- Email: [email protected]
Kinetic studies of dimensional aspects in thin film grain growth
PII: S1359-6454(98)00315-2
Acta mater. Vol. 46, No. 18, pp. 6397±6402, 1998 Acta Metallurgica Inc. Published by Elsevier Science Ltd Printed in Great Britain 1359-6454/98 $19.00 + 0.00
KINETIC STUDIES OF DIMENSIONAL ASPECTS IN THIN FILM GRAIN GROWTH M. A. PALMER{a, M. E. GLICKSMANb and K. RAJANb a
Virginia Commonwealth University, Department of Mechanical Engineering, Richmond, VA 232843015, U.S.A., bRensselaer Polytechnic Institute, Troy, NY, U.S.A. (Received 16 March 1998; accepted 1 September 1998)
AbstractÐIt has been shown that grain growth in succinonitrile thin ®lms is accurately described by the Mullins±Von Neumann grain growth law. However, there is a small fraction of grains showing signi®cant departures from ideal behavior (Palmer et al., Scripta metall. mater., 1994, 30, 633; Palmer et al., Metall. Mater. Trans. A, 1995, 26A, 1061). These grains are examined to see if they show departures from ideal behavior (the Uniform Boundary Model). It has been found that the vast majority of this small fraction of grains shows one of four distinct characteristics: they possess low-angle boundaries, they are small, they possess short boundaries, or they exhibit discontinuous boundary motion. With the exception of the lowangle boundaries all eects are due to the three-dimensional nature of the ®lm. These characteristics have been correlated with grain growth behavior: small grains tend to shrink faster than ideal grains, grains possessing short boundaries tend to stagnate, those undergoing discontinuous boundary motion show irregular behavior. Through these experiments real ®lms have been examined in the context of two-dimensional models. Acta Metallurgica Inc. Published by Elsevier Science Ltd
1. INTRODUCTION
A thin ®lm of succinonitrile as a collection of grains tends to follow the grain growth law advanced by Mullins [1] based on earlier work by Von Neumann [2]. This law predicts that the areal rate of change of a two-dimensional grain is given by dA p Mg N ÿ 6 dt 3
1
where M is the mobility of the boundary, g the grain boundary energy, and N the topological class (number of sides or vertices) of the grain. Figure 1 shows growth rate as a function of topological class. One will note the large spread in the growth rates shown in Fig. 1. In these experiments [3, 4] extremely pure succinonitrile was used eliminating the eect of trace impurities altering the motion of grain boundaries. The temperature was controlled to 21 mK which meant that temperature variations did not alter the grain growth. We have no reason to believe that the surface energy, and mobility are not uniform. However, our experiments show that there is a small fraction of grains which do not follow the Mullins±Von Neumann grain growth law equation (1). This further implies that there are individual grains which are not described by the Uniform Boundary Model. This model will only be applicable if: all grain boundary energies and mobilities are equal, the grains are two-dimensional, and the grain boundary motion is continuous. {To whom all correspondence should be addressed.
Not every grain can be described by the above three conditions, and those that cannot are not subject to Mullins analysis. There are a small number of grains possessing low-energy grain boundaries, which by de®nition, would not be expected to follow equation (1). Grains which change their topological class through neighbor switching, typically have one short boundary [3]. There are also small grains, those with diameters corresponding to the ®lm thickness. These grains cannot be considered two-dimensional. Thus, neither small grains nor grains which undergo neighbor switching, should follow equation (1). Mullins did not consider the annihilations of four- and ®ve-sided grains. In these cases, there are neighboring grains which do not undergo a topological transformation. However, as shown there are de®nite boundary changes as two grains which bordered the annihilated grain, now border each other [4]. Since the boundaries of these grains undergo a sudden change as a result of an adjacent topological event, they might not be expected to follow equation (1) throughout their lifetime. Mullins has stated that grains which experience a sudden change in the number of sides due to the disappearance of a neighboring grain are not expected to be governed by these equations [1]. 2. EXTREME VALUE ANALYSIS
To identify those grains showing unusual departures from ideal behavior we made use of extreme value analysis. This was done by analyzing the distributions of growth rates within each topological
6397
6398
PALMER et al.: DIMENSIONAL ASPECTS IN THIN FILM GRAIN GROWTH
A disproportionate number of gains was found to lie at the extreme of the distribution, and were classi®ed as extreme, warranting further investigation. The original data acquisition algorithm, used to determine the growth rate, was modi®ed so that the vertex positions of each grain were recorded in addition to the other data (size, perimeter, and topological class). A new algorithm mapped each grain on the screen, enabling further examination of the speci®c nature of each grain's growth. This procedure allowed us to examine grains classi®ed as ``extreme'' and determine if they showed anomalous behavior.
