Preface ........................................................ xv
Nomenclature ................................................. xvii
Introduction .................................................... 1
1 Kinematics, Conservation Equations, and Boundary Conditions
for Incompressible Flow .................................... 6
1.1 Fluid statics .............................................. 6
1.2 Kinematics of a fluid velocity field ....................... 7
1.2.1 Important geometric definitions ..................... 7
1.2.2 Strain rate and rotation rate tensors .............. 10
1.3 Governing equations for incompressible flow ............... 13
1.3.1 Conservation of mass: continuity equation .......... 13
1.3.2 Conservation of momentum: the Navier-Stokes
equations .......................................... 14
1.4 Constitutive relations .................................... 17
1.4.1 Relation between strain rate and stress ............ 17
1.4.2 Non-Newtonian fluids ............................... 19
1.5 Surface tension ........................................... 20
1.5.1 Definition of surface tension and interfacial
energy ............................................. 20
1.5.2 Young-Laplace equation ............................. 20
1.5.3 Contact angle ...................................... 21
1.5.4 Capillary height ................................... 22
1.5.5 Dynamic contact angle .............................. 24
1.6 Velocity and stress boundary conditions at interfaces ..... 24
1.6.1 Kinematic boundary condition for continuity of
normal velocity .................................... 24
1.6.2 Dynamic boundary condition for continuity of
tangential velocity ................................ 25
1.6.3 Dynamic boundary conditions for stresses ........... 26
1.6.4 The physics of the tangential velocity boundary
condition .......................................... 30
1.7 Solving the governing equations ........................... 32
1.8 Flow regimes ......................................... 33
1.9 A word on terminology and the microfluidics
literature ........................................... 33
1.10 Summary .............................................. 34
1.11 Supplementary reading ................................ 36
1.12 Exercises ............................................ 36
2 Unidirectional Flow ....................................... 41
2.1 Steady pressure- and boundary-driven flow through long
channels .................................................. 41
2.1.1 Couetteflow ........................................ 41
2.1.2 Poiseuille flow .................................... 46
2.2 Startup and development of unidirectional flows ........... 49
2.3 Summary ................................................... 50
2.4 Supplementary reading ..................................... 51
2.5 Exercises ................................................. 51
3 Hydraulic Circuit Analysis ................................ 60
3.1 Hydraulic circuit analysis ................................ 60
3.2 Hydraulic circuit equivalents for fluid flow in
microchannels ............................................. 62
3.2.1 Analytic representation of sinusoidal pressures
and flow rates ..................................... 68
3.2.2 Hydraulic impedance ................................ 69
3.2.3 Hydraulic circuit relations ........................ 70
3.2.4 Series and parallel component rules ................ 70
3.3 Solution techniques ....................................... 72
3.4 Summary ................................................... 74
3.5 Supplementary reading ..................................... 75
3.6 Exercises ................................................. 75
4 Passive Scalar Transport: Dispersion, Patterning, and
Mixing .................................................... 79
4.1 Passive scalar transport equation ......................... 80
4.1.1 Scalar fluxes and constitutive properties .......... 80
4.1.2 Scalar conservation equation ....................... 80
4.2 Physics of mixing ......................................... 82
4.3 Measuring and quantifying mixing and related parameters ... 84
4.4 The low-Reynolds-number, high-Peclet-number limit ......... 87
4.4.1 The high-Peclet-number limit ....................... 87
4.4.2 The low-Reynolds-number limit ...................... 87
4.5 Laminar flow patterning in microdevices ................... 88
4.6 Taylor-Aris dispersion .................................... 89
4.7 Summary ................................................... 91
4.8 Supplementary reading ..................................... 92
4.9 Exercises ................................................. 92
5 Electrostatics and Electrodynamics ........................ 97
5.1 Electrostatics in matter .................................. 97
5.1.1 Electrical potential and electric field ............ 97
