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On a stochastic theory of grain growth—III
Acta metall, mater. Vol. 39, No. 6, pp. 135%1365, 1991
0956-7151/91 $3.00 + 0.00 Copyright © 1991 Pergamon Press pie
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ON A STOCHASTIC THEORY OF GRAIN GROWTH--III C. S. P A N D E 1 and E. D A N T S K E R 2
INaval Research Laboratory, Washington, DC 20375-5000 and 2Geo-Centers Inc., Fort Washington, Md, U.S.A. (Received 10 April 1990)
Abstract--The stochastic theory of grain growth proposed previously [C. S. Pande, Acta metall. 35, 2671 (1987)] is examined and discussed in the light of a recent criticism [N. Ryum and O. Hunderi, Aeta metall. 37, 1375 (1989)]. We show that the theory is able to answer all major objections raised by Ryum and Hunderi. In particular we develop further the stochastic theory utilizing an N dimensional diffusion term, (N = 1, 2, or 3) and show mathematically that for N = 3 the volume of the specimen is strictly conserved. The grain size distribution is obtained for three dimensions in closed analytical form and is found to be approximately lognormal, in good agreement with experiments. R6sum~q:}n discute, fi la lumiere d'une critique recente IN. Ryum et O. Hunderi, Acta metall. 37, 1375 (1989}], la th6orie stochastique de la croissance des grains propos6e par C. S. Pande [Aeta metall. 35, 2671 (1987)], On montre que la th6orie est capable de r6pondre fi toutes les objections majeures formul6es par Ryum et Hunderi. En particulier, en developpe la th6orie stochastique en utilisant un terme de diffusion ~i N dimensions. (N = 1, 2 ou 3) et on montre math6matiquement que pour N = 3 le volume de l'6chantillon est strictement conserv6. On trouv6 que la distribution de taille des grains est obtenue pour trois dimensions sous une form6 analytique ferm6e et qu'elle est approximativement lognormale, en bon accord avec l'exp6rience. Zusammenfassung--Die vor kurzem vorgeschlagene stochastische Theorie des Kornwachstums [C. S. Pande, Acta metall. 35, 2671 (1987)] wird im Hinblick auf eine kiirzlich erschienene Kritik [N. Ryum and O. Hunderi, Acta metalL 37, 1375 (1989)] untersucht und diskutiert. Wir zeigen, dab die Theorie s/imtliche wesentlicheren von Ryum und Hunderi ge/iuBerten Einw/inde beantworten kann. Insbesondere entwickeln wir die stochastische Theorie weiter, indem wir einen N-dimensionalen Diffusionsterm benutzen (N = 1, 2, oder 3); wir zeigen mathematisch, dab das Volumen der Probe fiir N = 3 strikt erhalten bleibt. Die Verteilung der Korngrenzengr6Ben wird ffir drei Dimensionen in geschlossener analytischer Form erhalten; sie ist in guter Obereinstimmung mit den Experimenten etwa lognormal.
1. INTRODUCTION Grain growth is the increase in average grain size of a polycrystalline solid on annealing after recrystallization. (For a review see [1].) In recent years the phenomenon of grain growth has been the subject of considerable theoretical and computer investigations [2-8]. The experimental details of the process are more or less well understood. It is found that grains in a polycrystalline metal show a characteristic distribution in both size and shape [i.e. n u m b e r of facets (in three dimensions) or n u m b e r of sides (in two dimensions)]. Measurements of volume distribution in titanium by Okazaki and Conard [9] and by Rhines and Patterson [10] in aluminum clearly show that the distribution is characteristically lognormal. These distributions were obtained by separating and weighing individual grains, and thus are not subject to usual errors in the determination of the grain sizes. In addition, Rhines and Patterson have also determined the range of the standard deviation a of the lognormal distribution (approximately 0.4-2). A n y theory of grain growth must therefore be consistent with these results.
The complex nature of the problem has made an analytical theory of grain growth very difficult to formulate. Two basically different models of normal grain growth exist. One type of models consider grains moving in grain size space under the action of capillary forces. These models can be called drift models. Hillert's theory [11] is an example of this kind. A different model due to Louat [12] considers the grains to do random walk in grain size space, independent of any other forces. In this case grain growth becomes a diffusion-like process, with the distribution function F given by a diffusion type equation. It is shown by Ryum and Hunderi [13] that the drift model of Hillert has no solid physical basis, but the pure diffusion model of Louat cannot also be considered satisfactory, since it ignores capillary forces. It is also found [14] (henceforth called Part I) that Hillert's model gives grain size distributions which are too sharp, whereas Louat's model fits experiment only when standard deviation a is -~ 0.5. Saetre et al. [15] have proposed a model for normal grain growth where the rate of growth of each grain in the system is determined by the grain size distribution in its neighborhood. While the concept of
1359
1360
PANDE and DANTSKER: STOCHASTIC THEORY OF GRAIN GROWTH--III
such a model is probably more realistic, the treatment does not appear to be correct because the resultant grain size distribution is even sharper than Hillert's distribution, and clearly not in agreement with experiments. In an attempt to develop a realistic theory of normal grain growth it is was recognized recently [14] that the phenomenon of grain growth has characteristic features of a stochastic process, so that the mathematical apparatus of this process can be used for a precise description of grain growth. It is known that the rate of grain boundary migration in normal grain growth is proportional to its curvature, however a complete description of grain growth requires a consideration of all such boundaries or of the motion of the grains taking into account the surroundings and topology. Such a description is of course impractical even for a computer and a simpler description in terms of the stochastic motion of individual grains was proposed. In this description the grain boundaries move not only under the influence of capillary forces (curvature effects), but also under the influence of surroundings which adds a random or "noise" term. Validity of such a picture of grain growth is discussed in section (4) and the nature of the random "noise" term has been discussed in detail in [14]. Essentially it is argued that grains grow or shrink both under the drift and diffusion-like motion, the diffusion term modelling the effect of the topology and environment. Thus both drift and diffusion terms may