Post-pinch-off relaxation of two-dimensional droplets in a Hele-Shaw cell

We report on the shape relaxation of two-dimensional (2D) droplets, formed right after the spontaneous pinch-off of a capillary bridge droplet confined within a Hele-Shaw cell. An array of bridge droplets confined within a microchip device first undergoes neck thinning due to the evaporation-driven volume change. Subsequently, an abrupt topological change transforms each bridge droplet into a small central satellite droplet and the twin droplets pinned at the edges of the cell. We monitor the shape relaxation with high-temporal-resolution optical microscopy. Capillary action drives the 2D shape relaxation, while the viscous dissipation in the film retards it. As a result, the tip of the twin droplets exhibits a self-similar parabolic shape evolution. Based on these observations, the lubrication-approximation model accurately predicts the internal pressure evolution and the droplet tip displacement. The geometrical confinement substantially slows down the dynamics, facilitating visualization of the capillary-viscous regime, even for low-viscosity liquids. The characteristic relaxation timescale shows an explicit dependence on the confinement ratio (width/gap) and the capillary velocity of liquid. We verify the broad applicability of the model using different liquids.

Figure 1

(a) Schematic of the microchip. The top glass wafer contains reservoirs and the bottom glass wafer contains a set of parallel microchannels along with an array of bridging Hele-Shaw cells (top right inset). (b) SEM cross section of the U-shaped parallel microchannels (bottom left inset) as cut along B-${\mathrm{B}}^{\prime }$. (c) The U-shaped bridging Hele-Shaw cell as cut along A-${\mathrm{A}}^{\prime }$. (d) The inset of the Hele-Shaw cell edge.

Figure 2

(a) Liquid is first dispensed into the microchip (I). Evaporation begins at the reservoirs followed by the drainage of microchannels (II) and finally the array of confined bridge-droplets undergoes drainage and evaporation simultaneously (III). (b) Three-dimensional (3D) view of the array of confined capillary bridge droplets (the liquid is blue and the glass microchip structure is light gray). (c) Two consecutive pinch-off events in the nearest neighbor Hele-Shaw cells. Subsequently, the relaxed twin droplets pinned at the edge of Hele-Shaw cell (liquid is dark gray and vapor-filled cells are light gray).

Figure 3

(a) Bridge droplet before the pinch-off. [(b)–(d)] After pinch-off, the identical pointed droplets relax to a circular shape (increasing time from left to right). Liquid is dark gray. (e) The droplet volume (solid black symbols) and its liquid-vapor interface length (open blue symbols) evolution as a function of time. (f) Plot of $W/2r$ (open blue symbols) as a function of dimensionless time $t^{*}$. The droplet in-plane curvature is $1/r$ and the time axis is nondimensionalized using the ratio of half the Hele-Shaw cell width $W/2$ and the capillary velocity of the liquid $v_{Ca}$.

Figure 4

(a) The tip location evolution as a function of time. The inset highlights that the tip location is concurrent on to a single master curve when plotted as a function of nondimensionalized time $t^{*}$ (using the ratio of half cell width $W/2$ and capillary velocity of the liquid $v_{Ca})$. (b) The experimental relaxation time constant $\tau$ (left $y$ axis, black symbols) and the final equilibrium radius of droplets $R/(W/2)$ (right $y$ axis, solid symbols) vs the normalized reciprocal capillary velocity of liquids with regard to water. The blue line corresponds to the equilibrium radius for the droplets, $R/(W/2)=1.28$.

Figure 5

(a) The temporal evolution of the liquid-vapor interface is marked by different colors with $10\text{−}\mu \mathrm{s}$ temporal resolution. The data are marked by solid circle symbols and the lines correspond to the cubic spline-fit curves. The radius of curvature at the tip $r$ is marked by black circles. The tip region lies between $-W/4\le x\le +W/4$ as marked by the red box. (b) Liquid-vapor interface in the transformed coordinate system. The origin is shifted to the tip and the $y$ axis is mirrored. (c) Plot of $y$ vs $x^2/r$, where $r$ is the in-plane radius of curvature at the tip. The $(x,y)$ data in panel (c) corresponds to the transformed coordinate system as shown in panel (b). The liquid-vapor interface data as extracted from the images is marked by red solid symbols and the corresponding cubic spline-fit curves by the blue line. The green line indicates $y=x^2/r$. Thus, the transformation of liquid-vapor interface data from panels (a) to (c) demonstrates the self-similar parabolic tip evolution.