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Shape distinction of particulate materials by laser diffraction pattern analysis
Powder Technology 122 Ž2002. 205–211 www.elsevier.comrlocaterpowtec
Shape distinction of particulate materials by laser diffraction pattern analysis Hideo Yamamoto ) , Tatsushi Matsuyama, Masanori Wada Faculty of Engineering, Soka UniÕersity, 1-236 Tangi-cho, Hachioji, Tokyo 192-8577, Japan Received 3 July 2000; received in revised form 6 October 2000; accepted 6 November 2000
Abstract A pattern of a diffraction image depends on the particle shape, while the size of the pattern depends on the sectional area of the particle. In this work, the method to extract differences from the diffraction patterns due to different shapes of non-spherical particles was studied conceptually. In this respect, a radial segment Žwedge. photo-detector was assumed as a detector. Diffraction patterns and intensity patterns detected by the radial segment detector were calculated for many kinds of two-dimensional shapes, corresponding to the projections of particles, as a circle, ellipses, triangles, quadrangles, other anonymous shapes, also shapes extracted from real phytoplanktons. From these detected light intensity patterns, we extracted Žor define. two indexes: Acircular indexB and Apeak number.B It was shown that various shapes can be distinguished by means of two-dimensional mapping with these parameters. In addition, an applicability of a concentric detector was examined to estimate the particle size when the particle is non-spherical but is a single particle in the measurement. As a result, it was found that the circle equivalent diameter determined with usual scheme agreed well with the sectional area equivalent diameter of the original particle even in any cases of non-spherical samples. From these results, it was shown that the particle size and shape in wide range can be distinguished from the three-dimensional mapping with Acircular indexB, Apeak numberB and Aparticle sizeB. q 2002 Elsevier Science B.V. All rights reserved. Keywords: Particle characterization; Measurement of shape; Shape index; Fraunhofer diffraction
1. Introduction 1.1. Introduction In the case of Fraunhofer Žlaser. diffraction method at particle size analysis, an intensity pattern of diffraction from particles is measured with, usually, a concentric photo-detector: the intensity pattern obtained is assumed as a superimpose of contributions from each fraction of the size distribution: a reverse calculation with another assumption that the particle is spherical gives the size distribution. In this meaning, the result of the analysis is obtained as an equivalent diameter of spheres or its distribution. Even if the sample particle is non-spherical, the scheme of the detection and reverse calculation can work out in the very same way, and an equivalent diameter distribution is obtained although the diffraction pattern is not axi-symmetric in the case of non-spherical particle.
) Corresponding author. Tel.: q81-426-91-9454; fax: q81-426-919454. E-mail address: [email protected] ŽH. Yamamoto..
Actually, a pattern of a diffraction image depends on the particle shape, while the size of the pattern depends on the particle size. Only the latter information concerning particle size is used in the normal particle size analyzer. On the other hand, the fact that the pattern is not symmetry in the case of non-spherical particle implies also a possibility to recognize the shapes of particles from the diffraction pattern information. In this respect, there are two directions of studies: one is to make clear or interpret the relationship between the diffraction pattern obtained and the equivalent spherical size distribution in the case of non-spherical particle for a normal particle size analyzer, and the other is to use the non-symmetry pattern information from non-spherical particle directly to recognize the particle shape w1,2x. The latter subject, the shape distinction problem, is studied in this work, while the former was already discussed elsewhere w3x. Because of the range applicable for the normal Fraunhofer diffraction method at particle size analysis, from tens to hundreds of powder particles, crystals, a certain range of phytoplanktons will be supposed as samples to be distinguished. The problem addressed in this work will be discussed in detail in the next section.
0032-5910r02r$ - see front matter q 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 2 - 5 9 1 0 Ž 0 1 . 0 0 4 1 7 - X
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H. Yamamoto et al.r Powder Technology 122 (2002) 205–211
1.2. A concept of A appropriate extractionB of shape information There would be no necessity to state importance of the particle shape characterization. Every aspect at powder technology is affected by particle shape. Indeed studies on measurement of particle shape have a long history, and there are too many contributions concerning the field to have a brief review in this paper: proper reviews may be available elsewhere Že.g., Refs. w4,5x.. However, until now, powder dynamics influenced by particle shape quantitatively from such shape analysis cannot be predicted nor explained. Once we return to a conceptual discussion. For example, it can be stated that a complete shape information is, at least in the future, a set of all radial vectors representing particle surface for each particle w6x. However, this way requires a huge computational power. Our question is whether this is actually realistic or not. Here we would like to emphasize that it is important to discuss a proper balance between the amount of information and object of a quantitative discussion: how much information is necessary and satisfactory? The amount of information required may depend on each target, which could be discussions on static powder bed structure, particle dynamics in liquid and so on. Even if any kind of AcompleteB method to measure particle shape is established in the future, the information in actual use may yet be reduced into a few shape factors which contain a proper amount of information for each object under consideration. In this meaning, a concept of an appropriate extraction of shape information should be addressed for each purpose. Anyway, however, such total studies and discussions can be a big project in the future Žit is beyond the range of this paper.. In this work, the discussion is limited in a problem of shape distinction as an example realizing the concept. Suppose an on-line measurement for process control or a shape separation process. In some cases, it is necessary and satisfactory if some kinds of particle could be distinguished based on its shape. Even with such a limitation of the situation considered, the concept of an appropriate extraction works. In this paper, as mentioned in the Introduction, a shape distinction problem using Fraunhofer diffraction pattern is studied. If the Fraunhofer condition is satisfied, a light diffraction intensity pattern obtained from a particle is a two-dimensional Fourier transformation, square of its absolute value in strict, of a projection of the particle as well known Že.g., Refs. w7,8x.. Because the pattern is a kind of transformation, the total information held is almost the same with the direct imaging of the particle projection. Note that in the case of intensity pattern observation, its phase information is lost although the original Fourier transformation itself contains, therefore, direct reverse construction of the original image is not possible principally; yet a special numerical iteration technique to achieve the re-construction of the approximated image is studied w9x.
