Statistical theory of two-dimensional grain growth—II. Kinetics of grain growth

Acta metall, mater. Vol. 40, No. 3, pp. 533-542, 1992 Printed in Great Britain. All rights reserved

0956-7151/92 $5.00 + 0.00 Copyright © 1992 Pergamon Press plc

STATISTICAL THEORY OF TWO-DIMENSIONAL GRAIN GROWTH--II. KINETICS OF GRAIN GROWTH K. LQCKE t, I. H E C K E L M A N N t and G. A B B R U Z Z E S E 2 qnstitut ffir Allgemeine Metallkunde und Metallphysik, RWTH Aachen, Fed. Rep. Germany and 2Centro Sviluppo Materiali S.P.A., Roma, Italy (Received 14th August 1989; in revised form 12 June 1991)

Abstract-- A statistical theory of two-dimensional grain growth is derived from first principles. It describes the time evolution of the distribution function of the grain radii as a solution of a two-parametric continuity equation depending, besides on the time, on the radius R and on the number n of sides of the grains. Here beside the growth rate of an individual grain which is determined by the von NeumannMullins equation, also the changing rate of n (e.g. by neighbour switching) had to be evaluated. For solution of the continuity equation, additionally a topological relationship (mean number of sides as function of R) is required to be known. Since this relationship depends on the microstructure (see Part I), it causes a certain ambiguity in the final kinetic equations. For comparison, a simplified model using circular grains is treated which is based on heuristic assumptions but gives better insight into the "physical meaning" of the suppositions and results of the theory. The literature on this subject is thoroughly discussed. In particular, the physical principle of volume conservation during grain growth, which is connected with the fulfillment of the basic topological law ri = 6, was shown for several investigations not to be satisfied on the level of the basic equations. R~um~---A partir de principes de base, on 6tablit une thtorie de la croissance bidimensionnelle des grains. Elle dterit l'tvolutions dans le temps de la fonction de rtpartition des rayons de grains comme une solution de l'tquation de continuit6 ~i deux paramttres qui dtpend, en plus du temps, du rayon R et nombre n de cttts du grain. Dans notre cas, outre la vitesse de croissance d'une grain individuel qui est dttermin~e par l'tquation de Neumann et Mullins, la vitesse de variation de n (par exemple par ~ehange avec voisins) doit aussi ~tre 6valute. Pour rtsoudre l'tquation de eontinuitt, on doit aussi connaitre une relation topologique (le nombre moyen de c6tts en fonction de R). Puisque cette relation dtpend de la microstructure (voir partie I), ceci entrainee une certaine ambigui't6 dans les 6quations cinttiques finales. A titre de comparalson, un modtle simplifi6 utilisant des grains circulaires est traitt; il et bast sur les hypothtses heuristiques, mais donne une meilleure comprtbension de la "signification physique" des hypothtses et des rtsultats de cette thtorie. On donne une discussion dttalll~e de la litttrature sur ce sujet. En particulier, on montre que le principe physique de la conservation du volume pendant la croissance du grain, qui est li6 :~ la loi topologique r1 = 6, pour plusieurs exptriences n'est pas satisfait au niveau des 6quations fondamentales. Zusannnenfassung---Ausgehend von "first principles" wurde eine statistisehe. Theorie des zweidimensionalen Kornwachstums hergeleitet. Sie beschreibt die Zeitentwichlung der Kornradienverteilungsfuncktion als L6sung einer zwei-pararnetrigen Kontinuitfitsgleichung, die auBer vonder Zeit auch vom Radius R und v o n d e r Seitenzahl n der Ktrner abh~ngt. Dabei wurde neben der Wachstumsrate eines individueUen Korns, die durch die yon Neumann-Mullins Gleichung bestimmt wird, auch die .~[nderungsrate ri der Seitenzahl (z.B. durch 'neighbour switching') berticksichtigt. Zus~itzlich is ftir die Ltsung der Kontinuit~itsgleichung die Kenntnis einer topologischen Beziehung (mittlere Zahl der Seiten als Funktion von R) erforderlich. Da diese Beziehung eine Abh,~ngigkeit vonder Art des Geftiges zeigt (siehe Teil I), enthalten die resultierenden Kinetikgleichungen eine gewisse Unbestimmtheit. Zum Vergleich wurde ein vereinfachtes Modell kreisffrmiger K t m e r verwendet, alas auf heuristischen Annahmen basiert, jedoch ein besseres Verst/indnis in die physikalische Bedeutung der Voraussetzungen und Ergebnisse der Theorie liefert. Die Literatur tiber dieses Thema wurde eingehend diskutiert; iusbesondere konnte gezeigt werden, dab in vielen Untersuchungen dem physikalischen Prinzip der Volumenerhaltung im Kornwachstum, das mit der Erftillung des topologisehen Grundgesetzes a = 6 verkntipft ist, in den Grundgieichungen nicht Gentige geleistet wird.

quired for describing G G were reported. Based on these results, in the present Part II a statistical theory of 2D G G for the case of uniform GBs is designed. 3D G G will be treated in a subsequent paper [4]. In literature frequently theoretical works on G G are reported using approaches very different from the one of the present paper, namely the deterministic (e.g. [5, 6, 31, 32]) and the Monte Carlo (e.g. [7, 8]) approach. Both start from a given network of

1. INTRODUCTION Grain growth (GG) is the evolution of the microstructure by motion of grain boundaries (GBs) by the effect of GB energy [1, 2]. A quantitative description of G G must take into account a conservation o f certain topological relationships. In Part I of the present work [3], results of such topological investigations of two-dimensional (2D) microstruetures re533

