Direct determination of the dependence of the surface shear and dilatational viscosities on the thermodynamic state of the interface: Theoretical foundations

Recent developments in nonlinear optical techniques for noninvasive probing of a surfactant influenced gas/liquid interface allow for the measurement of the surfactant surface concentration, c, and thus provide new opportunities for the direct determination of its intrinsic viscosities. Here, we present the theoretical foundations, based on the Boussinesq- Scriven surface model without the usual simplification of constant viscosities, for an experimental technique to directly measure the surface shear (μ(s)) and dilatational (K(s)) viscosities of a Newtonian interface as functions of the surfactant surface concentration. This ability to directly measure the surfactant concentration permits the use of a simple surface flow for the measurement of the surface viscosities. The requirements are that the interface must be nearly flat, and the flow steady, axisymmetric, and swirling; these flow conditions can be achieved in the deep-channel viscometer driven at relatively fast rates. The tangential stress balance on such an interface leads to two equations; the balance in the azimuthal direction involves only μ(s) and its gradients, and the balance in the radial direction involves both μ(s) and K(s) and their gradients. By further exploiting recent developments in laser-based flow measuring techniques, the surface velocities and their gradients which appear in the two equations can be measured directly. The surface tension gradient, which appears in the radial balance equation, is incorporated from the equation of state for the surfactant system and direct measurements of the surfactant surface concentration distribution. The stress balance equations are then ordinary differential equations in the surface viscosities as functions of radial position, which can be readily integrated. Since c is measured as a function of radial position, we then have a direct measurement of μ(s) and K(s) as functions of c. Numerical computations of the Navier-Stokes equations are performed to determine the appropriate conditions to achieve the requisite secondary flow.