A GRAIN-BOUNDARY DIFFUSION MODEL OF DYNAMIC

GRAIN GROWTH DURING SUPERPLASTIC

DEFORMATION

BYUNG-NAM KIM1{, KEIJIRO HIRAGA1, YOSHIO SAKKA1

and BYUNG-WOOK AHN2

1National Research Institute for Metals,1-2-1 Sengen, Tsukuba, Ibaraki 305-0047, Japan and2Department of Mechanical Engineering, Taejon National University of Technology, Taejon 300-717,

South Korea

(Received 23 April 1999; accepted 16 June 1999)

AbstractÐDynamic grain growth during superplastic deformation is modelled on the basis of a grain-boundary di�usion mechanism. On the grain boundary where a static and a dynamic potential di�erencecoexist, matter transport along the boundary is assumed to contribute to dynamic grain growth throughdepositing the matter on the grain surface located opposite to the direction of grain-boundary migration.The amount of the di�usive matter during deformation is calculated for an aggregate of spherical grainsand is converted to the increment of mean boundary migration velocity. The obtained relationship betweenthe strain rate and the dynamic grain growth rate is shown to be independent of deformation mechanisms,provided that the grain growth is controlled by grain-boundary di�usion. The strain dependence, strain-rate dependence and temperature dependence of grain growth predicted from this model are consistentwith those observed in superplastic ZrO2-dispersed Al2O3. # 1999 Acta Metallurgica Inc. Published byElsevier Science Ltd. All rights reserved.

Keywords: Grain boundaries; Grain growth; Di�usion (interface); Mechanical properties (plastic); Ceramics(structural)

1. INTRODUCTION

The rate of grain growth during superplastic defor-

mation (dynamic grain growth) is generally higher

than that caused by annealing alone (static grain

growth). Such enhancement of grain growth rate

has been shown by the experimental fact that the

grain size of a superplastically deformed material is

larger than that of the material undergoing the

same thermal cycle without deformation [1±3]. As a

mechanism of dynamic grain growth, several theor-

etical models have been proposed. Clark and Alden

[1] assumed that grain-boundary sliding produces

an excess of vacancies in the boundary region

which increases the grain boundary mobility.

Wilkinson and Ca ceres [4] proposed a damage zone

formed around the triple points by grain-boundary

sliding, through which grain-boundary migration is

accelerated. Morral and Ashby [5] and Sato et al.

[6] pointed out that the switching of certain paired

grains can contribute to the dynamic grain growth.

Fridez et al. [7] explained the enhanced grain

growth by the reduced activation energy of grain-

boundary movement due to applied stress.

These models are based on the physically possible

mechanisms, respectively, and the predictions of the

models have been shown to reproduce the trend ofexperimental results. Thus, analyses based on thecharacteristic deformation mechanism of each ma-

terial seems to be essential for understandingdynamic grain growth behaviour. However, modelsof dynamic grain growth based on di�usion mech-

anisms are not found, although grain-boundary dif-fusion is regarded as a primary mechanism ofsuperplasticity, particularly in ®ne-grained ceramics.In this study, the enhanced migration of grain

boundaries is modelled on the basis of grain-bound-ary di�usion, and the quantitative relationship isderived between plastic deformation and dynamic

grain growth.

2. ENHANCED GRAIN GROWTH BY GRAIN-BOUNDARY DIFFUSION

2.1. Grain-boundary di�usion

Deformation by grain-boundary di�usion wasanalysed for a single spherical grain subjected to

unidirectional tensile stress [8, 9]. According to theanalysis, matter transport occurs owing to the po-tential di�erence generated along the grain bound-

ary by the applied stress, and the resultant shapevariation of the grain was converted to the macro-scopic plastic strain. When the uniaxial stress, s, isapplied to a single spherical grain of radius R, as

Acta mater. Vol. 47, No. 12, pp. 3433±3439, 1999# 1999 Acta Metallurgica Inc.

Published by Elsevier Science Ltd. All rights reserved.Printed in Great Britain

1359-6454/99 $20.00+0.00PII: S1359-6454(99)00201-3

{To whom all correspondence should be addressed.

