microfabricated rotor (a cylinder of density ρ , diameter D, and length L) compared to the pressure (p) acting on its projected surface area. This can be expressed as a non-dimensional load parameter: ζ ϭ ∝ (11.1) from which it can be seen that the load parameter decreases linearly as the device shrinks. For example, the load parameter due to the rotor mass for the MIT Microengine, which is a relatively large MEMS devices (measuring 4 mm in diameter and 300 microns deep), is approximately 10 Ϫ3 . The benefits of this scaling are that orientation or the freely suspended part becomes effectively irrelevant and that unloaded operation is easy to accomplish. In addition, since the gravity loading is negligible, the primary forces that one needs to support are pressure-induced loads and in a rotating device loads due to rotational imbal- ance. This last load is very important and will be discussed in more detail in connection to rotating lubri- cation requirements. The chief disadvantage of the low natural loading is that unloaded operation is often undesirable (in hydrodynamic lubrication where a minimum eccentricity is required for journal bearing stability), and in practice, gravity loading is often used to advantage. Therefore a scheme for applying an artificial load needs to be developed. This is discussed in more detail later in the chapter. 11.2.2 Applicability of the Continuum Hypothesis A common concern in microfluidic devices is the appropriateness of the continuum hypothesis as the device scale continues to fall. At some scale, the typical inter-molecular distances will be comparable to the device scales and the use of continuum fluid equations becomes suspect. For gases, this is measured by the Knudsen number (Kn) — the ratio of the mean free path to the typical device scale. Numerous experiments [Arkilic et al., 1997, 1993; Breuer et al., 2001] have determined that non-continuum effects become observable when Kn reaches approximately 0.1 and that continuum equations become meaning- less (the “transition flow regime”) at Kn of approximately 0.3. For atmospheric temperature and pressure, the mean free path of air is approximately 70nm. Thus, atmospheric devices with features smaller than approximately 0.2 microns will be subject to non-negligible non-continuum effects. In many cases, such small dimensions are not present, and the fluidic analysis can safely use the standard Navier–Stokes equa- tions (this is the case for the microengine). Nevertheless, in applications where viscous damping is to be avoided (for example in high-Q resonat- ing devices such as accelerometers or gyroscopes) the operating gaps are typically quite small (perhaps a few microns), and the gaps serve as both a physical standoff and a sense-gap where capacitive sensing is accomplished. In such examples one must work with the small dimension, and in order to minimize vis- cous effects, the device is packaged at low pressures where non-continuum effects are evident. For small Knudsen numbers, the Navier–Stokes equations can be used with a single modification — the boundary condition is relaxed from the standard non-slip condition to that of a slip-flow condition where the velocity at the wall is related to the Knudsen number and the gradient of velocity at the wall: u w ϭ λ Έ w (11.2) where σ is the tangential momentum accommodation coefficient (TMAC) which varies between 0 and 1. Experimental measurements [Breuer et al., 2001] indicate that smooth native silicon has a TMAC of approximately 0.7 in contact with several commonly used gases. Despite the fact that the slip-flow theory is valid only for low Kn, it is often used incorrectly with great success at much higher Knudsen numbers. Its adoption beyond its range of applicability stems primarily from the lack of any better approach short of solving the Bolzmann equation or Direct Simulation Monte Carlo (DSMC) computations [Beskok and Karniadakis, 1994; Cai et al., 2000]. For many simple geome- tries, the “extended” slip-flow theory works much better than it should and provides quite adequate results [Kwok et al., 2005]. This theory is demonstrated in the sections on Couette and squeeze-film damping later in the chapter. ∂u ᎏ ∂y 2 Ϫ σ ᎏ σ D ᎏ p ρπ LD 2 /4 ᎏ pLD Lubrication in MEMS 11-3 © 2006 by Taylor & Francis Group, LLC 11-4 