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MICROCOPY RESOLUTION TEST CHART
NATIONAL BUREAU OF STANDARDS - 1963
oroord-1922
Conf. 65/031-2
NOV 18 BSS
ORNI - AIC - OFFICIAL
RESLEASID FOR ANNOUNCEMENT
I NUCLEAR SCIENCE ABSTRACTS
1L - AEC - OFFICIAL

GRAIN BOUNDARY MIGRATION
by
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such employee or contractor of the Commission, or employee of such contractor prepares.
ployee or contractor of the Commission, or employee of such contractor, to the extent that
As usod in the above, "person acting on boball of the Commission" includes any on-
use of any information, apparitus, method, or process disclosed in this report.
B. Assumos any ilabilities with respoct to the use of, or for damages resulting from the
of any information, apparatus, method, or process disclosed in this report may not infringe
racy, completeness, or urefulness of the information contained in this report, or that the we
A, Makos any warranty or representatioa, expressed or implod, with respect to the accu-
States, por the Commission, nor any person acting on behalf of the Commission:
This report was preparad as an account of Goverament sponsored work. Neither the United
LEGAL NOTICE
Most metallic materials in their ultimate use form are polycrystalline.
The presence of the crystallite - or grain - boundaries has profound ef-
fects on the properties of the materials, and boundary migration may alter
these properties significantly. In attempts to understand these effects and
the underlying nature of grain boundaries, boundary migration studies have
played an increasingly important part. It is the purpose of this paper to
delineate the main results of these migration studies. This will be done in
two parts; in Part I, the effect of orientation and boundary structure factors
and their interdependence with impurity effects will be considered. In
Part II, investigations aimed at the quantitative understanding of the drastic
effect dissolved impurities have on boundary migration will be examined;
the results of these investigations will be compared with the predictions of
migration rate theories designed to account quantifatively for the impurity
effect.
In view of the facts that there have been a number of excellent re- .
views on various aspects of grain aggregates and grain boundaries (1-8) and
that there are several new,ones in the present volume, no attempt will be made to
cover the literature exhaustively, but rather only those works considered to
have the most direct bearing on the subjects under discussion will be referenced.
.
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7
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ORNL-AEC-OFFICIAL
* Professor, Department of Metallurgical Engineering, Illinois Institute of
Technologyä Chicago, Illinois.
# Metallurgist at the Oak Ridge National Laboratory, Oak Ridge,
Tennessee, which is operated by Union Carbide Coiporation for
the U. S. Atomic Energy Commission.
ORNI - AEC - OFFICIAL
Part I. Orientation and Structure
Low-angle and high-angle boundaries. A grain boundary may be
defined as a layer of distorted material created by the atomic mismatch be-
tween two differently oriented, adjacent crystals of the same phase. It is
customary to categorize grain boundaries as either low-angle or high-angle
boundaries. The dividing line is usually somewhat indefinitely placed at a
relative grain mismatch angle below which the boundary is in many respects
equivalent to a wall of discreet dislocations and above which the disloca-
tions are so close together as to lose their individual identities. This
generally means at mismatch angles of a few degrees. Thus, it has been
demonstrated (9,10) that low-angle - or sub-grain — boundaries can be
caused to migrate, presumably by dislocation glide, under the impetus of
uniaxial stress, reversing migration direction when the direction of the stress
is reversed, Figure 1. The migration rates of such boundaries are found to
be dependent nd only on the stress level but also on the temperature and the
angle of misfit between the bounding grains. In particular, the rates in-
crease with temperature and decrease with increasing misfit angle (11). The
latter effect, shown in Figure 2, is attributed to the mutual interference of
the dislocations when they are too close together.
In contrast to the relative unanimity of opinion as to the basic nature
of low-angle boundaries, there has been no single predominating concept
concerning the structure of high-angle boundaries; it is with this more con-
troversial field of high-angle boundaries that the remainder of this paper is
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concerned,
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Little direct evidence has as yet been obtained as to the detailed
nature of high-angle grain boundary structures. This is because such evidence
requires "seeing" individual atoms, and even the powerful electron micro-
scope has not been able to do that. The recent invention by E. W. Muller
of the field-ion microscope (12), however, with its capability of resolutions
down to 2-3 A along with readily obtained magnifications of 100 or more,
has now brought individual atoms within view. Investigations on grain
boundaries with the instrument are still fragmentary, but it is not unduly
optimistic to expect that the field-ion microscope will in the next few years
do for grain boundary structure ( and for other atomic-scale phenomena as
well) what the electron microscope has done for the understanding of dis-
locations in the past decade or so. An example of a grain boundary in
tungsten taken from the field-ion microscopy work of Brenner (13) is given
in Figure 3. It should be noted that each bright spot in this photograph is
the image of a single atom at a magnification of about 1,600,000 X!
Grain boundary models. Present day views of grain boundary structure
date from abc ut 1929 when Hargreaves and Hills (14) first visualized the
boundary as a region one to two atoms wide in which the atoms of the adjoin-
ing grains assume compromise positions between the two lattices. The models
which have since been proposed in attempts to describe in some detail the
references 1, 5, 6 and 7). Those which have been given the most attention
are:
(a) Mott's (15) "island" model. The boundary is considered to be
..----..
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made up of islands of good fit between the adjoining lattices surrounded by
regions of bad fit. Migration is viewed as a process involving the disordering,
or "melting," of a group of .n atoms from the matrix grain into the bad fit
boundary regions and the "crystallization" of these atoms onto the lattice of
the growing grain.
-na (b) Ke's (16) viscous boundary model. The boundary is looked
upon as a viscous region of disorder which can absorb disordered groups of
atoms - presumably resulting from relaxed vacancies -- from a grain,
thereby lowering the boundary viscosity and facilitating its migration. Im-
purity atoms are suggested to have an effect opposi te to that of vacancies.
(c) The dislocation model. The boundary is considered to be made up
of an array of closely spaced dislocations similar in many respects to low-
angle boundaries; the mod. has been proposed in various forms by many
authors, e.g., references 17, 18, 19 and 20.
As pointed out by Lucke and Stuwe (6), it has been difficult to make
choices between these models. In essence, the reason for this is that both the
theories and the available experimental evidence are still too limited to give
the kind of comprehensive description of grain boundaries necessary to ex-
plain all their major attributes. With respect to grain boundary migration,
these models have had some modest success. It is reasonably well establish-
ed that grain boundary diffusion rates are higher for boundaries between
grains with large misorientations (with the exception of twin orientations)
than for those where the misorientations are less than 15-20° (see, for example,
references 2 and 3). On the assumption that boundary migration is controlled
by boundary diffusion the above models then predict (21) that boundary
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migration should be more rapid for high-angle boundaries with large mis-
orientations than for those with smaller misorientations. There is much in-
direct evidence that this is true. However, recent measurements of the
migration rates of single boundaries have produced direct evidence that the
orientation dependence of high-angle boundary migration is a more specif-
ically selective effect -- i.e., that within the category of high-angle
boundaries with large misorientations there are certain special orientations
which under some conditions have much higher mobilities than others. The
three models for boundary structure listed above cannot account for such
specific selectivity. These problems are discussed in detail in the next two
sections.
Orientation dependence of migration. It has been known for many
years (7,8) that when deformed polycrystalline materials are annealed,
marked preferred orientation textures may be developed. These textures may
be similar to the deformation textures from which they form or may be quite
different. On further annealing, secondary recrystallization may take place,
and again marked, but different, textures can develop. It has been argued
(22,23), on the one hand, that oriented nucleation - i.e., the predominant
presence of growth nuclei in only certain orientation relations with the
matrix — plays a significant role in forming the annealing textures; on the
other hand, it has been maintained (21) that oriented growth -- high growth
rates associated with nuclei of certain orientations.: elative to the matrix
alone is sufficient to account for the annealing textures. This controversy
is, perhaps, not yet unambiguously resolved; be that as it may, however,
during the past decade studies of the growth of new grains in single crystals
have demonstrated quite convincingly that the orientation dependence of
growth is both real and marked. For example, Kohara, Parthasarathi and
Beck (24) found that in a single crystal of 99.994% purity aluminum cold roll-
ed to 80% reduction in area, rubbing with a fine abrasive on one side of the
sample produced many fine, randomly oriented grains on this side after short
subsequent annealing, but that on further anneciing only those grains having
orientation relationships with the matrix corresponding to approximately 40°
'rotations around a common (111) pole survived to grow completely across the
rolled crystal. A number of other studies have been made on single crystal
samples provided with a growth-supporting driving force either by deformation
or by a striation (or mosaic) structure resulting from solidification. Usually
the driving forces and annealing temperatures were selected in such a way that
spontaneous nucleation of new grains throughout the crystal was avoided.
Heavy local deformation introduced at one end of the crystal would then cause
new grains to nucleate "artificially" in this local region during annealing.
Most often, one or two of the grains so nucleared in a given crystal were
found to crowd out the others early in the competition to consume the matrix
crystal. One, or both, of two types of observation were then made: (1) As
in the work of Kohara, Parthasarathi and Beck (24), the orientation relation-
ships between the dominant new grains and the inatrix were determined and
expressed by giving for each dominant grain the rotation angle and the crystal-
lographic direction common to both new grain and matrix around which the
minimum rotation was necessary for the lattices of the two to coincide; (2)
The migration rates of the boundaries between the dominant grains and the
matrix were measured. Most frequently only type (1) measurements were made
(as examples, see references 24-33, 38). Both orientation and migration rate
measurements (i.e., types (1) and (2)) have been made only by Graham and
Cahn (34), by Aust and Rutter (35-37) and by Rath and Gordon (39). In
some cases (e.g., 31 and 40) orientation relation measurements were made on
new grains formed by spontaneous nucleation, that is, without the help of
localized heavy deformation. Finally, in two instances, (Graham and Cahn
(34) and Aust and Rutter (41)) migration rate experiments were carried out on
boundaries in bicrystals prepared with predetermined misorientations.
The single crystal measurements in which type (1) — orientation --
measurements were made established quite clearly that there is an ubiquitous
and strong tendency for grains with certain relative orientations with respect
to the matrix crystal to be preferred during annealing just as there is for
polycrystalline material. Further, many of these investigations (e.g., Kohara,
Parthasarathi and Beck (24), Liebmann, Lucke and Masing (25), Yoshida,
Liebmann and Lucke (26), Lucke and Ibe (33) and Aust and Rutter (29))
succeeded in proving that growth selection alone can produce these preferred
orientations*. For face-centered cubic crystals, 30° and 40° rotations about
(111), and for body-centered cubic crystals 25-50° rotations around (110),
have been most frequently, but not exclusively, found. An example from the
work of Yoshida, Liebmann and Lucke (26) is given in Figure 4. It is im-
portant to note that findings such as these do not mean that differences in
mobilities between various boundaries are necessarily responsible for growth
* Note that these experiments have not proved the general ab
oriented nucleation effect. Direct evidence on the latter, however, is
difficult to obtain if for no other reason than that it is difficult to define
the end of a nucleation event and the beginning of nucleus growth.
selection in single crystals or preferred orientation in general. As has been
demonstrated by Aust and Rutter (27-29) and by Lucke and I kve (33), twinning
during growth can result in the replacement of a migrating boundary by a
second boundary and a coherent twin boundary whenever thereby the total
interfacial energy is lowered -i.e., whenever the second boundary has a
specific interfacial energy lower than that of the first by an amount larger
than the specific interfacial energy of the coherent twin boundary. The
interfacial energy of grain boundaries varies with orientation factors and,
therefore, giains oriented so that they have low energy boundaries may well
Le favored by the interfacial energy factor during the nucleation or the growth
stage of any development of new grains. Thus, the observation of growth
selectivity based only upon orientation measurements means strictly only that
there is a strong orientation dependence of either -- or both — boundary
mobility or energy. It seems probable, however, that whenever there is a
strong mobility dependence this will be the dominating influence in growth
selection. This is because a new boundary produced to lower interfacial
energy would not be likely to survive unless its mobility is among the highest.
These orientation relation measurements on dominant grains growing
in single crystal matrices have been made largely on impure metals or on
alloys. In a few instances, zone-refined metals were used, however, and
these have made it evident that very high purity materials may behave dif-
ferently than less pure material.* In particular, it has been found that in

