- Email: [email protected]
Computer modelling of anisotropic grain microstructure in two dimensions
Acta metall, mater. Vol. 41, No. I, pp. 191-198, 1993
0956-7151/93 $5.00 + 0.00 Copyright © 1992 Pergamon Press Ltd
Printed in Great Britain. All rights reserved
COMPUTER MODELLING OF ANISOTROPIC GRAIN MICROSTRUCTURE IN TWO DIMENSIONS O S A M U ITO 1 and E. R, F U L L E R JR 2 ~Hitachi Research Laboratory, Hitachi Ltd, 3-1-1 Saiwai-cho Hitachi-shi, Ibaraki-ken 317, Japan and 2Materials Science and Engineering Laboratory, National Institute of Standards and Technology, Gaithersburg, MD 20899, U.S.A. (Received 20 August 1991; in revised form 8 June 1992) Abstract--Assuming an anisotropic factor for geometrical grain microstructure, known as the Voronoi or Johnson-Mehl model, several grain microstructures have been simulated in two dimensions employing nucleation seeds that are assumed to be elliptical and polygon like shapes. By varying the shape of these seeds, six different microstructures have been generated based on growth and impingement model in both simultaneous and continuous nucleation cases. In addition, morphology such as the aspect ratio distribution, grain size distribution and number of grain edges in each case have been analyzed and are correlated qualitatively.
l. INTRODUCTION One of the major obstacles to prevent better understanding of the physical property of ceramic materials is the geometrical complexity in grain microstructure. Since many physical properties of ceramics are determined by their grain microstructure, prediction and control of microstructural characteristics are important. So far, a number of computer simulations have been made for microstructures [1-3]. One of the methods for grain generation, considering grain geometry and topography is the model known as Voronoi polyhedra [4]. In the Voronoi model, two neighboring grains nucleate simultaneously, and their radii grow at the same constant rate. The grain boundary at which they meet will be a straight line and the perpendicular bisector between them. If two neighboring grains nucleate at different times, which means nucleation occurs continuously, and grow at the same rate, they will impinge along a boundary which forms a curve. This is known as the Johnson-Mehl model. These two models are briefly described in Table 1. Some microstructure simulations have been based on these two models employing an isotropic grain growth rate [5, 6], and this assumed isotropic grain growth is a common basis of them. However, in most ceramic materials, anisotropy has been observed in grain growth [7], and complex microstructures often result from it. This means that an anisotropic factor should be taken into account when simulating microstructures by computer. In this study, in order to generate a complex microstructure of the ceramics, these geometrical models have been modified assuming that growth rate depends on growth direction. In addition, we have analyzed morphology of microstructures such as aspect ratio distribution, grain size distribution and
number of grain edges in each case. These analyses are quantitatively evaluated and results are compared.
2. COMPUTATIONAL PROCEDURE 2.1. Generating microstructures First, the geometry that results from nucleation and growth to impingement is reviewed in the Voronoi and Johnson-Mehl models. Grain growth can be described regarding origin of coordinate as a nucleation site [8, 9]. In the Voronoi model, nucleation occurs simultaneously and all grains start to grow at the same time. The growing grain surface which corresponds to a line in two dimensions is specified by Xn=R.T where R is growth rate (constant), Xn is the surface of the growing grain n and T is time. In the case of continuous nucleation (Johnson-Mehl model), this equation is easily modified by Xn=R
" ( T - Tn)
Table 1. Conventional methods to generate grain microstructures n
Simultaneous
Continuous
Voronoi model
Johnson-Mehl
Isotropic Growth
191
model
192
OSAMU ITO and FULLER: COMPUTER MODELLING OF MICROSTRUCTURE
where Tn is time when nucleation occurs. Each nucleation site is randomly picked in a certain area. However, it should be noted that if there is already another grain which covers that site, then the site must be neglected. In these conventional models, growth rate is constant in every growth direction; i.e. rate is independent of the growth direction. Obviously, to assume an anisotropic crystal growth rate in these models, growth rate should be regarded as a function of growth direction. Therefore, the present anisotropic growth model with simultaneous nucleation (a variation of the Voronoi model) can be expanded by
J
YY k, y
.)
X
k(O):Length of vector xk = k(O)cosO yk = k(O)sinO
Fig. 1. Definition of grain shape with discrete points.
Xn = R(O). T where R(O) is the growth rate with a dependence on growth direction. In the anisotropic growth model with continuous nucleation (a variation of the Johnson-Mehl model), an equation is introduced in a similar manner by
Xn = R(O) . (T - Tn). In both the Voronoi and Johnson-Mehl models, growing grains eventually cover all over the plain and generate a structure which seems similar to the actual microstructure. It is obvious that these newly introduced models essentially have the same features as the conventional nucleation-growth models even if the anisotropic growth factor is added. Therefore, an anisotropic grain growth does not cause an unoccupied area in this spatial filling process. To deal with the anisotropic growth rate by computer, we employ an idea that the grain shape consists of discrete points and each discrete point is determined by the vector from the nucleation site. A schematic diagram is shown in Fig. 1. In this figure, the nucleation site is located at the origin of the coordinate and the growth rate is determined according to the growth orientation 0. In the case where growth is ellipse like, then k(O) is defined by
k(O ) = ~/r2 cos2(O;2+ sin2(O) where r is the aspect ratio of the ellipse.
