Gibbs elasticity, surface dilational modulus and diffusional relaxation in nonionic surfactant monolayers

The surface dilational modulus, which is closely related to the Gibbs elasticity of thin liquid films, has been measured with different experimental methods. We report on new measurements of the modulus for C12E6, a surfactant of the ethylene oxide adduct type, with a dynamic drop tensiometer. In this method, the surface of a small drop or bubble in the surfactant solution is periodically compressed and expanded at a low amplitude and a constant frequency. The modulus follows from the periodic changes in surface tension obtained by axisymmetric drop size analysis. The results are shown to match earlier measurements on the same system with the traditional method employing a barrier oscillating in the surface and a Wilhelmy plate. Both the old and the new data cover the concentration range from zero to the critical micellar concentration (c.m.c). The time scale of the dynamic experiments, which is of the order of the inverse frequency, was varied between 1 and 100 s. We present a quantitative interpretation of these results in terms of the surface equation of state, coupled with diffusional relaxation. The surface equation of state and the adsorption isotherm were derived on the basis of a 2-D solution treatment applied to a Gibbs dividing surface defined so as to account for the presence of solvent. This phenomenological approach, which is independent of molecular size and structure in the surface region, describes both nonideal entropy and enthalpy of the surface mixture to first order. In the entire range of concentrations and frequencies considered, we found almost quantitative agreement between theory and experiment, for both dynamic and equilibrium data. The numerical values found for the equation of state parameters indicate a surface solution of surfactant molecules with a molecular area twice as large as the solvent, and a negative heat of interaction, in contrast to fatty acids where this heat is positive. Relaxation of the surface tension appeared to be purely diffusion-controlled. In no part of the concentration range, did we find any indication for the kinetics of the final adsorption step from sub-surface to surface being rate determining.