J. Phys. II France
Volume 4, Numéro 7, July 1994
Page(s) 1135 - 1156
DOI: 10.1051/jp2:1994192
J. Phys. II France 4 (1994) 1135-1156

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

Oleg V. Bychuk1 and Ben O'Shaughnessy2

1  Department of Physics, Columbia University, New York, NY 10027, U.S.A.
2  Department of Chemical Engineering, Materials Science and Mining Engineering, Columbia University, New York, NY 10027, U.S.A.

(Received 23 February 1994, accepted 1 April 1994)

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, $\zeta(q)= (q + 1 )/2$ 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.

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