the effect of initial grain size distribution on abnormal grain growth in single-phase materials

THE EFFECT OF INITIAL GRAIN SIZE DISTRIBUTION

ON ABNORMAL GRAIN GROWTH IN SINGLE-PHASE

MATERIALS

W. E. BENSON{ and J. A. WERT

Department of Materials Science and Engineering, University of Virginia, Charlottesville,VA 22903, U.S.A.

(Received 2 December 1997; accepted 4 June 1998)

AbstractÐA model that describes the temporal evolution of an initial grain size distribution (GSD) hasbeen used to analyze the e�ect of initial GSD on grain growth. Based upon the work of Hillert, and Hun-deri and Ryum, this model uses a ®nite di�erence technique to numerically solve the continuity equation.A formal method has been developed to create synthetic GSDs. The model is used to study the evolutionof both continuous and discontinuous GSDs. For either case, two parameters describe the deviation fromsteady state: umax is the ratio of the largest grain radius to the critical grain radius, and Fo is the initialfraction of grains larger than twice the critical radius. The analysis is applied to lognormal GSDs obtainedfrom experimental results. It is shown that, for all initial GSDs with nonzero Fo, a transient period ofabnormal grain growth characterizes the initial stages of growth. Fo and umax allow approximate predictionof the duration and extent of abnormal grain growth. # 1998 Acta Metallurgica Inc. Published by ElsevierScience Ltd. All rights reserved.

1. INTRODUCTION

Grain growth is the process driven by reduction ingrain boundary energy that results from an increase

in the mean size of an array of grains. It can becharacterized as either normal or abnormal

growth [1]. In normal grain growth, the initial grainsize distribution (GSD) is relatively narrow and

evolves in a uniform fashion. Abnormal graingrowth is identi®ed by the presence of a few large

grains which grow at the expense of the smallergrains that surround them. The mean-®eldapproach has been shown to be a particularly suc-

cessful method for analyzing normal and abnormalgrain growth [2±6]. Initially developed by

Feltham [7], Hillert [1] and Louat [8], this set oftheories considers the size change of grains within

an environment which represents the averaged e�ectof the entire collection of grains [9]. Theoretical

results [10] supported by computer simulations [11]based on the mean-®eld approach have shown that

the growth rate of large grains is less than that forthe matrix of smaller grains. These studies conclude

that abnormal grain growth does not arise solelyfrom the presence of large grains in the initial GSD.

Only when other factors, such as di�erences ingrain boundary mobility or surface energy aniso-tropy, are considered is abnormal grain growth

possible in pure materials [10±13].

These studies are based upon the utilization of

Hillert's rarely observed steady state GSD as the

matrix GSD, a fact that Rios [14] determined to be

a major shortcoming. To understand the in¯uence

of the initial GSD on the growth of a single grain

larger than the maximum size of the GSD, Rios

applied Hillert's basic mean ®eld analysis to exper-

imentally measured GSDs with di�erent shapes. By

assuming that the matrix GSD remains approxi-

mately invariant during growth, he concluded that

there are cases in which the initially large grain

remains larger than the maximum size of the matrix

GSD. In keeping with Hillert's theory [1], Rios

noted this condition to be transient. He pointed out

that if the matrix GSD can evolve for very long

times, then the abnormally large grain should even-

tually approach the distribution. Rios proposed

that numerical solution of the continuity equation

might allow the description of the growth of an

abnormally large grain in the presence of an evol-

ving GSD. The present article describes e�orts

aimed at understanding the growth of ®nite frac-

tions of large grains in the presence of an evolving

GSD.

Following the numerical approach of Hunderi

and Ryum [4], Hillert's model has been im-

plemented to study the e�ect of the initial GSD on

grain size evolution in an ideal single-phase poly-

crystalline material. The initial deviation from

steady state is quanti®ed. It is shown that a GSD

Acta mater. Vol. 46, No. 15, pp. 5323±5333, 1998# 1998 Acta Metallurgica Inc.

