  How to make small liquid droplets move spontaneously and directionally on
solid surfaces is a challenge in lab-on-chip technologies, DNA analysis, and
heat exchangers. The best-known mechanism, a wettability gradient, does not
move droplets rapidly enough for most purposes and cannot move droplets smaller
than a critical size defined by the contact angle hysteresis. Here we report on
a mechanism using curvature gradients, which we show is particularly effective
at accelerating small droplets, and works for both hydrophilic and hydrophobic
surfaces. Experiments for water droplets on glass cones in the sub-millimeter
range show a maximum speed of 0.28 m/s, two orders of magnitude higher than
obtained by wettability gradient. From simple considerations of droplet surface
area change, we show that the force exerted on a droplet on a conical surface
scales as the curvature gradient. This force therefore diverges for small
droplets near the tip of a cone. We illustrate this using molecular dynamics
simulations, and describe nanometer-scale droplets moving spontaneously at over
100 m/s on nano-cones.
