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Investigation of a switching G mechanism for MEMS applications

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Investigation of a switching G mechanism for MEMS applications

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Content INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand comer and continuing from left to right in equal sections with small overlaps. ProQuest Information and Learning 300 North Zeeb Road, Ann Arbor, Ml 48106-1346 USA 800-521-0600 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. INVESTIGATION OF A SWITCHING MECHANISM FOR MEMES APPLICATIONS by Sangwook Kwon A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (AEROSPACE AND MECHANICAL ENGINEERING) May 2002 Copyright 2002 Sangwook Kwon Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 3074942 Copyright 2002 by Kwon, Sangwook All rights reserved. ___ ® UMI UMI Microform 3074942 Copyright 2003 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UNIVERSITY OF SOUTEi ER N CALIFORNIA TheGraduat: School University Park LOS ANGELES, CAUFOINIA 90089-1695 This dissertation, w ritten b y S/\rJ<Sr UOO i v lO JO A / Under th e direction o i h J S D isserta tio n Com m ittee, and approver, b y a il its members, has been presen ted to a i : d accepted b y The Graduate School, in p& tia l fulfillm ent o f requirem ents fo r th e degr, e o f DOCTOR OF PE LOSOPHY D a te a,..20flZ DISSERTATION COMM, TTEE Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Acknowledgments In the first place, I thank Dr. G.R. Shiflett for suggesting the initial direction of my research and whose guidance and encouragement have led me to produce much of this research and have been a great help in the progress. I would like to thank Dr. E. P. Muntz for his enormous support, guidance and helpful comments. I must also express my thanks to Dr. Firdaus Udwadia for his concise yet enlightening comments that improved the quality of this work. I would like express special thanks to Dr. Andrew Ketsdever for his support, assistance, helpful comment and discussion of experimental work. I express my appreciation for time and consideration given by Dr. Eun-Sok Kim. I express my thanks to Mark Young for his help with a high-speed camera and assistance in the experimental work Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Contents Acknowledgements.................................................................................... ii List of Tables............................................................................................vii List of Figures.........................................................................................viii Abstract.................................................................................................... xii Chapter 1 Introduction 1.1 Background and Motivation........................................................................ 1 1.1.1 Switching mechanism......................................................................... 5 1.1.2 Gas adsorption and desorption kinetics.............................................. 8 1.2 Literature review......................................................................................... 11 iii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 2 Analytical investigation 2.1 Lumped parameter model............................................................................. 17 2.1.1 Snap-through..........................................................................................17 2.1.2 Lumped parameter model.......................................................................21 2.2 Continuous model........................................................................................24 2.2.1 The geometry and boundary condition of clamped-clamped buckled beam.....................................................................................................24 2.2.2 Amplitude of buckled beam.................................................................. 25 2.2.3 Sample of mode shapes.........................................................................28 2.2.4 Non-dimensionalization....................................................................... 30 2.3 Strain energy................................................................................................30 2.3.1 Bending effects..................................................................................... 31 2.3.2 Longitudinal deformation effects.......................................................... 33 2.3.3 Total strain energy................................................................................ 36 2.4 Potential energy for various loading conditions......................................... 37 2.5 Solution under quasi-static loading............................................................37 2.5.1 Symmetric case.................................................................................... 41 2.5.2 Antisymmetric case..............................................................................42 2.6 Dynamic solution under impulsive loading................................................ 45 2.7 Generation of desorption forces.................................................................52 2.7.1 Magnitude of resulted switching force ............................................... 54 iv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.7.2 Estimated surface temperature increase....................................................57 2.7.3 The magnitude of impulse........................................................................58 2.7.4 Estimates of switching time......................................................................59 Chapter 3 Experimental investigation 3.1 Experimental goals and procedure...................................................................61 3.2 Experimental setup........................................................................................ 64 3.2.1 Lasers....................................................................................................... 65 3.2.2 Laser energy measuring device................................................................67 3.3.3 Neutral density filter............................................................................... 67 3.2.4 Vacuum chamber.................................................................................... 67 3.2.5 Vacuum pumping system........................................................................69 3.3.6 Pressure gauge........................................................................................ 70 3.2.7 Clamping system.................................................................................... 70 3.2.8 High speed camera.................................................................................. 71 3.2.9 Beam material.......................................................................................... 71 3.3 Beam designing consideration....................................................................72 3.3.1 Material limitation.................................................................................. 73 3.3.2 Geometric limitation................................................................................74 Chapter 4 Results and discussions 4.1 Predictive value......................................................................................... 75 v Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.1.1 Shock wave effects................................................................................. 86 4.1.1.1 Shock wave initiation....................................................................... 86 4.1.1.2 Shock wave influence....................................................................... 88 4.1.2 Photonic impact influence....................................................................... 91 4.2 Experimental results....................................................................................... 92 4.3 Observations and discussion.........................................................................106 4.4 Implications for Micro-bistable switches.....................................................110 4.4.1 Beam material........................................................................................112 4.4.2 Working gas.......................................................................................... 116 Chapter 5 Conclusions 5.1 Summary and concluding remarks...............................................................119 5.2 Suggestion for future work..........................................................................123 Bibliography......................................................................................................125 Appendix A Experimental Data......................................................................................... 131 Appendix B Tables........................................................................................................... 134 vi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List o f Tables Table 1: Numerical values of C02 and N2........................................................ 79 Table 2: Laser energy vs. ambient pressure data for C02.................................107 Table 3: Laser energy vs. ambient pressure data for N2...................................108 Table 4: Switching time vs. ambient pressure data............................................109 Table Al: Experimental data using carbon dioxide............................................131 Table A2: Experimental data using nitrogen..................................................... 132 Table A3: Experimental data of energy coupling efficiency effects..................133 Table B1: Mylar® data...................................................................................... 134 Table B2: Laser data.......................................................................................... 135 Table B3: Adsorption times for several gases on a representative surface (Muntz 1998)................................................................................................. 136 Table B4: Surface concentration and fractional coverage at atmospheric pressure (Muntz, 1998).................................................................................. 137 Table B5: Numerical data for several ambient gases..........................................138 Table B6: Physical properties of several materials common to MEMS.............139 Table B7: Number of molecules per monolayer and equivalent volumes..........140 vii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List of Figures Figure 1: Unstable equilibrium position............................................................... 4 Figure 2: Sketch of a switching mechanisms.........................................................6 Figure 3: Model to study snap-though.................................................................18 Figure 4: Load paths created by equilibrium state...............................................19 Figure 5: Load- deflection curve for snap-through model....................................21 Figure 6: A lumped parameter model of elastic buckled beam............................21 Figure 7: Load-deflection curve for lumped parameter model............................23 Figure 8: Geometry of the clamped buckled beam..............................................24 Figure 9: First buckling mode............................................................................. 29 Figure 10: Second buckling mode...................................................................... 29 Figure 11: Any potion of a bent beam................................................................ 31 Figure 12: Load-deflection curve for quasi-static loading..................................44 Figure 13: Deflection versus kinetic energy curve..............................................49 dKE Figure 14: Deflection curve.................................................................... 51 dax Figure 15-1: Schematic diagram of experiment...............................................64 Figure 15-2: Schematic diagram of experiment...............................................65 Figure 16: Picture of Nd:YAG Laser................................................................. 66 viii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 17: Picture of vacuum chamber.............................................................. 68 Figure 18: Schematic drawing of vacuum system.............................................69 Figure 19: Picture of clamping system..............................................................71 Figure 20: Laser energy per pulse versus temperature increase........................ 76 Figure 21: Pulse duration versus temperature increase..................................... 76 Figure 22: Energy coupling efficiency versus temperature increase................. 77 dKE Figure 23: Deflection versus curve......................................................... 77 dax Figure 24: Deflection versus required kinetic energy....................................... 78 Figure 25: Generated switching force per unit area ( C02)................................80 Figure 26: Generated switching force per unit area (N2)..................................80 Figure 27: Generated impulse per unit area (C 02).......................................... 81 Figure 28: Generated impulse per area (N2)................................................... 82 Figure 29: Generated impulse versus activation time....................................... 83 Figure 30: Generated impulse versus switching time....................................... 84 Figure 31: Laser energy versus switching time (C02)......................................84 Figure 32: Laser energy versus switching time (N2)....................................... 85 Figure 33: Energy coupling efficiency versus switching time (C02)...............85 Figure 34: Energy coupling efficiency versus switching time (N2).................86 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 35: Schematics of shock wave............................................................... 87 Figure 36: Pressure near the beam surface curve.............................................. 88 P T Figure 37: The time development of the profiles of pressure (— ) for — = 2 T 0 (K. Aoki. Et el, 1991)...................................................................... 89 Figure 38: Buckled state vs. time at 2.1 x 10'5 Torr ambient pressure (C02 )....93 Figure 39: Buckled state vs. time at 3.1 Torr ambient pressure (C02).............94 Figure 40: Buckled state vs. time at 10 Torr ambient pressure (C02)..............95 Figure 41: Buckled state vs. time at 50 Torr ambient pressure( C02).............. 96 Figure 42: Buckled state vs. time at 100 Torr ambient pressure (C02)........... 97 Figure 43: Buckled state vs. time at 500 Torr ambient pressure (C02)........... 98 Figure 44: Buckled state vs. time at 760 Torr ambient pressure (C02)........... 99 Figure45: Buckled state vs. time at 3 Torr ambient pressure (N2).................100 Figure 46: Buckled state vs. time at 5 Torr ambient pressure (N2).................101 Figure 47: Buckled state vs. time at 10 Torr ambient pressure (N2)...............102 Figure 48: Buckled state vs. time at 50 Torr ambient pressure( N2)............... 103 Figure 49: Buckled state vs. time at 100 Torr ambient pressure (N2)............ 104 Figure 50: Buckled state vs. time at 760 Torr ambient pressure( N2)............. 105 Figure 51: Schematic drawing of a micromechanical bistable system............ 112 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 52: Minimum necessary impulse and required temperature increase for seve ral representative materials...............................................................112 Figure 53: Energy coupling efficiency versus temperature increase of various mate rials................................................................................................... 113 Figure 54: Energy per laser pulse versus temperature increase of various materials... ..........................................................................................................114 Figure 55: Laser pulse duration versus temperature increase of various materials.... .......................................................................................................... 114 Figure 56: Time versus required gas desorption force....................................... 