The influence of spatial grain size correlation on normal grain growth in one dimension

Acta mater. Vol. 44, No. 4, pp. 1673-1680,

Pergamon 0!86-7151(!35)00261-8

1996

Elsevier ScienceLtd Copyright 0 1996Acta Metallurgica Inc. Printed in Great Britain. All rights reserved 1359-6454/96

$15.00 + 0.00

THE INFLUENCE OF SPATIAL GRAIN SIZE CORRELATION ON NORMAL GRAIN GROWTH IN ONE DIMENSION 0. HUNDERI’ and N. RYUM’ ‘Departments of Physics and *Department of Metallurgy, Norwegian Institute of Technology, 7034 Trondheim, Norway (Received 28 February

1995; in revised form 8 May

1995)

Abstract-In this paper we demonstrate that a size correlation between neighbouring grains will exist in a grain structure. Such a correlation is shown to prevail during coarsening of the grain structure and this will in turn influence the kinetics of the process and the size distribution. In this paper the phenomena has been investigated numerically in a one-dimensional grain structure. In the accompanying paper the treatment is extended to the two-dimensional case. In the one-dimensional case we demonstrate that a quasi stationary grain size function as well as a quasi stationary size correlation function exists in the system after a transient which depends markedly upon the initial conditions. The kinetics of the coarsening process has also been analysed analytically by means of a modification of Hillert’s mean field theory.

INTRODUCTION It is generally assumed reaches a quasi stationary

that normal grain growth grain size distribution after

a transient growth period. It must, however, be mentioned that such an assumption is not universally accepted [l]. A theory of normal grain growth should predict this quasi stationary distribution and the variation with time of the mean grain size. The transient behaviour in cases where the initial distribution differs from the quasi stationary distribution, is also of general interest. However, the grain structure of a material is not completely characterized by a grain size distribution function only. The grains may take up a mutual spatial arrangement that depends upon their relative sizes. Large grains may, for instance, preferentially be surrounded by small grains and vice versa. If such a spatial correlation between grains of different sizes exists, it will have an effect on the kinetics of grain growth. This possibility has so far received little attention. In fact, all the mean field theories of grain growth [24] assume explicitly that all grains in the assembly have the same environment. In the stochastic theories [5-81 the randomness of the grain assembly is a part of the framework of the theories and is thus not even an assumption. In the Monte Carlo treatment of grain growth no correlation needs of course to be assumed. Whether this treatment predicts such an effect or not, has been discussed by Ling et al. [9]. They state that the Aboav-Weaire’s law, which relates the average number of sides of the neighbouring grains to an n-sided grain, is sat-

isfied in their simulations. Also, the second momentum of the side distribution function obtained by the Monte Carlo simulation appears to be somewhat higher than what has been observed in froth and during two-dimensional grain growth [lo]. Quite recently it has been demonstrated by computer simulation, using a deterministic description of grain growth, that a characteristic spatial arrangement of the grains according to their relative sizes takes place during the process [l I]. These results have so far been described only rather superficially by the introduction of a spatial grain size correlation function and a more complete description is lacking. Such a correlation has been found experimentally in some two-dimensional network structures and is expressed by the Aboav-Weaire’s law mentioned above [12]. In the present investigation the preliminary results referred to above and related to this phenomenon, will be developed further [l I]. Only one-dimensional growth will be treated here. The reason for this is that the growth process in this case can be treated in an exact way without making any unsubstantiated assumptions (see below). The results obtained are believed to disclose some of the fundamental aspects of grain growth in general and thus to be of value also for the description of grain growth in systems of more practical relevance. Our one-dimensional model can be considered as a way of studying the effect of the local environment, but without the constraints imposed by space filling. In a separate publication the size correlation of grain sizes during two-dimensional grain growth, including the effect of topological constraints, will be described [13].

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MODELLING

In Hillert’s mean field theory of grain growth, the growth rate dR/dt of grains of size R is given by the equation:

M and e are the mobility and specific energy of grain boundaries and R, is a critical grain size that has to be chosen in such a way that the volume of the specimen remains constant during growth. CI is a dimensionless constant whose value is close to 0.5 in two-dimensional growth and close to 1.0 in a threedimensional system. The theory is based on the assumption that the rate of the process is determined by the rate of migration of the boundaries. The theory is a statistical one, thus no direct information of the rate of growth of one particular grain is given. The effect of the local environment of the grain R on the rate of its growth is not included. We now introduce the fictitious process of grain growth in one dimension by assuming that even in this case equation (1) is valid with its characteristic value of LX. In this case, since every grain has always two nearest neighbouring grains, a deterministic growth rate equation can also be given for each grain in the system. The effect of the local environment on the growth rate is thus included. Real grain growth must be a d > 2 process since there are no driving forces in one dimension (e.g. grains in a wire), but we believe that a study of this fictitious one-dimensional system can provide important insight into real grain growth. With this in mind we have studied grain growth in a space which is initially subdivided into grains in the following way: We distribute points randomly in space and construct Voronoi-cells around each point. In this way the space is divided into space-filling, non-overlapping crystals. In one-dimensional space this procedure is particularly simple because the grain size distribution obtained is a r-distribution and it can thus be expressed in an explicit, mathematical form. When treating the 2- or 3-dimensional spaces the grain size distribution has not been expressed in an exact way by a simple mathematical expression

IN ONE DIMENSION

We will return to this in a subsequent publication [13]. In one dimension G(Z/Z,) is simply the average size of all the neighbours to the grains in size class Z scaled by the arithmetic mean of all the grains in the ensemble, (Z,)/Z,,,. If no correlations exist, G(Z/Z,,,) will be equal to unity. The one-dimensional space was divided into lo5 grains (Voronoi-cells). We then induced grain growth in the system by allowing the grain boundaries to move according to the following deterministic rate equation:

f

=k{i(k+$)-k}

which automatically filling

satisfies

the condition

(W of space

CZ,= constant.

(3b)

Here, Zi+ , and Z,_, are the sizes of the two neighbouring grains to the ith grain which has the size Zi. k is a constant containing the grain boundary tension and the grain boundary mobility. The same equation was used in [l 11.As can be seen from equation (3a), the form of the growth equation is such that a grain of size 1 will shrink if the mean size of its two neighbouring grains is larger than Z and will grow if the opposite is the case. The growth equation was chosen to be of the same mathematical form as the analogous deterministic equations in two and three dimensions. It can thus be considered as a model where the demand for space filling has been relaxed. Growth is then not influenced by topology which is an unavoidable complication in higher dimensions. We find the model worthwhile studying in some detail since the effects of spatial size correlations can be treated in an exact way. The model is also similar to the linear bubble chain model previously studied by the authors [15] and by Mullins and Vinals [19]. We have, however, chosen to study a pure one-dimensional system instead of the linear bubble chain model in order to avoid the additional complications of 1.0

0.8

1141.

0.7

In such an assembly each grain has only two nearest neighbouring grains. For one particular grain of size I, we call the mean size of these two grains I,. The different grains of size I will of course have different values of I,,. We now define the grain size correlation function G(Z/Z,,,) in the following way:

0.6

^a z

q 3.0

0.5 2.0 $

0.4 0.3 0.2

1.0

0.1

G(Z/Z,) = (Z, >/mea”

(2)

where (Z,) is the arithmetic mean size of all nearest neighbours to grains of size 1, while meanis equal to the arithmetic mean grain size Z,, in higher dimensions this is, however, not the case.

0 0

1

2

3

4

5 P

Fig. 1. The grain size distribution function F(p) and the correlation function G(p) plotted against p = I/f, for the initial grain structure generated by a one dimensional Voronoi construction. The fully drawn curve is the corresponding r-distribution.

