Page 37 - Book Hosokawa Nanoparticle Technology Handbook

FUNDAMENTALS CH. 1 BASIC PROPERTIES AND MEASURING METHODS OF NANOPARTICLES

Fourier coefficients a n ,b n
Radius f( )

f ( )
Center of mass

0 2 Low High
Angle Frequency
a) Radius of perimeter b) Relation between rotation c) Fourier coefficient
angle and radius distribution
Figure 1.3.2
Fourier analysis of particle projective perimeter.
is obtained from inclination of the straight line using diameters obtained by different kinds of method such
the following equation [8]: as a particle size by dynamic-light-scattering method
and the specific surface diameter by the gas adsorption
P A D/2 (1.3.2) is used. These particle sizes include the influence of
particle shape and, so the ratio of these diameters can
The three-dimensional particle shape expressed by be used for a simple particle shape expression.
the fractal dimension is also possible from the log–log
plot of the surface area and volume of each particle,
instead of the perimeter and the projection area. References
[1] M. Suzuki, K. Kawabata, K. Iimura and M. Hitota:
1.3.5 Particle shape analysis by Fourier analysis J. Soc. Powder Technol., 41, 156–161 (2004).
[2] H. Furukawa, M. Shimizu, Y. Suzuki and H. Nishioka:
In the Fourier analysis method, the particle shape is
given as a function of the radius f( ) from the center of JOEL News, 36E, 50 (2001).
mass to perimeter as shown in Fig. 1.3.2a. The Fourier [3] J. Tsubaki, M. Suzuki and Y. Kanda: Nyumon
analysis is carried out by the following equation [9]: Ryushi Funtaikougaku, Nikkan Kougyo Shinbunsha, 8
(2002).

0 ∑ ⎛ 2 n 2 n ⎞ [4] J. Tsubaki, G. Jimbo: Powder Technol., 22, 161–169

f () a ⎜ ⎝ a cos b sin ⎟ ⎠ (1.3.3) (1979).
n
n

n 1 T T
[5] B.B. Mandelbrot: Fractal Kikagaku, Nikkei Sci. (1984).
The Fourier coefficients a and b represent the parti- [6] M. Suzuki, Y. Muguruma, M. Hirota and T. Oshima:
n
n
cle shape. J. Soc. Powder Technol., 25, 287–291 (1988).
a is the average radius of a particle image and T the [7] M. Matsushita, K. Itoh, M. Ohnishi, T. Ogawa, M. Kohno,
0
cycle of trigonometric functions. In the series of M. Tanemura, H. Honda, K. Miyamoto, K. Miyazaki and
Fourier coefficients a and b , the low-order coeffi- N. Yoshimoto: Katachi no kagaku, Asakura shoten (1987).
n
n
cients with small n value express the large scale sur- [8] M. Suzuki, S. Yamada, H. Kada, M. Hirota and
face roughness and the high order coefficients with T. Oshima: J. Soc. Powder Technol., 34, 4–9 (1997).
large n value express the small scale surface rough- [9] K. Otani, H. Minoshima, T. Uchiyama, K. Shinohara,
ness. This Fourier-analysis method has the merit to K. Takayashiki and T. Ura: J. Soc. Powder Technol., 32,
synthesize the original particle perimeter from the 151–157 (1995).
Fourier coefficients. Synthesizing the particle shape is
impossible by the other method. In order to rebuild the
original outline completely, the infinite number of 1.4 Particle density
Fourier coefficients would be required.
1.4.1 Density measurement of powders composed of
1.3.6 Particle shape analysis of nanoparticle
nanoparticles
A particle shape analyzer is not available for nanopar-
ticle, and so a method based on the microscopic parti- (1) Definitions of powder density
cle image is used usually. In order to obtain the average Powder is an inhomogeneous material in that there are
result about many particles, the ratio of two particle gaps between constituent particles, and there may be
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