Published by Elsevier Science Ltd. All rights reservedPrinted in Great Britain

1359-6454/98 $19.00+0.00PII: S1359-6454(98)00220-1

{To whom all correspondence should be addressed.

5323

evolves toward Hillert's steady state distributionregardless of the initial GSD, in agreement with

previous analytical results from both grain growthand particle coarsening simulations [15±17]. It isalso shown that the initial GSD can have a large

e�ect on the rate of change of critical grain size,and on the time required for the GSD to reachsteady state. Finally, two parameters that character-

ize the initial deviation from steady state are usedto make approximate predictions of the extent andduration of abnormal grain growth.

2. DEVELOPMENT OF THE MODEL

2.1. Finite di�erence model

In his classic paper, Hillert [1] applied the theoryof second phase particle coarsening [15, 16] to the

grain growth process and derived the well-knowngrowth equation for a grain with radius R:

dR

dt� K

�1

Rcÿ 1

R

��1�

where K is a constant that contains the grainboundary mobility and the grain boundary energy.In the present analysis, K= 1. Rc is the critical

grain radius, de®ned as the radius below whichgrains shrink and above which grains grow. For athree-dimensional analysis, Hillert showed that

Rc � 9=8�R, where �R is the average grain radius. LetP�R,t� dR represent the number of grains at time twith size between R and R� dR. In R±t space, thisdistribution function must satisfy the continuity

equation:

dPdt� ddR

�PdR

dt

�� 0: �2�

equations (1) and (2) explicitly describe the tem-

poral evolution of the initial GSD and Rc [4].GSD evolution was simulated by solving

equations (1) and (2) numerically. The integrations

were performed by means of the Crank±Nicholson®nite di�erence technique based on a central di�er-ence scheme [18] over a rectangular grid with con-

stant di�erences Dtk and DRh. At the tk + 1 timestep, the distribution function depends on the valueof Rc. An equation for Rc is implicit in the con-straint that the volume of the system remains con-

stant: �10

R2 dR

dtP�R,t� dR � 0: �3�

equation (3) was solved after each time step and the

new value of Rc was used in the numerical inte-gration to ®nd the GSD for the subsequent timestep. The simultaneous solution of equations (1)

and (2) requires that Rc be constant for each timestep. This approximation implies that the simulatedgrowth rate at intermediate times will be slowerthan the actual growth rate. Numerical results given

in Section 3 show that all initially non-steady stateGSDs converge to the asymptotic limit (Hillert's

steady state GSD), an observation that indicates thestability of the present method.The choice of an initial GSD remains an import-

ant aspect of the problem. A good test of the modelis to use an initial distribution for which the evol-ution is known. Hillert used the fact that during

steady state grain growth, Rc increases monotoni-cally with time. He found that the steady stateGSD in three dimensions is given by

P�u� �3u�2e�3 exp

� ÿ62ÿ u

��2ÿ u�5 �4�

where u � �R=Rc� is de®ned as the relative grainsize. Since this distribution exists at all times during

steady state grain growth, the distribution given interms of u should not change as a function of time,even though Rc increases. Steady state grain growth

simulation results, given in Section 3, show that Rc

follows the well-known parabolic grain growth law,indicating that the present numerical implemen-tation of Hillert's model accurately describes grain

growth kinetics.Equation (4) demonstrates that grains with radii

larger than or equal to 2Rc cannot exist during

steady state grain growth. Therefore, the evolutiontoward steady state of GSDs with some fraction ofgrains larger than 2Rc is of particular interest. For

initial GSDs of this type, Hillert predicts a transientperiod of abnormal grain growth during the earlystages of grain growth. To make a systematic study,

a consistent method for generating initial GSDs isnecessary. In the present work, two types of GSDwere studied: continuous GSDs in which all sizeclasses between the upper and lower size limits con-

tain grains, and discontinuous GSDs, in whichsmall and large size classes contain grains, whilesome intermediate size classes contain no grains.

