Acra mater. Vol. 45, No. 4, pp. 1785-1789, 1997 c 1997 Acta Metallurgica Inc.

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ABNORMAL GRAIN GROWTH DEVELOPMENT FROM UNIFORM GRAIN SIZE DISTRIBUTIONS

P. R. RIOS Universidade Federal Fluminense, Escola de Engenharia Industrial Metallirgica de Volta Redonda,

Av. dos Trabalhadores, 420, Vila Santa Cecilia, Volta Redonda, RJ 27260-740, Brasil

(Receiced 8 March 1996)

Abstract-The development of abnormal grain growth from uniform grain size distributions pinned by particles is examined. The main assumption is that a locally lower pinning force adjacent to a large grain can cause this grain to grow abnormally. The case of a matrix pinned by unstable particles is considered in detail. A quantitative criterion is derived which shows the interplay among the variables involved. From this it is shown how both deterministic and probabilistic factors contribute to abnormal grain growth. Practical implications are discussed with particular emphasis on why abnormal grain growth can be so difficult to control. The present treatment offers a new and plausible rationale for the development of abnormal grain growth from uniform grain size distributions containing unstable particles. 8 1997 Acta Metallurgica Inc

1. INTRODUCTION

One often distinguishes two kinds of grain growth [l]: normal and abnormal. In the former the grain size distribution remains uniform during grain growth whereas in the latter a few large “abnormal grains” grow, consuming a fine grained matrix which is often pinned by particles.

Abnormal grain growth has received increasing amounts of attention [l-13] but the development of abnormal grain growth from an initially uniform grain size distribution is not yet fully understood. The theoretical models normally assume the pre-existence of a grain with some sort of “advantage” as a condition for the occurrence of abnormal grain growth [l-l 61. An early model of this kind, which has been the basis of subsequent analytical treatments [4, 6, 7, 121, is that of Gladman [2]. This “advantage” is most commonly a larger size [l-l 31 but can also be a higher boundary mobility [6, 14-161 or in the case of thin films a higher surface energy [8].

It has been common knowledge for several decades that abnormal grain growth can develop in a matrix pinned by particles when the pinning force is somehow lowered [ 1, 2, 171. In practice one observes abnormal grain growth taking place from an initially uniform grain size distribution. Also there is some evidence that normal grain growth can occur prior to the development of abnormal grain growth [ 181. Thus it appears that an abnormal grain can develop from a matrix in which no grain has an initial size advantage that on its own would lead to abnormal grain growth. It is also well known that in these circumstances abnormal grain growth often appears to be “erratic”. This suggests that there is some probabilistic factor which influences abnormal grain

growth initiation. So a large grain which under ordinary circumstances would continue to grow normally might as a consequence of some chance factor become unstable and grow abnormally.

This work is an attempt at clarifying how deterministic and probabilistic factors can promote the development of abnormal grain growth from an initially uniform distribution in which the pinning force is lowered.

2. ABNORMAL GRAIN GROWTH WITH STABLE PARTICLES

One considers in the first place the simpler case of a matrix that is stagnant. No normal grain growth can occur because of a pinning force exerted by stable particles. Taking Hillert’s formalism [l] as the starting point one can write for the grain of maximum radius R,,,:

dRm,x dt

where K = aMy, M is the grain boundary mobility, y is the grain boundary energy per unit of area and r is a geometric factor approximately equal to unity for three-dimensional grains; Z is a pinning force or an opposing pressure related to the size and volume fraction of the particles; Rcr is the critical grain radius of the grain size distribution [l].

According to equation (1) abnormal grain growth could only take place if a large grain, RA, exists for which

+Kg>O. (2) nldl

1785

1186 RIOS: ABNORMAL GRAIN GROWTH DEVELOPMENT

Equation (2) was derived substituting Ra = R,,,+ ARA for R,,, in equation (1) and noting that ARAIR,,,<< 1.