Here high purity is taken to mean carefully zone-refirad, for it has be-
come clear that dissolved impurities in amounts as small as 0.1 to 1 atomic
ppm can have very large effects on boundary migration behavior (for ex-
amples, see references 29, 30, 35-39 and 42-47).
zone-refined lead (35-37) and zone-refined copper (30) the preferred orienta-
tions do not appear, at least at high temperatures. Some of the Aust-Rutter
results on zone-refined lead are reproduced in Figure 5 (35). It is apparent
from this figure that the rotation axes which relate the new grains to the matrix
crystals are grouped around the (771) and (001) poles in lead containing
0.0005 to 0.004 wgt % tin (Figure 5b), but that in the purer material (Figure
5a) no such grouping takes place. Aust and Rutter have described the arrange-
ment of rotation axes in Figure 5a as random; however, careful reexamination
of the figure engenders the impression that the rotation axes in Figure 5a are
random except that they have avoided the regions around which rotation axes
have clustered in Figure 5b. This impression is strengthened by a study of all
the Aust-Rutter data for materials in which they did not find preferred group-
ings such as those in Figure 5b. A composite plot of this data is shown in
Figure 6; included are not only the data on zone-refined lead (references 29 and
35) but also data from the alloys of this lead containing silver or gold in amounts
up to 0.83 at. ppm (the entire range studied )(37). The arcs around the
principle poles of the stereogram in Figure 6 enclose regions each of which
should contain approximately 5% of the rotation oxes if these are distributed
at random (48). Since the total number of rotation axes shown is 62, each
region should contain 3-4 axes but in fact contains only one (1-1/2 near the
(110) pole), and none of these is closer than about 4-5° to the principle pole
of concern. It may well be, then, that under certain conditions in par-
ticular, extremely high purity or with the presence of specific impurities -
a reverse kind of preferred orientation takes place in which certain orientations
are avoided. Grahams and Cahn (34) noted a similar phenomenon in quite
impure aluminum (99.6% pure — principle impurities 0.19% Fe and 0.12%
Si). This point needs some further experimental attention.
In view of the indicated importance of impurity effects in the develop-
ment of selective growth, a number of investigations based on zone-refined
materials have been carried out in which actual migration rate measurements
were made on single boundaries for which the misorientations of the bounding
grains were also measured. In each of these the investigators found definite
evidence that boundary mobility is orientation dependent.* The extensive
studies of Aust and Rutter (35-37, 41) huve dominated this area; they studied
boundary migration in striated crystals of zone-refined lead and very dilute
alloys of tin, silver and gold in this lead. Again, localized deformation at one
end of the striated crystals was used to provide, upon subsequent annealing,
many striation-free nuclei for growth into the striated crystals. Aust and
Rutter, found, as indicated above, that the dominant new grains had nearly
random orientation relationships with the matrix when grown in gold and silver
alloys (up to 0.83 at. ppm) and in unalloyed zone-refined lead at high tem-
peratures (sele, for example, Figure 5a); the boundaries between new grains and
matrix were in this case designated "random" boundaries. In alloys containing
9-80 at. ppm tin and in the unalloyed zone-refined lead at low temperatures
(e.g., Figure 5b), however, the dominant grains had certain unique relationships
-- rotations about low-index axes such as (111). (110) and (100) -- with
that
It should be noted that occasional observations have been made/the boundary
mobility may depend on the orientation of the boundary itself. For example,
it was pointed out by Kohara, Parthasarathi and Beck (24) that for the same
relative grain orientations, tilt boundaries appeared to be more mobile than
twist boundaries. This dependence on boundary orientation has been given
relatively little attention, however; most often it has been taken into account
only by making efforts to hold the effect constant.
the matrix crystals; the associated boundaries in this case were called
"special" boundaries. These unique relationships were not found for tin con-
tents above 80 at. ppm. Measurements of the migration rates of the boundaries
revealed that the rates of special and random boundaries were approximately
the same at high temperatures, but those of random boundaries were signifi-
cantly the lower at low temperatures. Increasing tin content markedly lowered
the migration rates and raised the apparent activation energies of random
boundaries (as did Ag and Au) but had only minor effects on the migration
rates, and none on the activation energies, of special boundaries. The latter
activation energies were the lowest of all the boundaries studied. These
effects are illustrated in Figures 7 (35) and 8 (36). Aust and Rutter also
found (41) that for tilt boundaries produced at predetermined misorientations
in bicrystals of zone-refined lead, those between crystals having the special
orientations found before showed higher migration rates and lower activation
energies than other boundaries. Similar, but more limited, results have been
obtained in very dilute alloys of copp er in zone-refined aluminum by Frois
and Dimitrov (38) and by Rath and Gordon (39).
An explanation of these results in terms of the differences in impurity
segregation at boundaries of different structures has been advocated by Aust
and Rutter. They concluded, however, that none of the models for boundary
W
very specific orientation selectivity of growth mobility found experimentally.
Instead, they based their interpretation on a concept of boundary structure first
advanced by Kronberg and Wilson (49) in a study of secondary recrystallization
in copper
Table 1. Kronberg-Wilson Relationships (from Ref. 50)
Axio or
rotation
Reciprocal of
Least.
Angle of
rotation
A
Α
Σ
Φ
Α
Σ
Φ
Α
Σ
ω
-
100
-
- -
-
-
180°
120.7°
10.1°
-
-
-
100
36.0°
22.09
28.1°
100
210 3
210 6
210 , 7
210
210 16
211 .. 3
211 6
211 7
211 10
211 10
181.8°
180°
73.40
96.49,
: 48.26
180ο.
101.00
136.6°
630
78.7°
70.6°
38.0°.
50.5o
86.09
20.5o
------
310
310
310
310
310
311
311
311
311
311
180
116.4°
141.0°
. 76.7•
03 .
146.40
164.2°
67.1
7
11
136
10o
3
6
9
11
18
321
421
321
322
322
922
41ο
410
410
18
ο
13
176
Φ.
136
17%
io
152.7
107.0°
180».
152.70
107.0°
180
10
10
.
10
311
180°
50.7°
117.00
411
Ο
180°
11
600
11
17A
106
120.0"
14°
111
153.50
11
38.2°
27.8°
46.8°
-
221
221
221
221
221
6
9
135
176
143.1°
90
180°
112.6°
61.0°
Οι
320
320
320
320
320
411
140° 411
411
100.6° -
180° 331
121.0°
331
1.80 381
331
7
11
13o
176
105
5
06.7°
881
10
17»
196
density of
common points
13η
17η
17»
10η
13ο
ΔΩ!»
110.00
82.1
63.00
180°
Kronberg and Wilson noted that the textures obtained in their samples
- rotations about common (111) and (100) axes -- were such that the
lattices of the new and old grains had a common sublattice of a special kind.
The type of sublattice referred to — labeled a coincidence lattice — consists
in general of an array of points which lie on the lattices of both grains (see
Figure 9a) and has the same symmetry as, but greater unit cell than, the basic
lattices. Coincidence lattices can be found for any orientation relationship
between two adjacent grains, but they have special significance only when, as
in the Kronberg-Wilson case, the density of coinciding sites is relatively high.
The orientation relationships leading to these special coincidence lattices
are now frequeritly referred to as Kronberg-Wilson (K-W) orientations. A
listing of some of these, where the coincidence-site density is 1-in-19 or
greater, is given in Table ! (50). The boundaries between grains of K-W
relationship are boundaries of relatively good fit, since some of the coinci-
dence sites lie in the boundary; the degree of good fit increases with the
density of coincidence sites. According to Kronberg and Wilson, the presence
of coincidence atoms may endow boundaries with high mobility and thus account
for the development of the particular textures they found. This view can be
applied to much of the preferred orientation work in both polycrystalline and
single crystal material, for the dominant grains observed usually can be shown
to have orientation relationships with the matrix which approximate K-W rela-
tionships. For example, the 30° and 40° (111) rotations frequently observed
inf.c.c. materials can be identified with the 27.80 and 38.20 (111) K-W
rotations shown in Table I, and the 25-30° (110) relation often found for
b.c.c. materials with the 26.50 (110) K-W relation in Table 1. The Aust
and Rutter "special" orientations were largely rotations of approximately
36-42° and 23° about (111) and 230, 28° and 37° about (100); the Frois
and Dimitrov preferred grains were approximately 40° about (111); and those
of Rath and Gordon were 35-37º about (100) and 39-44º about (111). All
of these may be seen from Table 1 to be roughly K-W relationships.
Aust and Rutter argued that K-W -- or special boundaries are
good-fit boundaries and, therefore, absorb considerably less impurity than !
bad-fit--- random — boundaries. The tendency to absorb impurities is also
affected by temperature; at low temperatures impurities segregate at boundaries
strongly, but at high temperatures the impurity atmosphere "evaporates." Thus,
at high temperatures there was little difference between the migration rates of ::
their special and random boundaries since neither absorbed significant amounts
of impurity. At low temperatures, however, the random boundaries become
impurity laden and the migration rates therefore fell well below those of the
relatively impurity-free special boundaries. The absence of special boundaries
in the Ag and Au alloys (and in the alloys with Sn above 80 ppm) was difficult
to explain. Evidence was found, however, that both Ag and Au had an
especially strong tendency to segregate to boundaries in lead; it was, there-
fore, suggested that the absence of special boundaries in these cases might mean
that the segregation tendency was strong enough so that even good-fit boundaries
were highly impurity-laden.
This interpretation is undoubtedly a reasonable one. It seems appro-
priate at this point, however, to extend it somewhat and to make a modifica-
tion in the viewpoint. The modification is indicated by a reinterpretation of
some of the experimental evidence and by a conceptual difficulty.
Kronberg and Wilson (49) originally suggested that high-density co-
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.
14
incidence-site boundaries - i.e., good fit boundaries — are intrinsically
high mobility boundaries. They reasoned that this may be so because the
atom movements necessary for migration of a coincidence type boundary may
be only fractions of an atomic distance – at least near the coincidence atoms
– and may also be partly cooperative in nature, almost martensite-like with-
out the macroscopic distortion, thus, perhaps, having low activation energies.
This view has also been expressed by others (e.g., ref. 7). In this connection,
however, it should be noted that though some of the atom movements for the
migration of a coincidence boundary may be cooperative in a given atom plane,
the sense of the ring-like movements in alternate planes would be in opposing
directions (49); this could conceivably reduce the effectiveness of any co-
operative component in the atom movements. More important, however, the
concept of lower activation energies in relatively "tight" boundaries is dif-
ficult to understand — both intuitively and because it appears to be contrary
to the well established fact of lower activation energies for diffusion in grain
boundaries as compared with grain bulk; the latter is presumably due to the
"looseness", or porosity, of the grain boundary structure as compared to the
"tight" bulk structure.
It is not clear whether Aust and Rutter explicitly advocated the
Kronberg-Wilson idea that good-fit boundaries are intrinsically high mobility
boundaries; their interpretation, however, certainly appears to imply that such
boundaries unimpeded by impurity atmospheres have mobilities which are at
least as high as - if not higher than those of more random boundaries. It
seems more natural: to adopt the view advanced by Li (20, 51) that boundaries
between grains having ideal high coincidence-site density relations are in
.
fact low mobility boundaries because their porosity is low and that only when
the grains deviate appreciably from the ideal K-W relations is the boundary
mobility high. This view is supported by the fact that essentially all of the
preferred growth relationships which have been observed are at best only
approximately K-W relationships. The rotation axes and rotation amounts
found are usually several degrees from the ideal values. The very thorough
statistical investigation by Lucke and Ibe (33) established this lack of ideality
quite clearly. Whereas previous studies were each limited to relatively small
numbers of samples, Lucke and I be examined the orientation relationships be- ;
tween some: 1200 recrystallized grains and the deformed single crystal matrices
in aluminum (99.998%) and in alloys of this aluminum with iron, silicon and
manganese in the range of about .2 to .5 wgt %. They found that although
certain boundaries, roughly of the K-W type, were indeed favored in the growth
process, there was a considerable scatter from ideal K-W relations. The rota-
tion axes of the dominant grains deviated by 12° on the average from the ideal
coincidence-site positions. As a matter of fact, Lucke and Ibe, along with
Lucke and Stuwe (6), questioned whether with such large deviations the K-W
concept could still have physical significance. It is believed, however, that
this objection is largely obviated by a model put forward by Brandon, Ralph,
Ranganathan and Wald (50). They proposed that deviations of several de-
grees in coincidence relationship could be accommodated by supposing disloca-
tions to be present in the coincidence sublattice along the boundary plane.
This is illustrated schematically for a two-dimensional, simple square lattice
in Figure 96. It may be seen in the figure that the introduction of disloca-
tions not only rotates the two lattices with respect to one another, but also
.
..
brives
:
widens the boundary (to perhaps 3 or 4 atom diameters) and introduces consider-
able porosity. Some experimental corroboration for the idea that boundaries
may still exhibit appreciable atomic fit even when they deviate several degrees
from ideal K-W relations has recently been found with the field-ion micro-
scope (13, 52). These same studies have also indicated that near-ideal K-W
boundaries have widths of no more than 1-2 atom diameters, whereas more
random high-angle boundaries — which still exhibit some good-fit regions -
have widths of 3 or 4 atom diameters.
The idea that good-fit boundaries are low mobility boundaries derives
further experimental support from the fact that, the most frequently observed
high mobility bound-ry types — 30° and 40° about (111) and 25-30° about
(110) - are not the highest coincidence-site density boundaries (see
Table 1). In addition, a coherent twin boundary may be considered to be a
K-W boundary of the highest coincidence-site density; yet, coherent twin
boundaries are known to be virtually immobile (see, for example, Graham and
Cahn (34) and Rathenau (53)).
Hypotheses for orientation-impurity effect on boundary mobility. In
the light of these considerations it seems worthwhile to consider modification
of the Aust-Rutter view on the orientation dependence of grain boundary
migration. This modification will be based on the following postulates:
(1) The mobility, M, of a grain boundary in a pure material depends
on the porosity — or vacancy concentration in the boundary; the mobility
is the higher the greater is the porosity because the energy barriers to atomic
diffusion are low when the porosity is high.
(2) The mobility in an impure material depends also on the segrega-
tion of solute impurities at the boundary, the greater the segregation the
17
lower the mobility tends to be.
(3) The boundary mobility in general may, thus, be represented by
the equation
M(p,c) = fy(p) x fg(1/cp)
where f, is an appropriate function of the boundary porosity, p, in the pure
material, and fy of the solute concentration at the boundary, Cg. Both f,
and fy may be temperature dependent.
(4) The porosity, and thus, the porosity effect on mobility, for varies
with the density of coincidence 'sites in the boundary and, consequently, with
the type of coincidence-site relationship, the perfection with which this
relationship is attained, and the orientation of the boundary with respect to
the planes of closest coincidence-site packing in a given coincidence lattice.
The latter two factors can be represented by an angular increment, 40, such
that 40 and porosity increase as the departure from ideal orientation, both
relative grain and boundary, increases.
(5) Solute segregation at a boundary follows the principles enunciated
by Cahn (54) and others (6, 42, 43, 55, 56) and discussed in detail later in
this paper*. Therefore, whether the solute tends to adsorb or to desorb at a
stationary boundary, for a moving boundary there will always be a solute
build-up at or near the boundary which can be represented by some effective
average boundary concentration, Cg:
(6) Because low porosity boundary structures are more like the bulk
grain structures than are high porosity boundary structures, for a given bulk
* See page 48
*
18
concentration c, the value of CR will be the higher the greater the boundary
porosity* .
On the basis of these six postulates, the curves in Figure 10 can be
constructed. Here, for convenience in plotting, the normalized functions
fo = fy(p) /f,(p)., f. = fg(1/cp)/f2(1/cp. and M/M, rather than the abso-
lute functions, are plotted against the incremental misfit angle 40. The
subscript o in each case indicates the value of the function in a completely
pure, randomt boundary.
The curves can be understood most readily by considering an absolutely
pure material first. In this case the value of the impurity function is by
definition unity, the mobilities of various boundaries are determined by the
porosity function alone, and, thus, the curves in Figure 100 essentially give
these mobilities. For a high-density coincidence-site boundary with ideal
relative grain orientation and boundary position, i.e., with 40 = 0), the
porosity function f, has the low value given at point c in Figure 10a, and the
mobility is very low. This would correspond to, say, a coherent twin boundary
or, perhaps, a K-W boundary with a l-in-5 coincidence-site density. For
some other K-W boundary, such as the commonly observed 1-in-7 boundary
represented by a 38° rotation around (111) the porosity function and the
mobility are higher, as indicated by the point b in Figure 10a. As the porosity
Electronic effects, as well as porosity, determine the amount of boundary
segregation, but for a given impurity these effects are undoubtedly
secondary to boundary porosity:
Random with respect to a boundary is taken in the Aust-Rutter sense to mean
a boundary with the lowest percent of good-fit regions possible.
in any boundary increases due to increase in 40, all boundaries increase their
mobilities as shown by curves A, B and C in Figure 10a. The shapes of these
curves have been approximated by assuming a sinesoidal variation of porosity
with 40 as derived by Li (20) for porosity versus misfit angle in tilt boundaries.
These curves indicate, thus, that for very pure materials random or
near-random boundaries, having the highest mobilities, should be preferred
rather than special boundaries of the K-W type; further, they indicate that
boundaries of the latter type should actually be avoided. Both of these con-
ditions have been observed as discussed in connection with Figures 5 and 6*.
In contrast, on the basis of the K-W concept of good-fit boundaries as high
mobilities boundaries, there appears to be no reason why such boundaries
should not still be preferred in very high purity materials.
The prediction of preference for non-K-W boundaries in high purity
materials appears at first glance to be in contradiction to the results of Aust
and Rutter for tilt boundaries in bicrystals of zone-refined lead (41). There
is, however, an alternative interpretation of these results to that given by Aust
and Rutter. The migration rates at 200° and 300°C and the temperature de-
pendence parameters, Q, for these boundaries are reproduced in Figures 11 and
12. The Q's particularly appear to indicate that K-W boundaries ought to
be preferred boundaries. Aust and Rutter have chosen to interpret these Q's
as real activation energies, that is, as the energy barrier to the basic atomic
process in the migration. If this is in fact true, then for each boundary the
*
It should be pointed out that if the predicted preference for highly porous
boundaries in pure materials is sharp enough, it could conceivably lead to
a marked preferred orientation in which the observed rotation amounts and
axes are only those corresponding to boundaries with structures of the
poorest fit possible.
migration rate, G, should be related to Q by an equation of the form
G = G, explained
A plot of the data in the manner shown in Figure 13 indicates that this equation
very probably does not hold for these boundaries. In Figure 13 the loga-
rithm of the various migration rates have been plotted at fixed temperature
against the corresponding Q's. In addition, the quantity exp ) has also
been plotted against Q. It may be seen that at each temperature the two
resulting lines, have widely different slopes -- G appearing to be relatively ::
insensitive to Q. For example, G at 300°C decreases by only a feictor of
about 4 when Q changes from 5 to 20 kcals/mole, whereas the exponential
quantity decreases by almost six orders of magnitude. This means either the
measuied Q's have little physical significance or there is some temperature-
independent terin in G which in every case almost exactly compensates for
the change in Q. The latter interpretation leads to the presumption of very
high entropies of activation (for a discussion of this point, see page 29
and references 36 and 57) for which there seems to be no satisfactory ex®
planation. On the other hand, theories of impurity effects do indicate that
impurities may readily lead to apparent activation energies of migration which
have no real relation to the underlying physical processes (see Part II).
Thus, it seems probable that at least many of the Q's in Figure 11 are only
apparent activation energies, and little physical significance should then be
attached directly to the variations shown.* The migration rates themselves,
on the other hand, are undoubtedly physically significant (although they may
in some cases be average rates). In interpreting these, however, it should
be kept in mind that variations in the striation structure probably introduced
considerable variations in driving force - Aust and Rutter did not give the
magnitude of such variations, but a factor of 2 or more would not seem un-
likely. In addition, because of the 3-4º mutual misorientations amongst the
striation subgrains, and other experimental limitations, the actual boundary
misorientations cannot be known very accurately. As a result, it becomes ..
apparent that a much larger number of rate measurements than shown in
Figure 12 is required to decide conclusively whether the data indicate higher
mobilities for K-W, near K-W or non-K-W boundaries.
: In materials containing dissolved impurities the effect of porosity on mi-
gration is considerably more complex than in pure materials. Here a change in
boundary porosity has an indirect effect on boundary mobility as well as the direct
effect already described. In a material of a specific bulk impurity concentration,
a boundary with a high porosity supports a higher impurity segregation than one
with low porosity; the boundary concentration CR is therefore higher in the former,
the impurity drag on the boundary is greater and, thus, the mobility tends to be
lower. As a result, increased 40, which corresponds to increased porosity,
*ends through this impurity drag effect to produce lower mobility. However, the
* It is worth noting that plots such as those in Figure 13 show similar results for
all of the single boundary migration data based on zone-refined material, in-
cluding other data of Aust and Rutter and those of Rath and Gordon. Thus, it
9. Chi
seems likely that all of the associated Q's are only apparent activation
anti
energies, with the probable exception of the lowest values found by Aust
and Rutter.
.
...
.
.
.
.
-
22
shapes of concentration function versus 40 curves and their relative positions
for different boundaries depend in a complex way on the magnitudes of bulk
impurity concentration, driving force and specific impurity-boundary inter-
action energy and also on the temperature, as discussed in detail later in
Part II of this paper). Briefly, in the mobility - Cg relationship there will
be a region at low co and high mobility where the mobility is almost unaffect-
ed by og because the segregated solute cannot keep up with the rapidly mov-
ing boundary. On the other hand, at high cg and low mobility the mobility
will depend more strongly on ce because the segregated solute now virtually
travels with the boundary. The region between the high and low mobility
segments will consist of a transition zone where (a) for boundaries subjected
to low driving forces the mobility changes smoothly from the high to the low
mobility region, but (b) for high driving forces the transition represents con-
ditions for which the boundary migrates part of the time in the high, and part
in the low, mobility regions, leading to an observed average apparent mobility
which drops very rapidly from one region to the other.
Since boundary concentration, GR, is a function of porosity, and
porosity of 40, the effects of concentration on mobility may be translated
into curves of concentration function versus 4,0, as has been done in Fig. 10b.
The schematic curves shown have been drawn for conditions such that for the
relatively low porosity boundaries, B and C, the virtually impurity-independent
region of high mobility is observed at low A0, but in the almost random
boundary, A, there is presumed to be enough porosity andthus, enough
bility, impurity-dependent region. At high 40 where all the boundaries
23
tend toward similar porosities, it has been assumed that there is enough segre-
gation at all these boundaries to place each of them in the impurity-dependent
region. These conditions have been chosen for illustrative purposes because,