Both types of initial GSD were generated usingequation (4) as a basis. By requiring that each in-itial GSD have equal Rc, reliable comparisons can

be made of the rate of evolution of a GSD towardsteady state, and of the corresponding increase ofRc. Therefore, throughout the present work, an

arbitrary initial value of Rc=50 has been used.

2.2. Initial grain size distribution development

To create continuous GSDs, two steady state

GSDs were combined in the following manner:

1. Two di�erent values for Rc are used inequation (4) to generate two initial GSDs.

2. The distribution with larger initial Rc is multi-plied by a relative amplitude factor, A< 1.

3. The two initial distributions are summed and theresulting distribution is normalized.

BENSON and WERT: EFFECT OF INITIAL GRAIN SIZE DISTRIBUTION5324

This method allows the creation of initial distri-

butions with tails that extend beyond 2Rc.

Continuous GSD development is simpli®ed by the

need for only two input parameters, the ratio of in-

itial values of Rc and A. Two important parameters

emerge from this construction: umax, the ratio of the

largest grain radius to Rc, and Fo, the initial frac-

tion of grains with radius larger than 2Rc. The

value of umax is determined by the largest input Rc,

and Fo is determined by both the ratio of the initial

values of Rc and A. Continuous GSDs can be

characterized as either unimodal or bimodal,

depending on the number of maxima present in the

initial GSD.

To generate discontinuous GSDs, a steady state

GSD was combined with a number of grains in a

larger size class in the following manner:

1. A steady state GSD is created with an initial

Rc1<50 and multiplied by an amplitude factor,

A1.

2. The size class at Rc2 corresponding to the umax

of interest is assigned an amplitude, A2.

3. The relative amplitude factor, A, is the ratio of

A2/A1.

4. The two distributions are summed and the

resulting distribution is normalized.

Because the number of large size classes that con-

tain grains can be precisely controlled, the develop-

ment of discontinuous GSDs is simpli®ed. The

input parameters include umax, so Fo is the only

parameter that emerges from the synthesis of dis-

continuous GSDs. Since all discontinuous GSDs

have two maxima, they can be characterized as in-

itially bimodal.

Several di�erent GSDs were created using the

methods described. GSD evolution was followed

through at least 5� 104 time steps. In all cases, this

allowed at least a four-fold increase in Rc. The par-

ameters associated with each initial GSD tested are

shown in Table 1. The values of Fo used in these in-

itial GSDs correspond to volume fractions between

0.01 and 0.1. These values were used so that reliable

comparisons with the evolution of the experimen-tally determined GSDs used in Rios' study [14]

could be obtained. GSD 1 is the steady state distri-bution, used here as a test of the model. Alsoshown is the nature of the initial con®guration: C/

U means the initial GSD is continuous/unimodal,C/B means the initial GSD is continuous/bimodaland D means the initial GSD is discontinuous.

3. RESULTS

3.1. Non-steady state GSD evolution

The evolution of an initially non-steady state con-

tinuous bimodal GSD is shown in Fig. 1. Figure1(a) shows P vs R as time progresses for GSD 2.The initial GSD is indicated by a heavy line andthe series of curves represents a set of periodic

snapshots of the GSD as it changes with time.Figure 1(b) shows P vs u for this GSD. For illustra-tive purposes, only the initial and ®nal distributions

are shown. While the number of large grains andthe size of the largest grains increase with time,umax decreases with time until the steady state GSD

is reached. The evolution of R2c for GSD 2 is shown

and compared with that of the steady state GSD inFig. 1(c). At the beginning of growth, a period of

rapid increase in R2c is evident. The vertical dotted

line indicates the time at which the slope of thecurve for GSD 2 becomes approximately constantand equal to that for the steady state GSD. At this

time, GSD 2 can be considered to have reachedsteady state. To determine more precisely the timeat which a GSD reaches steady state, the evolution

of F (the fraction of grains larger than 2Rc), is fol-lowed. As a GSD approaches steady state, Fapproaches zero. Figure 1(d) shows a plot of F as a

function of time for GSD 2. The value of Fincreases during the initial stages of grain growth.This is a characteristic of bimodal GSDs, as dis-

cussed below. Once a maximum value is attained, Fdecreases and reaches zero at time = 18 000, thesame time at which the steady state is indicated inFig. 1(b).

