Preface ........................................................ xv
1 Introduction and Overview .................................... 1
1.1 Micro- and nanofluidics ................................. 1
1.2 Some micro- and nanofluidic devices ..................... 3
1.3 What is it about the nanoscale? ......................... 7
1.4 Nanotechnology ......................................... 11
1.5 What is a fluid? ....................................... 13
1.6 Historical perspectives ................................ 14
1.6.1 Fluid mechanics ................................. 15
1.6.2 Heat and mass transfer .......................... 17
1.6.3 Electrokinetic phenomena ........................ 19
1.7 The thermal sciences ................................... 20
1.8 Electrostatics ......................................... 23
1.9 Electrolyte solutions .................................. 25
1.10 The electrical double layer ............................ 26
1.11 Colloidal systems ...................................... 29
1.12 Molecular biology ...................................... 32
1.13 The convergence of molecular biology and engineering ... 34
1.14 Design of micro- and nanofluidic devices ............... 35
1.15 Unit systems ........................................... 37
1.16 A word about notation .................................. 37
1.17 Chapter summary ........................................ 38
2 Preparatory Concepts ........................................ 40
2.1 Introduction ........................................... 40
2.2 Important constitutive laws ............................ 41
2.3 Determining transport properties ....................... 45
2.3.1 Viscosity ....................................... 45
2.3.2 Diffusion coefficient ........................... 48
2.3.3 Thermal conductivity ............................ 52
2.3.4 Electrical permittivity ......................... 54
2.3.5 Surface tension and wettability ................. 55
2.4 Classification of fluid flows .......................... 59
2.5 Elements of thermodynamics ............................. 62
2.6 The nature of frictional losses in channels and pipes .. 68
2.7 Chapter summary ........................................ 70
3 The Governing Equations for an Electrically Conducting
Fluid ....................................................... 74
3.1 Introduction ........................................... 74
3.2 The continuum approximation and its limitations ........ 75
3.3 Kinematics ............................................. 77
3.4 Surface and body forces ................................ 83
3.5 The continuity equation ................................ 87
3.6 The Navier-Stokes equations ............................ 88
3.7 Mass transport ......................................... 93
3.7.1 Definitions ..................................... 93
3.7.2 Governing equation .............................. 97
3.8 Electrostatics ........................................ 100
3.9 Energy transport ...................................... 102
3.10 Two-dimensional, steady, and incompressible flow ...... 106
3.11 Boundary and initial conditions ....................... 106
3.11.1 Velocity boundary conditions .................. 107
3.11.2 Mass transfer boundary conditions ............. 113
3.11.3 Electrostatics boundary conditions ............ 114
3.11.4 Temperature boundary conditions ............... 116
3.11.5 Other boundary conditions ..................... 117
3.12 Dimensional analysis and similarity ................... 117
3.13 Fluid, electrostatics, and heat and mass transfer
analogies ............................................. 123
3.13.1 Mole fraction and temperature similarity ...... 123
3.13.2 Velocity and electrical potential similarity .. 125
3.14 Other stress-strain relationships ..................... 126
3.15 Mathematical character of partial differential
equations ............................................. 128
3.15.1 Introduction ................................... 128
3.15.2 Mathematical classification of second-order
partial differential equations ................. 128
3.15.3 Characteristic curves .......................... 129
3.15.4 Boundary and initial conditions ................ 130
3.15.5 Classification of the governing equations of
micro- and nanofluidics ........................ 131
3.16 Well-posed problems ................................... 131
3.17 The role of fabrication, experiments, and theory in
micro- and nanofluidics ............................... 132
3.18 Chapter summary ....................................... 134
4 The Essentials of Viscous Flow ............................. 140
4.1 Introduction .......................................... 140
4.2 The structure of flow in a pipe or channel ............ 141
4.3 Poiseuille flow in a pipe or channel .................. 143
4.4 The velocity in slip flow ............................. 146
4.4.1 Gases .......................................... 146
4.4.2 Liquids ........................................ 147
4.5 Flow in a thin film under gravity ..................... 148
4.6 The boundary layer on a flat plate .................... 150
4.7 Fully developed suction flows ......................... 155
4.8 Developing suction flows .............................. 158
4.9 The lubrication approximation ......................... 162
4.10 A surface tension-driven flow ......................... 166
4.11 Stokes flow past a sphere ............................. 169
4.12 Sedimentation of a solid particle ..................... 172
4.13 A simple model for blood flow ......................... 173
4.14 Chapter summary ....................................... 174
5 Heat and Mass Transfer Phenomena in Channels and Tubes ..... 180
5.1 Introduction .......................................... 180
5.2 One-dimensional temperature distributions in channel
flow .................................................. 181
5.3 Thermal and mass transfer entrance regions ............ 184
5.4 The temperature distribution in fully developed tube
flow .................................................. 189
5.5 The Graetz problem for a channel ...................... 189
5.6 Mass transfer in thin films ........................... 192
5.7 Classical Taylor-Aris dispersion ...................... 194