115 Figure 57: Time versus required impulse...........................................................115 Figure 58: Exerted impulse force versus switching time.................................... 116 Figure 59: surface temperature increase versus exerted impulse force of various working gas.......................................................................................117 Figure 60: Initial surface temperature versus exerted impulse for AT = 300 ....117 Figure 61: Time versus exerted impulse for various working gas ( ^ G = l)....l 18 xi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Investigation of Switching Mechanism for MEMS Applications Sangwook Kwon (ABSTRACT) A pressure induced force due to rapid transient, thermally-induced gas desorption is an interesting mechanical actuation source for micromechanical systems. Buckled may be caused to change their state, using the transient pressure forces created by rapidly heating of the convex surfaces of the buckled beams. A MEMS switching model has been developed for the bi-stable, buckled micro-scale buckled beam. The transient forces are produced by both the heated ambient gas adjacent to the rapidly heated surface of the beam and the simultaneous ejection of adsorbed molecules from the surface. Fast heating of beam surfaces was produced by laser pulses with 4 to 6 nsec duration. The minimum quasi-static force necessary for snap-buckling and the minimum impulse necessary for dynamic snap-buckling of clamped-clamped shallow micro buckled beams was investigated analytically by an examination of unstable equilibrium states in the non-linear range. The analysis is based on the assumption of elastic behavior and the deformation is represented by a simple deflection xii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. function in a one-degree-of freedom system. The analytical results were compared to the results of switching experiments using meso-scale, clamped-clamped, shallow buckled Mylar® beams. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 1 Introduction 1.1 Background and Motivation Recently, increased attention has been given to microswitching devices utilizing the snap-through behavior of buckled beams. Microswitching devices are considered to play an important role in Micro Elecro-Mechanical Systems (MEMS). Such microswitches can be applied to acceleration sensors in electronic airbag systems, telecommunication systems, switching circuits in electronics, non-volatile memory cells, optical switches, microvalves microactuators, microtoggle switches, and so on. One well-known micro-switching device, developed by Texas Instruments Incorporated. Dallas, TX, is a digital micromirror device (DMD), for optical switching. The device has about a million micromirrors, each can tilt either +10° to 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. activate a pixel or a -10° to deflect light away (L.H. Hombeck, 1997). A simple mechanical bistable switching device can be developed by applying a transverse force on a long slender beam. If the applied force exceeds a minimum required conditions, the beam switches from one stable state to another. A pressure-induced force due to rapid transient, thermally induced gas desorption is an interesting source of mechanical actuation in micromechanical systems. Previous MEMS switching models, which used electrostatic forces, required the beam material to be electrically conductive. Our approach is somewhat less restrictive since it only requires good physical properties over a wide temperature range. Experiments using pulsed laser irradiation to deflect a thin film due to a thermally induced transient pressure pulse from accommodation of incident gas molecules on the heated surface were conducted to investigate some of the possibilities. In Pham-Van-Diep et el (1995), forces normal to the surface were produced by the combination of heated ambient gas at the beam surface and adsorbed molecules ejected from the beam surface as the surface is heated by the laser pulse. The desorption of mass from a surface and the heating of ambient gas close to the surface due to rapid surface heating at relatively high pressures are rarefied gas dynamics problems because of the necessarily very short transient heating times that are comparable to the mean collision time in the gas. The amount of gas desorbed and the resulting switching force exerted on the heated surface is 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. significant for micro-mechanical dimensions and practical in terms of energy for very short surface heating times. Experiments were carried out to investigate the functional dependence between the exerted switching force due to gas desorption and the accompanying local gas heating, and energy applied by the laser. Approximate methods for studying the nonlinear buckling problem are important for investigative and design purposes in micro-mechanics. Most structures and systems respond nonlinearly if external loads cause drastic changes in state. Clamped buckled beams have been widely used as micro structural elements. One important response of such elements, when loaded transversely, is snap-through buckling. This snap-buckling phenomenon is characterized by a sudden displacement from one equilibrium configuration to another distinctly different than the first. A complete theoretical and experimental analysis of a shallow clamped buckled beam under external transverse load is examined in order to illustrate the behavior of a nonlinear micro-structural system that undergoes snap buckling. In addition to the basic problems associated with handling nonlinear problems, additional complications arise in the transitional buckling modes. Because transitional buckling modes may be appreciably different from both the initial and final shapes, they present difficult obstacles in formulation and numerical solution of nonlinear problems. One possible approach to the analytical prediction of such dynamic instability levels deals only with the energy of the system. 3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Snap-through buckling occurs when the energy input is sufficient for the buckled beam to jump into a mode characterized by large deformation. During the jump, the buckled beam passes through an unstable equilibrium position. To determine the energy input necessary for snap-buckling to occur from a general energy view point, static unstable equilibrium states need to be examined. Unstable equilibrium position Deflection Figure 1: Unstable equilibrium position. A relative maximum in the state of unstable equilibrium can be used to define a dynamic snapping action criterion. A small increase in input energy beyond that needed to reach the unstable equilibrium point will produce snap-through action. The sufficient condition for snapping action, which has been derived from the principal of strain energy achievable in the system, will be equal to this input energy. The analytical work includes consideration of a number of symmetrical deflection modes as well as transitional nonsymmetrical buckling. The problem is then to find the lowest strain energy that is related to an unstable equilibrium state for 4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. impulsive loading conditions, but which is reasonable for practical loading conditions. For experimental purposes, a set of scaled buckled meso-switches are designed using various sized buckled meso-beams. An experiment was designed to generate a switching force on the heated beam surface from the combinations of rapid transient, thermally induced gas desorption and ambient gas heating. The data obtained also allowed us to examine performance issues of a pressure-driven micro-mechanical micro-switch. 1.1.1 Switching Mechanism It is important to understand the mechanisms which contribute to the switching force that cause the beam to snap through. The pressure of the gas near the beam surface is closely coupled to the beam surface temperature. During a time of increased surface temperature, there is a corresponding increase in the gas pressure. There are four mechanisms that combine to produce a net switching force on the beam surface as the beam surface is heated by a laser pulse (An alternative way to raise the surface temperature is through resistance heating.). These mechanisms are (G. Pham-Van-Diep, E. P. Muntz 1995; K. Aoki et el, 1991): 5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (A) The beam surface temperature increase suddenly as the surface is heated by the laser pulse. Heat is transferred to the gas causing a local pressure increase; (B) Coating particles or pigment may be sputtered or evaporated from the surface; (C) The pressure rise near the beam surface by sudden heating pushes the ambient gas molecules away and causes a compression wave in the ambient gas; (D) Ambient gas molecules initially adsorbed on the beam surface are ejected from the surface due to the sudden temperature increase, thereby applying an impulse to the beam surface. (E) Readsorption of ambient molecular as a cooling surface causes a reverse force to be exerted as the beam. These four mechanisms are illustrated in Figure 1. ) Sudden heating by Q Q laser pulse Thin coating (A) Figure 2: Sketch of switching mechanisms. 6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Compression wave (C) (D) Figure 2 - continued. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (E) Figure 2 - continued. 1.1.2 Gas adsorption and desorption kinetics All gases tend to adsorb below their critical temperature or above their critical pressures as a result of general van der Waals interactions with the solid surface. In • * the case of MEMS switching devices, interest centers on the size and nature of adsorbent-adsorbate interactions and on interactions between adsorbate molecules. On mechanically separating the solid and gas phases, there is a certain distribution of the adsorbate between them. In order to investigate the gas absorption or desorption by a given adsorbent, a definite weight or area of the area of the absorbent is first of all desorbed at as high temperature as practicable, in a high vacuum, in order to remove all adsorbed gas molecules. This may be expressed as v, the volume (cubic centimeters at STP) adsorbed per gram of solid versus the pressure P . The distribution is generally temperature dependent, so the complete 8 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. empirical description would be in terms of an adsorption function v = f(P,T) (S. Dushman, 1962). A useful way to consider the phenomenon of adsorption is from the point of view of the adsorption time, as discussed by de Boer (1953). Consider a molecule in the gas phase that is approaching the surface of the solid. If there were no attractive forces at all between the molecule and the solid, then the time of stay of the molecule in the vicinity of the surface would be of the order of molecular vibration time, or about 10_1 3 sec, and its accommodation coefficient would be zero. By this last, it is meant that the molecule retains its original energy. If attractive forces are present, then according to Frenkel (Coltharp, Ackerman, 1968), the average time of stay, t ,of the molecule on the surface will be t = v0 emim)T' (1) where z0 is the period of an oscillating molecule in the surface potential well and Q is the interaction energy, i.e., the energy of adsorption or desorption is frequently taken, if no other data is available, to be approximately the heat of vaporization of the liquid phase of the adsorbate. The other symbols have their usual designations with the subscript s referring to the surface. An additional empirical observation is that monolayer formation is usually achieved at around a few tenths of the vapor pressure of the absorbate at the absorbent’s surface Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. temperature (Adamson, 1976). The relationship between the average surface stay time t and the surface concentration, ns (in y 2), is / m ns = Si = N t (2) where N = ng (Sk Tg / mrt)^2 . The subscript g refers to gas phase quantities and 5, is an alternate nomenclature representing the number of occupied adsorption site per unit area. The fractional surface coverage can be described by ® = « A s S A (3) where Q s is the cross sectional area occupied by an adsorbed molecule. This is usually assumed to correspond to the size of the molecule, taken variously from kinetics or liquid densities (Muntz, 1998). Because this experiment will be conducted in a macroscopic scale, the fractional surface coverage of adsorbed gas has to manipulate as a dimension of a macroscopic condition, for the other conditions are the same; i.e., 0 @ A A (4) m fl T U where 0 is the fractional surface coverage of adsorbed gas, L is the length, b is the width of the specimen, m refers to macroscale and p refers to microscale. The scaling from macroscopic to microscopic dimensions is such that the most powerful 10 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. forces in the microdevice domain are expected to be electrostatic and pneumatic or pressure forces (Trimmer, 1989). Since response time also scales with size, the response time of micro-scale beam will be considerably shorter than- that of the meso-scale system used in the experiments reported herein. 1.2 Literature review The significance of snap-through buckling, insofar as it illustrates certain important features in more complicated problems of plates and shells, was pointed out by Marguerre (1938) who constructed a simplified mechanical model to demonstrate these features. The classic case of a bimetallic thermostat was studied by S. Timoshenko (1925) when he examined the bending of a beam due to the difference in the coefficients of thermal expansion between two jointed layers of dissimilar materials. Timoshenko (1935) obtained an approximate solution to the problem of a low arch under a uniformly distributed transverse load. This approximate solution avoids solving differential equations and becomes very useful when applied to systems with non-uniform stiffness - a case where the solution to the usual eigen- boundary-value problem is extremely difficult and in some cases impossible. Biezeno (1938) considered the problem of a low arch loaded transversely at the midpoint with a concentrated load. As mentioned in Theory o f Elastic Stability (Timoshenko and Gere), this problem was first discussed by Navier (1833). Fung and Kaplan (1952) investigated the problem of pinned low arches of various initial 11 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. shapes, and spatial distributions of the lateral load. They also considered the effect of prestress on the critical value of the load. Hoff and Bruce (1954) presented results for the pinned half-sine low arch under a half-sine distributed load, as a special case of their dynamic analysis of the buckling of laterally loaded low arches. The results of these two analyses show that a very shallow arch snaps through symmetrically (limit point instability), where a higher arch snaps through asymmetrically. Gjelsvik and Bodner (1962) obtained an approximate solution to the problem of a clamped low circular arch with a concentrated lateral load at the midpoint - a problem which was previously investigated by Timoshenko (1935) and Bizeno (1938) in an approximate manner but with different boundary conditions. Timoshenko’s solution only considered bending effects and Biezeno (1938) * considered different boundary conditions of the arch. Gjelsvik and Bodner (1962) also investigated the nonsymmetrical snap buckling of uniformly loaded clamped spherical caps by energy methods. Humphrey and Bodner (1962) examined the minimum impulse necessary for dynamic snap buckling of shallow spherical caps and circular cylindrical panels using unstable equilibrium states. Their analysis is based on the assumptions of elastic behavior, large deformations, and axially symmetrical deformations. Their results are limited by the fact that only the lowest 4 symmetrical mode was considered in their model. Schreyer and Masur (1966) obtained an exact solution to this problem (and to the case of uniform pressure) and *V.' 12 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. also investigated the concentrated load case that was presented by Gjelsvik and Bodner (1962). From Schreyer and Masur’s work, the arch snaps through q 2 r symmetrically regardless of the value of the rise parameter A, where A = — — , h P =half of the arch angle, R = radius of the arch centerline, and h = thickness of the arch, with assumption that bifurcation occurs in the stable equilibrium