Lubrication in MEMS

9 Lubrication in MEMS Kenneth S. Breuer 9.1 Introduction Brown University Objectives and Outline 9.2 Fundamental Scaling Issues The Cube-Square Law • Applicability of the Continuum Hypothesis • Surface Roughness 9.3 Governing Equations for Lubrication 9.4 Couette-Flow Damping Limit of Molecular Flow 9.5 Squeeze-Film Damping Derivation of Governing Equations • Effects of Vent Holes • Reduced-Order Models for Complex Geometries 9.6 Lubrication in Rotating Devices 9.7 Constraints on MEMS Bearing Geometries Device Aspect Ratio • Minimum Etchable Clearance 9.8 Thrust Bearings 9.9 Journal Bearings Hydrodynamic Operation • Static Journal Bearing Behavior • Journal Bearing Stability • Advanced Journal Bearing Designs • Side Pressurization • Hydrostatic Operation 9.10 Fabrication Issues Cross-Wafer Uniformity • Deep Etch Uniformity • Material Properties 9.11 Tribology and Wear Stiction • The Tribology of Silicon 9.12 Conclusions Acknowledgments 9.1 Introduction As microengineering technology continues to advance, driven by increasingly complex and capable micro- fabrication and materials technologies, the need for greater sophistication in microelectromechanical systems (MEMS) design will increase. Fluid film lubrication has been a critical issue from the outset of MEMS development, particularly in the prediction and control of viscous damping in vibrating devices such as accelerometers and gyros. Much attention has been showered on the development of models for accurate prediction of viscous damping and of fabrication techniques for minimizing the damping (which destroys the quality, or Q-factor, of a resonant system). In addition to the development and optimization of these oscillatory devices, rotating devices—micromotors, microengines, etc.—have also captured the attention of MEMS researchers since the early days of MEMS development, and there have been several demonstrations of micromotors driven by electrostatic forces [Bart et al., 1988; Mehregany et al., 1992; Nagle and Lang, 1999; Sniegowski and Garcia, 1996]. For the most part these motors were very small, © 2002 by CRC Press LLC with rotors of the order of 100 µm in diameter, and, although they had high rotational speeds (hundreds of thousands of revolutions per minute), their tip speeds and rotational energy (which scales with tip speed) were quite small (tip speeds of 1 m/s are typical). In addition, the focus of these projects was the considerable challenge associated with the fabrication of a freely moving part and the integration of the drive electrodes. Lubrication and protection against wear were low down on the “to-do” list and the demonstrated engines relied on dry-rubbing bearings in which the rotor was held in place by a bushing, but there was no design or integration of a lubrication system. The low surface speeds of these engines meant that they could have operated for long times using this primitive bearing; however, failure was observed frequently due to rotor/bearing wear. As MEMS devices become more sophisticated and have more stringent design and longevity requirements, the need for more accurate and extensive design tools for lubrication has increased. In addition, the energy density of MEMS is increasing. Devices for power generation, propulsion and so forth are actively under development. In such devices the temperatures and stresses are stretched to the material limits. Hence, the requirement for protection of moving surfaces becomes more than a casual interest—it is critical for the success of a “power MEMS” device (imagine, for comparison, an aircraft engine without lubrication—it would melt and fail within seconds of operation!). 9.1.1 Objectives and Outline The objective of this chapter is to briefly summarize some the issues associated with lubrication in MEMS. Lubrication is a vast topic, and clearly there is no way in which these few pages can adequately introduce and treat an entire field as diverse and complex as this. For this reason, some hard choices have been made. Our focus is to review the essential features of lubrication theory and design practice and to highlight the difficulties that arise in the design of a lubrication system for MEMS devices. A key feature of MEMS is that the fabrication, material properties and mechanical and electrical design are all tightly interwoven and cannot be separated. For this reason, some attention is devoted to the important issue of how a successful lubrication system is influenced by manufacturing constraints and material properties. Finally, one should always remember that MEMS is a rapidly developing and maturing manufacturing technology. The range of geometric options, available materials and