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having to define a contact area between the neighbours. Our main interest lies in trends when going from one to two and higher dimensions and this is best seen when studying a simple system. The results obtained are used in the accompanying paper to separate the effects of topology from the effects of local correlations in two dimensions. Having defined the model, we followed the variation with time of the mean grain size I,, the grain size distribution function F(I/Z,,,, t) and the correlation function G(l/l,, t). Cyclic boundary conditions were used in all simulations.

IN ONE DIMENSION 70

(a)

60 N=LE

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0 0

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30

40

50

TIME(arb ma)

3.5

RESULTS

(b)

In Fig. 1 the initial grain size distribution The size distribution of one-dimensional cells are given by a r-distribution:

is shown. Voronoi

g3 ' 2.5 2 1.5

F(I/I,)=4fexp

m

1

This is given by the fully drawn line in Fig. 1. The corresponding grain size correlation function is also shown in the same figure. As can be seen, in this initial grain structure, small grains have, on average, a surrounding of small grains and the large grains tend to be surrounded by large grains. The grain size correlation function G(r/Z,) is given by the following equation:

(5) ‘\

‘d

The grain size correlation function in this particular case can be easily derived theoretically, as shown in Appendix 1. Figure 1 shows that the points representing the simulated distribution function closely follow the theoretical curve. This demonstrates that the random number generator used gave statistically independent numbers and the accuracy of the simulation technique is good for ensembles of this size. In Fig. 2(a) the kinetics of grain growth is expressed as the variation of I,$ with time. As can be seen I,!,, increases linearly with time for large times, but has a small transient at the start of the growth process. This transient is shown more accurately in the inset, where also the derivative of the curve is given. The variation with time of the grain size correlation function is shown in Fig. 2(b). It is seen to change dramatically during grain growth: after a relatively short growth period there is now a tendency for the small grains to be surrounded by large grains and vice-versa. This is just the opposite of the correlation that existed in the initial grain system. The grain size distribution function also changes during grain growth and approaches a stationary shape when Z, has roughly tripled relative to its value at t = 0. The distribution functions after three relatively short growth periods are shown in Fig. 2(c). In Fig. 2(d) the final, quasi stationary shape of the grain size distribution function and the grain size corre-

0.5 0 I 0

1

2

3

4

I)

5

1 0.6 0.6 ; 0.4 0.2 0 0

2

1

3

4

P

2

1.6

~/ 0' 0

, 0.5

1

1.5

2

P

2.5

Fig. 2. (a) The variation of the mean grain size squared with time. The value of 1, at t = 0 is chosen to be 1.0.(b) The correlation function G(p) plotted against p = l/l,,, for three different growth times. The corresponding mean grain sizes are given on the graphs. (c) The normalized grain size distribution function F(p) plotted against p = l/l,,, for three growth times. The corresponding mean grain sizes are given on the graphs. (d) The quasi stationary size distribution function F(p) and the quasi stationary correlation function G(p) plotted against p = Z/l,,,.

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lation function are shown. The quasi stationary correlation function has better statistics than in Fig. 2(b), since we have averaged over a number of runs. It is straightforward to explain the general shape of the observed correlation function. Grain growth is a dynamic process. After quasi stationary condition has been reached, a small grain is, on the average, small because it has been eaten by its neighbours. And a large grain is, for the same reason, large because it has been eating its neighbours. This will lead to correlations in the system, a correlation where on average small grains are surrounded by large grains and vice versa. These results disclose the important fact that both the distribution function and the correlation function change during grain growth and that they both approach stationary conditions after extensive grain growth. It is reasonable to expect that the variation with time of these two functions are interrelated. This implies that the stationary condition of one of the functions can not be obtained when the other function is still changing. Also the growth rate, expressed as the increase with time of the mean grain size, is thus dependent upon both these functions. In an effort to separate these two effects, the grains in the initial grain system were shuffled by using a random number generator before grain growth was initiated. This means that no correlation existed at the start of the grain growth process, while the grain size distribution function was unchanged. The variation with time of 2; in these two cases, initially correlated and non-correlated, are shown in Fig. 3(a). As can be seen the growth rate dlk/dt is highest when no correlation is present at the start of the grain growth process. The two curves are seen to become gradually more parallel even though the approach is very slow. During grain growth the correlation function and the distribution function are found to change continuously for both starting conditions and approaching their stationary values gradually. The rates at which these changes take place are very different in the two cases. The change with time of the distribution function is in Fig. 3(b) characterized by the variation with time of the parameter (I). (l/Z) = 1,. (l/r). This parameter is a dimensionless measure of the shape of the distribution function. A quasi stationary state has been reached when this product is constant. As can be seen from Fig. 3(a) the two curves approach a common constant value, which is shown below to be 1.335. The approach is much more rapid when a correlation exists in the initial distribution than when such a correlation is absent. DISCUSSION

