Droplets in Microchannels: Dynamical Properties of the Lubrication Film

We study the motion of droplets in a confined, micrometric geometry, by focusing on the lubrication film between a droplet and a wall. When capillary forces dominate, the lubrication film thickness evolves nonlinearly with the capillary number due to the viscous dissipation between the meniscus and the wall. However, this film may become thin enough (tens of nanometers) that intermolecular forces come into play and affect classical scalings. Our experiments yield highly resolved topographies of the shape of the interface and allow us to bring new insights into droplet dynamics in microfluidics. We report the novel characterization of two dynamical regimes as the capillary number increases: (i) at low capillary numbers, the film thickness is constant and set by the disjoining pressure, while (ii) above a critical capillary number, the interface behavior is well described by a viscous scenario. At a high surfactant concentration, structural effects lead to the formation of patterns on the interface, which can be used to trace the interface velocity, that yield direct confirmation of the boundary condition in the viscous regime.

Figure 1

(a) Schematics of the experiment and notations used in the text. (b) Left: plot of the film thickness for $\mathrm{Ca}=3.1\times {10}^{-6}$. Right: plot of the film thickness for $\mathrm{Ca}=8.1\times {10}^{-5}$. Dotted rectangles: averaging area for the mean film thickness $h_{\infty }$.

Figure 2

Film thickness as a function of the capillary number. (red dots) Experimental data. Lines: theoretical predictions for different boundary conditions [10]: (dot-dashed) stress free, (dashed) sliding, (dotted) rolling. (solid blue) stress continuity at the interface with a viscous drop (${\mu }_d/{\mu }_f=25$) [3]. $h_{\mathrm{\Pi }}=\text{static}$ film thickness. Inset: static film thickness $h_{\mathrm{\Pi }}$ as a function of the Debye length ${\lambda }_D$.

Figure 3

(a) Local film thickness as a function of the capillary number. (red dots) Experimental data. For $3\times {10}^{-5}<\mathrm{Ca}<5\times {10}^{-5}$, two thicknesses coexist: (green squares) $h_{\max }$ and (blue lozenges) $h_{\min }$. Lines: see Fig. 2. Inset: sketch of the oscillatory disjoining pressure isotherm, according to a structural model [21]. The bold line corresponds to the common black film. (b) Spatial plot of the film thickness, for two capillary numbers in the transition zone (left, $\mathrm{Ca}=3\times {10}^{-5}$; right, $\mathrm{Ca}=3.4\times {10}^{-5}$). $h_{\max }$ is coded in red and $h_{\min }$ in light blue. Note that the color scales are adjusted in order to enhance the pattern contrast.

Figure 4

(a) Spatiotemporal plot of a droplet cross section (raw interference patterns). The section is taken along the direction of movement, as shown in the inset. Brighter colors correspond to higher thicknesses, in arbitrary units. The droplet and pattern velocities are measured from the angles between the dashed white lines: $U_d=345\mu \mathrm{m}/\mathrm{s}$ and $U_{\text{pat}}=27\mu \mathrm{m}/\mathrm{s}$. (b) Normalized interface velocity $(U_{\text{int}}/U_d)*100$ as a function of the lubrication film thickness $h_{\infty }$. (red dots) Experimental data. (solid line) Upper and lower estimations assuming a viscous model.