Alternatively, if a locally lower pinning force Z - AZ < Z is adjacent to a certain maximum grain RA = R,,,, then again one might have

+Ke>O. (3)

In this latter case, after the abnormal grain has grown beyond the region of the lower pinning force it will have gained a size advantage, RA = R,,, + ARA > R,,,. This grain will continue to grow abnormally only if this size advantage is such that equation (2) is obeyed.

One can regard equation (1) as an equilibrium condition for the maximum grain. The pinning force exerted by the particles balances the pressure difference between the maximum grain and the matrix. The critical equilibrium condition is

dR,,, -= dt

=O, (4)

Clearly equation (4) represents a condition of unstable equilibrium with respect to a displacement R, = Rmx i- ARA > Rm,, or Z-AZ < Z. Either displacement will result in dRJdt > 0 and abnormal grain growth will occur with increasing values of dRJdt. This instability is at the root of the “probabilistic” nature of abnormal grain.

In this case abnormal grain growth depends on the pre-existence of favorable conditions which are expressed by equations (2) and (3). If these conditions hold, abnormal grain growth takes place. If they do not hold and equation (1) is always obeyed, abnormal grain growth does not take place at all.

In those circumstances abnormal grain growth would take place immediately after the specimen was heated to a temperature high enough to make grain boundaries sufficiently mobile. If no abnormal grain growth occurs immediately then no grain growth, normal or abnormal, is likely to occur up to the melting point. It is worthy of note that if abnormal grain growth does occur it is going to be extremely severe since the matrix is pinned by stable particles and no normal grain growth takes place [13].

3. ABNORMAL GRAIN GROWTH WITH UNSTABLE PARTICLES

The situation becomes more complicated when the particles are unstable, i.e. when they can coarsen or dissolve. Since the pinning force is a function of particle size and volume fraction [4] one may expect that a certain rate of particle coarsening or dissolution will lead to a corresponding rate of decrease in the pinning force dZ/dt < 0.

As the pinning force decreases, normal as well as abnormal grain growth may become possible [9].

Suppose that a matrix is initially pinned by particles so that equation (1) is obeyed. Then as the particles coarsen or dissolve the pinning force, Z, decreases until the condition expressed by equation (4) is reached. This is the critical condition for the present analysis. Now if the pinning force continues to decrease both Rcr and R,,, may increase in such a way that equation (4) is still obeyed. This corresponds to the situation in which the grains undergo normal grain growth following particle coarsening [9]. The question is then whether such a situation is stable with respect to abnormal grain growth.

Admit that as one of the largest grains, R,,,, grows its boundary moves into a region in which the pinning force is lower than the average pinning force: Z - AZ. To avoid confusion let this particular grain be called “candidate grain”, RA, for abnormal grain growth. Inserting Z - AZ and R,,, = RA into equation (4) one obtains:

dRA p= KF= MyAZzO. dt

Based on equation (5) only, one cannot say whether abnormal grain growth will take place or not because normal grain growth is taking place and Rcr is increasing. One has two possibilities:

RA dF Z<O dt

or

d+ 2 > 0.

dt

(6)

In the first case, [equation (6)], the increase in the growth rate of the candidate grain is compensated for by the normal grain growth of the matrix. No abnormal grain growth will take place as the candidate grain cannot increase its relative size and the grain size distribution remains uniform. However, in the second case, [equation (7)], the candidate grain increases its relative size and moves away from the grain size distribution. This latter condition corresponds to abnormal grain growth.

