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The method to determine the optimum refractive index parameter in the laser diffraction and scattering method
Advanced Powder Technol., Vol. 12, No. 4, pp. 589– 602 (2001) Ó VSP and Society of Powder Technology, Japan 2001.
Translated paper The method to determine the optimum refractive index parameter in the laser diffraction and scattering method TAKESHI KINOSHITA Shimadzu Corp., 1 Nishinokyo-Kuwabaracho,Nakagyo-ku, Kyoto 604-8511, Japan Japanese version published in JSPTI Vol. 37, No. 5 (2000); English version for APT received 11 June 2001 Abstract—It is the most important and difcult problem to decide the refractive index of sample particles in particle size distribution measurements using the laser diffraction and scattering method. In this study, the method to decide both particle size distribution and refractive index of particle is investigated. The new method to determine the refractive index of particles is applied for several sample powders (polystyrene latex, glass beads, etc.) and the usefulness of this new method is conrmed. Keywords: Particle size distribution; particle size measurement; laser diffraction method; refractive index; light scattering.
NOMENCLATURE
s s¤ A f e μ
D
np Dap
Detected light intensity distribution vector Estimated light intensity distribution vector coefcient matrix to convert the particle size distribution vector to the light energy distribution vector particle size distribution vector error vector angle between s and s¤ difference vector between s and s¤ refractive index parameter arbitrary percent diameter
(—)
(—) (—) (—) (—) (—) (¹m)
1. INTRODUCTION
Particle size distribution analyzers that use the laser diffraction and scattering method as their measuring principle provide many excellent features such as high-
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speed measurement, excellent reproducibility and easy operation, in comparison with instruments using the conventional sedimentation method etc. Currently, the laser diffraction and scattering method is the mainstream measuring principle for particle size distribution analyzers. However, this measuring principle also involves the problem of the ‘refractive index’. The laser diffraction and scattering method, which is based on Fraunhofer diffraction theory and Mie scattering theory as indicated by its name, measures the particle size distribution by using the relationship between the particle diameter and the diffraction /scattering pattern. In the range of about 5 ¹m or larger particle diameter (diffraction region), the refractive index of particles is not a problem. In the range of smaller particles diameter (scattering region), however, the refractive index of particles (to be accurate, the relative refractive index to the surrounding medium) inuences the diffraction /scattering pattern. If sample particles belong to the scattering region and have different refractive indices, the scattered light intensity pattern varies, even if the particle diameter is exactly the same. Another problem is the inuence of the shape of particle on the measurement result. The shape of particle is assumed as a complete sphere in laser diffraction and scattering method. The particle size distribution analyzer calculates the particle size distribution from two different light intensity distributions. One is the measured light intensity distribution generated by the particles, the other is the theoretical scattered light intensity distribution calculated by using Fraunhofer diffraction theory or Mie scattering theory for particles with a known diameter. Here, we assume the case that there are two kinds of powders whose refractive indices are different but particle size distribution is the same, and the laser diffraction and scattering method is used to measure their particle size distribution. If the refractive index appropriate for each kind of powders is used for the measurement, the same particle size distribution will be obtained. However, if both kinds of particles are measured with a refractive index appropriate for either kind of particle, different particle size distributions will be obtained. As mentioned above, the laser diffraction and scattering method requires an accurate refractive index for accurate particle size distribution measurement. At present, however, it is very difcult to determine the accurate refractive index of particulate matter. Although the method of measuring refractive indices of particulate matter has been reported [1], this method has a limitation on the refractive index measuring range, and cannot cover all kinds of powders subjected to the laser diffraction and scattering method. Hayakawa et al. attempted to obtain accurate particle size distribution and refractive index simultaneously only from measurement results of the laser diffraction and scattering method without using any refractive index measuring methods [2, 3]. With their method, however, the applicable substances and the measuring range are limited to a certain extent. There is also another problem, i.e. the inuence of particle shape on the measurement result. The laser diffraction and scattering method assumes that the
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shape of sample particles is completely spherical. However, most actual sample particles have non-spherical shapes. Since the scattered light intensity distribution is affected not only by the particle size but also the particle shape, the particle size distribution obtained by the laser diffraction and scattering method is affected by the particle shape. For example, assume there are two kinds of powder materials. These two materials have the same particle size distribution and the same refractive index, but the shape of the particles differ. The measuring results of the particle size distribution by the laser diffraction method will differ. The refractive index of the particles used for the laser diffraction and scattering method is not a mere substance-specic constant, but must be regarded as a parameter needed for particle size distribution measurement. In this context, we should consider that the refractive index used for the laser diffraction and scattering method is different from the refractive index as a substance-specic constant. As described above, the particle size distribution measurement using the laser diffraction and scattering method requires a refractive index appropriate for the sample particles. Actually, however, there is no perfect method to determine the appropriate refractive index. Therefore, this is the most important problem in the laser diffraction and scattering method. Using the laser diffraction and scattering method, this study examines the method to determine the particle size distribution and the optimum refractive index simultaneously without using information other than the particle size distribution and the light intensity distribution. To avoid confusion, this paper clearly differentiates the refractive index as a substance-specic constant from the refractive index applicable to the laser diffraction and scattering method. Hereinafter, the former is simply referred to as the ‘refractive index’ and the latter is referred to as the ‘refractive index parameter’. Also, the refractive index becomes a problem only in the scattering region outside the diffraction region, but the name ‘laser diffraction method’ is generally given to this measuring principle. Therefore, we use this name in this paper.