By means of equation (2) the grain growth process in one dimension is described rigorously and no unsubstantiated assumptions are necessary. The description is deterministic because the behaviour of

IN ONE DIMENSION 80 70

- E 60 50 40 30 20 10 0 10

0

20

30

40

50

TIME(arb

umts)

1.6 (b)

4

1.5

c 6

1.4

“+.s_L b

1.3

. . . ..*..

. .. .

“a7_$.4.~

‘,

III

L/,..*.“ _ ,.

. . . . . .. . .: . i__ri_~

j

~

1.2 0

10

20

30

40 TIME(arb

50 units)

Fig. 3. (a) The variation

of the mean grain size squared with time for two different initial grain structures. Upper curve: Voronoi grain size distribution without correlation. Lower curve: Voronoi grain size distribution with correlation. The value of 1” at t = 0 is chosen to be 1.0 for both curves. (b) Variation of the product (I), (l/1) with time for the same initial grain structures as in (a). Upper curve: Voronoi grain size distribution without correlation. Lower curve: Voronoi grain size distribution with correlation.

each grain in the assembly is followed during the grain growth process. First of all we observe that a quasi-stationary grain size distribution is obtained after a transient period. Very often, similar behaviour is said to exist also for grain growth in two and three dimensions. However, in these cases an equation equivalent to equation (2) can not be given. In two dimensions the Mullins-von Neumann law [16, 171 is valid, but no exact relationship exists between the number of nearest neighbouring grains of one particular grain and the size of this grain. The mathematical treatment of the process is thus based on some assumptions of the statistical nature of the grain assembly. The grain growths in three and even in two dimensions are, however, geometrically extremely complicated processes and the assumptions made in order to make the mathematical treatment possible are not necessarily valid. In addition to this asymptotical approach with time to a quasi stationary grain size distribution, a spatial arrangement of the grains according to their relative sizes is shown to take place and a quasi stationary correlation between grains of different sizes is also approached asymptotically. The present investigation has demonstrated that for a true quasi stationary state to exist, both a quasi stationary grain

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size distribution and a quasi stationary spatial grain size correlation must exist. This implies that both these requirements must be taken into account in an analytic description of grain growth. In the present paper we have studied only the one-dimensional case. In the accompanying paper the two-dimensional case is treated in a similar, rigorous way and the correlation effect is shown to be present and of importance even in that case [13]. In this discussion we will make a first attempt to include the grain size correlation effect into an analytic theory of normal grain growth. We will do this by modifying the mean field theory of Hillert [2]. A more complete theory should introduce the correlation effects explicitly into the grain growth equation. By using the correlation function we include the effect of the surroundings of a grain on its growth rate in an average way only. As we will see this is, however, sufficient for obtaining a satisfactory description of grain growth in one dimension. In equation (2) the suffix i refers to the spatial position of the grains in the system. We now collect all grains of size I and write for the average growth rate dl/dt of these grains:

The term (l/Z,,) will in the general case, and as we have seen above, be a function of 1. In Hillert’s mean field theory it is assumed that all the grains in the assembly see the same environment. Thus, all grains of equal size will have the same growth rate and equation (6) takes the following simple form:

.