Fig. 1. Growth rate of grains in succinonitrile thin ®lms [3]. The lines indicate ideal behavior as predicted by equation (1).
class. If the growth rates, within a given topological class are normally distributed, then they can be described by 1 2 f z p eÿz =2 2p
2
where z is de®ned as z
x ÿ x : s
We sorted the previously collected [3] rate constant data for the two runs and determined the average growth rate, and the standard deviation of the growth rate, within each topological class. From this, z was de®ned as dA dA ÿ dt dt N z 3 sN where the subscript N denotes the speci®c topological class of the grain, dA=dt the growth rate of an individual grain, hdA=dtiN the mean growth rate for the topological class N, and sN the standard deviation of the growth rates within the topological class N.
3. RESULTS AND DISCUSSION
Our examination of anomalous grains included two aspects. Forty of the 800 grains examined were classi®ed as extreme grains, those showing (34 of the 40) one of four behaviors were classi®ed as anomalous. These were grains which possessed lowangle boundaries, small grains, grains with short boundaries, and grains exhibiting discontinuous boundary motion. Our results show that discontinuous boundary motion is the most common cause for a grain to be considered anomalous. 3.1. Low-angle boundaries One of the assumptions of the Mullins±Von Neumann analysis is that all boundaries are of uniform energy. In a real polycrystalline aggregate, there exists a small number of low-angle boundaries. Sometimes these boundaries are polygonalized dislocation arrays, and typically they have dihedral angles less than 158. In our experiment, the grain boundaries are decorated by thermal grooves. The depth of these grooves increases with increasing grain boundary energy. A low-angle boundary will therefore appear with much less optical contrast than does a ``normal'' grain boundary with average boundary energy. Five of the 34 anomalous grains possessed at least one low-angle boundary and were observed between the two runs. An example of grain growth in the presence of a low-angle boundary is shown in
Fig. 2. Grain growth with a low-angle boundary.
PALMER et al.: DIMENSIONAL ASPECTS IN THIN FILM GRAIN GROWTH
Fig. 2. The low-angle boundary disappears as grain growth progresses, and grains 42 and 43 essentially combine and form a single grain, which although not shown in Fig. 2, occurs between 100 and 120 s. Although the low-angle boundary is nearly invisible, its eect is evident on the other boundaries. In the ®gure, this is indicated by the arrows at 0 and 60 s. It appears as if the low-angle boundary retards the motion of the boundaries which it intersects. Note the low-angle boundary causes a local deviation of the curvature of the boundary. At 60 s, the in¯uence of this ``pulling'' becomes evident. The areas of grains 42 and 43 are plotted as a function of time in Figs 3 and 4, respectively. In each of these ®gures, the ideal growth rate, i.e. that which corresponds to equation (1), is plotted as a dashed line passing through the data at 0 s. The growth rate of both grains appears to be retarded by the low-angle boundary. If one considers the combined grain as a single grain, it would be a seven-sided grain for 80 s, and would transform into a six-sided grain as a neighbor is annihilated. The combined growth rate does not follow ideal behavior for either case. One can consider the low-angle boundary as a grain boundary dividing two grains or one could consider it as a ``¯aw'' in the combined grain. In either case the boundaries are clearly not of uniform energy and mobility and grains with low-angle boundaries are anomalous. That is, they tend not to conform to the assumptions of the Uniform Boundary Model and they show signi®cant departures from equation (1). 3.2. Small grains In Section 1 of this paper it was shown that the ®nite thickness of the ®lm alters the annihilation mechanisms responsible for grain growth, that is they do not occur exactly as predicted by the two-
Fig. 3. Area of grain 42 in Fig. 2. Grain 42 has seven sides. The line indicates ideal behavior as predicted by equation (1).
6399
Fig. 4. Area of grain 43 in Fig. 2. Grain 43 has four sides. The line indicates ideal behavior as predicted by equation (1).
dimensional theories [5]. Most of the grains characterized as anomalous were aected by the ®nite ®lm thickness. Grains de®ned as small were grains where the mean diameter was on the same length scale as the ®lm thickness. There are three types of anomalous behavior which can be attributed to ®nite ®lm thickness: accelerated shrinkage of small grains, stagnation of grains undergoing neighbor switching, and grains exhibiting discontinuous boundary motion. Mullins' analysis assumes that grains are twodimensional. Grains with three, four, or ®ve sides are observed to be annihilated in the grain growth process. As this happens, the grains shrink to essentially zero size, and they can no longer be assumed to be two-dimensional. An example of such a grain is shown in Fig. 5. This grain is small compared to its neighbors, and shrinks to a small area. This grain shrinks quickly. The area vs time curve for this grain is shown in Fig. 6. The slope of the dashed line represents the ideal growth rate of a ®ve-sided grain, were equation (1) to be followed. We observed that small grains tend to shrink faster than ideal grains. In this experiment, the ®ve grains that were identi®ed as small had growth rates less than predicted by equation (1), as shown in Table 1. In Table 1 hdA=dti refers to the ideal growth rate for the topological class. Similar results had been predicted earlier. According to Frost [5], normal grain growth stagnates when the average diameter of the grain is 2±3 times the ®lm thickness due to pinning at the surface. Our results show the converse behavior. Frost was referring to the retardation of grain growth and shrinkage once the grains exceeded a certain size. As the grains become three-dimensional they are no longer aected by this pinning, and the shrinkage rate is accelerated. Our earlier analyses [6] indicate that a grain need not be of uniform topological class throughout the
6400
PALMER et al.: DIMENSIONAL ASPECTS IN THIN FILM GRAIN GROWTH
Fig. 5. Accelerated shrinkage of small ®ve-sided grain.
