Scripta Materialia 50 (2004) 1241–1245
www.actamat-journals.com
Modeling of abnormal grain growth in textured materials
O.M. Ivasishin a, S.V. Shevchenko a,*, S.L. Semiatin b
a Department of strength and ductility of inhomogeneous alloys, Institute for Metal Physics, 36 Vernadsky Street, Kiev 03142, Ukraineb Air Force Research Laboratory, AFRL/ML, Wright-Patterson Air Force Base, OH 45433-7817, USA
Received 11 October 2003; received in revised form 31 December 2003; accepted 30 January 2004
Abstract
The effect of initial texture on the occurrence of abnormal grain growth (AGG) was modeled via a 3D Monte-Carlo approach. A
diagram of texture characteristics which give rise to AGG was derived. AGG was associated with periods of linear growth behaviour
of the largest grain within the microstructure.
� 2004 Published by Elsevier Ltd. on behalf of Acta Materialia Inc.
Keywords: Abnormal grain growth; Texture; Potts model
1. Introduction
Grain growth in polycrystalline solids such as that
which occurs during annealing treatments is a process
driven by the reduction of the overall grain-boundary
energy, thus yielding an increase in the average grain-size Dav. Usually, two limits of this process can be dis-
tinguished––normal grain growth and AGG. If there is
no anisotropy in the grain-boundary properties, normal
grain growth occurs, and the grain-size distribution
function remains uniform and self-similar. In the case of
AGG, grain-boundary motion is restricted by one or
several factors, and microstructure evolution proceeds
nonuniformly by the growth of several �abnormal’grains. A particular kind of AGG, sometimes called
‘‘catastrophic’’ AGG, results in the final volume of
abnormal grains more than three orders of magnitude
larger than that of the surrounding matrix grains [1].
Such heterogeneous microstructures may lead to poor
mechanical properties and thus should be avoided dur-
ing thermomechanical processing.
Following the initial fundamental work of Hillert [2]and Gladman [3], considerable theoretical attention has
been focused on the problem of AGG [4–10]. Most of
the models postulated that the initial grain-size distri-
bution was not uniform, but contained at least one large
*Corresponding author. Tel.: +380-444-240-120; fax: +380-444-243-
374.
E-mail address: [email protected] (S.V. Shevchenko).
1359-6462/$ - see front matter � 2004 Published by Elsevier Ltd. on behalf
doi:10.1016/j.scriptamat.2004.01.036
grain. The possible growth of such larger grains was
then analyzed. However, as shown recently by Rios
[10,11], AGG can develop even from a uniform grain-
size distribution for the case in which grain growth is
restricted by a pinning force which decreases slowly with
time. The Monte-Carlo (MC) method has been suc-cessfully applied to simulate AGG for different starting
conditions [12–15].
Based on recent numerical (MC) and analytical sim-
ulations of the AGG phenomenon [6,16–18], it appears
natural to consider both normal and AGG as limiting
cases within a unified framework in which different
kinds of anisotropy and grain-boundary pinning asso-
ciated with texture (i.e., spatial anisotropy of grain-boundary velocity) are included as factors. These factors
define the place of any particular microstructure in an
N -dimensional space. Hence, a processing map can be
developed to characterize the expected system behav-
iour. The objective of the present paper was to establish
the portion of such a map dealing with the effect of
texture-component width and intensity on grain growth
via 3D MC simulations. A method to define/recognizeperiods of AGG in MC simulations was also developed.
2. Theoretical aspects of abnormal grain growth
The theoretical model of AGG focuses on an isolated
abnormal grain A surrounded by normally growing
grains comprising the ‘‘matrix microstructure’’. In gen-
eral, the local grain-boundary velocity is proportional to
of Acta Materialia Inc.