5.1.2 Coulomb's law, Gauss's law for electricity in
a material, curl of electric field ................. 98
5.1.3 Polarization of matter and electric permittivity .. 100
5.1.4 Material, frequency, and electric-field
dependence of electrical permittivity ............. 102
5.1.5 Poisson and Laplace equations ..................... 104
5.1.6 Classification of material types .................. 105
5.1.7 Electrostatic boundary conditions ................. 105
5.1.8 Solution of electrostatic equations ............... 107
5.1.9 Maxwell stress tensor ............................. 107
5.1.10 Effects of electrostatic fields on multipoles ..... 108
5.2 Electrodynamics .......................................... 109
5.2.1 Charge conservation equation ...................... 110
5.2.2 Electrodynamic boundary conditions ................ 110
5.2.3 Field lines at substrate walls .................... 112
5.3 Analytic representations of electrodynamic quantities:
complex permittivity and conductivity .................... 112
5.3.1 Complex description of dielectric loss ............ 115
5.4 Electrical circuits ...................................... 116
5.4.1 Components and properties ......................... 117
5.4.2 Electrical impedance .............................. 119
5.4.3 Circuit relations ................................. 119
5.4.4 Series and parallel component rules ............... 120
5.5 Equivalent circuits for current in electrolyte-filled
microchannels ............................................ 122
5.5.1 Electrical circuit equivalents of hydraulic
components ........................................ 122
5.6 Summary .................................................. 126
5.7 Supplementary reading .................................... 127
5.8 Exercises ................................................ 127
6 Electroosmosis ........................................... 131
6.1 Matched asymptotics in electroosmotic flow ............... 132
6.2 Integral analysis of Coulomb forces on the EDL ........... 132
6.3 Solving the Navier-Stokes equations for electroosmotic
flow in the thin-EDL limit ............................... 135
6.3.1 Outer solution .................................... 136
6.3.2 Replacing the EDL with an effective slip
boundary condition ................................ 136
6.3.3 Replacing the Navier-Stokes equations with the
Laplace equation: flow-current similitude ......... 137
6.3.4 Reconciling the no-slip condition with
irrotational flow ................................. 138
6.4 Electroosmotic mobility and the electrokinetic
potential ................................................ 138
6.4.1 Electrokinetic coupling matrix representation of
electroosmosis .................................... 140
6.5 Electrokinetic pumps ..................................... 140
6.5.1 A planar electrokinetic pump ...................... 140
6.5.2 Types of electrokinetic pumps ..................... 143
6.6 Summary .................................................. 145
6.7 Supplementary reading .................................... 145
6.8 Exercises ................................................ 146
7 Potential Fluid Flow ..................................... 153
7.1 Approach for finding potential flow solutions to the
Navier-Stokes equations .................................. 153
7.2 Laplace equation for velocity potential and stream
function ................................................. 154
7.2.1 Laplace equation for the velocity potential ....... 154
7.2.2 No-slip condition ................................. 156
7.3 Potential flows with plane symmetry ...................... 156
7.3.1 Complex algebra and its use in plane-symmetric
potential flow .................................... 157
7.3.2 Monopolar flow: plane-symmetric (line) source
with volume outflow per unit depth Λ .............. 160
7.3.3 Plane-symmetric vortex with counterclockwise
circulation per unit depth Г ...................... 163
7.3.4 Dipolar flow: plane-symmetric doublet with
dipole moment к ................................... 165
7.3.5 Uniform flow with speed U ......................... 168
7.3.6 Flow around a corner .............................. 170
7.3.7 Flow over a circular cylinder ..................... 171
7.3.8 Conformal mapping ................................. 171
7.3.4 Potential flow in axisymmetric systems in
spherical coordinates ............................. 172
7.5 Summary .................................................. 173
7.6 Supplementary reading .................................... 173
7.7 Exercises ................................................ 174
8 Stokes Flow .............................................. 178
8.1 Stokes flow equation ..................................... 178
8.1.1 Different forms of the Stokes flow equations ...... 179
8.1.2 Analytical versus numerical solutions of the