The process to generate the microstructure is schematically shown in Fig. 2. Ellipses are employed for growing seeds. In the first stage of this generation, nucleation points are generated randomly in the two-dimensional space. Then each seed expands its size keeping its shape unchanging until it meets the surface of another grain where its growth terminates. In both simultaneous and continuous nucleations, in order to model the effect of the depletion region in which nucleation is excluded, a nucleation restriction zone is set up to exclude any nucleation event within a distance (1% of the unit length of the unit square) of the edges of any previous nucleated grains [5]. In continuous nucleation, to begin with, we select nucleation sites randomly, and then examine whether the nucleation site is already occupied by other growing grains or not. If occupied, nucleation is neglected and we continue to expand the grain that is already generated. Major factors which determine the morphology in continuous nucleation are nucleation rate and growth rate of seed. An important feature is that the nature of the structure does not change even if the rate between these factors varies. Structures for large and small grain sizes are equivalent in topology and differ only by a scaling factor. This feature has already been studied and the average grain size of the Johnson-Mehl model was given by Gilbert [10]. Based on this criteria, we employ an appropriate rate for seeds with different shapes in order to generate about hundred of grains in a continuous nucleation.
0~
(a) Seeds dispersed
(b) Expanding(initial)
(c) Expanding(final)
(d) Final structure
Fig. 2. Procedure to form grain microstructure in simultaneous nucleation (seed: ellipse, aspect ratio = 0.5).
OSAMU ITO and FULLER: COMPUTER MODELLING OF MICROSTRUCTURE
7-'
~ ~7'~/[~
193
OGn+t:CenterofGrainG.+t (Nucleation Site)
Distance between Grain Gn
andGrainGn+t
GrainGn
/ Fig. 3. Definition of distance between grains. Regarding termination of grain growth, we need to calculate the distance between the growing point and the other grain surface that the growing grain will meet. A schematic of the procedure to terminate growth is shown in Fig. 3. First, a position Gx is calculated, which is an intersection of the line Gn(/) - OGn + 1 and Gn + l(k) - Gn + l(k + 1). In this study, we regard line Gx - Gn(j) as the distance between grain Gn and grain Gn + 1. And we assume that growth of that point is terminated when this distance is 0.2% of the unit length of the unit square. 2.2. Analysis o f morphology
Various seeds employed in this study are shown in Table 2. Elliptical and polygon-like shape were employed and six different macrostructures were produced. In order to estimate the morphology of simulated microstructure quantitatively, we employed some parameters such as the aspect ratio distribution, grain size distribution and number of grain edges. A schematic diagram to determine the
aspect ratio of each grain is shown in Fig. 4. Regarding the nucleation site as the centre of a grain, the aspect ratio of the grain is the number got by dividing the minimum diameter by the maximum one. Regarding the number of grains, one may wonder how many grains would be enough so as to analyze microstructures quantitatively. In order to estimate the appropriate number, we examine the aspect ratio distribution of simultaneous nucleation case with ellipse-like seeds by increasing the number of grains. The aspect ratio of ellipse employed here is 0.3 and the number of grains ranges from 100 to 3500. Figure 5 shows the aspect ratio distribution when varying the number of grains. The number in parentheses indicates that of inner grains, while the number before the parenthesis is that of all grains generated. When analyzing morphology, we neglected grains facing outside and picked only inner grains for the quantitative calculation.
Table 2. Sha )e of seeds employed in this study ation Simultaneous
Continuous
Ellipse Aspect ratio= 1.0, 0.5, O.3 Nucleation Site
Octagon, Hexagon, Triangle
Aspect Ratio =
D min D ma~--~
Fig. 4. Calculation of aspect ration of each grain.
194
OSAMU ITO and FULLER: COMPUTER MODELLING OF MICROSTRUCTURE 30 25 20 t--
u.
.,I,.
15
10
100 (64) 500 (395) 1000 (813) 2500(2069) 3500(2899)
Aspect ratio =
00'0
0.2
0.4
0.6
0.8
0.3
1.0
Aspect Ratio Distribution
Fig. 5. Aspect ratio distribution in simultaneous nucleation of ellipse. In the case when the number of grains is more than 1000, no significant difference is seen between the distributions of aspect ratio. Based on this result, more than 1200 inner grains were generated for the following quantitative analysis of microstructures.