be important in the description of grain growth; the relative proportion of the two is an independent parameter in the theory, which is determined from experiments. Indeed, that is the only unknown parameter in the theory. A rigorous theory of isothermal grain growth for metals and alloys was developed using grain size (or grain boundary) as the main variable, the other variable (as is usual in simple stochastic processes) is time. Several constraints were imposed on the system in obtaining a grain size distribution, one of the most important is the constancy of the specimen size of volume (this will be discussed in much greater detail in Section 2.1). In a recent paper [13] Ryum and Hunderi have reviewed th e main analytical theories of normal grain growth in metals and alloys. Based on their own considerable research in the area of grain growth, they provide a deep insight into the complexity of the problem and shortcomings of the many attempts to provide a realistic theory of grain growth. Their review is thus certain to stimulate further interest into this fascinating and technologically important phenomenon. While we agree with many of their assessments on grain growth in general, we find some of their views inexact, and even erroneous. One aim of this paper is to provide an opposing point of view to some of the issues raised in their paper. Specifically, we argue that their criticism of our stochastic theory [14] of grain grown is unjustified and to
contend that, at this stage, this theory provides a most realistic description of normal grain growth. Indeed, most of the objections raised by them against this theory can be easily answered (see below). 2. STOCHASTIC THEORY OF GRAIN GROWTH FOR THREE DIMENSIONS One of the main criticisms of the stochastic theory by Ryum and Hunderi is that it does not satisfy the requirement of volume conservation during grain growth, and hence by implication it does not represent a quasistationary (i.e. self similar) distribution. They believe that the theory only satisfies the boundary condition for growth in one dimension. This criticism was anticipated in [14], and that paper specifically answers this issue qualitatively. Briefly, the grain size distribution function F in our model is given by a second order partial differential equation as t3F
--a
D 02F
Ot = O----RIf(R, t)F] +-~ OR---I
(1)
where D is the diffusion constant. The second term is the well-known diffusion term for one dimension. As we mentioned earlier, treating N dimensional (N = 1, 2, or 3) grain growth using a one dimensional diffusion equation is permissible, because this is equivalent to monitoring grain size by the linear intercept method. In grain size space, an individual grain is characterized by a point which moves to the left or right during grain growth. In such a situation the conservation equation should also be concerned only with N = 1, i.e. in conservation of a given length L, wh!ch is broken up into various grain intercepts. To obtain a solution which satisfies volume conservation, Ryum and Hunderi [13] transform a one-dimensional diffusion term to provide a three dimensional grain size distribution. (It also has no deterministic term.) This procedure is obviously flawed, and so as expected the resultant distribution function bears no resemblance to experimental distribution, it is nearly symmetrical with a cut off at R (2/~ = mean grain size). As pointed out in [14], if one insists on considering grain growth in higher dimensions, the corresponding diffusion term should be used, and not D/2(O2F/OR 2) which is valid only for one dimension (although the drift term remains unchanged). A detailed analysis of the partial differential equation using an N dimensional diffusion term will be considered in the next section. As in [14] we take 0t
f ( R ) = ---R
(2)
where ~ is a positive constant, and has the same meaning as in [14].
PANDE and DANTSKER: STOCHASTIC THEORY OF GRAIN GROWTH--III We also show that, as in [14], the theory is in good accord with experiments in the sense that the predicted distributions are approximately lognormal and the average grain size increases as p/2 (t is the time of annealing). Finally the applicability of the stochastic process to model grain growth is discussed in Section 4.
where Co is a normalizing constant and • is the degenerate hypergeometric function defined by the series .4z • (A, B; Z ) = 1 + ~
Based on the arguments given in Section (2) the differential equation representing grain size distribution in arbitrary dimensions is D
1
The specific case of grain growth in three dimensions should now be focused upon. With N = 3, equation (5) gives
k = ~ [ - (l + 2~) -t- /(1+ 2~12 + 8_~3. (13)
O ( O F )
+ 2RU_,O R RU-'ff-~
(3)
where N refers to the number of dimensions. A key step in solving equation (3) is writing
F(R, t) = RkC(R, t).
Figure 1 shows the relationship between k and ot/D to be hyperbolic. The condition requiring constant specimen size, namely
o~ F(R, t) RU dR
(4)
Substituting equation (4) into equation (3) and choosing k such that k:+
2~) k - - ~2c( =O
(5)
N--2+-~
= constant, independent of time
k n =2+~.
OC ~' OC D 02C
(6)
ot = ~ o--k + 5 ----~ OR where D a' = :¢ + -~ (2k + N - 1).
(7)
c(R,t)=i;
R2
R3
0+A,~+.4~+`4~+---
]
(8/
(15/
The constancy of the specimen size is thus assured by a proper choice of n, which in fact fixes n in terms of k (or ct/D and N). The distribution function for grain growth in three dimensions is then C0(
R
)k
F(R' t ) = -iT ~ - - ~ t , ]
If the solution is assumed to be self-similar as in [14], then C must have the form R
(14)
where N = 3, 2, or 1, then gives the correct relationship between n and k. For N = 3, from (14) one must have
a new equation is obtained. It is
i[A
(12)
3.2. N = 3
3.1. The N-dimensional Fokker-Planck equation
O (F)
t----
B(B + ])(B + 2) 3?
3. ANALYSIS
Ot=ct~
`4(`4 + l ) Z 2
-~ B(B + ]~) 2.t
A(A + 1)(.4 + 2 ) g 3
+
OF
1361
(
q
2 + ~ k, gs+ ~ + k3;
×e
-2b-f
Of course ifNis set to 1, then k = 1 and equation (11) reduces to the solution obtained in Part I [14] namely
CoR (~ e 3 F(R't)="~ i-~ ' D + 2 ;
where R~ is defined by
R 2c_-_ Kt
(9)
. (16)
R E) ~t "
(17)
2O
and n is to be determined by the specimen-size conservation requirement. The relation (8) is essential for self similarity [see 14, 22]. It can be shown from (8) that 0
OC Ot
nKC R~
RK OC 2R~OR"
(10) -10
Using this result, equation (6) is converted into an ordinary second-order differential equation, which can be solved by standard methods. The solution is
-20
i -10
F(R,t)=Co~q~ n, 54 D,
,
,
,
i -5
.
.
.
.
i O
.
.
.
.
i
.
.
.
.
~
i 10
Fig. 1. The dependence ofk on a/D. Note that lim k = 1.
