Advances in Multiphysics Simulation and Experimental Testing of MEMS

Contents......Page 12
Preface......Page 6
1. Challenges in Modeling Liquid and Gas Flows in Micro/Nano Devices M. Gad-el-Hak......Page 14
1.1. Introduction......Page 15
1.2. Fluid Mechanics Issues......Page 17
1.3. Fluid Modeling......Page 19
1.4. Gas Flows......Page 22
1.5. Liquid Flows......Page 25
1.6. Molecular Dynamics Simulations......Page 29
1.7. A Typical MD Result......Page 31
1.8. Hybrid Methods......Page 37
1.9. Surface Phenomena......Page 39
1.10. Conclusions......Page 44
References......Page 46
2. Using the Kinetic Equations for MEMS and NEMS C. Cercignani, A. Frezzotti and S. Lorenzani......Page 50
2.1. Introduction......Page 51
2.2. The Boltzmann Equation......Page 52
2.3. The Linearized Boltzmann Equation and the BGK Model......Page 55
2.4. The Macroscopic Balance Equations......Page 56
2.5. Boundary Conditions......Page 60
2.6. The Modified Reynolds Equation......Page 62
2.7. The Reynolds Equation and the Poiseuille-Couette Problem......Page 64
2.8. The Generalized Reynolds Equation for Unequal Walls......Page 68
2.9. A Kinetic Approach for the Evaluation of Damping in MEMS......Page 75
2.10. Kinetic Theory Extension to Dense Fluids......Page 80
2.10.1. The Mathematical Model......Page 82
2.10.2. Fluid-Wall Interaction and Boundary Conditions......Page 85
2.11. Numerical Results......Page 87
References......Page 90
3. Applying the Direct Simulation Monte Carlo (DSMC) Method to Gas-Filled MEMS Devices M. A. Gallis......Page 94
3.1. Introduction......Page 95
3.2. Basic Method......Page 97
3.2.1. Statistical Error......Page 100
3.2.4. Cell Size......Page 101
3.2.5. Time Step......Page 102
3.2.6. Comprehensive Study of Discretization Error......Page 103
3.3.1. Theoretical Results......Page 104
3.3.2. Fourier and Couette Flow......Page 106
3.3.3. DSMC Results......Page 107
3.4. Comparison to Moment-Hierarchy Theory......Page 110
3.4.1. Theoretical Results......Page 111
3.4.2. Simulation Results......Page 112
3.5.1. Heat Transfer at Arbitrary Knudsen Numbers......Page 115
3.5.2. Heat Transfer in a Microgap......Page 116
3.5.3. Thermal Actuation......Page 119
3.5.4. Heat Transfer from a Microbeam to the Substrate......Page 120
3.5.5. Gas Damping......Page 122
3.5.6. Gas Damping of a Cantilevered Microbeam......Page 124
3.5.7. Thermally Driven Flows......Page 125
3.6. Conclusions......Page 128
References......Page 129
4. New Approaches for the Simulation of Micro-Fluidics in MEMS T. Y. Ng, H. Li, L. S. Pan, D. Xu and K. Y. Lam......Page 134
4.1.1. Micro-Fluidic Gas Flows— Introduction and Review......Page 135
4.1.2. Micro-Fluidic Liquid Flows — Introduction and Review......Page 137
4.2.1. The Molecular Block Direct Simulation Monte Carlo (MB-DSMC) Method......Page 139
4.2.1.1. Basic Model and Assumptions......Page 140
4.2.1.2. A Discussion on the Macro Quantities......Page 142
4.2.1.3. Molecular Block DSMC Algorithm......Page 143
4.2.2.1. Decrease in Statistical Error......Page 145
4.2.2.2. Consistency of the Mean Free Path......Page 147
4.2.2.3. Consistency of the Dynamic Viscosity......Page 149
4.2.2.4. Validation of Temperature Field Simulation......Page 150
4.2.3. Conclusion......Page 152
4.3.1. A Modified One-Equation Model for Micro-Scale Liquid Flow......Page 153
4.3.2. Simulation Results for Liquid Flow in Micro-Tubes......Page 155
4.3.3. Conclusions......Page 161
References......Page 163
5. Evaluating Gas Damping in MEMS Using Fast Integral Equation Solvers A. Frangi, W. Ye and J. White......Page 166
5.1. Introduction......Page 167
5.2.1. Governing Equations......Page 168
5.2.2. Integral Formulation......Page 169
5.2.3. Null Space Problem......Page 170
5.2.4. Numerical Implementation......Page 171
5.2.5. Fast Solvers......Page 172
5.3. Extension to the Slip Flow Regime and Validation......Page 174
5.3.1.1. Biaxial Accelerometer......Page 176
5.3.1.2. Tang Resonator......Page 181
5.4. Extension to High Frequency Oscillatory Flow......Page 183