Table 1. Parameters used to create initial grain size distributions

GSD Rc2/Rc1 A umax Fo Con®guration

1 1 1 2 0 C/U2 1.82 0.05 3.2 0.009 C/B3 2.63 0.05 3.6 0.028 C/U4 2.13 0.008 4 0.006 C/U5 2.27 0.02 4 0.017 C/U6 2.55 0.03 4 0.028 C/U7 3.09 0.03 4 0.035 C/U8 5.81 0.007 6 0.028 C/B9 6.06 0.037 4 0.015 D10 6.16 0.025 4 0.02 D11 4.08 0.005 4 0.0015 D12 4.17 0.009 4 0.003 D13 7.78 0.006 7 0.002 D14 15.15 0.005 10 0.002 D

BENSON and WERT: EFFECT OF INITIAL GRAIN SIZE DISTRIBUTION 5325

To understand the relative importance of the par-

ameters umax and Fo, the model was used to investi-

gate the evolution of continuous GSDs with equal

umax and di�erent Fo, and equal Fo and di�erent

umax. The results of these analyses are shown in

Fig. 2. In Fig. 2(a), R2c is plotted against time for

GSDs 4±7 with umax=4.0 and Fo=0.006, 0.017,

0.028 and 0.035, respectively. For reference, R2c

against time for the steady state GSD is included.

As Fo increases, the transient period of abnormal

grain growth becomes shorter, as evidenced by the

time required for the slope of each curve to become

constant and approximately equal to that of the

steady state GSD. Also, the increase in R2c is much

greater and occurs more rapidly for GSDs with lar-

ger Fo. Figure 2(b) shows a plot of F as a function

of time. The ®gure shows that F declines much

more rapidly for GSDs with higher Fo. The

decrease in F throughout grain growth is a charac-

teristic of unimodal GSDs, as discussed below. The

time required for these GSDs to reach steady state

varies from 12 000 for Fo=0.035 to 25 000 for

Fo=0.006.

Next, the evolution of continuous GSDs with

equal Fo and di�erent values of umax was analyzed.

Figures 2(c) and (d) show results for GSDs 3 and 8

with Fo=0.028 and umax=3.6 and 6.0, respectively.

Figure 2(c) shows R2c as a function of time for these

GSDs, with the plot for the steady state GSD

included for reference. As umax increases, R2c

increases more rapidly during the initial stages of

growth and the slope of the R2c vs time plot

becomes constant and approximately equal to that

of the steady state GSD at earlier times. Figure 2(d)

shows a plot of F as a function of time. The ®gure

shows how F evolves di�erently for initially unimo-

dal and bimodal GSDs with the same Fo. As umax

increases, the time required for these GSDs to reach

steady state (for F to reach zero) is reduced.

To explore further the e�ects of initial GSD, the

evolution of discontinuous GSDs was modeled. The

results for GSD 9 (umax=4, Fo=0.015) are given in

Fig. 3(a), where P is shown vs R and u for

time = 0 to 5� 104. The ®gure shows a rapid shift

of the GSD towards the largest initially occupied

grain size class and a corresponding rapid decrease

in umax. The P vs R plot shows how the two initial

maxima of this bimodal GSD change with time.

The magnitude of the maximum associated with the

large grain size classes continually increases and

that associated with the small grain size classes con-

tinually decreases. The intermediate grain size

Fig. 1. Evolution of a non-steady state continuous bimodal GSD. (a) P vs R for GSD 2. The initialGSD is indicated by a heavy line and the direction of time is indicated. (b) P vs u for GSD 2. The in-itial distribution is shown by a heavy line and the ®nal steady state distribution is shown. (c) R2

c as afunction of time for GSD 2 and the steady state GSD. The vertical dotted line indicates the time at

which the distribution reaches steady state. (d) F as a function of time for GSD 2.