Note that our aim here is not such a re-construction of the original image. If so, direct imaging should be adopted because of its simplicity and directness. The point is which type of information obtained is proper or easier for the appropriate extraction of the shape information against its aim, diffraction pattern or direct imaging. Because the aim in this work is shape distinction, the point can be re-stated as which information is proper to extract differences of the shape feature: for a distinction, it is enough to detect differences. To realize the feature extraction from a light diffraction pattern, in this work, a radial segment Žwedge. photo-detector with 32 elements is mainly assumed to be adopted. The detail will be described in the main part of this paper. If the detector, and analysis after the detection, works well for the aim to extract features successfully, the number of the elements, i.e., the necessary number of a set of raw data for one measurement, should be noted as small enough. In contrary, if we compare, direct imaging requires over hundred pixels for a single particle detection. Even if an image analysis after the detection reduces the information into a few shape factors, a certain amount of computation is required. For a quick measurement or a quick distinction, which can be integrated with an automatic shape sorting or separation, it is obviously preferred that the number of the elements or pixels and necessary calculation is reduced as small as possible. In this respect, the measurement of a light diffraction pattern with a small number of elements has a significant merit. In this work, an applicability of the method to extract differences from the diffraction patterns due to different particle shapes was studied conceptually. Diffraction patterns and intensity patterns detected by the radial segment detector supposed were calculated numerically for many kinds of two-dimensional shapes corresponding to the projections of particles, as a circle, ellipses, triangles, quadrangles, other anonymous shapes supposed as real powder, also shapes extracted from real phytoplanktons. From these detected light intensity patterns, two shape indexes, Acircular indexB and Apeak number,B were defined. It will be discussed whether such various shapes can be distinguished by means of two-dimensional mapping with these indexes. In this meaning, although the range of this work is limited in a conceptual design, a target of the work in the future is an integration of this principle with such as a flow cytometry or a cell-sorter assembly. As such a method for quick distinction of particle shapes, with Fourier optics, matched spatial filter method can be another direction w10x. 2. Calculations 2.1. Fraunhofer diffraction intensity pattern It is known that a diffraction pattern given by an opaque particle is approximately equivalent to that given by an aperture with the same shape of cross-section of the parti-
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the proportional factor is negligible. The intensity patterns were realized by numerical computation with parameters, l s 630 nm and f s 300 mm, for various particle shapes. In typical discussions, particle size was assumed as around 50 mm. 2.2. Detector
Fig. 1. A schematic illustration of the radial segment Žwedge. photo-detector.
cle in a Fraunhofer diffraction region w7,8x. Let g 0 Ž x 0 , y 0 . represent a transparency distribution of an aperture on an x 0 –y 0 plane, where 0 - g 0 - 1 represents from complete opaque to transplant. The Fraunhofer diffraction intensity pattern, I Ž x, y ., from the aperture is, with an optical system using Fourier lens of focus f, expressed as:
To extract features into shape factors from the diffraction patterns, a radial segment Žwedge. photo-detector with 32 elements w1,2x was supposed instead of a concentric circular detector which is used in usual particle size analyzer. A schematic view of the detector is shown in Fig. 1. In the main part of discussions, inner and outer diameter of the detector was supposed as 2 and 4 cm, respectively. Afterwards, a more detailed discussion on the specification, the size of the detector, will also be given. Each element detects an integrated photo intensity corresponding to its area on the diffraction pattern. Diffraction patterns and intensity patterns detected by the radial segment detector were calculated for many kinds of two-dimensional shapes, corresponding to the projections of particles.
Ž 1.
2.3. Definition of circular index and peak number as shape factors
where G 0 is a Fourier transform of g 0 . Because light intensity detected with elements supposed is normalized by their maximum in the actual analysis, the absolute value of
Fig. 2 shows typical examples of diffraction pattern and detected light intensity of the wedge detector for apertures of circle and square as simple geometric shapes. As shown,
I Ž x , y . A G0
ž
x
y
lf lf
2
/
Fig. 2. Typical examples of diffraction pattern and detected light intensity distribution of the wedge detector: Ža. circle and Žb. square as simple geometric shapes.
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a circle gives a concentric pattern while a square does not although it is point-symmetry. The photo intensity patterns on the radial segment detector was normalized by their maximum. From these light intensity pattern detected, we extracted and define two indexes: Acircular indexB and Apeak number.B ACircular indexB is determined as a sum, normalized by the number of the segment detectors, of the normalized intensities detected by each segment. Definitely, in the case of circle, the pattern is uniform and the index becomes unity ŽFig. 2.. For the other shapes, it varies from 0 to 1. From the definition, the circular index is given, in an actual measurement, as a ratio of the total intensity detected by all segments to the production of the maximum intensity measured by a certain segment and the number of the segments as: Circular index s
Ž total intensity detected by all segments. . Ž maximum intensity. Ž segment number. Ž 2.
Note that Eq. Ž2. implies very speedy detection and calculation in the actual measurement because of the simplicity of the index definition. APeak numberB can be recognized from the angular pattern of the intensity on the segment detector with a certain threshold. For example, 0 for a circle, 2 for ellipses, 4 for quadrangles, 6 for triangles.