Stokes flow equations ............................. 180
8.2 Bounded Stokes flows ..................................... 180
8.2.1 Hele-Shaw flows ................................... 181
8.2.2 Numerical solution of general bounded Stokes
flow problems ..................................... 182
8.3 Unbounded Stokes flows ................................... 182
8.3.1 Stokes flow over a sphere in an infinite domain ... 182
8.3.2 General solution for Stokes flow over a sphere
in an infinite domain ............................. 187
8.3.3 Flow over prolate ellipsoids ...................... 188
8.3.4 Stokes flow over particles in finite domains ...... 189
8.3.5 Stokes flow over multiple particles ............... 189
8.4 Micro-PIV ................................................ 189
8.4.1 Deterministic particle lag ........................ 191
8.4.2 Brownian motion ................................... 191
8.5 Summary .................................................. 191
8.6 Supplementary reading .................................... 192
8.7 Exercises ................................................ 193
9 The Diffuse Structure of the Electrical Double Layer ..... 199
9.1 The Gouy-Chapman EDL ..................................... 199
9.1.1 Boltzmann statistics for ideal solutions of ions .. 200
9.1.2 Ion distributions and potential: Boltzmann
relation .......................................... 201
9.1.3 Ion distributions and potential: Poisson-
Boltzmann equation ................................ 202
9.1.4 Simplified forms of the nonlinear Poisson-
Boltzmann equation ................................ 203
9.1.5 Solutions of the Poisson-Boltzmann equation ....... 204
9.2 Fluid flow in the Gouy-Chapman EDL ....................... 208
9.3 Convective surface conductivity .......................... 210
9.4 Accuracy of the ideal-solution and Debye-Hьckel
approximations ........................................... 211
9.4.1 Debye-Hьckel approximation ........................ 212
9.4.2 Limitations of the ideal solution approximation ... 213
9.5 Modified Poisson-Boltzmann equations ..................... 213
9.5.1 Steric correction to ideal solution statistics .... 213
9.5.2 Modified Poisson-Boltzmann equation ............... 215
9.5.3 Importance and limitations of Poisson-Boltzmann
modifications ..................................... 216
9.6 Stern layer .............................................. 217
9.7 Summary .................................................. 217
9.8 Supplementary reading .................................... 218
9.9 Exercises ................................................ 218
10 Zeta Potential in Microchannels .......................... 225
10.1 Definitions and notation ................................. 225
10.2 Chemical and physical origins of equilibrium
interfacial charge .................................. 226
10.2.1 Electrochemical potentials ................... 226
10.2.2 Potential-determining ions ................... 227
10.2.3 Nernstian and non-Nernstian surfaces ......... 230
10.3 Expressions relating the surface charge density,
surface potential, and zeta potential ............... 232
10.3.1 Extended interface models: modifications to
pu .......................................... 234
10.3.2 Fluid inhomogeneity models: relation
between фо and ζ ............................. 234
10.3.3 Slip and multiphase interface models:
hydrophobic surfaces ......................... 236
10.4 Observed electrokinetic potentials on microfluidic
substrates .......................................... 237
10.4.1 Electrolyte concentration .................... 237
10.4.2 pH dependence ................................ 238
10.5 Modifying the zeta potential ........................ 238
10.5.1 Indifferent electrolyte concentrations ....... 238
10.5.2 Surface-active agents ........................ 239
10.5.3 Chemical functionalizations .................. 240
10.6 Chemical and fluid-mechanical techniques for
measuring interfacial properties .................... 240
10.6.1 Charge titration ............................. 240
10.6.2 Electroosmotic flow .......................... 241
10.6.3 Streaming current and potential .............. 242
10.7 Summary ............................................. 245
10.8 Supplementary reading ............................... 246
10.9 Exercises ........................................... 247
11 Species and Charge Transport ............................. 250
11.1 Modes of species transport ............................... 250
11.1.1 Species diffusion ................................. 250
11.1.2 Convection ........................................ 250
11.1.3 Relating diffusivity and electrophoretic
mobility: the viscous mobility .................... 252
11.2 Conservation of species: Nernst-Planck equations ......... 253