3. RESULT AND DISCUSSION 3.1. Simultaneous nucleation
Figure 6 shows one of the simulated microstructures in simultaneous nucleation with ellipse-like seeds. The aspect ratio of the seeds ranges from 0.3 to 1.0. The numbers in parentheses indicates that of inner grains, while the number before the parenthesis is that of all grains generated within the square in this figure. Figure 7 represents the frequency distribution for aspect ratio, grain size and number of grain edges. Number indicating here is that of inner grains. In terms of the grain size distribution, average grain size
is normalized according to the grain number and the value is fixed at one in every case. In the left figure, as the aspect ratio of the seeds decreases, its peak frequency of the aspect ratio moves left to a lower value. However, regarding grain size distribution, no significant difference in distribution is seen among figures. In addition, the same tendency is seen in the distribution of grain edge number even when the aspect ratio is varied. These results indicate that, in a microstructure with ellipse-type seeds, the aspect ratio distribution can be controlled but not the distribution of grain size and number of grain edges even when the aspect ratio of seeds are different. Figure 8 shows microstructures with polygon type seeds. Octagons and hexagons are slightly isotropic seeds and microstructures generated are similar to the Voroni case to some extent. A specific feature in this case is that they have a wavy boundary. In the triangle type of seeds, the situation is rather different and the grain has some acute angles. As shown in Fig. 9, roughly speaking, morphologic parameters do
(I) r=l.0
(2) r=0.5
N=100 (66)
N= 100 (66)
(3) r=0.3 N= 100 (67)
Fig. 6. Microstructure in simultaneous nucleation of ellipses.
r=l.0 N=1260 r=0.5 N=1265 r=0.3 N=2069
•
~ ~
Octagon N=1261 -.--o--- Hexagon N--1271 Triangle N=1664
r:Aspect ratio N:Number of inner grains
35
35
30
30
25
25
2o
20
~- 15 "
12r
15
u_
10
10
N:Number of inner grains
5
5 -
0.4 0.6 0.8 Aspect Ratio Distribution [a) ~.2
0#.0
1-.0
0
0
06
08
10
Aspect Ratio Distribution (o)
25
25
15
5 UI w
1
AA n
2
3
4
5
GS--
Grain Size Distribution
1
2
3
4
5
Grain Size Distribution
(b)
(b)
35
35
30
30
~
,~ 25
25
~ 20
2O
u_ 10 5
5
0 0 "~ ~ 3 4 5 6 7 8 9 101112131415
00 1 2 3 4 5 6 7 8 9 -1011 = 121314 1
Number of Grain Edge
Number of Grain Edge (c)
(c) Fig. 7. Quantitative estimation of grain morphology in simultaneous nucleation of ellipse.
(1) Octagon
Fig. 9. Quantitative estimation of grain morphology in simultaneous nucleation of polygons.
(2) Hexagon
©
N=100(67)
(3) S I c
N= 100(66)
Fig. 8. Microstructure in simultaneous nucleation of polygons. 195
N= 100(66)
196
OSAMU ITO and FULLER: COMPUTER MODELLING OF MICROSTRUCTURE
(1) r=l.0
(2) r=0.5
N=97(63)
N= 131(90)
(3) r=0.3 N=126(83)
Fig. 10. Microstructure in continuous nucleation of ellipses. In the grain size distribution, it is obvious that distribution range is much wider than that of the simultaneous nucleation. An important result in this figure is that no significant difference appears even if the aspect ratio of seeds is changed. The same is true of the grain edge number. When employing polygon type of seed in Fig. 13, no significant difference is seen between different shape of seeds as well as the case in the simultaneous nucleation. The reason is attributed in the same way as the simultaneous nucleation.
not show any particular difference in microstructures with polygon type seeds. The reason is that these seeds have no specific preferential direction in growing and the factor which mainly affects on the morphologic parameters employed in this study is anisotropic growth of seeds.
3.2. Continuous nucleation In continuous nucleation, the same type of seeds in simultaneous nucleation are employed. In Fig. 10, seeds are ellipses and in Fig. 11 seeds are polygons. In continuous nucleation, the number of grains can not be fixed before the simulation. Therefore, we used an appropriate initial parameter that determines grain number in order to generate approximately 1500 grains. Analysis of morphology was carried out within the inner grains in the same way as the simultaneous nucleation. Results are summarized in Figs 12 and 13. Regarding distribution of the aspect ratio, the distribution can be easily controlled in case of the seed ellipses.