3433

shown in Fig. 1, the migration velocity of the grainboundary in the normal direction vn(y) is obtainedby

vn�y� � 2�3cos2yÿ 1�DbdOskBTR2

�1�

and the strain rate, eÇ , in the stressed direction(y=0) becomes [9]

_e � vn�0�R� 4

DbdOskBTR3

�2�

where y is the angle from the stress axis, Db is the

grain-boundary di�usion coe�cient, d is the grain-boundary thickness, O is the atomic volume, kB isBoltzmann's constant and T is the absolute tem-

perature. Here, the mean normal stress wasassumed to be zero, as in Herring [10]. Equation (2)holds for a single grain of radius R. For actual ma-terials with grain size distributions, equation (2) can

be represented by

_�e � cbDbdOs

kBT �R3

�3�

where _�e is the mean strain rate, �R is the meangrain radius and cb is a constant. However, theabove model of grain-boundary di�usion assumes

the conservation of volume as a boundary con-dition, so that the matter transport occurs on theown grain boundary and no changes occur in grain

size.In actual fact, a grain boundary is the boundary

between two adjacent grains. Hence the matter of

the two grains is involved in the stress-directedgrain-boundary di�usion as depicted in Fig. 2. On

the grain boundary of higher potential, matter dis-solves from grain surface into grain boundary. The

dissolved matter di�uses through the grain bound-ary according to the potential gradient, and isdeposited on the grain surface of lower potential. If

the amount of matter dissolving into the grainboundary (I1, I2) and the amount of matter deposit-ing on the grain surface (O1, O2) are identical

between Grains 1 and 2 as I1=I2 and O1=O2,there would be no dynamic e�ects on grain growth.However, in most cases, at temperatures at which

grain-boundary di�usion occurs, static grain growthalso occurs. The migration of grain boundaries indi-cates the existence of a potential di�erence and mat-ter transport across the grain boundaries. Hence,

the amount of the dissolving matter would bedi�erent at the adjoining grain surfaces (I1 $ I2).This would also be the case for the depositing mat-

ter (O1$O2).For example, let us consider that the grain

boundary in Fig. 2 moves toward the upper direc-

tion and a dynamic potential gradient is exerted onthe boundary by applied stress. Then, I1>I2 and O1

< O2 holds due to the static potential di�erence,

and the dynamic potential di�erence along thegrain boundary controls the di�usion through theboundary. Considering only the net transport ofmatter, I2 and O1 are negligible for the simplest

case. This situation can be accomplished when thepotential m of the two adjacent grain surfaces are inthe order of m(surface of Grain 1) > m(grainboundary) > m(surface of Grain 2). The dissolutionfrom the grain surface of lower potential to thegrain boundary (I2) and the deposition from the

grain boundary to the grain surface of higher po-tential (O1) can thus be neglected. As a result, mat-ter transport between the two adjacent grains isaccelerated by the additional path of grain-bound-

ary di�usion, leading to enhanced grain growth.

2.2. Dynamic components of grain growth

The amount of di�usive matter can be calculatedfrom equation (1). When the spherical grain isdeformed by grain-boundary di�usion under the

condition of volume conservation, the amount ofmatter contributed to the di�usion is half of thevolume swept by the grain boundary. Therefore, the

amount of di�usive matter per grain per unit timeis

_V g � 1

2

�p0

2pRsinyds vn�y� � 3:08pDbdOskBT

�4�

where s is the distance from the stress axis along

the grain boundary, as shown in Fig. 1. Multiplyingthe number of grains per unit volume by equation(4), we obtain the amount of the di�usive matterper unit volume per unit time as

_V � 2:31DbdOs

kBT �R3

�5�

Fig. 1. Schematic description of a spherical grain stressedin the uniaxial direction.

Fig. 2. Schematic illustration of grain-boundary di�usion.I is the amount of matter dissolving into the grain bound-ary and O is the amount of matter depositing on the grain

surface.

KIM et al.: DYNAMIC GRAIN GROWTH3434

where the dimension of _V is timeÿ1. Since the areaof grain boundaries per unit volume is proportional

to 1�

�R , the mean migration velocity of grainboundaries �v is given by

�v � a1 _V �R � 2:31�a1�cb� �R _�e �6�

where a1 is a constant. Finally, by using a linear re-

lationship between the migration velocity and thegrain growth rate � �v � a2

_�R �, the contribution ofgrain-boundary di�usion to the grain growth d �R d is

represented by

d �R d � a �Rd�e �7�where a is 2.31a1/a2cb. In the following, the mean

grain radius and the mean strain rate are simplyrepresented as R and e, respectively.