1 -

L

0 1

.

On the basis of these equations Hillert’s theory predicts a quasi stationary grain size distribution for one dimensional grain growth which is given by the following equation: F(n) = 2e(&)exp(

-A)

(9)

with I = Z/l,. The growth rate of the critical size in the quasi stationary state is given by the following equation:

3-1-2k. dt

1617

shown by the variation of the grain size correlation function in Fig. 2(b). This means that a “local” critical grain size, Z,(Z), that depends on the grain size, must be introduced. A consequence of this is that the small grains will shrink, and the large grains will grow, with higher rates than in the mean field approximation, equation (7). We now modify the growth equation to include a functional form of this “local” critical size [equation (6)] which is consistent with what is found from the numerical simulation. We write the modified growth equation in the following way: ;=k{[m+,!?(;)+y(;~++;]

(11)

with TV+ p + y +. + = 1. Z, still has the same meaning as in the mean field theory, i.e. it is the grain size which has zero growth rate. When Zl = y = . . . = 0 and c( = 1 the equation is identical to equation (7). The parameters tl, /?, y, . will vary during grain growth and reach constant values when the quasi stationary state is obtained. We have used the results obtained by the simulations to determine the values of TV,/?, y in the quasi stationary state. All other coefficients in the expansion in equation (11) are so small that they can be neglected. When presenting the results from the simulation the most convenient scaling grain size was the mean grain size I,,, We can thus write an equation equivalent to equation (11)

where

Here I,, the critical grain size, has the same value for all grain sizes and is seen to be equal to the grain size that has zero growth rate. The requirement of a constant total volume of the assembly gives the following equation for I, in one dimension: 1 -=

IN ONE DIMENSION

(10)

During grain growth the small grains tend to be surrounded by large grains and vice versa and this tendency approaches a quasi stationary state, as

~‘++)+y’(;~=(;)i=G,(;)

(13)

GL’/J , ’and y ’are found by plotting the average inverse nearest neighbour size vs size. G, is in the same way as G a function which describes the correlations which exist in the system. It is the appropriate function to use in further modelling in view of equation (6). G,(Z/Z,) is plotted in Fig. 4(a). This function is of course not simply the inverse of G(Z/Z,). G, varies more rapidly with Z/Z, than l/G. This is substantiated by plotting the full size distribution function of the neighbours to grains of a given size class. These we might appropriately call partial distribution functions. In Fig. 4(b) we have plotted three of these distributions against pn = Z,/Z,,,. The three functions differ considerably in shape, reflecting that strong correlations exist in the system. The biggest differences between the curves are found for small values of Z,/I,,, The function G, weighs the small Z,/Z,,, values stronger than G, leading to the observed difference between G, and l/G. By curve fitting the data in Fig. 4(a) the best values for a’, jl ’ and y ’ were found to be:

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IN ONE DIMENSION -2

Y+B A,=-.

Lx3 27

0

0.5

1

1.5 D

2

2

0.8 0.6 0.4 0.2 0 0

0.5

1

1.5

2

pn

2.5

Fig. 4. (a) The quasi stationary correlation function G,(p) plotted against p = l/Z,,, (b) Three partial distribution functions F( p,) plotted against pn = 1,/I,,,. The size of the grains to which these partial distributions are neighbours are given on the graphs.

a’ = 0.862; /I ’= 0.430, y ’= 0.024. These values correspond to a scaling factor I,,$, = 1.229. Substituting this value into equation (12) we obtain the following values for CI, p and y : c( = 0.702; 1 = 0.285, y = 0.013. Knowing the growth rate dl/dt, it is simple to find the growth rate of the critical size, dlz/dt and the grain size distribution function F(1) with 1 = l/l, [2, 181. The expression takes the simplest forms when y = cr3/27. This is, within the accuracy of the simulation, satisfied by the values of TVand y found. Solving for the growth rate of the critical grain and the distribution function we find:

The parameter (I). (l/l) introduced above was calculated for this distribution function and found to be 1.335. This is in good agreement with the results obtained from Fig. 3(b). In Fig. 5 the result of the modified theory is compared with the result of the simulations. The curve in Fig. 5 was calculated from equation (14b) but with GIand /I found, not from a fit to G, (l/l,,,), but from a best fit of equation (14b) to the results of the simulation. The agreement between the simulation results and the modified theory, a theory which we might call correlation field theory, is excellent. The values of tl and fi obtained from a best fit of the distribution function were: GI= 0.862 and /I = 0.447. These values are in excellent agreement with the values found from a fit of the correlation function, showing a complete internal consistency in the theory we have developed. The good agreement also shows that it is sufficient to describe the effect of the environment on the growth rate of one particular grain size class by the correlation function G,,which is a function of the mean neighbour grain size only. The growth rate given by equation (12) increases linearly with l/Z, for large values of l/l,,,. This might be considered unphysical, but looking at the corresponding distribution function given by equation (14b) we observe that it has a cut-off at 1 = 3/c(. No grains with l/l, larger than 3/a exist in the system and G, is thus undefined for l/l,,, > 31~. In Fig. 6(a) and (b) the results obtained by the mean field and the correlation field theories are compared. In the mean field theory the critical radius is given by l/l, = (l/I ), the average taken over the whole ensemble of grains. In Fig. 6(a) we compare the mean grain sizes as obtained by the two simulations. In both these simulations the starting size distribution was the analytic mean field result, equation (9), and without any spatial size correlation at t = 0. This was 1

fi

1

\

0.8

/

g 0.6 !J. 0.4

/ /

0.2 -i'

J

OY

0 (14b)

0.5

1

1.5

2

P

2.5

Fig. 5. The quasi stationary size distribution function F(p) plotted against p = l/l,. The fully drawn curve for the distribution function is the best fit of the modified theory to the results of the simulation as shown by the points.

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Acknowledgement-We would like to thank Professor Hillert for many stimulating discussions. LOCAL FIELD SIM”LATIOi/

50 40

t

REFERENCES

/

20 10 0 10

20

30

50

40

TIME(arb. mts)

1.4 ,

I

1.2 2 E

1 0.6 0.6

0.2 0 0

0.5

Fig. 6. (a) Comparison

1

1.5

Mats

f

30

0

1679

2

h

2.5

of I:(t) for the mean field theory

and the local field theory. The value of I, chosen to be 1.0. (b) Comparison of the mean bution function and the modified (correlated bution function. The distribution functions against i = l/l,.

at t = 0 is field distrifield) distriare plotted

done in order to minimize effects of the initial transients. We find that the mean field theory predicts a growth rate which is approx. 1.8 times lower than that obtained by the correlated mean field theory. This is in excellent agreement with what is predicted from comparing equations (10) and (14a). When comparing the distribution functions predicted by the two theories, Fig. 6(b), marked differences are also found. The mean field theory gives a distribution which is much more peaked and narrower than what is predicted by the modified theory. The relatively large differences between the predictions shows that spatial correlation between grains of different sizes plays an important role in the grain growth process. By including the correlation effect we can give a complete theoretical description of grain growth in one dimension. In higher dimensions two factors must be included in order to describe grain growth; these are spatial size correlation and topological constraints. Several attempts have been made to include the constraints imposed by topology into theories of grain growth in two dimensions, but the influence of spatial size correlation has been neglected. We feel that this paper demonstrates that spatial size correlation is also important for the understanding of the process of grain growth. In the accompanying paper [14] a treatment similar to the one carried out here for the one-dimensional case has been carried out for grain growth in two dimensions.