2-~
(ll)
D
PANDE and DANTSKER:
1362
STOCHASTIC THEORY OF GRAIN GROWTH--III
where a is that cut-off value (determined experimentally). With condition (19) imposed, the grain-size distribution can certainly be
1.4 1.2 olD
"~I.0
=
0.0
0.8
~30.6
where C1 and C2 are constants. For a complete solution the value of a must be known in order to obtain CI/C 2. a is not likely to be found from pure theory. All that can be inferred
+ 0.4 b_ 0.2 0.0
1
0
2 R/
3
(from the expected shapes of the distributions) is that
4
Ct > 0 and C 2 < 0. Thus, until experiment provides us
Fig. 2. Comparison of F÷ grain size distributions for various values of ct/D. Each distribution is normalized. When ct/D = O, F is reduced to the classical Rayleigh distribution in I-D but does not give a realistic distribution in 3-D.
with an appropriate non-zero cut-off value, we shall
If the distribution given by (17) is considered, it must be understood that grain size is monitored by the linear intercept method.
With condition (18) imposed, F(R, t ) = F+(R, t). In Part I, a distribution was obtained using a one-dimensional diffusion term in equation (3). In
set a = 0 and employ condition [18]. 3.4. Comparison with Part I and lognormal distribution
Fig. 3.3. Grain size distribution
A grain size distribution for three dimensional grain growth can now be obtained for any value of ¢t/D using equations (16) and (13). The mathematics,
4,
the
plus sign in equation (13) (hereafter called F÷). F -
~
0.8
is shown in Fig. 3. If the boundary condition
is insisted on, then F -
0
(18)
,~0.6
must be discarded. However,
>, o 0.4
experimental grain size distributions are known to have a non-zero cut-off size, i.e. grains with radii less
than a certain cut-off value are not found in a given
3-dimensional
(a)
Figure 2 shows the distributions obtained using the
=
and
walk. It is seen that the 1-D and 3-D distributions
however, suggests two possible solutions as there are two possible values of k for each ot/D (see Fig. 1).
F(0, t)
1-dimensional
grain size distributions are plotted for different values of ct/D, where ct/D represents the relative strength between force driven motion and random
u/O • 0.50
-D
.o b-
I
1-0
~'\
0.2
distribution. The equivalent boundary condition is
J
I
F(a, t)
=
0
(19)
0,0 J
0
~ /
1.2 1.0
0.8
~
0.4
,
,
,
i
,
1
,
,
,
2 R/
3
4
Cb) .0
0.6
a / D - 1.00
3-D
~" 0 . 2 0.0 -0.2
0.5
1.0
1.5
2.0
R/
Fig. 3. Comparison of normalized F_ grain size distributions for various values of e/D. Note that Jim F _ = -o~. = -40 D
When o~/D >i 1.0, F_ is entirely negative.
0.0[
-
-
R/
Fig. 4. (a,b) Comparison of one and three dimensional grain size distributions (F+) for various values of ~/D. The agreement is better for larger ot/D.
PANDE and DANTSKER: STOCHASTIC THEORY OF GRAIN GROWTH--III are qualitatively similar and agree more with increasing ~,/D, O.e. with decreasing random component) which is to be expected since dimension N enters only in the diffusion term in equation (3) and with increasing =/D, the influence of the diffusion term is reduced. In Fig. 5 the 3-D grain size distributions are compared with the corresponding lognormal distributions. The agreement between the two distributions are satisfactory, indicating that equation (16) gives an approximate lognormal distribution. (o)
•
.
.
.
,
.
.
.
.
a,g:o:::
0.8
.•o.6 S
>, 0
0
.
4
"0.2
0.o 0
1
2 R / < R>
3
4
(b) ff'~
~/D • 1.00
i ,\
0.6
o.o.
g
'\\
0~,0.4
i b. 0.2
Lognormo
J
0.0
0
1
(c)
2 R/
•
0.5
,
3
.
,
/h~\
,
o l D - 2.00
\\
o- -
o.~9
0.4
.$ >,0.3
~ 0.2 Z o.11 O.C
]
l l
t.og
....
t
~
_/ 0
1
2 R/
3
4 " " "
Fig. 5. (a,b,c) Comparison of some three dimensional grain size distributions with corp~'ponding lognormal distributions. Experimentally, ¢rlies between 0.4 and 2 (see text).
1363
4. APPLICABILITY OF THE STOCHASTIC PROCESS The other major objection of Ryum and Hunderi [13] to the stochastic theory of grain growth is the applicability of the stochastic process itself to grain growth. Before coming to the mathematics of the process we mention the physical basis of the stochastic process. (I) The computer simulation of grain growth by Srolovitz et al. [16] clearly show that when dR/dt is plotted as a function of R, one does not obtain a smooth curve, rather it is a broad curve, with points scattered over a wide band. The width of the band is not due to inaccuracy of the data, since it is several times the width expected from the inaccuracies of computation alone. (2) The experimental evidence briefly mentioned in Part I [14], where the grain growth has been observed/n situ using a photo emission microscope, clearly show that grain boundaries do not move on the basis of curvature alone, (i.e. directed motion), larger grains especially grow or shrink, apparently at random. It should be stated that unlike Louat [12] we do not postulate that the directed motion due to curvature effects are unimportant. What we do state is that the two components viz directed motion as well as random motion arc both present. The proportion of the two is given by a single term i.e. ~,/D. This is a parameter of the theory, whose value has not yet been determined from fundamental principles. Ryum and Hunderi state that random component if it exists must be small. We do not find any theoretical or experimental evidence for this statement. Our analysis is valid irrespective of the proportion of the deterministic to random component. (3) The third evidence for a random component comes from studying the Ostwald ripening effect for large volume fraction. As mentioned in part I [14] the phenomenon of grain growth has great similarity with another phenomenon viz "Ostwald ripening". Indeed the mathematical analysis of Ostwald ripening by Lifshitz and Slyozov [17] and by Wagner [18], i.e. LSW theory provided a basis for the grain growth theory by Hillert. As Ryum and Hunderi [13] show, Hillert's theory is based on the consideration of a single grain growing in the "smeared out" or "average" surrounding of all other grains. The similarity between this model and LSW model is thus made even more obvious. It should be noted that LSW model is based on precipitate growth in a dilute system, i.e. when the precipitates are widely separated. What happens when the volume fraction of the precipitates are increased to say about 10%? The precipitates are now separated on the average
1364
PANDE and DANTSKER: STOCHASTIC THEORY OF GRAIN GROWTH--III by a distance less than their average size (it should be noted that this is an intermediate situation and does not correspond exactly to a grain ensemble where grains are touching each other (i.e. share common boundaries). Marder [19] for example shows that in such a situation the precipitates are found to grow at a variety of rates for a given size depending on the environment, a fluctuation in the growth rate of one precipitate affects those surrounding it. It should be noted that Marder does not start with a stochastic or random term, rather he considers the growth of the precipitates obeying the usual diffusion laws and boundary constraints. From these considerations Marder obtains a stochastic equation containing both the deterministic and diffusion term. Similar results have also been obtained by Tokuyama and Kawasaki [20]. Thus the use of the stochastic process to model effects of the surroundings is based on sound basis in theory, computation, and experiments. In every case where the effect of the surroundings has been taken into account a random component is present in the boundary motion and the size distribution is always found to be less sharp. Any other result should be questionable for its experimental or mathematical validity.