Fig.2.(a),(b)Model

resultsforumax=

4.0

anddi�erentFo.(a)R

2 casafunctionoftimeforGSDs4±7.Theplotforthesteadystate

GSD

isshownforcomparison.(b)Fas

afunctionoftimeforGSDs4±7.Forillustrativepurposes,thehorizontalaxisis

shownto

time=

25000.(c),(d)Model

resultsforFo=0.028andumax=3.6

and6.0.(c)R

2 casafunctionoftimeforGSDs3and8.(d)FasafunctionoftimeforGSDs3and8.


classes, initially unpopulated, gradually become

occupied. Eventually, the two parts of the distri-

bution join and the distribution becomes a continu-

ous/bimodal GSD. As grain growth continues, the

maximum at the small size classes disappears and

one maximum remains that is associated with the

initially populated large grain size class. At this

point, the GSD is characterized as continuous/

unimodal with non-zero F. As grain growth con-

tinues, F approaches zero and the GSD approaches

steady state.

The evolution of F in all bimodal GSDs studied

shows an increase during the initial stages of

growth. Figure 3(b) is a plot of F as a function of

time for GSD 9. Also shown on the ®gure are the

GSDs at times associated with certain critical fea-

tures of the plot. The evolution of F can be divided

into two stages. In Stage I, F increases to some

Fig. 3. Model results for discontinuous GSD evolution. (a) P vs R and u for GSD 9, time = 0±5�104.The initial GSD is indicated by a heavy line and the direction of time is indicated. (b) F as a functionof time for GSD 9 with the GSD at di�erent critical times shown. There are two stages of evolution.Stage I is characterized by an increase in F to some maximum value. At this point, the GSD becomescontinuous/unimodal. Stage II shows a decrease in F. From this point, gradual evolution to the steady

state occurs.


maximum value. This period of increase is associ-

ated with the development of the bimodal GSD

into a continuous/unimodal GSD, rapid increase in

R2c , and rapid decrease in umax. Stage II begins

when F reaches its maximum value. While R2c still

increases and umax still decreases, they do so at

diminishing rates. F decreases rapidly to a small

value. Continued GSD evolution is then character-

ized by a prolonged decline in F to zero as the

GSD approaches the steady state.

Simulations were performed for discontinuous

GSDs to evaluate the separate e�ects of umax and

Fo. The results for GSDs 10±12 with umax=4.0 and

Fo=0.02, 0.0015 and 0.003, respectively, are shown

in Figs 4(a) and (b). In Fig. 4(a), R2c as a function

of time is shown. There appears to be little di�er-

ence in the increase in R2c for GSDs 11, 12 or 1.

However, the plots for GSDs 11 and 12 diverge

with time from that of GSD 1. Only at times in

excess of 35 000 do the slopes of these two plots

become nearly constant and equal to that of GSD

1. The plot of R2c as a function of time for GSD 10

shows a rapid increase during the early period of

growth followed by a decline in the rate of increase.

At time = 12 000, the R2c curve for GSD 10

becomes constant and approximately equal to that

of GSD 1, indicating that it has reached the steady

state. To more accurately determine the time at

which a GSD reaches steady state, the change in F

with time is followed. Figure 4(b) shows the two-

stage evolution of F for GSDs 10±12. The ®gure

shows that as Fo increases, the maximum value

attained by F is greater and is reached earlier. Once

this maximum value is reached, F declines faster

and reaches zero earlier.