534

LI~CKE et al.: STATISTICAL THEORY OF GRAIN GROWTH--II

GBs and simulate the motion of the individual GB numerically. In the deterministic model the motion of the GBs resulting from macroscopic laws for the rate of GB motion, and in the Monte Carlo method the GB motion resulting from random jumps of the individual atoms considering the GBs as energy are calculated. Both methods have the advantage of being based on simple elementary processes and being rather free from additional assumptions. However, by describing only individual cases, they give less insight into the underlying general suppositions and physical mechanisms. Moreover, they require extremely large computer expenditures. Particularly in the Monte Carlo method, size and number of the grains which can be handled by the present computer facilities may be too small so that the resulting GG behaviour may deviate from that predicted by the macroscopic laws and important rare events (e.g. in secondary recrystallization) may be missed. In the present paper, in contrast, a statistical approach is applied. Here it is not the aim to describe the changes of the individual grains during GG, but to only reveal the changes of the grain size distribution. Since it is just this distribution by which a grain structure is generally characterized and which is usually measured in GG experiments, the statistical approach is appropriate. Based on the LifshitzSlyozov-Wagner theory [9] of Ostwald ripening the statistical method was first introduced into GG by Hillert [9] and improved and extended by the present authors in earlier work [10-13]. It stresses the analytical description of the GG problem and yields a continuity equation for the grain size distribution which can be solved numerically. Of all methods this statistical method so far yielded the most detailed interpretations of GG behaviour, also in the presence of textures, particles and solute drag, and has made the most far reaching predictions in good agreement with experiments [10, 13-16]. For the sake of simplicity, in the present paper uniform values for GB energy and GB mobility (e.g. the average values) are assumed, but this simplification will be dropped in a subsequent paper [4]. In the earlier statistical treatments of GG largely heuristic assumptions concerning driving force etc. have been applied. The present paper, however, gives a derivation of the equations for the kinetics of GG based on first principles and thus represents the solution of the classical problem of the time evolution of the grain size distribution during 2D GG. Several authors contributed to this result: (i) Based on simple physical assumptions von Neumann [17] and later MuUins [18] gave a theoretical expression for the rate of growth or shrinking of an isolated grain. This process of growth or shrinking is the main process in GG and, according to the von Neumann expression, its rate is determined only by the number of sides of the grain. (ii) Second processes consist in changes of the number of sides of the grains, e.g. by vanishing

of small grains or by neighbour switching. These processes were not appropriately treated before, yon Neumann only warned that they may disturb the growth and shrinking kinetics. They will be considered in the present work, but similar arguments as here are presented simultaneously and independently by Fradkov et al. [23]. (iii) Finally, the topological relationships between number of sides of the grains (which theoretically determines the growth rate) and the grain size (which is measured) need to be known and combined with the more physical assumptions (i) and (ii). These relationships which are of equal importance as the physical assumptions are established in Part I of the present work. Furthermore, in the present paper expressions for the rate of G G are derived also on the basis of a simplified model using circular grains. Such a model being applied in Part I to topological problems was already assumed in foregoing heuristic treatments of GG [10-13], but has led there to certain contradictions which have here been removed. But since this model possesses only one parameter for the characterization of the grains, namely the grain radius, it is well suited for providing a quantitative description of GG where also only this one parameter is taken into account. The results will be shown to be in agreement with those of the exact treatment. Moreover, this model, due to its simplicity, also contributes to clarify the "physical understanding" of the equations of the exact treatment. At the end of the present paper, a thorough discussion of the problems related to 2D GG and of treatments of these problems in the literature will be given. 2. GENERAL CONCEPT AND SUPPOSITIONS 2. I. The distribution functions As already pointed out in Part I, the two for the present purposes most important parameters for characterizing a grain are the grain size which in the 2D case is described either by the grain area A, by the equivalent grain radius R or by the normalized radius r defined by A =TzR2; r = R / t ~

(1)

and the number n of sides, i.e. the number of nearest neighbours of the grain (coordination number). Correspondingly, the microstructure is statistically described by the grain size distribution function (SDF), the coordination distribution function (CDF) and the two-parametric combined distribution function (SCDF). The CDF ¢p, is a discrete function representing the fraction of the number of grains with the number of sides n. Since R is a continuous parameter, one has for the SDF a continuous representation O(R)dR being the fraction of the number of grains in the

LOCKE et al.: STATISTICAL THEORY OF GRAIN GROWTH--II radius range dR and for the SCDF a mixed representation ~b,(R) dR being the fraction of grains in such a radius range dR and additionally in the coordination class n. Here, however, also for the SDFs mostly the discrete representation in the form of grain size classes i = 1,2,3... of the width AR will be used which is related to the continuous functions by tpi = tk(R,)AR;

ep,, = c~,(R,)AR.