MEMS: Introduction and Fundamentals 11.2.3 Surface Roughness Another peculiar feature of MEMS devices is that the surface roughness of the material used can become a significant factor in the overall device geometry. MEMS surface finishes are quite varied, ranging from atomically smooth surfaces found on polished single-crystal silicon substrates to the rough surfaces left by different etching processes. The effects of these topologies can be important in several areas for microde- vice performance. Probably the most important effect is the way in which the roughness can affect struc- tural characteristics such as crack initiation, yield strength, etc., although this will not be explored in this chapter. Secondly, the surface finish can affect fluidic phenomena such as the energy and momentum accommodation coefficient, and consequently, the momentum and heat transfer. Lastly, the surface char- acteristics (not only the roughness, but also the surface chemistry and affinity) can strongly affect its adhesive force. This is not treated in detail in this discussion, although it is mentioned briefly at the end of the chapter in connection with tribology issues in MEMS. 11.3 Governing Equations for Lubrication With the proviso that the continuum hypothesis holds for micron-scale devices (perhaps with a modified boundary condition), the equations for microlubrication are identical to those used in conventional lubrication analysis and can be found in any standard lubrication textbook [Hamrock, 1984]. We present the essential results here, but the reader is referred to more complete treatments for full derivations and a detailed discussion of the appropriate limitations. Starting with the Navier–Stokes equations, we can make a number of simplifying assumptions appro- priate for lubrication problems. These are itemized here: Inertia: The terms representing transfer of momentum due to inertia may be neglected. This arises because of the small dimensions that characterize lubrication geometries and MEMS in particular. In very high speed devices such as the MIT Microengine, inertial terms may not be as small as one might like, and corrections for inertia may be applied. However, preliminary studies suggest that these corrections are small [Piekos, 2000]. Curvature: Lubrication geometries are typically characterized by a thin fluid film with a slowly varying film thickness. The critical dimension in such systems is the film thickness, and this is assumed to be much smaller than any radius of curvature associated with the overall system. This assumption is particularly important in rotating systems where a circular journal bearing is used. Assuming that the radius of the bearing R is much larger than the typical film thickness c (i.e., c/R ϽϽ 1) greatly simplifies the governing equations. Isothermal: Because volumes are small and surface areas are large, thermal contact between the fluid and the surrounding solid is very good in a MEMS device. In addition, common MEMS materials are good thermal conductors. For both these reasons, it is safe to assume that the lubrication film is isothermal. With these restrictions, the Navier–Stokes equations, the equation for the conservation of mass, and the equation of state for a perfect gas may be combined to yield the Reynolds equation [Reynolds, 1886], written here for two-dimensional films: 0 ϭ Ϫ ϩ Ϫ ϩ ϩ ϩ ρ (w a Ϫ w b ) Ϫ ρ u a Ϫ ρ v a ϩ h (11.3) where x and y are the coordinates in the lubrication plane: u a etc. are the velocities of the upper and lower surfaces. An alternate and more general version may be derived [Burgdorfer, 1959] by non- dimensionalization with the film length and width (l and b), the minimum clearance h min , a characteristic ∂ ρ ᎏ ∂t ∂h ᎏ ∂y ∂h ᎏ ∂x ρ h(v a ϩ v b ) ᎏᎏ 2 ∂ ᎏ ∂y ρ h(u a ϩ u b ) ᎏᎏ 2 ∂ ᎏ ∂x ∂p ᎏ ∂y ρ h 3 ᎏ 12 µ ∂ ᎏ ∂y ∂p ᎏ ∂x ρ h 3 ᎏ 12 µ ∂ ᎏ ∂x © 2006 by Taylor & Francis Group, LLC shearing velocity u b , and a characteristic unsteady frequency ω . In addition, gas rarefaction can be incor- porated for low Knudsen numbers by assuming a slip-flow wall boundary condition: ΄ (1 ϩ 6K)PH 3 ΅ ϩ A 2 ΄ (1 ϩ 6K)PH 3 ΅ ϭ Λ ϩ σ (11.4) where A ϭ ; Λ ϭ ; σ ϭ (11.5) A is the film aspect ratio, Λ is the bearing number, and σ is the squeeze number representing unsteady effects. Solution of the Reynolds equation is straightforward, but not trivial. A chief difficulty is that gas films are notoriously unstable if they operate in the wrong parameter space. In order to determine the stabil- ity or instability of the numerically-generated solution, both the steady Reynolds equation and its unsteady counterpart need to be addressed with some accuracy. These issues are discussed more by Piekos and Breuer (1998) and others. 