region of large X . They concluded the existence of bifurcation of the equilibrium state is not an adequate condition for the use of the asymmetric buckling criterion. An additional requirement is that the bifurcation should occur at the state of equilibrium, which is stable according to the symmetric buckling criterion. Humphrey (1966) considered the general problem of large dynamic elastic deflections of a shallow circular arch under different loading conditions; specifically: cases of impulsive loading, step loading and rectangular pulse loading with various types of initial imperfections. Masur and Lo (1972) presented a general discussion of the behavior of the shallow circular arch regarding buckling, post buckling and imperfection sensitivity. The effects of inelastic material behavior have been considered by Franciosi, Augusti and Sparacio (1964), by Onat and Shu (1962), and by Lee and Murphy (1968). Said (1984) investigated the critical uniform lateral pressure of a initially buckled a pin-supported beam with various loading conditions for cases of static loading conditions, step loading, and impulse loading. Seid’s method of analysis is 13 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. similar to Hoff and Bruce (1954) who used an energy criterion to determine the critical pressures of laterally loaded shallow arches. Seid’s results for critical pressure for the buckled beam have been compared to those of Hoff and Bruce’s for the arch and found to vary in a similar fashion with buckling amplitude. The magnitude of the critical pressure for the initially buckled beam is less than that for the arch having the same amplitude. Haig (1990) illustrates the basic idea of the bistable micromechanical memory bridge consisting of a thin micromachined bridge elastically deformed in such a * way that it has two stable mechanical states. He also investigated the use of electrostatic forces for the changing the state of bridge. S.G. Jeong and Y.H. Cho (1996) designed and tested a prestressed bimorph beam for use in tunable acceleration switches, and investigated necessary and sufficient conditions for snap-through switching functions for a clamped-clamped shallow beam of prismatic cross section. Haig’s and Jeong’s models both used electrostatic forces for switching action; therefore, the beam material had to be electrically conductive. If Si02 and S i,which are widely used materials in MEMS, are used as the beam material, and electrostatic actuation is desired, the beam must be built in the form of a multi-layer structure in which one of the layers is a thin conductive film. Even if the thickness of the metal is small compared to the thickness of the whole beam, it will influence the mechani- cal properties and alter the snapping force. 14 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. M.Chiao and L.Lin (2000) investigated self-buckling behavior of micromachined beams described by an electromechanical model with experimental verification. This model consists of both electro-thermal and thermo-elastic analyses for beam shaped polysilicon microstructures. M.Taher A Saif (2000) examined a tunable micromechanical bistable system theoretically and experimentally. He applied an axial switching force for switching from one buckled state to another. He found good correspondence between theory and experiment. In his analytical study, he treats buckled states, and found that switching force increases linearly with buckled displacement. E.P. Muntz, G.R. Shiflett, D.A.Erwin, and J.A. Kune (1992) investigated a transient energy release, pressure driven microdevice. In this paper, a transient energy release followed by rapid cooling in a gas filled volume is discussed for high frequency switches and pressure driven microdevices. Such devices can produce 102 to 104 times more force per unit volume and, in devices which is operated at high cycle frequencies, 102 to 103 times more power per unit volume than comparably sized electrostatic devices. K. Aoki, Y. Sone, et al (1991) numerically studied the sudden heating ( or cooling) of a plane wall adjacent to a gas on the basis of the Boltzmann-Krook-Welander equation with diffuse reflections to obtain time dependent history of the gas behavior. This study reports that the sudden heating induces a shock wave that propagates through the gas. 15 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Various experimental results about snap buckling have been reported by Fung and Kaplan (1952), and by Roorda (1965). Snapping of low-pinned arches resting on an elastic foundation has been investigated by Simitses (1973). In another experimental study, Phan-Van-Diep, E.P.Muntz and D.C. Wadsworth (1995) measured the deflection of a micro-beam with an interferometric technique. The deflection force was created by desorption of adsorbed ambient molecules when the beam surface was rapidly heated. E.P. Muntz (1998) investigated the behavior of transient desorption and obtained a value or the generated normal force and impulse due to desorption. Most of the previous work on the nonlinear dynamics of buckling and snap- through concerned circular arch problems. A number of papers on microdevices that rely on a clamped boundary condition for their operation addressed linear mechanical issues. The method used to analyze snapping action of buckled beams in this study is partly similar to the energy method of Gjelsvik and Bodner’s circular arch model. However, Gjelsvik and Bodner’s arch model was only concerned with the central concentrated loading condition, whereas this study develops a more general solution for the snap-through switching action of nonlinear buckled beams with more varied transverse loading conditions. 16 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 2 Analytical investigation 2.1 Lumped parameter model 2.1.1 Snap-through To evaluate the concept of snap-though phenomena using energy methods, a simple snap-through buckling model is introduced (Simitises, 1976). The system consists of two pin-jointed, weightless rigid bars of each length L. One end of one bar is supported by a fixed pin joint, the two bars are pinned together, and the remaining end of the other bar is pinned to a linear horizontal spring. The rigid bars make an angle a with the horizontal when the spring is unstretched. The system is loaded 17 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. transversely through a force P applied quasistatically at the connection of the two rigid bars. As the load is increased quasistatically from zero, the spring will be compressed and the two bars will make an angle 0 with the horizontal (0 < a). The question then arises whether it is possible for the system to snap-through toward the other side at some value of the applied load. In seeking the answer to this question, we will first use the equilibrium approach and then analyze the system b y considering the character of the equilibrium position with an energy approach. Let the horizontal reaction of the spring be F. This force is equal to K times the compression in the spring, or p Figure 3: Model to study snap-buckling. F = 2Kl(cosO- cos a) (5) 18 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Furthermore, from symmetry the vertical reactions at the ends are ^ • Since no moment can be transferred through the middle joint, the equilibrium states are characterized by the following equation — cos0 = FI sin 6 (6) 2 Use of Eq. 5 yields P = sin 0 -co satan 0 (7) 4 Kl Note that “ < 0 < ^ . — f . t t P /2 P/2 Figure 4: Load paths created by equilibrium state. The equilibrium states, defined by Eq. 7, are plotted in Figure 4. Note that loading starts at point A ( P = 0, 0 - a ) and it is increased quasistatically. When point B is reached, we see that, with no appreciable change in the load, the system will tend to snap-through toward the CD portion of the curve. The load corresponding to 19 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. position B is a minimum required snap through force, and its magnitude may be obtained from the fact that — = 0 (8) dO Note that the right side of Eq. (7) is a continuous function of 6 with continuous first derivatives. If we denote by 0 B the angles corresponding to position B and B' , then 0B = ±cos~'[(cos«)^] (9) and the critical value is given by = |sin0B-c o s a ta n # J (10) Q v v j (A IC U 1 1/ g snap 4KI The total potential energy, UT, for the system, which is equal to the potential energy of the external force plus the energy stored in the spring, is given by 1 UT = — £ (2 /cos 0 - 2 / cos a ) - P(lsina-lsm O ) = 2A72(cos0 - cosa)2 - jP/(sina - sin#) (11) Static equilibrium positions are characterized by the vanishing of the first variation of the total potential, or dU7 — = 4A72 (cosa - cos0) sin # + PI cos# = 0 (12) dO which leads to the same equilibrium equation obtained earlier; i. e., 20 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. P . _ cosasinfl = sin 0 - - 4 KL C O S 0 (13) n / 2 - i t 12 Figure 5: Load-deflection curve for snap-buckling model. 2.1.2 Lumped parameter model The bar spring system shown in Figure 5 has many important features in common with a shallow buckled beam. Figure 6: A discrete model of elastic buckled beam. 21 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The model is considered as hinged in the middle and at the ends, and subjected to a single load p at the top. The investigation is restricted to a very shallow beam, that is lasina la cos a = ta n a « 1 (14) where la sin a is the initial unstrained height and 21 a cos a is the width of the beam. The total potential energy for the system, UT, is due to the external force and energy stored in the springs, from bending and axial deformation. From the geometry, the total strain energy equation is obtained by using Hook’s law to get UT = - n / . s m a - / „ S in (a -0 )]+ !A :r(20)! +AT[(/„ - / , ) !] (15) in which Kr is the rotational spring stiffness, (due to bending effects in the continuous real beam), and K is the longitudinal spring stiffness, (due to axial COS c c deformation of real beam). Since 21 g cos# = 21 a cos a , lg = la and Eq. 15 cos# becomes UT = -P la [sin a - cos a • tan(a - #)] + 2K .92 + Kl 2 [1 C °-Sa - ]2 (16) cos(a - 9) To obtain an unstable equilibrium position, set the equation ^ = 0 (17) d9 22 permission of the copyright owner. Further reproduction prohibited without permission. From above condition, equation becomes Placo sa -4 K r0cos2{ a -0 )-2 K la2cosa[sin(q-0)-cosatan{a-6)] = 0 (18) snap Figure 7: Load-deflection curve for model. Solving for the load required for snap-through leads to P s n a p = AKrQ cos2 (a - 9) / lacosa + 2Kla [sin(a - 6 ) -cos a tan(a - #)] = 0 (19) and the load-deflection curve is shown in Figure 6. As discussed in the previous snap-though model, loading is increased quasistatically until 0=0X . From the unstable equilibrium energy method, the equilibrium position between 6=9X and 0= 91 are unstable, and therefore the load at 9=9X is critical because the increasing the load at this equilibrium position will make the beam snap though. 23 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.2 Continuous model 2.2.1 The geometry and boundary condition of the clamped-clamped beam The lumped parameter system discussed in the previous section is a model of a continuous beam or bridge with clamped boundary conditions. The geometry of a clamped-clamped buckled beam is shown in Fig. 7. (0 ( X ) Figure 8: Geometry of a clamped buckled beam. For the case where both ends are clamped, the boundary conditions become dW I W = 0, — = 0 a tx = ±— (20) dx 2 The governing equation of motion for a buckled beam is a b o rd er nonlinear differential Equation (W. Y. Tseng, J. Dugundji, 1971). An undamped, unforced beam that undergoes stretching is described by 24 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where: A = cross sectional area; I = area moment of inertia of the beam cross section; E = Young's modulus; P, = applied axial load; / = beam length; x = distance along the undeflected beam; t = time; W = transverse deflection of the beam; m = beam’s mass per unit length; b = breadth of rectangular cross section; and h = thickness of rectangular cross section. 2.2.2 Amplitude of the buckled beam To obtain the amplitude of the buckled beam deformation, we assume a static buckled shape corresponding to the nth buckling mode and further assume that the contribution of all buckling modes other than nth mode are negligible. The 25 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. following linear differential equation describes a statically buckled beam (A.H. Neyfeh, 1994). Assume that P,^ is the nth critical axial buckling load and the post buckling displacement is bP (n ) < f > n, where bP (n ) is a dimensional scaling constant and is the nth buckling mode shape. Equating Eq. 21, the nonlinear equation of buckled beam, to Eq. 22 leads to Substituting the displacement into the above equation and dropping the time derivative, we obtain A A (24) for a statically buckled beam. To satisfy the above equation, either v / (25) 26 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. or d 2 6 h LJL = o nn) dx2 (26) The trivial solution for the dimensional constant bP (n ) = 0 corresponding to an unbuckled state. The nontrivial, buckled solution is found from Eq. 28; i.e., r a - V i dx v dx (27) or bp(n) - i 1 < N • A FA f ( A ^ d < f> n 2 dx i dx V J (28) Therefore initial amplitude of buckled beam can be obtained from bP m =± 2 /(/> P K en t) j 1 / ( A > 2 /2 FA f d ^ dx J -A dx V (29) 27 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. By assuming P, ) P ,{ c r it) where 'P,{ c r ^ , the critical axial buckling force for clamped-clamped boundary conditions is given by p °) Locking for the minimum strain energy for fixed initial amplitude, which is as much as shallow, the following results are obtained. As long as the initial amplitude is larger than a critical value of bp(C rit) ~ - 2 lP l(c r it) 1 / ( A ^ 2 / 2 E A f d<j>x d x - I d x (31) v / and this should be smaller than the amplitude of initial buckled beam. 2.2.3 Sample mode shapes The first and second buckling modes of the beam (W.Y. Tseng, J. Dugundji, 1971) are described by d > . = — (l + cos2/r— ) 2 K I (32) for the first and 28 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 9: First buckling mode shape < < A A < t > 2 = P k - - s \ n k - l I + cos k — — \ I . v y _ for the second. Figure 10: Second buckling mode shape 2 In both, k =8.986 and (3 = — k Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.2.4 Nondimensionalization The following notation is introduced for convenience. We nondimensionalize the spatial and temporal variable by letting 2.3 Strain energy If the stress-strain relation is the same during the loading and unloading processes, the behavior is called elastic and the specimen is perfectly elastic (Chen, 1987). A deformable body is said to be perfectly elastic if the state of stress and the corresponding state of strain are the same for the same level of the external forces. If a perfectly elastic body is under the action of external loads, the body deforms, and work is done by these external loads, and this work is stored in the system as strain energy. Because of the assumption that the material is perfectly elastic, the work done by the loads can regained if the loads are quasistatically decreased to zero. (34) Where, r is the radius of gyration of cross section, r = A 30 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.3.1 Bending effects As shown in Figure 10, the differential length of any portion of a curve is related to the related to the radius of curvature, p , by ds = pdO The curvature is simply the reciprocal of the radius of curvature: (35) ds p (36) x ax Figure 11: Any portion of a bent beam. dO In Cartesian components the curvature — of the deflected beam is positive or ds negative depending on whether the deflection curve y shows its concavity toward the positive or negative y axis; i. e., 31 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. dB L / * " t + t / j f (37) where the prime indicates the order of the derivative with respect to x. For a buckled beam, the variable y in Eq. 37 is replaced by the beam deflection W so the curvature is given by: K ’ = ±- d2W dx2 (38) The strain energy due to bending is given by (39) f l 2 -‘ A d2 W dx2 / fdW ) 2 \ 1 + V { dx J J % dx (40) dW which, when « 1 for very slightly buckled states, becomes dx 32 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.3.2 Longitudinal deformation effects The longitudinal strain of the centerline is geometric