thickness of the ®lm. For example, consider the annihilation of a three-dimensional ®ve-sided grain; this occurs in seven steps. As our experiments only observe the top surface of the grain, there is no way of telling if the grain is in one of these intermediate steps. Clearly, when a grain enters these intermediate steps it behaves anomalously, and should shrink more rapidly than would be predicted by equation (1). 3.3. Neighbor switching Neighbor switching is one of the two basic unit operations responsible for grain growth. When such an event occurs two grains lose a boundary, while two neighboring grains gain a boundary. We found four grains which reduced their topological class through neighbor switching. Such a sequence is shown in Fig. 7. The ®ve-sided grain indicated by an asterisk (*), appears to shrink normally for about 160 s. Then there is an 80 s period where the grain area barely changes. Finally, a new four-sided grain can be seen after 320 s. As shown earlier [6], this stagnation is the result of the neighbor switching occurring ®rst on one face of the ®lm, and then on the other. The area vs time plot for the neighbor switching sequence is shown in Fig. 8. The 80 s stagnation period is clearly evident. The dashed line
represents the ideal case where equation (1) is followed. Up until the stagnation period, the growth rate is close to ideal. Note that during the stagnation period the boundary between the two switching grains becomes extremely short. This short boundary moves in response to the twisting forces imposed by the new boundary on the other face of the ®lm. If this shrinkage scenario is indeed the case, then Mullins' analysis is not expected to hold. The stagnation of one boundary means that the motion of the other boundaries determine the growth rate of the grain. Consider that dA=dt for a six-sided grain as shown in Table 2 is close to the average or ideal growth rate, hdA=dti, for a ®vesided grain. The eect is even more pronounced for the ®ve-sided grains. 3.4. Discontinuous boundary motion The vast majority of those grains identi®ed as anomalous retained their topological class despite an adjacent topological event, and therefore demonstrated discontinuous boundary motion. An example of such a grain is shown in Fig. 9. In this case a seven-sided grain, indicated by an asterisk (*), remains seven-sided, despite the fact that three neighboring grains 1, 2, and 3, are annihilated. Consider Fig. 9 in the context of the sequence of events describing the annihilation of a three-dimensional ®ve-sided grain [6]. Three ®ve-sided grains were annihilated in 40 s. As explained in Section 1 of this paper, ®ve-sided grain annihilation takes place in several steps. The boundaries bordering the * grain at 40 s, therefore are dierent from those which border the grain at 0 s. In fact these new boundaries are probably combinations of the boundaries from the top face of the ®lm at 0 s, and the boundaries which extended through the thickness of the ®lm. This suggests that the boundary Table 1. Behavior of small grainsÐgrowth rates in mm2/s Run Grain Top. class dA/dt (mm2/s) hdA/dti (mm2/s)
Fig. 6. Growth rate of small ®ve-sided grain shown in Fig. 5. The line indicates ideal behavior as predicted by equation (1).
1 2 2 2 2
250 31 273 288 361
4 5 5 5 5
ÿ3
ÿ9.0 10 ÿ5.4 10ÿ3 ÿ5.4 10ÿ3 ÿ4.4 10ÿ3 ÿ5.2 10ÿ3
ÿ3
ÿ5.1 10 ÿ2.7 10ÿ3 ÿ2.7 10ÿ3 ÿ2.7 10ÿ3 ÿ2.7 10ÿ3
z ÿ1.54 ÿ2.48 ÿ2.58 ÿ1.65 ÿ2.3
PALMER et al.: DIMENSIONAL ASPECTS IN THIN FILM GRAIN GROWTH
6401
Fig. 8. Growth rate of neighbor switching grain in Fig. 7. The line indicates ideal behavior as predicted by equation (1). Fig. 7. Neighbor switching sequence of the lower ®vesided grain: * is the grain of interest. Table 2. Neighbor switching grains Run 1 2 2 2
Grain
N
dA/dt (mm2/s)
hdA/dti (mm2/s)
z
216 462 498 506
5 5 5 6
ÿ1.07 10ÿ3 ÿ1.02 10ÿ3 ÿ0.94 10ÿ3 ÿ2.06 10ÿ3
ÿ2.86 10ÿ3 ÿ2.70 10ÿ3 ÿ2.70 10ÿ3 0.00 10ÿ3
1.63 1.58 1.66 ÿ1.55
motion is discontinuous. Mullins has stated that his analysis will not describe discontinuous boundary motion. Because a discontinuous boundary motion involves an apparent discontinuity in the motion of a boundary, there is no reason to expect Mullins' analysis to describe accurately the behavior of these grains. They should be considered anomalous. Our data acquisition algorithm assigns each grain a new grain number only when a topological transformation takes place, and hence this eect is initially undetected.