5.5. Extension to Low Pressures by Means of the Corrected Viscosity Approach......Page 188
References......Page 191
6.1. Introduction......Page 196
6.2.1. Damping Mechanisms and their Parameters......Page 197
6.2.2. Measurement System Requirements......Page 199
6.2.3. Methods for Quality Factor Extraction......Page 200
6.2.3.1. Frequency Domain Analysis......Page 201
6.2.3.2. Transient Response Analysis......Page 204
6.2.4. Vibration Excitation Techniques......Page 205
6.3.1. General Features of Optical Measurement Techniques......Page 208
6.3.2.1. Image Processing Techniques......Page 210
6.3.2.2. Knife-Edge Techniques......Page 215
6.3.2.3. Grating Interferometry......Page 216
6.3.3. Damping Measurements of Out-of-Plane and Torsional Vibrations......Page 218
6.3.3.1. Optical Beam Deflection Techniques......Page 219
6.3.3.2. Interferometry Techniques......Page 220
6.3.3.3. Other Techniques for Out-of-Plane Vibration Measurements......Page 227
6.4.1. Electrical Techniques......Page 228
6.4.1.1. Damping by Electronics......Page 229
6.4.1.2. Standard Electrical Techniques for Quality Factor Measurements......Page 230
6.4.1.3. Specific Electrical Techniques for Capacitive MEMS Devices......Page 232
6.4.1.4. Architectures for Integrated Quality Factor Measurement......Page 234
6.4.2. Measurement Techniques for Nanoresonators......Page 236
6.4.2.1. Electron Microscopy Techniques......Page 237
6.4.2.3. Electromotive Measurements......Page 238
References......Page 239
7. Nonlinear Dynamics of Electrostatically Actuated MEMS S. K. De and N. R. Aluru......Page 248
7.1. Introduction......Page 249
7.2. Theory of MEMS Dynamics......Page 251
7.2.1. Mechanical Analysis......Page 252
7.2.3. Fluidic Analysis......Page 253
7.2.3.1. Compressible Reynold’s Squeeze Film Equation (CRSFE)......Page 254
7.2.3.2. Compressible Navier–Stokes Equations (CNSE)......Page 256
7.2.4. Electrostatic Analysis......Page 258
7.3. Dynamic Analysis in the Absence of Damping......Page 259
7.3.1. Numerical Simulations in the Absence of Damping......Page 261
7.4. Dynamic Analysis in the Presence of Fluid Damping......Page 265
7.4.1. Numerical Simulations in the Presence of Fluid Damping......Page 266
7.4.2.1. DC and AC Symmetry Breakings......Page 273
7.4.2.2. Complex Oscillations: M-cycles......Page 276
7.4.2.3. Period Doubling and Chaos: 2nM-cycles......Page 277
7.4.2.4. U(Universal)-Sequence: K-cycles......Page 279
7.4.2.5. UM-Sequence: KM-cycles......Page 280
7.5. Dynamic Analysis in the Presence of Fluid and Thermal Damping......Page 284
7.5.1. Classical Theory of Thermoelastic Damping......Page 287
7.5.2. Modified Theory of Thermoelastic Damping......Page 288
7.5.3. Numerical Simulations in the Presence of Fluid and Thermal Damping......Page 291
7.6. Conclusions......Page 296
References......Page 297
8. Coupled Deformation Analysis of Thin MEMS Plates S. Mukherjee and S. Telukunta......Page 300
8.1. Introduction......Page 301
8.2.1.1. Usual BIE — Indirect Formulation......Page 305
8.2.2. BIEs in Infinite Region Containing Two Thin Conducting Plates......Page 306
8.2.2.2. Gradient BIE — Source Point Approaching a Plate Surface s+1......Page 307
8.2.3. Boundary Integral Equations in the Lagrangian Formulation......Page 309
8.2.3.2. Lagrangian Version of the Gradient BIE......Page 310
8.2.3.3. Two Plates Close Together......Page 313
8.3.1. The Model......Page 315
8.3.2. FEM Model for Plates with Immovable Edges......Page 316
8.4.1. Lagrangian Relaxation Scheme for the Coupled Problem......Page 318
8.4.2. Newton Scheme for Solving the Coupled Problem......Page 320
8.4.2.1. Residuals and Their Gradients......Page 321
8.4.2.2. The Newton Algorithm......Page 324
8.5.1. Boundary Integral Equations for Two Plates Very Close Together......Page 325
8.5.2. Non-Linear Finite Element Analysis......Page 326
8.5.3. BEM/FEM Coupling......Page 327
8.6.1.1. BEM for Region Exterior to a Thin Flat Plate......Page 328
8.6.1.2. Lagrangian BEM for Region Exterior to a Curved Plate......Page 330