according to the model, they are the conditions which produce preference for
the special K-W boundaries of the type which have been most frequently ob-
served. This is shown in Figure 10c, in which the mobility of the near-random
boundary, A, and those of the medium-density and high-density coincidence-
site boundaries, B and C, have been obtained as appropriate products of the
two sets of curves in Figures 10a and b. It is seen that the K-W boundary of
intermediate coincidence-site density is indicated to have a higher mobility
than other boundaries when the orientations of the grains and the boundaries
are ideal, that is, when AO is zero. Furthermore, the mobilities of high and
medium coincidence-site density boundaries go through maxima as 40 in-
creases, the highest overall mobility corresponding to that of some intermediate
coincidence-site density boundary for which the relative grain and boundary
orientations are not quite ideal.
Other conditions of bulk concentration, driving force, impurity-
boundary interaction energy or temperature may be chosen to give different
concentration function -- A0 curves. For example, it may be supposed that
a given impurity has such a strong affinity for the boundary that even very
small absolute values of bulk concentration would effectively bring all the
boundaries for all values of 40 into the low mobility region where the impur-
ity atmosphere travels with the boundary. In this case the corresponding
concentration functions would be very nearly alike for all boundaries, as
shown in Figure 14a; multiplication by the porosity functions in Figure 10a
24
would then produce the mobility curves of Figure 146. It is seen that such
conditions could again bring about a preference for near-random boundaries
and the avoidance of ideal, high coincidence-site density boundaries just as
was indicated for pure materials (Fig. 10a); this might then account for the
Aust-Rutter observations on silver and gold alloys in zone-refined lead shown
in Figure 6. Since high values of bulk concentration for impurities with
lesser specific impurity-boundary interaction energies could presumably also
bring about the same condition, a possible explanation of the results of
Graham and Cahn (34) is suggested. The considerable quantities (0.4 wgt %)
and the particular nature of the impurities present in their aluminum may well
have been responsible for the avoidance of (111) rotation axes and the random
distribution of rotation axes observed.
31
.
Part II. Quantitative Impurity Effects, Theory and Experiment
It is evident from the discussion in Part I that small amounts
of soluble impurities can have a profound influence on the recrystalli-
zation of relatively pure metals. As early as 1940, Beck (58) observed
that the addition of 0.0083 wt % silver to lead greatly decreased the
rate of recrystallization of this metal. Subsequently, the work of
Smart and Smith (59) on copper and Lücke (60)(61) and co-workers on
aluminium showed that the addition of as little as 0.01 at. % of a
number of different impurities could raise the recrystallization tem-
perature of these metals by as much as 220°c. These experiments,
Shuninte!!
however, were only exploratory and were not complete enough to separate
the two component processes of nucleation and grain-boundary migration.
It was, therefore, not possible to determine whether the impurity effect
was due to a decrease in the rate of nucleation, in the rate of boundary
migration, or ab seemed more probable, in both. Thus, it became clear
that emphasis in subsequent work should be placed on obtaining quantita-
tive data which dealt with the nucleation and boundary migration processes
separately. Furthermore, the purity level of the parent metals used in
these early investigations was probably not high enough to insure that
residual impurity elements in the starting materials may not have masked
to some extent the full influence of the added solute element.
With the advent of zone-refining and the subsequent availability
of extremely pure materials, it became possible to make more careful
studies of the role of minute amounts of defined impurity additions.
Thus it was that Bolling and Winegard (44) (62) studied post-recrystallization
b
-
-
----
.
.
32
graln growth in polycrystalline, zone-refined lead as influenced by the
solutes tin, silver, and gold. Holmes and Winegard (45)(46) (63)(64)
conducted similar experiments on zone-refined tin to which they added
controlled quantities of either lead, bismuth, silver, or antimony. A
different approach to the impurity problem was carried out by Aust and
Rutter (35) (36) (37) who investigated the motion of individual grain
boundaries in bicrystals of zone-refined leed containing very small
additions of either tin, silver, or gold. In this case the driving
force for migration was the melt-grown lineage substructure which
remained stable during the experiments. This may be contrasted to the
ever-decreasing driving force, derived from the self-energy of the
grain boundaries, in the grain growth work. Gordon and Vandermeer (42)
studied the impurity problem from yet another experimental point of
view, one in which the driving forces for boundary motion were several
orders of magnitude larger than in either the grain growth or the single
boundary work. They measured the rates of growth of new grains during
recrystallization after moderate deformation in polycrystalline, zone-
refined aluminum and in this aluminum containing small amounts of
copper. Along this line also, Frois and Dimitrov (65) (66) Investigated
the influence of the solutes copper, magnesium, and silver on the
recrystallization of zone-refined aluminum after heavy deformation,
The one striking common feature in all of this work with zone-
refined metals was the large retarding effect the addition of only
small quantities of impurities exerted on boundary-migration rates.
For example, 0.006 wt% tin dēcreased the speed of boundary migration
of random boundaries at 300°C in lead by a factor of 1000 (35) while
as little as 0.0017 at. % copper reduced the mobility of boundaries
in zone-refined aluminum by something greater than 1000 at 139°c (42).
It was also found that in a given zone-refined metal, one solute would
retard boundary motion to a different degree than another. In lead,
for example, silver and gold showed a much larger effect per solute
atom than tin (37), while bismuth was more effective than lead in
retarding grain growth in tin (44).
Early Theories of Boundary Migration.
The first theories of
grain-boundary migration rates were put forth before the quantitative
measurements on zone-refined metals were available. Mott (15) and
Turnbull (67) derived expressions for the rate of motion of high angle
grain boundaries based on the formalism of absolute reaction rate
theory. Turnbull (67), defining the rate of boundary migration as the
product of the mobility of the rate determining step and the gradient
of the chemical potential in the direction of movement, assumed that
the atoms are transferred singly across the boundary during migration.
He obtained the expression
G - rate of grain boundary migration,
e = 2.718 (base of natural logarithms), ,
d = distance of boundary advance locally when an atom 18
transferred from one grain to the other,
.
3 4
h = Planck's constant,
N = Avogadro's number,
AP= driving free energy per mole,
as = entropy of activation for boundary motion,
R = wiversal gas constant,
Q = activation energy for migration,
T = absolute temperature (°K).
Expressed in terms of a diffusion coefficient for atom transport,
Eq. 1 can be rewritten in the equivalent form
(1-a)
where a lattice parameter,
D. - preexponential term,
and Q = the activation energy for the diffusion proce88
responsible for boundary migration.
Thus G 18 of the form
and therefore Eqs. 1 and 1-a predict that a plot of log G against 1/T
should give a straight line for which the slope 18 related to Q and
the intercept at 1/T = 0 18 C. Most, though not all, experimental
results are in good agreement with this straight line relationship, at
least for the rather limited temperature ranges over which migration
rate measurements can be made."
- --.--veronder eine nove program..
Turnbull (67) further suggested that there is reason to expect
the activation free-energy barriers which the atoms must surmount in
the elementary act of grain-boundary migration to be similar to those
surmounted in the elementary act of grain boundary self-diffusion.
Thus, be supposed, as did Beck, Sperry, and Hu (68), that the rate
determining step for grain-boundary mobility ought to be related to
the diffusion coefficient for atom transport along the grain' boundary.
Turnbull (67) offered some support for this point of view. He calcu-
lated the free energy of activation for boundary migration, AF, from
data obtained on metals of ordinary purity using Eq. 1.* On comparing
these AF, 's to the activation free energies, of lattice self-diffusion,
Turnbull concluded that the atomic mobility in grain-boundary migration
is many orders of magnitude larger than in lattice self-diffusion. In
spite of this, the very disconcerting fact remained that the activation
energy, Q, in these materials was frequently equal to or greater than
the activation energy for lattice self-diffusion (see Burke and Turnbull) (57).
Also, the As, '8, which, based on theoretical considerations, should be
nearly zero, were often very high. Turnbull (67) hypothesized that
these anomalies might be attributed to inclusions which, retarding
boundary migration, become less effective at elevated temperatures due
to their solution coalescence. Mott (15) also attempted to account for
the atoms are activated in group. He envisioned the atom transfer
process to be one where a group of atoms belonging to one crystal melt
* AFA 18 related to Q and A8A by the equation AFA - Q - TASA - RT.
- 36
and resolidify as a group onto the other crystal. On this basis, an
equation of the following form results:
where a = the number of atoms in a group,
T = the absolute melting temperature of the material,
Q = ni, L being the latent heat of fusion of the material.
There 18, however, no theoretical reason to suppose that atom transfer
ia boundary migration occurs in groups rather than singly.
Boundary Migration in Zone-Refined Metals. It seems to be clear
now that the early attempts at comparing experimental rates to
theoretical estimates proved unsatisfactory because the early theories
took no account of impurity effects, whereas the data used for compari-
son was obtained on "impure" materials. This has been brought to light
by the more recent work carried out on zone-refined metals. Aust and
Rutter (36) compared their experimental boundary migration rates in
zone-refined lead containing tin with the predictions of the theories
of Mott (15) and Turnbull (67). They found that with impurities present
· neither theory predicted the correct order of magnitude for boundary .
velocity. However, when Impurity effects were absent, the single pro-
cess théory of Turnbull (67) gave order of magnitude agreement with
experiment whereas the group process theory of Mott (15) was off by
about three orders of magnitude. A similar test of the theories was
10
.....
..................
.
.......
..
starten
Pusthave
made by Holmes and winegard (69) for migration rates observed during
grain growth in zone-refined lead and tin. The experimental data were
In excellent agreement with the single process theory predictions but
not those of the group process theory, and here again impurity effects
had been minimized. In a recent recrystallization study of zone-refined
nickel, Detert and Dressler (70) were, led to interpret their experi-
mental growth rates also in terms of the single process atom transport
-
-
V
..
.
T
..
.' A.
.
.
.
*Vane
1.
Idea.
}
*.*..*
It should be pointed out that the activation energy in the single
process theory cannot be calculated theoretically with any precision
and therefore when Eq. 1 is used to obtain theoretical estimates of G,
the value used for Q must be taken from experiment. However, the
magnitude of the preexponential factor, a Dor in Eg. 1-a can
be calculated sufficiently well to permit a meaningful comparison with
experiment provided the driving energy, i.e. AF, 18 known. Such a
comparison was made by Gordon and Vandermeer (42) for their boundary
AT
I
...
-.
.
- .
-"
...
come ..
.
.
migration rate data on zone-refined aluminum. Table II shows this
.
.
.,
.
.
.
.
.
.
.
wwtated
.
comparison and similar comparisons for experimental data obtained in
other recent studies on zone-refined metals. The comparisons between
calculated G'8 and experimental G's are given in terms of log Q.
since in view of the estimations of D. and Af in the calculated
quantities and the errors in the experimental quantities, only order of
magnitude agreement can be expected (42). The theoretical and experi...
mental values of Q, are seen to agree within approximately one order
. .
.
.-.
.
.
.
haberini istening this time.commons
.
.
.
.
.
. .
.
n
. .
. .
.
.
-
tamba
-
-
38
Table II. Comparison of Theoretical and Experimental
Preexponential Factors for Grain-Boundary Motion
in Zone-Refined Metals
Type
Metal
Driving
Force
(Cal/mole)
Log lo
Log G
Theoretical Experimental
Reference
Experiment
Aluminum
4.66
6.23
Aluminum
. Recryst.
Recryst.
28
Single boundary 2 x 10-3*
Recryst. 10* .
Single boundary 2 x 10-3
(42)
(65)
(71)
Aluminum
4.58
0.96
4.89
1.09
Nickel
.
(70)
6.55
- 1.57.
Lead
random
Lead
Single boundary 2 x 10°3
special
1.09
- 0.15 :
(36)
;
Lead
0.99
(62)
.
.
Grain growth
Grain growth
1.8 x 10°3*
1.6 x 10^3*
- 0.78
- 0.47
Tin
0.95
--
-
-
-
-
-
-
-
-
u
us.
--
---
Notes to Table I
. D. assumed = 0.5 cm/sec
d = lattice parameter a
* driving force estimated
.
. .
.
.
1
..
...
39
.
.
..
........
.
..
Hina
S.,
of magnitude. This check is considered to be reasonably good, partic-
ularly in view of the huge discrepancies--up to 10 or more orders of
magnitude as indicated by the high ASA's mentioned earlier--found for
similar comparisons on ordinary purity metals. Thus, there seems to be
à mounting body of evidence which strongly indicates that the elementary
process in grain-boundary migration involves the transfer of single
stoms across the boundary.
In all instances where the single process theory of Turnbull gives
reasonable agreement with experiment, the activation energies of
boundary migration are also much less than those of lattice self-diffusion.
As shown in Table III, they are much closer to the activation energies
found for grain boundary self-diffusion. Since the grain boundary is a
transition region where the atoms do not fit together very well, it may,
as pointed out in Part I, bė looked upon as a region containing excess
porosity or vacancies. · The boundary could then be thought to move by
means of an atom-vacancy Interchange. The activation energy of boundary
-
-
-
--
-
.
...
.
.
..
..
migration would in this case be approximately equal to the activation
energy for the motion of vacancies in the boundary and should correspond
to that for boundary self-diffusion. This has been found to be the case
in zone-refined aluminum and nickel as Table III 11lustrates. The
agreement 18 poor, however, for tin. Holmes and winegard (63) pointed
out that the activation free energies of boundary migration in zone-
refined lead and tin are also comparable to the activation free energies
of liquid self-diffusion in these retals. Thus, while there is some
.
Let u
Table III. Comparison of the Activation Energies for
Grain Boundary Migration with those of
Lattice and Grain Boundary Self-Diffusion
QUB
Metal
Measured
(Kcal/mole)
for Lattice
Self-Diffusion
(Kcal/mole)
Doponenden
Em
for Vacancy MI-
gration
(Kcal/mole)
for Grain Boundary
Self-Diffusion
(Kcal/mole)
Aluminum
34.08
16.0*
13.8 8
(42)
15.0
13.1
1.3.0
Nickel
30.1
66.86
26.6f
34.52
Lead
iesire
(Random bdy's)
6.0
(special bdy's)
25.7€
.
15.7°
6.1 -
Lead
6.7
mer
(62)
Tin
6.0
24.50
9.55€
12.21
(63)
See Reference (72)
(73)
(74)
8 "
h "
1 " ,"
*Est'd assuming
(75)
(76)
(77)
(78)(79)
(80)
(81)
= 0.47, see Ref. (42)
hii
PMI.
TRE
U
experimental support for the concept that the activation energy
barriers which the atoms must overcome are similar in grain-boundary ,
migration and grain boundary self-diffusion, the evidence 18 not
entirely in agreement on this point.
:
Impurity Drag Theories of Boundary Migration. When the
predictions of Turnbull's single process theory are tested with
data obtained on boundaries migrating in the presence of dissolved
impurities, hugie discrepancies are noted. This is, for example,
shown in Table IV, which compares the experimental G's for zone-
refined aluminum containing additions of copper with the Go
predicted by Eq. 1-a. Aust and Rutter (36) also found similar
discrepancies for grain boundaries migrating in lead containing
tin impurity atoms. It is evident from this and what has been said
earlier that impurities play an extremely important role in the
migration of grain boundaries--a role not explicitly taken into
account by the single process theory of Turnbull.
Theory of Lücke and Detert. Lücke and Detert (55) were the
first to attempt to formulate a quantitative atomistic theory
treating the impurity effect. The basic hypothesis of their theory
was that impurity atoms in solid solution tend to segregate to grain
boundaries because of interaction forces which exist between these
atoms and the boundary (1.e., the boundary 18 an energy sink for ...
Impurity atoms). They further assumed that the concentration of
impurity atoms at the grain boundary, whether stationary or moving,
1.
ot
ri
-
...
-
.:,:
.
.
Table IV. Comparison of G, from Turnbull Theory with Measured
G's of Dilute Aluminum-Copper Alloys
Log G.
Material
(ppm copper)
Log G.
(single process theory)
(measured)
6.23
Zone-Refined
Aluminum
4.66
2.1
14.3
4.66
4.66
4.3
17.5
11.3
4.66
17.0
34.0
10.5
4.66
68.0
4.66
4.66
124.0
258.0
4.66
18 given by the approximate equilibrium expression for a stationary
boundary
.
E
Co Cexpre
where
Ce grain boundary impurity concentration,
C = bulk impurity concentration,
E -= interaction potential between impurity atoms
and grain boundary. E is taken to be negative
if impurities are attracted to the boundary.
When a grain boundary laden with impurity atoms is caused to move,
the impurity at sphere creates a drag which, if not overcome by the
driving force, compels the boundary to carry the impurity atmosphere
along with it. In this situation the speed of the boundary is con-
trolled by the rate at which impurity atoms can diffuse along behind
the boundary. Ilicke and Detert obtained for the rate of migration
under these solute-controlled conditions an equation of the form
com-
where the quantities are as defined previously. Since the impurity
atoms are envisioned as moving behind the boundary and making their
diffusion jumps towards the boundary, the diffusion constants
D. and my were assumed to refer to lattice diffusion of the
Impurity.
44
According to this theory, if the driving force 1s high enough
to overcome the dragging effect of the impurities, then the
boundary breaks away from the impurity atmosphere. This will occur
at low impurity concentrations or at high temperatures where accord-
ing to Eq. 3, Co will be small. In this "broken away" state the
rate of boundary motion 18 assumed to be independent of impurities
and the activation energy for migration will be that for grain-
boundary diffusion, 1.e. Quapro
Figure 15 shows schematically how G should vary with temperature
for a given alloy composition based on the Ilcke-Detert theory. At
high temperatures G 1s solute-independent and determined by Eq. 1 (1-2),
whereas at low temperatures G depends on composition in the manner.
indicated by Eq. 4. According to the Lücke-Detert impurity drag
theory, the breakaway process should be very abrupt and should occur
at certain critical values of composition, driving force, and temper-
ature. It was recognized, however, that in reality the breakaway
would not be sharp but rather that a transition region, in which the
boundaries would be only partially broken away, should be expected,
as depicted in Fig. 15. The extent of this transition region can
not be ascertained from the theory, however.
Figure 16 illustrates schematically the temperature dependence
of boundary-migration rates predicted by the Lücke-Detert theory for
a series of impurity compositions. Experimentally it is not feasible
to study migration rates over as wide a temperature range as 18
4 5
.
MN.
--
---
..
.
-
-.-
-.
.
.
.
.
necessary to obtain any one complete curve in Fig. 16. As suggested
by this figure, however, it is possible to change C sufficiently 80
that at some composicions the measurable range of G lies in the
impurity controlled range (for example Cy, Cz, and Cz), at others in
the transition region (C. and Col, and at still others in the impu-
rity independent region (C). Thus, from theory the experimental
velocities would be expected to fall on those segments of the indi-
vidual curves which lie in the cross-batched area of Fig. 16. AB
may be seen in Fig. 17, the grain-boundary migration rates for alloys
of copper in zone-refined aluminum from the studies of Gordon and
Vandermeer (42) follow the pattern suggested by Fig. 16. The data
of Frois and Dimitrov (65) for copper in zone-refined aluminum also
show this same trend, though their data 18 not as extensive as that
of Gordon and Vandermeer in the high concentration region. Further-
more, it may be deduced from Fig. 16 that in the experimentally
measurable range of G, the "activation energy" (1.e. -RIIG I
*se a 1/1
ought to be low for low compositions and with increasing c go through
- Rd In G
a maximim and eventually level off. For low compositions
.......
.
....
.
.
.
..
.
.
.
.
.
....
.
...
.rar .
.
should equal Que and at high compositions Q-E. It should be pointed
out that all other "activation energies" are only apparent and do not
represent an actual energy barrier which must be overcome during the
movement of the boundary. The trend of the measured "activation
energies" for the Al-Cu alloys 18 just that predicted as can be seen
in Fig. 18. A similar trend 18 also noted in the data of Frois and
Dimitrov (65) for magnesium in zone-refined aluminum.
.
.
The excellent qualitative agreement between the Lucke-Detert
theory and experiment already noted for zone-refined aluminum was not
found in other investigations on impurity doped, zone-refined metals.
Figure 19 shows the composition variation of the activation energy
for grain growth in zone-refined tin to which lead has been added,
from the work of Holmes and Winegard (46) (64). At the higher concen-
trations of lead the activation energy, to be sure, is constant in
agreement with the Lücke-Detert idea. In addition, the activation
energy is low for zone-refined tin with no lead, which 18 as expected.
At intermediate lead levels, however, the activation energy curve
does not go through a maxima as Llcke and Detert would predict, a
behavior which Holmes and Winegard, (46) suggested was in contradic-
tion to the breakaway concept. Mkewise, Aust and Rutter (36) found
that the measured activation energies of random grain boundaries in
zone-refined lead alloyed with tin did not behave in a manner con-
sistent with the Ilcke-Detert breakaway process either. They observed,
as Fig. 8, page indicates, a steadily increasing activation energy
as impurities were added, with no obvious tendency to level off at
least up to tin concentrations of 0.0013 wt %. It would be interest-
ing to know whether or not at higher tin contents the upper curve of
Fig. 8 would go through a maximum and begin decreasing again.
Another aspect of boundary-migration behavior predicted by the
Jücke-Detert theory has been tested with experiment; this is the .
variation of G with C at constant temperature and driving force. The
.
:
-- :.
expected behavior is 1llustrated schematically in Fig. 20. At low .
impurity concentrations where Eq. 1 (1-a) holds,, the migration rate :
$should be high and independent of C as shown. The results from Frois.
and Dimitrov (65) for impurities in zone-refined aluminum are good
examples of this type of behavior as Fig. 21 indicates. At high C,
the rate is supposed to vary inversely with composition in accordance
with Eq. 4. The corresponding experimental data of Gordon and
Vandermeer (42) on migration rates in aluminum-copper alloys at
several different temperatures are plotted in Fig. 22. It is seen
that the data give curves of the predicted shape. Here also the
results of Frois and Dimitrov (65) are in qualitative agreement with
the predicted behavior.
When the Lücke-Detert equation for impurity controlled boundary
migration 18 expressed in terms of the rate of grain growth, it
.
becomes
11/2
Q.
20 Yan
RT
RT .
where A = average grain diameter,
t = annealing time,
a = grain growth exponent, usually = 2,
0 = specific grain-boundary energy per unit area, ...
Ve molar volume.
"Holmes and Winegard (46) found that in the composition range where
the activation energy 18 constant, the rate of graia growth for lead
,-:.
-
-
.
wa watu wote
:
in
&
in tin is proportional to cau/. According to Eq. 5 this aspect of
grain growth is in agreement with the Lücke-Detert theory.
Aust and Rutter (35) (36), on the other hand, had little success
in finding agreement between the Lucke-Detert theory and their experi-
mental results on the migration of single boundaries in zone-refined
tin alloys. They calculated that on the basis of the Lücke-Detert
theory, tin contents greater than 1 x 10° atom fraction and velocities
less than 11 mm/sec should give rise to impurity controlled boundary
migration behavior in zone-refined lead. Their data, shown in Fig.23
1llustrates that the variation of measured velocities with tin
content at 300°C for random boundaries does not in fact follow the
expected pattern. For example, at the higher tin contents, the
velocities are proportional to c-5/2 rather than cot.
It 18 necessary when considering only the role of impurities in
boundary migration as the Lücke -Detert theory does that all orienta.
tion variables be held constant. For single-boundary measurements
this would require that identical boundaries are studied in each of
the different alloys. Gordon (43) has suggested that the lack of
Agreement between the A ist and Rutter data and the Lücke-Detert
theory may be a result of the tremendous difficulty involved in
1solating orientation variables from impurity variables. Thus while
Aust and Rutter managed to separate their boundaries into three
groups with respect to orientation variables, there are probably
differences in orientation effects within each group. The
.
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1
49
Come
.
.
i in acest modern mit vero
experimental techniques used in polycrystalline boundary-migration
measurements, however, incorporate a statistical assurance that
only boundaries with approximately the same environment present
themselves for measurement. As a result, polycrystalline samples,
