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

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Abstract
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 ⟨rq⟩∼tζ(q), where ζ(q) = q for q < 1, ζ(q)= (q + 1 )/2 for q > 1 and ⟨r⟩∼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 ζ(q), and reproduce the observable c. We consider a simple example where end-functionalised macromolecules adsorb at a solid surface, finding c ∼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.

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