Table 3. Outermost grains as identi®ed by normal distribution Grains showing: Low-angle boundary Small size Neighbor switching Adjacent topological event
Run 1
Run 2
4 1 1 9
1 4 3 12
There is no typical manner in which these grains deviate from the predictions of equation (1). This is not surprising as it appears that the boundaries ``suddenly jump''. As shown in Table 3 the discontinuous boundary motion accounts for most of the anomalous grains. This is not surprising, given the fact that the most likely topological transformations are the annihilations of four- or ®ve-sided grains. This suggests that discontinuities in boundary motion, as cited by Mullins, should occur most frequently without a change in topological class by a neighboring grain. Because the boundary motion is to some extent discontinuous, there is no clear pattern to the anomalous behavior.
Fig. 9. Grain growth of seven-sided grain with adjacent topological events.
6402
PALMER et al.: DIMENSIONAL ASPECTS IN THIN FILM GRAIN GROWTH Table 4. Summary of growth rates (mm2/s)
Top. class
4 5 6 7 8 KN
All grains Fig. 1
MVN grains only Fig. 10
Run 1
Run 2
Run 1
Run 2
ÿ5.1 2 2.5 10ÿ3 ÿ2.8 2 1.1 10ÿ3 0.1 21.0 10ÿ3 5.1 22.2 10ÿ3 6.3 22.2 10ÿ3 3.0 20.1 10ÿ3
ÿ3.7 2 1.7 10ÿ3 ÿ2.7 2 1.1 10ÿ3 0.0 21.2 10ÿ3 3.7 22.1 10ÿ3 5.9 22.9 10ÿ3 2.5 20.1 10ÿ3
ÿ4.6 2 1.1 10ÿ3 ÿ2.7 2 0.7 10ÿ3 0.3 2 0.9 10ÿ3 4.6 2 1.4 10ÿ3 5.5 2 1.6 10ÿ3 2.9 2 0.1 10ÿ3
ÿ3.8 21.7 10ÿ3 ÿ2.5 20.6 10ÿ3 0.0 2 0.8 10ÿ3 3.3 2 1.2 10ÿ3 5.2 2 2.8 10ÿ3 2.5 2 0.1 10ÿ3
Fig. 10. Growth rate of Mullins±Von Neumann grains only. The lines indicate ideal behavior as predicted by equation (1).
3.5. Ideal (Mullins±Von Neumann) grains As a result of this re-examination, 34 of the 800 grains shown in Fig. 1 were eliminated as anomalous. Since these grains do not satisfy the requirements of the Mullins analysis, they should not be used to assess its validity. A revised plot of growth rate vs topological class is shown in Fig. 10. The agreement is improved by eliminating merely 5% of the grains. Table 4 shows the eect of the average growth rates and the spread within each topological class, as well as KN. The values of the average growth rates are unchanged, but their standard deviations are reduced. 4. CONCLUSION
Succinonitrile is shown to be a good system for modeling grain growth in thin ®lms, allowing us to examine the behavior of grains in a thin ®lm which
are aected by the ®nite ®lm thickness. In any real system there will be grains that do not satisfy the conditions set forth by Mullins, and these grains should not be considered when evaluating the Mullins±Von Neumann rate constant, or when assessing the validity of equation (1). This work also indicates that there are only a small fraction of such grains within the grain population, and that their eect on the overall behavior of the ®lm is negligible. The anomalous grains either have low-angle boundaries or are eected by the ®nite thickness of the ®lm. Neighbor switching, small grain size, and adjacent topological events all cause departures from ideal behavior, due to ®nite ®lm thickness. Boundary rearrangements are associated with grain annihilation. While the topological class of the grain may remain unchanged, the grain changes, as the individual boundaries change due to this rearrangement. The overwhelming majority of the grains in a succinonitrile thin ®lm appear to be close to ideal. There appears to be a slight variation in the growth rates. This could be caused by the system departing from local equilibrium, or from a natural variation in the grain boundary energy. REFERENCES 1. Mullins, W. W., J. appl. Phys., 1956, 27, 900. 2. Von Neumann, J., Metal Interfaces. ASM 1952, p. 108. 3. Palmer, M. A., Fradkov, V. E., Glicksman, M. E. and Rajan, K., Scripta metall. mater., 1994, 30, 633. 4. Palmer, M. A., Fradkov, V. E., Glicksman, M. E., Rajan, K. and Nordberg, J., Metall. Mater. Trans. A, 1995, 26A, 1061. 5. Palmer, M. A., Glicksman, M. E. and Rajan, K., Scripta mater., submitted. 6. Frost, H. J., Simulation of Microstructural Evolution in Polycrystalline Thin Films. MRS, 1990, p. 81.
