Analytical modeling of electrostatic membrane actuator for micro pumps

A large class of micro pumps consists of a cavity in a substrate, a diaphragm that seals the cavity, and inlets and outlets for the cavity which are controlled by valves. The cavity is initially filled with a fluid. The diaphragm is then actuated by a voltage between the diaphragm and the cavity floor to compress the fluid. When the pressure exceeds a certain value, the fluid is expelled. During actuation, the electrostatic attractive force of the substrate and the pressure rise in the fluid lead to the bending and stretching of the diaphragm. Thus, the prediction of the pump performance (e.g., fluid pressure, diaphragm stresses) requires the solution of a coupled nonlinear elasto-electro-hydrodynamic problem. In this paper, a simplified analytical model is developed to predict the state of an electrostatically actuated micro pump at equilibrium. The state includes the deformed shape and the internal stresses of the diaphragm and the pressure of the fluid when the actuator is subjected to a given applied de voltage. The model is based on the minimization of the total energy consisting of the capacitive energy, the strain energy of the diaphragm, and the energy of the fluid which is considered to be an ideal gas. The method is employed to study two pumps, one with an axisymmetric single cavity, and the other with an axisymmetric annular cavity (a cavity with an island in the middle). In the former case, upon actuation, the diaphragm contacts the cavity door from the outer periphery. Thus, energy is a function of the radius of the contact front, and equilibrium configuration is achieved at a radius where the derivative of the energy with respect to the radius vanishes. In the latter case, upon actuation, the diaphragm contacts the cavity from both of the inner and the outer peripheries. Here, equilibrium is reached when the derivatives of the energy with respect to the radii of both of the inner and outer periphery contact points vanish. It is expected that most practical pumps can be analyzed by one of the two formulations presented in the paper. Our analyses of both pumps indicate that the pressure of the gas at equilibrium increases only slightly when the stiffness of the diaphragm is increased, whereas it changes nearly inversely with the thickness of the dielectric between the diaphragm electrode and the cavity floor. Also, as expected, the pressure increases as the initial volume of the cavity (i.e., the volume of the gas to be compressed) is decreased. Furthermore, we find that the calculated stresses in the diaphragm do not exceed the typical yield stress values of many glassy polymers, a candidate material for the diaphragm. Therefore, the assumption of a linear elastic diaphragm employed in the proposed model does not put a limitation on the predictions. Dielectric breakdown may be a limiting factor for the maximum attainable pressure rather than the mechanical strength of the diaphragm material. Although stresses are low, they may be severe enough to cause delamination between different layers in the diaphragm, too.