11.4 Couette-Flow Damping The viscous damping of a plate oscillating in parallel motion to a substrate has been a problem of tremen- dous importance in MEMS devices, particularly in the development of resonating structures such as accelerometers and gyros. The problem arises because the proof mass, which may be hundreds of microns in the lateral dimension, is typically suspended above the substrate with a separation of only a few microns. A simple analysis of Couette-flow damping for rarefied flows is easy to demonstrate by choosing a model problem of a one-dimensional proof mass (i.e., ignoring the dimension perpendicular to the plate motion). This is shown schematically in Figure 11.1. The Navier–Stokes equations for this geometry reduce to: ϭ µ (11.6) in which only viscous stresses due to the velocity gradient and the unsteady terms survive. This can be solved using separation of variables and employing a slip-flow boundary condition [Arkilic and Breuer, 1993] yielding the solution the drag force experienced by the moving plate: D ϭ ΄ ΅ (11.7) where β ϭ Ί (11.8) ω h 2 ᎏ µ sinh β ϩ sin β ᎏᎏᎏ (cosh β Ϫ cos β ) ϩ D R 4 π U 2 ᎏ β ∂ 2 u ᎏ ∂y 2 ∂u ᎏ ∂t 12 µω l 2 ᎏ p a h 2 min 6 µ u b l 2 ᎏ p a h 2 min l ᎏ b ∂(PH) ᎏ ∂T ∂(PH) ᎏ ∂X ∂P ᎏ ∂Y ∂ ᎏ ∂Y ∂P ᎏ ∂X ∂ ᎏ ∂X Lubrication in MEMS 11-5 U sin t h FIGURE 11.1 Schematic of Couette-flow damping geometry. The upper plate vibrates with a proscribed amplitude and frequency. For most MEMS geometries and frequencies, the unsteadiness can usually be neglected. © 2006 by Taylor & Francis Group, LLC 11-6 MEMS: Introduction and Fundamentals is a Stokes number, representing the balance between unsteady and viscous effects, and D R is a correction due to slip flow at the wall: D R ϭ 2Kn β (sinh β ϩ sin β ) ϩ 2Kn 2 β 2 (cosh β ϩ cos β ) (11.9) Atypical MEMS geometry might have a plate separation of one micron and an operating frequency of 10 kHz. With these parameters, the Stokes number is very small (approximately 0.1), and the flow may be considered quasi-steady to a high degree of approximation. In addition, the rarefaction effects, indicated by D R ,are also vanishingly small at atmospheric conditions. 11.4.1 Limit of Molecular Flow Although the slip-flow solution is limited to low Knudsen numbers, the damping due to a gas at high degrees of rarefaction can be computed using a free-molecular flow approximation. In such cases the friction factor on a flat plate is given by Rohsenow and Choi (1961). C f ϭ Ί (11.10) where γ is the ratio of specific heats and M is the Mach number. It is important to recognize that the damping (and Q) in this case is provided, not only by the flow in the gap, but also by the flow above the vibrating plate. However, it is unlikely that the fluid damping provides the dominant source of damping at such extremely low pressures. More likely, damping derived from the structure (e.g., flexing of the sup- port tethers, non-elastic strain at material interfaces, etc.) will take over as the dominant energy-loss mechanism. Kwok et al. (2005) compared the continuum, slip-flow, and free molecular flow models for Couette damping with data obtained by measuring the “ring down” of a tuning fork gyroscope fabricated by Draper Laboratories. Figure 11.2 shows the measurements and theory confirming the functional behavior of the damping as the pressure drops (Kn increases) and the unexpected accuracy of these rather simple models. Although the trends are well-predicted, the absolute value of the Q-factor is still in error by a factor of two, suggesting that more detailed computations are still of interest. 