On the grain growth exponent of pure iron

Acta metall, mater. Vol. 42, No. 9, pp. 3071-3075, 1994

~

Pergamon

0956-7151(94)E0077-T

ElsevierScienceLtd. Printed in Great Britain

O N THE G R A I N G R O W T H E X P O N E N T OF P U R E IRON R. A. VANDERMEER1 and HSUN HU2 INaval Research Laboratory, Washington, DC 20375-5343 and 2University of Pittsburgh, Pittsburgh, PA 15261, U.S.A. (Received 16 September 1993; in revised form 20 January 1994)

Almract---Grain growth data on zone-refined iron were reanalyzed quantitatively to take into account the initial grain size and specimen thickness effect. The deviations from parabolic growth noted in an earlier analysis [Can. Metall. Q. 13, 275 (1974)] are attributed to the use of the approximate Beck formula (2 _~K. t") for grain growth kinetics which does not take thesetwo important factors into account. Based on the new analysis, grain growth in zone-refined iron over the entire range of temperatures studied can be understood in terms of a single, thermally-activated rate process with an activation energy approximately the same as that for lattice self-diffusion. Furthermore, it is not necessary to invoke a non-linear velocity/driving force relationship to understand the kinetics. The high activation energy may be caused by an unknown impurity present in the iron which gave rise to an impurity drag effect.

INTRODUCTION

2 I!"- 2~/" = K ' t

The subject of isothermal grain growth kinetics in metals and ceramics has been studied intensely for over 40 years. From the earliest theoretical treatments of Beck et al. [I] and Burke [2] until present, it has been supposed that under ideal conditions grain growth kinetics should obey parabolic kinetics, namely 22 -- 22 = K ' t

(1)

where 2 is a length scale of the grain size after annealing for time, t, K is a temperature-dependent rate parameter and 20 is the length scale of the grain size at time equal to zero. In most experiments the grain size is measured using stereological methods so that 2 = 2/Sv where Sv is the total grain boundary area per unit volume. The two conditions under which the parabolic form of equation (1) should be followed during grain growth have been rigorously justified recently by Mullins [3]. These are (1) grain boundary velocity is linearly related to the capillary (or curvature)-driven driving force (reduction of grain boundary area) and (2) during growth, the grain size distribution behaves in a statistically selfsimilar manner, i.e. consecutive configurations of the system are geometrically similar in a statistical sense [3]. Many different grain growth models have been developed [2, 4-9], all of which lead to an equation of the form of equation (1). Computer simulations also seem to confirm the parabolic kinetics after initial transients have decayed [10-12]. There is an enormous body of experimental data on grain growth kinetics in various materials, however, that fails to conform to equation (1). This has prompted experimenters to analyze grain growth data according to the empirical relationship

(2)

where n ( vs 112) is an empirical constant determined experimentally. An even simpler relationship for grain growth kinetics was put forth by Beck et al. [1] and is given by 2 ~ K . t".

(3)

Equation (3) has been widely utilized because it has the advantage that when log 2 is plotted vs log t, a straight line of slope n results. It's use implies that 2O = 0 at t = 0 or at the very least 20 is negligible compared to 2. Thus, the application of equation (3) should be limited only to cases where ~, >>2o. This limitation has not always been carefully considered in grain growth studies, a point that has been aptly made, for example, by Mistier and Coble [13]. The failure of data to conform to n = 1/2 in equation (2) has been attributed to a number of material related factors: (1) Adherence to statistical self-similarity (SSS) in scaled grain size distribution has not yet been achieved, i.e. grain growth is in a transient period. (2) Pinning forces causing growth stagnation are present due either to (a) second phase precipitates [14], (b) the specimen thickness effect [2] or (c) intrinsic drag forces on the grain boundaries [15]. (3) The relationship between velocity and driving force is non-linear due, for example, to a possible solute drag effect [16]. (4) The onset of abnormal grain growth may occur and interfere due to a change in the texture of the material with time [17] or when the material has been lightly strained [18]. The inappropriate neglect of the 2o term can also give rise to interpretations where the n value seems to be less than 1/2 [13]. The kinetic equation given by equation (1) needs to be modified when pinning forces causing stagnation