Considering the above one can then derive the condition for which equation (7) will hold. Deriving equation (7) one has

(8)

Assuming that the grain size distribution of the matrix, excluding the abnormal grains, remains uniform then equation (4) still holds for the matrix and also Rmax/Rc, = n does not change. One can thus

RIOS: ABNORMAL GRAIN GROWTH DEVELOPMENT 1787

use equation (4) to obtain a relationship between Rcr and Z:

z l-l/n -= G! Rcr

Differentiating on both sides of equation (9) gives

W/cc) (1 - l/n) d&r ~=_~~ dt R& dt (10)

Inserting equations (5) and (10) into equation (8) gives

d$ c,=ez+RAdo>()~ (11)

dt ctRcr (1 - l/n) dt

Since one is considering the beginning of abnormal grain growth, RA is still equal to R,,,. Using this and rearranging equation (11) one obtains

AZ > R& (n ” 1> ---( -9)&. (12)

The pinning force decreases so that (- dZ/dt) > 0. For the following discussion it is better to use a

dimensionless quantity: the relative value, AZ/Z, instead of the absolute value, AZ. Dividing equation (12) by equation (9) results in

(13)

If a certain “candidate grain”, RA = R,,,, grows into a region in which the pinning force is lower, Z - AZ, than the average pinning force, Z, then this grain will become unstable and grow abnormally if equations (12) or (13) are satisfied. Otherwise the candidate grain will continue to grow normally.

It was shown above that if a candidate grain finds a region of lower AZ it may be able to grow abnormally, i.e. d(RA/Rc,)/dt or d(RJR,,,)/dt > 0. This particular grain, RA, will obtain a size advantage over the grain of maximum size: RA = R,,, + ARA. However, one must consider that this grain will eventually grow beyond the region of lower pinning force, Z - AZ. The question is then whether the size advantage of the candidate grain will be enough to assure that it will continue to grow abnormally. Its size advantage must be such that the growth rate, dRJdt, is equal to that given by equation (5):

(14)

Inserting R,, = R,,, + AR+, into equation (14), re- membering that equation (4) is followed and considering that AR+JR,,,K 1 one obtains

(15)

Because equations (8) and (12) are satisfied, equation (15) implies that d(RA/Rc,)/dt > 0 for AR.JR,,,K 1.

From equation (15) together with equation (9) it can be shown that the relative size advantage must be at least

ARA __ = 9 (n - 1). R,,, (16)

If ARJR,,, < AZ(n - 1)/Z then the candidate grain will not continue to grow abnormally.

Equation (16) shows that the relative size advantage is of the order of magnitude of the relative local decrease in the pinning force, AZ/Z. The lower the value of AZ/Z required, the smaller is the size advantage. A further point is that broader distri- butions require relatively larger size advantages.

In what follows equation (13) will be taken as the basis of the discussion. However, it must be borne in mind that in addition to satisfying equation (13) it is further required that the lower pinning force region must be such that the size advantage gained should be enough to maintain abnormal grain growth.

4. DISCUSSION

4.1. Implications of present result Equation (13) provides a deterministic criterion for

the abnormal growth of a certain candidate grain, i.e. whenever the inequality expressed by equation (13) is satisfied for a certain candidate grain, abnormal grain growth of this grain will follow.

The probabilistic (or “erratic”) component of abnormal grain growth is explained by the fact that in order to satisfy equation (13) a certain maximum grain must happen to be adjacent to a region in which the pinning force is less than the average pinning force by a sufficiently high amount. This clearly depends on chance. It may happen as the boundary moves during normal grain growth which occurs as the pinning force decreases at a certain rate.

The idea that a locally lower pinning force could cause unpinning and consequently abnormal grain growth has been previously proposed by Elst et al. [19]. Some experimental evidence for an effect of the particle distribution on the onset of abnormal grain growth has been recently reported [20, 211.

Normally one has a particle size distribution and different interparticle distances. Therefore a certain large grain may locally find a lower than average pinning force. One could call this a statistical effect. As it moves, the grain boundary “samples” the particle distribution. This is in essence analogous to the procedure employed in quantitative metallogra- phy in which one takes a certain area at random to determine particle characteristics. There is, of course, a statistical error associated with this procedure as the measurements fluctuate from area to area. Notice that this statistical error occurs even if the microstructure is uniform. Another source of fluctuation in the pinning force would be a “true”

1788 RIO!3 ABNORMAL GRAIN GROWTH DEVELOPMENT

microstructural heterogeneity. One may have a heterogeneous particle distribution inherited from a previous heterogeneous precipitation process, e.g. large particles on grain boundaries separated from fine particles in the grain interior by a precipitate-free zone.