(2)

One then has T ~ 0 . ( R ) = O(R); n

Y~,p,. = ~0, n

E , , . = ~o.; E ~ o . = E ~ o , = 1 i

n

(3)

i

(4)

n

(the value of six is valid for all grain structures with only triple point intersections of the GBs [20]) and for the average number of sides of grains of the size class R~ or size R, respectively

r~i=

n

~0,

; tiR

n

0(R)

(5a,b)

Since in G G experiments one usually considers only the change of the size distributions and not the change of the size of individual grains, only a statistical [and not a deterministic] theory of GG is required describing the time (t) dependence of the SDF ~(R, t) or tp~(t). It can be calculated by (numerical) integration of a continuity equation (e.g. [10, 11]) which for convenience sake, will here be written in the form continuous in R =

~b(R, t)'--d-7 (R, t)

(6)

(for the discrete form see [10, 11]). It describes the flux of grains q~d R / d t in the R - t space, i.e. the changes of the frequency 4~(R, t) of the grains of the radius R with time t due to their growth or shirking (see Fig. 1)

[ Ri > Rc ~ Ri'< Rc /

---]

with the rate d R / d t as drift velocity. For reasons to be discussed in Section 5.2 the diffusion term [29] is here not applied. 2.2. The underlying suppositions

The following derivation of the rate of GG are based on two groups of assumptions: (a) Physical assumptions. They determine the rate of motion of individual GBs and will lead to an expression for the growth rate of an individual grain. A GB element is here assumed to move into the direction of the centre of its curvature experiencing the driving force p and a velocity v given by p = y/2 = ydO/ds; v = mp.

v = m p = my~2 = M/2.

I&RI

RC

Fig. 1. Grain size distribution ~oi vs Re. (Examples for the flux of grains during grain growth is indicated for the cases Ri > Rc and R; < Re.)

(8a,b)

Here 2 is the radius of curvature, y the specific GB energy, m the GB mobility and 0 the inclination of the GB element, s is the length coordinate along the GB and M = my will be denoted as GG diffusivity, since it has the dimension of a diffusivity. It is further assumed that the three boundaries only meet at triple points forming three 120° angles. This represents the equilibrium condition which in GG experiments should be well fulfilled. In deterministic treatments of GG the differential equation (8) is solved, either analytically for some simple boundary shapes [18], or, after rigorous simplifications, by voluminous numerical calculations for a given polycrystalline GB system [5, 6, 31, 32]. In the present statistical approach, however--and this is one of its advantages--this equation of motion [equation (8)] leads to simple analytical expressions for the average growth rate dR/dt and thus to a rather simple numerical integration of the continuity equation (6) (see Section 3). (b) Topological assumptions. These have the purpose of transforming the n-dependence of the growth rate of individual grains following from the physical assumptions into an R-dependence necessary for a description of GG. They will be shown to consist in a relationship between the average number of sides ~ii of grains of the radius class i and the radius Ri. As thoroughly discussed in Part I, this relationship depends on the nature of the grain structure. Most important for GG is the "special linear relationship" (SLR) [Part I, equation (12a)] ni = 3 +

'l

(7a,b)

Both expressions together give the equation of motion of a GB

i

and for the total averages of r and n Y = E ~0/ri= 1; ~ = E ~ o , n = 6

535

3r i

(9)

which is experimentally verified especially for microstructures resulting from GG. Its approximate derivation (Part I, Section 3.4) is shown to be based on three topological assumptions: (i) The assumption of a quasi-circular grain structure, i.e. of well equiaxed grains. Because of local minimization of GB energy, this is generally fulfilled for grain structures formed by GG. (ii) The principle

LOCKE et al.: STATISTICAL THEORY OF GRAIN GROWTH--II

536

I

"13

l

~' 11 0

Fig. 2. Grain of size class i surrounded by grains of size class j. of maximum surface covering of grains. It says that for quasi-circular grain structures the number of grains of the size j which can be arranged around a grain of the size i with all these grains being in contact with each other is approximately given by [Part I, equation (23)]

n#=

2n(R,+Rj) { +~ 2R/ ,~ 3 1 rj )

(10)

(see Fig. 2). (iii) The principle of random surface covering. It says that the fraction of the average perimeter S~ of the i grains covered b y j grains is in the average [Part I, equation (20)] w#=w/=

q,~Sj

=¢pjrj.

(11)

Y

The left hand part of this equation means that the available length cpyS/of the perimeters of grains of radius j is distributed randomly over the total length Z%Sj of the perimeters of all grains, independent of their size i. The right hand part is valid for circular grains for which one has Sj = 2nRj. 3. GRAIN GROWTH IN POLYGONAL MICROSTRUCTURES

3.1. The two-parametric continuity equation For polygonal grains a continuity equation as given by equation (6) cannot directly be applied, since not only changes in R, but also such in n have to be taken into account. In this case it is useful to consider grains with different n separately and to write for each number of side class n a separate continuity equation

ot

=

i I

I

vl

,_ 10

l I

s z

l

9 8

I I

I I

~

I

7 6 5 ~,. 3

C

I r

i•1 d

It I I I l l,l ib

I.I id

I I

II I,I In

ic

Radius Ri Fig. 3. Changes of the two-parameteric distribution function ~b,(Ri) during a side switching process [cf. Fig. 5(a)]. Fig. 3). The total change of ~ is then obtained by summing 4. over the various n at fixed R

=~JS(n,R,t)+~J"(n,R,t). n

(13)

n

These two terms will now be looked at.

3.2. The changes of grain size at constant n By applying the physical assumptions of Section 2 to the GBs of an individual grain v, a simple expression for the average growth rate of this grain can be derived. The change A0, of the inclination 0 of the GB elements occurring by going along all boundaries around this grain is obtained by subtracting from the total change 2n for each corner an angle of 60 ° -- n/3 (which is due to the 120 ° intersections, see Fig. 4). This yields AO, = 2n - n,n/3, and, with the perimeter S, and the average curvature AOJSv, the average driving force for this grain A0v = n 7 (nv - 6). P, = 7 St 3 S,

(14)

(Here the sign is reversed in order to make pv positive for an expanding grain.)

