Static structure factor of dilute solutions of polydisperse fractal aggregates

We have calculated the z-average static structure factor [S(q)z], z-average radius of gyration (Rgz), and the weight average aggregation number (mw) of fractal aggregates with an aggregate number distribution given by N(m){∝}m^-τf(m/m*). Here, q is the scattering wave vector and for the cutoff function f(m/m*) at a characteristic aggregation number m* we have chosen a stretched exponential function. We derive expressions for the prefactors of the scaling relations mw{∝}Rgzdf* and S(q)z{∝}(qRgz{)}^{{{-}}df*}$ and make explicit the conditions for their validity at finite values of ${m}w$ and ${qR}gz$. For values of {τ} close to two these conditions are shown to be very difficult to meet in practice. For the case of aggregates formed by a percolation process where {τ}=2.2 it is shown that attempts to measure df* directly from the slope of log₁₀(mw) vs log₁₀(Rgz) or log₁₀(S(q)z) vs log₁₀(q) are biased by the effect of the internal and/or external cutoff of the fractal regime.