Role of substrate aspect ratio on the robustness of capillary alignment

Abstract

Capillary forces associated with liquid bridges formed across solid substrates are routinely exploited to align and assemble micro- and nanoscale devices. The magnitude of these forces plays a critical role in minimizing substrate misalignment and therefore should be controlled for robust and reliable fabrication process. We explore through simulations and experiments the role of the substrate aspect ratio (L/W) on capillary restoring forces and torques. Simulations show that increasing the aspect ratio of the substrates increases the capillary torques and forces when the substrates are misaligned through either lateral or rotational perturbations. The effect of substrate area, perimeter, and liquid volume are also systematically explored to show that the increase in restoring torque is caused by an increase in aspect ratio. A simple theoretical model based on the geometry of the system shows excellent agreement with Surface Evolver simulations. Finally, parameters from experimental flip-chip devices [Josell, D. Wallace, W.E. Warren, J.A. Wheeler, D. Powell, A.C. J. Electron. Packag. 124, 227, (2002)] are used in our simulations to show how current capillary self-alignment schemes could benefit from using rectangular substrate shapes with aspect ratio greater than one.