~.(R, t)-~ (n, R, t ) . On +~---~{~.(R,t)~(n,R,t)} R. (12)

The first term being denoted as JR describes the rate of change of ~.(R, t) at fixed n by growing or shrinking of the grains with the rate (OR/Ot). (horizontal change in Fig. 3), and the second term called J" gives the rate ofchange of ~b,(R, t) by changing the number n of sides at fixed R by vanishing of small grains or by neighbour switching (vertical change in

Fig. 4. Example of an individual grain whose boundaries meet under 120 °.

LOCKE et al.: STATISTICAL THEORY OF GRAIN GROWTH--II With the average rate of GB motion v, [Equation (8)] it leads to an expression for the rate of increase of the grain area A,, first given by von Neumann [17] and discussed by Mullins [18] 7~

dA-----z= v, S, = rap, S, = -~ M(nv - 6). dt

(15a)

This equation exhibits the well known result that dAv/dt does not depend on shape or size of the grain, but only on the number n, of its sides: a grain with n, > 6 will grow, one with n, < 6 will shrink and one with n v = h = 6 will not change its area. With equation (1) it follows from equation (15a)

dR, 1 dA, mpvS, M dt -2nRv dt = 2nR, = ~ - ~ ( n , - 6 ) .

(15b)

[Note that here 2nRv is not longer the true perimeter S~ of the grain but only a quantity following from the definition of R by equation (1).] By inserting equation (15b) into equation (13) one obtains the first term of the continuity equation, equation (12) Ej,_M .

- M( r iOR 'O- 6{) }~ 6

grains ~t, fl, ~ then lose a side without change of their sizes so that again the sums over q~,(R~) at fixed R,, R~, R v are not influenced. In contrast, the small grain 0 after having reached its final number of sides n = 3 will experience only changes in Ri, so that by its further shrinking again no changes in the sums over the tk,(Ri) at fixed Rt occur. As the result of the above considerations it can be concluded, that for all these processes one has

~ J"(n, R, t) = 0 .

(17)

n

This means that the second term in the continuity equation (13) which is due to changes of number of sides of the grains at constant size can completely be omitted. The kinetics of creation and vanishing of sides needs to be investigated only if the time evolution of the CDF (and not of the SDF) is to be considered.

3.4. The final one-parametric continuity equation Inserting equations (16) and (17) into (13) yields an expression corresponding to equation (6)

O ~'E~b,(n_6)} 60R[.

537

R

(16)

3.3. The changes of the number of sides at constant R Let us now consider the second term in equations (12) and (13). In the case of neighbour switching which is schematically shown in Fig. 5(a) one sees that there in the moment of switching the grains a and c gain and b and d lose a side without change of the area of these grains. This means for the schematic n-R~ - diagram of Fig. 3 that these grains stay within their size classes R,, Rb, Re, Ra, respectively, and change only in n (by _+ 1). This leads to changes of the 4~,(R~), but not to changes of the total number of grains n

of a fixed size class R~. For the vanishing of a small grain [e.g. triangular grain 0 in Fig. 5(b)] the results are similar. The three (a)

(b)

Fig. 5. (a) Process of side switching. (b) Process of grain vanishing.

Odp(R, t) M c3 ((~(R, t) ,_ ] . 0t . . . 6 OR l----~ tn, - 6)S.

(18)

This is the most general form of the continuity equation. It can be specified by introducing an appropriate topological relationship ~R(R). Using the special linear relationship equation (9) best suited for the description of G G structures finally leads to the continuity equation . Ot . .

. 2 dR c~(R, t). ~ -

(19)

which can immediately be numerically integrated, if the starting grain size distribution 4~(R, 0) is known. One sees that equations (18) and (19) are identical to the one-parametric continuity equation (6) into which an "effective" growth rate dRj

M

M I ~1- ~ ] 1

(20)

is inserted as drift velocity (using again the discrete notation R = R~). This means equation (19) could formally be obtained also by assigning to each grain of the size R~ the effective growth rate equation (20) derived by averaging equation (15b) over all n at fixed R~. However, setting for all these grains of the same size R~ the same, namely this averaged growth rate dR~/dt would be an additional assumption and, in reality, not correct. It would correspond to the homogeneity assumption which, repeatedly being applied in heuristic approaches, postulates that equally sized grains also exhibit equal (namely the average) growth rates. However, in the preceding procedures here it was not necessary to apply this additional

538

LOCKE et al.: STATISTICAL THEORY OF GRAIN GROWTH--II

assumption so that, in this sense, the resulting equations (19), (20) are exact. 4. APPLICATION OF THE MODEL OF CIRCULAR GRAINS 4.1. Derivation of the growth rate

In an array of circular grains consisting of a grain of the radius R i surrounded by n¢ closely packed grains of the radius Rj (see Fig. 2, of. Part I, Section 3.4), all the GBs of the inner grain i experience the driving force Pu given by equation (22). By substituting Ri/Rj by equation (10) and setting R~ = S~/27t valid for circular grains, one obtains for the driving force acting along the perimeter of grain i from equation (22)

The fact that grain growth for its description needs only one parameter, namely the grain radius R, suggests to apply also for G G the one-parametric circular grain model in which the polygonal grain structure [e.g. Fig. 6(a)] is approximated by an assembly of circles having the same area as the original polygonal grains [Fig. 6(c)]. As shown in Part I (Section 3.4) this model fulfills very well the topological conditions leading to the experimentally well proved SLR. For describing GG, however, it must additionally satisfy the physical condition of yielding the same growth rate as the polygons, i.e. of yielding a driving force leading to the same dA~/dt as the von Neumann-Mullins equation (15a). This will now be checked. Considering again Fig. 6(c), one has first to realize that here the driving force for G G is exerted by the surface energy of the circular grains which must be distinguished from the GB energy applied in the polygonal structure [Fig. 6(a)]. This can best be seen in Fig. 6(b) where the polygonal grains are slightly disconnected along the GB. Since the total interracial energy as source of the driving force may not be changed by this procedure and since the total surface length after separation of the grains [Fig. 6(b)] is twice as large as the total length of the GBs before separation [Fig. 6(a)], a specific surface energy 7' of only half the size of the GB energy must be assigned to the new surfaces, i.e. ~ ' = 7/2. Due to its surface energy E s = 2~tR~3,', each circular grain of the size i experiences a shrinking pressure

is obtained. This equation which has already used before by the present authors [10, l 1] leads, for the average growth rate of the grains i, to equation (19) which also is the result of the exact treatment of the polygonal structure. Since ~ gives the driving force averaged over the various grains i, equations (22, 23) also mean that here in the circular grain model grains with a radius R~ <>~, will grow or shrink, respectively, only in the average and not individually (Section 3).

(21)

4.2. Comparison to the results for polygonal grains

The driving force Pu on a GB zj between a grain of size i and a grain of size j is now assumed (heuristically) to be given by the difference p~ - p j of the shrinking pressures of the two neighbouring grains [11], i.e.

In Sections 3 and 4 the expression for the growth rate dR~/dt of the grains of the size R; is derived in two different ways: (i) In the "exact" approach for polygonal grains the considered elementary unit is an individual grain of size i and coordination n which is treated as being isolated from the rest of the structure. The interaction with its neighbourhood is taken into account only by introducing the 120° angles. Then, as can be seen from equation (15a), the resulting growth rate (dA/dt), for such a grain with fixed n is no longer influenced by the neighbouring grains. The calculations on this basis finally yield an effective growth rate [equation (20)] to be used in the continuity equation [equation (19)] which turns out to be the average over all grains with fixed i but varying n. (ii) In the model of circular grains the considered elementary unit is a pair of grains of the sizes i and A i.e. an individual GB i, A which is treated as being

p~ = -dES/dA~ = -7'/R~ = -7/2R~.

pij = p i - pj = -~

=

(22)

Equation (22) can also be interpreted as a superposition of the curvatures 1/R~ a n d I/Rj of the surfaces of the two involved grains forming a curvature 1/20 of the GB as already postulated in [I0].

Fig. 6. (a-c) Replacement of the polygonal grain structure by a circular grain structure with equal grain areas.

pc = ~ ~ (n• - 6).

(23)

Since grain i corresponds to a regular polygon of n U sides, equation (23) exactly agrees with the von Neumann-Mullins expression equation (15a). For the case of the neighbouring grains having different sizes, with equations (11) and (22) and by averaging the driving force Pu over all pairs /j formed by all grains i and all their neighbours j the expression Wjp¢ ~,= J

y wj J

),

),

1

1

LOCKE et al.: STATISTICAL THEORY OF GRAIN GROWTH--II isolated from the rest of the structure. The calculations on this basis yield a driving force Po acting on such a G B / j [equation (22)] and finally, an effective driving force [equation (24)] to be used in the continuity equation (19) which turns out to be the average over all grain pairs /j with fixed i but varying j. It is interesting to note that both models yield the same expression for the mean growth rate [equation (20)] although it is derived for polygonal grains by application of the SLR, whereas for the circular grain model no use of this relationship is made. The reason for this agreement is that for deriving the SLR and the growth rate for the circular grain model the same suppositions are used, namely that of circular grains and of maximum and random surface covering. For the same reason, other correspondences between the two models occur. In both models the mean value R represents a critical radius in the sense that grains R ,~ R, respectively, grow or shrink in the average (but not necessarily individually). Or a large radius Ri of the central grain i corresponds to a large number of sides n which both cause a large growth rate of grain i. Or, vice versa, a large n of the central grain yielding a large growth rate corresponds to small surrounding grains which, due to their large shrinking pressure, cause a large driving force for growth of the central grain. Compared to the polygonal grain model the circular grain model has the disadvantage of being only an approximation, but--as for the interpretation of the SLR (Part I, Section 3.4)---it has the advantage of yielding a more direct physical understanding; e.g. it directly demonstrates the effects of the principles of maximum [equation (10)] and of random [equation (11)] surface covering. Also the physical understanding of the meaning of the expression equation (14) for the driving force is facilitated by using circular grains. By approximating the polygonal grain by a circular one with equal area A and n outside GBs running radially into it (Fig. 7), the total driving force acting on the surface of this grain becomes equal to the difference of the pulling force due to the n radial GBs and the shrinking force due to the circular GB n~, P~ = 2nR~

~ 7 (n - 2~). Rv = 2~R-----~

(25)

Fig. 7. Circular grain with n radial grain boundaries.

