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

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coefficient, and particle size, cannot be determined unambiguously from 
the decay rate of the autocorrelation function. The net effects of multiple 
scattering are that the instrument factor B/A decreases, and the 
autocorrelation factor decays faster, leading to too low an estimate for 
particle size. Thus, multiple scattering limits the application of the 
technique to low concentration dispersions (< 0.01% by volume), although 
techniques have been developed to overcome this condition. 
10.14.6 Particle interaction 
Since most colloidal dispersions are stabilized by particle interactions, the 
use of equation (10.51) may lead to biased estimates of particle size that 
are often concentration dependent. The effect may be taken into account 
by expanding the diffusion coefficient to a concentration power series that, 
at low concentrations, gives: 
D,^D^{\-^kj,c) (10.41) 
The equation reduces to the Stokes-Einstein equation for spherical 
particles. Since the friction coefficient for a non-spherical particle always 
exceeds the friction coefficient for a spherical particle, over estimation of 
particle size will occur if equation (10.41) is applied. 
The virial coefficient kD is positive for repulsive particle interaction 
and negative for attractive interaction. Thus if particle interaction is 
neglected the apparent size will be concentration dependent, increasing 
with increasing concentration for attractive interactions and decreasing 
with repulsive interactions. In such cases, the diffusion coefficient should 
be determined at a range of concentrations and D^ determined by 
extrapolating to zero concentration. 
Field scanning methods 591 
The effect of particle interaction is proportional to the average interparticle 
distance that, for a fixed volume concentration, decreases w i^th particle 
size. Hence, the effect of interaction reduces as particle size increases. 
However, small particles scatter much less light than large particles and it 
is necessary to use a higher concentration for reliable PCS measurements. 
In these cases the concentration needs to be increased to volume fractions 
up to 0.1% and, again, particle sizes can only be determined from 
extrapolations to zero concentration. 
10.14.7 Particle size effects 
PCS relies on uneven bombardment of particles by liquid molecules that 
causes the particle to move about in a random manner and this limits the 
technique to particles smaller than 2 or 3 |im. 
In order to avoid bias due to number fluctuations, it is necessary that 
there is at least 1000 particles present in the measuring volume and, for a 
typical value of the scattering volume of 10&quot;^ cm^, effects of number 
fluctuations are to be expected for particle diameters greater than around 
0.5 |Lim. Number fluctuations lead to an additional time decaying term in 
the autocorrelation function. Since the characteristic decay time of this 
additional term is usually much slovs^ er than the decay attributed to 
Brownian motion, the average particle size, which is proportional to the 
average decay time, will be overestimated if the effect of number 
fluctuations is neglected [277]. 
Loss of large particles due to sedimentation effects can usually be 
considered negligible. Stokes' law predicts that a 1 ^m particle of density 
3000 kg m'^ sediments in water at a rate of about 1 |Lim s&quot;^ so that, in 3 min, 
there will be no particles larger than 1 |am at a depth of 0.2 mm below the 
surface. Since the measuring volume is usually situated several mm below 
the surface, this effect is only important for unduly protracted 
measurement times. 
10.14.8 Polydispersity 
For monodisperse samples, a plot of G(r) against r gives a straight line 
with a constant slope which is inversely proportional to particle size. For 
polydisperse samples, the relationship is multi-exponential and a plot of 
G{T) against racquires curvature, the degree of which increases with 
increasing polydispersity [278]. 
592 Powder sampling and particle size determination 
The autocorrelation function for a polydisperse system represents the 
weighted sum of decaying exponential functions, each of which 
corresponds to a different particle diameter. For such a system: 
G ( r ) - J F ( r ) e x p ( - r r ) d r (10.42) 
F{r) is the normalized distribution of decay constants of the scatterers in 
suspension. Given G{T) it is necessary to invert equation (10.42) in order 
to determine F{r), Unfortunately, the inversion is ill-posed in that there 
are an infinite number of distributions which satisfy this equation within 
the experimental error to be found in G{T). A large number of algorithms 
have been suggested for the inversion and an evaluation of their 
performance can be found in Stock and Ray [279]. 
The autocorrelation function can also be analyzed by the method of 
cumulants. In this approach G(r) is fitted to a low order polynomial. For a 
third order cumulants fit: 
r 1 A 1 
^ \ ^ 
G(r) = - r r + - W^^\ - U^T^ (10.43) 
An average particle size is obtained from the average decay rate 7&quot; using 
equations (10.41-10.43) and an indication of spread (or polydispersity) is 
given by o^. 
An advantage of the cumulants approach is that it is computationally 
very fast. A chi-squared fitting error parameter serves to test whether the 
assumed Gaussian shape in diffusivities is reasonable. The calculated 
values of mean size and polydispersity are reasonable (chi-squared 
approaching unity) for approximately symmetrical distributions having a 
coefficient of variation within 25% of mean size. 
Commercially available instruments usually employ both approaches. 
For highly skewed distributions or distributions having more than one 
mode, an inversion algorithm must be used [280] whereas for narrowly 
classified mono-modal distributions the cumulants approach is satisfactory. 
The relative second moment, ^^2//^^ , a dimensionless quantity, is a 
measure of polydispersity. It is the intensity-weighted variance divided by 
the square of the intensity-weighted average of the diffusion coefficient 
distribution. The relative second moment is also called the polydispersity 
Field scanning methods 593 
index that characterizes the spread of the decay rates and hence the spread 
of particle size about the average value. 
Most inversion methods (e.g. Contin [281] and maximum entropy 
method) [282], require prior knowledge of the distribution. 
The singular value analysis and reconstruction method (SVR) reduces 
the inversion problem to a well-conditioned problem, thus eliminating the 
need for prior knowledge [283]. 
Other methods of translating the polydispersity index into size 
distribution information have been proposed [284]; but the reliability of the 
transformations is in question. Finsey details these procedures in a review 
containing 67 references [285]. A later review contains 292 references 
10.14.9 The controlled reference method 
In the controlled reference method laser light is guided into the sample cell 
by an optical waveguide. Particles within 50 \xxn of the tip of the wave 
guide (a fiber optic probe) scatter light, some of which is reflected back 
into the fiber and transmitted back through the guide. The reflected light 
from the interface between the guide tip and the suspension is also 
transmitted back. If these two components are coherent they will interfere 
with each other and result in a component of signal that has the difference 
or beat frequency between the reflected and scattered components. The 
difference frequencies are the same as the desired Doppler shifts. The 
received signal resembles random noise at the output of the silicon 
photodiode as a result of the mixing of the Doppler shifts from all the 
particles scattering the laser light. The photodiode output is digitized and 
the power spectrum of the signal is determined using fast Fourier transform 
techniques. The spectrum is then analyzed to determine the particle size