Acta mater. Vol. 46, No. 18, pp. 6397±6402, 1998 Acta Metallurgica Inc. Published by Elsevier Science Ltd Printed in Great Britain 1359-6454/98 $19.00 + 0.00
KINETIC STUDIES OF DIMENSIONAL ASPECTS IN THIN FILM GRAIN GROWTH M. A. PALMER{a, M. E. GLICKSMANb and K. RAJANb a
Virginia Commonwealth University, Department of Mechanical Engineering, Richmond, VA 232843015, U.S.A., bRensselaer Polytechnic Institute, Troy, NY, U.S.A. (Received 16 March 1998; accepted 1 September 1998)
AbstractÐIt has been shown that grain growth in succinonitrile thin ®lms is accurately described by the Mullins±Von Neumann grain growth law. However, there is a small fraction of grains showing signi®cant departures from ideal behavior (Palmer et al., Scripta metall. mater., 1994, 30, 633; Palmer et al., Metall. Mater. Trans. A, 1995, 26A, 1061). These grains are examined to see if they show departures from ideal behavior (the Uniform Boundary Model). It has been found that the vast majority of this small fraction of grains shows one of four distinct characteristics: they possess low-angle boundaries, they are small, they possess short boundaries, or they exhibit discontinuous boundary motion. With the exception of the lowangle boundaries all eects are due to the three-dimensional nature of the ®lm. These characteristics have been correlated with grain growth behavior: small grains tend to shrink faster than ideal grains, grains possessing short boundaries tend to stagnate, those undergoing discontinuous boundary motion show irregular behavior. Through these experiments real ®lms have been examined in the context of two-dimensional models. Acta Metallurgica Inc. Published by Elsevier Science Ltd
1. INTRODUCTION
A thin ®lm of succinonitrile as a collection of grains tends to follow the grain growth law advanced by Mullins [1] based on earlier work by Von Neumann [2]. This law predicts that the areal rate of change of a two-dimensional grain is given by dA p Mg N ÿ 6 dt 3
1
where M is the mobility of the boundary, g the grain boundary energy, and N the topological class (number of sides or vertices) of the grain. Figure 1 shows growth rate as a function of topological class. One will note the large spread in the growth rates shown in Fig. 1. In these experiments [3, 4] extremely pure succinonitrile was used eliminating the eect of trace impurities altering the motion of grain boundaries. The temperature was controlled to 21 mK which meant that temperature variations did not alter the grain growth. We have no reason to believe that the surface energy, and mobility are not uniform. However, our experiments show that there is a small fraction of grains which do not follow the Mullins±Von Neumann grain growth law equation (1). This further implies that there are individual grains which are not described by the Uniform Boundary Model. This model will only be applicable if: all grain boundary energies and mobilities are equal, the grains are two-dimensional, and the grain boundary motion is continuous. {To whom all correspondence should be addressed.
Not every grain can be described by the above three conditions, and those that cannot are not subject to Mullins analysis. There are a small number of grains possessing low-energy grain boundaries, which by de®nition, would not be expected to follow equation (1). Grains which change their topological class through neighbor switching, typically have one short boundary [3]. There are also small grains, those with diameters corresponding to the ®lm thickness. These grains cannot be considered two-dimensional. Thus, neither small grains nor grains which undergo neighbor switching, should follow equation (1). Mullins did not consider the annihilations of four- and ®ve-sided grains. In these cases, there are neighboring grains which do not undergo a topological transformation. However, as shown there are de®nite boundary changes as two grains which bordered the annihilated grain, now border each other [4]. Since the boundaries of these grains undergo a sudden change as a result of an adjacent topological event, they might not be expected to follow equation (1) throughout their lifetime. Mullins has stated that grains which experience a sudden change in the number of sides due to the disappearance of a neighboring grain are not expected to be governed by these equations [1]. 2. EXTREME VALUE ANALYSIS
To identify those grains showing unusual departures from ideal behavior we made use of extreme value analysis. This was done by analyzing the distributions of growth rates within each topological
6397
6398
PALMER et al.: DIMENSIONAL ASPECTS IN THIN FILM GRAIN GROWTH
A disproportionate number of gains was found to lie at the extreme of the distribution, and were classi®ed as extreme, warranting further investigation. The original data acquisition algorithm, used to determine the growth rate, was modi®ed so that the vertex positions of each grain were recorded in addition to the other data (size, perimeter, and topological class). A new algorithm mapped each grain on the screen, enabling further examination of the speci®c nature of each grain's growth. This procedure allowed us to examine grains classi®ed as ``extreme'' and determine if they showed anomalous behavior.