The deterministic factors represented on the right-hand side of equation (13) are a measure of the susceptibility of the material to abnormal grain growth. They determine the magnitude of AZ/Z. This makes abnormal grain growth more or less probable as it should in principle be easier to find a locally smaller value of AZ/Z, say 0.01, than a higher value, say 0.1.

4.2. Detailed discussion of factors that influence abnormal grain growth

4.2.1. Critical radius, R,. The critical radius, which is roughly equal to the mean grain radius, has a strong influence since it appears as a factor R& in equation (13). The smaller the critical radius the smaller the AZ/Z necessary for abnormal grain growth. The implication is that very fine grain sizes pinned by very fine particles may significantly increase the susceptibility to abnormal grain growth. This latter conclusion is consistent with a previous study [l 11. It is interesting to point out that abnormal grain growth models, which are essentially based on the ratio of an abnormal grain to the mean or critical grain, cannot usually predict this effect [2,4, 6, 7, 121.

4.2.2. Distribution width, n = Rma,/Rc,. The effect of the distribution width is given by the term n’/(n - l)*. This term has a minimum for n = 3. This somewhat unexpected result implies that broader distributions will be less susceptible to abnormal grain growth. A similar conclusion was reached in a previous study which considered abnormal grain growth in pure materials [lo]. On the other extreme as n -+ 1, R,,, --+ Rcr and dR,,,/dt = 0 for Z - 0. When all grains are equal no grain growth occurs even in the absence of a pinning force and therefore the discussion concerning AZ/Z is meaningless.

The value of n = 3 for which the material is most susceptible to abnormal grain growth is roughly the experimental value for the width of the grain size distribution usually obtained in metals [IO, 19, 20, 221. Notice that the value of n discussed here is only meaningful if no abnormal grain growth has occurred. Obviously if abnormal grain growth has occurred or if large grains are inherited from previous processing the grain size distribution is not uniform and the present discussion does not

apply. 4.2.3. Rate of decrease in pinning force, - d(Z/cc)/

dt > 0. One can identify two limiting cases. The first is when -d(Z/cc)/dt is very large. Equation (13) suggests that the material is not susceptible to abnormal growth. The reason is that the matrix becomes free to “react” by undergoing fast normal

grain growth [9, 131. Consider the extreme circum- stance in which all the particles were instantaneously removed, -d(Z/r)/dt + co. The matrix would then become free of particles and previous work has shown [lo] that abnormal grain growth is unlikely.

The other limiting case is when - d(Z/a)/dt is very small. Equation (13) suggests that in this case the material would be highly susceptible to abnormal grain growth. However, because the pinning force decreases very slowly, the boundary also moves very slowly and the possibility of finding a region with a value of AZ/Z low enough for abnormal grain growth within a reasonable time scale decreases. The extreme case would be when d(Z/cc)/dt - 0. The particles would then be stable, a situation already discussed in Section 2. According to equation (13) a material with stable particles would be the most susceptible to abnormal grain growth, but as discussed above if there is no grain for which equation (5) is satisfied then no abnormal grain growth will ever occur because the boundaries cannot move.

In summary, equation (13) does suggest that a slow decrease in pinning force, e.g. because of slowly coarsening particles, makes the material more susceptible to abnormal grain growth than a rapid decrease in pinning force due to, e.g. particles coarsening or dissolving fast. This conclusion is consistent with previous work [9, 12, 131.