3. Results and discussions 3.1. Shape distinction with two-dimensional mapping 3.1.1. Distinction of simple geometrical shape At first, distinction of rather simple shapes was examined. Simple shape here means geometrical shapes as a circle, ellipses, triangles, quadrangles, and their modifications. Fig. 3 shows two-dimensional mapping with circular index and peak number for the shapes. Horizontal axis and vertical axis represent circular index and peak number, respectively. In the figure, the shapes examined were illustrated as each plot. As mentioned, in the case of circle, circular index is unity and peak number is zero. In the
Fig. 4. Two-dimensional mapping with circular index and peak number for projections of real powder particles.
cases of ellipses, circular index decreased with increase of its aspect ratio, while peak number was constant as 2. In the cases of quadrangles, circular indexes were from 0.2 to 0.3 and peak number was 4. As shown, the shapes were distinguished successfully in the two-dimensional mapping according to their feature. 3.1.2. Shape distinction of real powder particle As a next step, shape distinction of projections of real powder particles was examined. Fig. 4 shows the result in the same way with Fig. 3. Also in this case, the shapes were distinguished successfully in the two-dimensional mapping according to their feature. In the cases of rod or ellipse like particles, peak number was 2. Even if a projection is similar with a circle, peak number did not became zero due to its small variation. Roundness of a particle seems to be determined with the circular index rather than peak number. If the particle shape is far from a circle, especially in the case of thin particle, the circular index takes smaller value. 3.1.3. Distinction of phytoplanktons As a trial, the two-dimensional mapping for projections of real phytoplanktons were also examined. Many kinds of phytoplanktons are in the range of its size from tens to hundreds of micrometer; therefore, applicability of this shape distinction method for plankton sorting based on its shape can be a kind of application and, consequently, an interesting issue. Fig. 5 shows the result. Many kinds of planktons could be distinguished successfully with the mapping. However, some planktons were attributed in a same range if that peak number was 4 and circular index was around 0.2–0.3. To separate these particles, another parameter may be available: it is simply the size of the particle. 3.2. Size measurement for a single particle
Fig. 3. Two-dimensional mapping with circular index and peak number for simple geometrical shapes.
As mentioned in the Introduction in detail, the particle size analysis with a laser diffraction method is influenced by particle shape. One major reason for this is that the
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Fig. 5. Two-dimensional mapping with circular index and peak number for projections of real phytoplanktons.
sample is supposed as a certain amount of group of particles with a size distribution: the size distribution is obtained with a reverse calculation. Here, suppose that a single particle is measured with a concentric detector as the same as normal particle size. Let us discuss how the size can be determined and how the result is influenced by the sample shape in this case. Because each element of concentric detector detects light intensity integrated with angle from 0 to 2p , detected in actual measurement is Žmean. radial distribution function as: dr
2p
H0
I Ž r ,u . d u .
Ž 3.
In the case of a spherical particle, the first peak of this distribution is maximum and the position, r, of it is determined by the particle diameter. This is a primitive reason why a particle size can be determined based on a measurement of the laser diffraction pattern w11x. In the case of non-spherical particles, if the measurement is carried out for a single particle, a kind of equivalent particle diameter for sphere may also be determined from the measurement of the position which gives the first peak in the radial distribution. Fig. 6 shows a relationship between the equivalent diameter for sphere determined in this way and Heywood diameter for some kinds of nonspherical particles. Their agreement is very well. Accordingly, a particle size can be also determined from the diffraction pattern measurement for non-spherical particles in this way. Fig. 7 shows a two-dimensional mapping of phytoplanktons, which is the same sample with that of Fig. 5, with circular index and equivalent particle diameter. With another mapping parameter, another distinction was achieved. The result indicates applicability of particle distinction with three-dimensional mapping, or two-dimensional mapping of a proper combination of parameters.
Fig. 6. A relationship between the equivalent diameter for sphere determined from the position of the first peak of radial distribution of light intensity and Heywood diameter for some kinds of non-spherical particles.
diameter and of the particle Žprojection. itself w4,5x. Fig. 8 shows a comparison between the conventional roundness factor and the circular index defined in this work for some geometrical shapes. While factors have a certain correlation, the circular index defined in this work has a wider range for the same sample. This may imply an effectiveness of this new shape factor because of its sensitivity. 3.3.2. Size of the segment detector As mentioned, a pattern of a diffraction image depends on the particle shape, while the size of the pattern depends on the sectional area of the particle. A smaller particle gives a larger pattern. Accordingly, even if the particle shape is the same, if the size of the particle is different, the
3.3. Detailed discussions on some specifications 3.3.1. Comparison of circular indexes A typical and conventional circular index or roundness factor is a ratio of perimeters of a circle with Heywood
Fig. 7. A two-dimensional mapping of phytoplanktons, which is the same sample as that of Fig. 7, with circular index and equivalent particle diameter.
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According to the original aim of this work, shape distinction, this stability can be within an allowable range. Also, it is implied that some options of detectors of different size are applicable to adopt or optimize the distinction process in an actual measurement.
