On the partial differential equations of electrostatic MEMS devices III: Refined touchdown behavior

AbstractThis paper is a continuation of [N. Ghoussoub, Y. Guo, On the partial differential equations of electrostatic MEMS devices: Stationary case, SIAM J. Math. Anal. 38 (2007) 1423–1449] and [N. Ghoussoub, Y. Guo, On the partial differential equations of electrostatic MEMS devices II: Dynamic case, NoDEA Nonlinear Differential Equations Appl. (2008), in press], where we analyzed nonlinear parabolic problem ut=Δu−λf(x)(1+u)2 on a bounded domain Ω of RN with Dirichlet boundary conditions. This equation models a simple electrostatic Micro-Electromechanical System (MEMS) device consisting of a thin dielectric elastic membrane with boundary supported at 0 above a rigid ground plate located at −1. Here u is modeled to describe dynamic deflection of the elastic membrane. When a voltage—represented here by λ—is applied, the membrane deflects towards the ground plate and a snap-through (touchdown) must occur when it exceeds a certain critical value λ∗ (pull-in voltage), creating a so-called “pull-in instability” which greatly affects the design of many devices. In an effort to achieve better MEMS design, the material properties of the membrane can be technologically fabricated with a spatially varying dielectric permittivity profile f(x). In this work, some a priori estimates of touchdown behavior are established, based on which the refined touchdown profiles are obtained by adapting self-similar method and center manifold analysis. Applying various analytical and numerical techniques, some properties of touchdown set—such as compactness, location and shape—are also discussed for different classes of varying permittivity profiles

Surface micromachined electrostatically actuated micro peristaltic pump

An electrostatically actuated micro peristaltic pump is reported. The micro pump is entirely surface micromachined using a multilayer parylene technology. Taking advantage of the multilayer technology, the micro pump design enables the pumped fluid to be isolated from the electric field. Electrostatic actuation of the parylene membrane using both DC and AC voltages was demonstrated and applied to fluid pumping based on a 3-phase peristaltic sequence. A maximum flow rate of 1.7 nL min^–1 and an estimated pumping pressure of 1.6 kPa were achieved at 20 Hz phase frequency. A dynamic analysis was also performed with a lumped-parameter model for the peristaltic pump. The analysis results allow a quantitative understanding of the peristaltic pumping operation, and correctly predict the trends exhibited by the experimental data. The small footprint of the micro pump is well suited for large-scale integration of microfluidics. Moreover, because the same platform technology has also been used to fabricate other devices (e.g. valves, electrospray ionization nozzles, filters and flow sensors), the integration of these different devices can potentially lead to versatile and functional micro total analysis systems (µTAS)

A stationary free boundary problem modeling electrostatic MEMS

A free boundary problem describing small deformations in a membrane based model of electrostatically actuated MEMS is investigated. The existence of stationary solutions is established for small voltage values. A justification of the widely studied narrow-gap model is given by showing that steady state solutions of the free boundary problem converge toward stationary solutions of the narrow-gap model when the aspect ratio of the device tends to zero

No Touchdown at Zero Points of the Permittivity Profile for the MEMS Problem

[[abstract]]We study the quenching behavior for a semilinear heat equation arising in models of micro-electro-mechanical systems (MEMS). The problem involves a source term with a spatially dependent potential, given by the dielectric permittivity profile, and quenching corresponds to a touchdown phenomenon. It is well known that quenching does occur. We prove that touchdown cannot occur at zero points of the permittivity profile. In particular, we remove the assumption of compactness of the touchdown set, made in all previous work on the subject and whose validity is unknown in most typical cases. This answers affirmatively a conjecture made in [W. Guo, Z. Pan, and M. J. Ward, SIAM J. Appl. Math., 66 (2005), pp. 309--338] on the basis of numerical evidence. The result crucially depends on a new type I estimate of the quenching rate, that we establish. In addition we obtain some sufficient conditions for compactness of the touchdown set, without a convexity assumption on the domain. These results may be of some qualitative importance in applications to MEMS optimal design, especially for devices such as microvalves.[[notice]]補正完畢[[journaltype]]國外[[incitationindex]]SCI[[ispeerreviewed]]Y[[booktype]]紙本[[countrycodes]]US