dimensional control is continually expanding and improving. This chapter necessarily focuses on the current state of the art in MEMS fabrication, and for this reason it tends to favor silicon-based fabrication processes and the constraints that lithographic-based batch fabrication techniques place on lubrication system design and performance, but all of this may be rendered obsolete in 5 or 10 years! Examples are drawn from several sources. In the sections on translational and squeeze-film damping examples are drawn from the extensive literature associated with accelerometer and gyro design. The rotating system lubrication discussion draws heavily on the MIT Microengine project [Epstein et al., 1997], which, to our knowledge, is the only MEMS device to date with rotating elements using a fluid- film lubrication system. This device is described in some detail further on in the chapter, and the analysis and examples of thrust bearings and journal bearings are drawn from that device. 9.2 Fundamental Scaling Issues 9.2.1 The Cube-Square Law The most dominant effect that changes our intuitive appreciation of the behavior of microsystems is the so-called cube-square law, which simply reflects the fact that volumes scale with the cube of the typical length scale, while areas (including surface areas) scale with the square of the length scale. Thus, as a device shrinks, surface phenomena become relatively more important than volumetric phenomena. The most important consequence of this is the observation that the device mass (and inertia) becomes negligibly small at the micro- and nanoscales. For lubrication, the implication of this is that the volumetric loads that require support (such as the weight of a rotor) quickly become negligible. As an © 2002 by CRC Press LLC example, consider the ratio of the weight of a 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 nondimensional load parameter: ρπLD2/4 D ζ = ---------------------- ∝ ---- pLD p from which it can clearly 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 device, measuring 4 mm in diameter and 300 µm deep) is approximately 10 3. The benefits of this scaling are that orientation of the freely suspended part becomes effectively irrelevant and that unloaded operation is easy to accomplish, should one desire to do so. In addition, because the gravity loading is negligible, the primary forces to be supported are pressure-induced loads and (in a rotating device) loads due to rotational imbalance. This last load is, in fact, very important and will be discussed in more detail in connection to rotating lubrication 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); in practice, gravity loading is often used to advantage. Thus, a scheme for applying an artificial load must be developed. This is discussed in more detail later in the chapter. 9.2.2 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 intermolecular distances will be comparable to the device scales, and the use of continuum fluid equations becomes suspect. For gases, the Knudsen number (Kn) is the ratio of the mean free path to the typical device scale. Numerous experiments (e.g., see Arkilic et al., 1993; 1997; Arkilic et al., 2001) have determined that noncontinuum effects become observable when Kn reaches approximately 0.1, and continuum equations become meaningless (the “transition flow regime”) at Kn of approximately 0.3. For atmospheric temperature and pressure, the mean free path (of air) is approximately 70 nm. Thus, atmospheric devices with features smaller than approximately 0.2 µm will be subject to nonnegligible, noncontinuum effects. In many cases, such small dimen- sions are not present, thus the fluidic analysis can safely use the standard Navier–Stokes equations (as is the case for the microengine). Nevertheless, in applications where viscous damping is to be avoided (for example, in high-Q resonating devices, such as accelerometers or gyroscopes) the operating gaps are typically quite small (perhaps a few microns) and the gaps serve double duty as both a physical standoff and a sense gap where capacitive sensing is accomplished. In such examples, one is “stuck” with the small dimension, so, in order to minimize viscous effects, the device is packaged at low pressures where noncontinuum effects are clearly 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: σ ∂ = λ2 – u uw ------------σ -----∂ - y w where σ is the tangential momentum accommodation coefficient (TMAC) which varies between 0 and 1. Experimental measurements [Arkilic et al., 2001] indicate that smooth native silicon has a TMAC of approximately 0.7 in contact with several commonly used gases.

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