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Remarks on “the energy-misorientation relationship of grain boundaries”
Scripta
METALLURGICA
V o l . 14, pp. 1 1 8 7 - 1 1 6 0 , 1980 Printed in t h e U . S . A .
REMARKS ON "THE ENERGY-MISORIENTATION OF GRAIN BOUNDARIES"
Pergamon P r e s s Ltd. All rights re~erved
RELATIONSHIP
A. H. King and R. W. Balluffi Department of Materials Science and Engineering Center for Materials Science and Engineering Massachusetts Institute of Technology Cambridge, Massachusetts 02139
(Received
June
23,
1980)
Introduction Kuhn, Baero and Gleiter, in their recent paper "On the Energy-Misorientation Relationship of Grain Boundaries" (i), have reported an attempt to measure the rate of rotation of single crystal balls of copper with respect to single crystal plates to which they are being sintered. They claim evidence for a reduction in the rate of rotation as a low energy misorientation is approached. They then go on to assume that the rate of rotation is proportional to the rate of change of energy with misorientation and interpret the claimed experimental observation to mean that the energy (y) vs. misorientation (@) plot is not sharply cusped at the low energy misorientation. It is the purpose of this note to suggest that this conclusion of Kuhn, et al. is highly questionable for two reasons: firstly, their data, based on the observation of only two grain boundaries (i.e., rotating spheres) are insufficient for the purposes at hand; and secondly, the assumptions ,on which the interpretation of the results is based are most probably unwarranted. Gleiter (2) has drawn heavily on the conclusions of Kuhn, et al. during a recent exchange in the literature regarding a postulated relationship between the shape of minima in the y vs. @ plot and the core widths of secondary grain boundary dislocations. It is not the purpose of this paper to comment in any way on that exchange, but rather to comment on the experiments and interpretation presented by Kuhn, et al. (i). Experimental Data The data presented by Kuhn, et al. consist of time-dependent measurements of the misorientations of two single crystal balls [Table i of (i)]. Only one set of data is presented graphically [Fig. 2 of (i)]. The authors seek to suggest that the rate of rotation is reduced as a low energy orientation is approached. In the figures and (apparently) also in the analysis, the misorientation of the ball with respect to the plate is represented by a single parameter, which is clearly inadequate. The energy of a grain boundary is a function of five macroscopic degrees of freedom and three microscopic ones. It is probably reasonable to expect that the grain boundary energy is always minimized with respect to the three microscopic degrees of freedom, which correspond to small rigid body translations of one grain with respect to the other. The five macroscopic degrees of freedom are associated with three direction cosines defining the relative misorientation of the grains (i.e., the angle and axis of misorientation) and two direction cosines defining the orientation of the grain boundary plane. So the rotation of a ball with respect to a plate represents the minimization of grain boundary energy with respect to five degrees of freedom, i.e., a six dimensional minimization. Kuhn, et al. present their data and analysis in terms of a two dimensional minimization, which therefore involves the implicit assumption that the system is fixed with respect to four degrees of freedom. These four degrees of freedom are associated with the axis of rotation and the orientation of the grain boundary plane. The authors (i) state that the axes of rotation changed during the experiments but imply that these changes were small, and hence, negligible. For Specimen i, the angle of rotation varied through 7.4 ° , while the axis varied through 2.5 ° . For Specimen 2, the angle varied through 5.6 ° , and the axis through 3.4 ° . In neither case, therefore, was the change in the axis small compared with the change in the angle.
1157 0036-9748/80/1O1157-04502.00/0 Copyright (c) 1 9 8 0 P e r g a m o n Press
Ltd.
1158
ENERGY-MISORIENTATION
RELATIONSHIP
Vol.
14,
No.