1242 O.M. Ivasishin et al. / Scripta Materialia 50 (2004) 1241–1245
grain-boundary curvature. Hence, the abnormal grain
will grow faster than the surrounding grains because the
motion of the abnormal grain-boundary is directed en-
tirely outside the grain. Following Hundery [19,20], thecontact area of neighboring grains A and j (i.e., grain-
boundary common for A and j) is given approximately
as SAj ¼ pR2j . Let B represent the coefficient that defines
the dependence of grain-boundary velocity on grain-
boundary curvature. Then, the growth velocity of
abnormal grain A is given by the following relation
[19,21]:
dRA
dt¼ � B
R2A
Xj
SAj1
RA
�� 1
Rj
�
¼ � BR2A
Xj
pR2j
1
RA
�� 1
Rj
�: ð1Þ
0 10 20 30 40 50 600.0
0.2
0.4
0.6
0.8
1.0
Rel
ativ
e gr
ain
boun
dary
mob
ility
Grain boundary misorientation
Fig. 1. Relative grain-boundary mobility as a function of misorien-
tation.
The number of grains j neighboring the abnormal
grain is proportional to its surface area 4pR2A. Due to the
relatively large number of j-grains, for the case of an
equiaxed microstructure, the summation using an aver-
age Rj can be performed to obtain the following
expression:
dRA
dt�
BR2A4p
2R2jAV
R2A
1
RjAV
�� 1
RA
�
¼ B4p2R2jAV
1
RjAV
�� 1
RA
�: ð2Þ
As AGG occurs, the ratio RA=RjAV increases, and the
AGG rate approaches asymptotically its limit, i.e.,
limRA!1
dRA
dt� B4p2R2
jAV
1
RjAV
� �¼ B4p2RjAV ;
limRA!1;t!1
dRjAV
dt¼ 0: ð3Þ
Eqs. (2) and (3) thus indicate that the diameter of a
large-enough abnormal grain increases linearly with
time.One can distinguish three stages of AGG. Stages 1
(initial) and 2 (linear, or steady growth) in the growth
behaviour of an individual abnormal grain correspond
to the kinetics described by Eqs. (1)–(3). The final stage
(i.e., Stage 3) corresponds to the situation when the
abnormal grain reaches the sample surface or when
several abnormal grains meet each other and form a
coarse microstructure.In the case of real materials, only ‘‘catastrophic’’
AGG can be observed experimentally [14,22–26].
However, the initial stage of AGG often cannot be
recognized on 2D metallographic images. By contrast,
the size of every grain can be tracked separately during
numerical simulations. Thus, every abnormal grain can
be detected in Stage 2 and studied during the simulation.
For this reason, computer simulation can be an espe-
cially fruitful tool for the investigation of AGG.
3. Simulation technique
Grain growth was simulated using the MC (Potts)
model described in Ref. [27]. Thus, only a brief
description is given here. The model domain was formed
by a 3D cubic array of model units (MU), each of whichrepresented a point in a cubic lattice. The size of indi-
vidual grains was taken to be equal to the diameter of a
sphere containing the same volume. The length measure
was assumed to be equal to 1 MU, and the time measure
was 1 MCS. During one MCS, the number of elemen-
tary flip-simulation trials was equal to the number of
sites in the model domain. All simulation cases utilized a
model domain of 2503 MU and started from the sameequiaxed initial microstructure with Dav ¼ 5:4 MU.
4. Results and discussion
The main results of the present work consisted of a
number of MC simulations used to establish the effects
of texture on AGG. These simulations were also used to
develop a map describing the effects of anisotropy on the
occurrence of AGG. A relative GB mobility parameter
M was introduced as a normalization factor for the
dependence of the elementary orientation-flip probabil-
ity on intergranular misorientation. M was assumed tobe small (0.05) for low-angle boundaries and equal to
unity for high-angle boundaries (Fig. 1).
A reference simulation, Case R, was run for an un-
textured material. As expected, normal grain growth
was predicted for this case (Fig. 2(a), solid line).
Fig. 2. Comparison of MC-predicted: (a) grain growth kinetics and (b), (c) grain-size distribution and microstructures after 50 MCS for several Case
1 simulations. The solid line in (a) corresponds to normal grain growth.
