Equilibrium shapes and locations of axisymmetric, liquid drops on conical, solid surfaces

Micro- and nano-scale liquid droplets must be precisely placed in proximity to conical solid tips in applications that include AFM tip dip-pen nanolithography. This seemingly simple task is strongly dependent on wetting conditions of the drop on the tip surface. Over a wide range of wetting conditions and drop volumes, drops can situate far from a conical needle apex at equilibrium, which can hinder the effectiveness of the respective applications. Needle geometry also affects drop location. On a right conical needle, liquid drops that wet the needle surface locate away from the apex. However, by careful consideration of their geometry, needles can be engineered to retain liquid drops near the tip or some other region of interest. In this work, a theoretical model is developed that predicts axisymmetric, equilibrium drop locations and shapes on conical, solid surfaces as a function of drop volume, needle geometry/shape, needle surface wettability (contact angle), liquid surface tension, line tension, and gravity.