Modeling and Simulations of Collapse Instabilities of Microbeams due to Capillary Forces

We present modeling and analysis for the static behavior and collapse instabilities of doubly-clamped and cantilever microbeams subjected to capillary forces. These forces can be as a result of a volume of liquid trapped underneath the microbeam during the rinsing and drying process in fabrication. The model considers the microbeam as a continuous medium, the capillary force as a nonlinear function of displacement, and accounts for the mid-plane stretching and geometric nonlinearities. The capillary force is assumed to be distributed over a specific length underneath the microbeam. The Galerkin procedure is used to derive a reduced-order model consisting of a set of nonlinear algebraic and differential equations that describe the microbeams static and dynamic behaviors. We study the collapse instability, which brings the microbeam from its unstuck configuration to touch the substrate and gets stuck in the so-called pinned configuration. We calculate the pull-in length that distinguishes the free from the pinned configurations as a function of the beam thickness and gap width for both microbeams. Comparisons are made with analytical results reported in the literature based on the Ritz method for linear and nonlinear beam models. The instability problem, which brings the microbeam from a pinned to adhered configuration is also investigated. For this case, we use a shooting technique to solve the boundary-value problem governing the deflection of the microbeams. The critical microbeam length for this second instability is also calculated.