Viscoelastic Drag Forces and Crossover from No-Slip to Slip Boundary Conditions for Flow near Air-Water Interfaces

The “free” water surface is generally prone to contamination with surface impurities, be they surfactants, particles, or other surface active agents. The presence of such impurities can modify flow near such interfaces in a drastic manner. Here we show that vibrating a small sphere mounted on an atomic force microscope cantilever near a gas bubble immersed in water is an excellent probe of surface contamination. Both viscous and elastic forces are exerted by an air-water interface on the vibrating sphere even when very low doses of contaminants are present. The viscous drag forces show a crossover from no-slip to slip boundary conditions while the elastic forces show a nontrivial variation as the vibration frequency changes. We provide a simple model to rationalize these results and propose a simple way of evaluating the concentration of such surface impurities.

Figure 1

Sketch of the experimental apparatus. The liquid-gas interface was prepared by deposing a spherical bubble on a PS surface. A vibrating glass sphere glued at the end of the AFM cantilever is used to induce the hydrodynamic drag force.

Figure 2

(a) Shows the viscous ${\mathrm{\Gamma }}_{\text{vis}}$ and the elastic part ${\mathrm{\Gamma }}_{\mathrm{el}}$ of the hydrodynamic drag coefficient for the sphere vibrating at frequency (200 Hz) in water close to a mica surface. (b) The hydrodynamic drag coefficient versus the distance for different vibration frequencies. The solid dark line is the theoretical drag coefficient ${\mathrm{\Gamma }}_0=(6\pi \eta R^2/d)$ calculated using a no-slip boundary condition on both surfaces (glass sphere and mica substrate).

Figure 3

(a) The two components of the drag coefficient measured on the bubble surface (sphere vibration frequency 100 Hz). (b) Viscous component for different frequencies of vibrations. The calculated drag coefficient corresponding to full slip and no-slip boundary conditions on the bubble surface are represented by the gray and dark line, respectively.

Figure 4

(a) $(F_{\text{vis}}/F_0)$ and $(F_{\mathrm{el}}/F_0)$ versus the vibration frequency, for the data of a first experiment. The data of a second run, obtain one month later under similar conditions, are shown in the inset. The theory (solid line) agrees very with the data, with fitted equilibrium surfactant pressure ${\mathrm{\Pi }}_0=0.25$ and ${\mathrm{\Pi }}_0=0.35\mathrm{mN}/\mathrm{m}$, respectively. (b) Data of a control experiment on a $60\mu \mathrm{M}$ SDS solution fitted with a surface pressure ${\mathrm{\Pi }}_0=1.0\mathrm{mN}/\mathrm{m}$.

Figure 5

Viscoelastic data on “pure” water (first and the second run) and on $60\mu \mathrm{M}$ SDS, as a function of the reduced frequency $(\omega /{\omega }_0)$. The dashed lines are the fitting curve using numerical calculation, and the continuous ones are given by Eqs. (10) and (11).