, entail a balancing point defect flux if the diffusion occurs by a defect mechanism. The surface pits observed at [OOl] tilt, nearcoincidence boundaries probably result from the vacancy flux induced by such a GB Kirkendall effect, although it is also possible that they are formed as a result of an interstitialcy mechanism of Cd diffusion. Computer simulation experiments [lo] indicate the likelihood of a vacancy mechanism of diffusion in coincidence boundaries. General (or non-
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coincidence) boundaries were not observed to form these pits in our experiments and we surmise that diffusion in these interfaces may proceed by a mechanism that differs in some detail from that in the coincidence boundaries, that does not result in a vacancy flux to the specimen surface. The difference between the two cases may be as drastic as a completely different mechanism of atomic motion, not involving point defects in the general boundaries, or it may derive from the poorer ability of coincidence boundaries to absorb vacancies, which must then be conducted to the surface. Aperiodic boundaries are unfortunately not amenable to study by computer simulation to clarify this issue. We believe that our observation is the first direct indication of a difference of diffusion mechanism between coincidence and general boundaries. 5.4. Some comments about DIGM tems
in hexagonal sys-
The present study is based on the elastic properties of the hexagonal system, which presents special advantages to the experimenter. An elastic isotropy condition obtained within the basal plane and many elastic properties depend only on the angle between an arbitrary reference direction and the crystallographic c-axis. These properties allow us to determine the coherency strains for any GB plane and, potentially, to vary the misorientation while maintaining a constant coherency strain. On the other hand, there are disadvantages to the use of hexagonal systems for DIGM studies because of the low solubilities in hexagonal solvents. In most cases, the hexagonal phase remains stable only up to one or one-and-a-half weight percent of solute. This means less free energy of mixing is available as the driving force for DIGM. From a technical point of view, quantitative compositional analysis of the alloyed zone is more difficult because of the very small solute concentrations involved.
6. CONCLUSIONS
DIGM is found to occur in the ZnCd system, with phenomenology similar to that observed in cubic metal systems, although there is less homogeneity of migration in the hexagonal bicrystal specimens. The progress of GB migration appears to scale with the driving force derived from coherency strain energy. There is evidence in support of a point defect mechanism of GB diffusion in coincidence boundaries, but not more general cases. Acknowledgements-This work was supported by the U.S. National Science Foundation, grant number DMR9404589. Professor R. W. Balluffi provided insightful comments upon the question of the GB diffusion mechanism.
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REFERENCES 1. A. H. King, Znt. Mater. Rev. 32, 173 (1987). 2. M. Hillert, Scrinta metall. 17, 237 (1983). 3. C. Y. Ma, W. Gust, R. A. Fournklle and B. Predel, Mater. Sci. Forum 126-128, 317 (1993). 4. J. W. Cahn and R. W. Balluffi, ‘Acta’metall. 29, 493 (1981). 5. W. H. Rhee, Y. D. Song and D. N. Yoon, Acta metall. 35, 57 (1987). 6. F.-S. Chen, G. Dixit, A. J. Aldykiewicz and A. H. King, Metall. Trans. A 21, 2363 (1990). 7. F.-S. Chen and A. H. King, Acta metall. 36,2827 (1988). of Grain and 8. I. Kaur and W. Gust, Fundamentals Interphase Boundary Diffusion. Ziegler Press, Stuttgart (1989). 9. V. Y. Aristov, V. L. Mirochnik and L. S. Shvindlennan, Fiz. Tverd. Tela. 18, 137 (1976). 10. R. W. Balluffi, T. Kwok, P. D. Bristowe, A. Brokman, P. S. Ho and S. Yip, Scripta metall. 15, 951 (1981). 11. D. A. Smith, C. M. F. Rae and C. R. M. Grovenor, in Grain Boundary Structure and Kinetics (edited by R. W. Balluffi). ASM, Materials Park, OH (1980). 12. C. A. Handwerker, J. W. Cahn, D. N. Yoon and J. E. Blendell, in Atomic Transport in Alloys: Recent Develop-
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