Finally we comment on equation obtained by us for equation was obtained by the stochastic theory starting from
the Fokker-Planck grain growth. This standard methods of a Langevin equation
dR
d--t =f(R, t) + r(t)
(21)
where f(R, t) is the deterministic term and T is the noise term. We assumed T to have properties similar to those in a classical Langevin equation. In the mathematical study of random walk or Brownian motion T(t) is assumed to be a Gaussian random variable with zero mean and with a correlation function which is proportional to a delta function 6A i.e.=0,
=BtSA(t-t')
(22)
where D is a constant and 6A is the Dirac delta function. Physically T(t) is thus a random fluctuating term acquiring values which are both positive and negative at various times with a mean value of zero. The analogy with Brownian motion is clear. In Brownian motion the apparent randomness of the motion is characterized by assuming that each collision moves the particle independently although the motion could be the combined effect of many, many collisions. Such an assumption is permissible and is usually justified (in case of Brownian motion) by using the notion that even a small variation in the velocity of the particle is much larger than the aftereffects of each collision leading to Brownian
motion. It should be noted that Einstein [21] in his analysis of Brownian motion had to make two assumptions which though reasonable, are necessary for the mathematical analysis. (1) Each particle is assumed to execute a motion which is independent of the motion of all other particles, also (2) that movements of any particle at different time intervals are independent processes. Obviously these may not be exactly true as in the case of grain growth, but could be justified by a similar argument. The plausible argument as in the case of Brownian motion is that any small increment in dR/dt is the combined effect of many small individual increments, such that each increment in dR/dt could effectively be treated as random and independent of others. Thus the Fokker-Planck equation for grain growth is based on sound physical and mathematically plausible principles. Further experimental and theoretical work may however be necessary to check the basic principles, and obtain more precise knowledge about drift and diffusion terms. 5. CONCLUSIONS The stochastic theory of grain growth described by Pande in [14] (Part I) is discussed and developed further in this paper, in the light of recent criticism of that paper by Ryum and Hunderi. Specifically we answer two main criticisms of the theory and show (1) In stochastic theory, the volume of the specimen is strictly conserved. This is proven rigorously by further analysis using "diffusion" in one and three dimensions as suggested previously [14]. (2) The stochastic nature of the grain growth is shown to be supported by available experimental, computer, and theoretical calculations. The mathematical idealizations needed are similar to those for Brownian motion and are based on plausible arguments. Further experimental and theoretical calculations however may be needed to establish the details of stochastic behavior. An example of such detail would be the value of ~t/D which determines the proportion of the deterministic to random component. So far, the value of this parameter is estimated by obtaining the best fit of the predicted grain size distribution with the experimental distribution. However the analysis presented is valid for any value of ct/D i.e. for any proportion of deterministic to random component. While we agree with Ryum and Hunderi that the diffusion term alone (as in Louat's model) is not sufficient to account for grain growth, we disagree with their statement that the diffusion term added as a perturbation to the Hillert-like drift term will lead to a realistic model of grain growth (this was
PANDE and DANTSKER:
STOCHASTIC THEORY OF GRAIN GROWTH--III
rigorously shown to be not so, in Part II [22]). It is our view that both drift and diffusion terms are important for grain growth as postulated in [14]. In view of the good agreement between theory and experiment, and also because the theory rigorously meets all the objections of R y u m and Hunderi we believe that the stochastic theory of grain growth is essentially sound and deserves further consideration. Indeed it is our expectation that further progress in theoretical description of grain growth is most likely to occur on the basis of such a theory. REFERENCES
I. H. V. Atkinson, Acta metall. 36, 469 (1988). 2. P. Feltham, Acta metall. 5, 97 (1957). 3. S. K. Kurtz and F. M. Carpay, J. appl. Phys. 51, 5725 (1980). 4. V. Y. Novikov, Acta metall. 26, 1739 (1978). 5. M. P. Anderson, G. S. Grest and D. J. Srolovitz, Scripta metall. 19, 225 (1985). 6. V. E. Fradkov, L. S. Shrindleman and D. G. Udler, Scripta metall. 19, 1285 (1985).
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7. 1-Wei Chen, Acta metall. 35, 1723 (1988). 8. M. P. Anderson, G. S. Grest and D. J. Srolovitz, Phil. Mag. B 59, 293 (1989). 9. K. Okazaki and H. Conrad, Trans. Japan Inst. Metals 13, 198 (1972). 10. F. N. Rhines and B. R. Patterson, MetalL Trans. 13A, 985 (1982). 11. M. Hillert, Acta metall. 13, 227 (1965). 12. N. P. Louat, Acta metall. 22, 721 (1974). 13. N. Ryum and O. Hunderi, Acta metall. 37, 1375 (1989). 14. C. S. Pande, Acta metall. 35, 2671 (1987). 15. N. Ryum and O. Hunderi, Acta metall. 37, 1380 (1989). 16. D. J. Srolovitz, M. P. Anderson, P. S. Sahni and G. S. Grest, Acta metall. 32, 793 (1984). 17. I. M. Lifshitz and V. V. Slyozov, J. Phys. Chem. Solids 19, 35 (1961). 18. C. Wagner, Z. Elektrochem. 65, 581 (1961). 19. M. Marder, Phys. Rev. A 36, 858 (1987). 20. M. Tokuyama and K. Kawasaki, Physica .4 123, 186 (1984). 21. A. Einstein, quoted in: C. W. Gardiner, Handbook o f Scientific Methods, 2nd edn, p. 3. Springer, Berlin (1985). 22. C. S. Pande and E. Dantsker, .4cta metall, mater. 38, 945 (1990).