Fig. 8. A comparison between the conventional roundness factor and the circular index defined in this work for some geometrical shapes.
detector measures a different part of the diffraction pattern with a fixed optical system. Consequently, the circular index defined in this work takes different value for the different size particle even if the shape is the same. This is a weak point of this shape factor. In this meaning, this shape factor cannot be defined in a strict sense, but it is an operational definition. However, according to our first aim, which is a shape distinction with a factor mapping method, a stability of the factor against particle size and applicable size range in this method should rather be discussed. Fig. 9 shows a relationship between Heywood diameter of a square and the circular index. A square aperture was adopted as a test example because it gives a typical anisotropic pattern as shown in Fig. 2. Here, some different sizes of the segment detector, with 10 or 20 mm inner diameter and with outer diameter from 40 to 100, were also examined. To compare the size, a schematic illustration showing the detector size was drawn in the figure. Although a square aperture is a typical example resulting anisotoropic diffraction pattern, a central area of the pattern is rather uniform as shown in Fig. 2. This is generally true for any shape aperture. Because a smaller particle gives a larger pattern as mentioned, the detector takes this rather uniform region in the cases of smaller particles. Consequently, as indicated in Fig. 9, the circular index takes a larger value in the cases of smaller particles. Because the detectors with a smaller inner diameter, 10 mm in this discussion, can detect this uniform region for a wider range of the particle size by the same reason, they give a larger circular index up to a bigger particle than the detectors with a bigger inner diameter. As discussed, the difference or perturbation of the circular index is resulted in by which part of the diffraction pattern is detected by the detector. A larger detector, therefore, was rather more stable for the change of the particle size. From the figure, the index was almost stable for each detector over 40 mm particle in this calculation while the absolute value of the index depended on the detector size.
3.3.3. Element number of the segment detector A stability of the circular index against particle rotation is discussed here. If a particle has a three-dimensional rotation, the shape of its projection itself changes. This problem is beyond the discussion here: it should be discussed elsewhere. Even if a projection does not change, and the projection rotates on the plane, the circular index varies in some cases. Especially in the case of a square aperture, this variation is worst big. This typical example is discussed here. Corresponding to the rotation of the particle Žprojection., the diffraction pattern rotates on the segment detector. According to Eq. Ž2., the circular index is, in this case, influenced by the maximum intensity detected on a certain element of the detector because the total intensity is the constant. As seen in Fig. 2, the diffraction pattern from a square aperture has four bright radials. In this case, the maximum intensity depends on how the bright radial is divided on the neighbor elements. Especially in the case of the strict square, the bright radials are orthogonal against each other. And note that 32, the number of the elements supposed here, is a multiple of 4, the number of the arms as bright radials. By this special situation, the division of the bright arms on the elements takes place completely the same for every arm. This causes the perturbation of the circular index of 50% in the worst case. However, this was not the case for, e.g., triangle. Because, in the case of triangle, the diffraction pattern has six bright arms, and 32 is not a multiple of 6. Therefore, some of the arms can be on an element fully, and consequently, the perturbation of
Fig. 9. A relationship between Heywood diameter of a square and the circular index. Some different sizes of the segment detector were also examined. To compare the size, a schematic illustration showing the detector size was drawn. Parameters indicate winner diameterrmm, outer diameterrmmx.
H. Yamamoto et al.r Powder Technology 122 (2002) 205–211
the maximum intensity is very little. For any other anonymous shapes, the situation is similar with the case of triangle. Additionally, if we adopt the element number 31, note that 31 is a prime number, it was confirmed that such perturbation does not take place also for a square. As a result, it was confirmed that the perturbation due to rotations of a projection can be avoided.
4. Conclusions A pattern of a diffraction image depends on the particle shape, while the size of the pattern depends on the sectional area of the particle. In this work, the method to extract differences from the diffraction patterns due to different shapes of non-spherical particles was studied conceptually. A radial segment Žwedge. photo-detector with 32 elements was assumed instead of a concentric circular detector, which is used in usual particle size analyzer. Each element detects an integrated photo intensity corresponding to the element area on the diffraction pattern. Diffraction patterns and intensity patterns detected by the radial segment detector were calculated for many kinds of two-dimensional shapes, corresponding to the projections of particles, as a circle, ellipses, triangles, quadrangles, other anonymous shapes, also shapes extracted from real phytoplanktons. The photo intensity patterns on the radial segment detector can be normalized by the maximum intensity. In the case of circle, the intensity pattern is uniform for the angle definitely. On the other hand, non-spherical Žnon-circular. particle causes various diffraction patterns. From these detected light intensity patterns, we extracted Žor define. two indexes: Acircular indexB and Apeak number.B ACircular indexB is determined as a sum, normalized by the number of the segment detectors, of the normalized intensities of each segment of the detector. Definitely in the case of circle, the pattern is uniform and the index becomes unity. For the other shapes, it varies from 0 to 1. APeak numberB can be recognized from the angular pattern of the intensity on the segment detector with a certain threshold. For example, 0 for a circle, 2 for ellipses, 4 for quadrangles, 6 for triangles. It
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was shown that various shapes can be distinguished by means of two-dimensional mapping with these parameters. In addition, an applicability of a concentric circular detector was examined to estimate the particle size when the particle is non-spherical but is a single particle in the measurement. As a result, it was found that the circle equivalent diameter determined with usual scheme agreed well with the sectional aria equivalent diameter of the original particle even in any cases of non-spherical samples. From these results, it was shown that the particle size and shape in wide range can be distinguished from the three-dimensional mapping with Acircular index,B Apeak numberB and Aparticle size.B In this respect, a concept of an appropriate extraction of shape information has been examined successfully.
Acknowledgements The authors are most grateful to Professor B. Scarlett, Delft University of Technology, for his kind and deep discussion, and also to Mr. T. Moriya, Soka University, for his technical support.
