Field Scanning Methods of particle size measurement
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Field Scanning Methods of particle size measurement

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been presented by Witt and Rothele [138]. 
10.5.1 Theoretical basis for MALLS instruments 
The angular intensity distribution of scattered light \{0,/J) at angle 6, for 
particles of diameter d and refractive index // is given by Mie scattering 
theory. For a polydisperse distribution: 
l{e,n) = ^^h{a,9,^}x'^nda (10.27) 
which is also known as a non-homogeneous Fredholm equation of the first 
kind. 7(0) is the intensity of the incident beam, n{a) is the number of 
particles in the laser beam with sizes between a and a+da, where 
a = nD/A, X is the wavelength of light in the surrounding medium. The 
solution depends on the exact form of the kemal function h{a,6,ij). 
For a ring detector the scattered light arriving at the /th ring of the 
detector is given by: 
/(^^,;/) = Zv^ \h{a,e^^)x^nda (10.28) 
Similar equations apply to each ring of the detector. This results in a linear 
set of / equations with k (discrete size classes) unknowns. The vector of 
scattered energy can be expressed as a matrix equation: 
S=CW (10.29) 
where S is the real vector of scattered energy, W is the real vector of size 
distribution of particles with diameters between the limits for the discrete 
size classes and C is the coefficient matrix of / rows and k columns. The 
element cik indicates that unit mass of particles in the Ai:h size interval 
produces a scattering signal on the /th ring of the detector. For this reason 
the particle sizes are divided into k discrete size classes that are determined 
by the following equation [139]: 
afi' =1.357 (10.30) 
548 Powder sampling and particle size determination 
where a = nDj/Ji, 6^ = rjf, is the maximum radius of each ring of the 
detector and/is the focal length. 
Equation (10.30) indicates that the maximum of scattered energy is 
reached only in one position of the detector, for representative diameters of 
each size class. The scattering matrix depends upon two parameters, the 
focal length of the receiving lens and the relative refractive index. 
Equation (10.30) is ill-posed so that the scattering matrix C is highly ill-
conditioned. This implies high-frequency oscillations in the solution W. 
Thus more sophisticated methods are required in order to find a solution. 
Integral transformations are used successfully [140] giving close solutions 
to equation (10.30). Alternatively a function constrained inversion in 
which the size distribution is assumed to follow a given form with two 
degrees of freedom or a model independent function can be employed. 
Alvares et. al. [141] successfully applied a method known as the 
Tikhonov Regularization method and L-curve criterion to generate data in 
close accord with the Malvern software. 
Fraunhofer theory applies to the scattering of light in the near forward 
direction by large particles. The scattering pattern for a single spherical 
particle consists of a series of light and dark concentric rings that decrease 
in intensity with increasing angular position. These rings are produced 
through constructive and destructive interference of light diffracted from 
the edge of the particle with changing light path length. The angular 
distribution of light flux I{d) for a single opaque spherical particle, as 
given by the Fraunhofer equation, is shown in equation (10.31) in terms of 
the first order spherical Bessel function J^iO). 
I{9) = I{Q) 2Ji(ad)' 
For a distribution of particle sizes this becomes: 
/(,)=/(o)|MM) 2 f{D)AD (10.32) 
Figure 10.5 presents a two-dimensional graph of the pattern for latex beads 
in air (/I = 1.55) of diameter 50 and 25 |Lim. The diffraction pattern is 
independent of the optical properties of the particles and is a unique 
function of particle size being defined by the size parameter a=^nD/X, 
where D is the particle diameter. The scale of the curve on the vertical 
Field scanning methods 549 
(0) degrees 
Fig. 10.5 The forward intensity distributions for single particles: note that 
84% of the energy lies within the first minimum. 
2 4 6 
Scattering angle (degrees) 
Fig. 10.6 Intensity distribution from three particles of size 10, 60 and 
200 |Lim. The intensity scale is times 1 for the 200 |Lim particle, times 100 
for the 60 |Lim particle and times 105 for the 10 |Lim particle [142]. 
550 Powder sampling and particle size determination 
axis decreases with increasing particle size, compressing to smaller angles 
on the horizontal axis, so that the 50 |Lim particle results in a curve 
compressed by a factor of two compared to the 25 \im particles. Over a 
particle size range of 1000:1 the scale changes a thousand fold. 
Figure 10.6 shows the relative intensities from three opaque particles, in 
air, of sizes 10, 60 and 200 iiim; this illustrates how the scattered flux in the 
forward direction falls off rapidly with decreasing particle size. The effect 
is also illustrated in Figure 10.7a together with the resulting diffraction 
pattern for a monosize distribution (Figure 10.7b). 
Information on the particle size distribution can be found by measuring 
the scattered light flux at several radial locations to characterize the series 
of annular rings from the particle's diffraction pattern. It is then necessary 
to invert the relationship between the scattering pattern and the particle 
size distribution. This can be done using iterative methods or analytical 
inversion techniques. 
In early instruments, the detectors consisted of a series of half rings 
[143,144] (Figure 10.8) so that a matrix equation developed. Sliepcevich 
and co-workers [145,146] inverted this equation to obtain the particle size 
distribution. The equation was solved by assuming the distribution fitted a 
standard equation and carrying out an iteration to obtain the best fit. A 
matrix inversion was not possible due to the large dynamic range of the 
coefficients and experimental noise that could give rise to non-physical 
results. An inversion procedure that overcame these problems was 
developed by Philips [147] and Twomey [148] that eliminated the need to 
assume a shape for the distribution curve. 
rx-rr . A^u. detector Diffracted light 
Laser beam 
(a) (b) 
Fig. 10.7 (a) Particle size determines diffraction angle (b) the diffraction 
pattern from a monosize distribution. 
Field scanning methods 551 
Fig. 10.8 Representation of a typical photosensitive silicon detector. The 
thin lines represent insulating gaps. 
The mathematics of the single-event Fraunhofer diffraction of an on-axis 
laser beam by a sphere was used in the development of this technique 
[149]. The assumption of single scattering is adequate for accurate 
measurement so long as the light obscuration by the particle field lies 
within the range 5-50%. Although measurement at low concentration is 
desirable it is not always possible; with industrial sprays, for example, size 
measurements may have to be made for light obscuration values in the 
range 90-99%. The more that multiple scattering occurs, the more the 
particle size distribution is biased to smaller sizes if single event theory is 
wrongly used. 
Boxman et. al suggest that more information can be obtained if the 
fluctuations in the signals from each detector are examined together with 
the mean values [150]. They note that this approach can identify whether 
the inaccuracy is due to insufficient sampling of the detector array or 
imperfections in the optical model. More recently Knight et. al [151] 
developed an analytical inversion method that gave improved resolution 
and accuracy in size distribution measurement. 
Considerable differences exist between instruments, both in hardware 
and software, so that there is a lack of agreement in data generated by 
different instruments. 
A commercially used means of measuring the scatter pattern is with a 
logarithmic line array detector, which has detector elements in a geometric 
size progression with each element larger than the preceding one by a