imperfection, and u represents the longitudinal displacement. Again, primes indicate derivatives with respect to the Cartesian coordinate x. By substituting these expressions we can rewrite es as Expanding the above equation to second order using a Taylor series expansion and dropping terms higher than the second order leads to (43) es = u’ + fV'fV'-h^fV'2+ - (44) 33 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where u'1 is assumed to be small compared to W '2. If we assume no initial geometric imperfection (W0 =0), then s, = — + -------- dx l y dx du 1 (dW 1 ---- (45) and the strain energy due to the longitudinal deformations is given by (46) -¥i 2 I l { f (47) The longitudinal displacement function u can be eliminated from the energy expression. The variation of the total energy with respect x must be zero for ds equilibrium; i.e. — - = 0. Therefore,^ is a constant with respect to x or es = C dx in which C is constant. Substituting es =C into Eq. 45 leads to (48) (49) 34 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. or, after rearranging d u = c _ }_(dw ' 2 dx 2 (50) Integrating over the span gives ^ - d x / ] * dx * -y2 -y2 , 1 fdW ^ dx j dx (51) The boundary conditions on u are w = 0 at x = ± ~ . The left side of Eq. 51 is therefore zero. Solving the remainder for the constant C yields o . W | E ] -a ' dx (52) 2 -A ' m ' \ dx dx (53) 2 iJyVdx) (54) Since s s = C , we have - 4 dW_ dx dx (55) 35 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Therefore, Um becomes EA f 2 , um = — 2 -K Yx f l 2 - ' A 21 1 1 ( f ) ' d x d x (56) (57) EA^l 0/2 J 8/ 1 ® * (58) 2.3.3 Total strain energy By summing the strain energy due to the bending and the axial deformation, we obtain the total strain energy as u slain= ub+um (59) U strain El Yi -Yi 'd2W~ * ♦ " 1 l ( dx2 _ e/2 J 8/ -Yl _~^2 , dx ) d x (60) 36 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.4 Potential energy for various loading conditions The potential energy for several loading conditions is given by % UP = ^PW{x)dx for quasistatic distributed loading -Yl Up = PW(x) a, where - ^ < a < -^, for concentrated loading U = I for impulse loading 2.5 Solution under quasi-static loading Let the deflection W(x) be represented by the sum of the first two mode shapes w (x) = = a\h + a (6l> n = l where < j > x and < f > 2 are the symmetric and antisymmetric components and a, and a2 are non-dimensional deflection functions, where, a, = bm ~bx/bm , a2 = b2(0)-b 2/b2 (0 ) (ft,, ft2 are the symmetric and antisymmetric amplitudes and ft1 (0 ), ft2 (0 ) are their initial amplitudes respectively). 37 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The total strain energy can be written U total = U b + U m + U p where the strain energy due to the bending is Ub = El % fd 2W^ dx2 ! // 2 -a dx' dx d2h d2fa dx2 2 dx2 dx = a 2 E I ,/] ( d 2< p x'\ ' 2 W & ’ J dx + < * 2 2^~ J 2 _ dx dx El r + 2 a,a2— I 2 -K y2 ( st2j V a 2 j > \ dy, \d x A -'* j V±2 dx2 dx and the strain energy due to the longitudinal deformation is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. h ) 'A f d*i~ + a 2 tic 8,2 -k J I dx _ d x (67) d x + 2 a , a d x + a d x d x + a 2 2 EA +4a> w A J -y2 A \ ( d < j > x ^ A d x dfa I dx d x dx 2_ i EA 8/ + 2a, a2 2 1 -y > ^2 / J J . A ( J/L /X fd h \ & t(d+z -‘ A -A' dx , dx d x 3 EA a, — 8/ + 2 a i a 2 o;2 1 -x d x A l ^ 3 EA ? + 2 C ‘1 °2 -^2 J A f , , K d x y \ - J < M Y d < t > 2 \ c ic J V < £ c J < & tic (69) 39 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. For convenience, we will define the following constants: 4 - f J - ' A A ( rilx \ - d fa dx2 dx (70) r i 'A f I 2 ( d2 fa N (71) *-f1 ( rfl’ik \ d fa f 5 V 2) l & 2 JU 2 J etc (72) R 4 812 •» 'YlfAJL* dx r ' A dx (73) Yi -Yl ~ 8 l2 J K 'Y dfa_ dx dx dx r ' A f # 2_ ' dx \dx dx (74) (75) B - M 1 ■a tl dfa dx B - I ‘ ~ W etc dx dx dx ^ d c ) _ p yjx dx A dx dx (76) (77) 40 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. With these definitions and assuming quasistatic distributed loading, Eq. 62 becomes Ulo la l = a 2 Bx + a2 B2 +2ala2B3 + a,4fl4 +a2 Bs +4 a 2a2 B6 + 2 a 2a 2B1 + 2a 3a2Bi + 2 ax a2 B9 A ~ p W \ + a 2(/> 2 ) i x -A 2.5.1 Symmetric case Under symmetric buckling, a2 - 0 and Eq. 79 becomes Y i U l0,ai = a 2Bi +a^B4- P Ja ^ d x -A dU Since, for equilibrium, — — = 0, we have 5a, m & v U total = 2a,5, + 4a,35 4 - P f^,dfc = 0 da, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Thus, the symmetric snap-buckling load is given by p = 2a , B + 4 a , \ (g2) A j^i dx -A 2.5.2 Antisymmetric case Assuming a very shallow buckled beam, L Jm is negligible and Eq. 79 becomes. Ulolal=Ub+Up (83) A = a zB} + a22B2 + la ^ B ^ - P J(a,^, + a2< p 2)dx (84) - Y A but J^ 2 « 0 , because of the antisymmetry. -Y i Therefore Y U 'otai = a\ B\ + ai Bi + 2ai°2B3 - P ja ^ d x (85) -Y i dU Since, for equilibrium, — — = 0, we have da2 = 2 a2B2 + 2 axB, = 0 (8 6 ) da2 42 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. or a2 = ! k B , (87) Substituting Eq. 87 into Eq. 79 gives ,2d 2 532 4 1, . _4 B 4 3 By B, Ulo la l = - a ^ + a^B, + a ; ^ B 5 +4 a ^ B 6-2a,3- ^ B , - 2 a,4 -^-5, 5, B. 5, B, 5, z> 3 ^2 f D ~ 2a i4 - T T B 9 ~ J P\ a A ~ a \ B 2 - l / 2 V 2 (88) dU, For equilibrium, apply — — = 0 to get 9a, at/, 9a, 2 4 2 2 to M / = 2a,5, -2 a ,-^ - + 4a,354 + 4 a ^ B 5 + 12a,2 -^ -5 6 -6 a ,2^ - 5 7 5, 8a,3 | ^ 5 8 -8 a ,3^ 5 9- 5. £ 5. 5. -Y i B. dx (89) 2 o r , after grouping powers of a ,, 2 5 ,-2 3 5 a, + 2 y a,2 + D 4 D D ^ ) 454 + 4 — ^-5S - 8 — 58 - 8 — ^-5, a' B2 b 2 B2 9 J' Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The antisymetric snap buckling load is therefore given by (91) The load-deflection curve of Eq (91) is shown in Figure 11. As for the lumped parameter anaysis, equilibrium positions between A and B are unstable, and therefore the load at P is critical because the slightest possible disturbance at this equilibrium position will make the beam snap toward a stable equilibrium position. That means, if the loading is increased quasistatically and reaches or exceeds that given Eq. 91, the beam will tend to snap-through toward the opposite stable state. Force P 2 a Deflection Figure 12: Load- deflection curve for quasi-static loading. 44 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.6 Dynamic solution under impulsive loading In the previous section, we’ve shown that snap through can occur in a buckled beam and the conditions under which it does occur under quasistatic loading conditions. However, we’re interested in snap through under dynamic loading conditions. From a design perspective, we are particularly interested in how the dynamic snap through force compares to our experimental conditions. The straight forward solution to the problem would be to solve the governing non-linear differential equations for the response of the buckled beam. These equations offer considerable difficulties, especially if an attempt is made to consider more than one deformation mode as well as possible asymmetrical modes. Approximate solutions of the dynamic equations using only a single symmetrical deformation mode have been obtained by Volmir (A.S.Volmir, 1958) andGrigoliuk (E.J.Grigoliuk, 1960). If the point of interest is solely the determination of the impulse conditions or the energy input necessary for snap through to occur, then the problem can be approached from a general energy viewpoint, in which only static equilibrium states need be examined. Snap-through would occur when the energy input is sufficient for the buckled beam to jump into a mode characterized by deformations; that is, the beam passed through an unstable equilibrium position. To begin, we can make a few simplifying assumptions. The deformation is assumed representable by a simple deflection function such that the beam can be treated as a one-degree-of 45 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. freedom system, as with the previous quasi-static solution. For impulsive loading conditions, the problem becomes one of finding the lowest strain energy associated with an unstable equilibrium state for an initially stable structure. This energy would represent the energy barrier or, assuming no damping, the minimum kinetic energy input required for snap buckling to occur. To begin, we can make a few simplifying assumptions. The deformation is assumed representable by a simple deflection function such that the beam can be treated as a one-degree-of freedom system, as with the previous quasi-static solution. The impulse is assumed to be distributed uniformly over and normal to the lateral surface. Every particle of the beam is assumed to be instantaneously accelerated to a finite velocity before any displacement occurs, and for all times t > 0 there is no further external loading. This would be a reasonable approximation to our experimental loading condition (gas desorption force) if the loading time is short compared to the time required for the beam to snap through. Because during the motion of the beam there is no external load and therefore no external work done, the statement of the equivalence of maximum kinetic and potential energy is simply KE = Ub+Um (92) where U b Js/dtc as before. Therefore, 46 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. in which KE is the initial kinetic energy and the u terms refer to a state in which the beam velocities are zero and the displacements are maximum in time. The longitudinal displacement u appear in Um though not in Ub, but by using the boundary conditions, it is possible to eliminate u and express L Jm in terms of the single function W in the following general way. If T is the kinetic energy and V the potential energy for any arbitrary state having two degrees of freedom, W and u, the general Lagrange equation of motion in the variable u is d f d(T - V) dt _ du Because the beam is very shallow, we can assume that it is very small compared to W , T becomes a function of W only and, since Um is the only term in V that depends on u , the Lagrange equation immediately reduces to ^ - = 0 (95) du Thus, the integral expression for Um must be a minimum with respect to u variations. This is simply equivalent to writing Newton’s law for motion in the 47 d(T-V) du = 0 (94) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. tangential direction, neglecting the tangential acceleration term. This also implies that the tangential motion is uncoupled to that in the transverse direction. Standard calculus of variations theory then leads to the conclusion that the Euler equation dg, d du dx j du dx = 0 (96) ds must be satisfied. This reduces to —L = 0. Because tangential acceleration has dx been neglected, this is the same expression obtained for static buckling by minimizing the energy integral term based on finding static equilibrium states. Using the above work, the kinetic energy becomes e i 'i KE= — ~d2W~ 2 > / *♦"1 [ 1 rdW) 2 dx 1 J 2 dx2 8 ' i i ( dx dx (97) Substituting the displacement function Eq. 61 and using the constants introduced in Section 2.5 leads to 2 4 2 2 3 °2 *2 ^2 *2 (98) 48 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. This quartic equation continuously increases with a ,. For certain portion of a,, this quartic expression has a peak near the origin. This peak indicates a state of unstable static equilibrium, and it seems reasonable to suggest that a buckling criterion can be established on this basis. This curve should not be confused with the more familiar and similar curve for static load versus deflection, which represents a succession of adjacent equilibrium states. Here there are only three equilibrium states defined by the points at which the energy curve has a horizontal tangent. KE critical KE Deflection Figure 13: Deflection - kinetic energy curve That is, when the kinetic energy reaches or exceeds the value corresponding to this peak, the possibility exists that snap-through will occur. It remains to locate this peak and compute the associated critical value of initial impulse above which buckling is likely. The extremes of the curve are found by setting the derivative of KE with respect to ax equal to zero. 49 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. dKE dax f t > 4 d d ^ ^ 4 5 .+ 4 ^ r5s-8 ^ -5 8- 8 ^ 5 < B. B-, B- 3 9 a, + 2 \ B. a. 2 ' R 2^ 2 5 ,-2 3 B. 2 y (99) For convenience, we will define the following constants: 5 4 „5 , „ „ 5 , 3 „ _ 5 , 2 „ , 5 ,2 a, = 454 + 4 — ^-5s - 8 — 5g- 8 — ^ 5 9 , a, = 12— ~B 6 - 6 — ^ 5 7, and 1 4 5 2 5 2 5 2 52 6 5 2 a 3 = 25, - 2——. With these definitions, Eq. 99 becomes 5, dKE 3 2 = a, a, + a 2 a, + a 3 a, dax (100) The expected and obvious solution a, = 0 corresponds to the initial equilibrium state. The other solutions are found by factoring Eq. 100 and solving the quadratic in a, to get a, = _ - a 2±-yja2 - 4 a,a 3 2 a, (101) The lowest root of the quadratic will correspond to value a, at the peak and that is a, - a2 - ^ a 2 -4 a ,a 3 2 a, (102) 50 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. dKE Deflection Figure 14: Deflection - curve da. Substitution of ap into kinetic energy equation gives the energy KE. To compare A with experimental data, let KE change to dimensional KE that is defined as in Section 2.2.4. By summing the kinetic energy acquired by each element, the total A kinetic energy KE can be expressed by a _ i KE = ^ - x m a s s of element x (initial element velocity) 2 r l , ,, , / lmPbdx^ pbhdx = | ~{pbhdx) bl 2 ph Im, (103) (104) (105) where p , h, b are the density of beam, the thickness of beam and the width of beam respectively. Imp is the impulse applied to the beam. The assumption that the 51 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. initial impulse Im? acts uniformly over the surface, and that the energy is completely transformed into uniform material velocity implies that Im„2 = 2— KE (106) P bl Therefore, the minimum impulse required for snap-though may be obtained from (HL - -a/ f - +aP A ^ B s +4V^B 6 + V ^ B 7-2a;^-Bs- 7 a ;^ TB 9) \ br B 2 B2 B2 B 2 (107) Where, 5,, B2, fi3, B4,B5, B6, B7, Bs and B9are defined as in Section2.5. 2.7 Generation o f desorption forces Focusing laser energy into a small volume, causing breakdown and relying on inverse bremsstrahlung for absorption is frequently used to deposit energy in a gas (Muntz, Shiflett, 1992). Typically, the conversion of lpJ into molecular motion will raise the temperature of 10"6 cm3 (1 0 0pm cube) of gas at atmospheric pressure through one to two thousand Kelvin. However, for the system under study, the switching action is caused by rapidly heating the surface of a beam with a laser. The beam deflections are primarily a result of adsorbed ambient molecules that rapidly desorb when the beam surface is heated. The desorbed molecules are 52 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. replaced by molecules in the ambient gas on a short time scale due to the high ambient pressure. For the case of a single laser shot, decreasing the pool of adsorbed molecules is not an issue. Molecules which are re-adsorbed after the laser pulse as the surface cools can contribute to a reverse thrust to damp the beam’s movement and disturb the switching action. Ablated coating or membrane particles might be a source of extra thrust. However it is doubtful that ablated particles are indeed responsible for the beam displacement based on the previous Pham-Van-Diep’s experiments (G. Pham-Van- Diep, 1995). Molecules of ambient gas desorbed during sudden heating remains the most plausible explanation for the observed beam displacements. To investigate switching action, laser pulses were applied to a buckled beam in various ambient pressures and with various gases. The laser pulses with nanosecond pulse duration produce sudden heating at the beam surface. Sudden heating is required to observe the ambient pressure increase which only exists for time scale on the order of the mean molecular collision time in a gas. Microdevices have dimensions on the order of micron. The smallest characteristic dimensions can be on the order of the local mean free path and their dynamic time response can easily be less than a microsecond at atmospheric pressure. For instance, an Si02 disc with 1 pm thickness and 10 pm diameter subjected to a sudden pressure difference of atmospheric pressure across its two surfaces will move it’s thickness in 2xl0 -7 sec. (G.Pham-Van-Diep, E.P. Muntz and D.C. Wadsworth, 1995) 53 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.7.1 Magnitude of switching forces To investigate the behavior of transient desorption, a kinetic description of the adsorption and desorption is required. Based on Langmuir kinetics, the rate of desorption and adsorption per unit area can be written as (Muntz, 1998) s (t) s (t)e~Q I{k/m )T A ,) ^ ) = M = W (108) * = ng{U T g/ m y _ ^ (iQ 9) 4 where, sr = total number of adsorption sites available per unit area -n ;1; Qs = cross sectional area occupied by an adsorbed molecule (assumed size of molecule); r0 = period of an oscillating molecule in the surface potential = 1 0 '1 2 to 1 0"13s; k = Boltzmann’s constant =1.3806 xlO-23 J/g ; 54 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. m = the mass per molecule; Q = heat of adsorption or desorption; ng = number density - N A p J ) \ R A R = universal gas constant; and Tg = gas temperature. Using the Eqs. 108 and 109 and assuming (sT - s,) = 1 for the unoccupied sites and (sT - 5,) = 0 for the occupied sites, the rate of change of occupied sites is given by dS'W =asT-C(l)s,(t) (HO) dt where, 2nm c m = r0 - y s'< * '" w « + » /% O T I) V ; If a step function increase in surface temperature, Ts is considered from the initial temperature (i.e., T = T to T = T P ), the solution to Eq. 110 is 5 ,0 £ S J , r = (H D '-'/T 55 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where, CF=C(t) at Ts = Ts F ; C0 = C(0 at Ts =Tso=Tg',and s = initial value of the number of occupied adsorption sites Therefore, evaporation rate can be rewritten as ~ ~ r = ~ asr )e_c>' (112^ at and the maximum evaporation rate occurs at t=0 , the time of the step change in surface temperature from Tso=Tg to TsF- The generated switching force per unit area by the gas desorption is (Muntz, 1998) P = (— —^-) m c j (H3) dt where: 7 Ce v = mean speed of desorbing molecules normal to the surface = ( * % / • 56 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Thus, the generated switching force per unit area in Pa is k x -t V 2 (114) 2.7.2 Estimated surface temperature increase An estimate of the surface temperature increase is (G. Pham-Van-Diep, 1995) where, 77 = energy coupling efficiency of surface; El = energy per laser pulse; Ah = heated area; Clh = thermal conductivity, K = thermal diffusivity p = density; c = specific heat at constant pressure; t = time; initial (115) 57 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. For efficiency, short exposure times will be required. Notice also that the maximum surface temperature excursion, - Tin ilia l is proportional to 4 t , so that for a given laser energy, EL, higher impulses can be achieved in the microvolume with shorter exposure times. 