Substrate Curvature Gradient Drives Rapid Droplet Motion

Making small liquid droplets move spontaneously on solid surfaces is a key challenge in lab-on-chip and heat exchanger technologies. Here, we report that a substrate curvature gradient can accelerate micro- and nanodroplets to high speeds on both hydrophilic and hydrophobic substrates. Experiments for microscale water droplets on tapered surfaces show a maximum speed of $0.42\mathrm{m}/\mathrm{s}$, 2 orders of magnitude higher than with a wettability gradient. We show that the total free energy and driving force exerted on a droplet are determined by the substrate curvature and substrate curvature gradient, respectively. Using molecular dynamics simulations, we predict nanoscale droplets moving spontaneously at over $100\mathrm{m}/\mathrm{s}$ on tapered surfaces.

Figure 1

Experimentally observed fast spontaneous motion of water droplets on a tapered glass tube with radius range $0.002–0.8\mathrm{mm}$, half-apex angle $\alpha =3.2°$, and different surface conditions. (a) Video frames of a $1{\mathrm{mm}}^3$ droplet moving on the tapered glass surface with the advancing and receding contact angles ${\theta }_{\mathrm{a}}=10.8°$, ${\theta }_{\mathrm{r}}=9.3°$. (b) Velocities, $v$, as functions of local radius, $R$, of the glass tapered tube, for droplets moving on the same tapered surface with three different surface conditions: untreated (${\theta }_{\mathrm{a}}=43.2°$, ${\theta }_{\mathrm{r}}=39.5°$) for a $1{\mathrm{mm}}^3$ droplet (green circles), plasma-treated (${\theta }_{\mathrm{a}}=16.3°$, ${\theta }_{\mathrm{r}}=13.6°$) for a $1{\mathrm{mm}}^3$ droplet (purple diamonds), and plasma-treated plus MTS nanotextured (${\theta }_{\mathrm{a}}=10.8°$, ${\theta }_{\mathrm{r}}=9.3°$) for a $1{\mathrm{mm}}^3$ droplet (blue squares) and a $0.0115{\mathrm{mm}}^3$ droplet (red triangles, but $\alpha =5.1°$). See the videos in the Supplemental Material [20] for more details. The red dashed and the blue dash-dotted lines are theoretical predictions for the $0.0115{\mathrm{mm}}^3$ and the $1{\mathrm{mm}}^3$ droplets (the fitting parameters are chosen as $\alpha =5.1°$, $\overline{\theta }=58.0^{\circ }$, and $\mathrm{\Delta }\theta =10°$ for the $0.0115{\mathrm{mm}}^3$ droplet, and $\alpha =3.2°$, $\overline{\theta }=40.0^{\circ }$, and $\mathrm{\Delta }\theta =10°$ for the $1{\mathrm{mm}}^3$ droplet).

Figure 2

Total free energy and driving force of a droplet as functions of substrate curvature $\kappa$ measured directly below the center of mass of the droplet. (a) Numerical results (dots) based on finite-element analysis for normalized total free energy $u$ of a water droplet on a conical surface and analytical approximation (solid lines). Results for water contact angles $\theta =120°$, 90°, 60°, and 30° are shown. The circle, triangle, and square symbols represent results for half-apex angles of the cone $\alpha =0°$ (cylinder), 19.5°, and 30°, respectively. (b) The corresponding driving force $F_{\mathrm{c}}=-dU/ds$ is calculated using the analytical approximation, where $s$ is a coordinate along the generatrix of the cone ($\alpha =19.5°$) pointing in the direction of increasing substrate curvature.

Figure 3

The dependence of the critical size $R_{\text{cr}}$ upon the contact angle $\theta$, hysteresis $\mathrm{\Delta }\theta$, and half-cone angle $\alpha$ for droplets placed on: (a) outer conical surface and (b) inner conical surface.

Figure 4

Molecular dynamics simulations of spontaneous motion of a droplet consisting of 386 water molecules on the outer surface (a), (b) and inner surface (c), (d) of a nanoscale carbon cone. In (b), the velocity of the droplet, $v$, which starts near the tip of the cone, is plotted as a function of the cross section radius of the cone, $R$, until it reaches the large end of the cone, where it briefly stops. The red dashed and the blue dash-dotted lines correspond to water contact angles of 51° and 138°, respectively. In (d), the corresponding velocity of the droplet moving inside the cone is plotted, from a starting point near the large end. Illustrations in (a), (c) show droplet positions at three different moments, $t_1$, $t_2$, and $t_3$, that are noted in the graphs (from the MD simulation movie, see the Supplemental Material [20]).