TOPOLOGICAL FOUNDATION AND KINETICS OF TEXTURE CONTROLLED GRAIN GROWTH

in the present paper first a statistical theory of 2-dimensional grain growth for the textureless case based on first principles the von Neumann Mullins equation and the topological grain size grain sides relationship is des- cribed. Then it is shown that the latter relationship follows from two fundamental topological principles, the principles of complete and random surface cove- ring, which are shown to be responsible also for other empirical topological 2-D and 3-D relationships (e.g. Weaire equation). Finally, textures are introducext into the topological discussion.


INTRODUCTION
In order to completely describe grain growth one has to know the true evolution of a function p(M, M2, ...M0 which gives the distribution of grain shape and size identified by the parameters M, M2, M k. In the rather common case of equiaxed grains it is usual to make an assumption on the shape of the grains which allows, with a good approximation, to treat the grains by only a single parameter namely the grain size, here expressed by the grain radius R. In this case only the time evolution of a size distribution o(R,t) must be considered. In the most general case of texture presence this requires to know the time evolution of the different grain size distributions corresponding to the various orientation classes.
There are two types of assumptions one needs for obtaining a quantitative description of grain growth kinetics: (a) Physical assumptions as those for the expressions of the driving force and the rate ofgrain boundary motion. They. were applied to an individual grain in the textureless case by yon Neumann using first principlest'2. It turned out that here the number n of grain sides plays an important role in the kinetics.
(b) Tooologica! assumptions which are required to transform the n-dependence of the krain growth kifietics into an R-delndence. These are as important as the hysical assumptions but treated less thoroughly in literature. This point will ere be emphasized.

2-DIMENSIONAL CASE WITHOUT TEXTURE
The physical expressions for driving force and grain boundary velocity are: (1) where 3' specific _grain boundary_ energy; m grain boundary mobility; k radius of curvature; O-inclination of the grain boundary element. It is further assumed that these boundaries only meet at triple points where they form 120 angles. This represents the equilibrium condition which in grain growth should be fulfilled quite well.
For a polygonal grain one can obtain a simple expression for the growth rate of the grain area by calculating the mean curvature with the total change AO of the inclination 0 occmTing by going along all boundaries around this grain. AO is obtained by substracting .from the total change 2r an angle of 60 for each corner (which is due to the 120 intersections, see also (3) This "von Neumann-Mullins equation" contains two parameters describing a grain, namely n and R, and in order to treat the time evolution of the bidimensional distribution o(n,R), one has to have an indendent equation for the rate of changing of the number of corners dn/dt based on first principles. However, at present such equation is not available which is also due to the discontinuous character of this parameter.
Therefore, in order to solve the problem of grain growth without further assumptions one has to look for a correlation between n and R. However, since one of these parameters changes discontinuously and the other continuously, this relationship cannot have unique but only statistical character. Dividing the grains in size classes one can substitute for each class i the average value of the scattered n values (see In order to apply Eq.(4) the fight expression of fii(l), must be known. In the literature often the existence of a linear relationship like n ao + a.r r R/R is mentioned. Since in a 2-D network like a grain microstructure where only three lines meet at a vertex the average number of comers per grain over the whole microstructure must be six, one has =_, q=6=ao.a (6) with ,p as the normalized frequency of grains belonging to the size class i. In the literature very many results were presented which do not fulfill this condition or use the wrong type of averages, e.g. the most probable numbers of sides. Therefore, here Eq.(5) has been checked by thorough own experimental investigations for very large numbers of grains (see Fig.2). Application of the I_east Square Method yielded a linear equation given very accurately by 3 + 3r (7) One can see that it fulfills Eq.(6) and that for the smallest grains (r=0) f=3 is obtained reflecting the fact that the disappearing grains are 3-sided ones.
Introducing Eq. (7) in the averaged von Neumann Mullins equation (4) leads to the equation dRil [1 1] (8) given by HillerP for 2-dimensional grain growth, but derived on a more heuristic basis. Now it shows more clearly the physical meaning of its origin. This equation together with a continuity equation can be used to describe the time evolution of the grain size distribution to(R,t) in the textureless case4: Eqs. (8) and (9) give a complete solution of the problem of quantitative description of 2-D grain growth. R is derived from first principles except for the assumption of the empirical linear relationship Eq. (7). In the next section, however, it will be shown that also this relationship is based on fundamental topological principles.

