© 2002 by CRC Press LLC N A Avogadro’s number P perimeter Q flux PO Poiseuille number R flow flow resistance R elec electrical resistance Re Reynolds number Re crit critical Reynolds number St Strouhal number T temperature U s characteristic velocity scale area-averaged velocity V voltage V 1 molar volume V i time-dependent voltage applied to a reservoir “well” W channel width STP standard temperature and pressure References Adamson, A.W., and Gast, A.P. (1997) Physical Chemistry of Surfaces, sixth edition, John Wiley & Sons, New York. Alarie, J.P., Jacobson, S.C., Culbertson, C.T. et al. (2000) “Effects of the Electric Field Distribution on Microchip Valving Performance,” Electrophoresis 21, pp. 100–106. Anderson, J.L., and Idol, W.K. (1985) “Electroosmosis Through Pores with Nonuniformly Charged Walls,” Chem. Eng. Commun. 38, pp. 93–106. Arkilic, E.B., Schmidt, M.A., and Breuer, K.S. (1997) “Gaseous Slip Flow in Long Microchannels,” J. MEMS 6, pp. 167–178. Baker, D.R. (1995) Capillary Electrophoresis, Techniques in Analytical Chemistry Series, John Wiley & Sons, New York. Bharadwaj, R., and Santiago, J.G. (2000) unpublished results, Stanford University, Stanford, CA. Bianchi, F., Ferrigno, R., and Girault, H.H. (2000) “Finite Element Simulation of an Electroosmotic- Driven Flow Division at a T-Junction of Microscale Dimensions,” Anal. Chem. 72, pp. 1987–1993. Bosse, M.A., and Arce, P. (2000) “Role of Joule Heating in Dispersive Mixing Effects in Electrophoretic Cells: Convective-Diffusive Transport Aspects,” Electrophoresis 21, pp. 1026–1033. Brandner, J., Fichtner, M., Schygulla, U., and Schubert, K. (2000) “Improving the Efficiency of Micro Heat Exchangers and Reactors,” in Proc. 4th Int. Conf. Microreaction Technology, AIChE, March 5–9, Atlanta, GA, pp. 244–249. Branebjerg, J., Fabius, B., and Gravesen, P. (1995) “Application of Miniature Analyzers: From Microfluidic Components to µTAS,” in Micro Total Analysis Systems, eds. A. van den Berg and P. Bergveld, Kluwer Academic, Dordrecht, pp. 141–151. Branebjerg, J., Gravesen, P., Krog, J.P., and Nielsen, C.R. (1996) “Fast Mixing by Lamination,” in Proc. 9th Annual Workshop of Micro Electro Mechanical Systems, February 11–15, San Diego, CA, pp. 441–446. Bridgman, P.W. (1923) “The Thermal Conductivity of Liquids Under Pressure,” Proc. Am. Acad. Arts Sci. 59, pp. 141–169. Brody, J.P., Yager, P., Goldstein, R.E., and Austin, R.H. (1996) “Biotechnology at Low Reynolds Numbers,” Biophys. J. 71, pp. 3430–3441. Burgreen, D., and Nakache, F.R. (1964) “ElectroKinetic Flow in Ultrafine Capillary Slits,” J. Phys. Chem. 68, pp. 1084–1091. Chen, Z., Milner, T.E., Dave, D., and Nelson, J.S. (1997) “Optical Doppler Tomographic Imaging of Fluid Flow Velocity in Highly Scattering Media,” Opt. Lett. 22, pp. 64–66. U © 2002 by CRC Press LLC N A Avogadro’s number P perimeter Q flux PO Poiseuille number R flow flow resistance R elec electrical resistance Re Reynolds number Re crit critical Reynolds number St Strouhal number T temperature U s characteristic velocity scale area-averaged velocity V voltage V 1 molar volume V i time-dependent voltage applied to a reservoir “well” W channel width STP standard temperature and pressure References Adamson, A.W., and Gast, A.P. (1997) Physical Chemistry of Surfaces, sixth edition, John Wiley & Sons, New York. Alarie, J.P., Jacobson, S.C., Culbertson, C.T. et al. (2000) “Effects of the Electric Field Distribution on Microchip Valving Performance,” Electrophoresis 21, pp. 100–106. Anderson, J.L., and Idol, W.K. (1985) “Electroosmosis Through Pores with Nonuniformly Charged Walls,” Chem. Eng. Commun. 38, pp. 93–106. Arkilic, E.B., Schmidt, M.A., and Breuer, K.S. (1997) “Gaseous Slip Flow in Long Microchannels,” J. MEMS 6, pp. 167–178. Baker, D.R. (1995) Capillary Electrophoresis, Techniques in Analytical Chemistry Series, John Wiley & Sons, New York. Bharadwaj, R., and Santiago, J.G. (2000) unpublished results, Stanford University, Stanford, CA. Bianchi, F., Ferrigno, R., and Girault, H.H. (2000) “Finite Element Simulation of an Electroosmotic- Driven Flow Division at a T-Junction of Microscale Dimensions,” Anal. Chem. 72, pp. 1987–1993. Bosse, M.A., and Arce, P. (2000) “Role of Joule Heating in Dispersive Mixing Effects in Electrophoretic Cells: Convective-Diffusive Transport Aspects,” Electrophoresis 21, pp. 1026–1033. Brandner, J., Fichtner, M., Schygulla, U., and Schubert, K. (2000) “Improving the Efficiency of Micro Heat Exchangers and Reactors,” in Proc. 4th Int. Conf. Microreaction Technology, AIChE, March 5–9, Atlanta, GA, pp. 244–249. Branebjerg, J., Fabius, B., and Gravesen, P. (1995) “Application of Miniature Analyzers: From Microfluidic Components to µTAS,” in Micro Total Analysis Systems, eds. A. van den Berg and P. Bergveld, Kluwer Academic, Dordrecht, pp. 141–151. Branebjerg, J., Gravesen, P., Krog, J.P., and Nielsen, C.R. (1996) “Fast Mixing by Lamination,” in Proc. 9th Annual Workshop of Micro Electro Mechanical Systems, February 11–15, San Diego, CA, pp. 441–446. Bridgman, P.W. (1923) “The Thermal Conductivity of Liquids Under Pressure,” Proc. Am. Acad. Arts Sci. 59, pp. 141–169. Brody, J.P., Yager, P., Goldstein, R.E., and Austin, R.H. (1996) “Biotechnology at Low Reynolds Numbers,” Biophys. J. 71, pp. 