2. BASIC CONCEPT
In this section, the basic concept to determine the optimum refractive index parameter is explained. The basic optical system of the measuring instrument is shown in Fig. 1. The measuring instrument irradiates a laser beam to the sample particles and measures the distribution of the scattered light intensity. Inherently, the scattered light intensity distribution is shown as a consecutive function. However, the scattered light intensity distribution is herein shown as a scattered light intensity distribution vector s, because the light intensity distribution is detected by a ring-shaped photodiode array. For the same reason, the particle size distribution is also inherently shown as a consecutive function, but it is shown here as particle size distribution vector f .
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Figure 1. Basic optical system of the laser diffraction and scattering method.
These two vectors are related according to: s D Af :
(1)
Here, A is the matrix that converts the scattered light intensity distribution into the particle size distribution. This matrix is calculated from Fraunhofer diffraction theory and Mie scattering theory. Of course, the scattered light intensity distribution vector s includes an error because it is a measured value. Therefore, (1) should be shown as follows: s D Af C e;
(2)
where e is an error vector. To obtain the particle size distribution vector f that minimizes the square sum of the error, the following normal equation is solved: .A T A /f D A T s:
(3)
Finally, the particle size distribution vector f is shown as follows: f D .A T A /¡1 A T s;
(4)
where ( )¡1 shows an inverse matrix and ( )T shows a transposed matrix. This is the normal process to determine the particle size distribution by the laser diffraction and scattering method [4]. In this process, although the refractive index parameter does not appear, it is already set as a value in the procedure to calculate matrix A . By using (1), another scattered light intensity distribution vector s ¤ can be calculated from the particle size distribution vector f determined by (4). s ¤ is shown as follows: s ¤ D A .A T A /¡1 A T s:
(5)
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s is actually the detected light intensity distribution by the laser beam irradiating the sample, herein referred to as the ‘detected light intensity distribution’. s ¤ is calculated by (5), herein referred to as the ‘estimated light intensity distribution’. When the optimum refractive index parameter is selected, the optimum particle size distribution is obtained and, then, it is expected that s ¤ will be completely same as or very closed to s. Here, a criterion to evaluate the conformity of s and s ¤ is necessary. These two vectors are light intensity distribution vectors. The distribution pattern or direction of the vector is important in particle size distribution measurement by the laser diffraction method. Therefore, the conformity of s and s ¤ can be evaluated by the value of the angle between s and s ¤ (cross-angle). The cosine of cross-angle μ is shown as follows by the denition equation of the inner product of the vector: cos μ D
s · s¤ ; jsj js ¤ j
(6)
where s · s ¤ is the inner product and jsj is the absolute value of the vector s. In this study, the cosine of the cross-angle, cos μ is adopted as a evaluation function to evaluate the conformity between s and s ¤ . The closer the value is to 1, the closer the cross-angle μ is to 0. When cos μ is the closest to 1, s ¤ is the closest to s. Also, both the refractive index parameter and the particle size distribution becomes the most optimum values. The particle size distribution becomes the most accurate one. Figure 2 shows this process. Hereinafter, the ‘cosine of the crossangle between light intensity distribution vectors’ is simply referred to as the ‘cosine of the cross-angle’. In addition, the cosine of the cross-angle has another meaning. Now, vector s and s ¤ are normalized by each absolute value. The difference between the normalized s and s ¤ is as follows: s s¤ DD ¡ ¤ : (7) jsj js j The square sum of the difference between each elements of s=jsj and s ¤ =js ¤ j is D · D . It is shown as follows: ´ ´ ´ s s¤ s s¤ s · s¤ : (8) D·DD ¡ · ¡ D2 1¡ jsj js ¤ j jsj js ¤ j jsj js ¤ j In (8), the term 1¡s ·s ¤ =jsjjs ¤ j is the same as cos μ in (6). When the cross-angle of s and s ¤ becomes close to 1, the square sum of the difference between each element of s=jsj and s ¤ =js ¤ j becomes minimum. As s and s ¤ are light intensity distribution vectors, each element of s and s ¤ does not take a negative value. Therefore, the cosine of the cross-angle of s and s ¤ move between 0 to 1. More generally, in the previous particle size distribution measurement (or calculation) method based on the laser diffraction principle, the particle size distribution vector f , that minimizes the error, was calculated by using (2). It was premised that
