Micro- and Nanoscale Fluid Mechanics: Transport in Microfluidic Devices

Half-title......Page 3
Title......Page 5
Copyright......Page 6
Introduction......Page 7
Preface......Page 17
Nomenclature......Page 19
1.1 Fluid statics......Page 30
1.2.1 Important geometric definitions......Page 31
GENERAL STRAIN RATE FOR THREE-DIMENSIONAL FLOWS......Page 34
EXTENSIONAL STRAIN RATE......Page 35
VORTICITY......Page 36
1.3.1 Conservation of mass: continuity equation......Page 37
1.3.2 Conservation of momentum: the Navier–Stokes equations......Page 38
GENERAL NEWTONIAN RELATION FOR SHEAR STRESS IN A 3D FLOW......Page 41
1.4.2 Non-Newtonian fluids......Page 43
1.5.2 Young–Laplace equation......Page 44
1.5.3 Contact angle......Page 45
1.5.4 Capillary height......Page 46
1.6.1 Kinematic boundary condition for continuity of normal velocity......Page 48
1.6.2 Dynamic boundary condition for continuity of tangential velocity......Page 49
1.6.3 Dynamic boundary conditions for stresses......Page 50
SLIP OF AN ISOLATED FLUID MOLECULE AT AN IDEAL, PERFECTLY FLAT SOLID SURFACE......Page 54
THE NAVIER SLIP BOUNDARY CONDITION......Page 55
1.7 Solving the governing equations......Page 56
1.9 A Word on Terminology and the Microfluidics Literature......Page 57
1.10 Summary......Page 58
1.12 Exercises......Page 60
2.1.1 Couette flow......Page 65
2.1.2 Poiseuille flow......Page 70
2.2 Startup and Development of Unidirectional Flows......Page 73
2.3 Summary......Page 74
2.5 Exercises......Page 75
3.1 HYDRAULIC CIRCUIT ANALYSIS......Page 84
3.2 HYDRAULIC CIRCUIT EQUIVALENTS FOR FLUID FLOW IN MICROCHANNELS......Page 86
3.2.1 Analytic representation of sinusoidal pressures and flow rates......Page 92
3.2.2 Hydraulic impedance......Page 93
3.2.4 Series and parallel component rules......Page 94
3.3 SOLUTION TECHNIQUES......Page 96
3.4 SUMMARY......Page 98
3.6 EXERCISES......Page 99
4 Passive Scalar Transport: Dispersion, Patterning, and Mixing......Page 103
4.1.2 Scalar conservation equation......Page 104
4.2 Physics of Mixing......Page 106
4.3 Measuring and Quantifying Mixing and Related Parameters......Page 108
4.4.2 The low-Reynolds-number limit......Page 111
4.5 Laminar Flow Patterning in Microdevices......Page 112
4.6 Taylor–Aris Dispersion......Page 113
4.7 Summary......Page 115
4.9 Exercises......Page 116
5.1.1 Electrical potential and electric field......Page 121
5.1.2 Coulomb's law, Gauss's law for electricity in a material, curl of electric field......Page 122
5.1.3 Polarization of matter and electric permittivity......Page 124
5.1.4 Material, frequency, and electric-field dependence of electrical permittivity......Page 126
5.1.5 Poisson and Laplace equations......Page 128
5.1.7 Electrostatic boundary conditions......Page 129
5.1.9 Maxwell stress tensor......Page 131
5.1.10 Effects of electrostatic fields on multipoles......Page 132
5.2 Electrodynamics......Page 133
5.2.2 Electrodynamic boundary conditions......Page 134
5.3 Analytic Representations of Electrodynamic Quantities complex permittivity and conductivity......Page 136
5.3.1 Complex description of dielectric loss......Page 139
5.4 Electrical Circuits......Page 140
5.4.1 Components and properties......Page 141
5.4.3 Circuit relations......Page 143
5.4.4 Series and parallel component rules......Page 144
5.5.1 Electrical circuit equivalents of hydraulic components......Page 146
5.6 Summary......Page 150
5.8 Exercises......Page 151
6 Electroosmosis......Page 155
6.2 Integral Analysis of Coulomb Forces on the EDL......Page 156
6.3 Solving the Navier–Stokes Equations for Electroosmotic Flow in the Thin-EDL Limit......Page 159
6.3.2 Replacing the EDL with an effective slip boundary condition......Page 160
6.3.3 Replacing the Navier–Stokes equations with the Laplace equation: flow–current similitude......Page 161
