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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"^ 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"^ 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 2! ^ \ ^ v3/y G(r) = - r r + - W^^\ - U^T^ (10.43) An average particle size is obtained from the average decay rate 7" 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 [286]. 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