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matrix A was known or xed. In this case, the accuracy of the resulting particle size distribution was not guaranteed because it was not guaranteed that matrix A was optimized with regard to refractive index. In this procedure, through the process to determine the optimum refractive index parameter, it is quantitatively evaluated that matrix A is optimized with regard to the refractive index. Then, the particle size distribution, which minimizes the error, is determined. That is to say, this procedure is a direct extension of the previous procedure and it is expected that a more accurate particle size distribution can be determined. Of course, there is the possibility that an unsuitable particle size distribution is selected as the optimum one because s ¤ becomes very close to s through the effects of other elements (e.g. quantization error or calculation method, etc.). However, it is impossible to examine all values of the refractive indices. Realistically, an appropriate range of the refractive index is set up and the optimum refractive index parameter is sought in this range. Therefore, when the optimum refractive index parameter is sought by using this method, we avoid selecting a value far from the optimum refractive index value as the optimum value to a certain extent. It is also possible to select a more accurate particle size distribution by ex-
Figure 2. Method to determine the optimum refractive index parameter.
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amining the relationship between the particle size distribution and the light intensity distribution [6].
3. EXPERIMENT METHOD
As a measuring instrument for the experiment, a laser diffraction type particle size distribution analyzer SALD-2000J (Shimadzu Corp.) was used. As samples for the experiment, the following powders were used. The particle size distribution of these samples was measured by a method other than the laser diffraction and scattering method. (i) Polystyrene latex spheres (ii) Glass beads (iii) White fused alumina In the laser diffraction and scattering method, the refractive index is shown as a complex number. In this study, however, the imaginary part was assumed to be ‘0’ and only the real part was examined.
4. RESULTS AND DISCUSSION
4.1. Polystyrene latex spheres The refractive index parameter determination method described previously was applied to polystyrene latex spheres whose nominal diameter was 1.11 ¹m. The examined refractive index was in the range 1.4– 1.8. The relationship between the refractive index and the cosine of the cross-angle is shown in Fig. 3. According to the result, the optimum value for the refractive index parameter proved to be 1.55. This value seems to be appropriate, in comparison
Figure 3. Variation of the cosine of the angle between the detected light intensity vector and the estimated light intensity vector versus the refractive index parameter of the latex sample.
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Figure 4. Particle size distribution of polystyrene latex when the refractive index parameter was 1.55.
Figure 5. Variation of 10 and 50% diameter versus the refractive index parameter of the latex.
with the refractive index of polystyrene latex, which is known to be 1.59 from certication. On the other hand, the particle size distribution for the latex particles, as shown in Fig. 4, also seems to be appropriate, relative to the nominal value of 1.11 ¹m. Furthermore, as shown in Fig. 5, the refractive index parameter value of 1.55 conforms to the optimum value determined by the method of Hayakawa et al., which is a useful method for refractive index parameter determination. 4.2. Glass beads The refractive index parameter determination method described previously was also applied to glass beads whose nominal particle size distribution was 1– 10 ¹m. The examined refractive index range was 1.55– 2.00.
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Figure 6. Variation of the cosine of the angle between the detested light intensity vector and the estimated light intensity vector versus the refractive index parameter of glass beads.
Figure 7. Particle size distribution of glass beads when the refractive index parameter was 1.75.
The relationship between the refractive index and the cosine of the cross-angle is shown in Fig. 6. According to the result, the optimum value for the refractive index parameter proved to be 1.75. On the other hand, the particle size distribution for the glass beads, as shown in Fig. 7, also seems to be appropriate, as compared with the nominal particle size distribution of 1 – 10 ¹m. Furthermore, as shown in Fig. 8 by the method of Hayakawa et al., the refractive index parameter value of 1.75 is the value ranked seventh. However, if 1.60, the value ranked rst, is applied, the particle size distribution becomes sharper than the nominal value, showing non-conformity with the actual value.
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Figure 8. Variation of 10 and 50% diameter versus the refractive index parameter of glass beads.
Figure 9. Particle size distribution of glass beads when the refractive index parameter was 1.60.
