pp. 33-47 in: "Science of Whitewares"; eds. V.E. Henkes et al; The American Ceramic Society, Wesrerville, OH (1996)
LIMITS IN PARTICLE SIZE CHARACTERIZATION
Herbert Giesche, NYS College of Ceramics at Alfred University
ABSTRACT
The limitations and specific requirements of different particle size characterization
methods are discussed. The theoretical background of each method is shortly summarized
and limits in terms of minimum or maximum measurable particle size are discussed
together with several experimental parameters which can influence the accuracy or which
can cause systematic errors. The paper focuses on sedimentation and optical (light
scattering) characterization methods.
INTRODUCTION
Particle size is one of the most important characteristics in ceramic processing. A precise
analysis as well as control throughout all processing steps is essential for high quality
production and low loss rates. A large number of publications dealt with specific aspects
in sample preparation and the analysis itself. The precise and controlled preparation of a
representative sample is the first and essential step in getting reliable analysis results.
This sample preparation has to be as closely related as possible to the actual processing
procedure or vice versa to the information and conclusions to be drawn from those
measurements. Despite the importance of those steps, the present paper will not address
those aspects but rather focus on general limitations and possible errors of the various
groups of instruments or techniques.
DEFINITION OF PARTICLE SIZE
In most cases particle size is described as the size of a spherical particle, which would
show the same behavior as the actual particle in the specific analysis. As shown below, a
variety of size descriptions exist and one has to specify in each presentation of the results,
which kind of particle size and how it was determined.
Geometric properties:
- Diameter, Chord Length (Si ),
Feret- (XF ), or Martin- (XM ) Diameter
- Surface Equal Sphere
X = Surface
S
π
- Projected Area
X PM = 4 Area π
(! non spherical particles are not
always in a random position)
- Volume Equal Sphere
X V = 3 6 Volumeπ
18 η w g
Sedimentation Equal Sphere (Stokes):
XW =
(ρ p − ρ fl ) g
Si
10 0
10 0
80
80
60
60
40
40
20
20
0
1
10
10 0
m ass or volume %
number %
Moreover not only the precision or reliability of the analysis is important but also the
way those results are presented. Some are quite obvious, like the difference between a
cumulative and differential distribution, or using a linear or logarithmic scale on the size
axis. However, there are some hidden differences which are not always indicated or
realized. For example, the average particle size will change depending on whether a
number-, area-, or volume- (mass-) distribution is shown. There is no general rule, which
of those presentations to recommend, this very much depends on the preferences of the
operator or the specific requirements of the process. As shown in fig. 1 the sample will
have a vastly different mean particle size when plotted as a number- or volumedistribution.
0
10 0 0
Pa rt ic le S iz e / um
Fig. 1: Micrograph (left) and cumulative size distribution (right)
The situation gets even more complicated, since most real powders are not spherical. A
shape factor can be introduced to account for this. One possible definition was provided
by Wadell.1 the shape factor, Ψ, is defined as the surface-ratio of a volume equal sphere
2
XV
⎛
⎞
over a surface equal sphere. Ψ = ⎜
X S ⎟⎠
⎝
As can be shown easily the following rank of order is valid for non-spherical particles.
XS = XPM ≥ XV ≥ XW
IMAGE ANALYSIS
Optical- or electron-micrographs can be used to determine the particle size of a sample
according to the various definition as mentioned earlier. Even so this technique may
provide much more information than just the particle size, there are a number of errors
resulting from sample preparation or the analysis set-up. Fig. 2 demonstrates some of
those errors, like the under-cut or the random or stable orientation of particles. Especially
small particles may not be counted when they are covered by larger particles, or the
measured particle size is larger for a sample with powder particles in a stable position
compared to randomly oriented particles. In addition computerized image analysis can
give faulty results depending on a correct alignment of the threshold value and/or
different orientations of particles towards the screen pixels. The latter is especially true
for needle like particles as demonstrated in Fig. 3.
under cut, random, or stable
Fig. 2 Errors due to particle position
Influence of orientation: a) 22 pixels b) 13 pixels
Fig. 3 Computer assisted image analysis
SIEVE ANALYSIS
Even with such a simple technique as the sieve analysis a number of errors have to be
considered. Some of those errors are quite obvious, like damaged or uneven sieves, a too
short analysis time, insufficient tapping, abrasion, breaking of particles, humidity,
clogging, or agglomeration of the powder. Yet, in case the tapping frequency becomes
too high, the passing-% of the sample likewise decreases. In the latter situation particles
do not have enough time to pass through the sieve opening before the movement of the
sieve will bounce them back. Thus for a fixed analysis time the amount of sample,
remaining on the sieve, passes through a minimum, as shown in fig. 4. Further
information about the correct sieve analysis is given in several reference papers.1-5
Fig. 4 Influence of time and vibration frequency on sieve analysis after F. G. Carpenter
and V. R. Deitz2
SEDIMENTATION
Using
Stokes
law,6
sedimentation
experiments will provide information about
the particle size distribution. Stokes law
relates the gravitational, the buoyancy, and
the drag force acting on the moving
(settling) particle in a medium, as shown
below.