2.3. Grain growth during superplastic deformation

Assuming that the mechanisms of static anddynamic grain growth work independently of each

other, we can estimate the grain growth duringsuperplastic deformation by adding these contri-butions. The static contribution dRs can beobtained by di�erentiating the rate equation of sta-

tic grain growth with respect to time t

Rns ÿ Rn

0 � kt �8�where R0 is the initial radius and k is a rate con-stant. Accordingly, the grain growth during super-

plastic deformation dR is represented by

dR � k

nR1ÿndt� aRde �9�

Providing the time history of strain during defor-mation, we can calculate the grain radius at any

time by the numerical integration of equation (9).However, the value of a has to be determined ex-perimentally. Since the value of a is a�ected by

such factors as the grain shape, the size distribution,the value of cb, etc., analytical determination is dif-®cult.Under the condition of constant strain rate,

equation (9) can be solved to be

Rn � k

na_e��Rn

0 �k

na_e

�exp�ane�: �10�

Equations (9) and (10) are just the same as thoseproposed empirically by Sato et al. [6]. They also

proposed an approximated equation

ln

�R

Rs

�� ae �11�

for the negligibly small contribution of static graingrowth. However, we propose a more accurate ap-proximation of equation (10) as

R � Rs � R0

�exp�ae� ÿ 1

� �12�

For a change in strain rate less than a factor of 10,

this equation gives a good approximation ofequation (10).

Regardless of the same ®nal form in the presentand the earlier study [6], it is distinctive in the pre-sent analysis that equations (9) and (10) are derived

from the physical mechanism of grain-boundary dif-fusion. Further discussion should be required onthe validity of the assumption that the mechanisms

of static and dynamic grain growth work indepen-dently. The following consideration may, however,give a rational basis to this assumption. Since the

static and the dynamic potential di�erences exist indirection parallel and normal to the grain bound-ary, respectively, the directions of di�usion aredi�erent to each other: atoms jump across the grain

boundary in static grain growth, while atoms di�usealong the grain boundary in dynamic grain growth.This suggests that the static and the dynamic com-

ponents can be treated independently for a ®rst ap-proximation.Another issue of di�usion-induced deformation

concerns an unusual assumption that grain bound-aries act as a perfect source and sink of vacancies.As discussed below, the present model is still valid

even though the grain boundaries are an imperfectsource and sink of vacancies. At high temperatures,the stress±strain rate relationship of polycrystallinematerials can be represented by

_eAs p

Rq�13�

where p and q are constants depending on the rate-controlling mechanisms. For example, the constantstake p=1 and q=3 for deformation controlled by

grain-boundary di�usion, and p=2 and q=1 fordeformation controlled by dislocation climb on thegrain boundary [11]. In the present model, however,

the dynamic grain growth is independent of suchrate-controlling mechanisms, because the amount ofthe di�used matter determines the deformation of

grains. This means that vn of equation (1) and _V ofequation (5) are proportional to s pR1ÿq ands pRÿq, respectively. Hence, the mean migration vel-ocity of the grain boundaries is represented by aR_e ,that is, equation (6). Consequently, the dynamicgrain growth rate of equation (9) is valid indepen-dently of the mechanism of superplastic defor-

mation, provided that the di�used mattercontributes to dynamic grain growth.

3. EXPERIMENTAL COMPARISON ANDDISCUSSION

To verify the present model, the grain growthrate of equation (9) was compared with that of10 vol.% zirconia-dispersed alumina which has

large tensile deformation exceeding 500% at 15008C[12]. The material was prepared by using colloidalprocessing [13] followed by sintering at 14008C inair. The relative density of the sintered body is

KIM et al.: DYNAMIC GRAIN GROWTH 3435

99.6%. As shown in Fig. 3, the sintered microstruc-ture consisted of equiaxed alumina grains with an

average diameter of 0.45 mm and uniformly dis-persed zirconia particles with an average diameterof 0.08 mm. Static and dynamic grain growth beha-viour was examined at 1400±15008C. Constant dis-

placement rate loading was applied to the materialin tension at initial strain rates _e 0 between 1:67�10ÿ4=s and 1:67� 10ÿ3=s. After tensile strain

reached the prescribed strains, loading was inter-rupted for microstructural examination. The otherdetails of the tensile tests are described elsewhere

[12]. The grain size of alumina was determined as1.56 times the average linear intercept length cor-rected for second-phase dispersion [14].

