Intermittent Brownian dynamics over a rigid strand: Heavily tailed relocation statistics in a simple geometry

We analyze the intermittent Brownian dynamics (a succession of adsorption and bulk relocation steps) of a test particle over a single strand. We propose an analytic expression of the relocation time distribution at all times. We show that this distribution has a nontrivial heavily tailed statistics at long time with a diverging average relocation time. In order to experimentally probe this first passage statistics, we follow the intermittent Brownian dynamics of water molecules over long and stiff imogolite mineral strands, using a field cycling NMR dispersion technique. Our analytic derivation is found to be in good agreement with experimental data on a large domain of observation. Implications for the efficiency of a search strategy on a single filament are then discussed and the importance of the confinement and/or the finite size effect is emphasized.

Figure 1

(Color online) Evolution of the relocation time statistics over a single filament of radius $r_{\mathit{cyl}}=1.6\mathrm{nm}$, starting at distance $\delta =0.3\mathrm{nm}$ from the boundary of the cylinder. The self-diffusion coefficient of the particle is taken as $D={10}^{-9}{\mathrm{m}}^2∕\mathrm{s}$. Cross symbols: numerical inversion of Eq. (1). Solid line: evolution of Eq. (2) for $t<t_{\mathit{cyl}}$ and Eq. (6) for $t>t_{\mathit{cyl}}$. Dashed line: evolution of Eq. (7) at long time, running as $1∕\left[t{\ln }^2(t∕t_{\mathit{cyl}})\right]$.

Figure 2

Left: SAXS pattern of a flow-aligned nematic sample of imogolite suspension. The flow direction is horizontal. Right: Position of the first maximum in the radial distribution function $g\left(r\right)$ as a function of ${\varphi }^{-1∕2}$, obtained from the structure factors (symbols) and linear fit (line).

Figure 3

Frequency dependence of water spin-lattice relaxation rate $R_1\left(f\right)$ in isotropic imogolite colloidal suspensions at $298\mathrm{K}$. Proton NMRD at $\varphi =1.8\times {10}^{-3}$ (full squares) and at $\varphi =0.9\times {10}^{-4}$ (full triangles). Inset, frequency derivatives of the dispersion curves, normalized by the volume fraction. Open square: Deuterium NMRD in ${\mathrm{D}}_2\mathrm{O}$ suspension at $\varphi =1.8\times {10}^{-3}$. In continuous lines: analytical model computed with Eqs. (8, 10).