2.7.3 The magnitude of impulse The impulse per unit area generated by gas desorption up to a time t is (Muntz, 1998) Im = j^Pdt = mCev(si0-a sT/CF)(\-e~C F ') (116) where, s, 0 = a ^ - , sT = Q ;1, a = ng( k ^ - ) ^ 2Qs,m d Ce v = i ^ / m ^ 2 “ before. Thus, the estimated impulse per unit area can be written in the form >m — <n7) yC0 CF j To be useful for examining actual situations, the expression for Im must be modified to include a geometric surface area multiplier, ^ . Since the Langmuir kinetics do not permit more than one layer of adsorbate, the only way to do this 58 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. conveniently is to multiply sT by ^ . An approximate value for can be obtained using Eqs 106 and 116 as 2 i * K E E c = — = ? — L J ---------------------------- (118) m C ev ( * 1 .0 - a S T / C F )(1 “ e ~Ch<) The quantities CF, C0 and a are all unchanged and the effective initial number of occupied sites i s T h u s , Im = m C„' - a ^ -X l - e‘C " ) 019) = « C „ ' A . X I - e ' c'‘) (120) ° 2/zm C0 CF = m(— / 2n J k ^ - ) /2Qs Y — ( - -)(1 - e’c'') (121) 2m g Inm S^ GQS C0 C/ 2 .7 .4 Estimates o f the switching time Muntz (Muntz, 1998) defined the activation time, tacl, as the time needed for a plate to move a distance equal to its own thickness under the influence of desorption forces. According to Muntz, ta c l is given by 59 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where / is a multiplicative scale factor for the scaled system, h is the thickness, p is the density of the specimen, and Im is the impulse per unit area applied to the surface. Since S0, the initial deflection at the mid-point, is approximately equal to the thickness of a specimen for the very shallow buckled beam used in the experiments, the activation time for the buckled beam may be estimated by replacing h2 in Eq. 122 with hS0 and the total time required to switch states, t^ ,, becomes twice the activation time or h . P - ' 5'' (IM) Im 60 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 3 Experimental investigation 3.1 Experimental goals and procedure Experiments using pulsed laser irradiation to deflect a thin Mylar® film due to a thermally induced transient pressure pulse on the rapidly heated surface were conducted. Theory indicates that the normal forces will be produced by the combination of heated ambient gas at the beam surface and adsorbed molecules ejected from the beam surface as the surface is heated by the laser pulse. The desorption of mass from a surface due to rapid surface heating at relatively high pressure becomes a rarefied gas dynamics problem because the very short transient time scales are usually comparable to the mean collision time in the gas. The amount of gas desorbed and the resulting switching force exerted on the 61 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. heated surface are significant for micromechanical dimensions and practical in terms of energy for very short surface heating times. The Experimental part of the study was concerned with obtaining the minimum necessary laser energy, switching force, and switching time for the geometry of a clamped-clamped buckled beam. The geometry of the buckled beam is determined by the span, the beam thickness, and the central height. Various experiments were carried out to investigate the functional dependence between the switching force exerted by the gas desorption and their beam response: (1) A laser pulse applied to the beam surface with no ambient gas in the vacuum chamber. This experiment investigated whether force due to the photonic impact can cause a change in beam state. For this experiment, the vacuum chamber was pumped down to the low 10-5 Torr range before applying a laser pulse. One of the most important problems in low ambient pressures (high vacuum condition) is the removal of gases which are present on the surface of beam. To remove all the remaining adsorbed molecules, several lower laser pulses were applied on the both sides of the beam before measurements were made. To maintain the lower-pressure range in the vacuum chamber, mechanical and turbo molecular pumps were both kept working after the gas-evacuation valve was closed. 62 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2) A laser pulse was applied to the beam with various ambient gas pressures in the vacuum chamber. This experiment examined the effects of ambient gas pressure on switching action (activation time, switching time). The vacuum chamber was evacuated as low as possible and a working gas gradually inserted to find the critical ambient pressure needed for switching action to occur and to explore differences in switching action as the ambient gas pressure increased. (3) Several different ambient gases (carbon-dioxide and nitrogen) were used to examine the effect of the gas itself on the switching action. (4) The optical absorbtivity of the beam surface at the laser wavelength was varied to explore whether energy coupling efficiency to the surface greatly influences a beam switching action. For instance, the absorptivity of the beam material, (Mylar® film) used in this experiment is reported to be 30% (Dupont Lab). If a very thin black coating is applied to the beam surface, the energy coupling efficiency to the surface goes up to over 90%. Thermal resistive paint was used for the coating. (4) Finally, to determine a more realistic multiplier, , a range of laser energy pulses were employed using the various neutral density filters to find the minimum laser pulse energy needed for switching to occur. 63 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.2 Experimental setup Schematics of the experimental apparatus are shown in Figure 14 -1,2 and a picture of the experimental setup is given in Figure 15. N&ouum chanrber Ftessue Gauge N1YA3 Laser Laser head Figure 15-1: Schematic diagram of the experiment 64 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Laserhead specimen Neutral desity filter Figure 15 -2: Schematic diagram of the experiment 3.2.1 Lasers The laser pulse can rapidly increase surface temperature of the beam target. The laser energy rapidly increases the temperature of the beam roughly between 200K and 1200K and the temperature increase lasts for approximately the duration of the laser pulse. The laser used for this experiment was the Minilite I-a pulsed solid state Q-switched Nd: YAG laser made by Continuum Electro Optics, Inc with a base 1064nm wavelength. The laser can produce various outputs with second, third, and fourth harmonic crystals. Using these nonlinear crystals, the base 1064nm output can be frequency doubled with about 50% power conversion efficiency yielding an output at 532nm. In order to produce light in the UV, the 65 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 532nm output is mixed with the residual 1064nm light or frequency doubled again. The resulting outputs at 355nm or 266nm are produced with the respective efficiencies of ~20% and -15% relative to the fundamental. The laser system consists of the power supply and the optical head. This system can achieve a pulse repetition rate between 1 to 15 Hz and produces 3mm beam size with less than 3mrad divergence. Minilite I specifications are presented in Table B2. Figure 16: Picture of Nd: YAG Laser The laser wavelength comes into play primarily in the effectiveness of the absorption of the laser energy into the beam material. The primary effect of the laser wavelength on generating a switching force is most likely due to the difference in the adsorption rate when different laser wavelengths are used. Our investigation on a Mylar® film indicate that the adsorption rate decreases with 66 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. increasing wavelength; i.e. laser beams of 532nm and 355nm wavelengths are absorbed less by the Mylar® than 266nm wavelength. 3.2.2 Laser energy measuring device Laser energy was measured by a 2-watt broadband power/energy meter, the Melles griot/13PEM 001/J. The instrument can measure the pulsed energy with its thermopile detector from 10pm - 2 J. The noise level of measurement is 1 pJ. 3.2.3 Neutral density filter Neutral density filters were used for attenuating the laser beam. Neutral density filters are designed to reduce to transmission evenly across a potion of the spectrum. There are two types: absorptivity and reflectivity. The absorptive types absorb light that is not transmitted, while reflective types reflect the non-transmittal light. Four different (90%, 80%, 70% and 60%) absorptive types filter were used in this experiment. 3.2.4 Vacuum chamber The vacuum chamber is one of the crucial components in the experimental set up. A picture of the vacuum chamber is shown in Figure 16. The vacuum chamber required a pumping port, a gas inlet, pressure gauges, and view-ports in addition to 67 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the standard ports. The vacuum chamber must have ports for the laser beam with an unobstructed path to the target. The beam was sealed in the vacuum chamber so that the working gas could be deliberately controlled. Various ambient gases may be admitted to the chamber after it has been evacuated. Since some gases are more suited to this requirement than others, suitable gas selection is important. Values of dwell time of a molecule on a surface and approximate fractional surface coverage for various gases are presented in Tables B3 and B4. We selected C02 and N2 as a working gas to examine differences in behavior causing from gas properties. Properties of various are also list in B5 and B7. Figure 17: Picture of vacuum chamber 68 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.2.5 Vacuum pumping system No single pump can evacuate gas from atmospheric pressure down to the high vacuum range so a mechanical pump (Varian/SD-450) and turbo pump (Leybold) were used in series to remove gas molecules from the gas-filled vacuum chamber and to maintain the required degree of gas rarefaction in the volume. Figure 17 shows a schematic of the vacuum system that was used in this experiment. Baratron C T urbc Pump Punp Figure 18: Schematic drawing of vacuum system. 69 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.2.6 Pressure gauge Two types (1 Torr range and 1000 Torr range) of baratron (MKS Instrument) were used to determined the pressure and could be useful over wide range of pressure- measurement from 1 Torr to 1000 Torr. We also used a hot filament ion gauge from Kurt J. Lesker Co to determine low pressures. The hot filament ion gauge is intended for use in vacuum systems operating at pressures from 10~3 Torr to 10"9 Torr. 3.2.7 Clamping system The physical setup in the vacuum chamber involves a beam, two clamps, and two auxiliary rods. Figure 18 show the clamping apparatus. The beam is clamped between blocks, one at each end of the beam. One of the blocks is fixed, whereas the other is allowed to slide by turning an adjustment micrometer screw. This permits variation of the beam length / central height ratio to cover a range of geometries. Auxiliary guide rods extend from the fixed block through holes on the movable block. The adjustment micrometer screw provides a controlled method for sliding one block relative to the other and buckling the beam. After the beam is buckled, the sliding block is fastened to the auxiliary guide rod by two bolts on the bottom of the block to complete the clamped-clamped boundary conditions. 70 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 19: Picture of clamping system 3.2.8 High speed camera A high-speed video camera (SONY DCR Series) was used for recording the switching operation and switching time. The camera was located above the top of the vacuum chamber and operated with a 1 /2 0 0 0 sec shutter speed 3.2.9 Beam material The most important parameters of the switching device are the temperature dependence of the switching action and the corresponding laser energy input. The 71 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. switching action is influenced by the thermal properties and the elastic modulus of the beam materials used and the geometry of structure. Using previous theories, the thickness of the beam should be as small as possible to increase the temperature dependent beam action at our boundary conditions. In our case we decided to use haze, low gloss type Mylar© polyester film for the beam material. This film has excellent temperature resistance, good physical properties over an exceedingly wide range of temperature, and low elastic modulus. Properties of Mylar® polyester film are shown in Table B1. 3.3 Beam design considerations 3.3.1 Material limitation Because the analysis is based on the assumption of elastic behavior and the deformation of the buckled beam is represented as a one-degree of freedom system, we should consider several things in designing the buckled beam. For instance, to minimize twisting of the beam (Ira Cochin, 1997), — >3 (124) h where, h = thickness of the beam; and b = width of the beam. 72 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The snap-through analysis is only valid if the local deformation in the beam material remains within the range of elasticity. If plastic deformation takes place, the results will be affected. It is clear that the built in longitudinal stress, er0, in the beam has to be below the limiting stress for elastic deformation, crljm , of the beam material; i.e., 0 -0 < 0 -iim (125) In addition, the maximum local stress, achieved during the snap-through should not exceed the limiting stress crH m ; i.e., 0- m a x < tflim (126) The maximum local stress takes place during the snapping process when the beam assumes the second mode shape < j )2, at which point the maximum local stress reaches a value of c r _ » 9 .2/(£-<7„)>^ (127) With the above Eq. 121 h2 < 0 .0 1 • l2 f y / )C'™/E) (128) So, for beam to be within the elastic range during snap-through, Eq. 125 and Eq. 128 should be satisfied (B. Haig, 1990). 