This award supports a coherent research program of modeling, analysis, and numerical simulation of micro- and nanoelectromechanical systems (MEMS and NEMS). The focus is on the theory of electrostatic-elastic and electrostatic-fluidic systems. The electrostatic-elastic problem of interest concerns the electrostatic deflection of elastic membranes. The study of steady-state deflections leads to the analysis of semi-linear elliptic partial differential equations. The solution set to such equations can depend heavily on the domain. In this work we will extend the theory of electrostatic deflection of membranes to include complications such as partially supported boundaries and domains which are not simply connected. Additionally, we will extend the theory of electrostatic-elastic systems to the nanoscale. In particular, we investigate the electrostatic deflection of carbon nanotubes. This requires a modification of the standard theory to handle the unique nanotube geometry. The electrostatic-fluidic problem of interest concerns induces instabilities on the surface of thin liquid films. The analysis of this problem leads to the study of thin film type equations with a novel nonlinear term due to the electrostatic pressure. Under this award, we will develop a nonlinear stability theory for such problems and extend the basic theory to the study of other electrostatic-fluidic systems of interest in MEMS and NEMS.
The advent of microelectromechanical systems has revolutionized numerous branches of science and industry. The rapidly developing field of nanoelectromechanical systems promises even more radical change. From biotechnology to materials science, from microelectronics to aerospace engineering, the impact of MEMS and NEMS cannot be overstated. However, in order to fully realize the potential of these systems, a theoretical understanding is necessary. This requires the construction and analysis of mathematical models of MEMS and NEMS systems. Such analysis allows for improved device design and often suggests new avenues for exploration. Interestingly, the ultra-modern technology of MEMS and NEMS often relies upon a phenomenon familiar to anyone who has walked across a carpet on a cold, dry, winter's day; namely, the phenomenon of electrostatics. Many MEMS and NEMS devices use the attraction between electrostatic charges to generate forces. In this project, we study such devices. In particular, we study the use of electrostatic forces to move mechanical components and we study the use of electrostatic forces in moving fluids. This analysis will yield insight into the operation and future design of efficient MEMS and NEMS devices. Additionally, as part of this project our research in MEMS and NEMS will be incorporated into the curriculum at the University of Delaware. Utilizing our new applied mathematics laboratory, students of mathematics will broaden their educational experience and try their hand at comparing their theories to the results of experiments they perform themselves.
