Dynamical scaling in fractal structures in the aggregation of tetraethoxysilane-derived sonogels

Dynamical scaling properties in fractal structures were investigated from small-angle X-ray scattering (SAXS) data of the kinetics of aggregation in silica-based gelling systems. For lack of a maximum in the SAXS intensity curves, a characteristic correlation distance ξ was evaluated by fitting a particle scattering factor model valid for polydisperse coils of linear chains and f-functional branched polycondensates in solution, so the intensity at q = ξ⁻¹, I(ξ⁻¹, t), was considered to probe dynamical scaling properties. The following properties have been found: (i) the SAXS intensities corresponding to different times t, I(q, t), are given by a time-independent function F(qξ) = I(q, t)ξ^−D/Q, where the scattering invariant Q has been found to be time-independent; (ii) ξ exhibited a power-law behavior with time as ξ ≃ t^α, the exponent α being close to 1 but diminishing with temperature; (iii) I(ξ⁻¹, t) exhibited a time dependence given by I(ξ⁻¹, t) ≃ t^β, with the exponent β found to be around 2 but diminishing with temperature, following the same behavior as the exponent α. In all cases, β/α was quite close to the fractal dimension D at the end of the studied process. This set of findings is in notable agreement with the dynamical scaling properties.