Capillary force between a probe tip with a power-law profile and a surface or a nanoparticle

This paper presents a method to exactly calculate the capillary bridge profile and capillary force between a probe tip with a power-law profile and a surface or nanoparticle at various humidities. The shape of the probe tip is denoted by a power-law profile including a shape index, which is a general extension of the usual parabolids. The accuracy of the usual circular approximation of the capillary bridge is investigated, and it is found that great deviations may occur for capillary bridges and capillary forces at nanoscales, especially for the probe tip with a small shape index. The proportion of the capillary pressure force and the surface tension force which constitute the capillary force is also considered. The results show that the surface tension force can even be dominant at nanoscales. The influences of various parameters, including the shape index, equivalent radius of the tip, relative humidity, contact angles and separation distances, on the capillary force between a probe tip with a power-law profile and a surface or nanoparticle are studied. The rupture distances of the liquid bridges at condensation equilibrium and constant volume conditions are also compared.