properly prepared and tested, automatically maintain the orientation
-..
--
variables approximately constant and may offer a readier means for
----
-
-
...
..
.
...........
.
...
.
.
.
testing the Lucke-Detert theory quantitatively.
When further attempts were made to test the Lücke-Detert theory
quantitatively with those experiments where the predicted trends
were observed, areas of disagreement were uncovered. Theory suggests
that in the impurity-dependent range, the activation energy for
migration should be equal to @-E. For copper in aluminum @ze
31, 120 cal/mole (82). Assuming E to be about -2280 cal/mole (1),
Q4 - E 18 33,400 cal/mole. Gordon and Vandermeer (42), however,
observed an activation energy for migration of only 28,000 - 29,000
cal/mole for copper in zone refined--about 5 to 6000 cal/mole too low.
A similar effect was found by Holmes and Winegard (46) in grain growth.
Thus it was that when Holmes and Winegard calculated solute diffusion
coefficients from their growth-rate data, using the Illcke-Detert
theory, the calculated rates appeared to be about two orders of
magnitude too large.
.
.
.
.
.
.
.
.
.
.
..
..-_.,......
Soinin il serience der
mir
contra os indication Landing
*
.
:
cal
.
.
.
Modifications to the Lucke-Detert Theory. Gordon and Vandermeer (42) :
tried to account for these discrepancies. They took issue with the
Lücke-Detert assumption that in impurity-controlled migration the atoms i.
. .
. . .
.
in Chem
. .
.
.
artawan dan mening
. .
.
w
..
.
.
may be treated as diffusing in undistorted bulk material, which led
to the insertion of D. and Q, into Eq. 4. It was pointed out by
Gordon and Vandermeer that, strictly speaking, the impurity atoms
can only exert a dragging force on the boundary in a region where
the lattice is still distorted to some extent by the presence of the
grain boundary (42). They argued that since the distortion of the
lattice can be expected to lower diffusion activation energies, Quy
should be substituted for Q in Eq. 4, where Qy may be significantly
less than QzIt was not possible, however, to decide how much
lower Qz should be than , but it was suggested that it could never
be less than Qce and that it was probably much closer to Q than QGB
One further modification in the Lücke-Detert theory was also
introduced by Gordon and Vandermeer (42). They suggested as McLean (1)
nad pointed out, that the boundary concentration given by Eq. 3 above.
is only an approximation and did not take into account alterations
in the vibrational frequencies of the impurity atoms when they are
i placed into a grain boundary. They proposed that the Ilicke-Detert
formula be modified to
..
-.
.
-..
AF
1
com
(
RT
AC
RT
where A 18 a vibrational frequency factor approximately equal to
1/4 and D. and af are impurity diffusion constants referring to
impurity diffusion in the distorted grain-boundary region. With these
modifications in the theory, the data on copper-doped zone-refined
NEX
VO
1
.
.
T
..
.
..
.
.
.
.
..
.
simon.eran..
n
ä.
E
,
........
ir prisimine
t
e
le this band and cansations are presenti
n
renant
-!
aluminum were found to be in excellent quantitative agreement with
theory. For example, when the preexponential portion of Eq. 6, i.e.
Go, was calculated from theory and compared with experiment, quite
good checks were obtained except for alloys in the transition region
where agreement 18 not expected. Table V shows this comparison.
Subsequently, Gordon (43), also showed that the Winegard and
Holmes data on grain growth in alloys of zone-refined tin contain-
ing lead and bismuth could also be satisfactorily explained by the
modified theory. Gordon pointed out that because of the low-driving
forces present in grain-growth studies, the "breakaway". behavior will
manifest itself in a way different from that in recrystallization
studies where the driving forces are much higher. This is schemati-
cally illustrated in Fig. 2A for log G vs 1/T. If the breakaway
temperature, T, 18 much less than Tes defined as the temperature
where Eqs. l-a and 6 cross, then the behavior shown in Fig. 24-a is
expected. If, however, T, is approximately equal to T., then the
behavior depicted in Fig. 24-18 likely. Gordon (43) showed that
in the Holmes and Winegard grain-growth data T. 4 T. and the latter
behavior is expected. It can also be seen from Fig. 24 that in going
from the solute independent to solute dependent regions, in the first
case the "activation energy" increases, goes through a maximum and
decreases again before leveling off. In the second case, however,
the "activation energy" increases and gradually levels off without
going through a maximm. The experimental behaviors shown in Figs. 18
and 19 are therefore both consistent with the modified Ilicke-Detert
impurity drag theory.
L
W
..
.
"::..
.
.
. .
52
Table V. Comparison of Theoretical G's with Experiment
for Impurity-Controlled Boundary Migration in
Zone-Refined Aluminum
Material
pom cu
Log G.
Meas:
Log G.
Calc from°EC. 6
2.2
4.3
17.0
34.0
68.0
124.0
258.0
(14.3)
(17.5)
11.3
10.5
9.6
11.0
10.7
10.1
9.8
9.4
9.3
9.2
8.8
9.1
Concerning the relative effects of different solute atoms on
boundary-migration rates, some qualitative comparisons have been made
with the predictions of the Lücke-Detert theory. Arct and Rutter
---.
---rend the windy..'-
found that per impurity atom, gold and silver were more effective in
retarding single-boundary migration in lead than was tin. This they
claimed was in contradiction to the predictions of the theory because
gold and silver being faster diffusers in lead than tin should,
according to Eq. 4, not have as large a retarding effect as tin.
This claim is certainly valid if all three solutes have about the
same interaction potential with the grain boundary. It is probable,
however, that solutes which diffuse rapidly and thus have low QE
will also have large, negative interaction energies, E. Since the
quantity Q - E, rather than Qy alone, 18 controlling in impurity
dependent boundary migration, it follows that Qe - E may be larger
for fast diffusing, than for slow diffusing solutes. Thus, the Aust
and Rutter results could be consistent with the Lücke-Detert theory
as modified by Gordon and Vandermeer.
The need to modify the Ilcke-Detert theory to obtain satis-
factory quantitative agreement with experiment has emphasized the
desirability of a more sophisticated theory. The main assumptions,
approximations and limitations of the Illcke-Detert theory were:
.: a) The prediction that breakaway from an impurity atmosphere
18 an abrupt process.
.-
.
this
thin
chambre ja teises staan en diensten toiminnan toiminnasta mielenkiin section
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basement des
Formation on
54
b) The assumption that the velocity of the grain boundary 18,
equal to the velocity of the impurity atoms.
c) The implicit assumption that Cg 18 given by the stationary
equilibrium value and that it does not change when the boundary
moves.
a) The consideration only of segregation in which impurity
atoms are absorbed at boundaries, but not that in which they are
desorbed from boundaries, 1.e., E. positive.
e) The assumption that diffusion of the impurity atoms in
undisturbed bulk material controls the solute-dependent migration
behavior.
1) The lack of consideration that the boundary-impurity
interaction energy and the diffusion coefficient of the impurity
vary as a function of distance across the grain boundary.
3.
The Impurity-Drag Theory of Cahn. Recently Cahn (54) attempted
& more rigorous formulation of the impurity drag problem in grain-
boundary migration. He sought to remove some of the limitations
of the Ilcke-Detert theory retaining, however, their fundamental
assumption that it is the force exerted by an impurity atmosphere
which is responsible for the observed phenomena. Simultaneously
and independently of Cahn, Lücke and Stuwe (6) albo treated the
Impurity drag problem in more detail. Their treatment was in many
respects identical to that of Cahn. A brief review of Cahn's
.
.
formulation is as follows. He supposed that as far as the impurity
atoms were concerned, the grain boundary could be represented as a
planar discontinuity characterized by an interaction energy, E(x),
and by a diffusion coefficient, D(x), for motion normal to the
boundary. Both E(x) and D(x) are functions only of the distance x
normal to an arbitrarily chosen center plane of the boundary. By
.
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.warw.rairera
considering in detail the diffusion or the impurity atoms in the
.
mm
---
.-.
-
..-.--.
potential field of the boundary, Cahn was able to derive an expres-
sion for the composition profile across a boundary moving with a
constant velocity, G, explicitly in terms of integrals involving
E(x) and D(x). Due to the complexity of this expression and because
of a lack of knowledge about the functions E(x) and D(x), Cahn was
unable to obtain a general solution for the composition profile and
was forced to consider only certain limiting cases. He was able to
solve for the total force exerted on the boundary by all the impurity
atoms under conditions such that the velocity of the grain boundary
was either much higher or much lower than the drift velocity of the
impurity atoms. Cahn then constructed an approximate equation for
the impurity drag force, Pe, which fit both these high and low-
velocity extremes. Finally, by suming all the forces acting on the
grain boundary, Cahn obtained the following expression relating the
velocity of boundary migration to the driving force, composition,
. and indirectly to temperature:
-".
ini
а са
AF = NG 4
56
In Eq. 7, 18 the reciprocal of the boundary mobility in pure
specimens. Alpha and beta are referred to as impurity drag parameters
and are related to E(x), D(x) and temperature through the integrals
Sinha E(x)
RT
2 RT
et
dx
aux
(8)
RT
D(x) dx ,
and
D(x) dx ,
()
where R = universal gas constant,
T = absolute temperature,
Ve molar volume of parent material,
D(x) = diffusivity profile of the impurity atoms,
E(x) = boundary-impurity atom Interaction energy profile.
The drag parameter a can be thought of as the reciprocal of the boundary
mobility at unit concentration of impurity while B represents the
reciprocal of the average drift velocity of the impurity atoms across
the grain boundary.
In very pure materials, i.e. C20, Eq. 7 becomes
AF » G
(10)
which 18 of the form of the single process theory of Turnbull, i.e.,
Eq. 1(1-2). In view of the fact that this theory has been shown to
yield satisfactory agreement with experiments on zone-refined metals
(see earlier discussion), it is possible to evaluate , by comparing
.
: 57
demers
Eq. 10 with Eq. 1-a.
Thus ^ can be written as
nibüste
naar het
Na
GB
RT expan
RT
.
(11)
eren
"
t
.
--..
'-.
.... :
nis
in i
i
-ni-.
merkintäm
..
--
.
.
.
,
.
-
.
...: ""
Since the important basic factors comprising the impurity :
drag parameters a and B are D(x) and E(x) and these are involved as
integrals, it le far more difficult to obtain a and B in terms of
fundamental quantities. This is brought about mainly because the
profile shapes of D(x) and E(x) across grain boundaries are not
known. Furthermore, the magnitudes of D(x) and E(x) are known, if
at all, only approximately. Nevertheless, it 18 possible to devise
some simple, albeit crude, profiles for D(x) and E(x) which allow
integration of Eqs. 8 and 9; this permits the derivation of relation-
ships between the theoretical drag parameters, a and a/b', on the one
hand, and the impurity diffusion coefficients, Dr. and DER, the . .
grain boundary half-width, 6, and an interaction potential at the
boundary center, E., on the other. This has been done for the
assumed profiles shown in Fig. 25. The profiles are defined only
for x positive, but it should be kept in mind that they are symmetric
about the grain boundary center. The interaction potential was
arbitrarily chosen as negative indicating absorption of impurity
atoms at the grain boundary. The solutions of Eqs. 8 and 9 for
various combinations of D(x) and E(x) are listed in Appendix A. The
combinations of E(x) and D(x) chosen were ones which approximate the
.....
.
...
...
...
.
..
..
....
......
.
..
.
T
mim
. -- Ir...
erw
..
.
.
.
. .
.
.
.
. .
.
58
types of boundaries distinguished by Cahn (54) (1.e., flat or
pointed extremms in E(x) at the boundary center, and different
relative ranges over which E(x) 18 altered compared to D(x)).
In Cahn's theory, the expression which relates the velocity
to driving force, composition and temperature, 1.6., Eq. 7, 18
unfortunately a cubic algebraic expression in G. Thus it is diffi-
cult to determine from it the velocity for a given set of conditions.
The situation becomes somewhat less complicated when certain extreme
cases are examined. If, for example, the velocity of the boundary
18 less than the drift velocity of the impurity atoms in the
boundary, 1.e., G < 1/8, then Eq. 7 simplifies to .
AF = NG + a CG
. (12)
a + QC
AF
This 16 considered a low velocity in the sense of the Cahn theory
and 18 attained either by a low-driving force, 1.e., AF/A < 1/8, or
a high-impurity content - . Equation 12 is equivalent to
the older Lücke-Detert formula 1f QC >> A, D(x) 16 constant and
equal to a lattice diffusion coefficient and E(x) 18 large and
ac
B
negative (absorption of impurities). Thus the Lücke-Detert theory
appears as a limiting case of the much more general theory of Cahn.
..
V
.
..
1..
Another limiting cabe occurs when the velocity of the boundary
,18 much greater than the drift velocity of the impurity atoms, i.e.,
G>> 1/B. In this high-velocity region
+
RU
f
(13)
closer-menomis
Equation 13 reveals that in the high-velocity range the velocity
18 not strictly proportional to driving force except under condi-
tions for which the second term is negligible, 1.e., for low C or
high AF. Also, since a/82 18 proportional to a diffusivity (see :
Eqs. in Appendix A), Eq. 13 indicates that impurities having
greater diffusivity will show a greater retarding effect, provided
the other terms that make up a/B2 are the same.
There are, according to Cann's theory (Eq. 7), three types of
transition to be expected in going from pure to impure material.
These three types of transition behavior are 1llustrated in Fig. 26,
which is essentially a plot of boundary velocity versus impurity
concentration on a log-log scale. As indicated, it is the magnitude
of the driving force which determines which one of these transitions
ought to be observed. At low driving forces the transition 18 smooth
and continuous suggesting a gradual capture of the grain boundary by
impurity atoms. For intermediate driving forces the transition 18
still continuous but there is an inflection in the variation of
velocity with composition. The range of driving forces over which
this second kind of transition occurs is, however; quite limited and
thus it would be observed only infrequently. At high driving forces
a third type of transition behavior 18 predicted. In this case there
appears to be, for a certain composition range, three velocities
..
...
M
--- -
tier
. -
mana mes.com.
-
mana
Se
mere
e
.
S
omein
tantummod
den
n
risi
which satisfy Eq. 7. Cahn pointed out that the middle velocity
represents a physically unstable situation and that only the highest
and lowest velocities in this composition range can be expected to
persist for finite periods. Thus, for high driving forces there is
a range of composition where the theory predicts that there are two
possible velocities for grain-boundary migration, with a discontinuous
transition between them.
This raises the question as to what kind of
experimental behavior is expected for boundaries falling in the
composition range where these two branches of the velocity-composition
curve overlap. It should be remembered in this connection that when
the velocity of a grain boundary is measured experimentally, the
macroscopic, and hence only the average, behavior of the boundary is
really observed. In fact, boundary velocities can be measured metallo-
graphically only after the boundaries have moved tens to hundreds of
thousands of atom distances. As a result, localized fluctuations in
migration rates would go unnoticed and only an average velocity
would be observed. If it may be assumed that there are perturbations
present which can alter the boundary velocity to and from the
extremums in the two-velocity range, then it can be imagined that a
given boundary segment might oscillate between these extremas. The
perturbations could undoubtedly arise from local fluctuations in
impurity concentration and/or in driving force. Macroscopically,
therefore, an apparent velocity at a value between the two actual
extremum velocities would be measured. This situation is schematically
ressions to
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1:2lustrated in Fig. 27-2 where a growth curve representing the plots
made during recrystallization studies of grain-boundary migration
16 shown. The stepwise behavior represents the Oscillations between
actual velocities, whereas the macroscopic resultant is indicated
by the heavy line through the steps.
This bebavior may even be more complicated. For example,
suppose a local segment of a grain boundary moving with a low velocity
encounters a region free of impurities in which it 18 temporarily
able to move at a high velocity. The fast moving segment then loops
out ahead of the slower moving portions of the boundary, introducing
local curvatures in the boundary. If these curvatures are high
enough, the driving force on the boundary will be altered by a term
involving the product of the grain boundary self energy and the
curvature. This surface tension contribution would act to slow
down the fast moving portion of the loop; buit, since inverse curva-
tures would be generated at positions where the loop joins the slower-
moving boundary segments, thua augmenting the driving force there, .
these segments would move at a slightly increased speed. On the
average, however, the overall boundary velocity will be increased
to a value above that given by the low-velocity branch. Thus, on a
local scale, boundary energy considerations may well be important in
deciding what average velocity the grain boundary assumes in the two-
velocity regions. In any event, the average velocity should be
intermediate in value between the high- and low-velocity extremes
predicted by the theory.
Wu...
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62
The consequences of this rationlization of the behavior of
boundaries in the two-velocity region are 1llustrated in Fig. 20-6.
The heavy line indicates the sort of behavior expected of the
experimental rates. The similarity between this and the curve
predicted by Gordon and Vandermeer from the Lücke-Detert theory
(see Fig. 20) should be noted. Thus, for high-driving forces, the
ndivision
measured velocities can be regarded as falling in one of three regions
corresponding to either the high-velocity branch, low-velocity branch,
or a transition region (two-velocity region) with velocities inter-
mediate to these. Apparently the two-velocity region in the Cann
theory corresponds to the partial breakaway (transition) regions in
the Lücke-Detert theory. There was, however, no way to decide from
from the Lücke-Detert theory how wide the transition region should be.
In this respect the Cam theory offers improvement for it specifies
that the transition region should be as wide as the overlapping of
the high and low branches of the velocity versus composition curves.
Thus, with reference to Fig. 29-the limits of the transition region
can be defined by the outermost extremity, Cy, of the high-velocity
branch and the innermost extremity, Cg of the low-velocity branch.
These extremities can be estimated in terms of AF, 1, a, and B. The
procedure 18 given in Appendix B. The result 18 that alloys with
compositions in the range
ntissimenticiniame stomasacinio teikiamavaraistai turnirwani
Vn
.
.
.
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(14)
can be expected to show transition region behavior.
.
Sam
63:
There are two other aspects of the Cana theory which differ
from the Llcke-Detert theory. First, Cahn's theory predicts that
impurities which avoid the grain boundary, 1.e., E(x) positive, will
produce the same drag effect as those which are absorbed, i.e., E(x)
negative. The former are pushed ahead of the boundary, the latter
dragged along by the boundary. The Lücke - Detert theory did not
explicitly consider the case of impurities which avoid the boundary.
Second, Cahn's theory specifies which diffusion coefficient is
appropriate 1f D(x) 18 not constant and equal to Dp: Thug, one of..
the modifications Introduced by Gordon and Vandermeer (42) 18 automaa:
tically incorporated into the Cahn theory.
While Cahn's theory has taken care of many of the limitations
of the Lücke --Detert theory, it still does not explicitly take
orientation effects into account. It should be pointed out, though,
that there are provisions in the Cahn theory for crientation effects
to be included. This could be accomplished 1.f the profiles of E(x)
and D(x) were known for all the different orientation relationships
between grains. However, this does not appear to be possible at the
present time due to insufficient knowledge about the atomic structure
of grain boundaries.
Although the Cahn theory makes certain predictions with regard
to boundary-migration behavior, it is restrictive in the sense that
it still does not allow a decision from first principles as to
whether a measured velocity 18 a high velocity or a low velocity, or
whether a given driving force 16 a high one or a low one. This 18
2 OF 2
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MICROCOPY RESOLUTION TEST CHART
NATIONAL BUREAU OF STANDARDS - 1963
.
r.
fr
grounds. They also pointed out that in the composition range where
G 18 proportional to 1/23, the activation energy for migration
should be constant, whereas Aust and Autter observed an ever-
increasing activation energy. This led them to suggest that the
apparent agreement mentioned above may be only fortuitous.
(20) viewed the impurity effect in another way. He assumed
that impurity atoms segregate to the boundary in order to reduce
the porosity of the boundary. His approximate treatment led to
the following expression for the velocity.
G (C = 0)
(Avº - AV*)
KC
ln -
(15)
RT
1 Kc
Where r 18 a proportionality constant, avo 18 the porosity of the
boundary with no impurity atoms, AV* 18 the porosity of the boundary
when it is completely saturated with impurity atoms, K is the equili-
brium constant for segregation and C is the composition. This theory
leads to a saturation effect in the velocity at high compositions,
an effect which is not considered in either the Tücke - Detert or
Cahn theories. Li (51) compared the predictions of Eq. 15 with the
experimental data of Aust and Rutter (35) for random boundaries in
alloys of lead containing tin at 300°c. He pointed out that the
variation of in
hon with concentration for this data followed
the trend predicted by Eq. 15. He was also able to calcula te a
standard free energy of absorption for tin to the grain boundaries
of-lead and found a value of 11,500 cal/mole. However, in order to
make a significant quantitative check of the theory of L (20) more
Information 18 required regarding the porosity of the boundary with
and without impurity atoms.
and comprehensive
Correlation of Cahn's Theory with Experiment. No one has yet
tried to correlate Cahn's theory quantitatively with experiment. In
this section of the paper an attempt is made to analyze some of the
available experimental data on grain-boundary migration rates in
terms of the theory with the aim of calculating the parameters a,
B, and a. It is hoped this analysis will indicate just how detailed
experimuntal rate studies must be to determine the se
parameters. At the present time, only two sets of experimental data
are extensive enough to make these calculations: the recrystallization
study of Gordon and Vandermeer (42) on copper in aluminum and the
grain growth study of Holmes and winegard (46) on lead in tin. To a
limited extent, the data of Frois and Dimitrov (65) can be used to
supplement that of Gordon and Vandermeer. Since for grain growth
work, the equations of Cahn must be modified, it is most convenient
to discuss these two sets of experiments separately.
1.
i
Boundary Migration in Recrystallization; Copper in Zone-Refined
Aluminum. If the driving force is known, I can be calculated very
simply from Eq. 10 and the measured migration velocities of the
zone-refined metal. This assumes, of course, that the zone-refining
has produced metal pure enough so that residual impurity effects are
negligible. Figure 28 summarizes the temperature dependence of 1/4
for zone-refined alminum using the velocity measurements of several
different investigators (42)(65)(71) (83) and the driving forces in
Table II. The 1/2's calculated from the Gordon and Vandermeer (42),
Dimitrov and Frois (65), and the Gordon and El Babayonni (83)
recrystallization data agrer quite well and can be represented by
a single equation of the form
1 = 4, exp
(16)
with the constants 1. and L equal to 1.5 x 10°3 cal/mole/cm/sec and
14,000 cal/mole respectively. The 1/2's estimated from data of Aust (71)
cannot be fitted to Eq. 15 with these same constants; they seem to
yield the same activation energy but the preexponential factor falls
about 1 to 2 orders of magnitude too low. Since the Aust data was
obtained on single boundaries, this discrepancy may reflect an unknown
inherent difference between single-boundary migration and the growth
of grains during bulk recrystallization. In succeeding calculations
it will be assumed that , is given by Eq. 16 with the constants
listed above.
The calculation of a requires that some of the experiinental
velocities fall entirely within the low velocity region of the Cahn
theory. Such conditions will prevail at high-impurity levels; in
this case Eq. 12, which predicts that the reciprocal of velocity be.
a linear function of solute composition at constant driving force s ,
and temperature, should apply. As Fig. 29 shows, again from the
data of Gordon and Vandermeer, such a dependency of velocity on
composition does in fact exist for zone-refined aluminum alloys
containing copper in excess of 17 x 10*° atomic fraction. The
vertical lines through some of the data points represent the range
ما
م انی
of estimated uncertainties in the measured velocities. Within these
uncertainties, straight lines can fairly well describe the data.
Equation 12 also predicts that the intercept at zero composition
for the plots shown in Fig. 29 should equal 1/AF. However, it
appears experimentally that this intercept is zero, a result implying
that ^<<a C for the compositions invoived here. Numerical calcu-
lations do indeed veryify this condition. Thus, for these alloys
Eq. 12 reduces to
(17)
ac
The parameter a has been calculated for a number of different
temperatures using Eq. 17 and the values of the slopes or the straight
