Materials Transactions, Vol. 48, No. 10 (2007) pp. 2762 to 2767 #2007 The Japan Institute of Metals
EXPRESS REGULAR ARTICLE
Evaluating Spatial Distribution Randomness of Overlap Permissive Second Phase in Three- and Two-Dimensions Kenjiro Sugio*1 , Yasutaka Momota*2 , Di Zhang*2 , Hiroshi Fukushima and Osamu Yanagisawa Mechanical System Engineering, Graduate School of Engineering, Hiroshima University, Higashi-Hiroshima 739-8527, Japan Three-dimensional local number, LN3D, and two-dimensional local number, LN2D, were defined as the number of gravity centers (GCs) of second phase particles in the measuring sphere and circle with specially determined radiuses, respectively, whose centers were put on GCs of noticed particles. LN3D and LN2D represent local number density including a noticed and its neighboring particles and each particle has a specific value. We suggested the quantitative method to evaluate the particle spatial distribution using the relative frequency distributions of LN3D and LN2D, and this method was examined by computer experiments using overlap permissive spheres. It was shown that randomness of second phase particles was correctly evaluated in 3- and 2-dimensions by this method, and the average and variance of LN3D and LN2D are proper descriptors to evaluate the spatial distribution randomness of second phase particles. It was also shown that spatial distribution randomness of 2-dimensional particles appeared on cut planes of 3-dimensional particles having uniform random arrangement can be evaluated by this method regardless of both the particle volume fraction and the particle size distribution. [doi:10.2320/matertrans.MER2007125] (Received June 4, 2007; Accepted August 10, 2007; Published September 12, 2007) Keywords: spatial distribution, randomness, second phase, particle reinforced metal matrix composite, computer experiment
1.
Introduction
Particle reinforced metal matrix composites (PMMCs) are now emerging as an important class of engineering materials due to their superior mechanical properties and amenability to the conventional metal working processes. Over the last decade, experimental and numerical studies1–11) have been indicated that the spatial distribution of second phase particles played an important role on the mechanical properties. Tessellation techniques,12,13) correlation functions,14–16) nearest neighbor distances17–19) contact distribution20) and counting methods21) have been suggested to describe the spatial distribution of second phase particles. However, the developments of techniques for characterizing the spatial distribution of second phase particles have not been enough. The descriptors for the spatial distribution of particles have to be sensitive to their arrangements. Simple scalar descriptors such as volume fraction have been successful in understanding and predicting material behaviors of PMMCs using the simple formula, ‘rule of mixtures’. Therefore, simple scalar descriptors for the spatial distribution, which links to material behaviors, are desired. In general, the spatial distribution of second phase particles is experimentally evaluated in 2-dimension by analyzing scanning electron microscope (SEM) or optical microscope (OM) images. The spatial distribution in 3-dimension has been also evaluated by serial sectioning method10,22) or synchrotron X-ray micro-tomography,23) but these techniques are not common. Computer modeling is a powerful tool for analyzing the spatial distribution of second phase particles in 3-dimension, and some researchers14–17,24) have investigated the statistical relationship between 3- and 2dimensional spatial distributions. If the simple statistical relationship between 3- and 2-dimensional spatial distributions is derived, material behaviors of PMMCs may be *1Corresponding *2Graduate
author, E-mail:
[email protected] Student, Hiroshima University
predicted only by the 2-dimensional spatial distribution, which is evaluated by analyzing SEM or OM images. We have employed the average of 2-dimensional local number, LN2D, as a spatial distribution descriptor for aluminum–silicon carbide (Al-SiC) composites, and indicated that this descriptor was sensitive to the spatial distribution of second phase particles.25) We have also investigated the relation between the average of LN2D and tensile behavior. However, the formulation of this relation has not been completed because statistical properties of LN2D have not been interpreted. In the present work, we have individually defined 3dimensional local number, LN3D, and 2-dimensional local number, LN2D, and suggested the quantitative method to evaluate the particle spatial distribution using the relative frequency distributions of LN3D and LN2D. This method was examined by computer experiments using overlap permissive spheres of uniform random arrangement. 2.