0956-7151/91 $3.00 + 0.00 Copyright © 1991 Pergamon Press pie
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ON A STOCHASTIC THEORY OF GRAIN GROWTH--III C. S. P A N D E 1 and E. D A N T S K E R 2
INaval Research Laboratory, Washington, DC 20375-5000 and 2Geo-Centers Inc., Fort Washington, Md, U.S.A. (Received 10 April 1990)
Abstract--The stochastic theory of grain growth proposed previously [C. S. Pande, Acta metall. 35, 2671 (1987)] is examined and discussed in the light of a recent criticism [N. Ryum and O. Hunderi, Aeta metall. 37, 1375 (1989)]. We show that the theory is able to answer all major objections raised by Ryum and Hunderi. In particular we develop further the stochastic theory utilizing an N dimensional diffusion term, (N = 1, 2, or 3) and show mathematically that for N = 3 the volume of the specimen is strictly conserved. The grain size distribution is obtained for three dimensions in closed analytical form and is found to be approximately lognormal, in good agreement with experiments. R6sum~q:}n discute, fi la lumiere d'une critique recente IN. Ryum et O. Hunderi, Acta metall. 37, 1375 (1989}], la th6orie stochastique de la croissance des grains propos6e par C. S. Pande [Aeta metall. 35, 2671 (1987)], On montre que la th6orie est capable de r6pondre fi toutes les objections majeures formul6es par Ryum et Hunderi. En particulier, en developpe la th6orie stochastique en utilisant un terme de diffusion ~i N dimensions. (N = 1, 2 ou 3) et on montre math6matiquement que pour N = 3 le volume de l'6chantillon est strictement conserv6. On trouv6 que la distribution de taille des grains est obtenue pour trois dimensions sous une form6 analytique ferm6e et qu'elle est approximativement lognormale, en bon accord avec l'exp6rience. Zusammenfassung--Die vor kurzem vorgeschlagene stochastische Theorie des Kornwachstums [C. S. Pande, Acta metall. 35, 2671 (1987)] wird im Hinblick auf eine kiirzlich erschienene Kritik [N. Ryum and O. Hunderi, Acta metalL 37, 1375 (1989)] untersucht und diskutiert. Wir zeigen, dab die Theorie s/imtliche wesentlicheren von Ryum und Hunderi ge/iuBerten Einw/inde beantworten kann. Insbesondere entwickeln wir die stochastische Theorie weiter, indem wir einen N-dimensionalen Diffusionsterm benutzen (N = 1, 2, oder 3); wir zeigen mathematisch, dab das Volumen der Probe fiir N = 3 strikt erhalten bleibt. Die Verteilung der Korngrenzengr6Ben wird ffir drei Dimensionen in geschlossener analytischer Form erhalten; sie ist in guter Obereinstimmung mit den Experimenten etwa lognormal.
1. INTRODUCTION Grain growth is the increase in average grain size of a polycrystalline solid on annealing after recrystallization. (For a review see [1].) In recent years the phenomenon of grain growth has been the subject of considerable theoretical and computer investigations [2-8]. The experimental details of the process are more or less well understood. It is found that grains in a polycrystalline metal show a characteristic distribution in both size and shape [i.e. n u m b e r of facets (in three dimensions) or n u m b e r of sides (in two dimensions)]. Measurements of volume distribution in titanium by Okazaki and Conard [9] and by Rhines and Patterson [10] in aluminum clearly show that the distribution is characteristically lognormal. These distributions were obtained by separating and weighing individual grains, and thus are not subject to usual errors in the determination of the grain sizes. In addition, Rhines and Patterson have also determined the range of the standard deviation a of the lognormal distribution (approximately 0.4-2). A n y theory of grain growth must therefore be consistent with these results.
The complex nature of the problem has made an analytical theory of grain growth very difficult to formulate. Two basically different models of normal grain growth exist. One type of models consider grains moving in grain size space under the action of capillary forces. These models can be called drift models. Hillert's theory [11] is an example of this kind. A different model due to Louat [12] considers the grains to do random walk in grain size space, independent of any other forces. In this case grain growth becomes a diffusion-like process, with the distribution function F given by a diffusion type equation. It is shown by Ryum and Hunderi [13] that the drift model of Hillert has no solid physical basis, but the pure diffusion model of Louat cannot also be considered satisfactory, since it ignores capillary forces. It is also found [14] (henceforth called Part I) that Hillert's model gives grain size distributions which are too sharp, whereas Louat's model fits experiment only when standard deviation a is -~ 0.5. Saetre et al. [15] have proposed a model for normal grain growth where the rate of growth of each grain in the system is determined by the grain size distribution in its neighborhood. While the concept of
1359
1360
PANDE and DANTSKER: STOCHASTIC THEORY OF GRAIN GROWTH--III
such a model is probably more realistic, the treatment does not appear to be correct because the resultant grain size distribution is even sharper than Hillert's distribution, and clearly not in agreement with experiments. In an attempt to develop a realistic theory of normal grain growth it is was recognized recently [14] that the phenomenon of grain growth has characteristic features of a stochastic process, so that the mathematical apparatus of this process can be used for a precise description of grain growth. It is known that the rate of grain boundary migration in normal grain growth is proportional to its curvature, however a complete description of grain growth requires a consideration of all such boundaries or of the motion of the grains taking into account the surroundings and topology. Such a description is of course impractical even for a computer and a simpler description in terms of the stochastic motion of individual grains was proposed. In this description the grain boundaries move not only under the influence of capillary forces (curvature effects), but also under the influence of surroundings which adds a random or "noise" term. Validity of such a picture of grain growth is discussed in section (4) and the nature of the random "noise" term has been discussed in detail in [14]. Essentially it is argued that grains grow or shrink both under the drift and diffusion-like motion, the diffusion term modelling the effect of the topology and environment. Thus both drift and diffusion terms may be important in the description of grain