Intensity-weighted distribution

The particle size distribution, X(s), above is, unfortunately, not in a form which is useful for most applications. This is because it is a scattered intensity weighted distribution (for brevity, intensity distribution ) rather than a size distribution based on the volume (weight) or number of particles. The difference between distributions weighted in different ways can be most easily explained by relating the various distributions to a number distribution. 

The aim of this paper is to describe the experimental and numerical techniques that, when combined, provide a procedure that enables full particle-size distribution studies of sub-micrometer emulsion systems. We then present distribution results for several oil/water emulsions to demonstrate the ability of these techniques to monitor the effect of processing variables (such as surfactant concentration) on the final emulsion. Finally, we discuss some of the problems of converting the intensity weighted distribution to a mass weighted distribution and suggest methods for minimizing or eliminating some of these problems. 

The distribution obtained from the inversion of Equation 4 is an intensity weighted distribution. That is, it displays the percentage of light scattered from each particle size in the population. What is desired is the amount of mass at each particle size (mass distribution), or the number of particles at each particle size (number or frequency distribution). In principle, once any distribution is known, the others can be calculated. Because of finite detection limits and noise in actual experimental data, however, it is not that straightforward. 

A built-in microcomputer system performs rapid quadratic least squares fit to the data, yielding D, R, a (normalized standard deviation of the intensity weighted distribution of diffusion constants) and x squares goodness of fit. The greater the value of a the larger the degree of polydispersity present in the particle sizes -a values less than 0.2 are generally considered to correspond to pure monodisperse systems. A typical result obtained for 4 x 10 3M CdS-SDS is 5= 7.25 x 10"8 cm2/s, R = 300 A, a = 0.60 and y2... 

Principal reported quantity Volume Intensity weighted distribution Mass Number and summed volume Number Mass... 

The primary result of such a data analysis is the intensity weighted distribution 2int of the translational hydrodynamic diameter Instrament software usually allows for conversion in volume or number weighted distributions, but this requires a model on the relationship between xj, t and the scattering intensity. Furthermore, distribution details that do no contribute significantly to the correlation function (e.g. very fine size fractions) and that are consequently ignored in the measured Qin, also cannot be revealed by numerical conversion. 

Differences between the two methods exist with regard to particle property and type of quantity DUM only evaluates the translational diffusion (xh,t) and probes number frequencies, whereas DLS is also sensitive to the diffusive rotation (xh app) and yields intensity weighted distribution functions. Furthermore, the methods usually differ in sample size Typical sample concentrations in DLS are in the range of 0.01 vol%. These are, for instance, 50,000 particles a 100 nm in a measurement volume of 10 pm, which are all observed in the order of minutes (Willemse et al. 1997), whereas with DUM, the total number of traced particles is smaller by factor 10-100, with an observation time in the order of seconds for each. Last but not least, DLS allows for a temporal resolution in the range from ns to ms, whereas DUM is subject to video processing, typically with 30 frames per second, and is, therefore, not sensitive to very fast relaxation processes (like gradient diffusion). 

An instmmental alternative to microelectrophoresis is electrophoretic light scattering (ELS). The light scattering at migrating particles leads to phase shift (Doppler effect), which can be detected by a heterodyne DLS set-up (i.e. reference-beating with frequency shift). The method yields an intensity weighted distribution of the zeta-potential. 

Fig. 4.31 Left Intensity weighted distribution of the translational hydrodynamic diameter 9ext( h,t) and 9int( h,t) calculated from the measured size distributions shown in Fig. 4.30 right correlation between the median values of the intensity and number weighted distributions measured with 90° and 173° DLS instrumentation (original data in Fig. 4.30)...

Figure 5 Intensity-weighted distributions of relaxation times in DLS at scattering angle 0 = 90° for P( BA2o%- t-AA8o%)ioo- -PAA95 in aqueous solution at 5g/L, O.IM NaCl and for an overall ionization degree a of the AA units of (a) 0%, (b) 20%, (c) 50%, (d) 70%, (e) 90% and (f) 100%...

For n = 0, we recover the number-weighted distribution, namely po(R) — p R), while for n = 3 the mass-weighted distribution is obtained. Of course, other weights can be introduced, for example, n = 2 corresponds to the area-weighted distribution, while n = 6 is sometimes referred to as the intensity-weighted distribution for reasons to be discussed later. 

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