3071

3072

VANDERMEER and HSUN HU: GRAIN GROWTH EXPONENT OF Fe ISSUE

1000

t ~ 100

t

"5

10

e~ 10 4

10 "~

10 -s

tO "s

K. t / 2 ,

10 .4

10 .3

10 .2

10 "t

tO o

l0 t

normalized time

Fig. 1. Comparison of the Burke equation [equation (4)] and the Beck equation [equation (3)] for grain growth in the presence of pinning forces indicating the "safe region" where the simpler Beck equation falls within 10% of the Burke equation. are operative. The earliest such treatment was due to Burke [2], giving the formula /~m

+In

_

=K.t

(4)

where 2m is the maximum limiting length scale of the grain size that results due to the pinning force. Equation (4) was derived by assuming a linear velocity/driving force relationship where the pinning force, assumed to be a constant, is subtracted directly from the capillarity force term. It is relatively straightforward to derive this equation from several of the grain growth models [2, 4, 9]. (A similar equation which predicts a larger retarding effect and a slightly slower approach to stagnation may be derived from Hillert's theory [5].) In the case of pinning due to the specimen thickness effect, 2m in equation (4) becomes approximately equal to the smallest dimension of the specimen [1]. In Fig. 1 the Burke equation, equation (4), and the Beck equation, equation (3) with n equal to 1/2, are compared. The Beck formula deviates substantially from the Burke formula when 2 approaches either 20 or )'m" A "safe" region where the two curves nearly overlap is denoted on Fig. 1. In this region, defined by 320 < 2 < 2m/9, the two curves are within about 10% of one another. This "safe" region provides a criteria for the appropriate usage of the simpler Beck equation in analyzing experimental data.

For pure, zone-refined metals where effects of impurities and precipitates are expected to be minimal (pinning forces due to these are likely to be absent), the grain growth exponent, n, is not always observed to be 0.5. Table 1 catalogues the findings from grain growth studies on several metals. In most cases the values of n are close to 0.5; it is sometimes difficult experimentally to differentiate between a value of 0.4 and 0.5. The notable exceptions in Table 1 appear to be aluminum and iron at low temperatures. In the aluminum and iron studies, the Beck equation [equation (3)] was used to calculate the exponent graphically from log-log plots. Thus the 20 term was not explicitly taken into account, a consideration that was attended to in one way or another in the data analyses of all the other metals listed in the Table 1. Furthermore, in the iron studies, the grain size length scale approached the thickness of the specimen at the higher temperatures and longer annealing times. Those data were either ignored or in a qualitative way were given less weight in the calculation of n. It seems that in the case of iron at least, it is more appropriate to analyze the data with the Burke equation [equation (4)] rather than the Beck equation. Of the 42 data points in the study none fell within the "safe" zone defined earlier, where the Beck equation can be used without much cause for concern. It is also a paradox of the original interpretation of the data on iron that an apparent n < 0.5 was noted in spite of the fact that grain growth was documented to exhibit SSS and there was negligible preferred orientation in the iron [23]. For these reasons, it seemed proper to reexamine the grain growth data on zone-refined iron, analyzing it using the Burke equation rather than the Beck equation. Such is the purpose of this paper. The new analysis allows the 20 term and the specimen thickness effect to be accounted for; all o f the data are employed in the calculation of the isothermal rate constants, K. The new analysis describes the data better than the original one did and leads to a simpler mechanistic picture of grain growth in zone-refined iron. DATA ANALYSIS AND RESULTS The data were obtained from Fig. 1 of Ref. [23]. From a xerographic enlargement of the figure, the

Table 1. Grain growth exponents for zone-refinedmetals Metal Aluminum Cadmium Lead Lead Iron Tin Tin

Exponent n 0.29 0.50 0.40 0.41 0.2~0.50 0.50 0.43

Authors Gordon and E1 Bassyouni[19] Simpson, Aust and Winegard[20] Bollingand Winegard[21] Drolet and Galihois[22] Uu [23] Holmes and Winegard[24] Drolet and Galibois[22]

Remarks Unknown pinning effectacknowledged

The n variedwith temperature, approaching 0.5 at high temperatures

VANDERMEER and HSUN HU: GRAIN GROWTH EXPONENT OF Fe Table 2. Experimental grain g r o w t h kinetic parameters for zonerefined iron Temperature (K)

2m (ram)

2o (mm)

550 600 650 700 750 800 850

0.81 0.81 0.81 0.81 0.81 1.10 0.81

0.039 0.039 0.041 0.049 0.044 0.042 0.046

i!!i!!!ii!i!i!!!!!!!!i!i!!!!!!!!!!!!!! !!!!!!i!!!!!!!i!i!!!!ii!!!!/!!!!!!!!