Fig. 1. Growth rate of grains in succinonitrile thin ®lms [3]. The lines indicate ideal behavior as predicted by equation (1).
class. If the growth rates, within a given topological class are normally distributed, then they can be described by 1 2 f z p eÿz =2 2p
2
where z is de®ned as z
x ÿ x : s
We sorted the previously collected [3] rate constant data for the two runs and determined the average growth rate, and the standard deviation of the growth rate, within each topological class. From this, z was de®ned as dA dA ÿ dt dt N z 3 sN where the subscript N denotes the speci®c topological class of the grain, dA=dt the growth rate of an individual grain, hdA=dtiN the mean growth rate for the topological class N, and sN the standard deviation of the growth rates within the topological class N.
3. RESULTS AND DISCUSSION
Our examination of anomalous grains included two aspects. Forty of the 800 grains examined were classi®ed as extreme grains, those showing (34 of the 40) one of four behaviors were classi®ed as anomalous. These were grains which possessed lowangle boundaries, small grains, grains with short boundaries, and grains exhibiting discontinuous boundary motion. Our results show that discontinuous boundary motion is the most common cause for a grain to be considered anomalous. 3.1. Low-angle boundaries One of the assumptions of the Mullins±Von Neumann analysis is that all boundaries are of uniform energy. In a real polycrystalline aggregate, there exists a small number of low-angle boundaries. Sometimes these boundaries are polygonalized dislocation arrays, and typically they have dihedral angles less than 158. In our experiment, the grain boundaries are decorated by thermal grooves. The depth of these grooves increases with increasing grain boundary energy. A low-angle boundary will therefore appear with much less optical contrast than does a ``normal'' grain boundary with average boundary energy. Five of the 34 anomalous grains possessed at least one low-angle boundary and were observed between the two runs. An example of grain growth in the presence of a low-angle boundary is shown in
Fig. 2. Grain growth with a low-angle boundary.
PALMER et al.: DIMENSIONAL ASPECTS IN THIN FILM GRAIN GROWTH
Fig. 2. The low-angle boundary disappears as grain growth progresses, and grains 42 and 43 essentially combine and form a single grain, which although not shown in Fig. 2, occurs between 100 and 120 s. Although the low-angle boundary is nearly invisible, its eect is evident on the other boundaries. In the ®gure, this is indicated by the arrows at 0 and 60 s. It appears as if the low-angle boundary retards the motion of the boundaries which it intersects. Note the low-angle boundary causes a local deviation of the curvature of the boundary. At 60 s, the in¯uence of this ``pulling'' becomes evident. The areas of grains 42 and 43 are plotted as a function of time in Figs 3 and 4, respectively. In each of these ®gures, the ideal growth rate, i.e. that which corresponds to equation (1), is plotted as a dashed line passing through the data at 0 s. The growth rate of both grains appears to be retarded by the low-angle boundary. If one considers the combined grain as a single grain, it would be a seven-sided grain for 80 s, and would transform into a six-sided grain as a neighbor is annihilated. The combined growth rate does not follow ideal behavior for either case. One can consider the low-angle boundary as a grain boundary dividing two grains or one could consider it as a ``¯aw'' in the combined grain. In either case the boundaries are clearly not of uniform energy and mobility and grains with low-angle boundaries are anomalous. That is, they tend not to conform to the assumptions of the Uniform Boundary Model and they show signi®cant departures from equation (1). 3.2. Small grains In Section 1 of this paper it was shown that the ®nite thickness of the ®lm alters the annihilation mechanisms responsible for grain growth, that is they do not occur exactly as predicted by the two-
Fig. 3. Area of grain 42 in Fig. 2. Grain 42 has seven sides. The line indicates ideal behavior as predicted by equation (1).
6399
Fig. 4. Area of grain 43 in Fig. 2. Grain 43 has four sides. The line indicates ideal behavior as predicted by equation (1).
dimensional theories [5]. Most of the grains characterized as anomalous were aected by the ®nite ®lm thickness. Grains de®ned as small were grains where the mean diameter was on the same length scale as the ®lm thickness. There are three types of anomalous behavior which can be attributed to ®nite ®lm thickness: accelerated shrinkage of small grains, stagnation of grains undergoing neighbor switching, and grains exhibiting discontinuous boundary motion. Mullins' analysis assumes that grains are twodimensional. Grains with three, four, or ®ve sides are observed to be annihilated in the grain growth process. As this happens, the grains shrink to essentially zero size, and they can no longer be assumed to be two-dimensional. An example of such a grain is shown in Fig. 5. This grain is small compared to its neighbors, and shrinks to a small area. This grain shrinks quickly. The area vs time curve for this grain is shown in Fig. 6. The slope of the dashed line represents the ideal growth rate of a ®ve-sided grain, were equation (1) to be followed. We observed that small grains tend to shrink faster than ideal grains. In this experiment, the ®ve grains that were identi®ed as small had growth rates less than predicted by equation (1), as shown in Table 1. In Table 1 hdA=dti refers to the ideal growth rate for the topological class. Similar results had been predicted earlier. According to Frost [5], normal grain growth stagnates when the average diameter of the grain is 2±3 times the ®lm thickness due to pinning at the surface. Our results show the converse behavior. Frost was referring to the retardation of grain growth and shrinkage once the grains exceeded a certain size. As the grains become three-dimensional they are no longer aected by this pinning, and the shrinkage rate is accelerated. Our earlier analyses [6] indicate that a grain need not be of uniform topological class throughout the
6400
PALMER et al.: DIMENSIONAL ASPECTS IN THIN FILM GRAIN GROWTH
Fig. 5. Accelerated shrinkage of small ®ve-sided grain.