lines in Fig. 29 and other similar plots. Figure 30 shows the result-
ing temperature dependence of a to be exponential in as it was
for 1, at least over the temperature range covered. The resulta may
be described by the equation
(18)
RT
with the constants Q. = 5.6 x 10cal/mole/cm/sec/atomic fraction
and A = 28, 400 cal/mole.
For compositions of 17 x 10-0 atom fraction and less, however,
significant departures from the velocity-composition dependency
indicated by Eq. 17 were observed; the measured velocities were
always higher than those predicted. For the sake of clarity these
deviating data have not been included in Fig. 29. Since these
velcolties are also not high enough to be on the high velocity branch,
they may be regarded as falling in the transition region. This 18
1llustrated for a typical set of data in Fig. 31 which shows the
variation of velocity with copper content over the entire range oť
compositions studied at an annealing temperature of 139°c. The curves
shown in Fig. 31 were computed from Eq. 7 for various arbitrarily
assumed values of B. a and a were calculated from Eqs. 16 and 18;
AF was taken as 4 cal/mole (42).
It has been possible to calculate two of the three parameters in
Cahn's theory because certain of the experimental velocities could be
interpreted as falling in each of the two extremum regions of the
theory, and the simple expressions (Eqs. 10 and 17), each of which
contained only one drag parameter, could be used to represent the
data. The calculation of B 18 not nearly so straightforward. One
method of estimating B may be discussed with reference to Fig. 3).
It involves trial and error curve fitting to obtain the best
description of the experimental data by a theoretical curve. For
example, if a 8 of 10% sec/cm or greater were chosen to describe the
experimental data, then the velocities of the 2.1 and 4.2 x 10-6
atomic fraction alloys would be expected to fall on the high-velocity
branch. They, in fact, are almost two orders of magnitude too low.
On the other hand, if ß were taken to be 10% sec/cm or less, then
the velocities of these same alloys should fall on the low-velocity
branch. However, the velocities would then be an order of magnitude
greater than the theoretical curve would suggest. Since the velocities
Pall neither on the low- nor high-velocity branch but intermediate to
these, they may be regarded as transition region velocities and
based on our earlier discussion B must be limited to a value in the
vicinity of 10 sec/en at the temperature 139°C for which Fig. 31
applies.
An alternate means or calculating B is through the use of the
inequalities expressed in Eq. 14. In applying Eq. 14, it is
necessary tə kmow at a given temperature which alloys show transi-
tion region behavior. For example, the alioys containing 2.1, 4.2,
and 17 x 10°° atomic fraction copper in aluminum may be regarded as
transition region alloys at 139°c (See Fig. 31). Using these compo-
sitions, the known values of a, a, and AE and Eq. 14, it is possible
to calculate the values B can have to give agreement between theory
and experiment. Because of the restricted amount of available data,
such computations allow at best, only the calculation of upper and
lower bounds on B. The results of such an analysis appl.ied to the
Gordon and Vandermeer copper in aluminum data at several tempera-
tures are shown in Fig. 30, although at some temperatures the data
was extensive enough to give only an upper limit of 8. The tempera-
ture dependence of B may be approximately described by the equation
B = $. exp RT
(19)
with the constants 6. = 1 x 10-5 sec/cm and B = 27,000 cal/mole.
Also plotted in Fig. 30 18 the ratio o/82. Several points
resulting from the data of Frois and Dimitrov (65) are included.
These were obtained by plotting their data shown in Fig. 21 in the
į
way suggested by Eq. 13, 1.e., by plotting G versus c. Although the
data are scattered, they can be represented by a striaght line the
slope of which allows the calculation of a/82.
The drag parameters A, B, and I are the quantities in the overall
expression for boundary migration rate, 1.e., Eg. 7, which give rise
to the temperature dependency of G. Figure 32 illustrates the nature
zone-refined aluminum. The parameters used in ca).culating the se
curves were obtained from Eqs. 6, 18, and 19 with their appropriate
constants. For comparison purposes the actual data points for the se
compositions are also plotted in Fig. 32. The agreement between the
predictions of the theory and the experimental results is seen to be
excellent.
Boundary Migration During Grain Growth; Lead in Zone-Refined Tin.
The driving forces in grain growth are typically about 3 orders of
magnitude lower than those in recrystallization. Hence there is the
possibility, as Gordon (43) has pointed out, that the transition
from high to low velocities may be different in grain growth than it
is in recrystallization. Figure 33, which is a plot of some of Holmes
and Winegard's (46) grain growth data, shows when compared with Fig. 26
that the grain growth data do indeed fall in the category of low
driving force. The transition is continuous and there is no over-
lapping of the high-end low-velocity branches as in recrystallization.
This implies that all the measured boundary velocities are less than
the drift velocity of lead atoms across the boimdary, i.e. Q<
LE
*.1.
1
Therefore, all the data, if they follow the predictions of Cahn's theory,
should be described by Eq. 12.
To be usable here, Eq. 12 must be modified since in grain growth
AF continuously decreases with time and 18, hence, a function of G.
For grain growth the driving force may be written as
(20)
20V
where the symbols are as defined earlier. Experimentally, a can usually
be described for 18othe mal grain growth by the well-known equation
Arkt"
(21)
where t is time and k and n are constants. The rate of grain growth
may be defined as
G = TEM
(22)
Substituting Eq. 22 into Eq. 20 to eliminate A and substituting the resultant
into Eq. 12 for AF, the following expreseion is obtained or the velocity
in grain growth according to the Carn theory
6 - Copy
(23)
Thus, a plot of 1/G2 versus c should result in a straight line with slope
related to a and intercept related to .. gince grain growth rates may be
compared at constant time, the term inside the parentheses is then a
.",
constant.
..
..
The data of Fig. 33 bave been replotted in the form suggested by
.
.
..
Eq. 23. The results are shown in Fig. 34; they may be seen to be in
A
L
excellent agreement with theoretical prediction. Cabo's theory also can
predict the correct variation with composition of the apparent activation
energy for boundary migration in grain growsh. By taking the logarithm
of both sides of Eg. 23, and differentiating with respect to 1/T, the
following equation can be arrived at for the apparent activation energy:
Q app. - ( + aces)
(24)
A and L are defined as an Eqs. 18 and 16. Values of Q app. have been
calculated from Eq. 24 for seven alloys of lead in tin and are compared
in Table VI with the measured values given by Holmes and winegard (10).
a and 4 were calculated from the slopes and intercepts of Fig. 34.
L and A, determined from the data of Holmes and Winegard (46) were 6,000
and 18,000 cal/mole, respectively. The calculated values in Table VI
are within about 1, COO cal/mole of the measured values, which is probably
within experimental error of the measureà quantities.
The Extent of Boundary Segregetion. The basic hypothesis of the
Cahn theory (and also of the Llcke-Detert theory) is that solute atoms,
segregated at grain boundaries, are responsible for the retarding of grain
boundary migration. In a recent review article, Westbrook (4) has discussed
the status of lovestigations on grain boundary segregation. One of the
chief problems Westbrook delineated was with respect to the matter of grain
boundary width and 1ts importance in segregation. He referred to many of
the experimental measures of boundary segregation which seem to indicate
solute segregations distributed over boundary regions far broader than
the 1-4 atom diameters commonly considered normal for grain boundary
.
widths. For example, the micro-hardness measurements of Westbrook
-
P.
#
With
nh
TABLE VI
Comparison of Apparent Activation Energies for Boundary Migration
in Grain Growth in Tin Containing Lead
Alloy
Atomic Fraction
Lead
Q meas
cal/mole
(taken from data of
Holmes and Winegar! (10)
calc
cal/mole
(Uping Cobn's
theory)
2 x 10-5
5 x 10-5
1x 20-4
8,600
11,800
13,500
9,500
12,200
14,300
15,900
16,300
17,300
17,700
16,000
18,000
18,000
18,000
*10-4
* 20-4
* 20-3
**
*
.
.
75
and Aust (84) suggest that segregations at random high-augle grain
boạndaries in lead doped with tin extend as much as 50 u from the
boundary center into the bulk grain. - Such large distances are
difficult to understand both theoretically and experimentally in terms
of the 2 to 10 A widths of boundaries which soap bubble models, hard
sphere models (see Fig.9), thermodynamic considerations, and recent
field-ion microscopy work (see Fig. 3) seem to indicate....
The Cahn theory offers a means of checking whether or
not wide boundary segregations occur for migrating grain
boundaries in zone-refined aluminum containing cop-
per. The theoretical equations for a and a/A2 given in Appendix A all
contain a term 6 which can be thoughtof as the grain boundary half-width
and which essentially describes how far the segregation extends from the
boundary center. Unfortunately, the equations for a and alße also
contain two other quantities- which are not accurately known. These are
E, the interaction potential at the boundary center and Dep, the
impurity diffusivity in grain boundaries. Thus, for a given profile
combination of E(x) and D(x), there are two equations and three unknowns,
provided a and B are known from experiment. Lucke and Stuwe (6), however,
have offered several equations by which E. can be roughly estimated. One
such equation is
(25)
where Y 18 Young's modulus, o 18 Poisson's ratio, r is the radius of
the solvent atom and 1= where r' 18 the radius of the solute
nd
i
Arne
W
..
MALA..
LIS
"
LT.
atom. Equation 25 18 just an approximation because it was derived from
elastic energy considerations only and ignores any chemical or electronic
terms which might possibly contribute to E, If, however, it may be
absumed that an E. given by Eq. 25 18 approximately correct, thus
reducing the unknowns to two, then by using the experimentally determined
values of a and B in conjunction with the theoretical equations for a and
a! Be, both-6– and Des can be calculated for each of the assumed profiles
Bhown in Fig. 25. The quantit:.es 6 and Deo calculated in this manner
from Cabn's theory will be thus referred to as experimental values.
Furthermore, it is usually found from diffusion and quenching experiments
that the activation energy for boundary diffusion may be expected to be
about 0.4 to 0.5 that of lattice diffusion. Using this as a rough
approximation, it is possible to obtain an estimate of Dc/De which can
then be used as a check against the Dor calculated from the experimental
migration rate data and the Cahn equations.
Table VII Bumarizes the results of the calculations of 8 and the
ratio Dc/Dfor copper in aluminum using various assumed profile
combinations of E(x) and D(x) (see Fig. 25). The interaction potential,
E., calculated from Eq. 25, was -3080 cal/mole. The term m in profile
combination II essentially describes how far Dar extends relative to 8 (the
higher m 1s the narrower is the region of high diffusivity).
There are two profile combinations, I and V, which give answers
which are inconsistent with experiment. For example, 18 for combination I,
TABLE VII
0.0056
Experimental Grain Boundary Half-Width, 6, and Ratio
of Diffusion coefficients, Ded/Dr, Calculated with the Cahn Theory
for Aluminum Containing Coppere
log DGB/DL
Profile
(est'd from
Sexpt'i (108 D expt'1 diffusion data)
Combination
8-10
II, (n = 2)a 0.71
10.16
8-10
IL (n =
8-10
II. (n =
12.01
8-10
3.1
11.10 -
8-10
380
11.64
8-10
0.07
8-10
6.6
10.94
a. E. = -3080 cal/mole
b. ággumes D = DE
c. assumes Done F Doz and QGB = (0.4 – 0.5)&
d. m 18 a measure of how far Deo extends compared to 8.
78
8/D 18 calculated from Eq. Al and then compared to the same quantity
calculated from Eq. A2, there is a discrepancy between the two of eight
orders of magnitude. A like discrepancy 18 found for combination v. In
addition, the experimental 8'8 calculated for these combinations appear
to be much too small. Thus it is concluded that the profile combinations
I and V are not good representations of D(x) and E(x) for the migrating
grain boundaries considered here.
The experimental 8's calculated for the other profile combinations
and listed in Table VII suggest that the wide segregations apparently
found in some investigations (4) are not present in the grain boundary
.
....
....
.
.
migration experiments of aluminum-copper alloys. Only with profile
.
.
..
ita
combinations IIC and IV do the experimental s's appear to be much greater
*
*
A
'
.
,-
than a few Angstroms, but even in these cases they are still two orders
of magnitude smaller than those indicated by, for example, the Westbrook
and Aust (84) work. It should also be noted that the Doo', expressed as
the ratio Der/Dry calculated from experiment are in order of magnitude
agreement with what is expected from diffusiou data (compare columns 3 and
4 in Table VII), with the exception of IIc which is about two orders of
magnitude too high.
Finally, there 18 one trend that should be pointed out regarding
.
.."
--
:*••..
ins
*..*.
..
-
m
minimu
La
the calculated quantities 8 and DeRDI, which may have some further
bearing on the nature of grain boundaries. Consider the profile
combinations Ila, b, and c, where only the relative range over which
Dao extends compared to 8 18 varied. Both the experimental 8 and Doo/D
Increase in going from IIa to IIc but Dap/Dt increases to the point
where it is out of the range of values expected when the range of Dap
19
1s almost equal to the range of E(x) (combination IIC) This can be
interpreted to mean that the best profiles are ones in which the range
of DGR 18 less than the range of E(x) 1.e. the impurity-grain boundary
interaction potential extends out further from the boundary center than
does the highly disordered regim where the diffusivity is very bigh.
This may mean that E(x) 18 composed of contributions other than just
misfit energy (strain energy) whereas the diffusivity is determined
primarily by tails misfit. Thus the E. used in the calculations of 6 and
DER/D, may be too small. However, if a larger E. had been chosen for
calculation purposes, the resulting 8's would be smaller, but the
Dc/De would be very little affected. In either situation, wide
segregations are not predicted by the Cabn theory for grain boundaries
migrating during recrystallization of aluminum.
9A%4.<rs -
APPENDIX A
+,
+
•
Theoretical Equations for a and also
, prixmi.1
•ts
, y: • •
Profile Combination I
,
. , , , , , ,
.
* , .'
(A)
*
*
(A2)
*
Profile Combination II
a ## [alm (a-2) - : - 4) - (an-(43)
# - (A) 4(ge - ³)
(44)
Profile Combination III
- a g (as # - stub # )
* * * [eos #--i
ಇರಾನಾಗಿದ್ದು ಒಟ್ಟು ನಾದ
(48)
- fi
*
""
DIP
(46)
Profile Combination IV
c響​。
(47)
weream-
n。
(A8)
Profile Combination v
al8
Do
到​。
(A10)
品
​3.)
&
APPENDIX B
>
-
Calculation of the Extremities of the High and Low Velocity Branches
in the Cahn Theory for High Driving Forces
Equation 7 expressing the interrelationship between G, C, T,
etc. 18 rewritten in terms of the composition
OY
coate Srper G - Am B -
(BI.)
For the experimental results discussed in this paper the last term
(1/a) 18 always negligible. For high velocities the first term of
this equation can also be neglected.
mbus
:c- G - A e B .
(Be)
The high velocity extremum Cz can be calculated by differentiating
Eq., with respect to G and settiug the result equal to zero. We
obtain
(B3)
at this extremity. Subøtituting (B3) Into (B2) gives
(BH)
The other extremity Ca defining the macroscopic transition region can
be arrived at by considering Eq. Bl in the 11ght of low velocity
behavior. In this case, the third term in Eq. Bl becomes negligible
Bo that
Sande
(B5).
Again differentiating with espect to G and setting the result to zero
yields
(B6)
at Ca.
Substituting (B6) back into (B5) gives
Ca - Part 3
(B7)
Thus, alloys whose compositions fall in the range
|
<<
ple
(B8)
may be expected to show the macroscopic transition behavior.
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....
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Crystals of Aluminum, Technical Rep. to ARO(D) and ONR from
Illinois Institute of Technology, October 1962.
40) C. G. Dunn and P. K. Koh, Primary Recrystallization Textures in Cold
Rolled Si-Fe Crystals, Trans. AIME, 212 (1958).
41) J. W. Rutter and K. T. Aust, Migration of (100) Tilt Boundaries in
High Purity Lead, Acta Met., 13 (1965) 181.
42) Paul Gordon and R. A. Vandermeer, The Mechanism of Boundary Migra-
tion in Recrystallization, Trans. AIME, 224 (1962) 917.
43) Paul Gordon, A rationalization of the Data on Grain Boundary Migration
in Zone-Refined Metals as Influenced by Dissolved Impurities,
Trans. AIME, 227 (1963) 699.
44) G. F. Bolling and W. C. Winegard, Some Effects of impurities on Grain
Growth in Zone-refined Lead, Acta. Met. 6 (1958) 288.
45) E.L. Holmes and W. C. Winegard, Effects of Lead, Bismuth, Silver and
Antimony on Grain Growth in Zone-Refined Tin, J. Inst. Metals,
88 (1960) 468.
46) E. L. Holmes and W. C. Winegard, The Effect of Lead and Bismuth on
Grain Growth in Zone-Refined Tin, Trans. AIME, 224 (1962) 945.
47) Paul Gordon and T. A. El-Bassyouni, The Effect of Purity on Grain Growth
in Aluminum, Trans. AIME, 233 (1965) 391. "
48) J. K. MacKenzie, The Distribution of Rotation Axes in a Random Aggre-
gate of Cubic Crystals, Acta. Met., 12, (1964) 223.
49) M. L. Kronberg and F. H. Wilson, Secondary Recrystallization in Copper,
Trans. AIME, 185 (1949) 501.
50) D. C. Brandon, B. Ralph, S. Ranganathan and M. S. Wald, A Field-lon
Microscope Study of Atomic Configuration at Grain Boundaries,
Acta Met., 12 (1964) 813.
51) J. C. M. Li, Discussion to Grain Boundary Migration, by K. T. Aust
and J. W. Rutter in Recovery and Recrystallization of Metals,
Interscience Publishers, New York (1963) 160.
52) J. Hren, An Analysis of the Atomic Configuration of an Incoherent Twin
Boundary with the Field Ion Microscope, Acta Met., 13 (1965) 479.
53) G. W. Rathenau, L'Etat Solide, Solvay Conference Report, Brussels, 1952.
-
-
-
.
.
.
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-
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-
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.
.
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.
ir
1.
.
.
I
54) J. W. Cahn, The Impurity-Drag Effect in Grain Boundary Motion,
Acta. Met. 10 (1962) 789.
55) K. Lucke and K. Detert, A Quantitative Theory of Grain Boundary
Motion and Recrystallization in Metals in the Presence of Impurities,
Acta Met., 5 (1957) 628.
56) E. S. Machlin, Theory of Solute Atom Limited Grain Boundary Migration,
Trans. AIME, 224 (1962) 1153.
57) J. E. Burke and D. Turnbull, Recrystallization and Grain Growth,"
Progress in Metal Physics, 3 (1952) 220.
LIST OF REFERENCES
(continued)
58. P. A. Beck, Recrystallization of lead, Trans. AIME, Vol. 137,
(1940), p. 222.
59. J. 8. Smart, Jr. and A. A. Smith, Jr., Effect of Certain Fifth-
Period Elements on Some Properties of High-Purity Copper,
Trans . AIME, Vol. 152, (1943), p.103.
60. K. Lücke, G. Masing, and P. Nolting, Influence of Small Additions
of Mn, Ni, and Zn on the Recrystallization of Super Purity
Aluminum. Zeit. f. Metallk., Vol. 47, (1956), p. 64.
61. K. Detert and K. Lücke, Influence of Defined Small Amounts of
Impurities on the Recrystallization of Aluminum. Brown University
Report No. AFOSR-TN-103AD-82016 (1956).
62. G. F. Bolling and W. C. Winegard. Grain Growth in Zone-Refined
Lead. Acta Met., Vol. 6, (1958), p. 283.
63. E. L. Holmes and W. C. Winegard, Grain Growth in Zone-Refined Tin.
Acta Me:., Vol. 7 (1959), p. 411.
64. E. L. Holmes and W. C. Winegard, Effect of solute Atoms on Grain
Boundary Migration in Pure Metals. Canadian J. Phys., Vol. 39,
(1961), p. 1223.
65. C. Frois and 0. Dimitrov, Influence de Faibles Addition de
Cuivre et de Magnesium sur la Recristallisation de l'Aluminium
de Zone Fondue. Mem. Scient, Rev, Metallurg., vol. 59, (1962),
p. 643.
Shinin
-
..
Si
.
.
..
.
.
.
7.
66. C. Fro18 and 0. Dimitrov, 7 Colloque de Metallurgie, Ecrouissage,
Restauration, Recristallisation, Presses Universitaires de France,
Paris, France, (1963), p. 181.
67. D. Turnbull, Theory of Grain Boundary Migration Rates, Trans. AIME,
Vol. 191, (1951), p. 661.
68. P. A. Beck, P. R. Sperry and H. Hu, The Orientation Dependence of
the Rate of Grain Boundary Migration. J. Appl. Phys., Vol. 21,
(1950), p. 420.
69. E. L. Holmes and W. C. Winegard, Normal Grain Growth in Zone Refined
High-Purity Metals, Canadian J. Phys., Vol. 37, (1959), p. 496.
70. K. Detert and G. Dressler, Rekristallisationsverhalten von
Zonengeschmolzenum Nickel. Acta Met., Vol. 13, (1965), p. 845.
71. K. T. Aust and J. W. Rutter, ultra-High Purity Metals, American
Society for Metals, Metals Park, Ohio, (1962), p. 115.
72. T. S. Lundy, and J. F. Murdock, Diffusion of A126 and Mn54 in
Aluminum. J. Appl. Phys., Vol. 33, (1962), p. 1671.
73. R. E. Hoffman, F. W. Pikus, and R. A. Ward, Self-Diffusion in
• Solid Nickel, Trang. AIME, vol. 206, (1956), p. 483.
74. B. Okkerse, Self Diffusion in Lead, Acta Met., Vol. 2, (1954), p. 551.
75. J. D. Meakin and E. Kokholm, Self Diffusion in Tin Single Crystals,
Trans. Met. Soc. AIME, Vol. 218, (1960), p. 463.
76. W. Lange and D. Bergner, Messung der Kowngrenzenselbstdiffusion in
Polykristallinem Zinn, Phys. Status Solidi, vol. 2, (1962), p. 1410.
77. W. R. Upthegrove and M. J. Sinnot, Grain Boundary Self-Diffusion
of Nickel, Trans. ASM, Vol. 50, (1958), p. 1031. .
78. M. Wintenberger, Elimination des lacunes dans les Aluminiums Treis
Purs, Acta Met., Vol. 1, (1959), p. 549.
ttg
- pen
79. W. DeSorbo and D. Turnbull, Kinetics of Vacancy Motion in High-
Purity Aluminum, Phys. Rev., Vol. 115, (1959), p. 560.
80. D. Schumacher, W. Schule, and A. Seeger, Untersuchung Atomater
Fehlstellen in Verformten.. and Abgeschrecktem, Zeit, Naturforsch,
Vol. 17a, (1962), p. 228.
81.. A. C. Damask and G. Ji Dienes, Point Defects in Metals, Gordon
and Breach Science Publishers, New York, 1963, p. 260.
82. J. D. Murphy, Interdiffusion in Dilute Al-Cu Solid Solutions,
Acta Met., Vol. 9, (1961), p. 563.
83. P. Gordon and T. A. El-Bassyouni-Private Communication.
84. J. H. Westbrook and K. T. Aust, Solute Hardening at Interfaces
in High Purity lead - I Grain and Twin Boundaries, Acta Met., Vol. 11,
(1963), p. 1151.
1.:
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22
FIGURE TITLES
(continued)
Fig
15. Schematic variation of log migration rate with temperature
for constant driving force and composition.
Fig. 16. Schematic variation of log migration rate versus reciprocal
of absolute temperature for several values of composition.
17. Experimental data for recrystallization in aluminum and in
copper in aluminum alloys. From Gordon and Vandermeer (42).
Fig. 18. Apparent activation energy for boundary migration as a
function of atomic fraction of copper in zone-refined aluminum.
From Gordon and Vandermeer (42).
Fig. 19. Activation energy versus concentration for grain growth in
zone-refined tin with lead added. From Holmes and Wine gard (64).
Fig. 20. Schematic variation of log migration rate with composition for
constant driving force and temperature.
21. Variation of the rate of growth of grains during recrystallization
as a function of concentration for copper and magnesium in
aluminum. From Frois and Dimitrov (65).
Fig. 22. Experimental variation of log migration rate with log composition
for copper in aluminum. From Gordon and Vandermeer (42).
Fig. 23. Logarithmic plot of rate of grain boundary migration at 300°C
versus tin concentration for "random" grain boundaries in zone-
refined lead. From Aust and Rutter (35).
Fig. 24. Schematic 11lustration of migration rate-temperature curves in
breakaway region for (a) T3 «T, and (b) Tom TcFrom
Gordon (43).
-
-
-
-
-
Fig. 25. The interaction energy profiles E(x) and diffusivity profiles
D(x) used in calculating the impurity drag parameters a and
Appendix A.
rate
Fig. 26. Variation of boundary migration with composition for various
driving forces 11lustrating the three types of transition
behavior expected with the Cahn theory.
Fig. 27. (a) Schematic plot of the growth of recrystallized grains
as a function of time for compositions in the two-velocity
region of Cahn's theory.
(0) Variation of boundary migration with composition at
constant temperature and for a high driving force.
Fig. 28. Experimental plot of grain boundary mobility versus reciprocal
of temperature for zone-refined aluminum. Open circles -
Aust and Rutter (71). Open triangles - Frois and Dimitrov (65). Filled
triangles - Gordon and Vandermeer. (42). Filled circles – Gordon
and El Bassyonni (83).
rate
Fig. 29. Experimental variation of reciprocal of migration rate with
concentration at several temperatures for copper in zone-
.
.
.
.
.
.
.
.
.
.
. .
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2
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.
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.
4
refined aluminum.
Fig. 30. Teroperature dependence of the impurity drag parameters A, B,
and als for boundary migration in aluminum-copper alloys.
Fig. 31. Log migration velocity versus log concentration at 139°C.
. Theoretical curves calculated from Eq. 7 for various B.
Data points from Gordon and Vandermeer (42).
Fig. 32. Variation of log migration velocity with reciprocal of absolute
temperature for three alloys of copper in zone-refined aluminum.
Theoretical curves calculated from Cahn theory. Data points
from Gordon and Vandermeer (42).
29
Fig. 33. Grain growth rates at three temperatures for various alloys
of lead in zone-refined tin. Data taken from Holmes and
Winegard (46).
Fig. 34. Experimental variation of reciprocal of grain growth rate
squared with concentration at several temperatures for lead
in tin. Data taken from Holmes and Winegard (46).
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Figure 1. Movement of a 2° boundary in a zinc single crystal under
uniaxial stress. The boundary is the vertical change in
shading; the irregular horizontal line is a small step in the
sample surface which serves as a reference line. Top: original
position of boundary. Middle: boundary caused to move
0.1 mm to right. Bottom: boundary caused to move 0.4 mm
to left by stress reversal. From reference 9.
.
.
.
.
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Figunr 2 Displacement rate of edge dislocation bound.
aries as a function of the boundary angle under the action of
a constant shear stress of 9.19 psi at 350'C. Ref. 11.

