Anomalous diffusion in gelatin-surfactant solutions and gels

A photon correlation spectroscopy study carried out on semidilute 4% (wt./vol) aqueous gelatin sols and gels with the anionic surfactant sodium dodecyl sulphate (SDS) showed three relaxation processes in the dynamic structure factor data. First an exponential process $S(q,t)\mathrm{}\sim \exp \left(-D_f{q}^{2}t\right)$ at $t<~20\mu \mathrm{s},$ referred to as the collective mode, was observed, followed by an intermediate power-law regime $S(q,t)\mathrm{}\sim t^{-\alpha \mathrm{}\left(q\right)\mathrm{}}$ at $20\mathrm{\mu }\mathrm{s}>~t<~1\mathrm{ms},$ and finally a stretched exponential decay $S(q,t)\mathrm{}\sim \exp -(t/{\tau }_c{)}^{\beta }$ was observed for $t>~1\mathrm{ms}.$ The power-law exponent α and characteristic time ${\tau }_c$ in the stretched part were found to be $q$ dependent: $\alpha \sim q^x,$ with $x=1.80\mathrm{}\pm 0.09$ in the sol state and $x=0.98\mathrm{}\pm 0.05$ in the gel state, and ${\tau }_c\sim q^{-y},$ with $y=2.05\mathrm{}\pm 0.08$ and $3.0\pm 0.2$ in the sol and gel states, respectively, but independently of SDS concentrations. In the fast mode the relaxation time was measured to be the same in the pure gelatin sol and gel states. It is proposed that the different $q$ dependences observed in the gel and sol states with and without SDS indicate the presence of different characteristic length scales. Our results agree with the predictions of the anomalous Gaussian diffusion model in the sol state, but differ significantly in the gel state.