5.8 The stochastic nature of diffusion: Brownian motion ... 199
5.9 Unsteady mass transport in uncharged membranes ........ 201
5.10 Temperature and concentration boundary layers ......... 205
5.11 Chapter summary ....................................... 207
6 Introduction to Electrostatics ............................. 213
6.1 Introduction .......................................... 213
6.2 Coulomb's law: The electric field ..................... 214
6.3 The electric field due to an isolated large flat
plate ................................................. 216
6.4 Gauss's law ........................................... 218
6.5 The electric potential ................................ 219
6.6 The electric dipole and polar molecules ............... 221
6.7 Poisson's equation .................................... 222
6.8 Current and current density ........................... 225
6.9 Maxwell's equations ................................... 226
6.10 Chapter summary ....................................... 227
7 Elements of Electrochemistry and the Electrical Double
Layer ...................................................... 230
7.1 Introduction .......................................... 230
7.2 The structure of water and ionic species .............. 231
7.3 Chemical bonds in biology and chemistry ............... 233
7.4 Hydration of ions ..................................... 234
7.5 Chemical potential .................................... 236
7.6 The Gibbs function and chemical equilibrium ........... 240
7.7 Electrochemical potential ............................. 243
7.8 Acids, bases, and electrolytes ........................ 244
7.9 Site-binding models of the silica surface ............. 246
7.10 Polymer surfaces ...................................... 249
7.11 Qualitative description of the electrical double
layer ................................................. 251
7.12 Electrolyte and potential distribution in the
electrical double layer ............................... 253
7.13 Multivalent asymmetric mixtures ....................... 259
7.14 The f potential and surface charge density: Putting
it all together ....................................... 260
7.14.1 The classical liquid-side view for
a symmetric electrolyte ........................ 260
7.14.2 The solid-side view and connection to the
liquid side .................................... 262
7.15 The electrical double layer on a cylinder ............. 265
7.16 The electrical double layer on a sphere ............... 266
7.17 Electrical conductivity in an electrolyte solution .... 267
7.18 Semi-permeable membranes .............................. 270
7.19 The Derjaguin approximation ........................... 275
7.20 Chapter summary ....................................... 278
8 Elements of Molecular and Cell Biology ..................... 283
8.1 Introduction .......................................... 283
8.2 Nucleic acids and polysaccharides ..................... 285
8.3 Proteins .............................................. 287
8.3.1 Protein function ............................... 288
8.3.2 Protein structure .............................. 289
8.3.3 Some common proteins ........................... 292
8.3.4 Few polypeptide chains are useful .............. 295
8.4 Protein binding ....................................... 295
8.5 Cells ................................................. 298
8.6 The cell membrane ..................................... 300
8.7 Membrane transport and ion channels ................... 301
8.8 Chapter summary ....................................... 304
9 Electrokinetic Phenomena ................................... 306
9.1 Introduction .......................................... 306
9.2 Electro-osmosis ....................................... 307
9.2.1 The relationship between velocity and
potential ...................................... 307
9.2.2 The Debye-Hьckel approximation reviewed ........ 312
9.2.3 Another similarity revealed .................... 312
9.2.4 Asymptotic solution for binary electrolytes
of arbitrary valence ........................... 313
9.2.5 Walls with different Ј potentials .............. 316
9.2.6 Species velocities in electro-osmotic flow:
Electromigration ............................... 318
9.2.7 Current and current density in electro-
osmotic flow .................................. 320
9.2.8 Electro-osmotic flow in an annulus ............. 322
9.2.9 Electro-osmotic flow in nozzles and diffusers .. 324
9.2.10 Dispersion in electro-osmotic flow ............. 328
9.3 Electrophoresis: Single particles ..................... 331
9.3.1 Introduction ................................... 331
9.3.2 Electrophoretic mobility ....................... 332
9.3.3 Henry's solution ............................... 334
9.3.4 The full nonlinear problem ..................... 336
9.4 Streaming potential ................................... 338
9.5 Sedimentation potential ............................... 341
9.6 Joule heating ......................................... 342
9.7 Chapter summary ....................................... 344
10 Essential Numerical Methods ................................ 348
10.1 Introduction .......................................... 348
10.2 Types of errors ....................................... 350
10.3 Taylor series ......................................... 351
10.4 Zeros of functions .................................... 353
10.4.1 Numerical methods ................................... 353
10.4.2 Polynomials ......................................... 358
10.5 Interpolation ......................................... 359
10.5.1 Linear interpolation ........................... 360
10.5.2 The difference table ........................... 361
10.5.3 Lagrangian polynomial interpolation ............ 362
10.5.4 Newton interpolation formulas .................. 363
10.5.5 Matlab interpolation functions ................. 365
10.5.6 Cubic spline interpolation ..................... 366
10.6 Curve fitting ......................................... 370
10.7 Numerical differentiation ............................. 373
10.7.1 Derivatives from Taylor series ................. 373
10.7.2 A more accurate forward formula for the first