O.M. Ivasishin et al. / Scripta Materialia 50 (2004) 1241–1245 1243
4.1. Case 1: AGG in a material with a sharp, single-
component texture
For Case 1, the initial texture consisted of a single,
symmetric, Gaussian-shaped component (with half-
width W and times-random intensity I ¼ 20) and a
random background. The texture maximum was at
gð45; 45; 45Þ, using Bunge notation for Euler angles. The
intensities of Gaussian and background components are
unambiguously related. For a given half-width Wmaximum possible I can be determined as the times-random intensity of the ‘‘pure’’ Gaussian texture com-
ponent, when the random component is absent.
Simulations 1–30, 1–25, 1–20, 1–10, 1–05, and 1–02
were run with I ¼ 20 and W ¼ 30�, 25�, 20�, 10�, 5�, and2�, respectively. The predicted grain growth kinetics for
several of the initial textures (Cases 1–10 and 1–30) are
Fig. 3. (a) AGG for Cases 1–10 (1––average grain-size, 2––largest grain-size
growing grains are white, and normally-growing grains are grayscale.
shown in Fig. 2(a). Fig. 2(b) and (c) present the
respective simulated grain-size distributions and micro-
structures.It is apparent that the AGG phenomenon correlates
strongly with texture width. AGG was predicted to
start more quickly for the narrow textures. In addition,
the abnormal grains for all Case 1 simulations did not
belong to the initial texture component. It is also
apparent that rapid average grain growth corresponds
to the final stage of AGG. Moreover, AGG does not
affect the average grain growth kinetics from the MCsimulation until the middle part of Stage 2 because
only a few large abnormal grains are surrounded by
thousands of normal grains. Hence, the period of the
active AGG (i.e., Stage 2) does not coincide in time
with the period when the average grain growth is rapid
(Fig. 3).
) and (b) MC-predicted microstructures at various times; abnormally-
1244 O.M. Ivasishin et al. / Scripta Materialia 50 (2004) 1241–1245
4.2. Case 2: AGG in a material with a texture of various
strengths
For Case 2 simulations, similar Gaussian-shapedinitial textures were assumed as in Case 1, but the tex-
ture-component intensity was varied; viz., W ¼ 5� and
I ¼ 2, 5, 10, 30, 60, and 100 times random correspond-
ing to Cases 2–2, 2–5, 2–10, 2–30, 2–60, and 2–100. The
results indicated that the number of abnormal grains
decreases in the more-strongly textured modeling vol-
umes. However, the effect of AGG on average grain
growth rate was very pronounced in the heavily texturedmaterials. This result is in good agreement with the
other AGG simulations. The small number of abnormal
grains leads to a longer Stage 2, or linear AGG. At the
same time, the surrounding grains grow normally, i.e.,
they follow power-law growth with a time exponent
equal to or less then 0.5.
4.3. Criterion of the AGG occurrence within the 3D
Monte-Carlo modeling volume
As seen from Eq. (1), the largest abnormal grain has
the maximum growth rate. Hence, a simple criterion for
identifying AGG during numerical simulations can be
introduced; i.e., AGG is associated with the periods
when the largest grain growth rate is linear with a
velocity close to the maximum possible for a given
grain-boundary mobility. Fig. 4 illustrates this criterion.
It should be noted also that Stage 2 of AGG for Case 2–30 ended before the period of rapid average grain growth
had begun. The phenomena of the linear growth kinetics
of the abnormally growing grains was recently found
from the 2D MC simulations of AGG in [15]. It has also
0 10 20 30 40 50 60 70 80 90 10010
15
20
25
30
35
40
Aver
age
diam
eter
of t
he la
rges
t gra
in, M
U
Time, MCS
Fig. 4. MC-predicted growth kinetics for the largest grain in the
modeling volume. AGG occurred for Cases 2–30 (squares) between 10
and 30 MCS (dashed line). Results for Case 1–1 (normal grain growth)
and Case 2–02 (weak texture) are shown by circles. The solid line is the
best power-law fit to the normal grain growth case.
been shown in [15] that the formation of abnormal
grains had statistical origin.
