Langmuir 1994,10, 1596-1599
1596
Brownian Motion at Liquid-Gas Interfaces. 3. Effect of Insoluble Surfactants K. Dimitrov, M. Avramov, and B. Radoev* Department of Physical Chemistry, Faculty of Chemistry, St. Kliment Ohridski University of Sofia, 1126 Sofia, Bulgaria Received January 6,1994" The drag force of a translating sphere half immersed on a flat liquid-gas interface in the presence of insoluble surfactants is analyzed. Special attention is paid to some singularitiesof the perturbed adsorption r and of the drag force. It is shown that in the frame of Stokes hydrodynamicsthe steady-stateperturbation r does not vanish at infinity but tends to constant and the drag force due to the Marangoni effect is infinite. These discrepancies are explained considering the incorrectnessof the Stokes velocity field far from the sphere. The Oseen approximation is proposed as a better hydrodynamic model and, as a result, a finite force proportional to the logarithm of the Reynolds number is obtained. Introduction A significant feature of the mass transfer at fluid interfaces is its coupling with hydrodynamics. One of the effects governing this coupling (the Marangoni effect9 is due to the surface activity of the transferred substance (matter, electricity, heat) and is formulated by the hydrodynamic boundary condition P n s = V,y, where pns is the tangential component of the stress tensor on the surface and V,y is the gradient of the surface tension y. (The subscript s denotes the quantities refered to the surface.) In the case of transport of surfactant, the surface tension y is a function of its Gibbs adsorption r, so that the stress boundary condition relates the mass (r)and the velocity (v) distributions. Such coupling between the concentration and the velocity fields do not exist in the bulk, as it is strictly demonstrated for linear processes in homogeneous media (Curie theorem2). The driving force effect in the bulk, corresponding to the Marangoni effect at the surface ( d y l d r ) ,is the fluid compressibility (apldc). The other effect controlling the surface transport has kinematics nature and emerges in the mass balance
3 + v , q o + r)v, = D,V:r at
+ ...
with I'O and I? being the equilibrium and the perturbed adsorptions, respectively. Generally the convective flux (Fo r)v, defines the r-balance as a nonlinear equation (note, that r depends on v,), but for small perturbations (r