growth; the relative proportion of the two is an independent parameter in the theory, which is determined from experiments. Indeed, that is the only unknown parameter in the theory. A rigorous theory of isothermal grain growth for metals and alloys was developed using grain size (or grain boundary) as the main variable, the other variable (as is usual in simple stochastic processes) is time. Several constraints were imposed on the system in obtaining a grain size distribution, one of the most important is the constancy of the specimen size of volume (this will be discussed in much greater detail in Section 2.1). In a recent paper [13] Ryum and Hunderi have reviewed th e main analytical theories of normal grain growth in metals and alloys. Based on their own considerable research in the area of grain growth, they provide a deep insight into the complexity of the problem and shortcomings of the many attempts to provide a realistic theory of grain growth. Their review is thus certain to stimulate further interest into this fascinating and technologically important phenomenon. While we agree with many of their assessments on grain growth in general, we find some of their views inexact, and even erroneous. One aim of this paper is to provide an opposing point of view to some of the issues raised in their paper. Specifically, we argue that their criticism of our stochastic theory [14] of grain grown is unjustified and to
contend that, at this stage, this theory provides a most realistic description of normal grain growth. Indeed, most of the objections raised by them against this theory can be easily answered (see below). 2. STOCHASTIC THEORY OF GRAIN GROWTH FOR THREE DIMENSIONS One of the main criticisms of the stochastic theory by Ryum and Hunderi is that it does not satisfy the requirement of volume conservation during grain growth, and hence by implication it does not represent a quasistationary (i.e. self similar) distribution. They believe that the theory only satisfies the boundary condition for growth in one dimension. This criticism was anticipated in [14], and that paper specifically answers this issue qualitatively. Briefly, the grain size distribution function F in our model is given by a second order partial differential equation as t3F
--a
D 02F
Ot = O----RIf(R, t)F] +-~ OR---I
(1)
where D is the diffusion constant. The second term is the well-known diffusion term for one dimension. As we mentioned earlier, treating N dimensional (N = 1, 2, or 3) grain growth using a one dimensional diffusion equation is permissible, because this is equivalent to monitoring grain size by the linear intercept method. In grain size space, an individual grain is characterized by a point which moves to the left or right during grain growth. In such a situation the conservation equation should also be concerned only with N = 1, i.e. in conservation of a given length L, wh!ch is broken up into various grain intercepts. To obtain a solution which satisfies volume conservation, Ryum and Hunderi [13] transform a one-dimensional diffusion term to provide a three dimensional grain size distribution. (It also has no deterministic term.) This procedure is obviously flawed, and so as expected the resultant distribution function bears no resemblance to experimental distribution, it is nearly symmetrical with a cut off at R (2/~ = mean grain size). As pointed out in [14], if one insists on considering grain growth in higher dimensions, the corresponding diffusion term should be used, and not D/2(O2F/OR 2) which is valid only for one dimension (although the drift term remains unchanged). A detailed analysis of the partial differential equation using an N dimensional diffusion term will be considered in the next section. As in [14] we take 0t
f ( R ) = ---R
(2)
where ~ is a positive constant, and has the same meaning as in [14].
PANDE and DANTSKER: STOCHASTIC THEORY OF GRAIN GROWTH--III We also show that, as in [14], the theory is in good accord with experiments in the sense that the predicted distributions are approximately lognormal and the average grain size increases as p/2 (t is the time of annealing). Finally the applicability of the stochastic process to model grain growth is discussed in Section 4.
where Co is a normalizing constant and • is the degenerate hypergeometric function defined by the series .4z • (A, B; Z ) = 1 + ~
Based on the arguments given in Section (2) the differential equation representing grain size distribution in arbitrary dimensions is D
1
The specific case of grain growth in three dimensions should now be focused upon. With N = 3, equation (5) gives
k = ~ [ - (l + 2~) -t- /(1+ 2~12 + 8_~3. (13)
O ( O F )
+ 2RU_,O R RU-'ff-~
(3)
where N refers to the number of dimensions. A key step in solving equation (3) is writing
F(R, t) = RkC(R, t).
Figure 1 shows the relationship between k and ot/D to be hyperbolic. The condition requiring constant specimen size, namely
o~ F(R, t) RU dR
(4)
Substituting equation (4) into equation (3) and choosing k such that k:+
2~) k - - ~2c( =O
(5)
N--2+-~
= constant, independent of time
k n =2+~.
OC ~' OC D 02C
(6)
ot = ~ o--k + 5 ----~ OR where D a' = :¢ + -~ (2k + N - 1).
(7)
c(R,t)=i;
R2
R3
0+A,~+.4~+`4~+---
]
(8/
(15/
The constancy of the specimen size is thus assured by a proper choice of n, which in fact fixes n in terms of k (or ct/D and N). The distribution function for grain growth in three dimensions is then C0(
R
)k
F(R' t ) = -iT ~ - - ~ t , ]
If the solution is assumed to be self-similar as in [14], then C must have the form R
(14)
where N = 3, 2, or 1, then gives the correct relationship between n and k. For N = 3, from (14) one must have
a new equation is obtained. It is
i[A
(12)
3.2. N = 3
3.1. The N-dimensional Fokker-Planck equation
O (F)
t----
B(B + ])(B + 2) 3?
3. ANALYSIS
Ot=ct~
`4(`4 + l ) Z 2
-~ B(B + ]~) 2.t
A(A + 1)(.4 + 2 ) g 3
+
OF
1361
(
q
2 + ~ k, gs+ ~ + k3;
×e
-2b-f
Of course ifNis set to 1, then k = 1 and equation (11) reduces to the solution obtained in Part I [14] namely
CoR (~ e 3 F(R't)="~ i-~ ' D + 2 ;
where R~ is defined by
R 2c_-_ Kt
(9)
. (16)
R E) ~t "
(17)
2O
and n is to be determined by the specimen-size conservation requirement. The relation (8) is essential for self similarity [see 14, 22]. It can be shown from (8) that 0
OC Ot
nKC R~
RK OC 2R~OR"
(10) -10
Using this result, equation (6) is converted into an ordinary second-order differential equation, which can be solved by standard methods. The solution is
-20
i -10
F(R,t)=Co~q~ n, 54 D,
,
,
,
i -5
.
.
.
.
i O
.
.
.
.
i
.
.
.