539

The fact that this expression is very similar to equation (15b) (where S~ = 2nR, and only n must be replaced by 3), suggests that also in equation (15b) the term being proportional to n represents the pulling force due to the n outside GBs and the other one the capillary shrinking force. However, there exists a difference between the two models in the mean driving force. This is due to the fact that polygonal and circular grains of equal area possess different perimeters Sv (see Part I, Section 3.3), and that not Pv but only the product pvSv = vv/m is determined by the number of sides n [equation (15a,b)]. Then for the same v = dRv/dt the driving force is the larger the smaller Sv, i.e. it is larger for circular than for polygonal grains. A specific value for pv can only be obtained by specifying the perimeter Sv, e.g. by using a regular polygon as approximation (see Appendix (b)). But these differences are of little importance for the description of GG kinetics since the continuity equation [equation (6)] solely depends on the average growth rate dR~/dt and not on the average driving force Pi. 5. DISCUSSION

5.1. Discussion of the grain growth kinetics The continuity equation (18) represents the final, most general solution of the classical problem of calculating the time evolution of the grain size distribution under the effect of the GB energy for uniform boundaries. Its derivation shows that this equation follows directly from the von Neumann-Mullins equation (15a) without additional assumptions and thus is only based on the macro-physical laws equations (7, 8). This means the derivation of this equation is based only on first principles. But in (numerically) integrating this equation there remains a serious difficulty: the effective growth rate equation (20) still depends on the grain size R vs number of sides n relationship in the form as(R ). Depending on the nature of the polygonal grain structure quite different types of such relationships can occur, so that here an ambiguity of principle nature arises which, in the general sense, cannot be removed by any theoretical treatment. All that can be done is to try to insert into equations (18) and (19) that relationship tiR(R ) which is best suited for the considered case. As pointed out in Part I, this should be the special linear relationship (SLR) equation (9) which is well fulfilled particularly in structures formed by GG. It is based on the assumptions (i) of an equiaxed grain structure, (ii) of random neighbourhood correlations and (iii) of the smallest grains being triangular. Assumption (iii) can be expected to be fulfilled during GG since, as generally assumed in the literature and as will be confirmed in a subsequent paper [19], in the present case of uniform GBs the reduction of the number of grains during GG takes place predominantly by vanishing of triangular grains.

LOCKE et al.: STATISTICAL THEORY OF GRAIN GROWTH--II

540

Assumption (ii) implies that in the case of nonrandom surface covering (Section 2.2) a modified expression for ti~ [Part I, Equation (26)] instead of equation (9) must be introduced into equation (20). This leads to

dR,]

M

dt _1oo~= ~

(t~, - 6)

= M { ~ - - ~ (2 - ~ ¢pjrjk#)}

(26,

with the k Ubeing correlation factors. This means that the continuity equation [equation (19)] is not valid generally but only in the case of random surface covering. However, as discussed in Part I, it seems to be safe to assume that in most structures formed by G G this random case applies, with the k o being close to one. Of special importance, however, is point (i) being concerned with the equiaxed grain structure. It is obvious that such a structure will be generated in the course of G G due to the effect of GB energy to minimize the length of the GBs. Then for the G G process the very simple time law equation (19) would be obtained. But the starting structures for G G may be quite different. For instance, after primary recrystallization a kind of random ("Voronoi") structure is expected for which the SLR is not valid at all (see Part I). This means that e.g. directly after primary recrystallization strongly changing structure types and thus continuously varying relationships fij(rt) would occur so that a practically unpredictable time law would result. In any case, the ~j(r,.) relationship must be such that the ti -- 6 law is exactly fulfilled. This is of great importance, since any deviations from fi = 6 (even small ones) would automatically lead to a violation of the physical principle of area conservation during GG. This is shown by the equation dA, n ~-~ ffi 3 M ~ c#,(~, - 6 )

= 3 M ( a - 6) ffi 0

(27)

following from equation (15a). One sees that only in the cases of ~i = 6 (as e.g. in the present treatment) automatically a conservation of area occurs. Such area conservation is also obtained in the circular grain model for which with equation (24) one has

= n M ~ ~lRl

--

= 0.

(28)

In the simulations in [21] and [22], in contrast, it was tried to compensate such area changes (caused by violation of ri = 6) by restoring the original area after

each simulations step by inserting a kind of normalization factor. This, however, is not sufficient. Any G G theory, statistical or deterministic, which does not satisfy this physical requirement of area conservation on the level of the basic equations, but fulfils it only by a posteriori corrections, contains some intrinsic incoherences.

5.2. Discussion of the work of the literature A rate equation of the type of equation (19) and the corresponding expression equation (20) for the driving force were already proposed in the pioneering work by Hillert [9]. But there these expressions were not derived from first principles, but were based more on heuristic arguments as to choose the simplest expression fulfilling the condition of growth of the grain sizes R > Rc and shrinking of the sizes R < Rc with Rc being a critical radius. Moreover, no general explicit expressions for the critical radius Rc were given there. These were derived and applied to G G simulations by the present authors [10--12] and were found to yield R~ = R only for the 2D case (for the 3D case Rc = R211~ was obtained). Hillert [9] also succeeded in deriving an asymptotic solution ~(R, t) of the continuity equation (19) for large times t. This solution ~b(R, t) consists in a time-independent distribution function P(r) of the relative radii r = R/I~ with the mean radius R as sealing factor increasing with time (quasi-steady state or self-similar case). Instead of introducing the SLR into the continuity equation (18) as would be required for obtaining the Hillert distribution P(r), Fradkov et al. [23] introduced a relationship tis linear in the area A and thus in r 2 which they claimed to have found experimentally in an aluminium foil. An asymptotic integration would here lead to a quasisteady state distribution of the exponential type r . e x p { - r 2 } . A similar distribution would be obtained if one applies the Louat [29] approach using a diffusional term in the continuity equation (6). These facts lead to a controversial situation. The Hillert distribution P(r) which also follows from the present theory, has never been observed, although the SLR contained in its derivation is well established. The exponential distribution, in contrast, is close to the observed distributions, although a thorough checking by the present authors has yielded a better fitting of the measuring points tis(r ) [23,24] by the SLR than by a quadratic relationship. According to the opinion of the present authors the main reason for not observing the Hillert distribution P(r) is not related to such a non-linear ris(r) relationship, but to the fact that the supposition of uniform GBs is not fulfilled. As can be seen in the modifications of the statistical G G theory taking into account textures [11, 16] and thus GBs with different energies and mobilities, the kinetics of G G is there drastically changed. This demonstrates that a quantitative agreement between experiment