1 ᎏ M 2 ᎏ πγ Quality factor (fluid) in drive axis Knudsen number Measurement (LCCC 701) Continuum flow Slip flow Molecular flow 10 10 10 8 10 6 10 4 10 2 10 −2 10 0 10 0 10 2 10 4 10 6 FIGURE 11.2 (See color insert following page 10-34.) Theory and measurements of Couette damping in a tuning fork gyro (Kwok et al. [2005]). Note that in the high Knudsen number limit, the free molecular approximation pre- dicts the damping more closely, but that the slip-flow model, though totally inappropriate at this high Kn level, is not too far from the experimental measurements. © 2006 by Taylor & Francis Group, LLC 11.5 Squeeze-Film Damping Squeeze-film damping arises when the gap size changes in an oscillatory manner squeezing the trapped fluid (Figure 11.3). Fluid, usually air, is trapped between the vibrating proof mass and the substrate result- ing in a squeeze film, which can significantly reduce the quality factor of the resonator. In some cases this damping is desirable, but as with the case of Couette-flow damping, it is often parasitic, and the MEMS designer tries to minimize its effects and maximize the resonant Q-factor of the device. Common methods for alleviating squeeze-film effects are to fabricate breathing holes (“chimneys”) throughout the proof mass which relieve the build up of pressure and to package the device at low pressure. Both of these solutions have drawbacks. The introduction of vent holes reduces the vibrating mass, necessitating an even larger structure, while the low-pressure packaging adds considerable complexity to the overall device development and cost. Figure 11.4 illustrates a high-performance tuning fork gyroscope fabricated by Draper Laboratories. 11.5.1 Derivation of Governing Equations The analysis of the squeeze-film damping is presented in the following section. The Reynolds equations may be used as the starting point. However, a particularly elegant and complete solution was published by Blech (1983) for the case of the continuum flow and was extended by Kwok et al. (2005) to the case of slip-flow and flows films with vent holes. This analysis is summarized here. Lubrication in MEMS 11-7 h(t ) = h 0 (1 + sin t ) FIGURE 11.3 Schematic of squeeze-film damping between parallel plates. As with Couette damping, for most prac- tical embodiments of MEMS, the damping is quasi-steady. FIGURE 11.4 Photograph of a typical microfabricated vibrating proof mass used in a high-performance tuning fork gyroscope. (Reprinted with permission of M. Weinberg at Draper Laboratories.) © 2006 by Taylor & Francis Group, LLC 11-8 MEMS: Introduction and Fundamentals The Navier–Stokes equations are written for the case of a parallel plate vibrating sinusoidally in a pro- scribed manner in the vertical direction. If we assume that the motion, subsequent pressure, and velocity perturbations are small, a perturbation analysis yields the classical squeeze-film equation derived by Blech, with an additional term due to the rarefaction: ΨH 3 ϩ 6KH 2 ϭ σ (11.11) where the variables have been non-dimensionalized, so that H represents the film gap, normalized by the nominal film gap H ϭ h(x, y, t)/h o ; Ψ is the pressure, normalized by ambient pressure Ψ ϭ P(x, y, t)/P 0 ; X and Y are the coordinates, normalized by the characteristic plate geometry X ϭ x/L x , Y ϭ y/L y ; T is time, normalized by the vibration frequency T ϭ ω t; and the squeeze number σ is defined as before: σ ϭ (11.12) Assuming small amplitude, harmonic forcing of the gap H ϭ 1 ϩ ε sin T, and a harmonic response of the pressure, we can derive a pair of coupled equations describing the in-phase (Ψ 0 ) and out-of-phase (Ψ 1 ) pressure distributions in the gap representing stiffness and damping coefficients, respectively: ϩ Ψ 1 ϩ ϭ 0 ϩ Ψ 0 ϭ 0 (11.13) Note that these equations represent the standard conditions with the adoption of a modified squeeze number, σ m ϵ σ /(1 ϩ 6K). The solutions are achieved either by manual substitution of Fourier sine and cosine series or by direct numerical solution. The results for rectangular plates with no vent holes are shown in Figure 11.5. σ ᎏ 1 ϩ 6K ∂ 2 Ψ 1 ᎏ ∂x 2 σ ᎏ 1 ϩ 6K σ ᎏ 1 ϩ 6K ∂ 2 Ψ 0 ᎏ ∂x 2 12 µω L 2 x ᎏ P 0 h 2 0 ∂(ΨH) ᎏ ∂T ∂ Ψ ᎏ ∂X ∂ ᎏ ∂X ∂Ψ ᎏ ∂X ∂ ᎏ ∂X 0 0510 15 2520 30 Squeeze number Non-dimensional forces 0.5 0.4 0.3 0.2 0.1 Damping force Spring force FIGURE 11.5 (See color insert following page 10-34.) Solutions to the squeeze-film equation for a rectangular plate. The stiffness and damping coefficients are presented as functions of the modified squeeze number, which includes a correction due to first-order rarefaction effects [Blech, 