Self-alignment of micro-parts using capillary interaction: Unified modeling and misalignment analysis

•A novel generalized semi-analytical explicit capillary force model is established.•Fourier series expansion is introduced to fit for the plates with arbitrary shape.•Incredible ability to predict the standoff height when considering chip gravity.•Complex phase diagram is obtained to characterize varieties of system instability. Capillary forces induced by liquid meniscus are pivotal to many important technologies, including AFM measurement, powder technology, nano-objects manipulation and capillary driven self-assembly, etc. These forces (lateral, vertical and torque) are the object of substantial theoretical and numerical modeling, especially due to its significant influence on alignment reliability of fluidic self-assembly. To further understand the capillary interaction and misalignment mechanism, we propose a generalized semi-analytical capillary force model, extending the previous single-factor models to a unified one. With an ingenious meniscus deformation mode, surface energy and the associated capillary forces and torque are given simultaneously. The well accordance between model (when degenerated into previous single-factor modes) and Surface Evolver (SE) results indicates that the complex meniscus deformation process can simply be captured by translation and rotation of the “plate series” for small perturbations. Then the intercrossing effect on misalignment is systematically evaluated, from which we obtain a complex phase diagram (also confirmed by SE simulation) to characterize system local instability. It shows that angular misalignment is always associated with the lateral one, which is also observed in the existing experiment. Moreover, the presented model is suitable for other plates (binding sites) with arbitrary shapes.