Modeling of a Self-Oscillating Cantilever

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1 Moeling of a Self-Oscillating Cantilever James Blanchar, Hi Li, Amit Lal, an Doglass Henerson University of Wisconsin-Maison 15 Engineering Drive Maison, Wisconsin 576 Abstract A raioisotope-powere, self-oscillating cantilever beam has been evelope for small scale applications. A thin beam is place within a short istance from a raioisotope sorce an as the charge particles from the sorce are collecte on the beam, it is attracte towars the sorce. As it contacts the sorce, the beam is ischarge an retrns to it's initial position. The perio of oscillation is governe by the time it takes for the beam to contact the srface an ischarge. A moel has been evelope to provie nerstaning of the behavior of sch a evice, an to help explore potential applications. Base on a single imensionless parameter, this moel preicts the esign space for which the beam will self-oscillate, as well as regimes for which contact is never achieve. Initial benchmarks of the moel are encoraging. Introction Efficient on-boar power for MEMS evices will create an opportnity for a wie variety of applications. Previosly sggeste power sorces incle fossil fels, fel cells, chemical batteries, an solar energy, bt nclear power sorces provie significant avantages for applications reqiring long lives or high power ensity. Sch sorces have been extensively researche on fairly large scales [1-4], bt application to the microscale is jst beginning. One approach to harnessing raioisotope power for MEMS evices is to collect raiate charges across a capacitor. This is calle a irect conversion nclear battery []. Other options tilie the heat proce by raioactive ecay, along with thermoelectric or thermionic evices to proce electricity. In this paper we emonstrate a novel irect conversion battery in which one of the electroes is elastically eformable. Recently [5], feasibility an a preliminary moel were presente. In this paper we present an analytical moel that reslts in an elegant approach to nerstaning varios areas of oscillator operation. Potential for Isotope-Powere MEMS evices Some typical isotopes that one wol se in a nclear powere MEMS evice are presente in Table 1. Characteristics reqire for these applications incle low range (to avoi passing throgh the evice) an an absence of gamma emission (for safety reasons). The 1-Po isotope in this table is an alpha emitter, bt all the others are beta emitters an none exhibit gamma emission. Power ensities range from.6 to 17 W/g. The lifetime of these evices will epen on the half-life of the isotope, so one can easily get a battery which retains a sbstantial fraction of it's available power over ecaes. One way to assess the life of a power sorce is to compare the energy ensity, which integrates the power over the life (withot recharging). A typical chemical battery has an energy ensity on the orer of 1 kj/g, while a typical nclear battery will have contain well over 1, kj/g. Hence, power sorces fele by raioisotopes are ieal for applications reqiring high power ensity an a long life (withot refeling). The experiments escribe below employ 6-Ni to power an oscillator. Isotope Average energy Half life Specific activity Specific Power Estimate Range in C (KeV) (year) (Ci/g) (W/g) (microns) 6-Ni Sr H Po P TABLE 1. Some caniate raioisotopes for MEMS applications. The first three colmns are obtaine from Reference 6. A Nclear Powere Oscillating Actator A novel application of raioisotope power at small scales is the realiation of a self-reciprocating or oscillating actator that can generate forces for microscale systems. The central iea behin this oscillator is to collect the charge particles emitte from the raioisotope by a cantilever. By charge conservation, the raioisotope thin film will have opposite charges left as it raiates electrons into the cantilever. Ths an electrostatic force will be generate between the cantilever an the raioisotope thin film. The reslting force attracts the cantilever towar the sorce. With a sitable initial istance the cantilever eventally reaches the raioisotope an the charges are netralie via charge transfer. Althogh the exact mechanisms of charge transfer can be tnneling or irect contact, the time scale of the charge transfer is mch shorter than the reciprocation cycle, allowing the etails to be ignore for cantilever performance analysis. As the electrostatic force is remove, the spring