1983; Kwok et al., 2005]. © 2006 by Taylor & Francis Group, LLC 11.5.2 Effects of Vent Holes The equations as previously derived are made more useful by extending them to account for the presence of vent holes in the vibrating proof mass. In such cases the boundary condition at each vent hole is no longer atmospheric pressure (Ψ 0 ϭ Ψ 1 ϭ 0), but rather an elevated pressure proscribed by the pressure drop through the “chimney” which vents the squeeze film to the ambient. Kwok et al. (2005) demonstrate that this can be incorporated into the previous model (in the limit of low squeeze number) by a modi- fied boundary condition for the squeeze-film equations for Ψ 0 : Ψ ϭ ΄ 32 3 1 Ϫ 2 ΅ ΄ 1 ϩ 8 1 ΅ σ (11.14) 12 4 This boundary condition has three components: a geometric component dependent on the plate thick- ness t, length L x , hole size L h , and nominal gap size h 0 ;ararefaction component (here based on the hole size); and a time-dependent component — the squeeze number σ .Note that as the thickness of the plate decreases and the chimney pressure drop falls, the boundary condition approaches zero. Similarly, as the open area fraction of the plate increases (more venting), the boundary condition approaches that of the ambient. This boundary condition can be applied at the chimney locations and can accurately simulate the squeeze-film damping of perforated micromachined plates. 11.5.3 Reduced-Order Models for Complex Geometries Most devices of practical interest have geometries that are too complex to enable full numerical simula- tion of the kind described previously. Reduced-order models are of great value in such cases. Many such models have been developed, including those based on acousto-electric models [Veijola et al., 1995]. In the case of squeeze-film damping in the limit of low squeeze numbers, such models reduce to solution of aresistor network that models the pressure drops associated with each segment of the squeeze film. This is effectively a finite-element approach to the problem. Instead of modeling a large number of elements, as is generally the case in a numerical solution, arelatively small number of discrete elements can be used, if higher-order solutions can be employedtoconnect each element together. Kwok et al. (2005) demon- strate this approach and model the damping associated with an inclined plate with vent holes. More com- plex numerical solution techniques based on boundary integral techniques have also been presented [Aluru and White, 1998; Kanapka and White, 1999] providing a good balance between solution fidelity and required computing power. 11.6 Lubrication in Rotating Devices Rotating MEMS devices bring a new level of complexity to MEMS fabrication and to the lubrication con- siderations. As discussed in the introduction, many rotors and motors have been demonstrated with dry- rubbing bearings, and the success of these devices is due to the low surface speeds of the rotors. However, as the surface speed increases in order to get high power densities, the dry rubbing bearings are no longer an option, and true lubrication systems need to be considered. An example of “Power-MEMS” develop- ment is provided by a project initiated at the Massachusetts Institute of Te c hnology in 1995 to demon- strate a fully functional microfabricated gas turbine engine [Epstein et al., 1997]. The baseline engine, illustrated in Figure 11.6,consists of acentrifugal compressor, fuel injectors (hydrogen is the initial fuel, although hydrocarbons are planned for later configurations), a combustor operating at 1600 K, and a radial inflow turbine. The device is constructed from single crystal silicon and is fabricated by extensive and complex fabrication of multiple silicon wafers that are fusion bonded in a stack to form the complete L h ᎏ L x λ ᎏ L h L h ᎏ L x h 0 ᎏ L x t ᎏ L x Lubrication in MEMS 11-9 © 2006 by Taylor & Francis Group, LLC 11-10 MEMS: Introduction and Fundamentals device. An electrostatic induction generator may also be mounted on a shroud above the compressor to produce electric power instead of thrust [Nagle and Lang, 1999]. The baseline MIT Microengine has at its core a “stepped” rotor consisting of acompressor with an 8 mm diameter and a journal bearing and turbine with a diameter of 6 mm. The rotor spins at 1.2 million