Dimensionally Frustrated Diffusion towards Fractal Adsorbers

Diffusion towards a fractal adsorber is a well-researched problem with many applications. While the steady-state flux towards such adsorbers is known to be characterized by the fractal dimension ($D_F$) of the surface, the more general problem of time-dependent adsorption kinetics of fractal surfaces remains poorly understood. In this Letter, we show that the time-dependent flux to fractal adsorbers ($1<D_F<2$) exhibit complex “dimensionally frustrated” self-similar time response and is characterized by a simple scaling law ${\rho }_0{t}^{1/D_F}=c$ (${\rho }_0$ is the concentration of particles, $t$ is the time, and $c$ is a constant). Indeed our analysis establishes the time response of technologically relevant nanonet (or nanocomposite) biochemical sensors as a test bed of time-dependent adsorption on fractal surface, providing a novel experimental measure of $D_F$ and an obvious route to improved sensor design.