Anomalous diffusion of surface-active species at liquid-fluid and liquid-solid interfaces

We study the role of bulk-surface exchange in the density relaxation kinetics and selfdiffusion of surface-active molecules at liquid surfaces. In “ strongly adsorbing ” systems, relaxation occurs through bulk-mediated effective surface diffusion characterized by one-step distributions with long tails ; molecules execute Lévy walks on the surface. Correspondingly, at times before particles are finally lost to the bulk, surface displacement $r$ is non-Fickian and exhibits anomalous scaling : moments grow as $\langle r^q\rangle\sim t^{\zeta(q)}$, where $\zeta(q) = q$ for $q 1$ and $\langle r\rangle\sim t$ ln $t$. The width of an initially localized density disturbance increases linearly in time with a “ speed ” $c$ which is universally related to other observables. Numerical simulations confirm the family of exponents $\zeta(q)$, and reproduce the observable $c$. We consider a simple example where end-functionalised macromolecules adsorb at a solid surface, finding $c \sim 1/s$ where $s$ is the surface “ stickiness ” parameter. At liquid-fluid interfaces viscoelastic effects compete. For sub-micron scales, we argue that self-diffusion will typically remain dominated at high coverages by the anomalous bulk-mediated mechanism, while surface viscoelasticity will dominate the relaxation of density perturbations.