4.3. White fused alumina Then, the refractive index parameter determination method described previously was applied to two kinds of white fused alumina whose nominal diameters were 1 and 5 ¹m. The examined refractive index parameter range was 1.55– 2.0 (real number). The relationship between the refractive index parameter for alumina (5 ¹m diameter) and the cosine of the cross-angle is shown in Fig. 10. According to the result, the optimum value for the refractive index parameter proved to be 1.75. This value seems to be appropriate, as compared with the refractive index of alumina, which is known to be 1.76. On the other hand, the particle size distribution for alumina, as shown in Fig. 11, also seems to be appropriate, relative to the nominal value of 5 ¹m.
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Figure 10. Variation of the cosine of the angle between detected the light intensity vector and the estimated light intensity vector versus the refractive index parameter of aluminum oxide (5 ¹m).
Figure 11. Particle size distribution of aluminum oxide (5 ¹m) when the refractive index parameter was 1.75.
The relationship between the refractive index parameter for alumina (1 ¹m diameter) and the cosine of the cross-angle is shown in Fig. 12. According to the result, the optimum value for the refractive index parameter proved to be 1.80. In comparison with the documented value, and referring to the particle size distribution shown in Fig. 13, this value also seems to be appropriate, relative to the nominal value of 1 ¹m. However, for the above-mentioned alumina of 5¹m diameter, the optimum value was 1.75. Although these samples are substantially the same material, there is a difference of 0.05 in the refractive index. As the particle is not spherical, it is seems that the shape of the particle inuences the refractive
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Figure 12. Variation of the coside of the angle between the detected light intensity vector and the estimated light intensity vector versus the refractive index parameter of aluminum oxide (1 ¹m).
Figure 13. Particle size distribution of aluminum oxide (1 ¹m) when the refractive index parameter was 1.80.
index parameter more strongly. This difference is an example that shows the nonconformity between the refractive index and the refractive index parameter.
5. CONCLUSION
In the laser diffraction and scattering method, selection of an appropriate refractive index is a very important issue; however, a denitive method has not yet been achieved. In this study, the following new method to select the optimum refractive index is examined.
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(i) In this study, a new method to determine the optimum value of the refractive index parameter was examined. The detected light intensity distribution was compared with the estimated light intensity distribution for each refractive index parameter. The refractive index value that gives the highest conformity for these two light intensity distributions was selected, and determined as the optimum refractive index. The propriety of this procedure could be conrmed. (ii) As the evaluation function to evaluate the conformity between detected light intensity distribution and the estimated light intensity distribution, the cosine of the cross-angle between the light intensity vectors was used. This gives a quantitative criterion to determine the optimum refractive index. Furthermore, since the evaluation function is standardized to take a value between 0 and 1, this method can provide a unied criterion unaffected by any differences in the particle size distribution and the refractive index of the samples, as well as the difference in concentration of the sample being measured. (iii) The refractive index in the laser diffraction and scattering method should be considered as a character including the inuence of the shape of the particle, etc. This fact has been suggested in this paper. Actually, the inuence of the shape of the particle can be included in the ‘refractive index parameter’. The reason is not yet clear. In the case when the particle size distribution and refractive index are determined by using the method presented in this paper, the particle size distribution, that seems to be the most accurate, can be obtained. Therefore, the refractive index determined by this method should be distinguished from the original refractive index. Thus, we propose to call this new refractive index the ‘refractive index parameter’. (iv) As the result of experiments show, the cosine of the cross-angle, which is used to evaluate the conformity of s and s ¤ , sometimes takes a value very close to the optimum refractive index. One of the reasons is as follows: the cosine of crossangle is calculated with even weights for each element of the light intensity distribution vector; however, the inuence of the refractive index on the light intensity distribution vector concentrates upon a part of the elements (6). It is expected to give clearer difference between the value of the evaluation function for the optimum refractive index and for another refractive index by using, for example, the weight matrix for the light intensity distribution vector. This is a subject for future research. REFERENCES 1. H. Takano, Hikari sanran-ho ni yoru aerosol biryushi no kussetsuritsu-sokutei no kento, J. Shimadzu Scientic Instrum. 3, 466– 470 (1991). 2. O. Hayakawa, K. Nakahira and J. Tsubaki, Estimation of the optimum refractive index by laser diffraction and scattering method, J. Soc. Powder Technol., Japan 30, 652– 659 (1993). 3. O. Hayakawa, Y. Yashuda, M. Naitoh and J. Tsubaki, The effect of refractive index input value on particle size distribution measured by the laser diffraction & scattering method, J. Soc. Powder Technol., Japan 32, 796– 803 (1995).
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4. Y. Suemoto, Laser scattering measurements of nonspherical particles, Japan J. Optics 20, 564– 569 (1991). 5. K. Hayashida, T. Niwa and H. Shimaoka, Measurement of submicron particles with Shimadzu laser diffraction particle size analyzer SALD-2000, Shimadzu Rev. 48, 309– 318 (1991). 6. T. Kinoshita, The inuence of refractive index in laser diffraction/scattering particle size measurement, Powder Sci. & Engng. 25(10), 56– 62 (1993).