Simulations were also performed for discontinu-

ous GSDs with Fo=0.002 and umax=7 and 10. The

results for GSDs 13 and 14 are shown in Figs 4(c)

and (d). Figure 4(c) shows R2c as a function of time

for these GSDs. The plot for GSD 13 begins to

diverge from that of the steady state during the

early stages of growth. For GSD 13, the time

required for the slope of the R2c curve to become

constant and approximately equal to that of GSD 1

is near 75 000. The curve does not show a rapid

increase in R2c ; rather, there is a prolonged period

in which R2c increases at a slow rate. The plot for

GSD 14 shows a rapid increase in R2c followed by a

decline in the rate of increase. The slope of the

curve becomes constant and approximately equal to

Fig. 4. (a), (b) Model results for discontinuous GSDs with umax=4.0 and di�erent Fo. (a) R2c as a func-

tion of time for GSDs 10±12. Shown for comparison is the plot for the steady state GSD. (b) F as afunction of time for GSDs 10±12. (c), (d) Model results for discontinuous GSDs with Fo=0.002 anddi�erent umax. (c) R2

c as a function of time for GSDs 13 and 14. The plot for GSD 1 is included forreference. (d) F as a function of time for GSDs 13 and 14. For illustrative purposes, the time axis is

shown to 60 000.


that of GSD 1 by about 12 000. Figure 4(d) showsF as a function of time for GSDs 13 and 14. The

®gure shows that as umax increases, F reaches agreater maximum value at earlier times. As graingrowth continues, F decreases much more rapidly

and reaches zero earlier.

4. DISCUSSION

4.1. Evolution of non-steady state GSDs

For all initially non-steady state GSDs studied,the growth rate of R2

c was larger than that of thesteady state GSD during the early stages of evol-

ution. At some point, these growth rates becameessentially equal. This early period of rapid growthrate can be characterized as a transient stage of

abnormal grain growth, predicted by Hillert to be anecessary part of the evolution of an initially non-steady state GSD with nonzero Fo [1]. The e�ects

of Fo and umax on this period of abnormal graingrowth are shown for continuous GSDs in Figs 2(a)and (c) and for discontinuous GSDs in Figs 4(a)and (c). As either of the two parameters is

increased, the duration of the transient stage ofabnormal grain growth is reduced, while the extentof abnormal grain growth is increased. This con-

clusion is supported by following the evolution of Fand R2

c . It is seen that F reaches zero at earliertimes and R2

c attains larger values for continuous

and discontinuous GSDs with larger Fo and umax.As the greater number of large grains grow in

GSDs with large Fo or umax, they consume many

small grains during the early stages of grain growth.Thus, a rapid increase in R2

c could be expectedduring these early times. Because the number of thesmallest grains decreases so much more rapidly

than the number of largest grains increases, Fdecreases rapidly in continuous GSDs with large Fo

or umax and the GSD reaches steady state earlier.

Conversely, for continuous GSDs with small Fo orumax, as large grains grow during the early stages ofgrain growth, they consume fewer small grains.

Thus, R2c increases less rapidly. Because the rate of

decrease in the number of smallest grains is closerto the rate of the increase in the number of largestgrains, F falls less rapidly and the GSD reaches

steady state at later times. Therefore, as either Fo orumax is increased, the initial deviation from thesteady state is enhanced, and the conditions that

promote the transient stage of abnormal graingrowth are made more severe.The two-stage evolution of F for discontinuous

GSDs, illustrated in Fig. 3(b), shows how the pre-sence of a small number of large grains in a GSDa�ect abnormal grain growth. Stage I is character-

ized by an increase in F. There are two features ofthe initial GSD that contribute to this increase. The®rst contribution comes from the expansion of theupper part of the initial GSD seen in the P against

R plot in Fig. 3(a). The larger grains in this part ofthe GSD grow at the expense of smaller grains in

both parts of the distribution. The growth of thesegrains increases F because the number of occupiedsize classes greater than 2Rc increases with each

time step. The second contribution from the initialGSD occurs when the largest grains in the lowerpart of the GSD reach the size class that is just

greater than 2Rc. Now the two contributions actsimultaneously to raise the rate of increase in F. Asmore intermediate size classes become populated

with grains, the contribution from the upper part ofthe GSD diminishes, and the slope of the F againsttime curve is reduced. The initial rate of increase inF is directly related to the initial deviation from

steady state, as shown in Figs 4(b) and (d). As Fo

or umax increase, F reaches a greater maximumvalue earlier. Thus, discontinuous GSDs with larger

Fo or umax evolve to become continuous/unimodalGSDs faster. At the beginning of Stage II, theseGSDs deviate much more from steady state than in-

itial GSDs with smaller Fo or umax. Continued graingrowth proceeds in the manner described above forcontinuous/unimodal GSDs with large Fo.