References w1x C.M.G. Heffels, D. Heitzmann, E.D. Hirleman, B. Scarlett, Part. Part. Syst. Charact. 11 Ž1994. 194. w2x C.M.G. Heffels, D. Heitzmann, E.D. Hirleman, B. Scarlett, Appl. Opt. 34 Ž1995. 6552. w3x T. Matsuyama, H. Yamamoto, B. Scarlett, Part. Part. Syst. Charact. 17 Ž2000. 41. w4x S. Endo, J. Soc. Powder Technol., Jpn. 29 Ž1992. 854. w5x S. Endo, J. Soc. Powder. Technol., Jpn. 35 Ž1998. 383. w6x B. Scarlett, personal communication. w7x M. Born, E. Wolf, Principles of Optics. Pergamon, London, 1959. w8x K. Murata, Optics. Science-sya, Tokyo, 1979. w9x J. Miao, P. Charalambous, J. Kirz, D. Sayre, Nature 400 Ž1999. 342. w10x I. Shimizu, M. Ohtake, S. Asakawa, S. Kamemaru, Trans. Jpn. Soc. Mech. Eng., Ser. C 55 Ž1989. 1235. w11x S. Hayashi, Particle Size Analysis and Technology, ed. Soc. Powder Technol. Jpn., Nikkankogyoshinbun-sya, Tokyo Ž1994. 149.
Shape distinction of particulate materials by laser diffraction pattern analysis Hideo Yamamoto ) , Tatsushi Matsuyama, Masanori Wada Faculty of Engineering, Soka UniÕersity, 1-236 Tangi-cho, Hachioji, Tokyo 192-8577, Japan Received 3 July 2000; received in revised form 6 October 2000; accepted 6 November 2000
Abstract A pattern of a diffraction image depends on the particle shape, while the size of the pattern depends on the sectional area of the particle. In this work, the method to extract differences from the diffraction patterns due to different shapes of non-spherical particles was studied conceptually. In this respect, a radial segment Žwedge. photo-detector was assumed as a detector. Diffraction patterns and intensity patterns detected by the radial segment detector were calculated for many kinds of two-dimensional shapes, corresponding to the projections of particles, as a circle, ellipses, triangles, quadrangles, other anonymous shapes, also shapes extracted from real phytoplanktons. From these detected light intensity patterns, we extracted Žor define. two indexes: Acircular indexB and Apeak number.B It was shown that various shapes can be distinguished by means of two-dimensional mapping with these parameters. In addition, an applicability of a concentric detector was examined to estimate the particle size when the particle is non-spherical but is a single particle in the measurement. As a result, it was found that the circle equivalent diameter determined with usual scheme agreed well with the sectional area equivalent diameter of the original particle even in any cases of non-spherical samples. From these results, it was shown that the particle size and shape in wide range can be distinguished from the three-dimensional mapping with Acircular indexB, Apeak numberB and Aparticle sizeB. q 2002 Elsevier Science B.V. All rights reserved. Keywords: Particle characterization; Measurement of shape; Shape index; Fraunhofer diffraction
1. Introduction 1.1. Introduction In the case of Fraunhofer Žlaser. diffraction method at particle size analysis, an intensity pattern of diffraction from particles is measured with, usually, a concentric photo-detector: the intensity pattern obtained is assumed as a superimpose of contributions from each fraction of the size distribution: a reverse calculation with another assumption that the particle is spherical gives the size distribution. In this meaning, the result of the analysis is obtained as an equivalent diameter of spheres or its distribution. Even if the sample particle is non-spherical, the scheme of the detection and reverse calculation can work out in the very same way, and an equivalent diameter distribution is obtained although the diffraction pattern is not axi-symmetric in the case of non-spherical particle.
) Corresponding author. Tel.: q81-426-91-9454; fax: q81-426-919454. E-mail address: [email protected] ŽH. Yamamoto..
Actually, a pattern of a diffraction image depends on the particle shape, while the size of the pattern depends on the particle size. Only the latter information concerning particle size is used in the normal particle size analyzer. On the other hand, the fact that the pattern is not symmetry in the case of non-spherical particle implies also a possibility to recognize the shapes of particles from the diffraction pattern information. In this respect, there are two directions of studies: one is to make clear or interpret the relationship between the diffraction pattern obtained and the equivalent spherical size distribution in the case of non-spherical particle for a normal particle size analyzer, and the other is to use the non-symmetry pattern information from non-spherical particle directly to recognize the particle shape w1,2x. The latter subject, the shape distinction problem, is studied in this work, while the former was already discussed elsewhere w3x. Because of the range applicable for the normal Fraunhofer diffraction method at particle size analysis, from tens to hundreds of powder particles, crystals, a certain range of phytoplanktons will be supposed as samples to be distinguished. The problem addressed in this work will be discussed in detail in the next section.
0032-5910r02r$ - see front matter q 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 2 - 5 9 1 0 Ž 0 1 . 0 0 4 1 7 - X
206
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1.2. A concept of A appropriate extractionB of shape information There would be no necessity to state importance of the particle shape characterization. Every aspect at powder technology is affected by particle shape. Indeed studies on measurement of particle shape have a long history, and there are too many contributions concerning the field to have a brief review in this paper: proper reviews may be available elsewhere Že.g., Refs. w4,5x.. However, until now, powder dynamics influenced by particle shape quantitatively from such shape analysis cannot be predicted nor explained. Once we return to a conceptual discussion. For example, it can be stated that a complete shape information is, at least in the future, a set of all radial vectors representing particle surface for each particle w6x. However, this way requires a huge computational power. Our question is whether this is actually realistic or not. Here we would like to emphasize that it is important to discuss a proper balance between the amount of information and object of a quantitative discussion: how much information is necessary and satisfactory? The amount of information required may depend on each target, which could be discussions on static powder bed structure, particle dynamics in liquid and so on. Even if any kind of AcompleteB method to measure particle shape is established in the future, the information in actual use may yet be reduced into a few shape factors which contain a proper amount of information for each object under consideration. In this meaning, a concept of an appropriate extraction of shape information should be addressed for each purpose. Anyway, however, such total studies and discussions can be a big project in the future Žit is beyond the range of this paper.. In this work, the discussion is limited in a problem of shape distinction as an example realizing the concept. Suppose an on-line measurement for process control or a shape separation process. In some cases, it is necessary and satisfactory if some kinds of particle could be distinguished based on its shape. Even with such a limitation of the situation considered, the concept of an appropriate extraction works. In this paper, as mentioned in the Introduction, a shape distinction problem using Fraunhofer diffraction pattern is studied. If the Fraunhofer condition is satisfied, a light diffraction intensity pattern obtained from a particle is a two-dimensional Fourier transformation, square of its absolute value in strict, of a projection of the particle as well known Že.g., Refs. w7,8x.. Because the pattern is a kind of transformation, the total information held is almost the same with the direct imaging of the particle projection. Note that in the case of intensity pattern observation, its phase information is lost although the original Fourier transformation itself contains, therefore, direct reverse construction of the original image is not possible principally; yet a special numerical iteration technique to achieve the re-construction of the approximated image is studied w9x.