73 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.3.2 Geometrical limitation A necessary geometric condition for snap-through behavior from a buckling analysis of a clamped-clamped initially buckled beam has been obtained (J.S. Go et el. 1996). The condition is S 4 (129) h V3 where, S0 is the initial beam deflection at the mid-point and h is the thickness. Note, the above equation indicates that the necessary geometric condition for snap-through is decided by the thickness and initial deflection of the beam and is not affected by material properties. 74 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 4 Results and discussion 4.1 Predicted values The surface temperature increase for the beam may be estimated from Eq. 115. Using the values for Mylar® provided in Table Bl, a energy coupling efficiency of T J = 0.3, an laser pulse energy of EL = 0.7m^/p U ise > 311 3163 °f A = 0-075Cm2, pulse duration of r = 6 x l0 ' 9 sec, Eq. 115 predicts a surface temperature increase of 812.5 ^ /p U jse • The surface temperature increases as 4 t under constant laser illumination and the maximum temperature increase is reached when t is equal to pulse duration. Plots of AT versus pulse duration (nsec), AT versus energy per pulse (mJ), and AT versus energy coupling efficiency are shown in Figure 19, 20, and 21 respectively. 75 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. energy coupling efficiency=0.3, tau=6(nsec) 2500 2000 £ 1500 500 0.2 0.4 0.6 0.8 1 1.2 energy per pulse(mJ) 1.4 Figure 20: Laser energy per pulse versus temperature increase. energy coupling efficiency=0.30, energy per laser pulse=0.7(mJ) 1800 1600 § 1400 1200 1000 800 600 pulse duration(nsec) > 9 x 10‘ Figure 21: Pulse duration versus temperature increase. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3000 2500 energy per laser pulse-0.7(mJ),tau=6(nsec) * 2000 1500 I S 8. i 1000 500 S ' S ' S L , - ' s S ^ s f ' ' J * s ' . s' 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 energy coupling efficiency Figure 22: Energy coupling efficiency versus temperature increase. By setting derivative of Eq. 98 with respect to a,, we obtained three equilibrium states defined by the points at which the energy curve has a horizontal tangent. 70 60 5 0 40 1 a * •a 20 -10 0.2 0.5 0.6 Deflection 0.7 0.8 0.9 0.3 0.4 Figure 23: Deflection versus d K E d a x 77 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The required kinetic energy ( K E) for the beam deflection is obtained from Eq. 98 using the values for Mylar© provided in Table B1 and beam dimensions l = \0mm, b = 2.5mm, h = \2x\0~3mm, and initial amplitude is bm = 1 mm respectively. > » <D S 0.8 0 ) c 0.6 I i C E 0.4 0.2 0.6 0.7 0.8 0.2 0.4 Deflection 0.5 0.3 Figure 24: Deflection vs. required kinetic energy By substituting these values and the numerical data presented in Table B1 into Eq. 107, we found the minimum required impulse per unit area for snap though to be Im = 1.1 x 10"3 Pa- sec and the peak occurs at 0.337 as shown in Figure 22. 78 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Estimates of the pressure generated by rapid gas desorption may be made from Eq. 114. Results for carbon dioxide and nitrogen are shown in Figures 24 and 25 respectively using the following numerical values: co2 *2 q u i m 7 .6 x 1 0 % 6* ,0 % n,{m2) 2.481 xlO- '9 1.366xl0'1 9 (sec) 1 0 ‘ 1 3 k(J/K) 1.3806xl0-2 3 2.4464 xlO25 M {Kgf mole) 44.04 xlO-3 28.02x1 O ' 3 N a (mole) 6.022 xlO2 3 m = ^ - ( K g ) 7.31xl0-2 6 4.65 xlO-26 Table 1: Numerical values of C02 and N2. 79 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. < 0 a a T I 0.2 0.3 0.4 0.5 0.6 tim e(sec) 0.7 0.8 0.9 ii x 10 Figure 25: Generated switching force per unit area (C02). 2.5 t o CL " f f l a £ 0.5 0.2 0.3 0 4 0.5 0.6 tim e(sec) 0.7 0.8 0.9 11 x 10 Figure 26: Generated switching force per unit area ( N2) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission The impulse per unit area generated by the gas desorption may be predicted from Eq. 117. For an assumed value of ^ G = l with C02 and N 2 as the working gases, the predicted impulse per unit area are ImC O j = 1.6462 x 10-4 Pa • sec and ImW j = 7.9245 xlO-6 Pa -sec. To be more generally applicable for actual conditions, Im must be modified to allow for the geometric surface multiplier . Estimates for ^ G are obtained roughly using Eq.117 and shown in Figures 26 and 27. Impulse per unit area (Carbon dioxide) 3.5 tiplier=ko i mu 2.5 CL n i m u m n e c e s s a r y i m p u l s e > I H H I H t I I 111 1 1 i n i n h i 111 0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 time(sec) -10 x 10 Figure 27: Generated impulse per unit area ( C02). 81 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Impulse per unit area (Nitrogen) 1.4 B C O C D 0) w 3 Q. E Jill llflli m in im i i>i hi in ------------ n n e c e jJliUU.ll s s a r y ii I M 1 1 1 1 1 1 ----------------- n p u ls e (Mill 'II ------------1 III III III • I 111 1 1» a ----------------- II I I I ill 1ftrn trm tit in nr II III in 1 _ --------W- -.9- 3 .......E ir= 100 1 1 6 Pw Pw PQ Q fl rufflMW nultiDlic WjBmBW! o in IL P i UMU&UI igennoR m iltin lieir= in P S B flifiQ flB I j s W l s l : tG&Bm KESaSSB 3 S S I S S S S S ssm m s m m m w m ______I_____ 1 _____ 1 __________i - i ____i ^ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 time(sec) Figure 28: Generated impulse per unit area ( N2). 0.9 1 x 10' -10 The activation time, tacl, required to displace the beam from the initial shape and time to switch states, , as a function of impulse per unit area may be estimated from Eq. 122 and Eq. 123. The activation time with respect to several multipliers is shown in Figure 28 and estimated switching time is shown in the Figure 29 for a Mylar® beam with dimensions l = \Cm, h = 12x10~*Cm, £0 =lmm, p = 1.390 and a multiplicative scale factor, / = 1. Form this work, the estimated switching time is less than 0.0607 sec by applying an analytically 82 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. obtained minimum necessary impulse for snap-through from Eq. 123. The estimated switching time as a function of input laser energy, and energy coupling efficiency with respect to C02 and N2 are examined and results are shown in Figures 30 through 33. As may be seen, we can expect a shorter activation time and faster switching action using C02 as a working gas than for N2. Increasing the ambient pressure, laser energy per pulse, and energy coupling efficiency may be generally expected to induce faster switching action as shown by the theoretical predictions in Figures 28-33. 2.5 O a 42. a T E c o o C O 0.5 (^ m u lt plier=10 Impulse (Pa sec) ,-4 x 10’ Figure 29: Generated impulse force versus activation time 83 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.35 0.3 0.25 ■® 0.2 o > 0.15 0.1 0.05 0.5 1.5 2 2.5 3 Impulse (Pa sec) 3.5 4.5 x 10"4 Figure 30: Generated impulse versus switching time (assumed J ] G = 1 0 ) Carbon-dioxide, where, n=0.3 0.14 0.12 0.1 0.08 ^ 0.06 u 1 (A 0.04 multiplier2 20 0.02 mi itiplie^So®01*®®^ ^ * © o o < >0 © 0 < * e o © - o o © < > o o o < >© & © < > ■ < * © © © < >©oe< j©e©< >ee©< > 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 Laser energy (mJ) Figure 31: Laser energy versus switching time (C02) 84 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Nitrogen, where, n=0.3 o> ■ p 0.8 « 0.6 0.4 rhultiplier=50 0.2 ^ O O O O O O O O O O 1.4 1.6 0.2 0.4 0.6 0.8 1 1.2 Laser energy (mJ) Figure 32: Laser energy versus switching time (N2) Carbon-dioxide, where, EL=0.7 (mJ) 0.07 0.06( 0.05 tiplier=1 0.04 o > i 0.03( 0.02 0.8 0.9 0.4 0.5 0.6 0.7 enegy coupling efficiency 0.2 0.3 0.1 Figure 33: Energy coupling efficiency versus switching time (C 02) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Nitrogen, where, EL=0.7 (mJ) 1.4 1.2*1 0.8 O ) I 0.6 0.4 0.2 0.9 0.4 0.5 energy coupling efficiency 0.6 0.7 0.8 0.2 0.3 Figure 34: Energy coupling efficiency versus switching time (N2) 4.1.1 Shock wave effects 4.1.1.1 Shock wave initiation Typically the conversion of 1 pJ into molecular motion will raise the temperature of KT6 Cm3 (100 pm cube) of gas at atmospheric pressure through 1 to 2 thousand Kelvin. Focusing laser energy into a small volume, causing breakdown and relying on inverse bremsstrahlung for adsorption is frequently used to deposit energy in a gas (Pirri 1977; Caledonia, 1989; Mazumder, 1987; Cross, 1987; Muntz 1992). 86 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. However, the gas near the beam surface is closely coupled to the beam surface temperature so an alternative way to heat the gas near the beam surface is to raise the surface temperature for a short period, either by rapid resistive heating of a thin conducting film resting on an insulating layer or by photons impinging directly onto a thermally insulating surface. During the time of increased surface temperature the gas temperature rises and a transient, increased pressure results. ■=> Laser pulse Ambient gas O molecule « 0 . ^ compression wavef^w u/nvp^ c > o (shock wave) O O o o o o o o sudden / surface temperature increase Figure 35: Schematics of shock wave As the beam surface temperature increases suddenly by applying rapid heating such as laser pulse, a gas motion is induced by the sudden heating and the disturbance propagates into the gas. The pressure rise near the beam due to sudden heating pushes the gas away from the beam surface and sends a compression wave (shock wave) outward into the gas. The density near the beam begins to decrease 87 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. because there is insufficient mass ejection from the beam surface to compensate for the outward gas flow. This leads to a pressure decrease since the thermal energy supply from the beam, which decreases with distance from the temperature rise near the beam surface, is insufficient to compensate for the density decrease, and an expansion wave is sent out after the shock wave. The expansion wave finally overtakes the shock wave and weakens it as time goes on, eventually the gas return to a rest equilibrium state at the initial temperature and pressure. (K.Aoki et al. 1991; Muntz et al. 1992) Pressure — = 2 increase AT Distance from the beam Figure 36: Pressure near the beam surface curve 4.1.1.2 Shock wave influence K. Aoki et el (1991) studied an unsteady gas motion induced from a rarefied gas shock wave caused by sudden heating of a plane wall bounding the gas. The motion 88 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. was analyzed numerically on the basis of Boltzmann-Krook-Welander equation and diffuse reflections. A summary of the numerical results for sudden heating T (— = 2 ) is shown in Figure 36 below. — = 2 - = 0.2 =2 time development t P T Figure 37: The time development of the profiles of pressure (— ) for — = 2 (K. Aoki. et el, 1991) In Figure 36, T0 ,PQ , and t0 are the temperature, pressure, and mean free time at the initial rest equilibrium state. On the basis of their numerical results, the maximum pressure increase owing to shock wave effects near the beam surface is less than 1.3, ( shockwave '0 J 1.23) for — = 2 at - =2.0. 89 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. To compare this to a gas desorption force profile, consider the minimum case of generated gas desorption force by assuming the minimum geometric surface multiplier ( S G )=1, an ambient pressure of 10 Torr, and C02 as a ambient gas. T For these conditions and — = 2, gas desorption impulse is calculated using Eq. 121 as 4.168xl0"5 Pa-sec with the initial ambient temperature assumed as 300K. The generated impulse due to gas desorption can be expressed as Impulse = P d e s o r p tio n (A v e )x ^me • The mean free time at the initial equilibrium state is defined as / = mean free time = mean free Path of gas molecules (X) = 5 8 2 x l0 -9„ec » Q m c d ii lfv v iin iv — xiv/ jvv mean speed of gas molecules (C') , and d , n , k , and m are the 1 7 T , where, X = —j= ------ , C = ------ (V 2-7t-d •ng) \7t-m) diameter of the gas molecule, the gas number density, Boltzmann’s constant, and the mass of molecule respectively. Since the maximum pressure increase due to the shock wave occurred at —=2.0, The time duration, /, is found from t = 2.0 x /0 'o =2.0x— =1.164xlO-8 sec. The average pressure increase due to gas desorption is C . _ D Im 4.168xl0-5Pa-sec _c01 _ therefore Pd e s o r p U o n ( A v e ) = — ■ =- . ■- g = 3581 Pa, so the comparable t 1.164x10 sec p ratio of the average pressure increase to initial ambient pressure is d e s o r P " o n (A v e ' > P o 90 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. =---------- =2.69. Note that this value was obtained by assuming the minimum 1330 Pa possible geometric surface multiplier value, Z G. Although the experiments performed as part of the current work were not detailed enough to determine the actual value of XG > the data do suggest that Z G is quite a bit larger than unity and, therefore, gas desorption is likely the dominant cause of motion in the buckled beam. 4.1.2 Photonic impact influence To investigate whether force due to the photonic impact can caused beam movement, consider the magnitude of photonic impact. If laser energy EL is absorbed during some time period, the magnitude of the momentum per area Pp h o lo n delivered to the surface is given, according to Maxwell’s prediction, by £ P p h o io n = ~ ’ where c is the speed of light and A is the beam spot size. The A c direction of Pp h o lo n is the direction of the incident beam. If the laser energy E, is entirely reflected, the magnitude of the momentum delivered will be twice that 2e given above, or PD h o t o n = — - in the total reflection. If the laser energy E, is partly Ac E L adsorbed, the delivered momentum will be lie between —- a n d . The laser Ac Ac 91 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. used for the experiments described in this document could typically deliver 0.7 mJ. If this energy was deposited onto a spot of 2.83 x 10"5 m2, we find that 8.23 xlO"8 Pa- sec < / ^ 0 ( 0 n < 1.65 xlO"7 Pa -see This range is far below the impulse predicted for gas desorption therby indicating that photonic impact is relatively insignificant and gas desorption is the plausible explanation for motion in the buckled beam. 4.2 Experimental results Figure 37 through 49 show visualization results of the beam deflection at several points in time after the laser pulse was triggered. The laser pulse was applied from upper side of pictures. The results were taken with a high speed digital video camera operating at 60fps with a shutter speed of 1/2000 sec. The camera was located on the view port at the top of the vacuum chamber. The experiment was repeated several times and the collected digital images processed to get the better images. Figure 37 - 43 show results for various pressures of C02 while Figure 44 - 49 show results for various pressures of N 2. Each frame is separated from the next by 0.0167 sec; i.e., (A): t=0 sec, (B): t=0.0167 sec, (C): t=0.033, and (D): t=0.050 sec. 92 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. < P ) Figure 38: C02 @ 2.1x10"5 Torr ambient pressure 93 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (A) Figure 39: C02 @ 3.1 Torr ambient pressure Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (A) Figure 40: C02 @ 10 Torr ambient pressure Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (A) Figure 41: C 02 @ 50 Torr ambient pressure Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (A) Figure 42: C 02 @ lOOTorr ambient pressure Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (A) Figure 43: C02 @ 500 Torr ambient pressure Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (A) Figure 44: C 02 @760Torr ambient pressure Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (A) (D) Figure 45: N2 @ 3 Torr ambient pressure 100 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (A) (D ) Figure 46: N 2 @5 Torr ambient pressure Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (A) (D ) Figure 47: N2 @ 10 Torr ambient pressure Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (A) (D ) Figure 48: N 2 @ 50 Torr ambient pressure Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 49: N 2 @100 Torr ambient pressure Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (A) (D) Figure 50: N2 @ 760 Torr ambient gas 105 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.3 Observations and discussion To examine whether photonic impact alone can cause switching, the vacuum chamber was pumped down to 2.1xl0"s/o rr. One of the most important problems in low ambient pressures (high vacuum conditions) is the removal of gases which are present on the surface of beam. To remove the all the remaining adsorbed molecules, a series of low energy laser pulse were applied to both sides of the beam before measurements were made. Switching