Procedure of Computer Experiments
2.1 Definitions of LN3D and LN2D At first, let us assume that gravity centers (GCs) of second phase particles are arranged in the closed-packed, facecentered, structure in 3-dimension as shown in Fig. 1(a). We defined the measuring sphere with center at GC of a noticed particle and a radius, R3D , where R3D is given so that the measuring sphere includes GCs of the nearest neighbor particles. In this definition, the number of GCs included in the measuring sphere is equal to 13 and the number density per unit volume is equal to 13=ð4R3D 3 =3Þ. Then, the measuring radius, R3D , is determined so that the number density of the measuring spheres is equal to the whole number density, V , and R3D is represented by 1=3 13 13 1:459 ) R ¼ ¼ 1=3 : ð1Þ ¼ V 3D 4R3D 3 =3 4V =3 V
Evaluating Spatial Distribution Randomness of Overlap Permissive Second Phase in Three- and Two-Dimensions
(a)
(b)
(c)
(d)
2763
Fig. 1 Ordered and random arrangements of gravity centers of second phase particles, (a) the FCC structure and (b) a random arrangement in 3-dimension and (c) the closest hexagonal structure and (d) a random arrangement in 2-dimension.
In the present work, the number of GCs included in the measuring sphere with the radius, R3D , is defined as 3dimensional local number, LN3D. When GCs are randomly distributed with keeping the number density V , the identical measuring sphere with the radius, R3D , is used for measurements as shown in Fig. 1(b), putting the centers of measuring sphere on GCs of particles. Secondly, let us assume that GCs of second phase particles are arranged in the closest hexagonal structure in 2dimension as shown in Fig. 1(c). We defined the measuring circle with center at GC of a noticed particle and a radius, R2D , where R2D is given so that the measuring circle includes GCs of nearest neighbor particles. In this definition, the number of GCs included in the measuring circle is equal to 7 and the number density per unit area is equal to 7=R2D 2 . Then, the measuring radius, R2D , is determined so that the number density of the measuring circle is equal to the whole number density, A , and R2D is represented by 7 7 1=2 1:493 ¼ A ) R2D ¼ ¼ 1=2 : ð2Þ R2D 2 A A In the present work, the number of GCs included in the measuring circle with radius, R2D , is defined as 2-dimensional local number, LN2D. When GCs are randomly distributed with keeping the number density A , the identical measuring sphere with the radius, R2D , is used for measurements as shown in Fig. 1(d).
Since the center of the measuring window (sphere and circle) is located at GCs of particles both in 3- and 2dimensions, each particle has one specific value of LN3D or LN2D, which is regarded as one of characteristics of a noticed particle similar to particle-size or particle-shape factor and so on. Then we consider that the evaluation of spatial distribution is possible by observing the frequency distributions of LN3D or LN2D as well as using size distribution to characterize the over all particle characteristics, for example. 2.2 Measurements of LN3D and LN2D Figure 2 shows the 3-dimensional computational space to arrange second phase particles. In order to eliminate errors close to the outer surfaces of the space, this computational space consists of the measurement- and the experimental domain. The measurement domain is included in the experimental domain. The size of the measurement domain is L0 L0 L0 , where L0 is non-dimensional length. The size of the experimental domain was extended to ensure that the measuring spheres and circles on cut planes stayed within the experimental domain. The extended length was 1.2 times as large as the radius of the measuring sphere. Since the particles were distributed in the experimental domain but LN3D and LN2D were measured only for the noticed GCs in the measurement domain, this extension could eliminate errors near surfaces.
2764
K. Sugio, Y. Momota, D. Zhang, H. Fukushima and O. Yanagisawa
2.0 Ratio of particle number
1.8
z
1.6 1.4 1.2 1.0 0.8
SID 1 SID 2 SID 3
0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 d0 Ratio of particle size
y x
Fig. 3
Measurement domain
Experimental domain
Fig. 2 The 3-dimensional computational space to arrange second phase particles.