LOCKE et al.: STATISTICAL THEORY OF GRAIN GROWTH--II and theory cannot be expected as long as the differences between the various GBs and thus the textures are not adequately considered. Also the works by Novikov [21] and Bunge [22] being mainly concerned with texture influences on GG (as also [10, 11]) report a derivation of a rate equation and corresponding G G simulations, but both approaches do not fulfil the intrinsic area conservation requirement equation (27). Moreover, the derivations in [22] which are similar to those given in Appendix (b), are violating the randomness condition by simply using for the contact probability the volume fractions instead of the surface fractions wj = ~ojrj of the grains of the sizej [see equation (11)]. In [21] a quite different approach is used, but, as will be shown in a subsequent paper [25], it is based on some inadmissible assumptions. Recently some papers emphasizing geometrical aspects for modelling of GG were published by Ryum et al. [26, 27], but they arrived at some results not being in complete agreement with some topological principles. They yield a non-linear relationship between radius r and coordination number n [Part I, equation (32)] which does not fulfil the ~ = 6 law, and for the number of sides of grains neighbouring an n-sided grain an expression different from the experimentally proved Weaire-Aboav equation [3]. The reason for the latter difference seems to be that the equation given in [26] (which was not claimed to be valid generally) was derived for a kind of regular grain pattern whereas the validity of the WeaireAboav equation requires a random pattern [3]. In [26, 27] also some work of the present authors was criticized. It was stated there that no good physical foundation exists for the drift velocity given by equation (20). For the 2-dimensional textureless case this is presented now in the present paper: equation (20) follows directly from the yon Neumann-Mullins equation and the special linear relationship equation (9) (see Section 3). Furthermore, the theory of GG leading to an equation of the type of equation (20) was considered to be a "single grain theory" in the sense of the driving force being equal to the GB energy smeared out equally over the whole body. However, as shown in Appendix (C), smearing out of the GB energy leads to a driving force different from that given by equation (20) or (24). The main criticism in [26, 27] concerns the use of the factor 1/2 in equation (24). It shall again be pointed out that this equation derived from the circular grain model agrees with equation (20) from the exact treatment so that there is no question about the correctness of the factor 1/2, but only about how to make this factor plausible. It has been argued in [27] that replacing one grain boundary by two surfaces as done in Fig. 6 in order to interpret the rate equations by the model of circular grains is "unphysical". This argument would hold for problems concerned with the GB structure, such as e.g. the GB mobility. It is irrelevant, however, if only the driving

541

force is concerned which is completely determined by size and local distribution of the GB energy. But size as well as local distribution are retained by slightly separating the grains [Fig. 6(b, c)] and simultaneously splitting the GB energy. (A model of separated grains, but less explicit was already used by Hunderi et al. [5].) It should finally be pointed out that the von Neumann-Mullins equation (15a) represents the "mechanics" of growth of an individual grain and is strictly valid if the simple physical assumptions given by equation (7) are fulfilled. This means that contradictions to the yon Neumann-Mullins equation as reported from Monte Carlo simulations [7, 8] (e.g. that grains with n > 6 are shrinking), would imply that the macroscopic laws [equation (7)] are not valid. This, however, could only be the case when the chosen grains were so small that statistical fluctuations could come into play. Also the occurrence of a diffusion term in the continuity equation (6) as proposed by Louat [29] would imply a violation of the macroscopic physical laws equation (7) and this would only be possible by thermal fluctuations in the case of extremely small grains. For this reason the diffusion term is here neglected. The very important particle and solute atom drag during GG [30] and texture effects will be discussed in later papers [4]. Acknowledgements--The authors like to thank Dr V. E.

Fradkov for most interesting discussions. They gratefully acknowledge the financial support by the Deutsche Forschungsgemeinschaft and by the Commission of the European Community. REFERENCES

1. C. S. Smith, Trans. Am. Inst. Min. Metall. Engrs 5, 175 (1948). 2. P. A. Beck, Phil. Mug., Suppl. 3, 245 (1954). 3. G. Abbruzzese, I. Heckelmann and K. Liicke, Acta metall, mater. 40, 519 (1992). 4. I. Heckelmann, G. Abbruzzese and K. Liicke. To be published. 5. O. Hunderi and N. Ryum and H. Westengen, Acta metall. 27, 161 (1979). 6. O. Hunderi and N. Ryum, Acta metall. 29, 1737 (1981). 7. M. P. Anderson, D. J. Srolovitz, G. S. Grest and P. S. Sahni, Acta metall. 32, 783 (1984). 8. D. J. Srolovitz, M. P. Anderson, P. S. Sahni and G. S. Grest, Acta metall. 32, 793 (1984). 9. M. Hillert, Acta metall. 13, 227 (1965). 10. G. Abbruzzese and K. Lficke, Acta metall. 34, 905 (1986). l l. Abbruzzese, K. Liicke and H. Eichelkraut, Proc. ICOTOM 8, Santa Ft, p. 693 0988). 12. G. Abbruzzese, Acta metall. 33, 1329 0985). 13. H. Eichelkraut, G. Abbruzzese and K. Liicke, Acta metall. 36, 55 (1988). 14. H. Eichelkraut and K. Liicke. To be published. 15. H. Eichelkraut, dissertation, RWTH Aachen (1988). 16. I. Heckelmann, X. Zhao, K. Lficke and G. Abbruzzese, Proc. ICOTOM 9, Avignon (1990). 17. J. yon Neumann, discussion to Ref. [28]. 18. W. W. Mullins, J. appl. Phys. 27, 900 (1956). 19. V. Fradkov, I. Heckelmann and K. Liicke. To be published. 20. W. C. Graustein, Ann. Math. 32, 149 (1932).