did not occur at these very low ambient gas pressures; however, small beam motions were observed. We couldn’t characterize this phenomenon clearly with our instrumentation, but we hypothesize this resulted from a combination of photonic impact and remaining adsorbed molecules. The laser energy was varied by gradually attenuating the laser energy using various absorptive type neutral density filters. From the experiments, the minimum necessary laser energy was 0.7 mJ for the macroscopic beam used in the study. To examine of the effect of energy coupling efficiency to the surface, a thin coat of thermally resistive black paint was applied to the surface of the beam. We could observe the difference in beam behavior as a function of coupling efficiency differences. Beams to which the absorptive coating had been applied switched with lower energy pulses at all pressures and with all gases tested. Close examination of 106 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the coating surface after many pulse had been applied revealed little damage so it is unlikely that ablation of the coating had a significant impact. Several different working gases were used to examine differences in behavior due to gas properties. The vacuum chamber was evacuated and an ambient gas admitted gradually to find the critical pressure at which switching action could be initiated. Switching action occurred from about 3 torr on using C02 as the ambient gas. Switching became possible starting at about 9-10 torr with N 2 slightly higher than with C02. Data collected with respect to C02 and N2 are shown in the Tables 1 and 2. Although we expected differences in activation time and switching time between C02 and N2, we couldn’t resolve such differences above the critical ambient pressure due to instrumentation limitations. ' '— 0.6 mJ 0.7mJ 0.8mJ 0.9 mJ l.OmJ 2.0xlO”5 torr X X X X X 3 torr X 0 o o o 5 ton- X 0 0 o o 10 ton- X 0 0 o o 15 torr X 0 0 0 0 20 torr X 0 0 o o 25 torr X 0 o o o 30 torr X 0 0 o o 35 ton- X 0 0 o o 40 torr X 0 0 o o 50 ton- X 0 o o o 100 ton- X 0 0 0 0 760 Torr X o 0 o o Table 2: Laser energy vs. ambient pressure data for C02 ( o : switching occurred, x : no switching) 107 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - —- — _ _ _ _ _ _ 0.6 mJ 0.7mJ 0.9 mJ l.OmJ 2.0xlO"5 torr X X X X 3 torr X X X X 5 torr X X X X 7 ton X X X X 9 ton- X X 0 o 10 ton- X 0 0 o 25 torr X 0 o o 30 ton- X 0 o o 35 ton X 0 0 o 40 ton X 0 o o 50 ton X 0 0 o 100 ton X o o o 760 Ton X o o o Table 3: Laser energy vs. ambient pressure data for N2 ( o : switching occurred, x : no switching) Faster switching action with increasing ambient gas pressure was expected.alues. Although the image sequences that were obtained showed that the beam moved slightly faster with increasing ambient gas pressure, accurate differences could not be obtained due to limitations in camera performance. Rough estimates of beam switching time based on the observed data are presented in the Table 3. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C02 3 torr 0.033 sec X 5 torr 0.033 sec X 10 ton- 0.033 sec 0.033 sec 50 ton- 0.033 sec 0.017 sec 100 torr 0.017 sec 0.017 sec 200 torr 0.017 sec 0.017 sec 500 ton- 0.017 sec 0.017 sec 760 torr 0.017 sec 0.017 sec Table 4: Switching time vs. ambient pressure data Data shown in the Tables A1 and A2 indicate that switching action occasionally did not occur at high ambient pressure. This could be due to readsorbed molecules on the cooling surface of the beam after the laser pulse since such readsorbtion would have a damping effect on the beam’s movement and disturb the switching action. The three laser frequencies (266 nm, 355 nm, and 532 ran) available from the Minilite I were used to examine the effect of laser frequency on switch performance. Neutral density filters, along with convex and concave lenses were used to attenuate the beam and thereby vary the amount of energy deposited. Equal amounts of energy were applied at the 3 different frequencies. Surprisingly, only pulses at 266nm caused snap-though implying that coupling efficiency is dependent on wavelength and that the calculations used to predict temperature increase need to be modified to account for wavelength. 109 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. We observed that marks began to appear on the heated area of the specimen after 20 - 30 laser pulses were applied. Therefore, the beam specimen was replaced every 20 laser pulses so as not to influence the mechanical properties of the beam and bias the results. Other collected data are presented in Appendix A. 4.4 Implications for Micro-bistable switches In the design of a MEMS switch, decisions on beam material and ambient gas must be made first since theory indicates that the required switching force is dependent on the elastic modulus. The material should have a low elastic modulus E to get a low switching force and the ambient gas should have a large molecular weight with a small molecular volume. Figure 50 shows a sketch of a micromechanical bistable system. It consists of a long slender micromechanical beam attached to a spacer with clamped-clamped boundary conditions. The method which was introduced and confirmed by large scale experimental apparatus, has the potential to be fabricated in microscales. There are various methods for fabricating a micro bistable buckled beam. One possible way is to start with a thin film deposited or grown on spacer material above the substrate, shape the beam by photolithography, and release it by underetching (N. Maluf, 2000). Due to differences in coefficients of thermal expansion between the beam and spacer/substrate, stresses at the interface cause the beam to buckle. The amount of bending depends on the 110 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. difference in coefficients of thermal expansion and processing temperature. Since differences in thermal expansion coefficients between the substrate and the deposited films are a primary source for mechanical strain fields, a substrate-film combination that is annealed to zero strain at some process temperature would be expected to have a strain level that is proportional to the product of the thermal expansion coefficient difference and the process to ambient temperature change. The amplitude of the buckled beam is desired to be as small as possible while still satisfying all other design constraints to keep the built-in stress low and thereby reduce the required switching force. To investigate the effects of beam material and gas, several materials and gases were evaluated on the basis of the theoretical work presented in Chapter 2. For these calculations the following assumptions were made: (1) The beam has dimensions 1=10 pm, b=2.5 pm, h=300A and < /0=0.5pm. These dimensions satisfy the geometrical constraints discussed in section 3.4. (2) The geometric surface area multiplier is assumed to have a value of So =io- (3) C02 gas is assumed as a working gas. (4) The energy coupling efficiency to the surface is 90%. The following sections discuss the various effects of beam material and ambient gas. I ll Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. L Figure 51: Schematic drawing of a micromechanical bistable system 4.4.1 Beam material Materials such as A l, C r,Si02, Ti and W are frequently used for MEMS applications. Numerical values of these materials are presented in Table B6. Values of the minimum necessary impulse and the required temperature increase of several representative materials based on Eq. 107 and Eq. 115 are presented in Figure 51 for assumed laser pulse duration of 10 nsec/pulse. □ Minimum necessary ftpulse PO^-S g/sec m m ) ■ Required Temperature increase (K) Figure 52: Minimum necessary impulse and required temperature increase of several representative materials 112 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. As may be seen, Si02 is a good choice for the beam material since Si02 has excellent mechanical properties and good thermal behavior. A bistable switch built of Si02 would need a smaller switching impulse due to the low elastic modulus and the fact that the surface temperature increase needed to provide a switching impulse is also lower than that required for the other materials studied. 600 500 energy per laser pulse=2*10'9,pulse duration=10 nsec - 300 E 200 100 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 1 energy coupling efficiency to surface Figure 53: Energy coupling efficiency versus Temperature increase for various materials. 113 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 250 Energy coupling efficiency=0J9, pulse duratio i=10 nsec 200 a 150 I 1 0 0 0.2 0.4 energy per laser pulse(J/pulse) 0.6 per laser pulse(J/pulse) 0.8 -9 x 10' Figure 54: Energy per laser pulse versus temperature increase for various materials Energy coupling eliiciency=0.9 35 r \ * 3 0 Q > f/i ( O g ^ g 25 I-k 15 10 . Cr(EL=1.33*10'9(J)) 'v '-x^Tl(EL=5.78*ia9(^-, x - ' - x J & ' ( E t s < 7 ^ 1 * 1 0 ' 1 0 ( J ) ) x^L(EL=1.21*10-9(3j)’‘- S i0 2 (E i^ .-W ljQ 3 J |^ 1 1.5 2 2.5 pulse duration(nsec) x Figure 55: Laser pulse duration versus necessary temperature increase for various materials. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1400 1200 ■ 1000 - Necessary generated switching force per unit area + ' .,.Cr(EL=2.3*10'6 mJ/pulse, tempinc=36.3K) " “ K . C O C L 0 ) a £ 800 '+^^Ti(EL=1*10‘5 mJ/pulse, tempinc=29:2K)K Si(EL=Tr3MQ'6 mJ/pulse, tempinc=26.5K) '+'" "•'k. 600 - 400 200 ^AI(EL=2.ri0‘6 mJ/pulse!lernfilTte^.2j<) ~ + ~ - 0.2 0.4 0.6 time (sec) 0.8 x 10 Figure 56: Time versus required gas desorption force x 10 2 3 3.5 .-8 0.5 2.5 time(sec) x 10 Figure 57: Time versus required impulse Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. L = » 10*10'6m,h=300*1(T10m 0.8 1.2 0.4 0.6 Impulse (Pa.sec) 0.2 Figure 58: Exerted impulse versus switching time 4.4.2 Working gas Figures 58 and 59 illustrate the effects of the choice of working gas on the impulse exerted on the beam. Several gases were considered and the results shown in these Figures are based on Eq. 117 assuming an ambient temperature initially at 300 K. As shown in the figures, C02 and 0 2 are the most appropriate choices for ambient gas since C 02 and 0 2 have higher molecular weights with smaller volumes compared with other possible ambient gases. However, C02 gas is more desirable for both convenience and safety. 116 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. initial temp=300K 1 .2 in dioxide 0.8 C O < 0 & • 0 .6 Q ) jO 3 Q . E 0.4 0.2 M — *— H 200 250 300 0 50 100 150 Surface temperature increase(K) Figure 59: Surface temperature increase versus exerted impulse of various working gases 1.2 x 10 < 0 C O Si 0.6 0 ) v,w M 3 CL E 0.4 0.2 0 > temperature increase=300K . . . C a ( b 9 n _ s J . i < i x i d ^ _ , « —^xygen ^ H e li^ m 270 280 290 300 310 320 330 initial temperature(K) Figure 60: Initial surface temperature versus exerted impulse for AT = 300 117 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Temp increase=100K 4.5 3.5 3 Q. E 1.5 Oxygen Nitrogen 0.5 A g ee Helium 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 time (sec) x 1Q -8 Figure 61: Time versus exerted impulse for various working gas < I c = ') Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 5 Conclusions 5.1 Summary and Concluding Remarks This study was motivated by an interest in MEMS switching devices. Of particular interest was the bistable, buckled micromechanical beam that is forced to change from one stable configuration to the other due to an applied force. Normally such switches are driven by electrostatic forces. The interest in this study was to drive such a switch by a gas pressure pulse. In an analytical sense, this interest was translated into a study of the dynamic behavior of a buckled beam due to a pressure pulse applied to the surface of the beam. For quasi-static loading as well as for impulsive loading conditions, the problem is to find the lowest strain energy related to an unstable equilibrium state for a shallow buckled beam. 119 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In this study, an approximate analytical solution of the non-linear buckling problem utilizing energy methods was presented along with the results of the corresponding experimental investigations of a simple clamped-clamped shallow buckled beam driven by pressure. The results illustrate some of the factors that must be considered in the development of more complicated micro-switching devices. Gjelsvik and Bodner found that there is an appreciable difference between the load-deflection relationship for one deflection term and two deflection terms for their arch model, but little difference between the load-deflection relationship for two deflection terms and six deflection terms. Therefore, the approximate analytical solutions presented herein were obtained by considering that the deflection functions were represented by two terms throughout the loading history. Guided by the analytical predictions, an experiment was designed to obtain the required necessary switching force due to gas pressure. Experiments using pulsed laser irradiation of a thin macroscopic scale beam to thermally induce a transient pressure pulse due to accommodation of incident gas molecules on the rapidly heated surface were then conducted. The pressure was produced by the combination of heated ambient gas at the beam surface and adsorbed molecules ejected from the beam surface as the surface was heated by the laser pulse. A thin Mylar film was used for the beam material because Mylar has relatively good thermal and mechanical properties. 120 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The theoretical beam surface temperature increase, AT, could be calculated during the laser pulse durations. The surface temperature increases with the square root of time under constant laser intensity with the maximum occurring at t = r where t is the pulse duration. The theoretical impulse per unit area and activation time were predicted for various values of the geometric surface multiplier, £ c . The activation time is highly related to the beam thickness and beam density. In order to predict the impulse applied to an actual device based on the theoretical work, the geometric surface multiplier, Z G, had to be determined by experiment. Based on repeated experimental observations, we roughly obtained an approximate necessary value forZG. Theoretical predictions have been compared to the results of the experimental investigation and have shown a reasonable agreement. Experiments were also conducted to investigate the effect of ambient gas pressure on snap-through performance. Two potential complicating factors were the concerns that photonic impact alone could cause switching or that residual gases present on the surface of the beam prior to installation in the vacuum chamber would contribute to the pressure pulse. To resolve both these issues, several low power laser pulse were applied to the beam while the vacuum chamber was pumped down as low as possible to remove the residual gas molecules and then a series of pulses were applied to the beam in a high vacuum to see if photonic impact alone could cause the beam to switch. No switching under such conditions 121 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. was observed. In subsequent experiments with an ambient gas introduced to the chamber, we observed that snap-though occurred at ambient pressures above a certain critical pressure. Differences in switching action could not be observed after the critical gas pressure was reached because of the limitations of the instrumentation. To examine how the energy coupling efficiency was related to the beam switching action, a thin black coating was applied to several beam samples. The black coating raised the energy coupling efficiency to over 90%. As expected, switching occurred with lower laser energy when the coating was applied. Several different ambient gases at various ambient pressures were used to observe the differences due to the gases and ambient pressures. Switching required a higher pressure using N2 as the ambient gas compared to using C02. However, above the critical ambient pressure, no particular differences were observed as the ambient gas pressures were changed. To determine an adequate beam material for a bistable micro-switch, several candidate MEMS materials were compared. Of the materials surveyed, Si02 is to be the best choice for the beam material because it provides the best compromise between mechanical properties and the need for a small switching force. For the ambient gas, the results for various working gases applicable to our system were compared to get the most efficient results and for future design effects. From the comparison, C02 and 0 2 are well suited to the role required of the ambient gas 122 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. because they have a higher molecular weight with smaller volume compared to the other gases considered. 