r/min. FIGURE 11.6 (Seecolor insert following page 10-34.) Schematic of the MIT Microengine, showing the air path through the compressor, combustor, and turbine. Forward and aft thrust bearings located on the centerline hold the rotor in axial equilibrium, while a journal bearing around the rotor periphery holds the rotor in radial equilibrium. Forward thrust bearing D L R c Main turbine air path High pressure plenum Low pressure plenum Aft thrust bearing Journal bearing Rotor High pressure plenum Low pressure plenum Aft thrust bearing Forward thrust bearing Journal bearing Rotor Main flow path Axis of rotation FIGURE 11.7 Illustrating schematic and corresponding SEM of a typical microfabricated rotor, supported by axial thrust bearings and a radial journal bearing. © 2006 by Taylor & Francis Group, LLC Key to the successful realization of such a device is the ability to spin a silicon rotor at high speed in a controlled and sustained manner. The key to spinning a rotor at such high speeds is the demonstration of efficient lubrication between the rotating and stationary structures. The lubrication system needs to be simple enough to be fabricated but with sufficient performance and robustness to be of practical use in the development program and in future devices. Figure 11.7 illustrates a microbearing rig that was fabricated to develop this technology. The core of the rotating machinery has been implemented but without the substantial complications of the thermal environment that the full engine brings. The rig consists of a radial inflow turbine mounted on a rotor and embedded inside two thrust bearings that provide axial sup- port. A journal bearing located around the disk periphery provides radial support for the disk as it rotates. 11.7 Constraints on MEMS Bearing Geometries 11.7.1 Device Aspect Ratio Perhaps the most restrictive aspect of microbearing design is that MEMS devices are limited to rather shal- low etches, resulting in devices with low aspect ratio. Even with the advent of deep reactive ion etchers (DRIE) in which the ion etching cycle is interleaved with a polymer passivation step, the maximum prac- tical etch depth that can be achieved while maintaining dimensional control is about 500 microns. Even this has an etch time of about nine hours, which makes its adoption a very costly decision. In compari- son, typical rotor dimensions are a few millimeters. The result is that microbearings are characterized by very low aspect ratios (Length/Depth, or L/D). In the case of the MIT microturbine test rig, the journal bearing is nominally 300 microns deep while the rotor is 4 mm in diameter, yielding an aspect ratio of 0.075. To put this in perspective, commonly available design charts [Wilcox, 1972] present data for val- ues of L/D as low as 0.5 or perhaps 0.25. Prior to this work there was no data for lower L/D. The impli- cations of the low aspect ratio bearings are that the task becomes supporting a disk rather than a shaft. The low aspect ratio bearings do not have terrible performance by any standard. The key features of the low L/D bearings are: The load capacity is reduced compared to conventional designs. This is because the fluid leaking out of the ends relieves any tendency for the bearing to build up a pressure distribution. For a given geom- etry and speed, a short bearing supports a lower load per unit length than its longer counterpart. The bearing acts as an incompressible bearing over a wide range of operation. Pressure rises, which might lead to gas compressibility, are minimized by the flow leaking out of the short bearing. Incompressible behavior (without the usual fluid cavitation that is commonly assumed in incom- pressible liquid bearings) can be observed to relatively high speeds and eccentricities. 