4.2. Analysis of initially lognormal GSDs

Rios took experimental grain volume distri-butions measured by NunÄ ez and Domingo [19] and

converted them into grain radiusdistributions [14, 20]. Via this procedure, heobtained the following expression for the grain size

distribution referred to the critical radius:

P�u� ��

1

�2p�1=2��

1

us

�exp

�ÿ 1

2

�ln�eu�s

�2��5�

where the values of s and e determine the shape of

the distribution. For the NunÄ ez±Domingo GSD,s = 1/3, e = 1.1814 and umax=2.4. Rios varied thevalues of s and e in an e�ort to investigate the

e�ect of the shape of the initial GSD on the growthof an abnormal grain in single-phase materials. Hecreated a ``sharp'' GSD (umax=1.6) with s = 1/6

and e = 1.0426 and a ``broad'' GSD (umax=4.1)with s = 2/3 and e= 1.9152. Rios concluded that alarge grain in broad GSDs always had negativegrowth rates and so would always be absorbed into

the steady state GSD. He also concluded that thelarge grain in sharp distributions may have eitherpositive or negative growth rates, and that this

large grain tends to become trapped at a size largerthan the maximum relative grain size of the steadystate GSD, and so is never absorbed into the steady

state GSD.With equation (5) as a basis, the present model

was used to follow the temporal evolution of the

NunÄ ez±Domingo GSD, as well as Rios' broad andsharp GSDs. The results are shown in Fig. 5 forthese GSDs with initial Rc=50. Figure 5(a) showsP against R for the NunÄ ez±Domingo GSD and R2

c


against time for these three GSDs. Included on the

plot is R2c as a function of time for Hillert's steady

state GSD. GSD evolution in terms of R occurs in

the same general fashion as previously discussed.

For Rios' sharp GSD, the plot of R2c as a function

of time initially lies below that of the steady state

GSD because umax<2. At time = 8000, the R2c plot

for Rios' sharp GSD crosses that of the steady state

GSD. The slope of this plot becomes constant and

approximately equal to that of the steady state

GSD at time = 1.25� 104. The R2c plot for the

NunÄ ez±Domingo GSD deviates initially from

Hillert's steady state GSD, but its slope becomes

constant and approximately equal to the slope of

the steady state plot by about t= 7�104. The plot

for Rios' broad GSD, however, shows a large initial

deviation from Hillert's steady state GSD. At

time = 3.5�104, the slope of this plot becomes

constant and approximately equal to that of the

steady state GSD. Figure 5(b) shows the temporal

evolution of the NunÄ ez±Domingo GSD as well as

Rios' sharp and broad GSDs in terms of the rela-

tive grain size. The time at which each GSD

reached steady state is shown. The results for the

evolution of R2c as well as the relative grain size

show that these three GSDs span a range of initial

deviation from steady state. Rios' sharp GSD and

the NunÄ ez±Domingo GSD are initially close to

steady state, while Rios' broad GSD is initially far

from the steady state. As the initial deviation from

steady state is increased, the time required to reach

steady state is reduced, while the amount of abnor-mal grain growth is increased. These results followthose for Hillert-based initial GSDs and are in

agreement with previous investigations [10, 11].However, these results di�er for the evolution ofRios' sharp GSD. The disagreement arises from the

fact that his sharp GSD has an initial umax<2. Forthis GSD, there are no grains larger than 2Rc andFo=0. Evolution of initial GSDs of this type is

characterized by a broadening of the distributionuntil the steady state condition is met. The largestgrains in these initial GSDs are not large enough tobe considered abnormally large grains. The present

implementation of Hillert's model shows that thereare no cases in which an abnormally large grainwill never be incorporated into the steady state

GSD given su�cient time and sample volume.

