Anomalous logarithmic slow-dynamics behavior on hierarchical and random systems

A biological interpretation of transient anomalous subdiffusion. I. Qualitative model

Anomalous subdiffusion has been reported for two-dimensional diffusion in the plasma membrane and three-dimensional diffusion in the nucleus and cytoplasm. If a particle diffuses in a suitable infinite hierarchy of binding sites, diffusion is well known to be anomalous at all times. But if the hierarchy is finite, diffusion is anomalous at short times and normal at long times. For a prescribed set of binding sites, Monte Carlo calculations yield the anomalous diffusion exponent and the average time over which diffusion is anomalous. If even a single binding site is present, there is a very short, almost artifactual, period of anomalous subdiffusion, but a hierarchy of binding sites extends the anomalous regime considerably. As is well known, an essential requirement for anomalous subdiffusion due to binding is that the diffusing particle cannot be in thermal equilibrium with the binding sites; an equilibrated particle diffuses normally at all times. Anomalous subdiffusion due to barriers, however, still occurs at thermal equilibrium, and anomalous subdiffusion due to a combination of binding sites and barriers is reduced but not eliminated on equilibration. This physical model is translated directly into a plausible biological model testable by single-particle tracking.

Quasi-anomalous diffusion processes in entangled solutions of wormlike surfactant micelles

In this article, we present a detailed analysis of the dynamic properties of entangled solutions of semi-flexible, threadlike surfactant micelles. These aggregates were formed by self-association processes in aqueous solutions of cationic surfactants such as cetylpyridinium chloride (CPyCl) or cetyltrimethylammonium bromide (CTAB) after the addition of different amounts of sodium salicylate (NaSal). We performed dynamic light scattering (DLS) experiments in combination with rheological measurements in order to investigate the dynamic properties of these viscoelastic surfactant solutions. In all samples, we observed three distinct relaxation regimes: initial monoexponential decay, followed by a power-law behavior at intermediate observation times. A second monoexponential region was detected at very long times, and this terminal regime described the viscoelastic features of the samples. The fast decay mode was induced by local cooperative motions in the gellike network. The intermediate and slowest decay modes point to the existence of quasi-anomalous diffusion processes. These phenomena are characterized by linear-diffusion properties at long times, and they obeyed anomalous logarithmic slow-dynamics behavior at intermediate time zones. The anomalous diffusion properties at intermediate time scales can be induced by the bending motions of the rod-shaped micelles between two entanglement points. This regime, which was more extended at lower temperatures, was described by the power-law form of the correlation function. The power-law exponent depended on the chemical structure of the surfactants and the temperature. The power-law regime shifted toward earlier times as the gellike network evolved. The slowest mode of the correlation function coincided very well with the shear stress relaxation times of the three-dimensional, transient networks. We observed that the temperature dependence of the slowest mode followed Arrhenius laws. This result provides experimental evidence for thermally activated topological relaxation processes of random fluid phases. We obtained activation energies of approximately 30 kcal/mol, and these data coincided well with previously reported literature values, which were determined in similar surfactant solutions. Characteristic "screening lengths", over which viscous effects became important, could also be determined from the activation energy. The elastic modulus G0, calculated from the slowest mode of the correlation function, was in pretty good agreement with rheological data. The light-scattering spectra were consistent with the theoretical model of dynamical coupling of the concentration fluctuations to viscoelasticity. Since only minute sample volumes are required for advanced DLS experiments, this method to extract viscoelasticity is well suited for advanced studies of gellike biomaterials.