The greatest uncertainty in this equation is
related to the drag force. The drag
coefficient, Cw , the Reynolds number, Re,
and the projected area of the settling particle can not be described sufficiently in all
cases. The linear relation between
Reynolds number and drag coefficient,
Cw = 24/Re, is valid only in the laminar
flow regime at Reynold numbers below
0.25. This relation is used for the basic
Stokes equation as utilized in most sedimentation experiments. At higher Reynold
numbers a transitional region and finally
turbulent flow is observed as shown in fig.
5, where spherical particles were assumed.
FG = VP ρ P g =
π
x3 ρP g
buoyancy: FA = VP ρ fl g =
π
x 3 ρ fl g
weight:
drag force: FW = c W (Re) A P
6
6
ρ fl
2
with:
VP : volume of particle
ρ pl or ρ fl : density
g:
gravity constant
x: particle diameter
c W (Re): drag coefficient
AP :
projected area of particle
w g : sedimentation velocity
w 2g
1,000
100
transition range for:
0.25 < Cw < 2000
Cw
10
turbulent
flow for
Cw ≅ 0.45
laminar
flow for:
1 Re < 0.25
Cw ≅ 24/Re
0.1
0.1
1
10
100
1000
10000
100000
Reynolds number
Fig. 5 Drag coefficient as a function of Reynolds numbers for spherical particles
The limitation in the range of Reynolds numbers thus limits the maximum particle size to
be determined for a specific system, since ordinarily the basic Stokes equation, valid for
the laminar regime, is used without further corrections. Table 1 summarizes those
limitations for a variety of materials.
Table 1: Laminar flow restriction, maximum particle diameter at Re=0.25
material
X max = 3
4.5 η
(ρ p − ρ fl ) ρ fl g
2
flour
quartz
alumina
lead
density
[g/cm3]
1.5
2.65
3.96
11.4
Xmax
water
97.2
65.3
53.5
35.3
in
µm
air
43.7
36.2
31.9
22.2
Yet the measurable range can be extended by using various equations to correct for the
non linear relation between Cw and Re in the transition range. Table 2 provides a brief
overview of equations, the recommended range, and the associated error for Cw for each
equation over this range.
In order to expand the measurable range a different dispersion media could also be used.
Having a higher viscosity liquid expand the range to larger particle sizes. One example is
shown in the following diagram (fig. 6).
Table 2: Correction formulae for the transition range7
formula Cw = ...
Re-range
max. error %
(24/Re) +2
< 10
-4/+4
(24/Re) +1
<102
+14/-40
(24/Re) + 0.5
< 105
+32/-40
(24/Re)+(2.8/Re1/4)
0.1 < Re < 4 103
+2/-8
(21/Re)+(6/Re1/2)+0.28
0.1 < Re < 4 103
+4/-4
(24/Re)+(3.6/Re0.313)
0.1 < Re < 103
+8/-5
((4.8/Re)+0.63)2
0.1 < Re < 103
+20/-1
(24/Re)+(4/Re1/2)+0.4
< 2 105
+6/-6
Sieve Analysis
% Passing
150
100
50
0
0
200
400
600
Particle size / um
Volume %
100
80
60
40
20
0
0
160
320
480
640
size / um
Fig. 6 Sieve analysis of a spray-dried powder compared with sedimentation analysis in
ethylene glycol.
The sieve analysis and particle size distributions determined from sedimentation analysis
of a spray dried powder correlated well. Ethylene glycol was used as the dispersion
medium, having a viscosity of 16.1 mPa s and a density of 1.1101 g cm-3. The powder
immersed in the liquid had a density of 1.93 g cm-3, and thus particle sizes of up to about
0.5 mm could be measured in this way.
In addition to limitations by the laminar flow regime or the maximum particle size, the
shape of the settling particles has a noticeable influence on the sedimentation velocity.