4.2.4. Mobility, M, and grain boundary energy per unit area, y. Finally, equation (13) suggests that highly mobile boundaries with a high grain boundary energy per unit area make the material more susceptible to abnormal grain growth. It is important to mention here that the present approach is based on the original treatment of Hillert [l] in which differences in mobility and grain boundary energy for individual grain boundaries are not taken into account. So the mobility and grain boundary energy values here should be seen in principle as average values. A more detailed treatment of this problem will not be attempted in the present work [6, 71.

4.2.5. Most susceptible microstructure for abnor- mal grain growth. From the above one might infer that the microstructure most susceptible to abnormal grain growth would consist of very fine grains pinned by very fine fairly slowly coarsening particles. Moreover the distribution width RmaJRcr should have a value of -3. A more heterogeneous particle distribution would also be more favorable. The mobility should be high, which suggests that low temperatures are not favorable. The effect of temperature will be discussed below. A well known example of a susceptible system is fine grained austenite containing fine, sparingly soluble carbides/ nitrides. Examples of abnormal grain growth in such a system are numerous, e.g. [23]. Unfortunately, what has been said above shows that abnormal grain growth can be difficult to control when unstable particles are used.

RIOS: ABNORMAL GRAIN GROWTH DEVELOPMENT 1789

4.2.6. EfSect of temperature. The factors in equation (13) that are most temperature dependent are the mobility, which increases with temperature, and the rate of decrease in pinning force, which also increases with temperature. They appear in equation (13) as a ratio: ( - dZ/dt)/M. This might explain why some systems, notably austenite containing carbides/ nitrides, may exhibit an “abnormal grain growth temperature” or a “grain coarsening temperature” [18, 231 which is often located between 1000 and 1100°C. This temperature range would be the range within which the ratio (-dZ/dt)/M would pass through a minimum, thus being the range within which the material would be most susceptible to abnormal grain growth.

4.2.7. Grains that are able to grow abnormally. In his early work on abnormal grain growth Gladman [2] remarked that the largest grain should be unpinned first. However, the probabilistic nature of the phenomenon requires that a region of lower pinning force be adjacent to a large grain. So there is a probability that grains smaller than the largest grain will be unpinned and even unpinned first. The problem is that a smaller grain requires a locally lower pinning force than a larger grain and consequently abnormal grain growth is less likely to occur.

Another important point in this regard is that the number of grains growing abnormally per unit volume will depend on the probabilistic factor. Therefore controlling the number of growing grains per unit volume should be very difficult. This result could be important when abnormal grain growth is desirable as in the production of coarse grained iron-silicon for magnetic purposes.

4.2.8. Effect of specimen size. The probabilistic nature of abnormal grain growth implies the existence of a specimen size effect. The possibility of abnormal grain growth is given by the product of the probability that a large grain develops into an abnormal grain times the number of large grains. This number obviously increases with specimen size. So again one has a difficult problem. Even if one does not find abnormally large grains in a 1 cm’ laboratory specimen, this does not guarantee that a long wire or strip of steel will be free from abnormal grain growth.

5. SUMMARY AND CONCLUSIONS

1. A locally lower pinning force adjacent to a large grain can cause this large grain to become unstable and grow abnormally. This mechanism is able to explain how a uniform grain size distribution may initially increase its mean grain size by normal grain growth and then “unexpectedly” develop abnormally large grains.

2. A quantitative criterion has been derived and is expressed by equation (13). It shows the interplay among the several variables involved.

3. Equation (13) shows how both deterministic and probabilistic factors contribute to the develop- ment of abnormal grain growth, which explains why it is so difficult to control.

4. The present treatment offers a new and plausible rationale for the development of abnormal grain growth from an initially uniform grain size distri- bution in materials containing unstable particles.

Acknow,ledgements-This work was supported by Conselho National de Desenvolvimento Cientifico e Tecnol6gico (CNPq) and by Companhia Siderlirgica National (CSN) through an agreement with Universidade Federal Flumi- nense. The author would like to acknowledge the contribution of the late Professor J. C. Bruno to the present work.

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