Rate constant, K (mm2/s) 7.92 4.69 3.75 1.73 8.15 3.26 1.23

x × x x x x x

10 .3

10 - s 10 -7 10 -6 10 -~ 10 -~ 10 -4 10 -3

10 .4

~e 10 "s

O

data points on the graph were digitized and entered as a data base into a Macintosh II personal computer. At each temperature, the 2 vs time data were fitted mathematically to equation (4) using a non-linear, general curve fitting computer routine to determine best fit values for the parameters, K, 20 and 2~. The computed values are listed in Table 2. Figure 2 is a plot of 2 vs annealing time on a log-log scale demonstrating the quality of the data fit that was achieved with these parameters. In all cases the correlation coefficient, R, measuring goodness of fit was greater than 0.99 (a value of 1.00 is maximum). Treating the parameter K as a rate constant, a plot of In K vs 1000/T(K) is presented in Fig. 3. Clearly the parameter K can be described over the entire temperature range by an Arrhenius equation of the form

t2

K = K0" exp - - -

(5)

RT

3073

where R is the gas constant, T is the absolute temperature, K0 and Q are constants, the latter being the activation energy of grain growth. A least squares analysis of the data gave a value of 249 kJ/mol for Q which is in excellent agreement with the value calculated for the high temperature regime in the earlier analysis [23]. In the original study [23], a supplementary exper-

| e~ o) m e~

i0 "~

10 .7

10 4

0.8

0.9

1

1.1

1.2

1.3

1000/T(K) Fig. 3. Semi-log plot of rate constant, K, vs reciprocal of absolute temperature for grain growth in zone-refined iron. iment had been performed from which it was implied that 20 should be less than 0.060 mm, the grain size observed at the completion of recrystallization at 700°C. The present analysis leads to a calculated value consistent with that observation. The 20 appeared to be independent of temperature and its mean value was 0.039 mm. This finding suggests that the recrystallized grain size is probably little affected by recrystallization temperature. The pioneering experimental work of Beck et al. [1] on the specimen thickness effect in grain growth found that growth stagnated at a grain size between 1 and 1.2 times the specimen thickness. The explanation of this effect, due to Mullins [25], is that the grain boundaries intersecting the free surface form grooves that act to pin the grain boundaries (equivalent to a pinning force on the grain structure) and stop normal grain growth when the grains become the order of magnitude in size as the specimen thickness. The experimental study on iron exployed specimens 0.75 mm in thickness [23]. Thus, at all temperatures but 800°C, the 2m value of 0.81, deduced by analysing the data with the Burke equation, is well within the range noted by Beck et al. [1] and commonly accepted by the scientific community. The high value of ~'m calculated for 800°C seems to be anomalous and is largely the result of one or two data points that are slightly high.

0.1

DISCUSSION 0.01

10 e

l0 t

10 a

10 ~

10 ~

l0 s

10 ~

Annealing time (sec)

Fig. 2. Isothermal grain growth of zone-refined iron showing a log-log plot of the mean intercept grain length vs annealing time at various temperatures from 850°C (left) to 550°C (right) in 50°C increments. Solid lines are the calculated curves based on equation (4) and the constants listed in Table 2. Data points are from Ref. [23].

The new analysis of the existing grain growth data on zone-refined iron allows a simpler interpretation of the results than was previously rendered [23]. The new results challenge some of the earlier conclusions. The reason the grain growth exponent determined previously using the Beck equation was less than 0.5 is clearly the result of neglecting the 20 term and because the influence of the specimen thickness effect had not been given enough careful consideration. Use