5.2 Suggestions for future work Several areas of future study are still open for further theoretical and experimental investigation. In the analytical work, better approximations to the load-deflection relationships and the switching force for the buckled beam can be obtained by including more terms in the expression for the deflected shape (symmetric functions as well as anti-symmetric functions). Furthermore, the formulation and numerical solution to the more complex continuous non-linear buckling problem and for different boundary conditions would be of interest. Measuring gas desorption force directly through experiment by using an appropriate experimental setup is interesting since it would enable one to obtain a more exact value of a gas multiplier. One fertile field for investigation is the interaction of a working gas with the surface of the beam since it is apparent that the absorptive capacity of the beam depends largely upon the character of the beam surface. The amount of impulse should be much greater for porous substances than for those with smooth surfaces due to the increase in area for adsorbed gas molecules. More accurate data can be obtained in the future by using advanced diagnostic tools like extremely high speed video cameras to characterize the beam shape 123 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. during snap-though under various gas conditions. Investigations of the influence of pure photonic impact, the influence of any remaining absorbed molecules, and the range of operating temperatures remain to be done in the future. 124 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Bibliography Adamson, A.W., “Physical Chemistry of Surface”, John Wiley and Sons, Inc., New York, 1976. Aoki, K., Sone, Y., Mishino K., and Sugimoto, H., “Numerical Analysis of Unsteady Motion of a Rarefied Gas Caused by Sudden Changes of Wall Temperature”, in Rarefied gas Dynamics, ed. A. Beylich, VCH, Weinheim, 222- 231. 1991. Berman, A., “Vacuum Engineering Calculations, Formulas, and Solved exercises”, Academic Press, Inc. 1992. Bizeno, C. B., “Das Durchschlagen eines schwach Gekrummten Stabes,” Zeitschrift Ange. Math, und Mekh., Vol. 18, P21,1938. Boresi, A., “A Refinement of the Theory of Buckling of Rings under Uniform Pressure,” Appl. Mech., Vol.22, PP. 95-102, 1955. Caledonia, G.H., “Laboratory Simulations of Energetic Atom Interactions Occurring in Low Earth Obit”, In Rarefied Gas Dynamics, eds. E.P. Muntz, D.P. Weaver and D.H. Campbell, AIAA Progress in Aeronautics and Astronautics, Vol. 116, 129-142,1989. Campbell, AIAA Progress in Aeronautics and Astronautics, Vol. 116, 143-145 1989. Chiao, M., and Lin, L., “Self-Buckling of Micro machined Beams Under Resistive Heating,” J. microelectromechanical sys. Vol.9, PP.146-151,2000. Chrisey, D. B., Hubler, G. K., “ Pulsed Laser Deposition of Thin Films” PP. ISO- 188, John Wiley & Sons, Inc., New York, 1994. Coltharp, M.T. and Hackerman, N., J. Phys.Chem., 72,1171,1968. 125 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Cross, J.B., Blais, N.C., “High Energy/Intensity CW Atomic Oxygen Beam Source”, in Rarefied Gas dynamics, eds. E.P. Muntz, D.P. Weaver and D.H. De Boer, J. H., “The Dynamical Character of Absorption”, The Clarendon Press, Oxford, 1953. Dushman, S., “Scientific Foundations of VACUUM TECHNIQUE” , 2n d edition, John Wiely & Sons, Inc. New York, PP.376-434,1962. Fang, W. and Wickert, J. A., “Post buckling of micromachined beams,” J. Micromech.Microengineering, Vol.4, pp. 116-122.1994. Franciosi, V., Agusti, G., and Sparacio, R., “Collapse of Arches under Repeated Loading,” J. Struct. Div., ASCE, Vol.90, STI, P. 165,1964. Fung, Y. C., and Kaplan, A., “Buckling of Low Arches or Curved Beams of small Curvature,” NACA TN 2840,1952. Galonsky, A., leki, K., Kruse, Jon J., and Zecher, Philip D., “Optical transmission of Mylar and Teflon films”, Optical Engineering, Vol.33, No.9,1994. Gjeksvik, A., and Bodner, S. R., “The Energy Criterion and Snap Buckling of Arches,” J. Eng. Mech. Div., ASCE, Vol.88, EM5, P.87,1962. Gjeksvik, A., and Bodner, S. R., “Nonsymmetrical Snap Buckling of Spherical caps,” J. Eng. Mech. Div., ASCE, EM5, PP.135-165,1962 . Go, J. S., Cho, Y. H., Kwak, B. M., and Park, K. H., “Snapping micros-witches with adjustable acceleration threshold,” Sensor and Actuators,” PP.579-583,1996. HSlg, B., “ On a micro-Electro-Mechanical Nonvolatile Memory cell,” Trans. On Elect. Dev., IEEE, Vol.37, PP.2230-2235,1990. Hoff, N. J., and Bruce, V. G., “Dynamic Analysis of the Buckling of Laterally Loaded Flat Arches,” J. Math and Phys., Vol.XXXII, No.4,1954. Hombeck, L. J., Howell, T. R., Knipe R. L., and Mignardi, M. A., “Digital micromirror device-Commercialization of massiveky parallel MEMS technology,’ in proc. ASME Inc. Mech. Eng. Congr. Expo., vol. DSC-62, HTD-354,1997. 126 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Humphereys, J. S., and Bodner, S. R., “Dynamic Buckling of Shallow Shells Under Impulsive Loading,” J. Eng. Mech. Div., ASCE, EM2, PP. 17-36,1962. Humphereys, J. S., “On the Adequacy of Energy Criteria for Dynamic Buckling of Arches,” AIAA, Vol4, PP.921-923,1966. Humphereys, J. S., “On dynamic Snap Buckling of Shallow Arches.” AIAA, PP. 878- 886,1966. King, D. S., Cavanagh, R. R., “Molecular Desorption from Solid Surface: Laser Diagnostics and Chemical Dynamics.” Advances in Chemical Physics, ed K.P. Lawley, 45-89, John Wiley and Sons Ltd., New York, 1989. Lee, H. N., and Murphy, L. M., “Inelastic Buckling of Shallow Arches,” J. Eng. Mech. Div, ASCE, Vol.94, EMI, P.87, 1968. Lin, L , and Chiao, M , “Electrothermal responses of lineshape microstructures, “Sensors and Actuators, PP35-41, 1996. Lin, L , and Lin, S , “Vertically driven microactuators by electrothermal buckling effects,” Sensors and Actuators, A71, PP.35-39, 1998. Lin, S. C. H. and Pugacz-Muraszliwicz, I, “Local Stress Measurement in Thin Thermal Si02 Films on Si Substrates”, J. Appl. Phys, Vol43, No.l, 1972. Lindberg , U , Soderkvist , J , Lammerink, T , and Elwenspoek, M , “Quasi- buckling of micromachined beams,” J. Micromech, Microengineering, Vol.3, pp. 183-186,1993. Maluf, N , “ An Introduction to Microelectromechanical Systems Engineering”, ARTECH HOUSE, INC, Boston, PP. 87-156,2000. Masur, E. F , and Lo, D. L. C , ‘The shallow Arch-General Buckling, Post- Buckling, and Imperfection Analysis,” J. Struct. Mech, Vol.l, No.l, P.91, 1972. Muntz, E. P , Shiflett, G. R , Erwin, D. A , and Kune, J. A , “Transient Energy Release, Pressure Driven Microdevices,” J. Microelectromechanical system 1, No.3, PP. 155-162,1992 127 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Muntz, E. P., “Surface Dominated Rarefied Flows and the Potential of Surface Nanomanipulations,” 21s t Interational Symposium on Rarefied Gas Dynamics, Marseille, France, July, 1998. Mazumder, J., Rockstroh, T. J., and Krier, H., “Spectroscopic Studies of Plasma During CW Laser Gas Heating in Flowing Argon”, J. Appl. Phys., 62(12), 4712- 4718,1987. Nayfeh, A.H., Kreider, W., and Anderson, T. J., “Investigation of Natural Frequencies and Mode Shapes of Buckled Beams,” AIAA Vol.33, PP.l 121-1126, 1995. Nettesheim, S., and Zenobi, R., “ Pulsed Laser Heating of Surfaces: Nanosecond Timescale Temperature Measurement Using Black body Radiation,” Chemical Physics Letters, Vol.255, P39,1996. Onat, E. T., and Shu, L. S., “Finite Deformation of a Rigid Perfectly Plastic Arch,’ J. Appl. Mech., Vol.29, No.3, P.549,1962. Peterson, K. E., “Bistable micromechanical storage element in silicon”, IBM Tech. Disci. Bull., Vol. 20, no. 12, P. 5309, 1978. Pham-Van-Diep, G., Muntz, E. P., and Wadsworth, D. C., “Transient Normal Forces on Rapidly Heated Surfaces with Applications to Micromechanical Devices in Rarefied Gas Dynamics,” eds J. Harvey, G.Lord, Vol.2, P.701, Oxford University Press, Oxford, 1995. Pirri, A.N., Mansler, J.J., and Nebulsine, P.E., “Propulsion by Absorption of Laser Radiation”, AIAA, 12, 1259-1261,1977. Roorda, J., “Stability of structures with small Imperfections,” J. Eng.Mech.Div., ASCE, Vol91, EMI, P.87, 1965. Saif, M.Taher A., “On a Tunable Bistable MEMS-Theory and Experiment” J. Microelectromechanical systems, Vol.9, NO.2 PP. 157-169,2000. Sanders JR., J. L., “Nonlinear Theories for Thin shells,” Quart. Appl. Math, Vol.21 PP.21-36, 1963. Schreyer, H. L., and Masur, E. F., “Buckling of shallow Arches,” J.Eng.Mech.Div. , ASCE, Vol.92, EM4, P.l, 1966. 128 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Seid, P., “Dynamic Stability of Laterally Loaded Buckled Beams,” J. Eng. Mech. Div., ASCE, PP. 1556-2569, 1984. Simitises, G. J., “Snapping of Low Pinned Arches on an Elastic Foundation,” J. Appl. Mech., Vol.40, No.3, P.741, 1973. Simitises, G. J., An Introduction to the ELASTIC STABLITY OF STRUCTURES, PRENSENTICE-HALL, INC, New Jersey, 1976. Singer, J., and Babook, C. O., “On the Buckling of Rings under Constant Directional and Centrally Directed Pressure,” J. Appl. Mech., Vol.37, No.l, PP.215- 218, 1970. Smith, P. D. and Hetherington, J. G., “Blast and Ballistic Loading of Structure”, Butterworth-Heinemann Ltd, 1994. Smith JR., C. V., and Simitses, G. J., “Effect of Shear and Load Behavior on Ring Stability,” Proc. ASCE, EM3, PP.559-569,1969. Tabata, O., Kawahata, K., Sugiyama, S., and Igarashi, I., “Mechanical Property Measurements of Thin films Using Load- Deflection of Composite Rectangular Membrane”, IEEE Micro Electro Mechanical Systems, PP. 152-156, 1989. Timoshenko, S. P., “Analysis of Bi-Metal Thermostats,” Journal of the Optical Society of America, Vol.l 1, PP.223,1925. Timoshenko, S. P., “Buckling of Curved Bars with Small Curvature,” J. Appl. Mech., Vol.2, No.l, P. 17, 1935. Timoshenko, S. P. and Gere, J., Theory of Elastic Stability, McGraw-hill Book Co. New York, 1961. Tseng, W. Y., and Dugungi, J., “Nonlinear Vibration of Buckled Beam Under Harmonic Excitation,” J. Appl. Mech., PP467-476,1971. Trimmer, W. S., “Micro-robots and Micromechanical Systems”, Sensors and Actuators, 19,267-287, 1989. Vangbo, M., “An analytical analysis of compressed bistable buckled beam,” Sensors Actuators, Vot.69, pp212-216,1998. 129 permission of the copyright owner. Further reproduction prohibited without permission. Wasserman, E., “The Effect of the Behavior of the Load on the Frequency of Free Vibrations of a Ring,” NASA TT-F-52, 1961. Wempner, G., and Kesti, N., “On the Buckling of Circular Arches and Rings,” Proceedings, Forth U.S. National Congress of Applied mechanics, ASME, Vol.2, PP.843-852, 1962. Yamaki, N., and Mori, A., “Nonlinear Vibration of Clamped Beam with Initial Axial Displacement, PART I: Theory,” J. Sound and Vibration PP.333-346, 1980. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix A Table A1 Experimental data using C02 1st 2nd 3rd 4th 2.1 xlO"5 torr X X X X 3 torr X o X o 5 ton- o X X 0 10 ton- o 0 0 0 15 torr o 0 0 0 20 torr o o o 0 25 torr o o o 0 30 ton- o o o 0 35 ton- o 0 0 0 40 torr o o o 0 50 ton- 0 o o 0 100 ton- o o o 0 150 torr X o o 0 200 ton o o o 0 300 ton o o o X 400 ton o o o 0 500 ton o X 0 0 760 ton o o ° . _ 0 Where, E,= 0.7 mJ. o: switching occurred, x: no switching Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table A2 Experimental data using N 2 1st 2nd 3rd 4th 2.1 xlO-5 torr X X X X 3 torr X X X o 5 torr 0 X X X 10 torr o X o 0 15 torr 0 o o o 20 torr o o o 0 25 torr o o o o 30 torr o o o o 35 ton- o o o o 40 ton- o 0 o o 50 torr o o o o 100 ton- o o o o 150 ton- o o o o 200 ton- o o o o 300 ton- 0 o o o 400 ton- 0 o X o 500 torr o o o 0 760 torr o o 0 X Where, EL= 0.7 mJ. o: switching occurred, x: no switching Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table A3 Experimental data of energy coupling efficiency effect. ' — —------------- Mylar® Black coated Mylar® 6 mJ (532 nm) X 0 4 mJ (355 nm) X 0 2 mJ (266 nm) o 0 1.8 mJ (266nm) o 0 1.6 mJ (266nm) o 0 1.5mJ (266nm) 0 0 1.4mJ (266nm) o 0 1.2mJ (266nm) o 0 l.OmJ (266nm) 0 0 0.9mJ (266nm) 0 0 0.8mJ (266nm) o 0 0.7mJ (266nm) 0 0 0.6mJ (266nm) X 0 0.5mJ (266nm) X 0 Where, atmosphere ambient pressure. 133 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix B Tables Table B1 Numerical values of Mylar® Symbol Description Value Comment c „ Thermal conductivity 1.548x1 O'3 J/Cm sec K P density x M S / a f K Thermal diffusivity 9.51x10-*Cm2 / /sec Assume 300K n Energy coupling efficiency to surface 30% - 90% Depend on material adsorptivity c Specific heat 1.1715 J/g°C 0.28Cal/g°C E Elastic modulus 3.79 N/mm2 550 Kpsi 134 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table B2 Numerical value of laser Wavelength 1064 nm 532 nm 355 nm 266nm Energy 28 mJ 12 mJ 4 mJ 2 mJ Average power 375mW 150mW 60mW 30mw Pulse width 5 - 7 nsec 4 - 6 nsec 4 - 6 nsec 4 - 6 nsec Stability 2% 3% 4% 8% 135 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table B3 Adsorption times for several Gases on a Representative surface Assuming r0 = 10",35 (Muntz, 1998) Gas Q (kcal/mole) Q (J/kg) r(s) r(5) Ts =300K o o I I e - t CO , n 2, o 2 4 6 x l0 5 8.5x10"" 6 x l0 '5 He lxlO - 1 1x10s 1.2 xlO"1 3 1.6 xlO"1 3 h 2 1.5 3x l0 6 1.3xl0"1 2 2xlO”1 0 C02 8 7.6x10s X o i. 4xl0"s H 20 10 to 15 2.3 xlO6 -3 .5 x l0 6 3X10"6 -2 x l0 " 2 3xl0"9 -8 x l0 " 2 Ar 4 4x10s 6.3x1 O'9 3xl0"3 Where, r0 = the period of an oscillating molecule in the surface potential well (10~1 2 to 10"1 3 s) Q= the heat of adsorption or desorption 136 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table B4 Surface Concentrations and Fractional Coverage at Atmospheric pressure (Muntz, 1998) Gas ns {m 1) 0 Ts = 300K ts = 1 0 0 a: Ts = 300K Ts = 100 A c o , n 2,o2 3xl01 7 2 x l0 2 3 5xl0-2 3xl04 (multilayer) He lxlO1 5 1.4xl01 5 5xl0"5 8 x l0 '5 Hi lxlO1 6 2xl01 8 9x10“ " 2xl0-2 c o 2 2 x l0 2 0 2xl03 2 5x10' condensed (multilayer) h 2 o lxlO2 2 lxlO3 7 - 8 x l0 2 5 -3 x l0 4 8 condensed condensed Where, ns =surface concentration (# / m2 ), 0=fractional surface coverage Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table B5 Numerical values of several ambient gases Molecular mass (tmxIO2 7 ) Diameter (idx\0'°m ) Qs (im2) Q ( J / k g ) c o 2 73.1 5.62 2.481 xlO'1 9 7.6 x 10s Ar 66.3 4.17 1.366xl0"1 9 4xl05 *2 46.5 4.17 1.366xl0'1 9 6 x 1 0 s 0 2 53.12 4.07 1.301x1 O'1 9 6 x 1 0 s He 6.65 2.92 6.697 xlO'2 0 1 x 1 0 s Where Qs is the cross sectional area occupied by an adsorbed molecule. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table B6 Physical properties of several materials common to the MEMS Material E (Elim ) ( Gpa) C C ,h ^ /c m - sec- °C^ (C /W %ec> Al 70(0.17) 2.70 0.9 2.22 0.914 Cr 140(0.3) 7.19 0.46 0.67 0.203 Si 180(5.0) 2.33 0.678 0.84 0.532 Si02 75(8.4) 2.27 1.0 0.014 0.0062 Ti 110(0.4) 4.51 0.519 27.6 11.79 W 410(4.0) 19.3 0.14 1.66 0.61 Where, E : Elastic Modulus p : density C : specific heat Clh: thermal conductivity K : thermal diffusitivity 139 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table B7 Number of Molecules per Monolayer and Equivalent Volumes (S. Dushman,1962). GAS M M/(22.415) M/(24.050) £ ^ T 7 o io5 f2 0 io5 f0 h2 2.016 0.0900 0.08381 15.22 6.08 5.67 He 4.003 0.1790 0.1664 24.16 9.65 8.99 Ar 39.94 1.7820 1.661 8.54 3.41 3.18 *2 28.02 1.250 1.165 8.10 3.24 3.02 o2 32.00 1.428 1.330 8.71 3.48 3.24 CO 28.01 1.250 1.165 8.07 3.23 3.00 co2 44.01 1.963 1.830 5.34 2.13 1.99 h 2 o 18.02 0.8041 0.7492 5.27 2.11 1.96 Where, Ns is the number of molecule required to form monolayer per square centimeter, V0 is the volume of gas adsorbed monolayer per square centimeter at STP and V 2 0 = 1.074F0 140 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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Asset Metadata

Creator Kwon, Sangwook (author)

Core Title Investigation of a switching G mechanism for MEMS applications

School Graduate School

Degree Doctor of Philosophy

Degree Program Aerospace and Mechanical Engineering

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag engineering, mechanical,OAI-PMH Harvest

Language English

Contributor Digitized by ProQuest (provenance)

Advisor Shiflett, Geoffrey (committee chair), Ketsdever, Andrew (committee member), Kim, Eun Sok (committee member), Muntz, E. Phillip (committee member), Udwadia, Firdaus (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c16-236745

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