6.4 Electroosmotic Mobility and the Electrokinetic Potential......Page 162
6.5.1 A planar electrokinetic pump......Page 164
6.5.2 Types of electrokinetic pumps......Page 167
6.7 Supplementary Reading......Page 169
6.8 Exercises......Page 170
7.1 Approach for Finding Potential Flow Solutions to the Navier–Stokes Equations......Page 177
7.2.1 Laplace equation for the velocity potential......Page 178
7.3 Potential Flows with Plane Symmetry......Page 180
7.3.1 Complex algebra and its use in plane-symmetric potential flow......Page 181
7.3.2 Monopolar flow: plane-symmetric (line) source with volume outflowper unit depth......Page 184
7.3.3 Plane-symmetric vortex with counterclockwise circulation per unit depth lembda......Page 187
7.3.4 Dipolar flow: plane-symmetric doublet with dipole moment k......Page 189
7.3.5 Uniform flow with speed U......Page 192
7.3.6 Flow around a corner......Page 194
7.3.8 Conformal mapping......Page 195
7.4 Potential Flow in Axisymmetric Systems in Spherical Coordinates......Page 196
7.6 Supplementary Reading......Page 197
7.7 Exercises......Page 198
8.1 Stokes Flow Equation......Page 202
8.1.1 Different forms of the Stokes flow equations......Page 203
8.2 Bounded Stokes Flows......Page 204
8.2.1 Hele-Shaw flows......Page 205
8.3.1 Stokes flow over a sphere in an infinite domain......Page 206
8.3.2 General solution for Stokes flow over a sphere in an infinite domain......Page 211
8.3.3 Flow over prolate ellipsoids......Page 212
8.4 Micro-PIV......Page 213
8.5 Summary......Page 215
8.6 Supplementary Reading......Page 216
8.7 Exercises......Page 217
9.1 The Gouy–Chapman EDL......Page 223
9.1.1 Boltzmann statistics for ideal solutions of ions......Page 224
9.1.2 Ion distributions and potential: Boltzmann relation......Page 225
9.1.3 Ion distributions and potential: Poisson–Boltzmann equation......Page 226
SIMPLIFIED FORMS OF THE NONLINEAR POISSON–BOLTZMANN EQUATION: 1D, SYMMETRIC ELECTROLYTE......Page 227
9.1.5 Solutions of the Poisson–Boltzmann equation......Page 228
LINEARIZED 1D POISSON–BOLTZMANN SOLUTION – TWO PARALLEL PLATES......Page 230
9.2 Fluid Flow in the Gouy–Chapman EDL......Page 232
9.3 Convective Surface Conductivity......Page 234
9.4 Accuracy of the Ideal-Solution and Debye–Huckel Approximations......Page 235
9.4.1 Debye–Huckel approximation......Page 236
9.5.1 Steric correction to ideal solution statistics......Page 237
9.5.2 Modified Poisson–Boltzmann equation......Page 239
9.5.3 Importance and limitations of Poisson–Boltzmann modifications......Page 240
9.7 Summary......Page 241
9.9 Exercises......Page 242
10.1 Definitions and Notation......Page 249
10.2.1 Electrochemical potentials......Page 250
10.2.2 Potential-determining ions......Page 251
10.2.3 Nernstian and non-Nernstian surfaces......Page 254
10.3 Expressions Relating the Surface Charge Density, Surface Potential, and Zeta Potential......Page 256
10.3.2 Fluid inhomogeneity models: Relation between phi0 and zeta......Page 258
10.3.3 Slip and multiphase interface models: Hydrophobic surfaces......Page 260
10.4.1 Electrolyte concentration......Page 261
10.5.1 Indifferent electrolyte concentrations......Page 262
10.5.2 Surface-active agents......Page 263
10.6.1 Charge titration......Page 264
NEUTRAL MARKER ELUTION......Page 265
10.6.3 Streaming current and potential......Page 266
10.7 Summary......Page 269
10.8 Supplementary Reading......Page 270
10.9 Exercises......Page 271
11.1.2 Convection......Page 274
11.1.3 Relating diffusivity and electrophoretic mobility: the viscous mobility......Page 276
DIFFUSION......Page 277
11.2.2 Nernst–Planck equations......Page 278
11.3.1 Charge conservation equation......Page 280
11.4 Logarithmic Transform of the Nernst–Planck Equations......Page 282
11.6 Summary......Page 283