Note that our aim here is not such a re-construction of the original image. If so, direct imaging should be adopted because of its simplicity and directness. The point is which type of information obtained is proper or easier for the appropriate extraction of the shape information against its aim, diffraction pattern or direct imaging. Because the aim in this work is shape distinction, the point can be re-stated as which information is proper to extract differences of the shape feature: for a distinction, it is enough to detect differences. To realize the feature extraction from a light diffraction pattern, in this work, a radial segment Žwedge. photo-detector with 32 elements is mainly assumed to be adopted. The detail will be described in the main part of this paper. If the detector, and analysis after the detection, works well for the aim to extract features successfully, the number of the elements, i.e., the necessary number of a set of raw data for one measurement, should be noted as small enough. In contrary, if we compare, direct imaging requires over hundred pixels for a single particle detection. Even if an image analysis after the detection reduces the information into a few shape factors, a certain amount of computation is required. For a quick measurement or a quick distinction, which can be integrated with an automatic shape sorting or separation, it is obviously preferred that the number of the elements or pixels and necessary calculation is reduced as small as possible. In this respect, the measurement of a light diffraction pattern with a small number of elements has a significant merit. In this work, an applicability of the method to extract differences from the diffraction patterns due to different particle shapes was studied conceptually. Diffraction patterns and intensity patterns detected by the radial segment detector supposed were calculated numerically for many kinds of two-dimensional shapes corresponding to the projections of particles, as a circle, ellipses, triangles, quadrangles, other anonymous shapes supposed as real powder, also shapes extracted from real phytoplanktons. From these detected light intensity patterns, two shape indexes, Acircular indexB and Apeak number,B were defined. It will be discussed whether such various shapes can be distinguished by means of two-dimensional mapping with these indexes. In this meaning, although the range of this work is limited in a conceptual design, a target of the work in the future is an integration of this principle with such as a flow cytometry or a cell-sorter assembly. As such a method for quick distinction of particle shapes, with Fourier optics, matched spatial filter method can be another direction w10x. 2. Calculations 2.1. Fraunhofer diffraction intensity pattern It is known that a diffraction pattern given by an opaque particle is approximately equivalent to that given by an aperture with the same shape of cross-section of the parti-
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the proportional factor is negligible. The intensity patterns were realized by numerical computation with parameters, l s 630 nm and f s 300 mm, for various particle shapes. In typical discussions, particle size was assumed as around 50 mm. 2.2. Detector
Fig. 1. A schematic illustration of the radial segment Žwedge. photo-detector.
cle in a Fraunhofer diffraction region w7,8x. Let g 0 Ž x 0 , y 0 . represent a transparency distribution of an aperture on an x 0 –y 0 plane, where 0 - g 0 - 1 represents from complete opaque to transplant. The Fraunhofer diffraction intensity pattern, I Ž x, y ., from the aperture is, with an optical system using Fourier lens of focus f, expressed as:
To extract features into shape factors from the diffraction patterns, a radial segment Žwedge. photo-detector with 32 elements w1,2x was supposed instead of a concentric circular detector which is used in usual particle size analyzer. A schematic view of the detector is shown in Fig. 1. In the main part of discussions, inner and outer diameter of the detector was supposed as 2 and 4 cm, respectively. Afterwards, a more detailed discussion on the specification, the size of the detector, will also be given. Each element detects an integrated photo intensity corresponding to its area on the diffraction pattern. Diffraction patterns and intensity patterns detected by the radial segment detector were calculated for many kinds of two-dimensional shapes, corresponding to the projections of particles.
Ž 1.
2.3. Definition of circular index and peak number as shape factors
where G 0 is a Fourier transform of g 0 . Because light intensity detected with elements supposed is normalized by their maximum in the actual analysis, the absolute value of
Fig. 2 shows typical examples of diffraction pattern and detected light intensity of the wedge detector for apertures of circle and square as simple geometric shapes. As shown,
I Ž x , y . A G0
ž
x
y
lf lf
2
/
Fig. 2. Typical examples of diffraction pattern and detected light intensity distribution of the wedge detector: Ža. circle and Žb. square as simple geometric shapes.
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a circle gives a concentric pattern while a square does not although it is point-symmetry. The photo intensity patterns on the radial segment detector was normalized by their maximum. From these light intensity pattern detected, we extracted and define two indexes: Acircular indexB and Apeak number.B ACircular indexB is determined as a sum, normalized by the number of the segment detectors, of the normalized intensities detected by each segment. Definitely, in the case of circle, the pattern is uniform and the index becomes unity ŽFig. 2.. For the other shapes, it varies from 0 to 1. From the definition, the circular index is given, in an actual measurement, as a ratio of the total intensity detected by all segments to the production of the maximum intensity measured by a certain segment and the number of the segments as: Circular index s
Ž total intensity detected by all segments. . Ž maximum intensity. Ž segment number. Ž 2.