11.7.2 Minimum Etchable Clearance It is reasonable to question why one could not fabricate a 300 micron long “shaft”, but with a much smaller diameter, to greatly enhance the L/D. For example, a shaft with a diameter of 300 microns would result in a reasonable value for L/D. This raises the second major constraint on bearing design by current microfabrication technologies — that of the minimum etchable clearance. In the current microengine manufacturing process, the bearing and rotor combination is defined by a single deep and narrow etch, currently 300 microns deep and about 12 microns in width. No foreseeable advance in fabrication technology will make it possible to significantly reduce the minimum etchable clearance, and this has considerable implications for bearing design. In particular, if one were to fabricate a bearing with a diameter of 300 microns in an attempt to improve the L/D ratio, the result would be a bearing with a clearance to radius c/R of 12/300, or 0.04. For a fluid bearing, this is two orders of magni- tude above conventional bearings and has several detrimental implications. The most severe implication is the impact on the dynamic stability of the bearing. The non-dimensional mass of the rotor depends on (c/R) raised to the fifth power [Piekos, 2000]. Bringing the bearing into the center of the disk and raising the c/R by a factor of 10 results in a mass parameter increasing by a factor of 10 5 . This increase in effective mass has severe implications for the stability of the bearing. Lubrication in MEMS 11-11 © 2006 by Taylor & Francis Group, LLC 11-12 MEMS: Introduction and Fundamentals These reasons and others not enumerated here make the implementation of an inner-radius bearing less attractive. Therefore, the constraint of small L/D is unassailable as long as one requires that the microdevice be fabricated in situ. If one were to imagine a change in the fabrication process such that the rotor and bearing could be fabricated separately and subsequently assembled reliably, this situation would be quite different. In such an event, the bearing gap is not constrained by the minimum etch dimension of the fabrication process, and almost any “conventional” bearing geometry could be considered and would probably be superior in performance to the bearings discussed here. Such fabrication could be considered for a “one-off”device, but does not appear feasible for mass production, which relies on the monolithic fabrication of the parts. Lastly, the risk of contamination during assembly — a common con- cern for all precision-machined MEMS — effectively rules out piece-by-piece manufacture and assembly and constrains the bearing geometry as described. 11.8 Thrust Bearings Thrust bearings support any axial loads generated by rotating devices such as turbines, engines, or motors. Current fabrication techniques require that the axis of rotation in MEMS devices lie normal to the lithographic plane. This lends a significant advantage in the design and operation of thrust bearings because the area available for the thrust bearing is relatively large as defined by lithography, while the weight of the rotating elements will be typically small due to the cube-square law and the low thicknesses of microfabricated parts. For these reasons, thrust bearings are one area of microlubrication where solu- tions abound and problems are relatively easily dealt with. Two thrust bearing options exist: (a) hydrostatic (externally-pressured) thrust bearings, in which the fluid is fed from a high-pressure source to a lubrication film, and (b) hydrodynamic, where the support- ing pressure is generated by a viscous pump fabricated on the surface of the thrust bearing itself (see Figure 11.9). Hydrostatic bearings are easy to operate and relatively easy to fabricate. These have been suc- cessfully demonstrated in the MIT Microengine program [Frechette et al., 2005; Liu et al., 2003]. The thrust bearing in Figure 11.8 shows an scanning electron micrograph (SEM) of the fabricated device cut though the middle to reveal the plenum, restrictor holes, and the bearing lubrication gap, which is approximately 1 micron wide. Key to the successful operation of hydrostatic thrust bearings is the accurate FIGURE 11.8 Close-up cutaway view of micro thrust bearing showing the pressure plenum (on top), the feed-holes, and the bearing gap (faintly visible). (SEM reprinted with permission of Lin et al. [1999].) © 2006 by Taylor & Francis Group, LLC [...]... 