4.4. N -dimensional AGG diagram: I–W cross-section
The simulation results for Cases 1 and 2 demon-
strated that a continuous change in the characteristics of
abnormal growth is observed depending on the model-ing-volume spatial anisotropy. Hence, it is possible to
determine the full set of spatial anisotropy parameters
(such as texture intensity and halfwidth, grain-boundary
energy and mobility dependence on misorientation,
different kinds of grain-boundary pinning forces, etc.)
which control AGG. With an appropriate set of physical
or modeling experiments, an N -dimensional diagram of
AGG can thus be constructed in principle. For example,Case 1 corresponds to the 1D section of such an AGG
diagram along the W direction for a material with a one-
component texture + background and Case 2 to the
section of this diagram along the I direction.
To construct the I–W section of such a map (Fig. 5),
30 additional simulations based on the 1503 MU domain
were conducted. The initial textural states for all 40 runs
(including Cases 1 and 2) are marked in Fig. 5 by thesolid points. This set of simulations revealed the entire
spectrum of possibilities between the limiting cases of
normal grain growth and AGG. Region D, or the
‘‘catastrophic’’ AGG region, corresponds to those cases
for which the average grain-size increases more than five
times during Stages 1–3 of AGG. This type of AGG
occurs for strong and narrow textures with weak, ran-
dom backgrounds. Hence, the probability of AGG is
0 10 20 30 40 50 60 70 80 90 1000
5
10
15
20
25
30
35
G
G
TR
D
C
B
A
Tex
ture
Hal
fwid
th W
, deg
.
Texture Intensity I, x-random scale
Fig. 5. The I–W section of the AGG diagram based on 42 MC sim-
ulation runs. A: normal grain growth kinetics, B: kinetics with pro-
nounced slow and fast periods of ‘‘normal-like’’ grain growth, C: AGG
kinetics, and D: ‘‘catastrophic’’ AGG. The region marked TR con-
notes the transition between low- and high-angle boundaries in de-
grees. The G–G line corresponds to the maximum possible texture
intensities.
O.M. Ivasishin et al. / Scripta Materialia 50 (2004) 1241–1245 1245
greatest for a uniformly-textured material containing a
small number of randomly-oriented grains.
The analysis used to derive maps such as Fig. 5 is
valid only for the early stages of AGG, corresponding tothe start of microstructural evolution. The position of a
particular microstructure on the N -dimensional map
changes continuously because of the evolution of texture
(as well as other parameters) during grain growth. For
example, the weakening of a strong texture during AGG
can be associated with the migration of a microstructure
from regions D or C on the AGG map to region A, thus
describing the post-AGG microstructure. The oppositesequence may occur during the formation of a strong
annealing texture, thus resulting in later-time AGG.
Later-time AGG can also be caused by other factors,
whose influence increases during the normal grain
growth process. For instance, the delayed initiation of
abnormal growth following a normal growth period has
been reported for Al–1 wt% Ga [28]. Another example
of late stage AGG, caused by sulfur segregated at thegrain-boundaries during the early stage of grain growth,
has been observed during the isothermal annealing of
electrodeposited nanocrystalline Ni and Ni–Fe alloys
[29].
5. Summary and conclusions
The transition between normal and AGG in textured
materials was analyzed using a 3D Monte-Carlo (Potts)
model in which initial texture intensity/halfwidth char-
acteristics could be carefully quantified. It was demon-
strated that abnormal grain growth and rapid growth ofthe average grain-size do not coincide. Periods of AGG
are associated with periods of linear growth behaviour
of the largest grain within the microstructure. Periods of
normal grain growth are characterized by the similar
behaviour of the average grain-size and the largest
grain-size. Overall microstructure evolution in relation
to AGG can be expressed in terms of an N -dimensional
AGG diagram introduced in this work. An I–W cross-section of the AGG map was constructed based on 40
simulation runs.
Acknowledgements
The present work was supported by the Air Force
Office of Scientific Research (AFOSR) and the AFOSREuropean Office of Aerospace Research and Develop-
ment (AFOSR/EOARD) within the framework of
STCU Partner Project P-057A. The encouragement of
the AFOSR program managers (Drs. C.H. Ward and
C.S. Hartley) is greatly appreciated.
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