1. M. A. Fortes, in Grain Growth in Polycrystalline Materials (edited by G. A. Abruzzese and P. Brozz), p. 319. Trans. Tech Publications, Zurich (1992). 2. M. Hillert, Acta metall. 13, 227 (1965). 3. K. Liicke, G. Abruzzese and I. Heckelmann, in Grain Growth in Polycrystalline Materials (edited by G. A. Abruzzese and P. Brozz), p. 3. Trans. Tech. Publications, Zurich (1992). 4. V. E. Fradkov, L. S. Shvindlennan and D. G. Udler, Phil. Mag. LRtt. 55, 289 (1987). 5. N. P. Louat, M. S. Duesbery and K Sadananda, in Grain Growth in Polycrysialline Materials (edited by G. A. Abruzzese and P. Brozz), p. 67. Trans. Tech. Publications, Zurich (1992). 6. C. S. Pande, in Grain Growth in Polycrystalline Materials (edited by G. A. Abruzzese and P. Brozz), p. 94. Trans. Tech. Publications, Zurich (1992). 7. H. J. Frost, C. V. Thompson and D. T. Walton, Acta metal1 mater. 38, 1455 (1990). 8. A. Thorvaldsen, in Grain Growth in Polycrystalline Materials (edited bv G. A. Abruzzese and P. Brozz). ” p. 361. Trans. Tech. Publications, Zurich (1992). 9. S. Ling, M. P. Anderson, G. S. Grest and J. A. Glazier, in Grain Growth in Polvcrvstalline Materials (edited bv G. A. Abruzzese and-P..Brozz), p. 39. Trans. Tech. Publications, Zurich (1992). 10. D. A. Aboav, in &ain’ Growth in Polycrystalline Materials (edited by G. A. Abruzzese and P. Brozz), p. 275. Trans. Tech. Publications, Zurich (1992). 11. 0. Hunderi, and N. Ryum, in Grain Growth in Polycrystalline Materials (edited by G. A. Abruzzese and P. Brozz), p. 89. Trans. Tech. Publications, Zurich (1992). 12. C. J. Lambert and D. Wearie, Metallography 14, 307 (1981); Phil. Mag. B47, 445 (1983). 13. K. Martinsen, 0. Hunderi and N. Ryum, ibid. 14. A. Thorvaldsen, in Grain Growth in Polycrystalline Materials (edited by G. A. Abruzzese and P. Brozz), p. 307. Trans. Tech. Publications, Zurich (1992). 15. ‘0. Hunderi, N. Ryum and H. Westengen, Acta metall. 27, 161 (1979). Metal Interfaces, p. 108. ASM, 16. J. von Neumann, Cleveland, Ohio (1952). 17. W. W. Mullins, J. appl. Phys. 27, 900 (1956). 18. 0. Hunderi and N. Ryum, J. Mater. Sci. 15, 1104 (1980). 19. W. W. Mullins and J. Vinals, in Grain Growth in Polycrystaliine Materials (edited by G. A. Abruzzese and P. Brozz), p. 53. Trans. Tech. Publications, Zurich (1992).

APPENDIX We seek the average length of the neighbours to grains of length I. Consider four nucleation centres along a line. Assume that the distances between them are xi_ ,:I,xi.,+ , and x,+ ,,,+ Z. We now carry out a Voronoi construction around these centres. The length of grain i will then be:

and the length

of the neighbour: ~“=~(X,.,+I+Xi+l.,+Z).

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We now average the last expression with the restriction that 1/2(xi_ ,,, + x,,,+ ,) = I. Averaging over xi+ ,,i+2 carries no restrictions and the average is simply I,,,. When points are distributed randomly along a string, the size distribution of line intervals have an exponential distribution: f(x,)

=

Here

1 is the mean

length

distribution that:

.

of the intervals.

With

this

of interval

lengths it is straightforward

to show

X ,_I,, + x,,,+,=*,=fl. The average therefore:

exp(-xJ~) I

IN ONE DIMENSION

size of the neighbour

to grains

of size 1 is