0.075) typical of a deep reactive ion etched rotor such as the MIT microengine The bearing number is defined as: ΄ ΅ 6µω R Λϭ ᎏ ᎏ p c © 2006 by Taylor & Francis Group, LLC 2 (11.15) Lubrication in MEMS 1 1-1 5 100 10 1 10 2 8 =0 6 10 3 = 0 4 = 0 2 = 0 10 4 10 5 10 1 100 Λ 101 90 = 0.8 70 φ (deg) = 0.8 50 30 =0 9 10 0 10 1 100 Λ 101 FIGURE 11 .10 Static performance (eccentricity and attitude angle... frame shows the measurements (from a 26:1 scaled-up experimental rig) along with the theoretical predictions based on the assumed geometry The right-hand frame shows the same measurements compared with the same model but using slightly modified geometric parameters [Orr, 1999] © 2006 by Taylor & Francis Group, LLC 1 1-2 2 MEMS: Introduction and Fundamentals deriving from the turbine which drives the rotor... resistance The fluid then flows through the lubrication passage (the bearing gap) If the rotor moves to one side, the restrictor and lubrication film act as a pressure divider such that the pressure in the lubrication film rises, forcing the rotor back towards the center of the bearing The advantages of using hydrostatic lubrication in MEMS devices are that: The rotor tends to operate near the center of the. .. that are used for MEMS This alleviates many of the reservations and costs that might inhibit their adoption Because the MEMS constraint is the minimum gap dimension, the wave bearing in a MEMS machine can be implemented only by selectively enlarging the bearing gap Piekos (2000) analyzed the performance of the wave bearing for the microengine geometry and found (Figure 11.14) that while the load capacity... not moving The conventional inherent restriction of the flow entering the lubrication channel also enhances the stiffness The stiffness coupled with the rotor mass defines a natural frequency which was measured by Orr (1999) The presence of this frequency led to the discovery of the axial-through-flow mechanism Simple theory [Orr, 1999] was also able to predict the frequency in a scaled-up experimental... hydrodynamic operation The side load is developed by applying a differential pressure to the two plena located on the aft side of the rotor © 2006 by Taylor & Francis Group, LLC 1 1-2 0 MEMS: Introduction and Fundamentals them undesirable in a practical MEMS rotor system The primary difficulty is that, in order to satisfy the requirements of sub-synchronous stability, the rotor needs to operate at very high... (magnesium fluoride) stripes of an elsewhere hydrophobic (silicone rubber) substrate (a) Low-condensate-volume regime: the parallel channels of condensed water have a constant cross-section and a small contact angle Several droplets are also seen on the hydrophobic domains of the substrate (b) High-condensate-volume regime: some of the liquid channels develop a single drop, when the contact angle exceeds a... angle θ at the edges of the stripes where the liquid appears to be pinned The cross-section of these microchannels is position-independent and represents a circular cap Several drops of the condensate between the hydrophilic stripes are also observed in Figure 12.2(a) When the process of water condensation proceeds further the contact angle θ increases beyond a certain value, and the microchannels undergo... (2000) shows the time evolution (from top to bottom) of deposition of water condensing onto curved wettable patches of different width in its corners This width increases from channel (a) to channel (e) When water condenses and gradually fills the channels, the behavior of the liquid depends solely on the width of the corner If the latter is large, as in the case (e), the drop develops in the corner... actively developed and improved These techniques use molten solder and are based on the restoring force arising from surface tension that drives the misaligned solder joint to become well-aligned and minimizes the total interfacial energy of the system The final well-aligned configuration is then fixed by cooling down and solidifying the solder Figure 12.6(a) presents such a misaligned layout of the . used for MEMS. This alleviates many of the reservations and costs that might inhibit their adoption. Because the MEMS constraint is the minimum gap dimension, the wave bearing in a MEMS machine. the stability of the bearing. Lubrication in MEMS 1 1-1 1 © 2006 by Taylor & Francis Group, LLC 1 1-1 2 MEMS: Introduction and Fundamentals These reasons and others not enumerated here make the. 701) Continuum flow Slip flow Molecular flow 10 10 10 8 10 6 10 4 10 2 10 −2 10 0 10 0 10 2 10 4 10 6 FIGURE 11.2 (See color insert following page 1 0-3 4.) Theory and measurements of Couette damping