i0
Kuhn, et al. make the second implicit assumption that the grain boundary plane remains parallel to the surface of the plate, thus removing the necessity of considering the two degrees of freedom associated with the grain boundary plane. In a similar series of experiments (3), Mykura observed anomalous rotation of a large proportion of balls, associated with grain boundaries, lying in orientations not parallel to the plate, which become curved due to grain boundary sliding. The magnitude of the deviation of the grain boundary from its assumed position was as much as 15 ° in some of Mykura's observations. The possible variations in all five degrees of freedom are thus too large to warrant the use of only one of them, as done by Kuhn, et al. Indeed, this becomes a general criticism of all schematic diagrams of y vs. O, which must be regarded as being potentially highly misleading. The data from Specimen 2, which were not shown graphically by Kuhn, et al., show a behavior w h i c h m a y be well described as a linear decrease in mlsorientation, followed discontinuously by a slowly increasing misorientation, as we show in Fig. la. A curve with a similar discontinuity is suggested by Kuhn, et al. as a plausible representation of the data for Specimen i [see dashed curve in Fig. 2 of (i)]. In our Fig. I we also show two other conceivable curves, viz, one showing an increasing rate of rotation, leading to a sharp discontinuity at the minimum (Fig. ib), and one showing a continuously decreasing rate of rotation as the minimum is approached (Fig. Ic). It is obvious that in order to give any preference, more data points, particularly in the region of the minimum, are required. For Specimen i there are no experimental data taken within approximately lOh and 2 ° before the time of the possible discontinuity, and for Specimen 2 there are none within 5h and ½o. It is thus not possible to determine the shape of the curve in this region to any greater accuracy than this: indeed, Kuhn, et al. chose the less accurate set of measurements (exhibiting greater data scatter) for presentation and illustration of their interpretation. Data abstracted from such relatively large angular deviations from the misorientation of minimum energy cannot be used with any confidence to predict the form of the curve in that region. In addition, it should be mentioned that the slope of the ~ vs. 8 plot, based on a secondary grain boundary dislocation model (4) which involves small Burgers vectors, and a A01nA8 function (5) for the energy, can be quite small at relatively small deviations from the misorientation of minimum energy, rising sharply only at misorientations much closer to the minimum. (Here A0 is the angular deviation from the minimum.) Thus an energy cusp could be detected by the method attempted by Kuhn, et al. only if all five macroscopic degrees of freedom of the grain boundary were measured with a considerably higher density of data points than that obtained for the single degree of freedom which they did measure. Interpretation
of the Data
The discussion presented by Kuhn, et al. is based on the operation of a single mechanism for the rotation of the balls about an axis parallel to the grain boundary. It is not clear from the data presented, that the axis of rotation was at any time parallel to either the grain boundary or the plate. There are two known mechanisms for rotation about axes parallel to the grain boundary: these are the Shewmon mechanism (6) [as discussed by Kuhn, er al.] and the anomalous rotation mechanism presented by Mykura (3). In addition to these, a mechanism for rotation about an axis perpendicular to the grain boundary has been proposed by Pond and Smith (7). We note that the increasing misorientation observed in the later part of the experiment on Specimen 2 cannot be explained solely on the basis of the Shewmon mechanism, and the anomalous rotation mechanism must therefore be invoked: this mechanism provides for rotation of the sphere, the rate of which is not related directly driven by ~
.
Such anomalous
mechanism are specifically
to ~ LL~
dt
=
as well as any rotation occurring by the Pond and Smith
ignored in the discussion presented by Kuhn, et al.
Kuhn, et al. base their discussion d88
and which is in the negative sense compared with rotation
rotations,
on an equation having the following form:
M ( r ) • dy
dO
(i)
'
where M(r) is a rotational mobility appropriate to the Shewmon mechanism, and dependent on the neck radius, r. The reduction in rotation rate which they claim to have observed is then associated with a reduction
in driving force,
i.e., dy dO "
We wish to point out that the mobility given
Vol.
14,
No.
i0
ENERGY-MISORIENTATION
130
RELATIONSHIP
11159
130
130.
129
129-
(o)
129
(b)
128¸
128
~28
127 -
o z
127
~ 127 Z <[
z
126-
_~ i- 1 2 6
~) 126
~
125.
z
z
g
(c)
125.
125
~ o
124123 122
0
200
400
u) .~ 124.
124
123.
123
122
600
TIME (h)
0
200
400
600
TIME (h)
122
J
0
200
400 TIME
..J
j~e
600
(h)
FIG. 1 Three conceivable interpretations of the data given by Kuhn, et al. for their Specimen 2: (a) shows a constant rate of rotation as a discontinuity is approached; (b) shows an increasing rate; and (e) shows a decreasing rate approaching a smooth minimum. The data points are the same in each case. by Kuhn, et al. has the form: M(r)
=
24D~ ~kT[A(T).t]3/n
'
where D is the lattice self-diffusivity, and ~ is the atomic volume. The time dependence of M(r) depends on the neck growth rate, for which Kuhn, et al. assume an idealized theoretical model: in his experiments, Mykura (3) observed considerable variations in neck growth rates, and hence considerable deviations from any theoretical model. It is also important to realize that the use of the above mobility involves the assumption that the grain boundary always acts as a perfect source or sink for the point defects required for the associated diffusional flux, irrespective of the grain boundary parameters. Jaeger and Gleiter (8), however, have demonstrated that grain boundaries which lie within 2.5 ° of a low energy misorientation may act as highly inefficient sources and sinks for point defects. If the source or sink efficiency is reduced when the sphere is close to an energy minimum, then M(r) is also reduced, and the ball rotation rate will drop for any given driving force. We suggest that if the rotation rate of the spheres really does slow down as an energy minimum is approached, that this is an effect of reduced mobility and/or the operation of conflicting rotation mechanisms, rather than reduced driving force for a single mechanism.
