Analysis of the Dynamics and Touchdown in a Model of Electrostatic MEMS

We study a reaction-diffusion equation in a bounded domain in the plane, which is a mathematical model of an idealized electrostatically actuated microelectromechanical system (MEMS). A relevant feature in these systems is the “pull-in” or “jump-to contact” instability, which arises when applied voltages are increased beyond a critical value. In this situation, there is no longer a steady state configuration of the device where mechanical members of the device remain separate. It may present a limitation on the stable operation regime, as with a micropump, or it may be used to create contact, as with a microvalve. The applied voltage appears in the equation as a parameter. We prove that this parameter controls the dynamics in the sense that before a critical value the solution evolves to a steady state configuration, while for larger values of the parameter, the “pullin” instability or “touchdown” appears. We estimate the touchdown time. In one dimension, we prove that the touchdown is self-similar and determine the asymptotic rate of touchdown. The same type of results are obtained in a disk. We also present numerical simulations in some two-dimensional domains which allow an estimate of the critical voltage and of the touchdown time. This information is relevant in the design of the devices.