Determination of branch fraction and minimum dimension of mass-fractal aggregates

Abstract

Particles of micrometer to nanometer size often aggregate to form branched structures. Such materials include metals and metal oxides as well as biological and polymeric materials (considering the persistence length as a primary unit). Characterization of such structures is difficult since they typically display disordered, irregular features in three dimensions. Branched aggregates display two limiting size scales: that of the primary particle, R1 and that of the aggregate, R2. The mass-fractal model is often used to describe such structures where the aggregate mass, z=M2/M1, is related to the aggregate size, r=R2/R1, through a scaling relationship z=alpha r (d(f)), where the lacunarity alpha is close to 1 and may depend on the growth mechanism. Scattering of x rays, light and neutrons yields a direct measure of the mass-fractal dimension since I(q) approximately q(-d(f)) for 1/R2<q<1/R1 using scaling arguments. For linear, monodisperse aggregates with convoluted chain paths, analytic functions describing both the scaling and larger-size aggregate scattering regimes have been reported. For example, the Debye function for linear, Gaussian coils describes scattering when d(f)=2. Real, mass-fractal aggregates, however, can display variability from the linear chain, monodisperse model. Often the branch content is of vital importance to understanding both the growth of aggregates and their physical properties, especially dynamic properties. An approach is presented for the analysis of aggregate branching from static small-angle scattering. Comparison is made with analytic, simulation, and experimental results from the literature.

Colloidal stability of tannins: astringency, wine tasting and beyond

“…[15], dashed line: star polymer model [14]. The essential feature is that both fits are obtained assuming that the whole scattering profile is dominated by protein-tannin micelles.…”

Quantification of interaction and topological parameters of polyisoprene star polymers under good solvent conditions

“…The structure also displays topological connectivity that is independent of these thermodynamic and steric constraints. These features can be distinguished by considering the average minimum path of p Kuhn units through the structure [30][31][32]. One possible minimum path is shown in units with dark borders in Fig.…”

“…On the other hand, the connectivity path of s is related to the mass through the minimum dimension d min , which represents the mass fractal dimension for the minimum path. Two pairs, s : d min and p : c, work in tandem to represent the whole structure as a mutually conjugate set of parameters such that the mass z can be obtained by raising the connectivity path, s to the minimum dimension d min , or alternatively raising the minimum path p to the topological connectivity dimension c, giving, [31,33,34] …”

Small angle x-ray scattering and electron microscopy of nanoparticles formed in an electrical arc

“…For smooth particles p = 4, for aggregated particles or particles having a rough surface, p∼3 and p = 2 for discs or Gaussian chains. 15 All of the analysis of the scattering data has been performed using the Irena package of Macros developed by Jan Ilavski of the Advanced Photon Source. 16 This allows, for example, the measured data to be fitted to the Unified function and also particle size distributions to be determined.…”