Translated paper The method to determine the optimum refractive index parameter in the laser diffraction and scattering method TAKESHI KINOSHITA Shimadzu Corp., 1 Nishinokyo-Kuwabaracho,Nakagyo-ku, Kyoto 604-8511, Japan Japanese version published in JSPTI Vol. 37, No. 5 (2000); English version for APT received 11 June 2001 Abstract—It is the most important and difcult problem to decide the refractive index of sample particles in particle size distribution measurements using the laser diffraction and scattering method. In this study, the method to decide both particle size distribution and refractive index of particle is investigated. The new method to determine the refractive index of particles is applied for several sample powders (polystyrene latex, glass beads, etc.) and the usefulness of this new method is conrmed. Keywords: Particle size distribution; particle size measurement; laser diffraction method; refractive index; light scattering.
NOMENCLATURE
s s¤ A f e μ
D
np Dap
Detected light intensity distribution vector Estimated light intensity distribution vector coefcient matrix to convert the particle size distribution vector to the light energy distribution vector particle size distribution vector error vector angle between s and s¤ difference vector between s and s¤ refractive index parameter arbitrary percent diameter
(—)
(—) (—) (—) (—) (—) (¹m)
1. INTRODUCTION
Particle size distribution analyzers that use the laser diffraction and scattering method as their measuring principle provide many excellent features such as high-
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T. Kinoshita
speed measurement, excellent reproducibility and easy operation, in comparison with instruments using the conventional sedimentation method etc. Currently, the laser diffraction and scattering method is the mainstream measuring principle for particle size distribution analyzers. However, this measuring principle also involves the problem of the ‘refractive index’. The laser diffraction and scattering method, which is based on Fraunhofer diffraction theory and Mie scattering theory as indicated by its name, measures the particle size distribution by using the relationship between the particle diameter and the diffraction /scattering pattern. In the range of about 5 ¹m or larger particle diameter (diffraction region), the refractive index of particles is not a problem. In the range of smaller particles diameter (scattering region), however, the refractive index of particles (to be accurate, the relative refractive index to the surrounding medium) inuences the diffraction /scattering pattern. If sample particles belong to the scattering region and have different refractive indices, the scattered light intensity pattern varies, even if the particle diameter is exactly the same. Another problem is the inuence of the shape of particle on the measurement result. The shape of particle is assumed as a complete sphere in laser diffraction and scattering method. The particle size distribution analyzer calculates the particle size distribution from two different light intensity distributions. One is the measured light intensity distribution generated by the particles, the other is the theoretical scattered light intensity distribution calculated by using Fraunhofer diffraction theory or Mie scattering theory for particles with a known diameter. Here, we assume the case that there are two kinds of powders whose refractive indices are different but particle size distribution is the same, and the laser diffraction and scattering method is used to measure their particle size distribution. If the refractive index appropriate for each kind of powders is used for the measurement, the same particle size distribution will be obtained. However, if both kinds of particles are measured with a refractive index appropriate for either kind of particle, different particle size distributions will be obtained. As mentioned above, the laser diffraction and scattering method requires an accurate refractive index for accurate particle size distribution measurement. At present, however, it is very difcult to determine the accurate refractive index of particulate matter. Although the method of measuring refractive indices of particulate matter has been reported [1], this method has a limitation on the refractive index measuring range, and cannot cover all kinds of powders subjected to the laser diffraction and scattering method. Hayakawa et al. attempted to obtain accurate particle size distribution and refractive index simultaneously only from measurement results of the laser diffraction and scattering method without using any refractive index measuring methods [2, 3]. With their method, however, the applicable substances and the measuring range are limited to a certain extent. There is also another problem, i.e. the inuence of particle shape on the measurement result. The laser diffraction and scattering method assumes that the
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shape of sample particles is completely spherical. However, most actual sample particles have non-spherical shapes. Since the scattered light intensity distribution is affected not only by the particle size but also the particle shape, the particle size distribution obtained by the laser diffraction and scattering method is affected by the particle shape. For example, assume there are two kinds of powder materials. These two materials have the same particle size distribution and the same refractive index, but the shape of the particles differ. The measuring results of the particle size distribution by the laser diffraction method will differ. The refractive index of the particles used for the laser diffraction and scattering method is not a mere substance-specic constant, but must be regarded as a parameter needed for particle size distribution measurement. In this context, we should consider that the refractive index used for the laser diffraction and scattering method is different from the refractive index as a substance-specic constant. As described above, the particle size distribution measurement using the laser diffraction and scattering method requires a refractive index appropriate for the sample particles. Actually, however, there is no perfect method to determine the appropriate refractive index. Therefore, this is the most important problem in the laser diffraction and scattering method. Using the laser diffraction and scattering method, this study examines the method to determine the particle size distribution and the optimum refractive index simultaneously without using information other than the particle size distribution and the light intensity distribution. To avoid confusion, this paper clearly differentiates the refractive index as a substance-specic constant from the refractive index applicable to the laser diffraction and scattering method. Hereinafter, the former is simply referred to as the ‘refractive index’ and the latter is referred to as the ‘refractive index parameter’. Also, the refractive index becomes a problem only in the scattering region outside the diffraction region, but the name ‘laser diffraction method’ is generally given to this measuring principle. Therefore, we use this name in this paper.