thickness of the ®lm. For example, consider the annihilation of a three-dimensional ®ve-sided grain; this occurs in seven steps. As our experiments only observe the top surface of the grain, there is no way of telling if the grain is in one of these intermediate steps. Clearly, when a grain enters these intermediate steps it behaves anomalously, and should shrink more rapidly than would be predicted by equation (1). 3.3. Neighbor switching Neighbor switching is one of the two basic unit operations responsible for grain growth. When such an event occurs two grains lose a boundary, while two neighboring grains gain a boundary. We found four grains which reduced their topological class through neighbor switching. Such a sequence is shown in Fig. 7. The ®ve-sided grain indicated by an asterisk (*), appears to shrink normally for about 160 s. Then there is an 80 s period where the grain area barely changes. Finally, a new four-sided grain can be seen after 320 s. As shown earlier [6], this stagnation is the result of the neighbor switching occurring ®rst on one face of the ®lm, and then on the other. The area vs time plot for the neighbor switching sequence is shown in Fig. 8. The 80 s stagnation period is clearly evident. The dashed line
represents the ideal case where equation (1) is followed. Up until the stagnation period, the growth rate is close to ideal. Note that during the stagnation period the boundary between the two switching grains becomes extremely short. This short boundary moves in response to the twisting forces imposed by the new boundary on the other face of the ®lm. If this shrinkage scenario is indeed the case, then Mullins' analysis is not expected to hold. The stagnation of one boundary means that the motion of the other boundaries determine the growth rate of the grain. Consider that dA=dt for a six-sided grain as shown in Table 2 is close to the average or ideal growth rate, hdA=dti, for a ®vesided grain. The eect is even more pronounced for the ®ve-sided grains. 3.4. Discontinuous boundary motion The vast majority of those grains identi®ed as anomalous retained their topological class despite an adjacent topological event, and therefore demonstrated discontinuous boundary motion. An example of such a grain is shown in Fig. 9. In this case a seven-sided grain, indicated by an asterisk (*), remains seven-sided, despite the fact that three neighboring grains 1, 2, and 3, are annihilated. Consider Fig. 9 in the context of the sequence of events describing the annihilation of a three-dimensional ®ve-sided grain [6]. Three ®ve-sided grains were annihilated in 40 s. As explained in Section 1 of this paper, ®ve-sided grain annihilation takes place in several steps. The boundaries bordering the * grain at 40 s, therefore are dierent from those which border the grain at 0 s. In fact these new boundaries are probably combinations of the boundaries from the top face of the ®lm at 0 s, and the boundaries which extended through the thickness of the ®lm. This suggests that the boundary Table 1. Behavior of small grainsÐgrowth rates in mm2/s Run Grain Top. class dA/dt (mm2/s) hdA/dti (mm2/s)
Fig. 6. Growth rate of small ®ve-sided grain shown in Fig. 5. The line indicates ideal behavior as predicted by equation (1).
1 2 2 2 2
250 31 273 288 361
4 5 5 5 5
ÿ3
ÿ9.0 10 ÿ5.4 10ÿ3 ÿ5.4 10ÿ3 ÿ4.4 10ÿ3 ÿ5.2 10ÿ3
ÿ3
ÿ5.1 10 ÿ2.7 10ÿ3 ÿ2.7 10ÿ3 ÿ2.7 10ÿ3 ÿ2.7 10ÿ3
z ÿ1.54 ÿ2.48 ÿ2.58 ÿ1.65 ÿ2.3
PALMER et al.: DIMENSIONAL ASPECTS IN THIN FILM GRAIN GROWTH
6401
Fig. 8. Growth rate of neighbor switching grain in Fig. 7. The line indicates ideal behavior as predicted by equation (1). Fig. 7. Neighbor switching sequence of the lower ®vesided grain: * is the grain of interest. Table 2. Neighbor switching grains Run 1 2 2 2
Grain
N
dA/dt (mm2/s)
hdA/dti (mm2/s)
z
216 462 498 506
5 5 5 6
ÿ1.07 10ÿ3 ÿ1.02 10ÿ3 ÿ0.94 10ÿ3 ÿ2.06 10ÿ3
ÿ2.86 10ÿ3 ÿ2.70 10ÿ3 ÿ2.70 10ÿ3 0.00 10ÿ3
1.63 1.58 1.66 ÿ1.55
motion is discontinuous. Mullins has stated that his analysis will not describe discontinuous boundary motion. Because a discontinuous boundary motion involves an apparent discontinuity in the motion of a boundary, there is no reason to expect Mullins' analysis to describe accurately the behavior of these grains. They should be considered anomalous. Our data acquisition algorithm assigns each grain a new grain number only when a topological transformation takes place, and hence this eect is initially undetected.
