The profile of a capillary liquid bridge between solid surfaces

Scanning force microscopy, such as atomic force microscopy (AFM) is complicated by the capillary force of a water meniscus formed in air between the probe tip and the sample. This small liquid bridge between the hydrophilic sample and the sharp AFM tip can be formed by capillary condensation from the vapor phase. We present an analytical model that describes the shape of the meniscus in which the pressure difference across the curved liquid air interface is taken into account. The analysis is based on a minimization of the liquid surface energy, together with the boundary condition of a given pressure drop across the surface as determined by the relative humidity of the vapor. The capillary forces that the wetting liquid exerts on the AFM tip are derived from the model. The resulting expressions can be used to describe arbitrary axial symmetric liquid air interfaces with nonzero total curvature, such as fluid bridges between two surfaces and droplets under a uniform force. The model illustrates some of the basic concepts of capillarity, such as surface tension forces and interfacial pressure drops.