Gaussian size distribution

Figure 11.6 The effect of size dispersion on extinction of visible light by water droplets. Each curve is labeled with a, the standard deviation in the Gaussian size distribution.

A fit taking into account a Gaussian size distribution function g(D) with width a was also performed. Our results are in reasonable agreement with the TEM ones [3], The correction due to the paramagnetic contribution is applied hereafter in all the experiments considered. 

Despite the difficulties of determining the actual nature of the asphaltenes, Sughrue et al. [39] have used size exclusion chromatography to determine molecular sizes of molecules associated with vanadium. As would be expected, the size varies from feed to feed, but - in general - a small amount of vanadium is contained in molecules of ca 3 nm diameter, while most is contained in molecules with a Gaussian size distribution centred on 8-10 nm diameter species. Given the relative diameters of the molecules (ca 8-10 nm) and the pores (ca 20 nm) it is not surprising that vanadium does not penetrate far into the pellets. 

The effect of polydispersity can be seen directly when considering first a system of homogeneous spheres having a Gaussian size distribution [46]. Here the scattering curves are fully determined by the form amplitude BQ(q) as defined... 

Fig. 3. Form factor P(q) of homogeneous spheres with a Gaussian size distribution at constant number average diameter calculated for different standard deviations a. Solid line 0=0 nm dashed line o/Dj =7.5% dotted line o/D =15% (taken from Ref. [46])...

Fig. 4. Isoscattering point for a system of polydisperse core-shell spheres. A Gaussian size distribution with a standard deviation of 9% has been assumed for the cores whereas the thickness of the shell was kept constant. The inset gives the contrast p - Pm (electrons/nm ). The uppermost curve refers to homogeneous spheres with diameter 37 nm. The dashed line marks the isoscattering point which coincides with the minimum of the form factor of the homogeneous sphere...

Polydispersity plays an important role for S(q) as well [63]. Here a discussion of the alterations effected to S(q) in the case of hard sphere interaction will suffice. For this purpose S(q) of a system of hard spheres maybe obtained from the Percus-Yevick theory [66] generalized by Vrij and coworkers [67,68] to polydis-perse systems. The solid line in Fig. 6 displays S(q) resulting for a system of hard spheres with a Gaussian size distribution characterized by a standard deviation of 7.5%. The main feature induced by polydispersity is the much weaker side minimum of S(q) as compared to the monodisperse case. Hence, a finite width of the size distribution will tend to smear out the oscillations of S(q) at higher q. 

The scattering data thus corrected are solely due to the radial excess electron density of the particles. Fig. 14 displays the measured intensity (filled circles) of the polystyrene latex discussed in conjunction with Fig. 10. The solid line is the fit of the experimental data by a core-shell model and a slightly asymmetric size distribution ([46] see below) taken from the analysis by ultracentrifugation [87]. In terms of a Gaussian size distribution the polydispersity corresponds to a standard deviation of 4.2%. The thin shell having a higher electron density stems from the adsorbed surfactant used in the synthesis of the latex. This effect and its detection by SAXS will be discussed further below (see Sect. 4.4). 

Simulations on the effect of step free energy on grain growth behaviour have also been made. Figure 15.11 shows the result of a Monte Carlo simulation made by Cho. For the simulation, Cho assumed that the grain network was a set of grains with a Gaussian size distribution (standard deviation of 0.1) located on vertices of a two-dimensional square lattice. Deterministic rate equations, Eq. (15.15) for v/> and Eq. (15.29) for v j, were... 

Based on the theory of Mie modified concepts deal with non-spherical cluster shape and core shell particles. Clusters of a increasing eccentricities exhibit a splitting of the peak to two or three more or less distinct peaks. Moreover, the oblate cluster shape induces a well-defined splitting of the polarization. Thus, by choosing the appropriate polarization of the exciting beam selective excitation of modes can be done. Clusters randomly oriented with a Gaussian size distribution result in a flat and broad peak (Figure 7) [8]. 

If we fit some SANS data along the dilution line where m/s = 0.95, we find that a model of poly-disperse spheres fits the data best. By using a gaussian size distribution... 

Calculating Eq. (18.19) with a Gaussian size distribution for g(I) one obtains in agreement with Eq. (18.17) the following approximation... 

Fig. 6.9 Log-log plots of the scattering intensity ceilcuiated from (1) a single 35 nm-diametered ellipsoid particle (blue) and (2) spheres with a Gaussian size distribution (red). The sphericeil mean diameter is 20 nm...

If one can assume centro-symmetry, the numerical form of Eq. (6.14) can be used to calculate the scattering intensity from systems that are anisotropic. Significantly lower calculation times can also be achieved because only one summation over the real space is needed at each q value. To illustrate how a realistic system cau be modeled using this method, we will consider a system of spheres that have a Gaussian size distribution with a mean of 150 A and standard deviation of 15 A. Similar to Eq. 6.34, the scattered intensity of the systan can be calculated by the equations ... 

The data were fitted using a single thin shell vesicles model with radius R [67] and a Gaussian size distribution f x) to take into account polydispersity, according to the relation... 

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