8.6.2. MEMS Plates......Page 331
8.7. Pull-In Analysis......Page 336
8.8. Discussion......Page 337
References......Page 339
9. Pull-In Instability in Electrostatically Actuated MEMS due to Coulomb and Casimir Forces R. C. Batra, M. Porfiri and D. Spinello......Page 342
9.1. Introduction......Page 343
9.2.2. Mechanical Deformations......Page 347
9.2.3. Electrostatic Force......Page 349
9.3. The Casimir Effect......Page 351
9.4.1. Mathematical Model of a Micromembrane......Page 355
9.4.3. Discrete Non-Linear Formulation......Page 358
9.4.4. Pseudo-Arc-Length Continuation Method......Page 361
9.4.5. One Degree-of-Freedom Model......Page 363
9.4.6.1. Rectangular MEMS......Page 365
9.4.6.2. Circular Disk......Page 370
9.4.6.3. Elliptic Disk......Page 372
9.4.7. Pull-In Instability and Symmetry Breaking in an Annular Circular Disk......Page 375
9.4.8.1. Effect on Pull-In Parameters......Page 377
9.4.8.2. Effect on Symmetry Breaking......Page 379
9.5. Summary......Page 381
References......Page 382
10. Numerical Simulation of BioMEMS with Dielectrophoresis G. R. Liu and C. X. Song......Page 388
10.1. Introduction......Page 389
10.2. The Theoretical Background of BioMEMS with DEP......Page 390
10.3.1. Interpolation Formulations......Page 392
10.3.2. Gradient Smoothing......Page 394
10.3.3. Variational Form......Page 396
10.4. Results and Discussion......Page 397
10.4.1. Simulation of the DEP Array......Page 398
10.4.1.1. Case 1: Linear Potential Change in the Gap......Page 399
10.4.2.1. Case 3: Study of the Traveling Wave DEP Array......Page 404
References......Page 410
11. Continuous Modeling of Multi-Physics Problems of Microsystems for Topology Optimization G. K. Ananthasuresh......Page 412
11.1. Introduction......Page 413
11.2. Continuous Interpolation of Material Under Mechanical Loads......Page 416
11.3. Electro-Thermal Microactuation......Page 419
11.4. Electrostatic Microactuation......Page 422
11.5. Miscellaneous Microsystems Problems and Emerging Techniques......Page 426
11.5.1. Emerging Techniques......Page 427
11.6. Continuous Modeling of a Microfabrication Process......Page 428
11.7. Continuous Modeling of Protein Sequences......Page 431
Acknowledgments......Page 434
References......Page 435
12. Mechanical Characterization of Polysilicon at the Micro- Scale Through On-Chip Tests A. Corigliano, F. Cacchione, A. Frangi, S. Zerbini and M. Ferrera......Page 440
12.1. Introduction......Page 441
12.2.1. Fabrication Process......Page 443
12.2.2. Test Structures and Data Reduction Procedure......Page 445
12.3. Weibull Approach......Page 446
12.4.1.2. Data Reduction Procedure......Page 450
12.4.2.1. General Description......Page 452
12.4.2.2. Data Reduction Procedure......Page 453
12.4.3.1. Young’s Modulus......Page 455
12.4.3.2. Rupture Strength......Page 456
12.5.1. General Description......Page 458
12.5.3. Experimental Results......Page 462
12.6. Conclusions......Page 463
Acknowledgments......Page 464
References......Page 465
13. Nano-Scale Testing of Nanowires and Carbon Nanotubes Using a Micro-Electro-Mechanical System H. D. Espinosa, Y. Zhu, B. Peng and O. Loh......Page 468
13.2.1. Dynamic Vibration......Page 469
13.2.2. Bending......Page 470
13.2.3. Tensile Tests......Page 471
13.3. A MEMS-Based Material Testing Stage......Page 472
13.3.1. Device Description......Page 473
13.3.3. Analytical Modeling of the Thermal Actuator......Page 474
13.3.3.1. Electrothermal Model......Page 475
13.3.3.2. Thermomechanical Model......Page 478
13.3.3.3. Thermomechanical Response of Entire Loading Device......Page 481
13.3.4. Multiphysics FEA of the Thermal Actuator......Page 482
13.3.6. Evaluating the Analytic and Finite Element Models......Page 484
13.3.7. Load Sensor......Page 485
13.4. Design Criteria......Page 489
13.5.1. Sample Preparation......Page 492
13.5.2.2. Tensile Tests of Nanowires......Page 493
13.5.2.3. Tensile Tests of Carbon Nanotubes and the Effects of Irradiation......Page 494
References......Page 498
Editorial Board......Page 504