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Figure 3. Field-ion microscope picture of a 27° grain :
boundary in tungsten. Arrons indicate position
of boundary. From reference 13.
.
.
':.'.'. ---
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olok
010
100
j'111. A. Orientation relation Intworn artifirinlly marlontoed
grains and cloformiral mntrix in anung wirre the matrix Wein
oxtended 20 por cent. mori the murdention wahrnusel by cutting
only, Thin nntrix is plotted in Atiiniloral projection.
Orientation of thin 111) poles of now crymnis churnc.
ferized by 40° LIPID rotniona.
+ Orientation of the (111) polrk of now crgintola chiaror.
forizcil by 400 (111) rotation..
Jelenl ait untion of (111) polos nitor clockwino 40° 11111
rotntion.
A Helmond mituntion of (111) polna nitor countnr.clockwimu
40'1111) rotation.
13 Belonil nituntion of tho (111) polea nftor coumtor.clock wino
4029 [111rointion.
From reference 26.
1
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OTI O
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(n) %one-rcfined Icad with tin additions less than 0.0001
WI pet: preferred grains with 0 = 25 to 60 deg about axes
shown (0), and "next-to prcſerred" grains with = 16 to
55 deg about axes shown ().
e ooi
(6) Zone-refined lead with tin additions from 0.0005 to
0.004 wt pct: preferred grains with a primarily 30 to 50
deg about axos within 12 deg of <111), and 26 to 28 deg
about axes within 6 deg of < 100>.
Figure 5. Rotation axes in lead and lead-tin alloys. From reference 35.

