Modeling of a Self-Oscillating Cantilever

A radioisotope-powered, self-oscillating cantilever beam has been developed for small scale applications. A thin beam is placed within a short distance from a radioisotope source and as the charged particles from the source are collected on the beam, it is attracted towards the source. As it contacts the source, the beam is discharged and returns to it's initial position. The period of oscillation is governed by the time it takes for the beam to contact the surface and discharge. A model has been developed to provide understanding of the behavior of such a device, and to help explore potential applications. Based on a single dimensionless parameter, this model predicts the design space for which the beam will self-oscillate, as well as regimes for which contact is never achieved. Initial benchmarks of the model are encouraging. Introduction Efficient on-board power for MEMS devices will create an opportunity for a wide variety of applications. Previously suggested power sources include fossil fuels, fuel cells, chemical batteries, and solar energy, but nuclear power sources provide significant advantages for applications requiring long lives or high power density. Such sources have been extensively researched on fairly large scales [14], but application to the microscale is just beginning. One approach to harnessing radioisotope power for MEMS devices is to collect radiated charges across a capacitor. This is called a direct conversion nuclear battery [2]. Other options utilize the heat produced by radioactive decay, along with thermoelectric or thermionic devices to produce electricity. In this paper we demonstrate a novel direct conversion battery in which one of the electrodes is elastically deformable. Recently [5], feasibility and a preliminary model were presented. In this paper we present an analytical model that results in an elegant approach to understanding various areas of oscillator operation. Potential for Isotope-Powered MEMS devices Some typical isotopes that one would use in a nuclear powered MEMS device are presented in Table 1. Characteristics required for these applications include low range (to avoid passing through the device) and an absence of gamma emission (for safety reasons). The 210-Po isotope in this table is an alpha emitter, but all the others are beta emitters and none exhibit gamma emission. Power densities range from 0.006 to 137 W/g. The lifetime of these devices will depend on the half-life of the isotope, so one can easily get a battery which retains a substantial fraction of it's available power over decades. One way to assess the life of a power source is to compare the energy density, which integrates the power over the life (without recharging). A typical chemical battery has an energy density on the order of 1 kJ/g, while a typical nuclear battery will have contain well over 10,000 kJ/g. Hence, power sources fueled by radioisotopes are ideal for applications requiring high power density and a long life (without refueling). The experiments described below employ 63-Ni to power an oscillator. Isotope Average energy Half life Specific activity Specific Power Estimated Range in Cu (KeV) (year) (Ci/g) (W/g) (microns) 63-Ni 17.4 100.2 57 0.006 14 90-Sr 195.8 28.8 138 0.16 332 3-H 5.7 12.3 9664 32.5 3 210-Po 5304.3 0.38 4493 137 11 32-P 694.9 0.04 285700 1.18 1344 TABLE 1. Some candidate radioisotopes for MEMS applications. The first three columns are obtained from Reference 6. A Nuclear Powered Oscillating Actuator A novel application of radioisotope power at small scales is the realization of a self-reciprocating or oscillating actuator that can generate forces for microscale systems. The central idea behind this oscillator is to collect the charged particles emitted from the radioisotope by a cantilever. By charge conservation, the radioisotope thin film will have opposite charges left as it radiates electrons into the cantilever. Thus an electrostatic force will be generated between the cantilever and the radioisotope thin film. The resulting force attracts the cantilever toward the source. With a suitable initial distance the cantilever eventually reaches the radioisotope and the charges are neutralized via charge transfer. Although the exact mechanisms of charge transfer can be tunneling or direct contact, the time scale of the charge transfer is much shorter than the reciprocation cycle, allowing the details to be ignored for cantilever performance analysis. As the electrostatic force is removed, the spring force on the cantilever retracts it back to the original position and it begins to collect charges for the next cycle. Hence, the cantilever acts as a charge integrator allowing energy to be stored and converted into both mechanical and electrical forms. Based on this idea, a prototype cantilever device has been made and an analytical model is developed. A schematic of the device is shown in figure 2. Figure 2: Schematic of the oscillator Electromechanical Model A model has been developed to provide understanding of the behavior of this oscillator [5]. The radioisotope source is modeled as a current source. The cantilever/source gap is modeled as a time varying capacitor. A parasitic resistor is included to model possible leakage paths for the collected charge. Several physical mechanisms may contribute to this resistance. Both naturally occurring ions, and ions created by electronic collisions between emitted particles and gas molecules will constitute a current. Furthermore, secondary electrons emitted from the cantilever due to high-energy electron-substrate collisions may contribute to the leakage current with a polarity opposite to the emitted current. Charge conservation results in: 0 0 =       ∂ ∂ − − d V t A R V I ε α (1) where I is the total emitted current from the radioisotope, A is the area of the capacitor, R is the equivalent resistance, V is the voltage across the source and the cantilever, t is the time, d is the distance between the electrodes, and α is an empirical coefficient describing the portion of the total emitted current that gets collected by the cantilever. The first term is the emitted current; the second is the leakage current and the third is the displacement current. There are at least three reasons for imperfect charge collection (i.e., α < 1). First, the charged particles emitted from the source have an angular distribution and only the particles that fall in the solid angle formed by the intersection of the source and cantilever are collected. Second, some high energy particles can travel through the cantilever. Third, when secondary electrons are emitted from the cantilever, positive charges are left in the cantilever, reducing the net negative charges. The third term in equation1 is the displacement current of the capacitor. The electrical field E between the source and the cantilever has been approximated as uniform, i.e. E = V/d, because the angle of approach between the cantilever and the source is small allowing the approximation that an average gap d exists between the cantilever and the source. Assuming that the cantilever moves very slowly, an assumption that is verified by experiment, one can ignore the cantilever's inertia. In this quasi-static approximation, the spring force of the cantilever exactly balances the electrostatic attraction force acting on the cantilever. This can be written as: QE d d k = − ) ( 0 (2) where k is the spring constant, d0 is the initial distance, d is the distance between the cantilever and the source, Q is the total charges on the cantilever and E is the electric field. Assuming a uniform electric field, the capacitor can be modeled as a parallel plate capacitor C and the charge on it is: d AV CV Q 0 ε = = (3) Combining Equations 2 and 3 with the uniform electric field approximation gives: