Transient deformation of a droplet near a microfluidic constriction: A quantitative analysis

We report on experiments that consist of deforming a collection of monodisperse droplets produced by a microfluidic chip through a flow-focusing device. We show that a proper numerical modeling of the flow is necessary to access the stress applied by the latter on the droplet along its trajectory through the chip. This crucial step enables the full integration of the differential equation governing the dynamical deformation, and consequently the robust measurement of the interfacial tension by fitting the experiments with the calculated deformation. Our study thus demonstrates the feasibility of quantitative in situ rheology in microfluidic flows involving, e.g., droplets, capsules, or cells.

Figure 1

(a) Characterization chip. Streamlines (from left to right) are visible due to fluorescent tracers and show the divergent flow at the entrance of the wide chamber. Scale bar is $100\mu \mathrm{m}$. (b) Droplet entering the wide chamber. The droplet is first elongated along $y$ and then relaxes toward a spherical shape (stack of 3 images). The deformation of a few percent is barely visible by eye. (c) Schematics showing the parameters $a$ and $b$ describing the droplet shape. This configuration corresponds to a positive deformation of the droplet.

Figure 2

(a) Steady deformation [see Eq. (2)] as a function of position, as obtained from a finite-element simulation (see Sec. 2b3) of an experiment in Hudson et al.'s geometry [10] (modified viscosity ratio $\lambda =30$). The droplet is replaced by a phantom sphere that does not disturb the flow. For comparison, we plot Eq. (1) using Eq. (9) ($D_{{\mathrm{steady}}_{\mathrm{Taylor}}}+\dot{\epsilon }$). (b) Same as panel (a), but when the droplet is actually as a no-slip hard-sphere obstacle. For comparison, we plot Eq. (1) using Eq. (9) ($D_{{\mathrm{steady}}_{\mathrm{Taylor}}}+\dot{\epsilon }$).

Figure 3

(a) $y$ component of the steady viscous stress at the surface of the droplet, at both the entrance and in the middle of the wide chamber, as obtained from finite-element simulations (see Sec. 2b3) of an experiment in our geometry (viscosity ratio $\lambda =30$). The droplet is modeled as a no-slip hard-sphere obstacle. (b) Resulting steady deformation [see Eq. (2)] as a function of position. The steady deformation is presented here for a surface tension chosen equal to $72\mathrm{mN}/\mathrm{m}$.

Figure 4

(a) Calculated theoretical deformation $D_{\mathrm{th}}$ as a function of droplet position $x$, for various values of the droplet-medium interfacial tension $\gamma$ as indicated and an arbitrary initial condition for illustration. $D_{\mathrm{th}}\left(x\right)$ is obtained by integrating Eq. (3) along the trajectory of a droplet advected by the flow, and invoking the simulated $D_{\mathrm{steady}}\left(x\right)$ [Fig. 3]. (b) Fit of the experimental data $D_{\exp }\left(x\right)$ (dots, collected for 28 droplets) to the calculated theoretical deformation $D_{\mathrm{th}}\left(x\right)$, integrated with the first data point as an initial condition. The best-fit parameter ${\tau }_{\mathrm{ca}}$ [see Eqs. (3) and (4)] is indicated.