on 0 0
Olie
Figuro 6.
Composite stereographic plot of Aust and Rutter
data for "random" axes of rotation which relate each
now grain with matrix crystal by smallest amount. Note
regions of low axes density enclosed by arcs near
principle poles, Open circles - zone-refined lead, ref.35;
horizontal-line circles zone-refined lead, ref. 29;
vertical-line circles - silver alloys, and crossed-circles
- gold alloys in zone-refined lead, ref. 37.
Q
.
RATE OF GRAIN BOUNDARY MIGRATION
mm PER MIN. AT 300°C
000
لللللللللللاهما
0.0001
.002
boundaries and "special" grain boundaries. Ref. 36.
at 300°C (logarithmio) vo wt pot Sn for "random" grain
Fig. 7. A comparison of rate of grain boundary migration
WEIGHT PEACENT OF TH
"RANDOM
GRAN BOUNDARIES
20-21 (1000)
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TIN CONCENTRATION (WT. PCT)
Figure 8. Apparent activation energies for boundary migration versus
tin content in tin-in-lead alloys. Reference 36.
(010) POLES
(010) POLES
GRAIN BN
GRAIN 8
GRAIN A
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Fig. 9. Symmetrical high angle coincidence tilt boundaries in simple square lattice. Open-circle atoms form co-
incidence sublattice common to both grains. Left: Two grains related by 370 rotation around (001) (the
direction perpendicular to page) forming high density coincidence site boundary with l-in-5 caincidence
site density. Note minimal disturbance of lattices at boundary. Right: Two grains related by rotation
around 2001) of a few degress less than 37º. Coincidence sublattice dislocation at boundary superimposes
substructure on boundary. Note increased boundary width and porosity, particularily at dislocation core,
and creation of coherency strain well into both grains.