(1) Octagon
0
N-100(66)
3.3. Correlative relation of morphology parameters From these results, we concluded that the aspect ratio of each grain is generally controllable when varying the seed shapes. However, the distribution of grain size or grain edge number is not easily changed. Therefore, we focussed on the relationship of these morphologic parameters by picking a hundred grains. Figure 14 shows correlative relation of grain morphology in continuous nucleation of ellipses. No
(2) Hexagon
0
N= 100(66)
(3) Triangle
A
N= 105(73)
Fig. 11. Microstructure in continuous nucleation of polygons.
r=l.0 N=1324 r=0.5 N=1435 r=0.3 N=1731
•
g"
Octagon N=1340 HexagonN=1373 N:Numberof innergrains TriangleN=1796
r:Aspectratio N:Numberof inner grains
•
35
35
30 25
3O
20
cr
15
LJ -
10 5 O
°~.o
02
04
06
08
~
0.0
10
0.2
'
'
'
0.4
'
0.6
'
'
0.8
--~
1.0
Aspect RatioDistribution (a)
Aspect RatioDistribution (o) 20
20 15
m
5 --
0
2
3
4
~ .
O-
-
Grain Size Distribution (b)
~'4-
5
Grain Size Distribution (b) 25
25 i
20 ~
~ 20
~ 15
15
¢.)
U_
It
5
5 0 1 2 3 4 5 6 78
0 1 2 3 4 5 6 7 8 9 101112131415
9 10111213141
Number of Grain Edge
Number of Grain Edge
(c)
(c)
Fig. 13. Quantitative estimation of grain morphology in continuous nucleation of polygons.
Fig. 12. Quantitative estimation of grain morphology in continuous nucleation of ellipses.
14
14 ~) 12 o) 10 no 8 uJ 6 t'-(5
2 0
~
12
** o
e o o
o o
*
*
e*
~
~ e e
5
12 o) 10
r=l.0 • oe ooaee~ ~ ~ e ~ ooe * e
t" •~
;*
r=l.0 o
6 D~ 4 I* 0
o
12
r=0.3
.
•
,
o,
6
~ Z
6 4 2 0 0
..'+®°° ~..
•
=E
e--
i!.
::
, o a oo~p
il + ~
01
Aspect Ratio
1.0
0
1
2
r=0.3
.
(5
-**.~**-.- ,,
0.2 0.4 0.6 0.8
:. °.."
0!*- *'
r=0.3
.;:
. r.?0
3
Grain Size
4
0
~o
a
o o
o
,~*~ * ~,*,~
.~
0.2 0.4 0.6 0.8
Aspect Ratio
Fig. 14. Correlative relationships of grain morphology in continuous nucleation of ellipses. 197
1.0
198
OSAMU ITO and FULLER: COMPUTER MODELLING OF MICROSTRUCTURE
strong relationships are seen in grain edge number vs aspect ratio and grain size vs aspect ratio. No matter which aspect ratio they may have, the bigger the grain size becomes, the more the grain edge number increases. This is not a particular result in this study, the same tendencies are always seen in cases using polygons. Before the simulations, we expected a different grain shape might effect morphology parameters such as grain size distribution or grain edge number, especially with ellipse type of seeds. In spite of this expectation, there is no significant effect when varying the aspect ratio of ellipse types of seeds or employing different shape of polygons in both simultaneous and continuous nucleation cases.
even when seeds have different shape. Regarding correlative relation of grain morphology, strong correlation is recognized between grain size distribution and distribution of grain edge number. This means that the bigger the grain size becomes, the more the grain edge number increases, while no strong relationships are seen in grain edge number vs aspect ratio and grain size vs aspect ratio. Acknowledgements--This work was done when one of the
authors (O. I.) was a guest scientist in NIST. We would like to acknowledge the many who supported us and gave stimulating suggestions, especially, Mark Vaudin, John E. Blendell, Jay S. Wallace and W. Craig Carter.
REFERENCES
4. CONCLUSION In order to generate anisotropic grain microstructures, several microstructures based on geometrical microstructure known as the Voronoi and Johnson-Mehl models have been simulated in two-dimensions. In terms of anisotropic factor, various shapes of seeds, ellipse and polygon, were employed. In addition, morphology such as aspect ratio distribution, grain size distribution and number of grain edges in each case was analyzed. The results are summarized as follows. When varying the aspect ratio of seeds, the resultant aspect ratio distribution of microstructure can be controlled, but not grain size and grain edge number,
1. M. P. Anderson, D. J. Srolovitz, G. S. Grest and P. S. Sahni, Acta metall. 32, 783 (1984). 2. J. Wejchert, D. Weaire and J. P. Kermode, Phil. Mag. B 53, 15 (1986). 3. M. P. Anderson, G. S. Grest and D. J. Srolovitz, Phil. Mag. B 59, 293 (1989). 4. A. Getis and B. Boots, Model of Spatial Processes. Cambridge Univ. Press (1979). 5. K. W. Mahin, K. Hanson and J. W. Morris Jr, Acta metall. 28, 443 (1980). 6. H. J. Frost and C. V. Thompson, Acta rnetall. 35, 529 (1987). 7. K. Yamada, J. Am. Ceram. Soc. 73, 2103 (1990). 8. N. Ryun and O. Hunderi, Acta metall. 37, 1375 (1989). 9. T. O. Saetre, O. Hunderi and N. Ryun, Acta metall. 37, 1381 (1989). 10. E. N. Gilbert, Ann. Math. Statist. 33, 958 (1962).
0956-7151/93 $5.00 + 0.00 Copyright © 1992 Pergamon Press Ltd
Printed in Great Britain. All rights reserved
COMPUTER MODELLING OF ANISOTROPIC GRAIN MICROSTRUCTURE IN TWO DIMENSIONS O S A M U ITO 1 and E. R, F U L L E R JR 2 ~Hitachi Research Laboratory, Hitachi Ltd, 3-1-1 Saiwai-cho Hitachi-shi, Ibaraki-ken 317, Japan and 2Materials Science and Engineering Laboratory, National Institute of Standards and Technology, Gaithersburg, MD 20899, U.S.A. (Received 20 August 1991; in revised form 8 June 1992) Abstract--Assuming an anisotropic factor for geometrical grain microstructure, known as the Voronoi or Johnson-Mehl model, several grain microstructures have been simulated in two dimensions employing nucleation seeds that are assumed to be elliptical and polygon like shapes. By varying the shape of these seeds, six different microstructures have been generated based on growth and impingement model in both simultaneous and continuous nucleation cases. In addition, morphology such as the aspect ratio distribution, grain size distribution and number of grain edges in each case have been analyzed and are correlated qualitatively.
l. INTRODUCTION One of the major obstacles to prevent better understanding of the physical property of ceramic materials is the geometrical complexity in grain microstructure. Since many physical properties of ceramics are determined by their grain microstructure, prediction and control of microstructural characteristics are important. So far, a number of computer simulations have been made for microstructures [1-3]. One of the methods for grain generation, considering grain geometry and topography is the model known as Voronoi polyhedra [4]. In the Voronoi model, two neighboring grains nucleate simultaneously, and their radii grow at the same constant rate. The grain boundary at which they meet will be a straight line and the perpendicular bisector between them. If two neighboring grains nucleate at different times, which means nucleation occurs continuously, and grow at the same rate, they will impinge along a boundary which forms a curve. This is known as the Johnson-Mehl model. These two models are briefly described in Table 1. Some microstructure simulations have been based on these two models employing an isotropic grain growth rate [5, 6], and this assumed isotropic grain growth is a common basis of them. However, in most ceramic materials, anisotropy has been observed in grain growth [7], and complex microstructures often result from it. This means that an anisotropic factor should be taken into account when simulating microstructures by computer. In this study, in order to generate a complex microstructure of the ceramics, these geometrical models have been modified assuming that growth rate depends on growth direction. In addition, we have analyzed morphology of microstructures such as aspect ratio distribution, grain size distribution and
number of grain edges in each case. These analyses are quantitatively evaluated and results are compared.
2. COMPUTATIONAL PROCEDURE 2.1. Generating microstructures First, the geometry that results from nucleation and growth to impingement is reviewed in the Voronoi and Johnson-Mehl models. Grain growth can be described regarding origin of coordinate as a nucleation site [8, 9]. In the Voronoi model, nucleation occurs simultaneously and all grains start to grow at the same time. The growing grain surface which corresponds to a line in two dimensions is specified by Xn=R.T where R is growth rate (constant), Xn is the surface of the growing grain n and T is time. In the case of continuous nucleation (Johnson-Mehl model), this equation is easily modified by Xn=R
" ( T - Tn)
Table 1. Conventional methods to generate grain microstructures n
Simultaneous
Continuous
Voronoi model
Johnson-Mehl
Isotropic Growth
191
model
192
OSAMU ITO and FULLER: COMPUTER MODELLING OF MICROSTRUCTURE
where Tn is time when nucleation occurs. Each nucleation site is randomly picked in a certain area. However, it should be noted that if there is already another grain which covers that site, then the site must be neglected. In these conventional models, growth rate is constant in every growth direction; i.e. rate is independent of the growth direction. Obviously, to assume an anisotropic crystal growth rate in these models, growth rate should be regarded as a function of growth direction. Therefore, the present anisotropic growth model with simultaneous nucleation (a variation of the Voronoi model) can be expanded by
J
YY k, y
.)
X
k(O):Length of vector xk = k(O)cosO yk = k(O)sinO
Fig. 1. Definition of grain shape with discrete points.
Xn = R(O). T where R(O) is the growth rate with a dependence on growth direction. In the anisotropic growth model with continuous nucleation (a variation of the Johnson-Mehl model), an equation is introduced in a similar manner by
Xn = R(O) . (T - Tn). In both the Voronoi and Johnson-Mehl models, growing grains eventually cover all over the plain and generate a structure which seems similar to the actual microstructure. It is obvious that these newly introduced models essentially have the same features as the conventional nucleation-growth models even if the anisotropic growth factor is added. Therefore, an anisotropic grain growth does not cause an unoccupied area in this spatial filling process. To deal with the anisotropic growth rate by computer, we employ an idea that the grain shape consists of discrete points and each discrete point is determined by the vector from the nucleation site. A schematic diagram is shown in Fig. 1. In this figure, the nucleation site is located at the origin of the coordinate and the growth rate is determined according to the growth orientation 0. In the case where growth is ellipse like, then k(O) is defined by
k(O ) = ~/r2 cos2(O;2+ sin2(O) where r is the aspect ratio of the ellipse.
The process to generate the microstructure is schematically shown in Fig. 2. Ellipses are employed for growing seeds. In the first stage of this generation, nucleation points are generated randomly in the two-dimensional space. Then each seed expands its size keeping its shape unchanging until it meets the surface of another grain where its growth terminates. In both simultaneous and continuous nucleations, in order to model the effect of the depletion region in which nucleation is excluded, a nucleation restriction zone is set up to exclude any nucleation event within a distance (1% of the unit length of the unit square) of the edges of any previous nucleated grains [5]. In continuous nucleation, to begin with, we select nucleation sites randomly, and then examine whether the nucleation site is already occupied by other growing grains or not. If occupied, nucleation is neglected and we continue to expand the grain that is already generated. Major factors which determine the morphology in continuous nucleation are nucleation rate and growth rate of seed. An important feature is that the nature of the structure does not change even if the rate between these factors varies. Structures for large and small grain sizes are equivalent in topology and differ only by a scaling factor. This feature has already been studied and the average grain size of the Johnson-Mehl model was given by Gilbert [10]. Based on this criteria, we employ an appropriate rate for seeds with different shapes in order to generate about hundred of grains in a continuous nucleation.
0~
(a) Seeds dispersed
(b) Expanding(initial)
(c) Expanding(final)
(d) Final structure
Fig. 2. Procedure to form grain microstructure in simultaneous nucleation (seed: ellipse, aspect ratio = 0.5).
OSAMU ITO and FULLER: COMPUTER MODELLING OF MICROSTRUCTURE
7-'
~ ~7'~/[~
193
OGn+t:CenterofGrainG.+t (Nucleation Site)
Distance between Grain Gn
andGrainGn+t
GrainGn
/ Fig. 3. Definition of distance between grains. Regarding termination of grain growth, we need to calculate the distance between the growing point and the other grain surface that the growing grain will meet. A schematic of the procedure to terminate growth is shown in Fig. 3. First, a position Gx is calculated, which is an intersection of the line Gn(/) - OGn + 1 and Gn + l(k) - Gn + l(k + 1). In this study, we regard line Gx - Gn(j) as the distance between grain Gn and grain Gn + 1. And we assume that growth of that point is terminated when this distance is 0.2% of the unit length of the unit square. 2.2. Analysis o f morphology
Various seeds employed in this study are shown in Table 2. Elliptical and polygon-like shape were employed and six different macrostructures were produced. In order to estimate the morphology of simulated microstructure quantitatively, we employed some parameters such as the aspect ratio distribution, grain size distribution and number of grain edges. A schematic diagram to determine the
aspect ratio of each grain is shown in Fig. 4. Regarding the nucleation site as the centre of a grain, the aspect ratio of the grain is the number got by dividing the minimum diameter by the maximum one. Regarding the number of grains, one may wonder how many grains would be enough so as to analyze microstructures quantitatively. In order to estimate the appropriate number, we examine the aspect ratio distribution of simultaneous nucleation case with ellipse-like seeds by increasing the number of grains. The aspect ratio of ellipse employed here is 0.3 and the number of grains ranges from 100 to 3500. Figure 5 shows the aspect ratio distribution when varying the number of grains. The number in parentheses indicates that of inner grains, while the number before the parenthesis is that of all grains generated. When analyzing morphology, we neglected grains facing outside and picked only inner grains for the quantitative calculation.
Table 2. Sha )e of seeds employed in this study ation Simultaneous
Continuous
Ellipse Aspect ratio= 1.0, 0.5, O.3 Nucleation Site
Octagon, Hexagon, Triangle
Aspect Ratio =
D min D ma~--~
Fig. 4. Calculation of aspect ration of each grain.
194
OSAMU ITO and FULLER: COMPUTER MODELLING OF MICROSTRUCTURE 30 25 20 t--
u.
.,I,.
15
10
100 (64) 500 (395) 1000 (813) 2500(2069) 3500(2899)
Aspect ratio =
00'0
0.2
0.4
0.6
0.8
0.3
1.0
Aspect Ratio Distribution
Fig. 5. Aspect ratio distribution in simultaneous nucleation of ellipse. In the case when the number of grains is more than 1000, no significant difference is seen between the distributions of aspect ratio. Based on this result, more than 1200 inner grains were generated for the following quantitative analysis of microstructures.
