Page 36 - Book Hosokawa Nanoparticle Technology Handbook

1.3 PARTICLE SHAPE FUNDAMENTALS
nanoparticle. Quantification of particle shape is clas- same operation is repeated. If the relation between r
sified roughly into the shape index which is the ratio and N(r) shows a straight line on log–log paper as
of two kinds of different definition particle diameter shown Fig.1.3.1, the value corresponding to the incli-
and the other expression such as fractal dimension or nation of this straight line is defined as the fractal
Fourier analysis of particle perimeter. In the case of dimension D [6].
the ratio of two different definition particle diameter, D
there are many similar kinds of shape indices such as Nr() r (1.3.1)
degree of elongation (aspect ratio) major axis/minor A high value of fractal dimension D means a rough
axis, degree of flatness minor axis/thickness, degree rugged surface of a particle, and a value of fractal
of circular equivalent area diameter (Heywood dimension close to 1 means a smooth surface like
diameter)/equivalent perimeter diameter, degree of spherical beads. The fractal dimension is also
sphericity equivalent volume diameter/equivalent obtained from the number of adsorbed gas molecules
surface area diameter [3]. In addition, unidirectional with different adsorption area instead of a line seg-
maximum particle (Feret diameter)/equivalent ment. In this method, a powder with larger surface
perimeter diameter are also used as a particle shape area has higher measurement accuracy, and the aver-
index, and the value of the shape index of a particle age value showing the three-dimensional shape of
perimeter without concave shows near unity. Each of many particles can be obtained. Thus, this method is
these is a ratio of diameters of a particle, so these effective in shape analysis of a nanoparticle.
shape indices are non-dimensional values and should In the covering method, a particle projection image
not be influenced by the particle size. However, the is covered with the square of r instead of a line seg-
perimeter and the projection area of a fine particle ment, and the relation between the number of squares
have the tendency to decrease with the decrease in containing a projection image or an outline of a parti-
particle size, because the resolution of a particle cle N(r) and size r, is also used for the particle shape
image is getting worse. Therefore, strictly speaking, analysis. Moreover, the turning-radius method is well
the shape indices are influenced by the resolution of used to obtain the fractal dimension of an agglomer-
microscopic particle images. ate particle. In this method, the circle of a radius R is
In these indices, the degree of elongation or aspect drawn from the center of an agglomerate particle, and
ratio is an index with which a particle expresses long the number of the primary particles contained in the
and slender, and the index shows a large value for a circle N(R) is counted. In this method, the fractal
slim fiber particle. The high degree of flatness value dimension D is obtained from the log–log plot of R
means flakey shape. The degree of circularity and the and N(R) [7]. From the log–log plot of the perimeter
degree of sphericity are set to 1 of maximum when a P and the projection area A of each particle, the aver-
particle has a spherical shape, and the index shows a age fractal dimension of many particle perimeters
small value for a non-spherical particle.
The shape indices, expressed with ratios of two par- 3
ticle diameters, are divided into two categories. The 10
degree of elongation, the degree of flatness, etc. in the
one category show the overall shape of a particle, and
the degree of circularity and the degree of sphericity in
the other category show the surface roughness of a par-
ticle. Using the relation between two shape indices in
different categories, difference in particle shape can be
expressed more clearly in the two-dimensional figure
than the case using only one diameter ratio. As an exam-
ple of such a relation, the unidirectional maximum par- Number of lines N(r) (−) 10 2 D = 1.134
ticle diameter (Feret diameter)/diameter of equivalent 3
particle perimeter is plotted on the horizontal axis and 2 r
the degree of circular is plotted on the vertical axis [4]. 1

1.3.4 Particle shape expression by fractal dimension
The fractal dimension is a dimension, taking a real Profile of precipitated calcium carbonate particle
numerical value, proposed by Mandelbrot [5], and is
also used for particle shape expression. The divider 10 −2 −1
method is used to measure the fractal dimension of 10 10 1
particle projection image perimeters. In this method, Opening of divider r (−)
the number of the polygonal lines N(r) necessary to
go around a particle perimeter is measured r as shown Figure 1.3.1
in Fig. 1.3.1. The line of length r is changed and the Fractal dimension measurement using divider method.

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