Our v i e w may t h u s be summed u p a s f o l l o w s : [1] t h e d a t a p r e s e n t e d by Kuhn, e t a l . a r e insufficient i n t y p e and i n p r e c i s i o n t o e s t a b l i s h whether a reduction in the rotation rate r e a l l y d o e s o c c u r c l o s e t o a n e n e r g y minimum, and [2] i f s u c h a r e d u c t i o n a c t u a l l y d o e s o c c u r , t h e n i t may b e a t t r i b u t e d t o c h a n g i n g r o t a t i o n m e c h a n i s m s , a n d / o r t o e f f e c t s on t h e r o t a t i o n a l m o b i l i t y w h i c h a r e a l r e a d y d o c u m e n t e d (8) i n s t e a d o f t h e a b s e n c e o f a s h a r p c u s p a t t h e minimum. Acknowledgment The authors acknowledge Contract No. DMR-78-24185.
the support of the National Science Foundation MRL Program,
References 1. 2. 3. 4.
H. H. H. P.
Kuhn, G. B a e r o a n d H. G l e i t e r , A c t a Met. Gleiter, S c r i p t a M e t . 14, 569 ( 1 9 8 0 ) . M y k u r a , A c t a Met. 27, 243 ( 1 9 7 9 ) . H. P u m p h r e y , S c r i p t a Met. 9, 151 ( 1 9 7 5 ) .
2 7 , 959 ( 1 9 7 9 ) .
under
1160
5. 6. 7. 8.
ENERGY-MISORIENTATION
RELATIONSHIP
Vol.
14, No.
i0
W. T. Read and W. Shockley, Phys. Rev. 78, 275 (1950). P. G. Shewmon, in Recrystallization, Grain Growth and Textures, p. 165, ASM, Metals Park, Ohio (1966). R. C. Pond and D. A. Smith, Scripta Met. Ii, 77 (1977). W. Jaeger and H. Gleiter, Scripta Met. 12, 675 (1978).
METALLURGICA
V o l . 14, pp. 1 1 8 7 - 1 1 6 0 , 1980 Printed in t h e U . S . A .
REMARKS ON "THE ENERGY-MISORIENTATION OF GRAIN BOUNDARIES"
Pergamon P r e s s Ltd. All rights re~erved
RELATIONSHIP
A. H. King and R. W. Balluffi Department of Materials Science and Engineering Center for Materials Science and Engineering Massachusetts Institute of Technology Cambridge, Massachusetts 02139
(Received
June
23,
1980)
Introduction Kuhn, Baero and Gleiter, in their recent paper "On the Energy-Misorientation Relationship of Grain Boundaries" (i), have reported an attempt to measure the rate of rotation of single crystal balls of copper with respect to single crystal plates to which they are being sintered. They claim evidence for a reduction in the rate of rotation as a low energy misorientation is approached. They then go on to assume that the rate of rotation is proportional to the rate of change of energy with misorientation and interpret the claimed experimental observation to mean that the energy (y) vs. misorientation (@) plot is not sharply cusped at the low energy misorientation. It is the purpose of this note to suggest that this conclusion of Kuhn, et al. is highly questionable for two reasons: firstly, their data, based on the observation of only two grain boundaries (i.e., rotating spheres) are insufficient for the purposes at hand; and secondly, the assumptions ,on which the interpretation of the results is based are most probably unwarranted. Gleiter (2) has drawn heavily on the conclusions of Kuhn, et al. during a recent exchange in the literature regarding a postulated relationship between the shape of minima in the y vs. @ plot and the core widths of secondary grain boundary dislocations. It is not the purpose of this paper to comment in any way on that exchange, but rather to comment on the experiments and interpretation presented by Kuhn, et al. (i). Experimental Data The data presented by Kuhn, et al. consist of time-dependent measurements of the misorientations of two single crystal balls [Table i of (i)]. Only one set of data is presented graphically [Fig. 2 of (i)]. The authors seek to suggest that the rate of rotation is reduced as a low energy orientation is approached. In the figures and (apparently) also in the analysis, the misorientation of the ball with respect to the plate is represented by a single parameter, which is clearly inadequate. The energy of a grain boundary is a function of five macroscopic degrees of freedom and three microscopic ones. It is probably reasonable to expect that the grain boundary energy is always minimized with respect to the three microscopic degrees of freedom, which correspond to small rigid body translations of one grain with respect to the other. The five macroscopic degrees of freedom are associated with three direction cosines defining the relative misorientation of the grains (i.e., the angle and axis of misorientation) and two direction cosines defining the orientation of the grain boundary plane. So the rotation of a ball with respect to a plate represents the minimization of grain boundary energy with respect to five degrees of freedom, i.e., a six dimensional minimization. Kuhn, et al. present their data and analysis in terms of a two dimensional minimization, which therefore involves the implicit assumption that the system is fixed with respect to four degrees of freedom. These four degrees of freedom are associated with the axis of rotation and the orientation of the grain boundary plane. The authors (i) state that the axes of rotation changed during the experiments but imply that these changes were small, and hence, negligible. For Specimen i, the angle of rotation varied through 7.4 ° , while the axis varied through 2.5 ° . For Specimen 2, the angle varied through 5.6 ° , and the axis through 3.4 ° . In neither case, therefore, was the change in the axis small compared with the change in the angle.