UNDERLYING TOPOLOGICAL PRINCIPLES
The agreement of the linear relationship with the average values in Fig.2 shows that this is not only an empirical law but expresses some topological facts. Therefore an (approximate) derivation of Eq.(4) will be given on the basis of .fundamental topological principles. A model will be used in which polygonal grmns are substituted by circles of equivalent area. Two basic assumptions are made:  (Fig.3a). The grains j with radius P which can surround grain with radius are calculated by r is valid for grains being strictly circular whereas for polygonal grains the value 3 is more correct.
(ii) Random surface co.vedng (Fi.3b). The average number of neighbours of a grain i is given by (11) Here represents (12) the fraction of the total grain surface occupied by grains of the size . This means Eq.(11) is valid for the case that the surface of any grain i is ocupied by gra.ns j in a random way, i.e. only according to the surface fraction of the j-The above derivation reveals the two first principles which determine the validity of Eq. (7), namely the principles of total and random surface covering being approximately expressed by Eqs. (10) and (12). On the other hand if one wants to describe a micmstructure where topological correlations are present, i.e. that the principle of random surface covering is not fulfilled, the linear relationship is maybe no longer valid5.
These topological principles can also be applied to the_3-D case in order to derive a relationship between size Ri and number of faces fi of the grains.  (15) (o 2 distribution variance). One sees that the relationship Eq.(14) is no longer linear but quadratic in R and depends on the shape of the size distribution Fig.4 shows the so far most complete 3-D data, measured by Rhines and Patterso on Aluminium. Here it has to be pointed out that all the data for the 3-D case in the literature, also those from Rhines and Patterson are always discussed in terms of a linear relationship. However, even the rather rough straight line shown in Fig.4 represents a non-linear law in the (fi,R) space, namely fi =Rt. Moreover, the downward deviation of the experimental data for low fis values shows that the exponent of the re law should be higher than 1.33. In Fig.5 the data from Rhines and Patterson .are transformed to linearly divided axes. Here it can be seen easily that the relation is rather quadratic than linear. This is_ in agreement with the above predictions of a quadratic law (Eq.(14)) with fly (R=0) 4 (tetrahedron).
From the principles of total and random surface covering also the aver-average boundary length L of the grains as a function of the grain size i can be calculated. If a grain is surrounded only by grains of the size j, the length of its boundary is given with Eq.(lO) by L# = 2x R, = ___6R' 2 RtRI (16) and by applying the random surface coveting postulate it follows for the average length of grains i: 2 r, Such type of relationship (Eq. (17)) was also observed experimentally .
Finally it should be mentioned that the above topological principles are also determining other topological relationships, e.g. the correlation between number n of sides of a grain and the average number f of sides of the first neigbhours in a 2-D microstructure: 6+ (19) nffi5+ ( variance of the distribution of grgi_'n comers). This relationship has empirically been found by Aboav and W7, but no clear derivation based on fundamental assumptions, is reported yet in the literature. As will be shown in another ave' also this type of rather complex correlation can be derived from the fundamental topological principles. All these agreements demonstrate that the grain structure even in rather large detail is determined by the principles of complete and random surface coverages.

INTRODUCTION OF TEXTURE FOR THE 2-D CASE
In the case that different orientations are present the linear relationship and the fundamental law fi--6 for the whole microstructure are still valid. Even if one applies the above topological principles Eqs.(10) and (12) to a single orientation class (H) one still gets a linear relationship f 3 + 3 --=-_ ,R (20) R since these principles concern only the grain geometry independent on the orientation. But one sees that the average number of sides of the given orientation can be different from 6 as much as R w deviates from R: H = 3 + 3----6 (21) R IntegrateA over all orientation classes, one obtains again the fundamental law fir6 for the whole microstructure.
with 0 H being the relative frequency of grains belonging to the orientation class H. The contribution of different texture classes to the total linear relationship is schematically shown in Fig.6. Here the simple case of only two orientation classes with different sizes was chosen so that their radii do not superimpose. Such microstructures can be found e.g. during secondary recrystallization where one has primary and secondary grains of rather different sizes.
Re|ative radius rt.Rilfi F c s e s " 6: Conuution of two orientation L and K with different mean sizes IP and R t to the total linear relationship. The average number of sides of the grains belonging to different size and orientation classes are presented by Furthermore it follows for all other topological rules which are derived from these principles, e.g. the grain size delndence of the grain boun__ _ _ length or the Weaire-Equation (Eqs. (17), (19)) that they are not only valid for the whole microstructure, but also for grains of a single orientation class.

CONCLUSIONS
Any statistical theory of grain growth must contain some topological assumptions capable to take into account all the complex features or grain arrangement. Here it has been shown that important topological laws, as e.g. the relationship between grain size and number of sides (faces in 3-D), the dependence of the mean grain boundary length on the grain size and the Weaire-equation, can be derived from fundamental topological principles, namely the postulates of total and random surface covering. It is further shown that these relationships are also valid if, instead of the whole microstructure, only grains belonging to individual orientation classes are considered.