Note that Eq. Ž2. implies very speedy detection and calculation in the actual measurement because of the simplicity of the index definition. APeak numberB can be recognized from the angular pattern of the intensity on the segment detector with a certain threshold. For example, 0 for a circle, 2 for ellipses, 4 for quadrangles, 6 for triangles.
3. Results and discussions 3.1. Shape distinction with two-dimensional mapping 3.1.1. Distinction of simple geometrical shape At first, distinction of rather simple shapes was examined. Simple shape here means geometrical shapes as a circle, ellipses, triangles, quadrangles, and their modifications. Fig. 3 shows two-dimensional mapping with circular index and peak number for the shapes. Horizontal axis and vertical axis represent circular index and peak number, respectively. In the figure, the shapes examined were illustrated as each plot. As mentioned, in the case of circle, circular index is unity and peak number is zero. In the
Fig. 4. Two-dimensional mapping with circular index and peak number for projections of real powder particles.
cases of ellipses, circular index decreased with increase of its aspect ratio, while peak number was constant as 2. In the cases of quadrangles, circular indexes were from 0.2 to 0.3 and peak number was 4. As shown, the shapes were distinguished successfully in the two-dimensional mapping according to their feature. 3.1.2. Shape distinction of real powder particle As a next step, shape distinction of projections of real powder particles was examined. Fig. 4 shows the result in the same way with Fig. 3. Also in this case, the shapes were distinguished successfully in the two-dimensional mapping according to their feature. In the cases of rod or ellipse like particles, peak number was 2. Even if a projection is similar with a circle, peak number did not became zero due to its small variation. Roundness of a particle seems to be determined with the circular index rather than peak number. If the particle shape is far from a circle, especially in the case of thin particle, the circular index takes smaller value. 3.1.3. Distinction of phytoplanktons As a trial, the two-dimensional mapping for projections of real phytoplanktons were also examined. Many kinds of phytoplanktons are in the range of its size from tens to hundreds of micrometer; therefore, applicability of this shape distinction method for plankton sorting based on its shape can be a kind of application and, consequently, an interesting issue. Fig. 5 shows the result. Many kinds of planktons could be distinguished successfully with the mapping. However, some planktons were attributed in a same range if that peak number was 4 and circular index was around 0.2–0.3. To separate these particles, another parameter may be available: it is simply the size of the particle. 3.2. Size measurement for a single particle
Fig. 3. Two-dimensional mapping with circular index and peak number for simple geometrical shapes.
As mentioned in the Introduction in detail, the particle size analysis with a laser diffraction method is influenced by particle shape. One major reason for this is that the
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Fig. 5. Two-dimensional mapping with circular index and peak number for projections of real phytoplanktons.
sample is supposed as a certain amount of group of particles with a size distribution: the size distribution is obtained with a reverse calculation. Here, suppose that a single particle is measured with a concentric detector as the same as normal particle size. Let us discuss how the size can be determined and how the result is influenced by the sample shape in this case. Because each element of concentric detector detects light intensity integrated with angle from 0 to 2p , detected in actual measurement is Žmean. radial distribution function as: dr
2p
H0
I Ž r ,u . d u .
Ž 3.
In the case of a spherical particle, the first peak of this distribution is maximum and the position, r, of it is determined by the particle diameter. This is a primitive reason why a particle size can be determined based on a measurement of the laser diffraction pattern w11x. In the case of non-spherical particles, if the measurement is carried out for a single particle, a kind of equivalent particle diameter for sphere may also be determined from the measurement of the position which gives the first peak in the radial distribution. Fig. 6 shows a relationship between the equivalent diameter for sphere determined in this way and Heywood diameter for some kinds of nonspherical particles. Their agreement is very well. Accordingly, a particle size can be also determined from the diffraction pattern measurement for non-spherical particles in this way. Fig. 7 shows a two-dimensional mapping of phytoplanktons, which is the same sample with that of Fig. 5, with circular index and equivalent particle diameter. With another mapping parameter, another distinction was achieved. The result indicates applicability of particle distinction with three-dimensional mapping, or two-dimensional mapping of a proper combination of parameters.
Fig. 6. A relationship between the equivalent diameter for sphere determined from the position of the first peak of radial distribution of light intensity and Heywood diameter for some kinds of non-spherical particles.
diameter and of the particle Žprojection. itself w4,5x. Fig. 8 shows a comparison between the conventional roundness factor and the circular index defined in this work for some geometrical shapes. While factors have a certain correlation, the circular index defined in this work has a wider range for the same sample. This may imply an effectiveness of this new shape factor because of its sensitivity. 3.3.2. Size of the segment detector As mentioned, a pattern of a diffraction image depends on the particle shape, while the size of the pattern depends on the sectional area of the particle. A smaller particle gives a larger pattern. Accordingly, even if the particle shape is the same, if the size of the particle is different, the
3.3. Detailed discussions on some specifications 3.3.1. Comparison of circular indexes A typical and conventional circular index or roundness factor is a ratio of perimeters of a circle with Heywood
Fig. 7. A two-dimensional mapping of phytoplanktons, which is the same sample as that of Fig. 7, with circular index and equivalent particle diameter.
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According to the original aim of this work, shape distinction, this stability can be within an allowable range. Also, it is implied that some options of detectors of different size are applicable to adopt or optimize the distinction process in an actual measurement.