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VANDERMEER and HSUN HU: GRAIN GROWTH EXPONENT OF Fe

of the Burke equation allows all of the data to be included in a quantitative characterization of grain growth whereas use of the Beck equation required a subjective weighting and exclusion of certain data points in an attempt to account roughly for these two effects. From the prior interpretation of the iron grain growth data based on the Beck equation, it was concluded that because n < 0.5 at the lower temperatures, there existed a non-linear velocity vs driving force relationship at the low annealing temperatures. This was taken to imply that residual impurities which presumably could create a solute drag effect [26] were involved in the grain boundary migration process at low temperatures. Such a non-linear relationship need not be be invoked with the new analysis because implicit in the use of the Burke

equation is a linear velocity~driving force relationship. Also, from the new interpretation of the data it may be concluded that grain growth in zone-refined iron is a single, thermally-activated rate process over the entire temperature range studied. This finding is in contradiction to the conclusion arrived at in the earlier interpretation where a discontinuity in the Arrhenius plot was thought to exist. To understand the earlier result required a two-stage rate process which was rationafized by the existence of a solutedrag effect or possibly a grain boundary phase transformation [23]. The value of the activation energy, Q, estimated in the present analysis is essentially the same as was deduced for the high temperature regime in the earlier analysis. It corresponds closely to the activation energy for lattice self diffusion in iron. The value seems too high for a grain boundary migration process in a zone-refined metal where, compared to what is usually found, activation energies are closer to the grain boundary self diffusion activation energy [27]. This result suggests the possibility that an unknown or undetected impurity is present in the zone-refined iron giving rise to an impurity drag effect [26]. The presence of an impurity drag effect does not necessarily conflict with a linear velocity/driving force relationship since in grain growth the driving force for grain boundary migration is a low one in the context of the impurity drag theory. In this regime, velocity remains linearly related to driving force [27]. Grain growth experiments in zone-refined metals doped with small amounts of impurities seem to bear this out [28, 29]. The new analysis on iron leaves only aluminum as an anomaly among the zone-refined metals in which grain growth does not obey an essentially parabolic time dependency--see Table 1. The difficulty with aluminum may lie in the presence of an undetected fine precipitate which can cause pinning forces as the authors speculated [19]. Furthermore, because material was limited, the aluminum grain growth specimens were given multiple anneals and polished, etched and examined after each anneal. It is known

from the work of Fiset : e t a L [30], that in such repeated use specimens the grain growth rate can slow down. They noted that during the multiple annealing cycles, new dislocations were introduced which underwent polygonization and these dislocations tended to pin the grain boundaries. Clearly, more definitive experiments on zone refined aluminum are needed. Most previous grain growth studies used sheet specimens. With such materials, not only the sheet thickness effect would be involved, but some texture would also fikely to be introduced by the processing. Both of these would affect grain growth kinetics. Moreover, many of the past experiments used specimens in the as-deformed condition for grain growth anneals. Thus, the time for recrystallization was also involved, and the initial grain size for grain growth was not known. In view of the present findings, it would be desirable to process material for grain growth studies in such a way as to minimize texture and the specimen thickness effect, One way, following Kasen [31], would be to begin with a cube of material and process it by repeated compression along its three orthogonal directions at low temperatures, followed by rapid annealing for recrystallization to a uniformly fine initial grain size. From this cube stock, small specimens with sufficient thickness can be produced for grain growth studies with little complications of texture and specimen thickness effect.

CONCLUSIONS 1. Grain growth data on zone-refined iron were successfully reanalyzed quantitatively to take into account the initial grain size and the specimen thickness effect. 2. Deviations from parabolic growth noted in the earlier analysis [23] are attributed to the use of the approximate Beck formula for grain growth kinetics which does not take these two important factors into account. 3. It is not necessary to invoke a non-linear velocity/driving force relationship to understand grain growth in zone-refined iron. 4. Over the entire range of temperatures studied, grain growth in zone-refined iron can be understood in terms of a single, thermally activated rate process with an activation energy approximately the same as that for lattice self diffusion in iron. 5. An unknown impurity may be present in the iron which can give rise to an impurity drag effect causing the high activation energy.

Acknowledgements--The authors would like to thank Drs C. S. Pande, B. B. Rath and S. Marsh for helpful discussions and for reading the manuscript. This work was performed at the Naval Research Laboratory under the sponsorship of the Office of Naval Research of the U.S. Department of Navy whose support is gratefully acknowledged.

VANDERMEER and HSUN HU:

GRAIN GROWTH EXPONENT OF Fe

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