Abstract
Densely packed surface fractal aggregates form in systems with high local volume fractions of particles with very short diffusion lengths, which effectively means that particles have little space to move. However, there are no prior mathematical models, which would describe scattering from such surface fractal aggregates and which would allow the subdivision between inter- and intraparticle interferences of such aggregates. Here, we show that by including a form factor function of the primary particles building the aggregate, a finite size of the surface fractal interfacial sub-surfaces can be derived from a structure factor term. This formalism allows us to define both a finite specific surface area for fractal aggregates and the fraction of particle interfacial sub-surfaces at the perimeter of an aggregate. The derived surface fractal model is validated by comparing it with an ab initio approach that involves the generation of a "brick-in-a-wall" von Koch type contour fractals. Moreover, we show that this approach explains observed scattering intensities from in situ experiments that followed gypsum (CaSO4 ⋅ 2H2O) precipitation from highly supersaturated solutions. Our model of densely packed "brick-in-a-wall" surface fractal aggregates may well be the key precursor step in the formation of several types of mosaic- and meso-crystals.
Highlights
In colloid sciences fractal scaling concepts constitute an important formalism that provides for the statistical description of the properties of particles and their aggregates
Mass fractal scaling can be associated with the packing efficiency of an aggregate, which in turn depends on the type of aggregation, e.g., diffusion or reaction limited mechanism
We derived and validated a model for a structure factor expressed by Eq (15) which describes the scattering from “brick-in-a-wall” surface fractal aggregates build of primary particles
Summary
In colloid sciences fractal scaling concepts constitute an important formalism that provides for the statistical description of the properties of particles and their aggregates (e.g., morphologies, porosity, density, and specific surface area). Contrary to the mass fractal equation (Eq (8)) we did not include an upper cut-off value of the aggregate This is only useful when all features, including the primary particle form factor, the intermediate plateau regime, the surface fractal regime, and the upper cut-off regime fall within the measured q-range. Since such aggregates are typically orders of magnitude larger as the primary particles, we assume that these aggregates extend to macroscopic sizes and I(q → 0) → ∞
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