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A9 expresses the angles by which the relativo grain and the boundary
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ACTIVATION ENERGY, O
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ORIENTATION DIFFERENCE, I, IDEGS)
<100> TILT
Figure 11. Apparent activation energies for boundary migration,
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1000

0
0
300°C
RATE OF GPAIN BOUTDART MIGRATION (en bes kind
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Fig. 12. Migration rates of tilt boundaries in zone-refined lead.
Ref. 41.

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SCHEMATIC
· IMPURITY
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In VELOCITY
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REGION
IMPURITY
CONTROLLED

RECIPROCAL TEMPERATURE

ORNL-LR-DWG 64003R

SCHEMATIC
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MEASUREABLE
RANGE OF G
G, LOGARITHM OF BOUNDARY MIGRATION RATE
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Alloy
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SCHEMATIC
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. uri.....
.....
.:
:
Figas
v
o
,
W
.
*.
*;..
.-.
.-
.'-
.
-
...
-
-
-
- -.
ORNL-DWG 65-10141
HIGH DRIVING FORCE
- INTERMEDIATE DRIVING FORCE
.
..
LOW DRIVING FORCE
0.014
Fig 26
10-1
100
101
..
10²
103
104

.
.
.
.
ORNL-DWG 65–10436

MACROSCOPIC
VELOCITY G -
Ò, AVERAGE DIAMETER OF LARGEST
RECRYSTALLIZED GRAIN
To MICROSCOPIC VELOCITY GA
-MICROSCOPIC VELOCITY G2
SCHEMATIC
097
t, ANNEALING TIME
ORNL-DWG 65-10140

100
HIGH VELOCITY
REGION
GB
E
TRANSITION
+
011
Mit
time. Asismis-
LOW
VELOCITY
NREGION
0.01
-
104
10
..
og
.
in
with the
iis .
.
ORNL-DWG 65-10139

log † (com/otocie)
2100017 (OK)
in
Am
s
!: *venimente
i
ngredients
in
the media vinde inte minden rendin
vantagem
M
i nimalistinitati ....
daras
n
itaria
IP-
RECIPROCAL OF VELOCITY (cm / sec )**
(x106)
N
w
A
125°C
50 100
CONCENTRATION OF COPPER ( atomic fraction )
150
170°C
200
189°C -
250 (x10-6)
...
155°C
139°C
....
ORNL-DWG 65-10150

.

ORNL-DWG 65-10135
log DRAG PARAMETERS a AND B
Q/22 -
log DRAG PARAMETER *182

1
2.4
2.8
3.2
3.6
4.0
4000/ (°K)
Fig 30
.
. .
.
WY
***, London Dia
ORNL-DWG 65-10134
B=102
i B=103
'B=104
B=105
log VELOCITY. (cm/sec)
.
10-9
10-8
10-7 . 10-6 40-5 10-4 10-3
CONCENTRATION (atomic fraction),
10-2

ORNL-DWG 65-10142
log VELOCITY (cm/sec)
ZONE REFINED AL
A4.3 ppm Cu -
68 ppm cu
.
1.4
1.8
2.2
2.6
3.0
3.4
3.8

zabis
1000/700K)
...
..
.
.
.
.
.
..

to
..
.
13
V
r
.
.-
Y
ORNL-DWG 65-10143
-2.5
log GRAIN GROWTH RATE (cm/sec)
227 °C
188 °C
134 °C
-4.5
10-5
bania 33
40-4
10-3
CONCENTRATION OF LEAD (atomic fraction)
L
hawa
*Mieren
--
ORNL-DWG 65-10144
188 °C
(x109)
RECIPROCAL OF GRAIN GROWTH RATE
SQUARED (cm/sec)-2
0.4
0.3
34 °C
0.2
0.1
227 °C
. 2 . 4 . 6 8. 10 (x1024)
CONCENTRATION OF LEAD (atomic fraction)
5
4
.
.
.
.
A
.
A
.
Skr

.
.
.
END
.
DATE FILMED
12/ 7 /65