The following equation relates the sedimentation velocity with other particle parameters
by using the previously defined shape factor, (XV /Xp )2.
4 (ρ p − ρ fl ) g
Wg =
3
ρ fl
2
⎛ XV ⎞
XV
⎜⎜
⎟⎟
⎝ X P ⎠ CW (Re( X P ))
Several other factors are less easily described by correction formulae. The most
prominent effects are temperature and concentration or density gradients in the system, or
the increased Brownian motion of small particles. For example, the particle concentration
not only decreases the actual sedimentation velocity at high solid loading (hindered
sedimentation or particle particle interaction), at intermediate concentrations some
models even predict a faster sedimentation due to particles following the draft of other
particles and thus increased sedimentation velocity.8
In addition, several instrument specific factors have to be considered. In general the
particle concentration and vice versa the particle size distribution is determined as a
function of time and position by means of light or x-ray absorption, osmotic pressure, or
the weight of the settled material. However, the detector signal is not always linear
Fig. 7 Extinction coefficient of lime stone and quartz as a function of size for whithe light.
proportional to the particle size at a given concentration. Especially light detectors
require major correction at or particularly below particle sizes of 1 µm. Fig. 7 provides
one example of the latter effect for quartz and lime stone. For example, 6 µm quartz particles will cause a 3 times higher light absorption compared to the same amount of 2 µm
particles. As the particle size increases above 10 µm, those corrections become relative
insignificant. However in the micron and submicron size range the latter effect can cause
serious mistakes in the calculated particle size distribution. In order to avoid this effect
some instruments use x-ray detectors which are independent of the particle size in this
range. X-ray detectors, however will need higher particle concentrations than light detectors and this can influence the sedimentation velocity as demonstrated earlier. In case
the analysis-material contains strong x-ray absorber, like elements with a high atomic
number, the x-ray detection principle is ideally suited for those measurements down to
particle sizes of 10 nm. Besides the fat that Brownian motion becomes significant in this
range.
INTERACTION OF LIGHT WITH PARTICLES
Particles can interact with electromagnetic fields in various ways. The Coulter Counter is
one of those principle, however, it will not be covered here in further detail. It is based on
the change of electrical resistivity in a high frequency field when a particle passes
through a narrow connection hole placed between both electrodes. From the change in
resistivity the particle volume and thus the particle size can be calculated. The interaction
of light with particles is another principle which can be used to determine the particle
size.9,10 Depending on the size of the particle and the wavelength of the light, various
effects can be observed. Large particles with a large size-wavelength ratio α = π X / λ >>
1 display Fraunhofer diffraction, whereas Rayleigh-Scattering dominates for small
particles (α << 1). For the transition range of 0.1 < α < 10 the Mie theory11 will describe
the light scattering behavior, Yet the latter theory is relative complicated and has been
solved only for spherical particles. The scattering intensity varies largely with scattering
angle, particle size, refractive index of the materials, and the polarization direction of the
light-source / -detector. Fig. 8 shows one example indicating the complex behavior in the
intensity distribution.
Fig. 8 Mie Polarogram for λ = 0.4 µm, α =5, and refractive index m=1.46+0i
For relative small particles the Rayleigh- or dipole-scattering is observed, also called
Tyndall effect, and the corresponding formula is relative simple.
2
4
6
I =π X
I0
8 λ4
⎛ m2 −1 ⎞
X6
m
⎟⎟ 1 + cos 2 Θ ∝ 4 with m = p
⎜⎜ 2
m fl
λ
⎝m + 2⎠
The angular dependence of the intensity is symmetric and independent (within the given
limits) of the particle size. A particle size of 20 nm is about the upper limit when using
visible light.12
For larger particles the following formula will describe the scattering behavior:
I
(
)
X4
X2
⎛ π ⎞ 4 ⎛ J1 (α Θ ) ⎞
X ⎜2
=
⎟ ∝ 2 (for Θ = 0) or ∝ 2 (for Θ ≠ 0)
I0 ⎜⎝ 4 λ ⎟⎠
λ
λ
⎝ αΘ ⎠
with J1 (α Θ ) = Bessel function
2
2
Theory predicts very wavy curves and a larger opening of the aperture angle is to be used
in order to smoothen the data. The smaller the particles the more it is recommended to
measure in the forward direction since this will provide the highest scattering intensities.
