Capillary self-alignment of polygonal chips: a generalization for the shift-restoring force

Capillary-driven self-alignment using droplets is currently extensively investigated for self-assembly and microassembly technology. In this technique, surface tension forces associated to capillary pinning create restoring forces and torques that tend to bring the moving part into the alignment. So far, most studies have addressed the problem of square chip alignment on a dedicated patch of a wafer, aiming to achieve 3D microelectronics. In this study, we investigate the shift-restoring forces for more complex moving parts such as regular—convex and non-convex—polygons and regular polygons with regular polygonal cavities. A closed-form approximate expression is derived for each of these polygonal geometries; this expression agrees with the numerical results obtained with the Surface Evolver software. For small shifts, it is found that the restoring force does not depend on the shift direction or on the polygonal shape. In order to tackle the problem of microsystem packaging, an extension of the theory is done for polygonal shapes pierced with connection vias (channels), and a closed form of the shift-restoring force is derived for these geometries and again checked against the numerical model. In this case, the restoring force depends on the shift direction. Finally, a non-dimensional number, the shift number, is proposed that indicates the isotropic or anisotropic behavior of the chip according to the shift direction.