Number-weighted distributions

Figure 2 Overlay of MALDI TOF spectra and the calculated differential number weight distributions obtained via SEC analysis ofpolv(dimethylsiloxane)s I, II and III M (SEC) =1920, 5000, 10900 (SEC) =2470. 5920. 12700 M (MALDI TOF) =1290, 2720, 5350, M , (MALDI TOF) = 1460. 2890, 5530 g mol- respectively...

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. 

The electrophoretic motion is either measured microscopically or by light scattering. The former way is called microelectrophoresis and usually employs ultramicroscopes when dealing with colloidal particle systems. The optical instrumentation can be identical to that of DUM, while the software has to be modified because only the displacement in the direction of the electric field is relevant. The method yields a number weighted distribution of zeta-potentials. Similar to DUM, a sufficiently large number of trajectories has to be evaluated in order to keep the statistical uncertainty within an acceptable level. Moreover, the method may be insensitive to weak scatterers within a polydisperse colloidal suspension. 

Eventually the exemplary DLCA population is used to illustrate the effect of ignoring the aggregate structure when analysing DLS and OCA data. For that purpose the virtual measurement distributions in Fig. 4.28—fet(- stokes), int.goA- h.eff), gint,i73°(- h,eff)—2 6 Converted to number weighted distribution... 

Conversion from these distribution functions to number weighted distributions of Xh,t and N were mn by assuming ... 

On the other hand, a remarkably good agreement between the number weighted distributions o( h,t) qo(N) of the centrifugation and of the backscattering DLS is observed for almost all silica grades (Figs. 4.32 and 4.33). Severe deviations are... 

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)...

Fig. 4.34 Number weighted distribution of the translational hydrodynamic diameter o(aii,t) and number weighted distribution of the aggregation number qo(N) calculated from size distributions measured with OCA and DLS on p5uogenic silica suspensions (six different grades with BET = 50-300 m%)...

Fig. 4.35 Correlation between the medians of the number weighted distribution...

The simplest way to introduce the size distribution is by counting. Suppose that we measure the size of each particle (e.g, with a microscope), sort the particles according to their size into bins , and then count the particle number in each bin. This process leads to the number-weighted distribution. Let us introduce the particle number density, N R), where N(R)dR represents the number of particles per unit volume with radii between R and R -1- d/ , whereby we assume spherical particles for simplicity. The total particle concentration C, being defined as the number of particles per unit volume, is given by the following ... 

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. 

Other useful characteristics of any probability distribution are its moments. For the particle size distribution, we can introduce the moments with different weight, but for simplicity we shall only refer to moments defined with respect to the number-weighted distribution. The moment of order m is defined as follows ... 

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