2 force on the cantilever retracts it back to the original position an it begins to collect charges for the next cycle. Hence, the cantilever acts as a charge integrator allowing energy to be store an converte into both mechanical an electrical forms. Base on this iea, a prototype cantilever evice has been mae an an analytical moel is evelope. A schematic of the evice is shown in figre. the sorce is small allowing the approximation that an average gap exists between the cantilever an the sorce. Assming that the cantilever moves very slowly, an assmption that is verifie by experiment, one can ignore the cantilever's inertia. In this qasi-static approximation, the spring force of the cantilever exactly balances the electrostatic attraction force acting on the cantilever. This can be written as: k ) QE ( () where k is the spring constant, is the initial istance, is the istance between the cantilever an the sorce, Q is the total charges on the cantilever an E is the electric fiel. Assming a niform electric fiel, the capacitor can be moele as a parallel plate capacitor C an the charge on it is: Figre : Schematic of the oscillator Electromechanical Moel A moel has been evelope to provie nerstaning of the behavior of this oscillator [5]. The raioisotope sorce is moele as a crrent sorce. The cantilever/sorce gap is moele as a time varying capacitor. A parasitic resistor is incle to moel possible leakage paths for the collecte charge. Several physical mechanisms may contribte to this resistance. Both natrally occrring ions, an ions create by electronic collisions between emitte particles an gas molecles will constitte a crrent. Frthermore, seconary electrons emitte from the cantilever e to high-energy electron-sbstrate collisions may contribte to the leakage crrent with a polarity opposite to the emitte crrent. Charge conservation reslts in: V V I ε A R t α (1) where I is the total emitte crrent from the raioisotope, A is the area of the capacitor, R is the eqivalent resistance, V is the voltage across the sorce an the cantilever, t is the time, is the istance between the electroes, an α is an empirical coefficient escribing the portion of the total emitte crrent that gets collecte by the cantilever. The first term is the emitte crrent; the secon is the leakage crrent an the thir is the isplacement crrent. There are at least three reasons for imperfect charge collection (i.e., α < 1). First, the charge particles emitte from the sorce have an anglar istribtion an only the particles that fall in the soli angle forme by the intersection of the sorce an cantilever are collecte. Secon, some high energy particles can travel throgh the cantilever. Thir, when seconary electrons are emitte from the cantilever, positive charges are left in the cantilever, recing the net negative charges. ε AV Q CV () Combining Eqations an with the niform electric fiel approximation gives: V k( ) ε A (4) which can be rewritten as: k V A ε Sbstitting Eqation 5 into Eqation1 reslts in: t ε RA αi ( ) ε ka The behavior of the oscillator can be nerstoo throgh a stability analysis of this eqation. Before carrying ot this analysis, it is convenient to introce a few relevant imensionless variables. With the following efinitions: t ε RA (7) (5) (6) The thir term in eqation1 is the isplacement crrent of the capacitor. The electrical fiel E between the sorce an the cantilever has been approximate as niform, i.e. E V/, becase the angle of approach between the cantilever an eqation 6 becomes (1 ) 1 (8)

3 where αir / (9) k Hence, the oscillator behavior is controlle by this single imensionless parameter. The significance of this parameter can be nerstoo as the ratio of two crrents, as shown below. If there is large resistance between the beam an the sorce, then the first term on the right sie of eqation 6 can be ignore an the soltion is 1 (1) From this soltion, the time it takes for the beam to contact the sorce, fon by setting, is contact (11) or T ε Ak αi (1) where T is the time to contact. If sch a beam were fixe, the charge that accmlate on the beam ring time T wol proce a potential given by V Q C k (1) The leakage crrent then is jst V/R. The ratio of the crrent reaching the collector beam to this leakage crrent is the imensionless parameter. In this sense, is a competition between the leakage an storage crrents, with large inicating low leakage crrents. Hence, one wol expect the beams to work more efficiently for large vales of. The eqilibrim points of eqation 8 are fon by setting the time erivative in eqation 8 to an solving for the roots of the reslting fnction. There is always an eqilibrim point at 1 an a secon real root which is negative. This latter root will be ignore for the remainer of this iscssion. For >.8 (14) 7 the remaining two roots are imaginary an the beam will always contact the sorce becase the time erivative is always negative on <<1. For <.8 there are two real roots between an 1. The larger of these roots is a stable root, while the smaller is nstable. Hence, the beam will only be attracte to the sorce if the beam is somehow externally eflecte to a imensionless gap less than that given by the smaller root. By taking a Taylor series expansion in of this lower root, one fins that the smaller root is approximately (15) To smmarie, the beam will not self-oscillate for <.8 an will only contact the sorce if is externally ecrease to the vale given in eqation 15. The vale of this lower root is shown in Figre (sing the fll soltion, rather than jst the approximation) Figre : Smaller root of the right han sie of eqation 8 for small vales of. The beam will not contact the sorce nless an external force reces the gap to this vale (or below). A Variation on the Original Moel To the extent that the resistance R epens on the ion ensity between the beam an the sorce, one might expect R to be proportional to the gap with. This changes the natre of the eqilibria iscsse in the previos paragraph. To explore this, we repeat the analysis after sbstitting R ρ (16) into eqation 6. In this eqation, is the instantaneos gap between the beam an the sorce an ρ is a constant. Physically, ρ represents the resistance per nit gap with between the sorce an the beam. Using eqation 16, the imensionless variables become t ε ρa w αiρ k (17)