Table 1
Nine models evaluated in the present calculation. Extended volume fraction, Vex
Particle diameter, d0
Size distribution
Model 1 Model 2
0.01 0.4
0.015L0 0.05L0
SID 1 SID 1
Model 3
3.14
0.1L0
SID 1
Model 4
0.01
0.015L0
SID 2
Model 5
0.4
0.05L0
SID 2
Model 6
3.14
0.1L0
SID 2
Model 7
0.01
0.015L0
SID 3
Model 8
0.4
0.05L0
SID 3
Model 9
3.14
0.1L0
SID 3
We assumed that the shape of second phase particles was spherical and overlaps of particles were permitted. The number of second phase particles in the measurement domain, N0 , was set at 6000 as setting value. The GCs of second phase particles were positioned randomly in the experimental domain including the measurement domain by computer-generated random number, keeping the number density N0 =L0 3 . Three kinds of particle diameter, d0 , and three kinds of size distribution were evaluated as shown in Table 1, where d0 was given by the extended volume fraction, overlap permissive volume fraction, 3 4 d0 6Vex 1=3 3 N0 =L0 ) d0 ¼ L0 : ð3Þ Vex ¼ 3 2 N0 The shape of size distribution is shown in Fig. 3. The size distribution 1 (SID 1) is the mono-sized distribution and the diameter of all particles is set at d0 . The size distributions 2 and 3 (SID 2 and SID 3) have eleven classes of particle diameters, and the particle diameters are varied linearly. The maximum and the minimum diameters were set at 1:9d0 and 0:1d0 , respectively. The particle numbers of each class are
The size distributions of second phase particles.
also varied linearly, and the maximum and the minimum particle numbers were set at 1:9N0 =11 and 0:1N0 =11, respectively. Nine models (Model 1–Model 9) were totally evaluated in the present calculation, changing Vex and size distribution as shown in Table 1. Figures 4(a) and (b) show the particle configurations of Model 2 and Model 8. Mono-sized particles are randomly distributed in Fig. 4(a) and poly-dispersed particles are randomly distributed in Fig. 4(b). The measurement of LN3D was carried out in two iterations. In the first iteration, total number of GCs of second phase particles included in the measurement domain, NV , was counted. Whole number density per unit volume, V , was calculated by V ¼ NV =L0 3 , and R3D was calculated by eq. (1). In the second iteration, measuring sphere with the radius, R3D , was defined, and GCs included in each measuring sphere, whose center was positioned at the noticed GC, were counted. A given volume is not necessary in the measurement of LN3D because LN3D is measured only by GCs of second phase particles. Three series of the particle configurations (Config 1, Config 2 and Config 3) with different seeds of random number were evaluated in the present calculation. Because about 6000 data of LN3D were obtained in one calculation and eleven calculations were carried out for each series changing the seed of random number, data of LN3D more than 60000 were collected for each series. Then, the relative frequency distribution of LN3D was obtained finally. When Model 2 and Model 8 are cut by arbitrary planes, circles appear on the cut planes as shown in Figs. 4(c) and (d). We used these planes to measure LN2D. The measurement of LN2D was also carried out in two iterations. In the first iteration, total number of GCs of circles on cut planes, NA , and total area of cut planes, A, was measured, changing positions and normal vectors of cut planes randomly. Whole number density per unit area, A , was calculated by A ¼ NA =A, and R2D was calculated by eq. (2). In the second iteration, measuring circles with the radius, R2D , were defined for all GCs of circles on cut planes and GCs included in each measuring circle, whose center was positioned at the noticed GC, were counted. Positions and normal vectors of cut planes
Evaluating Spatial Distribution Randomness of Overlap Permissive Second Phase in Three- and Two-Dimensions
(a) Model 2
(b) Model 8
(c) Model 2 Cut plane
(d) Model 8 Cut plane
2765
Fig. 4 Particle configurations in the 3-dimensional computational space for (a) Model 2 and (b) Model 8, and circles on cut planes for (c) Model 2 and (d) Model 8.
used in the second iteration were identical to those in the first iteration. A volume is necessary for the measurement of LN2D. If particles don’t have a volume, circles will not appear on cut planes. As shown in Table 1, nine models were evaluated in the present calculation so as to investigate the effect of the particle volume on the relative frequency distribution of LN2D. The identical particle configuration in 3-dimension, Config 1, is employed for all models. Because about 6000 data of LN2D were obtained in one calculation and eleven calculations were carried out for each model, data of LN2D more than 60000 were collected for each model. Then, the relative frequency distribution of LN2D was obtained finally.
point number, k, is represented by
3.
The average and the variance of this modified Poisson distribution are 14 and 13, respectively. Figure 5 shows the probability distribution of eq. (5) and the relative frequency distributions of LN3D measured using three series of particle configurations (Config 1, Config 2 and Config 3), with changing the seeds of random number. Table 2 shows the
Results and Discussion
3.1 Relative frequency distribution of LN3D When the points are randomly distributed in 3-dimansion, the number of points in a window whose volume is WV at arbitrary position is Poisson distributed.26) Probability of the
PðkÞ ¼
ðV WV Þk expðV WV Þ k ¼ 0; 1; 2 : k!
ð4Þ
In the measurement of LN3D, we have defined so that the measuring sphere has the volume of WV ¼ 4R3D 3 =3, and then V WV ¼ 13 using eq. (1). When the centers of the measuring sphere are put on GCs of all particles, the measuring spheres always include one GC. Therefore, Poisson distribution shifts to the larger LN3D by 1, and we can expect that the probability of LN3D is represented by PðLN3D ¼ k þ 1Þ ¼
13k expð13Þ k!
k ¼ 0; 1; 2 :
ð5Þ
2766
K. Sugio, Y. Momota, D. Zhang, H. Fukushima and O. Yanagisawa
0.16 Theory Config 1 Config 2 Config 3
0.10 0.08
Relative frequency, Probability
Relative frequency, Probability
0.12
0.06 0.04 0.02 0.00
0
5
10
15
20
25
30
0.12 0.10 0.08 0.06 0.04 0.02 0.00
35
Theory Model 1 Model 2 Model 3 Model 4 Model 5 Model 6 Model 7 Model 8 Model 9
0.14
0
5
10
LN3D
15
20
LN2D
Fig. 5 The relative frequency distributions of LN3D measured for three particle configurations (Config 1, Config 2 and Config 3) comparing with the expected probability distribution.
Fig. 6 The relative frequency distributions of LN2D measured for nine models (Model 1–Model 9) comparing with the expected probability distribution.
Table 2 The average, LN3Dav , and the variance, LN3Dvar , of the expected probability distribution and the measured relative frequency distribution of LN3D.
Table 3 The average, LN2Dav , and the variance, LN2Dvar , of the expected probability distribution and the measured relative frequency distributions of LN2D. The whole number density of GCs of circles on cut planes, A , is also shown.
LN3Dav
LN3Dvar
Expected by eq. (5)
14
13
Config 1
14.030
12.914
Expected by eq. (7)
8
7
—
7.920
6.911
88.12
LN2Dav
LN2Dvar
A (number=L20 )
Config 2
14.004
13.015
Model 1
Config 3
13.988
12.987
Model 2
8.018
7.028
300.95
Model 3 Model 4
8.020 7.996
7.007 7.071
595.59 112.64
Model 5
8.024
7.068
384.56
Model 6
7.998
7.046
764.82
Model 7
7.850
6.942
58.57
Model 8
8.022
7.141
199.81
Model 9
8.007
6.977
397.99
average and the variance of the theoretical distribution expected by eq. (5) and the measured distributions. All the measured distributions agree with the distribution expected by eq. (5), the 3-dimensional uniform random arrangement. As a result, when GCs of second phase particles have the uniform random arrangement in 3-dimension, the relative frequency distribution of LN3D exhibits the modified Poisson distribution whose average and variance were 14 and 13, respectively. In other words, randomness of second phase particles in 3-dimension can be correctly evaluated by using the relative frequency distribution of LN3D, and the average and variance of LN3D are proper descriptors. 3.2 Relative frequency distribution of LN2D When the points are randomly distributed in 2-dimension, the number of points in a window whose area is WA at arbitrary position is Poisson distributed. Probability of the point number, k, is represented by PðkÞ ¼
ðA WA Þk expðA WA Þ k ¼ 0; 1; 2 : k!