1157 0036-9748/80/1O1157-04502.00/0 Copyright (c) 1 9 8 0 P e r g a m o n Press
Ltd.
1158
ENERGY-MISORIENTATION
RELATIONSHIP
Vol.
14,
No.
i0
Kuhn, et al. make the second implicit assumption that the grain boundary plane remains parallel to the surface of the plate, thus removing the necessity of considering the two degrees of freedom associated with the grain boundary plane. In a similar series of experiments (3), Mykura observed anomalous rotation of a large proportion of balls, associated with grain boundaries, lying in orientations not parallel to the plate, which become curved due to grain boundary sliding. The magnitude of the deviation of the grain boundary from its assumed position was as much as 15 ° in some of Mykura's observations. The possible variations in all five degrees of freedom are thus too large to warrant the use of only one of them, as done by Kuhn, et al. Indeed, this becomes a general criticism of all schematic diagrams of y vs. O, which must be regarded as being potentially highly misleading. The data from Specimen 2, which were not shown graphically by Kuhn, et al., show a behavior w h i c h m a y be well described as a linear decrease in mlsorientation, followed discontinuously by a slowly increasing misorientation, as we show in Fig. la. A curve with a similar discontinuity is suggested by Kuhn, et al. as a plausible representation of the data for Specimen i [see dashed curve in Fig. 2 of (i)]. In our Fig. I we also show two other conceivable curves, viz, one showing an increasing rate of rotation, leading to a sharp discontinuity at the minimum (Fig. ib), and one showing a continuously decreasing rate of rotation as the minimum is approached (Fig. Ic). It is obvious that in order to give any preference, more data points, particularly in the region of the minimum, are required. For Specimen i there are no experimental data taken within approximately lOh and 2 ° before the time of the possible discontinuity, and for Specimen 2 there are none within 5h and ½o. It is thus not possible to determine the shape of the curve in this region to any greater accuracy than this: indeed, Kuhn, et al. chose the less accurate set of measurements (exhibiting greater data scatter) for presentation and illustration of their interpretation. Data abstracted from such relatively large angular deviations from the misorientation of minimum energy cannot be used with any confidence to predict the form of the curve in that region. In addition, it should be mentioned that the slope of the ~ vs. 8 plot, based on a secondary grain boundary dislocation model (4) which involves small Burgers vectors, and a A01nA8 function (5) for the energy, can be quite small at relatively small deviations from the misorientation of minimum energy, rising sharply only at misorientations much closer to the minimum. (Here A0 is the angular deviation from the minimum.) Thus an energy cusp could be detected by the method attempted by Kuhn, et al. only if all five macroscopic degrees of freedom of the grain boundary were measured with a considerably higher density of data points than that obtained for the single degree of freedom which they did measure. Interpretation
of the Data
The discussion presented by Kuhn, et al. is based on the operation of a single mechanism for the rotation of the balls about an axis parallel to the grain boundary. It is not clear from the data presented, that the axis of rotation was at any time parallel to either the grain boundary or the plate. There are two known mechanisms for rotation about axes parallel to the grain boundary: these are the Shewmon mechanism (6) [as discussed by Kuhn, er al.] and the anomalous rotation mechanism presented by Mykura (3). In addition to these, a mechanism for rotation about an axis perpendicular to the grain boundary has been proposed by Pond and Smith (7). We note that the increasing misorientation observed in the later part of the experiment on Specimen 2 cannot be explained solely on the basis of the Shewmon mechanism, and the anomalous rotation mechanism must therefore be invoked: this mechanism provides for rotation of the sphere, the rate of which is not related directly driven by ~
.