.
~
i 10
Fig. 1. The dependence ofk on a/D. Note that lim k = 1.
2-~
(ll)
D
PANDE and DANTSKER:
1362
STOCHASTIC THEORY OF GRAIN GROWTH--III
where a is that cut-off value (determined experimentally). With condition (19) imposed, the grain-size distribution can certainly be
1.4 1.2 olD
"~I.0
=
0.0
0.8
~30.6
where C1 and C2 are constants. For a complete solution the value of a must be known in order to obtain CI/C 2. a is not likely to be found from pure theory. All that can be inferred
+ 0.4 b_ 0.2 0.0
1
0
2 R/
3
(from the expected shapes of the distributions) is that
4
Ct > 0 and C 2 < 0. Thus, until experiment provides us
Fig. 2. Comparison of F÷ grain size distributions for various values of ct/D. Each distribution is normalized. When ct/D = O, F is reduced to the classical Rayleigh distribution in I-D but does not give a realistic distribution in 3-D.
with an appropriate non-zero cut-off value, we shall
If the distribution given by (17) is considered, it must be understood that grain size is monitored by the linear intercept method.
With condition (18) imposed, F(R, t ) = F+(R, t). In Part I, a distribution was obtained using a one-dimensional diffusion term in equation (3). In
set a = 0 and employ condition [18]. 3.4. Comparison with Part I and lognormal distribution
Fig. 3.3. Grain size distribution
A grain size distribution for three dimensional grain growth can now be obtained for any value of ¢t/D using equations (16) and (13). The mathematics,
4,
the
plus sign in equation (13) (hereafter called F÷). F -
~
0.8
is shown in Fig. 3. If the boundary condition
is insisted on, then F -
0
(18)
,~0.6
must be discarded. However,
>, o 0.4
experimental grain size distributions are known to have a non-zero cut-off size, i.e. grains with radii less
than a certain cut-off value are not found in a given
3-dimensional
(a)
Figure 2 shows the distributions obtained using the
=
and
walk. It is seen that the 1-D and 3-D distributions
however, suggests two possible solutions as there are two possible values of k for each ot/D (see Fig. 1).
F(0, t)
1-dimensional
grain size distributions are plotted for different values of ct/D, where ct/D represents the relative strength between force driven motion and random
u/O • 0.50
-D
.o b-
I
1-0
~'\
0.2
distribution. The equivalent boundary condition is
J
I
F(a, t)
=
0
(19)
0,0 J
0
~ /
1.2 1.0
0.8
~
0.4
,
,
,
i
,
1
,
,
,
2 R/
3
4
Cb) .0
0.6
a / D - 1.00
3-D
~" 0 . 2 0.0 -0.2
0.5
1.0
1.5
2.0
R/
Fig. 3. Comparison of normalized F_ grain size distributions for various values of e/D. Note that Jim F _ = -o~. = -40 D
When o~/D >i 1.0, F_ is entirely negative.
0.0[
-
-
R/
Fig. 4. (a,b) Comparison of one and three dimensional grain size distributions (F+) for various values of ~/D. The agreement is better for larger ot/D.
PANDE and DANTSKER: STOCHASTIC THEORY OF GRAIN GROWTH--III are qualitatively similar and agree more with increasing ~,/D, O.e. with decreasing random component) which is to be expected since dimension N enters only in the diffusion term in equation (3) and with increasing =/D, the influence of the diffusion term is reduced. In Fig. 5 the 3-D grain size distributions are compared with the corresponding lognormal distributions. The agreement between the two distributions are satisfactory, indicating that equation (16) gives an approximate lognormal distribution. (o)
•
.
.
.
,
.
.
.
.
a,g:o:::
0.8
.•o.6 S
>, 0
0
.
4
"0.2
0.o 0
1
2 R / < R>
3
4
(b) ff'~
~/D • 1.00
i ,\
0.6
o.o.
g
'\\
0~,0.4
i b. 0.2
Lognormo
J
0.0
0
1
(c)
2 R/
•
0.5
,
3
.
,
/h~\
,
o l D - 2.00
\\
o- -
o.~9
0.4
.$ >,0.3
~ 0.2 Z o.11 O.C
]
l l
t.og
....
t
~
_/ 0
1
2 R/
3
4 " " "
Fig. 5. (a,b,c) Comparison of some three dimensional grain size distributions with corp~'ponding lognormal distributions. Experimentally, ¢rlies between 0.4 and 2 (see text).
1363
4. APPLICABILITY OF THE STOCHASTIC PROCESS The other major objection of Ryum and Hunderi [13] to the stochastic theory of grain growth is the applicability of the stochastic process itself to grain growth. Before coming to the mathematics of the process we mention the physical basis of the stochastic process. (I) The computer simulation of grain growth by Srolovitz et al. [16] clearly show that when dR/dt is plotted as a function of R, one does not obtain a smooth curve, rather it is a broad curve, with points scattered over a wide band. The width of the band is not due to inaccuracy of the data, since it is several times the width expected from the inaccuracies of computation alone. (2) The experimental evidence briefly mentioned in Part I [14], where the grain growth has been observed/n situ using a photo emission microscope, clearly show that grain boundaries do not move on the basis of curvature alone, (i.e. directed motion), larger grains especially grow or shrink, apparently at random. It should be stated that unlike Louat [12] we do not postulate that the directed motion due to curvature effects are unimportant. What we do state is that the two components viz directed motion as well as random motion arc both present. The proportion of the two is given by a single term i.e. ~,/D. This is a parameter of the theory, whose value has not yet been determined from fundamental principles. Ryum and Hunderi state that random component if it exists must be small. We do not find any theoretical or experimental evidence for this statement. Our analysis is valid irrespective of the proportion of the deterministic to random component. (3) The third evidence for a random component comes from studying the Ostwald ripening effect for large volume fraction. As mentioned in part I [14] the phenomenon of grain growth has great similarity with another phenomenon viz "Ostwald ripening". Indeed the mathematical analysis of Ostwald ripening by Lifshitz and Slyozov [17] and by Wagner [18], i.e. LSW theory provided a basis for the grain growth theory by Hillert. As Ryum and Hunderi [13] show, Hillert's theory is based on the consideration of a single grain growing in the "smeared out" or "average" surrounding of all other grains. The similarity between this model and LSW model is thus made even more obvious. It should be noted that LSW model is based on precipitate growth in a dilute system, i.e. when the precipitates are widely separated. What happens when the volume fraction of the precipitates are increased to say about 10%? The precipitates are now separated on the average
1364
PANDE and DANTSKER: STOCHASTIC THEORY OF GRAIN GROWTH--III by a distance less than their average size (it should be noted that this is an intermediate situation and does not correspond exactly to a grain ensemble where grains are touching each other (i.e. share common boundaries). Marder [19] for example shows that in such a situation the precipitates are found to grow at a variety of rates for a given size depending on the environment, a fluctuation in the growth rate of one precipitate affects those surrounding it. It should be noted that Marder does not start with a stochastic or random term, rather he considers the growth of the precipitates obeying the usual diffusion laws and boundary constraints. From these considerations Marder obtains a stochastic equation containing both the deterministic and diffusion term. Similar results have also been obtained by Tokuyama and Kawasaki [20]. Thus the use of the stochastic process to model effects of the surroundings is based on sound basis in theory, computation, and experiments. In every case where the effect of the surroundings has been taken into account a random component is present in the boundary motion and the size distribution is always found to be less sharp. Any other result should be questionable for its experimental or mathematical validity.