3.1. Static grain growth

In Fig. 4(a), the static grain growth behaviour isshown as a function of annealing time. The growth

behaviour of the present material is characterizedwith n=4 in equation (8). The slope of Fig. 4(a)corresponds to the rate constant k in equation (8),and an activation energy Q of 572 kJ/mol is

obtained from its temperature dependence, asshown in Fig. 4(b). It was reported in aluminamonolithics that the values of n and Q are 4 and

564 kJ/mol, respectively [15], similar to those of thepresent material. Cawley et al. [16] also reportedthat the activation energy for the lattice di�usion of

oxygen is 572 kJ/mol. Since the di�usion of the oxy-gen ion is slower than that of the aluminum ion, itis considered that the lattice di�usion of the oxygen

ion controls the static grain growth in alumina.Although the zirconia particles are dispersed andthey grow during annealing, it seems that the basiccontrolling mechanism of static grain growth in the

present material is identical to that of alumina

monolithics. The dispersion of zirconia particles is

thus supposed to play a role only in grain-boundary

pinning, lowering the growth rate.

Fig. 3. Microstructure of 10 vol.% ZrO2-dispersed Al2O3 prepared by colloidal processing.

Fig. 4. (a) Static grain growth behaviour and (b) tempera-ture dependence of the rate constant k.

KIM et al.: DYNAMIC GRAIN GROWTH3436

3.2. Dynamic grain growth

For the comparison of the present model with

the observed grain growth behaviour, we should

know the value of a in equations (9) and (10). Since

analytical evaluation is di�cult as mentioned in the

previous section, we determined the value by ®tting.

Figure 5 shows the dynamic grain growth behaviour

at 15008C as a function of deformation time. The

dotted line denotes the static grain growth calcu-

lated from equation (8) and the solid lines represent

the theoretical prediction from equation (9) for an avalue of 0.6. The ®gure demonstrates that for

a=0.6, the present model reproduces both strain-

rate dependence and time dependence of the

dynamic grain growth behaviour observed in this

material. The temperature dependence of dynamic

grain growth is shown as a function of deformation

time in Fig. 6. The theoretical predictions with the

same a value also reproduce the actual grain growth

at temperatures between 1400 and 15008C. Figure 7

represents an additional comparison for Zn±22%

Al alloy, which also illustrates that the a value is

the e�ective material constant. As shown in Fig. 7,

the theoretical prediction with a=0.45 is consistent

with the experimental grain sizes, reported bySenkov and Myshlyaev [3].Next, let us examine the approximation by

equation (12). Since equation (9) is dependent on

the strain rate, a in equation (12) should depend on_e . When the range of _e is small, however, a can beregarded as a constant, as noted before. All the ex-

perimental data for grain size are plotted as a func-tion of strain in Fig. 8. Applying equation (12) toFig. 8, we obtain 0.57 as the a value, which is

almost equivalent to the previous value of 0.6.Thus, it follows that the di�erence in the initialstrain rate of a factor of 10 and a decrease in thestrain rate during the tensile tests does not seem to

a�ect the grain growth behaviour signi®cantly.Figure 8 also shows the prediction from

R � Rs � aR0e �a � 0:89� �14�proposed by Wilkinson and Ca ceres [4]. Within theresults of the present experiment, equation (12)seems to give a more suitable approximation to the

dynamic grain growth behaviour in the 10 vol.%zirconia-dispersed alumina. The numerical inte-gration of equation (9) at each strain rate was

found to give much better agreement with the ex-perimental data. The numerical integration ofequation (9) would be particularly useful for a large

Fig. 5. Dynamic grain growth behaviour at 15008C. Thetheoretical prediction from equation (9) is represented by

solid lines for a=0.6.

Fig. 6. Dynamic grain growth behaviour at _e 0=1.67 �10ÿ4/s. The theoretical prediction from equation (9) is rep-

resented by solid lines for a=0.6.

Fig. 7. Dynamic grain growth behaviour in Zn±22% Alalloy [3].

Fig. 8. Approximated relationships between the graingrowth ratio (RÿRs)/R0 and the strain.

KIM et al.: DYNAMIC GRAIN GROWTH 3437

variation in strain rate during deformation and/orfor a strain rate range wider than a factor of 10.

3.3. Implication for e�ect of cavitation damage

Regardless of the general agreement between thedynamic grain growth behaviour predicted by the

present model and that of the observed one, closeinspection of the plots reveals that the presentmodel tends to underestimate the grain size at small

strains. That is, the predicted grain sizes are slightlysmaller than those of the observed ones for grainsizes smaller than 0.4 mm at the respective strain

rates. The prediction also tends to overestimate the®nal grain size at the high strain rate,_e 0 � 1:67� 10ÿ3=s. The trend of such a discrepancy

was found to depend strongly on the determinationof the a value: for example, a larger a value of 0.7gives a more accurate prediction to the small grainsizes, while the discrepancy in the ®nal grain size

