On the shape and size of liquid droplets on flat solid surfaces

Abstract

This article introduces two dimensionless positive geometric parameters that characterize the shape of a liquid droplet on a flat solid surface, which formed by the surface tension. The first parameter, “shape coefficient” K, is defined by the ratio of volume to surface and is always >3 (3 is the space dimension). The second parameter, “holding limit” κ0, is defined by the fraction of osculating surface and K and is <1. The ratio of the surface tension energy of a droplet attached to a substrate in zero gravity to the energy of the same droplet floating in zero gravity is presented through these parameters as 1-(K-3)(κ0-κ)/3(1-κ0), where the material parameter κ (which appears in the Young equation κ=cosθ) indicates the decrease in liquid surface tension by the solid The relative energy of the surface tension, K and κ0, are explicitly expressed for a droplet of an elliptical rounded segment (ERS) shape through its eccentricity e, relative height χ, and relative rounding radius η. It is shown that the Young equation is a self-consistent (i.e., leading to η=0) minimum condition of the energy only in the spherical (e=0) case. The rounding, either inner or outer, is specified by the legs of a triangle with zero angles and the median as a slope line. The main result obtained is the proof that the outer rounded ERS weighty droplets with inflection points, due to weight and hydrostatic forces, cannot exist if their radii larger than 2-4 capillary length. This proscription is absent in zero gravity.