FIGURE 10.11 Decay of the autocorrelation function with time interval due to Brownian motion of the particles. The decay occurs more rapidly for small particles because they move faster.

Here, n is the refractive index of the medium, X is the wavelength of light, and 8 is the scattering angle. By measuring the decay of the autocorrelation function with time, it is possible to determine the diffusion coefficient and hence the particle size from the Stokes-Einstein equation (Equation 10.2).

PCS is suitable for accurate determination of particle size in suspensions that are monodisperse or that have a narrow size distribution. It is less reliable for suspensions with broad size distributions because of difficulties associated with interpreting the more complex autocorrelation decay curves (Horne 1995). PCS can be used to monitor the flocculation of particles in suspensions because aggregation causes them to move more slowly (Dalgleish and Hallet 1995). It can also be used to determine the thickness of adsorbed layers of emulsifier on spherical particles (Dalgleish and Hallet 1995). The radius of the spherical particles is measured in the absence of emulsifier (provided they are stable to aggregation) and then in the presence of emulsifier. The difference in radius is equal to the thickness of the adsorbed layer, although this value may also include the presence of any solvent molecules associated with the emulsifier. PCS is restricted to the analysis of dilute suspensions of particles (0 < 0.1%).

Doppler Shift Spectroscopy. When a laser beam is scattered from a moving particle, it experiences a shift in frequency, known as a Doppler shift (Trainer et al. 1992, Horne 1995). The frequency of the scattered wave increases slightly when the particle moves toward the laser beam and decreases slightly when it moves away. Consequently, there is a symmetrical distribution of Doppler shifts around the original frequency of the laser beam. The magnitude of the Doppler shift increases with particle velocity, and hence with decreasing particle size. Consequently, there is a broad distribution of Doppler shifts in a polydisperse suspension that contains particles moving at different velocities (Figure 10.12). The particle size distribution is determined by analyzing this Doppler shift spectrum. Two types of measurement techniques are commonly used in DSS: homodyne and heterodyne. Homodyne techniques determine the Doppler shifts by making use of the interference of light scattered from one particle with that scattered from all the other particles, whereas heterodyne techniques determine frequency shifts by comparing the frequency of the light scattered from the particles with a reference beam of fixed frequency (Trainer et al. 1992). The heterodyne technique is capable of analyzing suspensions with much higher particle concentrations than is possible using the homodyne technique, and so only it will be considered here.

A typical experimental arrangement for making Doppler shift measurements is shown in Figure 10.13. A laser beam is propagated along an optical waveguide which is immersed in the sample being analyzed. Part of the laser beam is reflected from the end of the waveguide and returns to the detector, where it is used as a reference beam because its frequency is not

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