increases.The present model does not involve the e�ect of

cavitation damage. In the examined material, how-

ever, cavitation damage occurs during deformation.As typically shown in Fig. 9, although cavitationwas found to stay at the level of a few per cent up

to about half the failure strain, it increased expo-nentially at larger strains and reached a level of 30±40% at failure. Since such cavitation damage causesthe expansion of the deformed gauge portion, usual

measurement of true strain along the gauge lengthoverestimates the pure strain of the alumina matrix,in particular at larger tensile strains. This means

that an increase in cavitation damage has an e�ectof apparently increasing the strain and the strainrate in equations (7), (9) and (10), and thereby

leads to overestimation of the grain size in equation(10). For the true value of a, the theoretical predic-tion is thus expected to reproduce the actual beha-

viour accurately at smaller strains, where cavitationdamage can be neglected, and to give overestimatedgrain sizes at larger strains, where the e�ect of cavi-tation damage is pronounced. These considerations

suggest that a=0.6 is an apparent value and thetrue a value may be close to around 0.7.

For more accurate prediction of grain size in theentire range of tensile deformation, therefore, thee�ect of cavitation damage on strain rate and the

interaction between cavitation and grain growthhave to be included in the present model as a func-tion of time or strain. For further accuracy, the

change in the grain aspect ratio (equal to the majoraxis length/minor axis length) during deformationshould also be analysed. This is because an increase

in the aspect ratio, which was a maximum of 1.5 inthe present experiment, reduces the strain rate inequation (3), and thereby the relationship betweenthe strain and the amount of di�usive matter devi-

ates from that of the present model.

4. SUMMARY AND CONCLUSIONS

Dynamic grain growth during superplastic defor-mation was analysed on the basis of a grain-bound-

ary di�usion mechanism. In this analysis, thedynamic component of the grain growth is assumedto involve the following sequences:

1. applied stress exerts a potential di�erence alonggrain boundaries;

2. matter dissolves from grain surfaces of higherpotential;

3. the matter di�uses through the grain boundariesaccording to the potential gradient; and

4. the matter deposits on grain surfaces of lower

potential.

Since the static component of grain growth arises

from a potential di�erence existing normal to thegrain boundaries, the static and the dynamic com-ponents can be assumed to contribute to grain

growth independently and hence the sum of bothcomponents gives the total grain growth. A theor-etical model derived from this analysis is shown to

hold independently of the mechanisms of superplas-tic deformation. For a wide range of strain rate,tensile strain and temperature, the prediction fromthe derived model is also shown to agree well with

the dynamic grain growth behaviour observed ex-perimentally in superplastic zirconia-dispersedalumina.

REFERENCES

1. Clark, M. A. and Alden, T. H., Acta metall., 1973, 21,1195.

2. Ghosh, A. K. and Hamilton, C. H., Metall. Trans.,1979, 10A, 699.

3. Senkov, O. N. and Myshlyaev, M. M., Acta metall.,1986, 34, 97.

4. Wilkinson, D. S. and Ca ceres, C. H., Acta metall.,1984, 32, 1335.

Fig. 9. Variation of the cavitated volume fraction withrespect to strain.

KIM et al.: DYNAMIC GRAIN GROWTH3438

5. Morral, J. E. and Ashby, M. F., Acta metall., 1974,22, 567.

6. Sato, E., Kuribayashi, K. and Horiuchi, R., J. Japan.Inst. Metals, 1988, 11, 1043.

7. Friedez, J. D., Carry, C. and Mocellin, A., Adv.Ceram., 1985, 10, 720.

8. Coble, R. L., J. appl. Phys., 1961, 32, 793.9. Green II, H. W., J. appl. Phys., 1970, 41, 3899.

10. Herring, C., J. appl. Phys., 1950, 21, 437.11. Burton, B., Mater. Sci. Engng, 1972, 10, 9.

12. Nakano, K., Suzuki, T. S., Hiraga, K. and Sakka, Y.,Scripta mater., 1998, 38, 33.

13. Suzuki, T. S., Sakka, Y. and Hiraga, K., J. Japan.Soc. Powder Metall., 1997, 44, 356.

14. Wurst, K. J. C. and Nelson, J. A., J. Am. Ceram.Soc., 1972, 55, 109.

15. Besson, J. and Abouaf, M., Acta metall. mater., 1991,39, 2225.

16. Cawley, J. D., Halloran, J. W. and Cooper, A. R., J.Am. Ceram. Soc., 1991, 74, 2086.

KIM et al.: DYNAMIC GRAIN GROWTH 3439