4 an the ifferential eqation becomes (1 ) w 1 (18) We now have a new imensionless parameter w, which controls the behavior of the system. This system still has an eqilibrim point at 1. If w>1, there is no eqilibrim point on <<1, so the beam will always work. If w<1, then there is an eqilibrim point at 1-w. This is a stable point, so the beam will always eflect to this point an stop. It will never contact the sorce. Eqation 18 can be solve analytically. The time epenent soltion for the gap is given by ( / 1 w 1 e ) (19) This reslt is plotte for three representative vales of w in Figre 4. From eqation 19, we can etermine the time to contact for the beam. For w<1, there is never contact. Bt for w>1, we fin eqation, which is plotte in Figre 5. 1 ln 1 w contact (19) w w eta w1 Figre 4: Dimensionless gap with vs imensionless time for three vales of w. Negative vales of have been set to Figre 5: Dimensionless contact times as a fnction of w. Prototype Device A prototype evice has been mae to verify the cantilever operation an the moel [5]. A β sorce mae of 6 Ni is se as the raioisotope. The 6 Ni is electroplate as a 4 mm x 4 mm thin film on a 1 mm thick Al plate an the activity is 1 mci. The cantilever is mae of copper with imensions 5 cm x 4 mm x 6 microns. The thickness 6 microns was chosen to captre most of the electrons as the penetration epth of a 67 KeV electron in copper is approximately 14 microns [7]. The cantilever is clampe between two nonconctive Teflon blocks for electrical inslation. The sorce is clampe by two glass slies, which are monte on a Teflon base. The Teflon base is in trn monte on a linear motion stage se to control the initial istance between the sorce an the cantilever. The setp is place insie a vacm chamber with a glass top. A microscope connecte to a CCD camera otsie the chamber is se to the monitor the gap between the sorce an the cantilever. Experimental Reslts Figre 6 shows the istance verss time for an initial istance of microns with a perio of 6 mintes an 8 secons. Figre 7 shows the measre an calclate contact times for varios vales of. Each of these graphs se the moels in which the resistance R is proportional to the separation an the following vales for the constants: α.5 I 5 pa A 1 m k.16 N / m ρ A/ m w We have also seen the qalitative behavior erive in the previos sections, that is, we ve seen beams stop before reaching the sorce, bt this preiction remains to be verifie qantitatively.

5 gap (m).5e-5.e-5.5e-5.e-5 1.5E-5 1.E-5 5.E-6 Measre Calclate.E+ 1 4 time (s) Figre 6: Measre gap for the prototype oscillator Perio (secons) Measre 5 Calclate.E+ 5.E-6 1.E-5 1.5E-5.E-5 Initial Separation (m) Figre 7: Measre an calclate contact times for the prototype beam. Conclsions Raioisotopes offer an attractive power sorce for MEMS evices for applications that reqire high power ensity or long life (withot intervention). A irect charging nclear evice with a eformable electroe has been emonstrate, creating a nclear powere oscillator that can be sefl in small-scale applications. The esign isses associate with this oscillator have been explore an appropriate imensionless parameters have been erive to allow simple esign. These moels have been verifie qalitatively with preliminary experiments. Acknowlegment This work is spporte by the US DOE ner NEER grant DE-FG7-99ID1781. References 1. Thomas, Ncleonics, 1,19 (1955).. J. H. Coleman, Ncleonics, 11, 4 (195).. E. G. Liner an S.M. Christian, J. Appl. Phys.,, 11 (195). 4. J. Bran, L. Fermvik an AA. Stenback, J. Phys. E, 6, 77 (197). 5. Li, H., Lal, A., Blanchar, J., Henerson, D., Selfreciprocating Raioisotope-powere Cantilever, Digest of Technical Papers, International Conference on Soli State Sensors an Actators, Transcers 1, Mnich, pp G. Harer, Pocket Gie for Raiological Management, Perma-Fix Environmental Services, (1999). 7. Stopping Powers for Electrons an Positrons, International Commission on Raiation Units an Measrements, (1984).

Self-reciprocating radioisotope-powered cantilever

JOURNAL OF APPLIED PHYSICS VOLUME 92, NUMBER 2 15 JULY 2002 Self-reciprocating radioisotope-powered cantilever Hui Li and Amit Lal a) SonicMEMS Laboratory, Department of Electrical and Computer Engineering,