Capillary forces are very important as they exceed in general other adhesion forces. But at the same time the exact calculation of these forces is very complex, so often assumptions and approximations are used. Previous research was done with regard to micrometer sized particles, but the behavior of nanoscale particles is different. Hence, the results for micrometer sized particles cannot be directly transferred when considering nanoscale particles. Therefore, a simulation method was developed to calculate numerically the shape of a rotationally symmetrical capillary bridge between two spherical particles or a particle and a plate. The capillary bridge in the gap between the particles is formed due to capillary condensation and is in thermodynamic equilibrium with the gas phase. Hence the Kelvin equation and the Young–Laplace equation can be used to calculate the profile of the capillary bridge, depending on the relative humidity of the surrounding air. The bridge profile consists of several elements that are determined consecutively and interpolated linearly. After the shape is determined, the volume and force, divided into capillary pressure force and surface tension force, can be calculated. The validation of this numerical model will be shown by comparison with several different analytical calculations for micrometer-sized particles. Furthermore, it is demonstrated that two often used approximations, (1) the toroidal approximation and (2) the use of an effective radius, cannot be used for nanoscale particles without remarkable mistake. It will be discussed how the capillary force and its components depend on different parameters, like particle size, relative humidity, contact angle, and distance, respectively. The rupture of a capillary bridge due to particle separation will also be presented.
Dörmann, M., & Schmid, H.-J. (2014). Simulation of Capillary Bridges between Nanoscale Particles. Langmuir, 1055–1062. https://doi.org/10.1021/la404409k
Dörmann, Michael, and Hans-Joachim Schmid. “Simulation of Capillary Bridges between Nanoscale Particles.” Langmuir, 2014, pp. 1055–62, doi:10.1021/la404409k.