4.3. Extent and duration of transient abnormal graingrowth

The results from continuous, bimodal and log-normal GSD evolution simulations show that the

initial deviation from steady state determines howan initially non-steady state GSD will evolve tosteady state and how the increase of R2

c occurs.

Since all of the initial GSDs had the same initialRc, the amount of excess grain growth with respectto the steady state GSD can be determined for eachGSD. The excess grain growth is the di�erence in

Fig. 5. Model results for lognormal GSDs. (a) P vs R for the experimentally-measured NunÄ ez±Domingo GSD and R2

c as a function of time for Rios' broad, sharp and the NunÄ ez±Domingo GSDs.Also shown is the plot for Hillert's steady state GSD. For illustrative purposes, time is shown to

2.5� 104. (b) P vs u at di�erent times for these GSDs.


the value of R2c for a GSD at the time it has

reached steady state and the value of R2c for the

steady state GSD at that same time. This di�erence

is normalized by dividing by R2c at time = 0. Figure

6(a) shows excess grain growth plotted as a function

of umax. The straight line is a best-®t line for the

points on the plot. The plot shows that, as the rela-

tive size of the largest grains in the initial GSD

increases, the extent to which abnormal grain

growth occurs increases. There are no prior exper-

imental or modeling results available for compari-

son, but the low values of excess grain growth can

be most likely attributed to the e�ect of Fo on GSD

evolution. From the discussion in Section 4.1, the

large initial volume fraction of large grains limits

excess grain growth because the relatively small

volume fraction of small grains is consumed before

the large grains appreciably increase their size. This

conclusion is supported by following the e�ect of

Fo on the duration of the transient stage of abnor-

mal grain growth.

Figure 6(b) shows the relation between the time

required to reach steady state and Fo for initially

non-steady state GSDs. The straight line best-®t is

shown on the plot. The ®gure shows that, as Fo

increases, the time required to reach steady statedecreases. The correlation is good for both continu-

ous and discontinuous GSDs with large Fo. This in-dicates that, for these GSDs, the time required toreach steady state depends more on the fraction of

large grains initially present than the con®gurationof the GSD at time = 0. The ®gure shows a widerange of times to reach steady state for GSDs with

small Fo. The initial con®guration of these GSDshas a greater e�ect than Fo on the duration of thetransient abnormal grain growth stage. For GSDs

with small Fo, discontinuous GSDs with small umax

require longer times to reach steady state thanGSDs with large umax. In both ®gures, there isreasonable correlation so that predictions can be

made about the degree and duration of the transi-ent stage of abnormal grain growth from character-istics of the initial GSD.

5. CONCLUSIONS

A model based on Hillert's formulations has beenused to analyze the e�ect of initial GSD on grain

growth in single-phase materials. The evolution of

Fig. 6. Predicting the extent and duration of the transient stage of abnormal grain growth. (a) Excessgrain growth as a function of initial umax. (b) Time required for an initially non-steady state GSD toreach steady state as a function of Fo. In each plot, the best ®t line is shown. The legend applies to

both ®gures.


both continuous and discontinuous GSDs has beenanalyzed. The results show:

1. Indices can be used to quantify the deviationfrom steady state for a GSD by consideringeither the maximum relative grain size (umax) or

the fraction of grains larger than 2Rc (Fo).2. The extent to which abnormal grain growth

occurs during the early stages of growth in non-

steady state GSDs can be correlated with umax.3. The time required to reach steady state for in-

itially non-steady state GSDs depends on both

Fo and the initial con®guration of the GSDwhen Fo is small, but predominately on Fo whenFo is large.

AcknowledgementsÐThis work was supported by GeneralElectric Corporate Research and Development,Schenectady, NY 12301, U.S.A. The authors are gratefulto P. Follansbee, M. Henry and M. F. X. Gigliotti fortheir helpful comments and suggestions.

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the effect of initial grain size distribution on abnormal grain growth in single-phase materials

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