11.2.1 Species fluxes and constitutive properties ........ 253
11.2.2 Nernst-Planck equations ........................... 254
11.3 Conservation of charge .............................. 256
11.3.1 Charge conservation equation ...................... 256
11.3.2 Diffusivity, electrophoretic mobility, and molar
conductivity ...................................... 258
11.4 Logarithmic transform of the Nernst-Planck equations ..... 258
11.5 Microfluidic application: scalar-image velocimetry ....... 259
11.5.1 SIV using caged-dye imaging ....................... 259
11.5.2 SIV using photobleaching .......................... 259
11.6 Summary .................................................. 259
11.7 Supplementary reading .................................... 261
11.8 Exercises ................................................ 261
12 Microchip Chemical Separations ........................... 265
12.1 Microchip separations: experimental realization .......... 265
12.1.1 Sample injection .................................. 266
12.1.2 Resolution ........................................ 267
12.2 ID Band broadening ....................................... 268
12.2.1 Analyte transport: quiescent flow, no electric
field ............................................. 268
12.2.2 Transport of analytes: electroosmotic flow and
electrophoresis ................................... 269
12.3 Microchip electrophoresis: motivation and experimental
issues ................................................... 270
12.3.1 Thermal dissipation ............................... 270
12.3.2 Compact, folded, long-pathlength channels ......... 270
12.4 Experimental challenges .................................. 270
12.4.1 Pressure-driven flow .............................. 271
12.4.2 Analyte band dispersion in turns and expansions ... 271
12.5 Protein and peptide separation ........................... 273
12.5.1 Protein properties ................................ 273
12.5.2 Protein separation techniques ..................... 273
12.6 Multidimensional separations ............................. 275
12.7 Summary .................................................. 276
12.8 Supplementary reading .................................... 276
12.9 Exercises ................................................ 277
13 Particle Electrophoresis ................................. 281
13.1 Introduction to electrophoresis: electroosmosis with
a moving boundary and quiescent bulk fluid ............... 281
13.2 Electrophoresis of particles ............................. 283
13.3 Electrophoretic velocity dependence on particle size ..... 286
13.3.1 Smoluchowski velocity: large particles, small
surface potential ................................. 287
13.3.2 Henry's function: effect of finite double layers
for small φ0 ...................................... 287
13.3.3 Large surface potential - effect of counterion
distribution ...................................... 289
13.4 Summary .................................................. 292
13.5 Supplementary reading .................................... 294
13.6 Exercises ................................................ 295
14 DNA Transport and Analysis ............................... 298
14.1 Physicochemical structure of DNA ......................... 299
14.1.1 Chemical structure of DNA ......................... 299
14.1.2 Physical properties of dsDNA ..................... 300
14.2 DNA transport ............................................ 303
14.2.1 DNA transport in bulk aqueous solution ............ 303
14.3 Ideal chain models for bulk DNA physical properties ...... 308
14.3.1 Idealized models for bulk DNA properties .......... 309
14.3.2 Dependence of transport properties on contour
length ............................................ 320
14.4 Real polymer models ...................................... 320
14.5 dsDNA in confining geometries ............................ 323
14.5.1 Energy and entropy of controlled polymer
extension ......................................... 323
14.5.2 Energy and entropy of confinement for ideal
polymers .......................................... 326
14.5.3 DNA transport in confined geometries .............. 327
14.6 DNA analysis techniques .................................. 328
14.6.1 DNA amplification ................................. 328
14.6.2 DNA separation .................................... 328
14.6.3 DNA microarrays ................................... 329
14.7 Summary .................................................. 329
14.8 Supplementary reading .................................... 331
14.9 Exercises ................................................ 332
15 Nanofluidics: Fluid and Current Flow in Molecular-Scale
and Thick-EDL Systems .................................... 336
15.1 Unidirectional transport in infinitely long
nanochannels ............................................. 336
