Mass-fractal growth in niobia/silsesquioxane mixtures: a small-angle X-ray scattering study

The nucleation and growth of niobium pentaethoxide-derived clusters in ethanol was monitored at 298–333 K by small-angle X-ray scattering. The data were analyzed with a newly derived model for polydisperse mass-fractal-like structures.


Mass-fractal agglomerates with an exponential cut-off length ξ
The diffusion limited cluster aggregation (DLCA) mechanism typically leads to less polydisperse agglomerates, as implied by the exponentially decaying cut-off function (Sorensen & Wang, 1999). Sharper cut-off functions such as a Gaussian cutoff (h(r,ξ)= exp(-(r/ξ) 2 ) are realistic for a variety of aggregation mechanisms. It would be convenient to define a function of which the cutoff behavior can be related to the degree of polydispersity. To this end, firstly we introduce an infinitely sharp cut-off function, i.e. a unit step or Heaviside step function h(r,ξ)= H(ξ-r). The intensity function of a mass fractal with a hard cutoff function was described by a rotationally averaged Fourier transform: Herein, H(ξ-r) = 1 for r < ξ and H(ξ-r) = 0 for r > ξ. Since the volume or primary units was assumed infinite small S(q) was being normalized over its entire agglomerate volume V A .
Instead of using the unit step function we can also move the upper boundary of the sine transform from ∞ to ξ, as shown in the right hand side part of Equation (S1). Secondly, polydispersity is introduced by the integral: Herein, w(ξ) is an intensity weighted probability density function of the cutoff parameter ξ.
We applied a Schultz-Zimm distribution (Kotlarchyk & Chen, 1983), which was found to give realistic results in earlier studies on similar systems (Besselink et al., 2013;Stawski et al., 2011a,b;Pontoni et al., 2002): and μ is the intensity weighted average of ξ and the Z-parameter is related to the distribution of the cutoff distance, i.e., the variance of ξ corresponds to (σ ξ ) 2 = μ 2 /(Z+1). By combining Equation (S1)-(S3) we obtain: This integral can be evaluated as a Laplace transform and for integer values of Z. An analytical solution is given by where i is the imaginary number. The derivatives of a that are expanding with increasing Z can be generalized by the following Riemann's sum: Here, η is an integer variable that varies from 0 to Z. In analogy with the mass fractal structure function with an exponential cutoff Equation (S2), the function is normalized over its agglomerate volume V A such that S(q→0) = 1. The Porod volume of such agglomerate is described by: Then, after normalization of Equation (S6) with Equation (S8), and replacing complex elements with goniometric equations we obtain: Here S I and S F represent the contributions of the integer and fractional values of Z to S SC (q).

( )
where I 0 = N·(V A ) 2 ·(Δρ) 2 , which corresponds to the scattering intensity at q→0 (since S(q→ 0) = 1), N is the particle number density, V A the particle volume of the fractalic agglomerate (Equation (S8)) and Δρ and is the averaged difference in electron density between particles and their surroundings. For comparison of the Schultz cut-off model with the exponential cutoff model it is more convenient to express the size of a cluster by the radius of gyration that is derived from Feigin & Svergun (1987) and Porod (1982): Sinceγ(r) is essentially an auto-convolution product of Δρ(r) a hard cutoff function is not realistic. The relative variance of ξ that can be derived from the Z parameter is always larger, because the relative variance of R G and the relationship between Z and polydispersity depends on the geometry of the fractal. Alternatively, we may extract a polydispersity factor C P following the procedure described by Sorensen and Wang (1999). Provided that S(q) is normalized over the entire agglomerate volume Equation (S8), such that S(q→0) =1), the effective structure function in the fractal regime (q·R G >> 1) is described by: where the constant C is related to the geometry of the fractalic agglomerate and C P is a measure of the polydispersity (Sorensen and Wang, 1999). Experimental data revealed that C = 1.0 ± 0.05 for mass fractals with D f between 1.7 and 2.1 (Sorensen and Wang, 1999).
The C P value increases with increasing polydispersity and can be associated with a particular growth mode, i.e. C P~1 .5 for diffusion limited cluster aggregation (DLCA) and C P > 2 for reaction limited cluster aggregation (RLCA) (Sorensen and Wang, 1999). C P is a size independent measure of polydispersity and depends solely on Z and D f. . It can be derived from Equation (S9) by taking the limit (q·R G ) → ∞ of S SC (q·R G )·(q·R G ) Df , which corresponds to: Note that the Riemann's sum diminished since the limit was dominated by the η = 0 element of the Riemann's sum. As illustrated in Figure 2 by simulations of S SC (q,μ,D f ,Z) with R G =10 nm and D f = 2, the height of the fractal regime as characterized by C·C P decreases with increasing Z value.