2. BASIC CONCEPT
In this section, the basic concept to determine the optimum refractive index parameter is explained. The basic optical system of the measuring instrument is shown in Fig. 1. The measuring instrument irradiates a laser beam to the sample particles and measures the distribution of the scattered light intensity. Inherently, the scattered light intensity distribution is shown as a consecutive function. However, the scattered light intensity distribution is herein shown as a scattered light intensity distribution vector s, because the light intensity distribution is detected by a ring-shaped photodiode array. For the same reason, the particle size distribution is also inherently shown as a consecutive function, but it is shown here as particle size distribution vector f .
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Figure 1. Basic optical system of the laser diffraction and scattering method.
These two vectors are related according to: s D Af :
(1)
Here, A is the matrix that converts the scattered light intensity distribution into the particle size distribution. This matrix is calculated from Fraunhofer diffraction theory and Mie scattering theory. Of course, the scattered light intensity distribution vector s includes an error because it is a measured value. Therefore, (1) should be shown as follows: s D Af C e;
(2)
where e is an error vector. To obtain the particle size distribution vector f that minimizes the square sum of the error, the following normal equation is solved: .A T A /f D A T s:
(3)
Finally, the particle size distribution vector f is shown as follows: f D .A T A /¡1 A T s;
(4)
where ( )¡1 shows an inverse matrix and ( )T shows a transposed matrix. This is the normal process to determine the particle size distribution by the laser diffraction and scattering method [4]. In this process, although the refractive index parameter does not appear, it is already set as a value in the procedure to calculate matrix A . By using (1), another scattered light intensity distribution vector s ¤ can be calculated from the particle size distribution vector f determined by (4). s ¤ is shown as follows: s ¤ D A .A T A /¡1 A T s:
(5)
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s is actually the detected light intensity distribution by the laser beam irradiating the sample, herein referred to as the ‘detected light intensity distribution’. s ¤ is calculated by (5), herein referred to as the ‘estimated light intensity distribution’. When the optimum refractive index parameter is selected, the optimum particle size distribution is obtained and, then, it is expected that s ¤ will be completely same as or very closed to s. Here, a criterion to evaluate the conformity of s and s ¤ is necessary. These two vectors are light intensity distribution vectors. The distribution pattern or direction of the vector is important in particle size distribution measurement by the laser diffraction method. Therefore, the conformity of s and s ¤ can be evaluated by the value of the angle between s and s ¤ (cross-angle). The cosine of cross-angle μ is shown as follows by the denition equation of the inner product of the vector: cos μ D
s · s¤ ; jsj js ¤ j
(6)
where s · s ¤ is the inner product and jsj is the absolute value of the vector s. In this study, the cosine of the cross-angle, cos μ is adopted as a evaluation function to evaluate the conformity between s and s ¤ . The closer the value is to 1, the closer the cross-angle μ is to 0. When cos μ is the closest to 1, s ¤ is the closest to s. Also, both the refractive index parameter and the particle size distribution becomes the most optimum values. The particle size distribution becomes the most accurate one. Figure 2 shows this process. Hereinafter, the ‘cosine of the crossangle between light intensity distribution vectors’ is simply referred to as the ‘cosine of the cross-angle’. In addition, the cosine of the cross-angle has another meaning. Now, vector s and s ¤ are normalized by each absolute value. The difference between the normalized s and s ¤ is as follows: s s¤ DD ¡ ¤ : (7) jsj js j The square sum of the difference between each elements of s=jsj and s ¤ =js ¤ j is D · D . It is shown as follows: ´ ´ ´ s s¤ s s¤ s · s¤ : (8) D·DD ¡ · ¡ D2 1¡ jsj js ¤ j jsj js ¤ j jsj js ¤ j In (8), the term 1¡s ·s ¤ =jsjjs ¤ j is the same as cos μ in (6). When the cross-angle of s and s ¤ becomes close to 1, the square sum of the difference between each element of s=jsj and s ¤ =js ¤ j becomes minimum. As s and s ¤ are light intensity distribution vectors, each element of s and s ¤ does not take a negative value. Therefore, the cosine of the cross-angle of s and s ¤ move between 0 to 1. More generally, in the previous particle size distribution measurement (or calculation) method based on the laser diffraction principle, the particle size distribution vector f , that minimizes the error, was calculated by using (2). It was premised that
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matrix A was known or xed. In this case, the accuracy of the resulting particle size distribution was not guaranteed because it was not guaranteed that matrix A was optimized with regard to refractive index. In this procedure, through the process to determine the optimum refractive index parameter, it is quantitatively evaluated that matrix A is optimized with regard to the refractive index. Then, the particle size distribution, which minimizes the error, is determined. That is to say, this procedure is a direct extension of the previous procedure and it is expected that a more accurate particle size distribution can be determined. Of course, there is the possibility that an unsuitable particle size distribution is selected as the optimum one because s ¤ becomes very close to s through the effects of other elements (e.g. quantization error or calculation method, etc.). However, it is impossible to examine all values of the refractive indices. Realistically, an appropriate range of the refractive index is set up and the optimum refractive index parameter is sought in this range. Therefore, when the optimum refractive index parameter is sought by using this method, we avoid selecting a value far from the optimum refractive index value as the optimum value to a certain extent. It is also possible to select a more accurate particle size distribution by ex-
Figure 2. Method to determine the optimum refractive index parameter.
