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: shevchsv@imp.kiev.ua (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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