Passage of surfactant-laden and particle-laden drops through a contraction

We report on the experimental behavior of oil drops suspended in water and passing in a constricted capillary tube under an imposed pressure gradient. The surfaces of the droplets are covered either by colloidal solid particles or soluble surfactants. We investigate the coupling between the flow behavior and the concentration gradient in adsorbed species that are induced by surface expansion when a water lubrication film persists between the drop and the capillary walls. For both particle-laden and surfactant-laden drops, we evidence the formation of strong concentration gradients resulting in surface tension gradients. We show how local values of surface tension can be monitored using flow rate measurements. In the case of particle-laden drops, we demonstrate that the surface tension gradient is balanced by viscous friction in the lubrication film. In the case of surfactant-laden drops, we suggest a Marangoni flow opposes the decrease of surfactant concentration at the front of the drop, up to a threshold value of the surface expansion rate. Finally, we discuss how these effects increase the passage time of surfactant-laden drops in the constriction.

Figure 1

Schematic representation of the variation of interfacial tension $\gamma$ of particle-laden (black line) or surfactant-laden (red line) interfaces with, respectively, the covering rate or the interfacial concentration $\mathrm{\Gamma }$. Schematic representations of the situations corresponding to three points of the curves are shown for surfactant- (left) and particle- (right) laden interfaces.

Figure 2

(a) Schematic representation of the experimental setup. (b) Schematic representation of an oil drop in water crossing the contraction and definition of the notations.

Figure 3

Left axis: Normalized drop flow rate (red circles). Right axis: curvature radius of the drop back (black circles) and front (blue circles) as a function of time for a 125 μm radius surfactant-laden drop at an imposed pressure difference $\mathrm{\Delta }P=1202\mathrm{Pa}$.

Figure 4

View of a Pickering drop covered with silica microspheres (scale bar: 50 μm) and close view of its surface (inset, scale bar: 5 μm). The surface fraction covered with particles was measured to be $C=0.86\pm 0.04$ from close-up views.

Figure 5

Particle-laden drops. (a) Views of the key positions of the drop in the contraction. (i) Drop entering the contraction. Scale bar: 150 μm; (ii) drop front interface crossing the contraction center, $z_F=0$; (iii) drop mass center in the middle of the contraction,$z_G=0$. (b) Drop normalized flow rate as a function of the drop mass center position for a particle-laden drop. Symbols refer to experimental data, blue solid line to computed flow rate with ${\gamma }_B={\gamma }_F={\gamma }_{o/w}$, gray solid line to the computed flow rate with ${\gamma }_B=0$ and ${\gamma }_F={\gamma }_{o/w}$. The inset is a zoom at the minimum of flow rate corresponding to photograph (ii). The drop radius is 136 μm and the pressure difference $\mathrm{\Delta }P=4365\pm 20\mathrm{Pa}$.

Figure 6

(a) Photographs of the key positions of a surfactant-laden drop at 5 cmc. (i) Drop entering the contraction. Scale bar: 150 μm; (ii) drop front interface crossing the contraction center, $z_F=0$; (iii) drop mass center in the middle of the contraction,$z_G=0$. (b) Drop normalized flow rate as a function of the drop mass center position for a surfactant-laden drop. Symbols refer to experimental data, blue solid line to computed flow rate with ${\gamma }_B={\gamma }_F={\gamma }_{\mathrm{cmc}}$, gray solid line to the computed flow rate with ${\gamma }_B={\gamma }_{\mathrm{cmc}}$ and ${\gamma }_F={\gamma }_{o/w}$. The drop radius is 125 μm and the pressure difference $\mathrm{\Delta }P=1202\pm 20\mathrm{Pa}$.

Figure 7

Schematic representation of the velocity profiles in the lubrication film and in the drop. The velocity at the drop interface is denoted $U_{\mathrm{int}}$.

Figure 8

Schematic representation of the particle-laden interface; $\pi$ is the surface pressure ($\pi =\sigma zz$, the surface stress in the principal direction $z$), $h$ the water film thickness, $U_{\mathrm{int}}$ the interfacial velocity, and ${\eta }_w$ the water viscosity. The particle density gradient generates a surface pressure gradient along the drop, which is balanced by the hydrodynamic friction in the water film.

Figure 9

Interfacial tension at the back of a particle-laden drop computed with Eq. (9) as a function of the one computed with Eq. (5) at the minimum flow rate. Error bars result from uncertainty on interfacial velocity measurements (vertical), and flow rate measurements (horizontal). The only varying parameter is the imposed pressure difference across the contraction, $\mathrm{\Delta }P$.

Figure 10

Front surface tension of surfactant-laden droplets normalized by the surface tension of a bare oil-water interface as a function of the expansion rate of the drop at the front and for different surfactant concentrations: 1 cmc (green circles), 5 cmc (black circles), and 10 cmc (red circles). Surface tension is computed following Eq. (6) at the position corresponding to the minimum flow rate. The expansion rate is computed with Eq. (B5) at the same position.

Figure 11

Surface concentration normalized by its value at the cmc computed from the data of Fig. 10 using Eq. (2) for different surfactant bulk concentrations (same as Fig. 10). The solid line is a guide to the eye.

Figure 12

Passage time of a surfactant-laden drop as a function of the imposed pressure difference normalized by the clogging pressure given by Eq. (5) with $Q=0$. The time is the one needed by the drop to cross the contraction, i.e., to move from position $z_1$ to $z_2$ (see Fig. 2). Experimental data (symbols) are shown with the ones predicted for a constant surface tension at the front, respectively, ${\gamma }_{\mathrm{cmc}}$ (red line) and ${\gamma }_{ow}$ (blue line). The predicted crossing time when the front surface tension is modified by the surface expansion is shown as a black line.

Figure 13

Schematic representation of the drop at two successive instants in the constriction showing how the surface expands .

Figure 14

Surface expansion rate of surfactant-laden droplets as a function of their front velocity, computed following Eq. (B4) for surfactant concentrations of 1 cmc (green symbols), 5 cmc (black symbols), and 10 cmc (red symbols). Solid line represents Eq. (B5) with ${\rho }_F=25\mu \mathrm{m}$.