Table 3. Outermost grains as identi®ed by normal distribution Grains showing: Low-angle boundary Small size Neighbor switching Adjacent topological event
Run 1
Run 2
4 1 1 9
1 4 3 12
There is no typical manner in which these grains deviate from the predictions of equation (1). This is not surprising as it appears that the boundaries ``suddenly jump''. As shown in Table 3 the discontinuous boundary motion accounts for most of the anomalous grains. This is not surprising, given the fact that the most likely topological transformations are the annihilations of four- or ®ve-sided grains. This suggests that discontinuities in boundary motion, as cited by Mullins, should occur most frequently without a change in topological class by a neighboring grain. Because the boundary motion is to some extent discontinuous, there is no clear pattern to the anomalous behavior.
Fig. 9. Grain growth of seven-sided grain with adjacent topological events.
6402
PALMER et al.: DIMENSIONAL ASPECTS IN THIN FILM GRAIN GROWTH Table 4. Summary of growth rates (mm2/s)
Top. class
4 5 6 7 8 KN
All grains Fig. 1
MVN grains only Fig. 10
Run 1
Run 2
Run 1
Run 2
ÿ5.1 2 2.5 10ÿ3 ÿ2.8 2 1.1 10ÿ3 0.1 21.0 10ÿ3 5.1 22.2 10ÿ3 6.3 22.2 10ÿ3 3.0 20.1 10ÿ3
ÿ3.7 2 1.7 10ÿ3 ÿ2.7 2 1.1 10ÿ3 0.0 21.2 10ÿ3 3.7 22.1 10ÿ3 5.9 22.9 10ÿ3 2.5 20.1 10ÿ3
ÿ4.6 2 1.1 10ÿ3 ÿ2.7 2 0.7 10ÿ3 0.3 2 0.9 10ÿ3 4.6 2 1.4 10ÿ3 5.5 2 1.6 10ÿ3 2.9 2 0.1 10ÿ3
ÿ3.8 21.7 10ÿ3 ÿ2.5 20.6 10ÿ3 0.0 2 0.8 10ÿ3 3.3 2 1.2 10ÿ3 5.2 2 2.8 10ÿ3 2.5 2 0.1 10ÿ3
Fig. 10. Growth rate of Mullins±Von Neumann grains only. The lines indicate ideal behavior as predicted by equation (1).
3.5. Ideal (Mullins±Von Neumann) grains As a result of this re-examination, 34 of the 800 grains shown in Fig. 1 were eliminated as anomalous. Since these grains do not satisfy the requirements of the Mullins analysis, they should not be used to assess its validity. A revised plot of growth rate vs topological class is shown in Fig. 10. The agreement is improved by eliminating merely 5% of the grains. Table 4 shows the eect of the average growth rates and the spread within each topological class, as well as KN. The values of the average growth rates are unchanged, but their standard deviations are reduced. 4. CONCLUSION
Succinonitrile is shown to be a good system for modeling grain growth in thin ®lms, allowing us to examine the behavior of grains in a thin ®lm which
are aected by the ®nite ®lm thickness. In any real system there will be grains that do not satisfy the conditions set forth by Mullins, and these grains should not be considered when evaluating the Mullins±Von Neumann rate constant, or when assessing the validity of equation (1). This work also indicates that there are only a small fraction of such grains within the grain population, and that their eect on the overall behavior of the ®lm is negligible. The anomalous grains either have low-angle boundaries or are eected by the ®nite thickness of the ®lm. Neighbor switching, small grain size, and adjacent topological events all cause departures from ideal behavior, due to ®nite ®lm thickness. Boundary rearrangements are associated with grain annihilation. While the topological class of the grain may remain unchanged, the grain changes, as the individual boundaries change due to this rearrangement. The overwhelming majority of the grains in a succinonitrile thin ®lm appear to be close to ideal. There appears to be a slight variation in the growth rates. This could be caused by the system departing from local equilibrium, or from a natural variation in the grain boundary energy. REFERENCES 1. Mullins, W. W., J. appl. Phys., 1956, 27, 900. 2. Von Neumann, J., Metal Interfaces. ASM 1952, p. 108. 3. Palmer, M. A., Fradkov, V. E., Glicksman, M. E. and Rajan, K., Scripta metall. mater., 1994, 30, 633. 4. Palmer, M. A., Fradkov, V. E., Glicksman, M. E., Rajan, K. and Nordberg, J., Metall. Mater. Trans. A, 1995, 26A, 1061. 5. Palmer, M. A., Glicksman, M. E. and Rajan, K., Scripta mater., submitted. 6. Frost, H. J., Simulation of Microstructural Evolution in Polycrystalline Thin Films. MRS, 1990, p. 81.