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

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.