11.8 Exercises......Page 285
12.1 Microchip Separations: Experimental Realization......Page 289
12.1.1 Sample injection......Page 290
12.1.2 Resolution......Page 291
12.2.1 Analyte transport: quiescent flow, no electric field......Page 292
12.2.2 Transport of analytes: electroosmotic flow and electrophoresis......Page 293
12.4 Experimental Challenges......Page 294
12.4.2 Analyte band dispersion in turns and expansions......Page 295
CAPILLARY ZONE ELECTROPHORESIS......Page 297
ISOELECTRIC FOCUSING......Page 298
12.6 Multidimensional Separations......Page 299
12.8 Supplementary Reading......Page 300
12.9 Exercises......Page 301
13.1 Introduction to Electrophoresis: Electroosmosis with a Moving Boundary and Quiescent Bulk Fluid......Page 305
13.2 Electrophoresis of Particles......Page 307
13.3 Electrophoretic Velocity Dependence on Particle Size......Page 310
13.3.2 Henry's function: effect of finite double layers for small phi0......Page 311
13.3.3 Large surface potential – effect of counterion distribution......Page 313
13.4 Summary......Page 316
13.5 Supplementary Reading......Page 318
13.6 Exercises......Page 319
14 DNA Transport and Analysis......Page 322
14.1.1 Chemical structure of DNA......Page 323
14.1.2 Physical properties of dsDNA......Page 324
DEFINITION OF LINEAR (UNBRANCHED) POLYMER PROPERTIES AND LENGTH SCALES......Page 325
14.2.1 DNA transport in bulk aqueous solution......Page 327
14.3 Ideal Chain Models for Bulk DNA Physical Properties......Page 332
14.3.1 Idealized models for bulk DNA properties......Page 333
SUMMARY OF IDEAL MODELS......Page 342
14.4 Real Polymer Models......Page 344
EFFECT OF SELF-AVOIDANCE......Page 345
EFFECT OF CHARGE AND THE EDL ON POLYMER PROPERTIES......Page 346
14.5.1 Energy and entropy of controlled polymer extension......Page 347
14.5.2 Energy and entropy of confinement for ideal polymers......Page 350
14.5.3 DNA transport in confined geometries......Page 351
SANGER SEQUENCING......Page 352
14.7 Summary......Page 353
14.8 Supplementary Reading......Page 355
14.9 Exercises......Page 356
15.1 Unidirectional Transport in Infinitely Long Nanochannels......Page 360
15.1.2 Electrokinetic coupling matrix for thick-EDL transport......Page 361
ELECTROKINETIC COUPLING COEFFICIENTS FOR THICK DOUBLE LAYERS......Page 362
OBSERVED SYSTEM DEPENDENCE ON d AND d* WHEN FORCING FUNCTIONS ARE CONTROLLED......Page 365
OBSERVED SYSTEM DEPENDENCE ON d AND d* WHEN ONE FORCING FUNCTION AND ONE OUTCOME ARE CONTROLLED......Page 366
ELECTROVISCOSITY IN THICK-EDL SYSTEMS......Page 367
15.1.3 Circuit models for nanoscale channels......Page 368
15.2 Transport through Nanostructures with Interfaces or Nonuniform Cross-Sectional Area......Page 369
15.2.1 1D equilibrium model......Page 370
CONCENTRATION POLARIZATION AND CURRENT RECTIFICATION......Page 372
OGSTON SIEVING......Page 373
15.3 Supplementary Reading......Page 374
15.4 Exercises......Page 375
16 AC Electrokinetics and the Dynamics of Diffuse Charge......Page 379
16.2 Equivalent Circuits......Page 380
16.2.1 The double layer as a capacitor......Page 381
THE DOUBLE LAYER AS A CAPACITOR......Page 382
DOUBLE-LAYER CHARGE......Page 383
16.3.1 Induced-charge double layers......Page 387
16.3.3 Flow due to induced-charge double layers – AC electroosmosis......Page 388
16.4 Electrothermal Fluid Flow......Page 389
16.5 Summary......Page 391
16.6 Supplementary Reading......Page 392
16.7 Exercises......Page 393
17.1 Dielectrophoresis......Page 397
17.1.2 The force on an uncharged, uniform, isotropic sphere in a linearly varying electric field with uniform, isotropic phase......Page 399
17.1.3 Maxwellian equivalent body for inhomogeneous, spherically symmetric particles......Page 404
MAXWELLIAN EQUIVALENT BODY FOR THIN OUTER SHELLS......Page 405