15.1.1 Fluid transport ................................... 337
15.1.2 Electrokinetic coupling matrix for thick-EDL
transport ......................................... 337
15.1.3 Circuit models for nanoscale channels ............. 344
15.2 Transport through nanostructures with interfaces or
nonuniform cross-sectional area .......................... 345
15.2.1 ID equilibrium model .............................. 346
15.2.2 Large molecule and particle transport ............. 349
15.3 Supplementary reading .................................... 350
15.4 Exercises ................................................ 351
16 Electrokinetics and the Dynamics of Diffuse Charge ....... 355
16.1 Electroosmosis with temporally varying interfacial
potential ................................................ 356
16.2 Equivalent circuits ...................................... 356
16.2.1 The double layer as a capacitor ................... 357
16.3 Induced-charge flow phenomena ............................ 363
16.3.1 Induced-charge double layers ...................... 363
16.3.2 Flow due to induced-charge double layers -
induced-charge electroosmosis ..................... 364
16.3.3 Flow due to induced-charge double layers - AC
electroosmosis .................................... 364
16.4 Electrothermal fluid flow ................................ 365
16.5 Summary .................................................. 367
16.6 Supplementary reading .................................... 368
16.7 Exercises ................................................ 369
17 Particle and Droplet Actuation: Dielectrophoresis,
Magnetophoresis, and Digital Microfluidics ............... 373
17.1 Dielectrophoresis ........................................ 373
17.1.1 Inferring the Coulomb force on an enclosed
volume from the electric field outside the
volume ............................................ 375
17.1.2 The force on an uncharged, uniform, isotropic
sphere in a linearly varying electric field with
uniform, isotropic phase .......................... 375
17.1.3 Maxwellian equivalent body for inhomogeneous,
spherically symmetric particles ................... 380
17.1.4 Dielectrophoresis of charged spheres .............. 382
17.1.5 Dielectrophoresis of nonspherical objects or
objects in nonlinearly varying fields ............. 382
17.1.6 Nonuniform and anisotropic phase effects .......... 385
17.2 Particle magnetophoresis ................................. 389
17.2.1 Origin of magnetic fields in materials ............ 390
17.2.2 Attributes of magnetism ........................... 391
17.2.3 Magnetic properties of superparamagnetic beads .... 392
17.2.4 Magnetophoretic forces ............................ 392
17.2.5 DC magnetophoresis of spheres - linear limit ...... 393
17.3 Digital microfluidics .................................... 393
17.3.1 Electrocapillarity and electrowetting ............. 394
17.4 Summary .................................................. 395
17.5 Supplementary reading .................................... 396
17.6 Exercises ................................................ 397
Appendix A. Units and Fundamental Constants ................... 405
A.l Units .................................................... 405
A.2 Fundamental physical constants ........................... 406
Appendix B. Properties of Electrolyte Solutions ............... 407
B.l Fundamental properties of water .......................... 407
B.2 Aqueous solutions and key parameters ..................... 407
B.3 Chemical reactions, rate constants, and equilibrium ...... 408
B.3.1 Henderson-Hasselbach equation ..................... 409
B.3.2 Conjugate acids and bases; buffers ................ 411
B.3.3 Ionization of water ............................... 411
B.3.4 Solubility product of weakly soluble salts ........ 412
B.3.5 Ideal solution limit and activity ................. 412
B.3.6 Electrochemical potentials ........................ 413
B.4 Effects of solutes ....................................... 413
B.4.1 Dielectric increments .............................. 413
B.5 Summary .................................................. 415
B.6 Supplementary reading .................................... 415
B.7 Exercises ................................................ 416
Appendix C. Coordinate Systems and Vector Calculus ............ 418
C.l Coordinate systems ....................................... 418
C.l.l 3D coordinate systems .............................. 418
C.1.2 2D coordinate systems ............................. 419
C.2 Vector calculus .......................................... 420
C.2.1 Scalars, vectors, and tensors ..................... 420
C.2.2 Vector operations ................................. 423
C.2.3 Del or nabla operations ........................... 426
C.2.4 Biharmonic and E4 operators ....................... 431
