Furmidge k-Factors: From Liquid-like and Superhydrophobic to High Hysteresis and Wettability Patterned Surfaces

Recently, it has been shown droplet friction can be divided into static and kinetic regimes similar to solid-on-solid sliding friction and that an Amontons'-like law for contact line friction can be defined. It is also widely assumed that the frictional forces can be related to contact angles measured in side-profile view through the Furmidge-Kawasaki equation providing a constant of proportionality, the Furmidge k-factor, for the droplet-substrate system is known. Here, we consider under what conditions this assumption is valid and present a new method for determining the value of k based on a tilting-substrate experiment using a single droplet. We validate our method using droplets of water on hydrophobic and hydrophilic slippery covalently attached liquid-like surfaces (SCALS), which have ultralow contact angle hysteresis and extremely low contact line pinning. Remarkably, we find our methodology can be applied to nanoparticle-based superhydrophobic surfaces which have contact angles exceeding 160° and on which droplets roll at extremely low substrate tilt angles. We further show our methodology can be applied to surfaces with high contact angle hysteresis. In all these cases where the contact area is expected to be best approximated by a circular contact line, we find a numerical value for the Furmidge k-factor consistent with a theoretical prediction of k = π/4 ≈ 0.785. Finally, we show our methodology can be applied to droplets pinned onto contact area shapes defined using wettability contrasts.