3430–3441. Burgreen, D., and Nakache, F.R. (1964) “ElectroKinetic Flow in Ultrafine Capillary Slits,” J. Phys. Chem. 68, pp. 1084–1091. Chen, Z., Milner, T.E., Dave, D., and Nelson, J.S. (1997) “Optical Doppler Tomographic Imaging of Fluid Flow Velocity in Highly Scattering Media,” Opt. Lett. 22, pp. 64–66. U © 2002 by CRC Press LLC 7 Burnett Simulations of Flows in Microdevices 7.1 Abstract 7.2 Introduction 7.3 History of Burnett Equations 7.4 Governing Equations 7.5 Wall-Boundary Conditions 7.6 Linearized Stability Analysis of Burnett Equations 7.7 Numerical Method 7.8 Numerical Simulations Application to Hypersonic Shock Structure • Application to Two-Dimensional Hypersonic Blunt Body Flow • Application to Axisymmetric Hypersonic Blunt Body Flow • Application to NACA 0012 Airfoil • Subsonic Flow in a Microchannel • Supersonic Flow in a Microchannel 7.9 Conclusions Nomenclature 7.1 Abstract In recent years, interest in computing gas flows at high Knudsen numbers in microdevices has been considerable. At low Knudsen numbers, models based on the solution of compressible Navier–Stokes equations with slip-boundary conditions are adequate. At high Knudsen numbers, either higher order (beyond Navier–Stokes) continuum equations or the particle methods such as direct simulation Monte Carlo (DSMC) are employed to compute the flows. Higher order continuum approximations are based on the Chapman–Enskog expansion of the Boltzmann equation (leading to Burnett and super-Burnett equations), or moment methods based on taking the moments of the Boltzmann equation with flow variables (leading to Grad’s 13-moment equations or Levermore’ moment equations, for example). In this chapter, the history of the Burnett equations and a variety of Burnett approximations (conventional Burnett equations, augmented Burnett equations and BGK–Burnett equations) are presented. The phys- ical and numerical issues related to these approximations are discussed. Traditionally, Burnett equations have been employed to compute the high-altitude, low-density hypersonic flows in the continuum– transition regime. Therefore, some Navier–Stokes and Burnett solutions are presented for hypersonic shock structure and blunt-body flows and rarefied subsonic airfoil flow to provide some perspective on the applicability and suitability of Burnett equations for computing flows at high Knudsen numbers. Calculations are then presented for both subsonic and supersonic flows in microchannels. These com- putations are compared with Navier–Stokes solutions with slip-boundary conditions and DSMC solutions. Ramesh K. Agarwal Wichita State University Keon-Young Yun Wichita State University © 2002 by CRC Press LLC 8 Molecular-Based Microfluidic Simulation Models 8.1 Abstract 8.2 Introduction 8.3 Gas Flows Molecular Magnitudes • An Overview of the Direct Simulation Monte Carlo Method • Limitations, Error Sources and Disadvantages of the DSMC Approach • Some DSMC-Based Gas Microflow Results • Recent Advances in the DSMC Method • DSMC Coupling with Continuum Equations • Boltzmann Equation Research • Hybrid Boltzmann/Continuum Simulation Methods • Lattice Boltzmann Methods 8.4 Liquid and Dense Gas Flows Electric Double Layer and Electrokinetic Effects • The Electro- Osmotic Flow • Molecular Dynamics Method • Treatment of Surfaces 8.5 Summary and Conclusions 8.1 Abstract This chapter concentrates on molecular-based numerical simulation methods for liquid and gas micro- flows. After a brief introduction of molecular magnitudes, a detailed coverage of the direct simulation Monte Carlo method for gas flows is presented. Brief descriptions of other simulation methodologies, such as the Boltzmann and lattice Boltzmann methods and the molecular dynamics method, are also given. Throughout the chapter, extensive references to books, research and review articles are supplied with the perspective of guiding the reader toward recent literature. 8.2 Introduction Simulation of microscale thermal fluidic transport is gaining importance due to the emerging technol- ogies of the 21st century, such as microelectromechanical systems (MEMS) and nanotechnologies. Min- iaturization of device scales, for the first time, has enabled the possibility of integration of sensing, computation, actuation, control, communication and power generation within the same microchip. The small size, light weight and high-durability of MEMS, combined with the mass-fabrication ability, result in low-cost systems with a wide variety of applications spanning from control systems to advanced energy systems to biological, medical and chemical applications. Despite these diverse prospects and fast growth Ali Beskok Texas A&M University © 2002 by CRC Press LLC 9 Lubrication in MEMS 9.1 Introduction 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, Kenneth S. Breuer Brown University . super-Burnett equations), or moment methods based on taking the moments of the Boltzmann equation with flow variables (leading to Grad’s 1 3- moment equations or Levermore’ moment equations, for example) communication and power generation within the same microchip. The small size, light weight and high-durability of MEMS, combined with the mass-fabrication ability, result in low-cost systems. 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