542

LOCKE et al.: STATISTICAL THEORY OF GRAIN GROWTH--II

21. V. Yu. Novikov, Acta metall. 27, 1461 (1979). 22. H. J. Bunge and E. Dahiem, Proc. 7th lnt. Symp. on Metals, Mater. SCI. Ris~, p. 255 (1986). 23. V. E. Fradkov and D. G. Udler, Preprint, Inst. of Solid State Physics, Chernogulovka (1989). 24. V. E. Fradkov, A. S. Kravchenko and L. S. Shvindlerman, Scripta metall. 19, 1291 (1985). 25. G. Abbruzzese and K. Liicke, To be published. 26. T. O. Saetre, O. Hunderi and N. Ryum, Acta metall. 37, 1381 (1989). 27. N. Ryum and O. Hunderi, Acta metall. 37, 1375 (1989). 28. K. Liicke, I. Heckelmann and G. Abbruzzese. To be published. 29. N. P. Louat, Acta metall. 22, 721 (1974). 30. K. Liicke and G. Abbrnzzese, To be published. 31. H. J. Frost and C. V. Thompson, Proc. SPIE's 31st Syrup. on Optical and Optoelectronic Applied Science and Engineering, San Diego (1987). 32. H. J. Frost, C. V. Thompson, C. L. Howe and Junho Whang, Scripta metall. 22, 65 (1988). APPENDIX Approximations for the Driving Force (a) Considering a regular polygon of the class i, n with 120° angles and circular curvatures of the sides (Fig. AI), one obtains the perimeter

4x

so=

L6R"

-]cotn/n -x/~

fornf°rn=#66

yielding e.g.

pO~(n~oo)= ~---~l--~'ffr3~.

(A3) znt(~ t n j The same results would be obtained by calculating the radius of curvature 2 ° for the sides and inserting it into equation (7a). But it has to be noted that, in reality, the stationary shape of the moving boundary is not exactly circular. For some examples this shape is calculated by Mullins [18]. (b) Sometimes also the case of a regular polygon with straight sides (Fig. A1) is of interest for which the driving force cannot be calculated from equations (14) or (7a) (no 120° angles and no curvature). Then an approximate ex-

'x ...._ -,0

dx

"=

-

=

+

/

(A4)

Here dx = cos(n/n)dRl, is the displacement of the GB, R~, again the outer radius, dE,~ = - n ? dRy, the decrease in energy due to the annihilation of the external (radial) GBs during growth, dE, = 7 dS+ the increase in energy due to the growth of the surrounding GB, and S~ = 2nR + sin (n/n) the length of the surrounding GB. This leads to 7ndR~, y dS~

p+~

sg~

=

sg~

7

ctg E

2R~

n

2 sin(,,In)~ -

cos(7~/n)

(A5)

J

with p~+ (n = 6 ) = 0 ;

p+ (n ---,o0) -- 2gR,~n( f l - ~ } .

(A6)

One recognizes that the difference to the polygon with circular boundaries [equation (A2)], which is due to the deviations from the 120° angles, is small. (c) Finally, the other extreme case, namely that the GB energy is assumed to be smeared out continuously over the body, shall be looked at. Considering first circular grains in an array shown in Fig. 6(c), one obtains with equation (A4) and R* = R212R = R(1 + K2)/2

(A1)

with Ra being the outer radius. With equation (14) this leads to the driving force

p°(n=6)=0;

pression can be derived from the change of energy dE by growth of the grain by the area dA -- S + dx de I-de,x de, q

dRin

J

dA

2~

X ,:R~

J des ? 27t dR~ dA 2 2nRi dRi and thus

=

(A7)

2R*

y 2Ri

(A8)

de

P=

2 LR*

~

R,j'

(A9)

For polygonal grains [Fig. 6(b)] a similar result is obtained. Here one can use the relationship between the mean perimeter S~ of the grains and size Rt (Part I, Section 3.1) s, = 2~{1.04s, + 0.08R}.

(A10)

This yields dE,x +dEs __= dA dA

Zj %Sj~12 ~ % nR~

dS i 2 2nR~dR~

~~

(All)

J

and thus P=

dE,x + d e , dA

~, -1.12 2 r*

~

"

(AI2)

Since the variation coefficient K~~ 0.3 (see Part I, Section 2.3), one has • ~ R R R =~=~-(1 +~r 2} ~ 1.83' (AI3)

1200

[

Fig. A1. Regular polygons of class i, n with 120° angles and circular curvatures for the sides (full line) and straight sides (dotted line). The growth of the grain by dRy, and dx is indicated.

This means the expressions equations (A9) and (A12) do not agree with equation (24). Smearing out the energy of the GBs obviously leads to an energy E,x and thus to a first term in equations (A9, A12) being too large by 40-50%. The reason seems to he in the fact that the outer GBs being not completely radial may not fully be taken into account for calculating the driving force and thus E,x. That both terms in equation (AI2) are larger than those in equation (A9) is obviously due to the larger perimeter of polygons compared to circles.