Such anomalous
mechanism are specifically
to ~ LL~
dt
=
as well as any rotation occurring by the Pond and Smith
ignored in the discussion presented by Kuhn, et al.
Kuhn, et al. base their discussion d88
and which is in the negative sense compared with rotation
rotations,
on an equation having the following form:
M ( r ) • dy
dO
(i)
'
where M(r) is a rotational mobility appropriate to the Shewmon mechanism, and dependent on the neck radius, r. The reduction in rotation rate which they claim to have observed is then associated with a reduction
in driving force,
i.e., dy dO "
We wish to point out that the mobility given
Vol.
14,
No.
i0
ENERGY-MISORIENTATION
130
RELATIONSHIP
11159
130
130.
129
129-
(o)
129
(b)
128¸
128
~28
127 -
o z
127
~ 127 Z <[
z
126-
_~ i- 1 2 6
~) 126
~
125.
z
z
g
(c)
125.
125
~ o
124123 122
0
200
400
u) .~ 124.
124
123.
123
122
600
TIME (h)
0
200
400
600
TIME (h)
122
J
0
200
400 TIME
..J
j~e
600
(h)
FIG. 1 Three conceivable interpretations of the data given by Kuhn, et al. for their Specimen 2: (a) shows a constant rate of rotation as a discontinuity is approached; (b) shows an increasing rate; and (e) shows a decreasing rate approaching a smooth minimum. The data points are the same in each case. by Kuhn, et al. has the form: M(r)
=
24D~ ~kT[A(T).t]3/n
'
where D is the lattice self-diffusivity, and ~ is the atomic volume. The time dependence of M(r) depends on the neck growth rate, for which Kuhn, et al. assume an idealized theoretical model: in his experiments, Mykura (3) observed considerable variations in neck growth rates, and hence considerable deviations from any theoretical model. It is also important to realize that the use of the above mobility involves the assumption that the grain boundary always acts as a perfect source or sink for the point defects required for the associated diffusional flux, irrespective of the grain boundary parameters. Jaeger and Gleiter (8), however, have demonstrated that grain boundaries which lie within 2.5 ° of a low energy misorientation may act as highly inefficient sources and sinks for point defects. If the source or sink efficiency is reduced when the sphere is close to an energy minimum, then M(r) is also reduced, and the ball rotation rate will drop for any given driving force. We suggest that if the rotation rate of the spheres really does slow down as an energy minimum is approached, that this is an effect of reduced mobility and/or the operation of conflicting rotation mechanisms, rather than reduced driving force for a single mechanism.
Our v i e w may t h u s be summed u p a s f o l l o w s : [1] t h e d a t a p r e s e n t e d by Kuhn, e t a l . a r e insufficient i n t y p e and i n p r e c i s i o n t o e s t a b l i s h whether a reduction in the rotation rate r e a l l y d o e s o c c u r c l o s e t o a n e n e r g y minimum, and [2] i f s u c h a r e d u c t i o n a c t u a l l y d o e s o c c u r , t h e n i t may b e a t t r i b u t e d t o c h a n g i n g r o t a t i o n m e c h a n i s m s , a n d / o r t o e f f e c t s on t h e r o t a t i o n a l m o b i l i t y w h i c h a r e a l r e a d y d o c u m e n t e d (8) i n s t e a d o f t h e a b s e n c e o f a s h a r p c u s p a t t h e minimum. Acknowledgment The authors acknowledge Contract No. DMR-78-24185.
the support of the National Science Foundation MRL Program,
References 1. 2. 3. 4.
H. H. H. P.
Kuhn, G. B a e r o a n d H. G l e i t e r , A c t a Met. Gleiter, S c r i p t a M e t . 14, 569 ( 1 9 8 0 ) . M y k u r a , A c t a Met. 27, 243 ( 1 9 7 9 ) . H. P u m p h r e y , S c r i p t a Met. 9, 151 ( 1 9 7 5 ) .
2 7 , 959 ( 1 9 7 9 ) .
under
1160
5. 6. 7. 8.
ENERGY-MISORIENTATION
RELATIONSHIP
Vol.
14, No.
i0
W. T. Read and W. Shockley, Phys. Rev. 78, 275 (1950). P. G. Shewmon, in Recrystallization, Grain Growth and Textures, p. 165, ASM, Metals Park, Ohio (1966). R. C. Pond and D. A. Smith, Scripta Met. Ii, 77 (1977). W. Jaeger and H. Gleiter, Scripta Met. 12, 675 (1978).