ð6Þ
In the measurement of LN2D, we defined so that the measuring circle has the area of WA ¼ R2D 2 , and then A WA ¼ 7 using eq. (2). When the centers of the measuring circles are put on GCs of all particles, the measuring circle always includes one GC. Therefore, Poisson distribution is expected to shift to the larger LN2D by 1, and the probability of LN2D is expected to be PðLN2D ¼ k þ 1Þ ¼
7k expð7Þ k!
k ¼ 0; 1; 2 : ð7Þ
The average and the variance of this modified Poisson distribution are 8 and 7, respectively. Figure 6 shows the probability distribution of eq. (7) and the relative frequency distributions of LN2D measured using nine models (Model 1–Model 9). Table 3 shows the average and the variance of the theoretical distribution and the measured distributions. The whole number density of circles on cut planes per unit area, A , is also shown in Table 3. When the size of second phase particles is smaller, A decreases. When the size of second phase particles is larger, A increases. In spite of the difference in A , all the measured distributions agree with the expected eq. (7), the 2-dimensional uniform random arrangement. In the present method, the effects of particle-volume fraction or particle-size distribution do not appear on LN2D but appear on A , because size of the measuring window has been defined based on A , which has already included their effects. By observing the frequency of LN2D defined in this study, spatial distribution randomness of 2-dimensional particles appeared on sectional planes of 3-dimensional particles having the uniform random arrangement can be evaluated regardless of both the particle-volume fraction and particlesize distribution. Thus the randomness of second phase particles was correctly evaluated in 2-dimension using the
Evaluating Spatial Distribution Randomness of Overlap Permissive Second Phase in Three- and Two-Dimensions
relative frequency distribution of LN2D, and the average and variance of LN2D are proper descriptors. Since the present method of evaluating 3- and 2-dimensional spatial distributions has no concern with both the particle-volume fraction and particle-size distribution and there is the correspondence of parameters each other in the 3and 2- dimensions, we can simply judge the spatial distribution randomness only by analyzing SEM or OM images. 4.
Conclusions
Three-dimensional local number, LN3D, and two-dimensional local number, LN2D, were defined as the number of GCs of second phase particles in the measuring sphere and circle with specially determined radiuses, respectively, whose centers were put on GCs of noticed particles. The radiuses of the measuring sphere and circle were determined so that the number densities in the sphere and the circle, including GCs of 13 and 7 particles, correspond to the whole number density in 3-dimension and 2-dimension, respectively. One specific value of LN3D or LN2D is given to each particle and this is regarded as one of characteristics of a noticed particle similar to particle-size or particle-shape factor and so on. The quantitative method was proposed to evaluate the particle spatial distribution using the relative frequency of LN3D and LN2D, in 3- and 2-dimensions, respectively. This method was examined by computer experiments using overlap permissive spheres of uniform random distribution. When GCs of overlap permissive second phase particles have uniform random arrangement in 3-dimension, the relative frequency distribution of LN3D exhibits modified Poisson distribution with average of 14 and variance of 13 and that of LN2D for the 2-dimensional particles on cut planes exhibits modified Poisson distribution with average of 8 and variance of 7. Randomness of second phase particles was correctly evaluated in 3- and 2-dimensions using the relative frequency distribution of LN3D and LN2D, respectively, and the average and variance of them are proper descriptors for judgment of the randomness. Spatial distribution randomness of 2-dimensional particles appeared on cut planes of 3-dimensional ones having uniform random arrangement can be evaluated by the proposed
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