Fitting of spherically shaped particles
The intermediate phase seen in the SAXS pattern prior and parallel to the mesostructure formation was assumed to originate from particles being formed by surfactant and condensing silica species. In a first approximation these particles were assumed to be spherical. Since the characteristic slope is accessible only in a small q-region and moreover obscured by other scattering contributions, no precise conclusion is possible. As shown in Figure 3.4.13 and Figure 3.4.19 the characteristic slope changes in the course of the in-situ experiments, equivalent to a change in the shape of the particles. Since the phase seems to trigger the evolving mesostructure, a transformation from spherical to cylindrical structures is likely. No oscillations of the scattered intensity are observed in the Porod regime because of the polydispersity of the particles. As approximation in case of the phenylene bridged precursor a Debye function was used [140], where the constant 2/3 provides an asymptotic approach to the form factor of a perfect sphere for large scattering vectors. In an other attempt the radius was aproximated using Guinier´s law. The obtained radius of gyration, Rg, is linked to the radius of spheres by Rg,spheres = 5/3Rg. Examples for both fits are shown in Figure 3.7.1 for the EGMS/P123/1 M HCl system. The radii resulted in Rd=5.2 nm, respectively, Rg=7.8 nm (Rg, sphere~10 nm).
Figure 3.7.1. Experimental scattering curve of the EGMS/P123/1 M HCl synthesis, 50 minutes after mixing the components, and the corresponding fit curves (a) Debye, (b) Guinier approximation.
APPENDIX A
0,1 1
Form factor of spheres (R=10 nm, σ1=0, σ2=1) Debye approximation (Rd=10 nm) Fit (Rd -> 5.9 nm)
q-4 - decay
log Intensity / a.u.
log q / nm-1
0,1 1
Form factor of spheres (R=10 nm, σ1=0, σ2=1)
Form factor of spheres (R=10 nm, σ1=0, σ2=1) Debye approximation (Rd=10 nm) Fit (Rd -> 5.9 nm)
q-4 - decay
log Intensity / a.u.
log q / nm-1
0,1 1
Form factor of spheres (R=10 nm, σ1=0, σ2=1)
Figure 3.7.2. Form factor for monodisperse and polydisperse spheres with the corresponding (a) Debye curve and (b) Guinier curve.
In Figure 3.7.2 (a) the scattering intensity for monodisperse and sligthly polydisperse spheres (σ=1 nm) with a radius of 10 nm is plotted together with the corresponding Debye function for spheres. Additionally, the curve derived by fitting the scattering from the monodispere spheres is shown. The same approach is plotted in Figure 3.7.2 (b) applying Guinier´s law. In case of the Debye approximation it is clearly visible, that the function underestimates the exact solution [136]. In the fits of the experimental scattering curves, the obtained radii are smaller than the expected ones. The opposite applies for the Guinier fit.
For better understanding of the fitted radii obtained by using Guinier´s law (Rg), respectively, Debye´s approximation (Rd) in the kinetic measurements, scattering curves of spherical particles were simulated with different radii. The values chosen are comparable to the expected values (range of 5 nm–12 nm). Scattering profiles were generated using the classical expressions for the form factor of spheres (Eq. 2.5) and a size distribution function to smear out the minima/maxima (σ=pR) (Eq. 2.8). The resulting curves were then fitted with the Debye approximation for spherical particles and Guinier´s law in the region accessible to the experiment as well as for a larger q-range (0.01 to 3 nm-1).
In Figure 3.7.3 the scattering curve for polydisperse spherical particles with radius R=9.9 nm and a standard deviation of σ=0.4 R is shown together with the obtained fit curves. For the experimentally investigated q-regime both approximations agree well, but give different radii, namely, Rd=7.5 nm and Rg=10.5 nm (leading to a radius for spheres of 13.6 nm). Both obtained radii differ significantly from the employed radius R=9.9 nm.
Despite the observed deviation, the fit values follow the behaviour of the “true” radius, as is shown in Figure 3.7.4 a), and therefore give information about growth/shrinkage of the particle diameter.
APPENDIX A
Generated curves (R=Rsphere): 1 - 30
0,2 0,4 0,6 0,8 0,940,960,981,001,021,04
Rd/R (Experimentally accessible)
Rd/R
Rg,sphere/R (Experimentally accessible)
Rg,sphere/R
Rd/Rg,sphere(Experimentally accessible)
Rd/Rg,sphere
Generated curves (R=Rsphere): 1 - 30
0,2 0,4 0,6 0,8 0,940,960,981,001,021,04
Rd/R (Experimentally accessible)
Rd/R
Rg,sphere/R (Experimentally accessible)
Rg,sphere/R
Rd/Rg,sphere(Experimentally accessible)
Rd/Rg,sphere
Measure of polydisperity p (σ=pR) / a.u.
Value actual fit
(a) (b)
Figure 3.7.3. Simulated curve with radius R=9.9 nm and polydispersity p=0.4 R and the corresponding fits following Eq. 2.3 (Guinier´s law) and Eq. 2.9 (Debye approximation). The q-range which was investigated in the experiment is highlighted and was also used as data in the fits. Both fits optically agree quite well with the smeared form factor of spheres, although they lead to differing radii.
Figure 3.7.4. (a) Resulting fit values for thirty simulated curves. Rsphere is the radius used in the simulation.
(b) Ratios of the obtained values for Rd (Debye) and Rg, sphere (Guinier) from fitting the curves generated from different R, plotted against polydispersity. Fits were performed on the experimentally accessible q-range as well as a q-range ranging from 0.01 to 3 nm-1. A sample curve is shown in Figure 3.7.3. The values have been averaged over thirty curves. Fitting was performed with Mathematica.
From the normalized fit results of the simulated scattering curves of spherical particles, shown in Figure 3.7.4 (b), one can see that polydispersity has an influence on the deviation of the obtained values. Radius Rg, sphere which is derived from the Guinier fit is always larger than the “real” mean radius R. The factor between the values for the radii obtained by Debye and Guinier approximation (Rd/Rg,sphere) was found to be approximately 0.56≤0.02 in the simulated curves with tedency to increase with higher polydispersity of the particles. The value derived from the fits of the experimentally obtained scattering curves for the EGMS-kinetic gives a factor of 0.62≤0.1. In conclusion the approximation functions such as Eq. 2.9 (Debye) and Eq. 2.3 (Guinier) can not give the exact values but both reflect the tendency of particle evolution equally well.
APPENDIX B