Determination of Interfacial Concentration of a Contaminated Droplet from Shape Oscillation Damping

We present a method to measure the very small interfacial concentration of a contaminant that is irreversibly adsorbed on the interface of a bubble or droplet. It is an application of the linear theory of shape oscillation which relates the Gibbs elasticity to the damping, extended by numerical simulations to deal with moving droplets. It explains previous unexpected observations on the effect of contamination at various oscillation wavelengths. The experimental procedure is easy to implement and can thereby deeply enhance the analysis of most systems involving uncontrolled contamination.

Figure 1

Computed shapes of an oscillating rising droplet. Time sequence during the transient stage and comparison with the experimental drop shapes (red lines). Color levels correspond to the computed surface concentration of surfactant, normalized by ${\mathrm{\Gamma }}_{\mathrm{eq}}$. Distance between shapes are not to scale. $g$ is the gravity acceleration.

Figure 2

Simulation results (parameters from Tables 1 and 2). (a) Time evolution of the rise Reynolds number $\mathrm{Re}(t)=2{\rho }_{o}V(t)R/{\mu }_o$, with $V(t)$ the instantaneous droplet velocity, normalized by the Reynolds number at terminal stage ${\mathrm{Re}}_{\infty }$, and of the oscillation amplitudes $a_n(t)$ for modes 2–5 normalized by $R$; (b) comparison of angular frequency ${\omega }_n$ and damping rate ${\beta }_n$ between simulation and theory [21].

Figure 3

Experimental values ${\omega }_{n,\exp }$ and ${\beta }_{n,\exp }$ for oscillation modes $n=2–5$ from [20], compared to the theoretical predictions ${\omega }_{n,\mathrm{th}}$ and ${\beta }_{n,\mathrm{th}}$ from [21] with ${\sigma }_{\mathrm{eq}}=0.0475$ N/m and $E=0.1$ and to the theoretical values ${\omega }_{n,\mathrm{clean}}$ and ${\beta }_{n,\mathrm{clean}}$ ($E=0$, same ${\sigma }_{\mathrm{eq}}$).

Figure 4

Evolution of ${\omega }_{n,\mathrm{th}}$ (dashed lines) and ${\beta }_{n,\mathrm{th}}$ (continuous lines), normalized by their respective values ${\omega }_{n,\mathrm{clean}}$ and ${\beta }_{n,\mathrm{clean}}$, as a function of ${\mathrm{\Gamma }}_{\mathrm{eq}}$. Parameters are in Table 1.