3. RESULT AND DISCUSSION 3.1. Simultaneous nucleation
Figure 6 shows one of the simulated microstructures in simultaneous nucleation with ellipse-like seeds. The aspect ratio of the seeds ranges from 0.3 to 1.0. The numbers in parentheses indicates that of inner grains, while the number before the parenthesis is that of all grains generated within the square in this figure. Figure 7 represents the frequency distribution for aspect ratio, grain size and number of grain edges. Number indicating here is that of inner grains. In terms of the grain size distribution, average grain size
is normalized according to the grain number and the value is fixed at one in every case. In the left figure, as the aspect ratio of the seeds decreases, its peak frequency of the aspect ratio moves left to a lower value. However, regarding grain size distribution, no significant difference in distribution is seen among figures. In addition, the same tendency is seen in the distribution of grain edge number even when the aspect ratio is varied. These results indicate that, in a microstructure with ellipse-type seeds, the aspect ratio distribution can be controlled but not the distribution of grain size and number of grain edges even when the aspect ratio of seeds are different. Figure 8 shows microstructures with polygon type seeds. Octagons and hexagons are slightly isotropic seeds and microstructures generated are similar to the Voroni case to some extent. A specific feature in this case is that they have a wavy boundary. In the triangle type of seeds, the situation is rather different and the grain has some acute angles. As shown in Fig. 9, roughly speaking, morphologic parameters do
(I) r=l.0
(2) r=0.5
N=100 (66)
N= 100 (66)
(3) r=0.3 N= 100 (67)
Fig. 6. Microstructure in simultaneous nucleation of ellipses.
r=l.0 N=1260 r=0.5 N=1265 r=0.3 N=2069
•
~ ~
Octagon N=1261 -.--o--- Hexagon N--1271 Triangle N=1664
r:Aspect ratio N:Number of inner grains
35
35
30
30
25
25
2o
20
~- 15 "
12r
15
u_
10
10
N:Number of inner grains
5
5 -
0.4 0.6 0.8 Aspect Ratio Distribution [a) ~.2
0#.0
1-.0
0
0
06
08
10
Aspect Ratio Distribution (o)
25
25
15
5 UI w
1
AA n
2
3
4
5
GS--
Grain Size Distribution
1
2
3
4
5
Grain Size Distribution
(b)
(b)
35
35
30
30
~
,~ 25
25
~ 20
2O
u_ 10 5
5
0 0 "~ ~ 3 4 5 6 7 8 9 101112131415
00 1 2 3 4 5 6 7 8 9 -1011 = 121314 1
Number of Grain Edge
Number of Grain Edge (c)
(c) Fig. 7. Quantitative estimation of grain morphology in simultaneous nucleation of ellipse.
(1) Octagon
Fig. 9. Quantitative estimation of grain morphology in simultaneous nucleation of polygons.
(2) Hexagon
©
N=100(67)
(3) S I c
N= 100(66)
Fig. 8. Microstructure in simultaneous nucleation of polygons. 195
N= 100(66)
196
OSAMU ITO and FULLER: COMPUTER MODELLING OF MICROSTRUCTURE
(1) r=l.0
(2) r=0.5
N=97(63)
N= 131(90)
(3) r=0.3 N=126(83)
Fig. 10. Microstructure in continuous nucleation of ellipses. In the grain size distribution, it is obvious that distribution range is much wider than that of the simultaneous nucleation. An important result in this figure is that no significant difference appears even if the aspect ratio of seeds is changed. The same is true of the grain edge number. When employing polygon type of seed in Fig. 13, no significant difference is seen between different shape of seeds as well as the case in the simultaneous nucleation. The reason is attributed in the same way as the simultaneous nucleation.
not show any particular difference in microstructures with polygon type seeds. The reason is that these seeds have no specific preferential direction in growing and the factor which mainly affects on the morphologic parameters employed in this study is anisotropic growth of seeds.
3.2. Continuous nucleation In continuous nucleation, the same type of seeds in simultaneous nucleation are employed. In Fig. 10, seeds are ellipses and in Fig. 11 seeds are polygons. In continuous nucleation, the number of grains can not be fixed before the simulation. Therefore, we used an appropriate initial parameter that determines grain number in order to generate approximately 1500 grains. Analysis of morphology was carried out within the inner grains in the same way as the simultaneous nucleation. Results are summarized in Figs 12 and 13. Regarding distribution of the aspect ratio, the distribution can be easily controlled in case of the seed ellipses.
(1) Octagon
0
N-100(66)
3.3. Correlative relation of morphology parameters From these results, we concluded that the aspect ratio of each grain is generally controllable when varying the seed shapes. However, the distribution of grain size or grain edge number is not easily changed. Therefore, we focussed on the relationship of these morphologic parameters by picking a hundred grains. Figure 14 shows correlative relation of grain morphology in continuous nucleation of ellipses. No
(2) Hexagon
0
N= 100(66)
(3) Triangle
A
N= 105(73)
Fig. 11. Microstructure in continuous nucleation of polygons.
r=l.0 N=1324 r=0.5 N=1435 r=0.3 N=1731
•
g"
Octagon N=1340 HexagonN=1373 N:Numberof innergrains TriangleN=1796
r:Aspectratio N:Numberof inner grains
•
35
35
30 25
3O
20
cr
15
LJ -
10 5 O
°~.o
02
04
06
08
~
0.0
10
0.2
'
'
'
0.4
'
0.6
'
'
0.8
--~
1.0
Aspect RatioDistribution (a)
Aspect RatioDistribution (o) 20
20 15
m
5 --
0
2
3
4
~ .