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amining the relationship between the particle size distribution and the light intensity distribution [6].
3. EXPERIMENT METHOD
As a measuring instrument for the experiment, a laser diffraction type particle size distribution analyzer SALD-2000J (Shimadzu Corp.) was used. As samples for the experiment, the following powders were used. The particle size distribution of these samples was measured by a method other than the laser diffraction and scattering method. (i) Polystyrene latex spheres (ii) Glass beads (iii) White fused alumina In the laser diffraction and scattering method, the refractive index is shown as a complex number. In this study, however, the imaginary part was assumed to be ‘0’ and only the real part was examined.
4. RESULTS AND DISCUSSION
4.1. Polystyrene latex spheres The refractive index parameter determination method described previously was applied to polystyrene latex spheres whose nominal diameter was 1.11 ¹m. The examined refractive index was in the range 1.4– 1.8. The relationship between the refractive index and the cosine of the cross-angle is shown in Fig. 3. According to the result, the optimum value for the refractive index parameter proved to be 1.55. This value seems to be appropriate, in comparison
Figure 3. Variation of the cosine of the angle between the detected light intensity vector and the estimated light intensity vector versus the refractive index parameter of the latex sample.
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Figure 4. Particle size distribution of polystyrene latex when the refractive index parameter was 1.55.
Figure 5. Variation of 10 and 50% diameter versus the refractive index parameter of the latex.
with the refractive index of polystyrene latex, which is known to be 1.59 from certication. On the other hand, the particle size distribution for the latex particles, as shown in Fig. 4, also seems to be appropriate, relative to the nominal value of 1.11 ¹m. Furthermore, as shown in Fig. 5, the refractive index parameter value of 1.55 conforms to the optimum value determined by the method of Hayakawa et al., which is a useful method for refractive index parameter determination. 4.2. Glass beads The refractive index parameter determination method described previously was also applied to glass beads whose nominal particle size distribution was 1– 10 ¹m. The examined refractive index range was 1.55– 2.00.
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Figure 6. Variation of the cosine of the angle between the detested light intensity vector and the estimated light intensity vector versus the refractive index parameter of glass beads.
Figure 7. Particle size distribution of glass beads when the refractive index parameter was 1.75.
The relationship between the refractive index and the cosine of the cross-angle is shown in Fig. 6. According to the result, the optimum value for the refractive index parameter proved to be 1.75. On the other hand, the particle size distribution for the glass beads, as shown in Fig. 7, also seems to be appropriate, as compared with the nominal particle size distribution of 1 – 10 ¹m. Furthermore, as shown in Fig. 8 by the method of Hayakawa et al., the refractive index parameter value of 1.75 is the value ranked seventh. However, if 1.60, the value ranked rst, is applied, the particle size distribution becomes sharper than the nominal value, showing non-conformity with the actual value.
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Figure 8. Variation of 10 and 50% diameter versus the refractive index parameter of glass beads.
Figure 9. Particle size distribution of glass beads when the refractive index parameter was 1.60.
4.3. White fused alumina Then, the refractive index parameter determination method described previously was applied to two kinds of white fused alumina whose nominal diameters were 1 and 5 ¹m. The examined refractive index parameter range was 1.55– 2.0 (real number). The relationship between the refractive index parameter for alumina (5 ¹m diameter) and the cosine of the cross-angle is shown in Fig. 10. According to the result, the optimum value for the refractive index parameter proved to be 1.75. This value seems to be appropriate, as compared with the refractive index of alumina, which is known to be 1.76. On the other hand, the particle size distribution for alumina, as shown in Fig. 11, also seems to be appropriate, relative to the nominal value of 5 ¹m.