derivative ..................................... 375
10.8 Numerical integration ................................. 376
10.8.1 The trapezoidal rule ........................... 377
10.8.2 Simpson's rules ................................ 380
10.8.3 Matlab integration functions ................... 382
10.8.4 The indefinite integral ........................ 382
10.8.5 Other formulas ................................. 383
10.8.6 Grid (mesh) size ............................... 383
10.8.7 Singularities .................................. 384
10.9 Solution of linear systems ........................... 386
10.9.1 Solving sets of linear equations in Matlab .... 389
10.9.2 Iterative solution to linear systems ........... 390
10.9.3 Tridiagonal systems ............................ 393
10.9.4 Ill-conditioning and stability ................. 396
10.10 Solution of boundary value problems .................. 398
10.10.1 Introduction .................................. 398
10.10.2 Linear equations .............................. 399
10.10.3 Nonlinear equations ........................... 403
10.10.4 Systems of ordinary differential equations .... 405
10.10.5 Derivative boundary conditions ................ 407
10.10.6 Convergence tests and Richardson
extrapolation ................................. 409
10.10.7 Solving boundary value problems with Matlab
functions ..................................... 410
10.11 Solution of initial value problems ................... 411
10.11.1 Introduction .................................. 411
10.11.2 Taylor series method .......................... 413
10.11.3 Euler methods ................................. 414
10.11.4 Runge-Kutta methods ........................... 416
10.11.5 Adams-Moulton methods ......................... 419
10.11.6 Symplectic integrators ........................ 419
10.11.7 Stiff equations and stability ................. 424
10.11.8 Solving initial value problems using Matlab
functions ..................................... 428
10.12 Numerical solution of the PNP system ................. 428
10.13 Partial differential equations ....................... 430
10.13.1 Elliptic equations ............................ 431
10.13.2 Parabolic equations ........................... 432
10.13.3 The Matlab PDE solver ......................... 435
10.14 Verification and validation of numerical solutions ... 435
10.15 Chapter summary ...................................... 438
11 Molecular Simulations ...................................... 447
11.1 Introduction .......................................... 447
11.2 The molecular world ................................... 449
11.3 Ensembles ............................................. 451
11.4 The potentials ........................................ 451
11.5 Using the Lennard-Jones potential ..................... 453
11.6 Molecular models for water ............................ 456
11.7 Periodic boundary conditions .......................... 457
11.8 The Ewald sum ......................................... 460
11.9 Numerical issues ...................................... 463
11.9.1 Time integration ............................... 463
11.9.2 Truncation of interactions ..................... 464
11.9.3 Boundary conditions ............................ 465
11.10 Postprocessing ....................................... 465
11.11 Nonequilibrium molecular dynamics .................... 467
11.11.1 Introduction .................................. 467
11.11.2 Poiseuille flow ............................... 468
11.11.3 Electro-osmotic flow .......................... 469
11.12 Molecular dynamics packages .......................... 471
11.12.1 Introduction .................................. 471
11.12.2 What MD/NEMD simulators do .................... 471
11.13 Summary .............................................. 472
12 Applications ............................................... 475
12.1 Introduction .......................................... 475
12.2 DNA transport ......................................... 476
12.2.1 How does DNA move? ............................. 477
12.2.2 Mathematical model ............................. 479
12.2.3 Results ........................................ 481
12.2.4 DNA current .................................... 482
12.2.5 Comparison with experiment ..................... 483
12.3 Development of an artificial kidney ................... 484
12.3.1 Background ..................................... 484
12.3.2 The nanopore membrane for filtration ........... 486
12.3.3 Hindered transport ............................. 487
12.4 Biochemical sensing ................................... 491
12.4.1 Introduction ................................... 491
12.4.2 What is a biosensor? ........................... 492
12.4.3 Receptor-based classification of biosensors .... 493
12.4.4 Transducer-based classification of biosensors .. 494
12.4.5 Evaluation of biosensor performance ............ 495
12.4.6 Nanopores and nanopore membranes for
biochemical sensing ............................ 496
12.5 Chapter summary ....................................... 498
Appendix A Matched Asymptotic Expansions ..................... 501
A.l Introduction .......................................... 501
A.2 Terminology ........................................... 501
A.3 Asymptotic sequences and expansions ................... 502
A.4 Regular perturbations ................................. 503
A.5 Singular perturbations ................................ 504
Appendix В Vector Operations in Curvilinear Coordinates ...... 508
B. 1 Cylindrical coordinates ............................... 508
B.2 Spherical coordinates .................................. 508
B.3 Rectangular coordinates ............................... 509
Appendix C WebSites ........................................... 510
C.l Fluid dynamics and micro-and nanofluidics .............. 510
C.2 General nanotechnology ................................ 511
C.3 Wikipedia ............................................. 511
Appendix D A Semester Course Syllabus ........................ 512
Bibliography .................................................. 515
Index ......................................................... 533
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