O-
-
Grain Size Distribution (b)
~'4-
5
Grain Size Distribution (b) 25
25 i
20 ~
~ 20
~ 15
15
¢.)
U_
It
5
5 0 1 2 3 4 5 6 78
0 1 2 3 4 5 6 7 8 9 101112131415
9 10111213141
Number of Grain Edge
Number of Grain Edge
(c)
(c)
Fig. 13. Quantitative estimation of grain morphology in continuous nucleation of polygons.
Fig. 12. Quantitative estimation of grain morphology in continuous nucleation of ellipses.
14
14 ~) 12 o) 10 no 8 uJ 6 t'-(5
2 0
~
12
** o
e o o
o o
*
*
e*
~
~ e e
5
12 o) 10
r=l.0 • oe ooaee~ ~ ~ e ~ ooe * e
t" •~
;*
r=l.0 o
6 D~ 4 I* 0
o
12
r=0.3
.
•
,
o,
6
~ Z
6 4 2 0 0
..'+®°° ~..
•
=E
e--
i!.
::
, o a oo~p
il + ~
01
Aspect Ratio
1.0
0
1
2
r=0.3
.
(5
-**.~**-.- ,,
0.2 0.4 0.6 0.8
:. °.."
0!*- *'
r=0.3
.;:
. r.?0
3
Grain Size
4
0
~o
a
o o
o
,~*~ * ~,*,~
.~
0.2 0.4 0.6 0.8
Aspect Ratio
Fig. 14. Correlative relationships of grain morphology in continuous nucleation of ellipses. 197
1.0
198
OSAMU ITO and FULLER: COMPUTER MODELLING OF MICROSTRUCTURE
strong relationships are seen in grain edge number vs aspect ratio and grain size vs aspect ratio. No matter which aspect ratio they may have, the bigger the grain size becomes, the more the grain edge number increases. This is not a particular result in this study, the same tendencies are always seen in cases using polygons. Before the simulations, we expected a different grain shape might effect morphology parameters such as grain size distribution or grain edge number, especially with ellipse type of seeds. In spite of this expectation, there is no significant effect when varying the aspect ratio of ellipse types of seeds or employing different shape of polygons in both simultaneous and continuous nucleation cases.
even when seeds have different shape. Regarding correlative relation of grain morphology, strong correlation is recognized between grain size distribution and distribution of grain edge number. This means that the bigger the grain size becomes, the more the grain edge number increases, while no strong relationships are seen in grain edge number vs aspect ratio and grain size vs aspect ratio. Acknowledgements--This work was done when one of the
authors (O. I.) was a guest scientist in NIST. We would like to acknowledge the many who supported us and gave stimulating suggestions, especially, Mark Vaudin, John E. Blendell, Jay S. Wallace and W. Craig Carter.
REFERENCES
4. CONCLUSION In order to generate anisotropic grain microstructures, several microstructures based on geometrical microstructure known as the Voronoi and Johnson-Mehl models have been simulated in two-dimensions. In terms of anisotropic factor, various shapes of seeds, ellipse and polygon, were employed. In addition, morphology such as aspect ratio distribution, grain size distribution and number of grain edges in each case was analyzed. The results are summarized as follows. When varying the aspect ratio of seeds, the resultant aspect ratio distribution of microstructure can be controlled, but not grain size and grain edge number,
1. M. P. Anderson, D. J. Srolovitz, G. S. Grest and P. S. Sahni, Acta metall. 32, 783 (1984). 2. J. Wejchert, D. Weaire and J. P. Kermode, Phil. Mag. B 53, 15 (1986). 3. M. P. Anderson, G. S. Grest and D. J. Srolovitz, Phil. Mag. B 59, 293 (1989). 4. A. Getis and B. Boots, Model of Spatial Processes. Cambridge Univ. Press (1979). 5. K. W. Mahin, K. Hanson and J. W. Morris Jr, Acta metall. 28, 443 (1980). 6. H. J. Frost and C. V. Thompson, Acta rnetall. 35, 529 (1987). 7. K. Yamada, J. Am. Ceram. Soc. 73, 2103 (1990). 8. N. Ryun and O. Hunderi, Acta metall. 37, 1375 (1989). 9. T. O. Saetre, O. Hunderi and N. Ryun, Acta metall. 37, 1381 (1989). 10. E. N. Gilbert, Ann. Math. Statist. 33, 958 (1962).