Static structure factor of dilute solutions of polydisperse fractal aggregates

We have calculated the z-average static structure factor [S(q${)}_{\mathit{z}}$], z-average radius of gyration (${\mathit{R}}_{\mathit{g}\mathit{z}}$), and the weight average aggregation number (${\mathit{m}}_{\mathit{w}}$) of fractal aggregates with an aggregate number distribution given by N(m)∝${\mathit{m}}^{\mathrm{-}\mathrm{\tau }}$f(m/${\mathit{m}}^{\mathrm{*}}$). Here, q is the scattering wave vector and for the cutoff function f(m/${\mathit{m}}^{\mathrm{*}}$) at a characteristic aggregation number ${\mathit{m}}^{\mathrm{*}}$ we have chosen a stretched exponential function. We derive expressions for the prefactors of the scaling relations ${\mathit{m}}_{\mathit{w}}$∝${\mathit{R}}_{\mathit{g}\mathit{z}}^{\mathit{d}\mathit{f}\mathrm{*}}$ and S(q${)}_{\mathit{z}}$∝(${\mathit{qR}}_{\mathit{g}\mathit{z}}$ ${)}^{\mathrm{-}\mathit{d}\mathit{f}\mathrm{*}}$ and make explicit the conditions for their validity at finite values of ${\mathit{m}}_{\mathit{w}}$ and ${\mathit{qR}}_{\mathit{g}\mathit{z}}$. 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 ${\mathit{df}}^{\mathrm{*}}$ directly from the slope of ${\log }_{10}$(${\mathit{m}}_{\mathit{w}}$) vs ${\log }_{10}$(${\mathit{R}}_{\mathit{g}\mathit{z}}$) or ${\log }_{10}$(S(q${)}_{\mathit{z}}$) vs ${\log }_{10}$(q) are biased by the effect of the internal and/or external cutoff of the fractal regime.