Fig. 8. A comparison between the conventional roundness factor and the circular index defined in this work for some geometrical shapes.
detector measures a different part of the diffraction pattern with a fixed optical system. Consequently, the circular index defined in this work takes different value for the different size particle even if the shape is the same. This is a weak point of this shape factor. In this meaning, this shape factor cannot be defined in a strict sense, but it is an operational definition. However, according to our first aim, which is a shape distinction with a factor mapping method, a stability of the factor against particle size and applicable size range in this method should rather be discussed. Fig. 9 shows a relationship between Heywood diameter of a square and the circular index. A square aperture was adopted as a test example because it gives a typical anisotropic pattern as shown in Fig. 2. Here, some different sizes of the segment detector, with 10 or 20 mm inner diameter and with outer diameter from 40 to 100, were also examined. To compare the size, a schematic illustration showing the detector size was drawn in the figure. Although a square aperture is a typical example resulting anisotoropic diffraction pattern, a central area of the pattern is rather uniform as shown in Fig. 2. This is generally true for any shape aperture. Because a smaller particle gives a larger pattern as mentioned, the detector takes this rather uniform region in the cases of smaller particles. Consequently, as indicated in Fig. 9, the circular index takes a larger value in the cases of smaller particles. Because the detectors with a smaller inner diameter, 10 mm in this discussion, can detect this uniform region for a wider range of the particle size by the same reason, they give a larger circular index up to a bigger particle than the detectors with a bigger inner diameter. As discussed, the difference or perturbation of the circular index is resulted in by which part of the diffraction pattern is detected by the detector. A larger detector, therefore, was rather more stable for the change of the particle size. From the figure, the index was almost stable for each detector over 40 mm particle in this calculation while the absolute value of the index depended on the detector size.
3.3.3. Element number of the segment detector A stability of the circular index against particle rotation is discussed here. If a particle has a three-dimensional rotation, the shape of its projection itself changes. This problem is beyond the discussion here: it should be discussed elsewhere. Even if a projection does not change, and the projection rotates on the plane, the circular index varies in some cases. Especially in the case of a square aperture, this variation is worst big. This typical example is discussed here. Corresponding to the rotation of the particle Žprojection., the diffraction pattern rotates on the segment detector. According to Eq. Ž2., the circular index is, in this case, influenced by the maximum intensity detected on a certain element of the detector because the total intensity is the constant. As seen in Fig. 2, the diffraction pattern from a square aperture has four bright radials. In this case, the maximum intensity depends on how the bright radial is divided on the neighbor elements. Especially in the case of the strict square, the bright radials are orthogonal against each other. And note that 32, the number of the elements supposed here, is a multiple of 4, the number of the arms as bright radials. By this special situation, the division of the bright arms on the elements takes place completely the same for every arm. This causes the perturbation of the circular index of 50% in the worst case. However, this was not the case for, e.g., triangle. Because, in the case of triangle, the diffraction pattern has six bright arms, and 32 is not a multiple of 6. Therefore, some of the arms can be on an element fully, and consequently, the perturbation of
Fig. 9. A relationship between Heywood diameter of a square and the circular index. Some different sizes of the segment detector were also examined. To compare the size, a schematic illustration showing the detector size was drawn. Parameters indicate winner diameterrmm, outer diameterrmmx.
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the maximum intensity is very little. For any other anonymous shapes, the situation is similar with the case of triangle. Additionally, if we adopt the element number 31, note that 31 is a prime number, it was confirmed that such perturbation does not take place also for a square. As a result, it was confirmed that the perturbation due to rotations of a projection can be avoided.
4. Conclusions A pattern of a diffraction image depends on the particle shape, while the size of the pattern depends on the sectional area of the particle. In this work, the method to extract differences from the diffraction patterns due to different shapes of non-spherical particles was studied conceptually. A radial segment Žwedge. photo-detector with 32 elements was assumed instead of a concentric circular detector, which is used in usual particle size analyzer. Each element detects an integrated photo intensity corresponding to the element area on the diffraction pattern. Diffraction patterns and intensity patterns detected by the radial segment detector were calculated for many kinds of two-dimensional shapes, corresponding to the projections of particles, as a circle, ellipses, triangles, quadrangles, other anonymous shapes, also shapes extracted from real phytoplanktons. The photo intensity patterns on the radial segment detector can be normalized by the maximum intensity. In the case of circle, the intensity pattern is uniform for the angle definitely. On the other hand, non-spherical Žnon-circular. particle causes various diffraction patterns. From these detected light intensity patterns, we extracted Žor define. two indexes: Acircular indexB and Apeak number.B ACircular indexB is determined as a sum, normalized by the number of the segment detectors, of the normalized intensities of each segment of the detector. Definitely in the case of circle, the pattern is uniform and the index becomes unity. For the other shapes, it varies from 0 to 1. APeak numberB can be recognized from the angular pattern of the intensity on the segment detector with a certain threshold. For example, 0 for a circle, 2 for ellipses, 4 for quadrangles, 6 for triangles. It
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was shown that various shapes can be distinguished by means of two-dimensional mapping with these parameters. In addition, an applicability of a concentric circular detector was examined to estimate the particle size when the particle is non-spherical but is a single particle in the measurement. As a result, it was found that the circle equivalent diameter determined with usual scheme agreed well with the sectional aria equivalent diameter of the original particle even in any cases of non-spherical samples. From these results, it was shown that the particle size and shape in wide range can be distinguished from the three-dimensional mapping with Acircular index,B Apeak numberB and Aparticle size.B In this respect, a concept of an appropriate extraction of shape information has been examined successfully.
Acknowledgements The authors are most grateful to Professor B. Scarlett, Delft University of Technology, for his kind and deep discussion, and also to Mr. T. Moriya, Soka University, for his technical support.
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