Local Modes and the Surface-Bulk Exchange Rate at a Liquid Interface

The small-scale motions of a liquid surface due to thermally excited capillary waves are dominated by over-damped local modes of oscillation, which for a water surface at 300 K occupy the range of wave vectors between the critically damped k c = 1709 cm -1 and k max ≈ 10 -8 cm -1 . Values of k below k c correspond to under-damped, normal modes of oscillation. For values of k near k max, the peak displacement produced by a given increment of surface energy, at constant surface tension, increases sharply with increasing k, an effect which probably contributes to the reported apparent decrease in surface tension at high k . The radial dependence of a local mode is of the form exp( − kr ) J 0 ( kr ), where J 0 ( kr ) is a Bessel function and the exponential factor arises from the finite rate at which the displacement spreads out from the site of the initial disturbance. The time dependence of a local mode consists of a rapid initial displacement of the surface followed by a very slow recovery. Any element of the surface spends most of its time in the recovery phase, and the rate of recovery is independent of k when k ≫ k c . The recovery process transfers molecules out of the surface layer; hence the first-order rate constant for transfer of a surface molecule into the bulk liquid is found to be k B T /2 h σ 2 s -1, where η is the viscosity and σ is the molecular size.