17.1.5 Dielectrophoresis of nonspherical objects or objects in nonlinearly varying fields......Page 406
MULTIPOLE EXPANSION......Page 407
DEP OF ELLIPSOIDS......Page 408
17.1.6 Nonuniform and anisotropic phase effects......Page 409
ELECTROROTATION......Page 410
TRAVELING-WAVE DEP......Page 411
ELECTRODE CONFIGURATIONS......Page 412
17.2 Particle Magnetophoresis......Page 413
17.2.1 Origin of magnetic fields in materials......Page 414
17.2.2 Attributes of magnetism......Page 415
17.2.4 Magnetophoretic forces......Page 416
17.3 Digital Microfluidics......Page 417
17.3.1 Electrocapillarity and electrowetting......Page 418
17.4 Summary......Page 419
17.5 Supplementary Reading......Page 420
17.6 Exercises......Page 421
A.1 Units......Page 429
A.2 Fundamental physical constants......Page 430
B.2 Aqueous solutions and key parameters......Page 431
B.3 Chemical reactions, rate constants, and equilibrium......Page 432
B.3.1 Henderson–Hasselbach equation......Page 433
B.3.3 Ionization of water......Page 435
B.3.5 Ideal solution limit and activity......Page 436
B.4.1 Dielectric increments......Page 437
B.6 Supplementary reading......Page 439
B.7 Exercises......Page 440
C.1.1 3D coordinate systems......Page 442
C.1.2 2D coordinate systems......Page 443
C.2.1 Scalars, vectors, and tensors......Page 444
C.2.2 Vector operations......Page 447
CROSS PRODUCT......Page 449
GRADIENT OPERATOR......Page 450
DIVERGENCE OPERATOR......Page 451
CURL OPERATOR......Page 453
LAPLACIAN AND E2 OPERATORS......Page 454
C.2.5 Vector identities......Page 455
C.4 Supplementary reading......Page 457
C.5 Exercises......Page 458
D.1 Scalar Laplace equation......Page 460
D.3 Continuity equation......Page 461
D.4 Navier–Stokes equations......Page 462
D.5 Supplementary reading......Page 463
E.2.1 Nondimensionalization of the Navier–Stokes equations: Reynolds number......Page 464
E.2.2 Nondimensionalization of the passive scalar transfer equation: Peclet number......Page 467
E.2.3 Nondimensionalization of the Poisson–Boltzmann equation: Debye length and thermal voltage......Page 469
E.3 Summary......Page 471
E.5 Exercises......Page 472
THE GENERAL LEGENDRE POLYNOMIAL SOLUTION TO THE AXISYMMETRIC LAPLACE EQUATION......Page 474
CREATING HIGH-ORDER MULTIPOLE SOLUTIONS FROM LOWER-ORDER MULTIPOLES......Page 476
MONOPOLE......Page 477
THE GENERAL HARMONIC SOLUTION TO THE PLANE-SYMMETRIC CYLINDRICAL LAPLACE EQUATION......Page 480
LINEAR 2D QUADRUPOLE......Page 481
F.2.1 The Green's function for Stokes flow with a point source......Page 482
F.3 Stokes multipoles: stresslet and rotlet......Page 483
F.4 Summary......Page 485
F.6 Exercises......Page 486
G.1 Complex numbers and basic operations......Page 489
G.1.2 Calculus operations......Page 490
G.2 Using complex variables to combine orthogonal parameters......Page 492
G.3 Analytic representation of harmonic parameters......Page 493
G.3.2 Mathematical rules for using the analytic representation of harmonic parameters......Page 494
G.4 Kramers–Kronig relations......Page 495
G.5.1 Joukowski transform......Page 496
G.7 Supplementary reading......Page 497
G.8 Exercises......Page 498
H.1 Thermodynamics of intermolecular potentials......Page 499
H.1.1 Monopole pair potentials......Page 500
LENNARD–JONES POTENTIAL......Page 501
H.2 Liquid-state theories......Page 502
DISTRIBUTION AND CORRELATION FUNCTIONS......Page 503
MAYER f FUNCTIONS AND THE POTENTIAL OF MEAN FORCE......Page 504
CORRELATION FUNCTIONS......Page 505
H.2.3 Total correlation functions and the Ornstein–Zernike equation......Page 506
H.4 Atomistic simulations......Page 507
H.4.2 Water models......Page 509
MULTIPOINT MODELS......Page 510
H.4.3 Nondimensionalization in MD simulations......Page 514
H.5 Summary......Page 515
H.7 Exercises......Page 516
Bibliography......Page 519
Index......Page 529