Omnidirectional droplet propulsion on surfaces with a Pac-Man coalescence mechanism

Dispensing, transporting, and manipulating minute liquid volumes is important to a wide range of scientific and application areas, spanning from biology and medicine to chemistry and materials. Although digital microfluidics are emerging as a popular methodology in the lab-on-a-chip field, important obstacles to its broad adoption, such as those related to reusability, reconfigurability, and susceptibility to fouling, persist. In addition, as liquid volumes decrease in size, their sustenance in an open atmosphere becomes practically impossible due to high volatility, leading rapidly to complete evaporation. Here we introduce and demonstrate a microfluidic platform based on the coalescence-induced self-propulsion of sessile microdroplets on unpatterned substrates. We employ controlled dropwise liquid ejection by electrohydrodynamic printing to create, actively sustain (despite high volatility), and freely translate femtoliter-sized droplets (∼2–5 μm radius) under open-atmosphere conditions. The omnidirectional planar movement of the droplets is achieved by a precisely controlled, on-demand sequence of coalescence events, where the directed motion of an already deposited droplet is dictated by the positioning of the subsequent droplet printed adjacent to it. We studied the transport mechanism experimentally and theoretically and found that, for short time scales (relative to the substrate translation and droplet evaporation), the radial growth of the new smaller droplet being printed can greatly exceed the retraction of the neighboring, already deposited droplet. This rapid growth is exploited to create a liquid meniscus bridging the so generated droplet doublet, and merging it into a single droplet by the action of capillary pressure differences. Evaporative losses ensure that the moving droplets are kept at a constant size after merging, in each coalescence cycle. We observe this coalescence whenever the interspacing between two sequentially printed droplets is below a critical value, and show that we can control this with the underlying substrate velocity. Armed with this transport mechanism, we then demonstrate the utility of this approach by tasking the coalescing self-propelled droplet doublet to perform microfluidic operations such as collecting, transporting, and merging solid residues on a surface, in an on-demand, Pac-Man type motion, exemplifying capabilities for digital microfluidic applications.

Figure 1

Formation of a droplet pair: (a) Schematic of the droplet pair formation. The droplet pair is formed by shifting the nozzle-substrate axis (by moving the substrate) beyond the perimeter of the large, initial “mother” droplet. This leads to the formation of a “daughter,” adjacent droplet, which grows through the continuous addition of printed smaller droplets by the nozzle. The pair reaches a quasiequilibrium state once the continuously added printed liquid equals evaporative losses of the pair. (b) Bottom view, optical micrograph of two adjacent sessile droplets formed by the process shown in (a), before coalescence. Scale bar: (b) 10 μm.

Figure 2

Droplet movement initiation: droplets 1 and 2 coalesce to form a composite droplet 3, which in turn relaxes to a hemispherical cap shape, and is translated by a distance $\mathrm{\Delta }y$ as its contact line retracts, from the point of origin marked in red (and placed at the edge of the large radius $R_{1,y}$ of the mother droplet contact line), while a new droplet is growing at a distance $d$ from it. The coalescence-induced dislocation of the composite droplet 3 is shown in the schematic illustrations in (a), from a side-view perspective, and in the optical micrographs in (b), from a bottom-view perspective. Scale bar: (b) 10 μm.

Figure 3

(a) Optical microscope images showing the relaxation of the merged composite droplet from a hemiellipsoidal shape, right after coalescence, to its equilibrium spherical cap shape. (b) Time evolution of the composite droplet contact radii along the major ($R_{\mathrm{y}}$) and minor ($R_{\mathrm{x}}$) axes from one coalescence event. The curves are fits to Eqs. (1) and (2). The inset (upper right) shows a schematic of the coalescence of two sessile droplets viewed from the top; the red arrows denote the direction of liquid motion along the $x$ and $y$ axes. (c) Variation of the characteristic relaxation time, ${\tau }_R$ with the equilibrium droplet contact radius, $R_{3,\mathrm{eq}}$ for ten coalescence events. The line is the best fit to Eq. (3). Scale bar: (a) 10 μm.

Figure 4

(a) Optical microscope images showing the growth of the daughter droplet (droplet 2), with contact radius $R_2$, to its equilibrium size of radius $R_{2,\mathrm{eq}}$. (b) Temporal evolution of the contact radius $R_2$ of droplet 2. The inset (lower right) shows a schematic of the growing droplet, with the red arrows denoting the direction of liquid motion along the $x$ and $y$ axes. The black curve is a fit to Eq. (4), and the red curve is a power law fit, such that $R_2\sim t^{1/3}$. Scale bar: (a) 10 μm.

Figure 5

(a) Optical micrographs showing droplet pairs interacting with a stage velocity of increasing magnitude, from 0.001 to $0.1\mathrm{mm}{\mathrm{s}}^{-1}$. As the stage velocity reaches $0.021\mathrm{mm}{\mathrm{s}}^{-1}$, an intermittency in coalescence events is observed. The white arrows indicate the solute residues left behind by single evaporated droplets that did not coalesce. At the bottom micrograph, one can observe a series of individually separated droplets, within a time frame (here, 40 ms), as the stage is translated at a relatively high velocity of $0.1\mathrm{mm}{\mathrm{s}}^{-1}$. (b) Plot showing the interaction type of two sessile droplets separated by a distance $d$, for a given stage velocity, with $C=\mathrm{coalescence}$ and $N=\mathrm{noncoalescence}$. The distance $d$ is measured center-to-center, as shown in the inset, and the value of $\frac{d}{R_{1,y}+R_2}$ is measured directly from experiments, for each coalescence and noncoalescence event. (c) Plot showing a simulation (solid line), and experimental data (symbols) of the change in the contact line positions $Y_1$ and $Y_2$ with time, for a coalescence event, with $\frac{d}{R_{1,y}+R_2}=0.98$ and a stage velocity of $0.015\mathrm{mm}{\mathrm{s}}^{-1}$, (d) similarly for a non-coalescence event with $\frac{d}{R_{1,y}+R_2}=1.2$ and a stage velocity of $0.025\mathrm{mm}{\mathrm{s}}^{-1}$. Scale bar: (a) 10 μm.

Figure 6

Sequence of optical micrographs showing the self-propelled droplet collecting up an array of residues. At $t_0$, the array of residues is intact. From $t_1$ to $t_7$, the droplet pair is photographed picking up the solid particles, cleaning the surface along its way. The black arrows illustrate the movement of the nozzle-substrate axis. Scale bar:10 μm.