Finally we comment on equation obtained by us for equation was obtained by the stochastic theory starting from
the Fokker-Planck grain growth. This standard methods of a Langevin equation
dR
d--t =f(R, t) + r(t)
(21)
where f(R, t) is the deterministic term and T is the noise term. We assumed T to have properties similar to those in a classical Langevin equation. In the mathematical study of random walk or Brownian motion T(t) is assumed to be a Gaussian random variable with zero mean and with a correlation function which is proportional to a delta function 6A i.e.
(22)
where D is a constant and 6A is the Dirac delta function. Physically T(t) is thus a random fluctuating term acquiring values which are both positive and negative at various times with a mean value of zero. The analogy with Brownian motion is clear. In Brownian motion the apparent randomness of the motion is characterized by assuming that each collision moves the particle independently although the motion could be the combined effect of many, many collisions. Such an assumption is permissible and is usually justified (in case of Brownian motion) by using the notion that even a small variation in the velocity of the particle is much larger than the aftereffects of each collision leading to Brownian
motion. It should be noted that Einstein [21] in his analysis of Brownian motion had to make two assumptions which though reasonable, are necessary for the mathematical analysis. (1) Each particle is assumed to execute a motion which is independent of the motion of all other particles, also (2) that movements of any particle at different time intervals are independent processes. Obviously these may not be exactly true as in the case of grain growth, but could be justified by a similar argument. The plausible argument as in the case of Brownian motion is that any small increment in dR/dt is the combined effect of many small individual increments, such that each increment in dR/dt could effectively be treated as random and independent of others. Thus the Fokker-Planck equation for grain growth is based on sound physical and mathematically plausible principles. Further experimental and theoretical work may however be necessary to check the basic principles, and obtain more precise knowledge about drift and diffusion terms. 5. CONCLUSIONS The stochastic theory of grain growth described by Pande in [14] (Part I) is discussed and developed further in this paper, in the light of recent criticism of that paper by Ryum and Hunderi. Specifically we answer two main criticisms of the theory and show (1) In stochastic theory, the volume of the specimen is strictly conserved. This is proven rigorously by further analysis using "diffusion" in one and three dimensions as suggested previously [14]. (2) The stochastic nature of the grain growth is shown to be supported by available experimental, computer, and theoretical calculations. The mathematical idealizations needed are similar to those for Brownian motion and are based on plausible arguments. Further experimental and theoretical calculations however may be needed to establish the details of stochastic behavior. An example of such detail would be the value of ~t/D which determines the proportion of the deterministic to random component. So far, the value of this parameter is estimated by obtaining the best fit of the predicted grain size distribution with the experimental distribution. However the analysis presented is valid for any value of ct/D i.e. for any proportion of deterministic to random component. While we agree with Ryum and Hunderi that the diffusion term alone (as in Louat's model) is not sufficient to account for grain growth, we disagree with their statement that the diffusion term added as a perturbation to the Hillert-like drift term will lead to a realistic model of grain growth (this was
PANDE and DANTSKER:
STOCHASTIC THEORY OF GRAIN GROWTH--III
rigorously shown to be not so, in Part II [22]). It is our view that both drift and diffusion terms are important for grain growth as postulated in [14]. In view of the good agreement between theory and experiment, and also because the theory rigorously meets all the objections of R y u m and Hunderi we believe that the stochastic theory of grain growth is essentially sound and deserves further consideration. Indeed it is our expectation that further progress in theoretical description of grain growth is most likely to occur on the basis of such a theory. REFERENCES
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7. 1-Wei Chen, Acta metall. 35, 1723 (1988). 8. M. P. Anderson, G. S. Grest and D. J. Srolovitz, Phil. Mag. B 59, 293 (1989). 9. K. Okazaki and H. Conrad, Trans. Japan Inst. Metals 13, 198 (1972). 10. F. N. Rhines and B. R. Patterson, MetalL Trans. 13A, 985 (1982). 11. M. Hillert, Acta metall. 13, 227 (1965). 12. N. P. Louat, Acta metall. 22, 721 (1974). 13. N. Ryum and O. Hunderi, Acta metall. 37, 1375 (1989). 14. C. S. Pande, Acta metall. 35, 2671 (1987). 15. N. Ryum and O. Hunderi, Acta metall. 37, 1380 (1989). 16. D. J. Srolovitz, M. P. Anderson, P. S. Sahni and G. S. Grest, Acta metall. 32, 793 (1984). 17. I. M. Lifshitz and V. V. Slyozov, J. Phys. Chem. Solids 19, 35 (1961). 18. C. Wagner, Z. Elektrochem. 65, 581 (1961). 19. M. Marder, Phys. Rev. A 36, 858 (1987). 20. M. Tokuyama and K. Kawasaki, Physica .4 123, 186 (1984). 21. A. Einstein, quoted in: C. W. Gardiner, Handbook o f Scientific Methods, 2nd edn, p. 3. Springer, Berlin (1985). 22. C. S. Pande and E. Dantsker, .4cta metall, mater. 38, 945 (1990).