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Figure 10. Variation of the cosine of the angle between detected the light intensity vector and the estimated light intensity vector versus the refractive index parameter of aluminum oxide (5 ¹m).
Figure 11. Particle size distribution of aluminum oxide (5 ¹m) when the refractive index parameter was 1.75.
The relationship between the refractive index parameter for alumina (1 ¹m diameter) and the cosine of the cross-angle is shown in Fig. 12. According to the result, the optimum value for the refractive index parameter proved to be 1.80. In comparison with the documented value, and referring to the particle size distribution shown in Fig. 13, this value also seems to be appropriate, relative to the nominal value of 1 ¹m. However, for the above-mentioned alumina of 5¹m diameter, the optimum value was 1.75. Although these samples are substantially the same material, there is a difference of 0.05 in the refractive index. As the particle is not spherical, it is seems that the shape of the particle inuences the refractive
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Figure 12. Variation of the coside of the angle between the detected light intensity vector and the estimated light intensity vector versus the refractive index parameter of aluminum oxide (1 ¹m).
Figure 13. Particle size distribution of aluminum oxide (1 ¹m) when the refractive index parameter was 1.80.
index parameter more strongly. This difference is an example that shows the nonconformity between the refractive index and the refractive index parameter.
5. CONCLUSION
In the laser diffraction and scattering method, selection of an appropriate refractive index is a very important issue; however, a denitive method has not yet been achieved. In this study, the following new method to select the optimum refractive index is examined.
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(i) In this study, a new method to determine the optimum value of the refractive index parameter was examined. The detected light intensity distribution was compared with the estimated light intensity distribution for each refractive index parameter. The refractive index value that gives the highest conformity for these two light intensity distributions was selected, and determined as the optimum refractive index. The propriety of this procedure could be conrmed. (ii) As the evaluation function to evaluate the conformity between detected light intensity distribution and the estimated light intensity distribution, the cosine of the cross-angle between the light intensity vectors was used. This gives a quantitative criterion to determine the optimum refractive index. Furthermore, since the evaluation function is standardized to take a value between 0 and 1, this method can provide a unied criterion unaffected by any differences in the particle size distribution and the refractive index of the samples, as well as the difference in concentration of the sample being measured. (iii) The refractive index in the laser diffraction and scattering method should be considered as a character including the inuence of the shape of the particle, etc. This fact has been suggested in this paper. Actually, the inuence of the shape of the particle can be included in the ‘refractive index parameter’. The reason is not yet clear. In the case when the particle size distribution and refractive index are determined by using the method presented in this paper, the particle size distribution, that seems to be the most accurate, can be obtained. Therefore, the refractive index determined by this method should be distinguished from the original refractive index. Thus, we propose to call this new refractive index the ‘refractive index parameter’. (iv) As the result of experiments show, the cosine of the cross-angle, which is used to evaluate the conformity of s and s ¤ , sometimes takes a value very close to the optimum refractive index. One of the reasons is as follows: the cosine of crossangle is calculated with even weights for each element of the light intensity distribution vector; however, the inuence of the refractive index on the light intensity distribution vector concentrates upon a part of the elements (6). It is expected to give clearer difference between the value of the evaluation function for the optimum refractive index and for another refractive index by using, for example, the weight matrix for the light intensity distribution vector. This is a subject for future research. REFERENCES 1. H. Takano, Hikari sanran-ho ni yoru aerosol biryushi no kussetsuritsu-sokutei no kento, J. Shimadzu Scientic Instrum. 3, 466– 470 (1991). 2. O. Hayakawa, K. Nakahira and J. Tsubaki, Estimation of the optimum refractive index by laser diffraction and scattering method, J. Soc. Powder Technol., Japan 30, 652– 659 (1993). 3. O. Hayakawa, Y. Yashuda, M. Naitoh and J. Tsubaki, The effect of refractive index input value on particle size distribution measured by the laser diffraction & scattering method, J. Soc. Powder Technol., Japan 32, 796– 803 (1995).
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4. Y. Suemoto, Laser scattering measurements of nonspherical particles, Japan J. Optics 20, 564– 569 (1991). 5. K. Hayashida, T. Niwa and H. Shimaoka, Measurement of submicron particles with Shimadzu laser diffraction particle size analyzer SALD-2000, Shimadzu Rev. 48, 309– 318 (1991). 6. T. Kinoshita, The inuence of refractive index in laser diffraction/scattering particle size measurement, Powder Sci. & Engng. 25(10), 56– 62 (1993).