C.2.5 Vector identities ................................. 431
C.2.6 Dyadic operations ................................. 433
C.3 Summary .................................................. 433
C.4 Supplementary reading .................................... 433
C.5 Exercises ................................................ 434
Appendix D. Governing Equation Reference ...................... 436
D.l Scalar Laplace equation .................................. 436
D.2 Poisson-Boltzmann equation ............................... 437
D.3 Continuity equation ...................................... 437
D.4 Navier-Stokes equations .................................. 438
D.5 Supplementary reading .................................... 439
Appendix E. Nondimensionalization and Characteristic
Parameters .................................................... 440
E.1 Buckingham П theorem ..................................... 440
E.2 Nondimensionalization of governing equations ............. 440
E.2.1 Nondimensionalization of the Navier-Stokes
equations: Reynolds number ........................ 440
E.2.2 Nondimensionalization of the passive scalar
transfer equation: Peclet number .................. 443
E.2.3 Nondimensionalization of the Poisson-Boltzmann
equation: Debye length and thermal voltage ........ 445
E.3 Summary .................................................. 447
E.4 Supplementary reading .................................... 448
E.5 Exercises ................................................ 448
Appendix F. Multipolar Solutions to the Laplace and Stokes
Equations ..................................................... 450
F.l Laplace equation ......................................... 450
F.l.l Laplace equation solutions for axisymmetric
spherical coordinates: separation of variables
and multipolar expansions ......................... 450
F.l.2 Systems with plane symmetry: 2D cylindrical
coordinates ....................................... 456
F.2 Stokes equations ......................................... 458
F.2.1 The Green's function for Stokes flow with
a point source ..................................... 458
F.3 Stokes multipoles: stresslet and rotlet .................. 459
F.4 Summary .................................................. 461
F.5 Supplementary reading .................................... 462
F.6 Exercises ................................................ 462
Appendix G. Complex Functions ................................. 465
G.l Complex numbers and basic operations ..................... 465
G.l.l Arithmetic operations .............................. 466
G.1.2 Calculus operations ............................... 466
G.2 Using complex variables to combine orthogonal
parameters ............................................... 468
G.3 Analytic representation of harmonic parameters ........... 469
G.3.1 Applicability of the analytic representation ....... 470
G.3.2 Mathematical rules for using the analytic
representation of harmonic parameters ............. 470
G.4 Kramers-Krönig relations ................................. 471
G.5 Conformal mapping ........................................ 472
G.5.1 Joukowski transform ............................... 472
G.5.2 Schwarz-Christoffel transform ...................... 473
G.6 Summary .................................................. 473
G.7 Supplementary reading .................................... 473
G.8 Exercises ................................................ 474
Appendix H. Interaction Potentials: Atomistic Modeling of
Solvents and Solutes .......................................... 475
H.l Thermodynamics of intermolecular potentials .............. 475
H.l.l Monopole pair potentials ........................... 476
H.l.2 Spherically symmetric multipole pair potentials .... 477
H.2 Liquid-state theories .................................... 478
H.2.1 Integral techniques for concentration profiles ..... 479
H.2.2 Why the direct correlation function does not
describe concentration profiles .................... 482
H.2.3 Total correlation functions and the Ornstein-
Zernike equation ................................... 482
H.3 Excluded volume calculations ............................. 483
H.4 Atomistic simulations .................................... 483
H.4.1 Defining atomic forces and accelerations .......... 485
H.4.2 Water models ...................................... 485
H.4.3 Nondimensionalization in MD simulations ........... 490
H.5 Summary .................................................. 491
H.6 Supplementary reading .................................... 492
H.7 Exercises ................................................ 492
Bibliography .................................................. 495
Index ......................................................... 505
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