However, the X4 dependency will limit the measurable range in the forward direction. In
addition the scattering intensity will become independent of the refractive index of the
material for measurements in the forward direction. Fig. 9 shows the light scattering
intensity as a function of particle size for monochromatic and white light.7
Two facts become evident from those graphs. First, a monochromatic light source and a
small aperture size will cause a very noisy curve due to the many maxima and minima in
the diffraction curve (left diagram). Using a broader wave length spectrum and a larger
aperture size will smoothen the data, as shown in the right diagram, since the peaks of the
different wavelength and for the different observation angles will cancel out those sharp
maxima. Secondly it is apparent that the refractive index of the material has only a minor
effect on the curves, however, absorption properties of the material, indicated by the
imaginary part in the refractive index (curve 2), will cause a pronounced down shift of
the scattering intensity curve. The extinction coefficient, which is the combination of
light scattering and absorption has the same behavior. Thus in order to smoothen the data
a large aperture and a broad wavelength range is recommended. An example for quartz
and lime stone was shown earlier in fig. 6.
Another effect which has to be considered seriously in light scattering experiments is the
coincidence error. In principle the diffraction pattern and intensity distribution is
determined for single particles. Thus, in case the particle concentration becomes too high,
several particles will be measured at the same time and correspondingly critical errors
can occur. Fig. 10 demonstrates this relation in a graph.7 The importance of small
measuring volumes and/or low particle concentrations become evident. A coincidence
error of 5% should be seen as an upper limit. Most light scattering instruments limit the
actual measurement volume by focusing the light source and reducing the detector
aperture as shown in the schematic diagram in fig. 10.
10
10
9
Intensity
0 deg
7
15
10
5
30
90
10
3
10
1
10
= 0.4 m m = 1.46
-1
0.1
1
10
100
particle size /um
Fig. 9 Light scattering intensity for monochromatic- (left) and white-light (right)
Fig. 10 Maximum number of particles, Cn , as a function of the coincidence error, fk , for
various measuring volumes.
Moreover each sample has to be prepared and analyzed carefully and the results should
be critically evaluated. Fig. 11 gives an example of a spray dried powder. Since it
disintegrated into the primary particles when dispersed in water, propanol or other
organic liquids were used as dispersion medium. Experiments in test tubes indicated that
the powder was well dispersed and that it did not disintegrate with time. However, when
measured with a light scattering instrument, Microtrac FRA, Leeds Northrupp, a constant
shift towards smaller particle sizes was observed, clearly an indication of powder
disintegration. The pumping unit of the instrument caused such high shear forces that the
particles broke apart.
100
3 min.
80
60
40
20
0
100
80
21 min.
60
40
20
0
100
80
37 min.
60
40
20
0
100
80
62 min.
60
40
20
0
0.13
1.06
8.5
68
543
Particle size / um
Fig. 11 Disintegration of spray-dried powder under shear. Particle size as a function of
time.
CONCLUSIONS
Various particle size characterization techniques have been described and critically
analyzed. The limitations of each method were demonstrated. However a proper
knowledge of those limitation can be used to modify the analysis conditions in such a
way to accomplish a particle size analysis even so it is out side of the regular range.
Examples also demonstrated that one should always try to characterize samples with
different techniques, since systematic errors are not always obvious. Finally the operator
has to judge, which sample preparation will provide the most reliable results. In addition
it has to be stressed that different method will present results in different ways and
especially the kind of distribution, e.g. a number-, area-, or volume-distribution should
always be clearly labeled. Again, the operator has to decide, which presentation is the
most useful to characterize or control the process.
REFERENCES
1.
2.
3.
H. Wadell; J. Geology, 40 (1932) 443-51
F. G. Carpenter, V. R. Deitz; J. Res. of NBS, 45 (1950) 328-45
K. T. Whitby; Symp. on Particle Size Measurement, Spec. Tech. Pub. No.234,
ASTM 1958, 3/25
4. H. Rumpf; Staub 20 [8] (1960) 253-66
5. K. Leschonski; Proc. of 3rd Particle Size Analysis Conf., (1978) 205-17, HeydenVerlag, London
6. C. G. Stokes; Trans. Cam. Phil. Soc. 9 (1951)
7. K. Leschonski; Grundlagen und moderne Verfahren der Partikelmeßtechnik, GVT
Kurs, Clausthal (1986)
8. B. Koglin; Chem.-Ing.-Techn. 44 [8] (1972) 515-21
9. M. Kerker, The scattering of light, Academic Press, New York (1969)
10. H.C. van de Hulst; Light scattering by small particles, Wiley, NY (1957)
11. G. Mie; Ann Phys 25 (1908) 377-445
12. R. Brossmann, Ph.D. Thesis, Karlsruhe (1966)