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Cambridge University Press 978-0-521-11903-0 - Micro- and Nanoscale Fluid Mechanics: Transport in Microfluidic Devices Brian J. Kirby Index More information

Index

acid hydraulic, 62, 66 conjugate, 411 capacitor, 117, 122, 357 strong, 410 capillary ﬂows, 29 acid dissociation constant, 409 capillary height, 22 activity, 413 capillary zone electrophoresis, 241, 273 adjusted distribution function, 480, 481 Cartesian coordinates, 418 adverse pressure gradient, 140 curl, 429 ampholyte, 275 divergence, 428 analytic representation of functions, 68, 469 gradient, 427 annealing, 300 Cauchy momentum equations, 15 aqueous solution, 407 Cauchy–Riemann equation, 467 asymptotic, 132 CCD camera, 189 autocorrelation, 189 chaotic mixing, 329 Avogadro’s number, 112, 200, 406 characteristic length of covalent bond, 477 bad solvent, 300, 304 of Navier–Stokes equations, 440 baker’s transformation, 84 charge band broadening, 268, 271 bound, 97 base conservation of, 110 conjugate, 411 density, 99, 201 strong, 410 density on surface, 105, 199 basis vector, 418 double layer, 359, 360 bead–spring chain model, 317 of an electron, 201, 251, 406 end-to-end length, 317 free, 97 persistence length, 317 number (valence), 99, 201, 251, 408 biharmonic equation, 180, 431 charge-determining ion, 229 biharmonic operator, 180, 431 chemical potential, 227, 413 billiard-ball model, 213, 477 circuit equivalent, 356 Bingham plastic, 20 circulation, 13, 156 Bjerrum length, 478 circumferential velocity, 9, 159 Boltzmann Clausius–Mossotti factor, 377, 393 constant, 104, 191, 200, 406 closure relation, 483 statistics, 200, 201, 213, 214, 475 coion, 199 bond length, 477 complex bound charge, 97 conjugation, 465 boundary conditions, 24, 155 differentiation, 466 boundary layer, 156 distance, 157 Brillouin function, 104 number, 465 Brownian motion, 191, 199 number, polar form, 465 Buckingham- theorem, 440 plane rotation, 158 buffers, 411 representation of functions, 68, 469 Butler–Volmer equation, 111 velocity, 159 velocity potential, 158 caged dye, 259 compliance, 62, 66 capacitance, 62, 117, 123, 357, 394 concentration, 407 differential, 357, 359 logarithmically transformed, 259

505

© in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-11903-0 - Micro- and Nanoscale Fluid Mechanics: Transport in Microfluidic Devices Brian J. Kirby Index More information

506 Index

conductance, electrical, 117 del operator, 426 conductivity, 110, 123, 241, 256 density, 6, 22 complex, 114 depolarization factor, 384 effective surface, 210 developing flows, 50 molar, 256, 258 diamagnetism, 390 conductor, 105 dielectric conjugate acid, 411 constant, 101 conjugate base, 411 ideal, 105 conservation equation increment, 413 charge, 110 dielectrophoresis, 3, 373, 391 current, 119 dielectrophoretic mobility, 374, 379 mass, 13, 62, 153, 155, 437 electrodeless, 389 momentum, 13, 133, 153, 438, 440 insulative, 389 constitutive relation, 16, 17 negative, 374 for Newtonian fluid, 17 positive, 373 contact angle, 21, 394 diffusion hysteresis, 24, 395 equation, 268 continuity equations, 13, 62, 153, 155, thermodiffusion, 261 437 diffusive flux, 80, 253, 255 continuum approximation, 484 diffusivity for liquids, 484 DNA in bulk solution, 304 contour length, 301, 302 species, 252, 255 convective flux, 80, 254, 255 digital microfluidics, 393 coordinates dilute solution (ideal solution), 226, 412 Cartesian, 418 dipole, 452, 454, 457 curvilinear, 419 force on, 109 cylindrical, 418 torque on, 109 rectangular, 419 dipole moment, 104, 414 spherical, 418 direct correlation function, 482 correlation function dispersion, 79, 89, 91, 271 direct, 482 distribution function, 217 total, 482, 483 adjusted, 480, 481 Couette flow, 41 divergence operator, 426 counterion, 199, 281, 363 Cartesian, 428 couplet (rotlet), 461 cylindrical, 428 cross product, 423, 425 spherical, 428 cross-correlation, 189 divergence theorem, 428 curl operator, 426 DNA, 298, 299 Cartesian, 429 annealing, 300 cylindrical, 429 base pair spacing, 301 spherical, 430 bending energy per unit length, 310 current, conservation of, 119 bulk diffusivity, 304 current density, ohmic, 62, 110 bulk electrophoretic mobility, 306 curvilinear coordinate system, 419 chemical structure, 299 cylindrical coordinates, 418 contour length, 301, 302 curl, 429 double-stranded, 299 divergence, 428 end-to-end length, 303 gradient, 427 gel electrophoretic mobility, 331 Laplacian, 430 melting temperature, 300 persistence length, 299, 302, 303 Dean flow, 59 radius of gyration, 299, 303 Debye (unitofdipole moment), 406 sequencing, 328, 331 Debye layer, 132, 133, 281, 355, 363 single-stranded, 300 charging time, 356 dot product, 423 Debye length, 32, 123, 202, 287, 292, 446 double layer, 132, 133, 281, 355, 363 Debye screening parameter,288 charge in, 359, 360 Debye–Falkenhagen equation, 264 charging time, 356 Debye–Huckel¨ approximation, 205, 211, 212, Gouy–Chapman model, 199, 213 359 hypernetted-chain model, 479, 483

507 Index

overlap, 336 inner solution, 132, 208 potential drop across, 133 outer solution, 132 Stern model, 217, 234 electroosmotic doublet mobility, 138, 139, 241 in potential flow, 165 velocity, 140 strength, 165 electrophoresis, 251, 265, 269, 363, 391 drag coefficient, 188 correction factor, 287 ellipsoid in Stokes flow, 188 particle, 281 sphere in Stokes flow, 186 electrophoretic Dukhin number, 263 mobility, 252, 265, 273, 288 dyadic tensor, 433 velocity, 255 dynamic similitude, 443 electrophoretic mobility dynamic viscosity, 17 DNA in bulk solution, 306 DNA in gels, 331 E4 operator, 431 electrorotation, spectroscopy, 387 E2 operator, 431 electrowetting, 393 eccentricity, 188 end-to-end length, 303 EDL (electrical double layer), 132, 133, 281, 355, of bead–spring chain polymer, 317 363 of freely jointed chain polymer, 313 effective pair potential, 476 of freely rotating chain polymer, 316 effective zeta potential, 291 equation(s) electric displacement, 100 biharmonic, 180, 431 electric field, 97, 102, 104 Butler–Volmer, 111 extrinsic, 133 Cauchy momentum, 15 intrinsic, 133 Cauchy–Riemann, 467 electric force field, 251 charge conservation, 110 electric polarization, 100, 102 continuity, 13, 62, 153, 155, 437 electric potential (voltage), 97, 154 current conservation, 119 electric stream function, 469 Debye–Falkenhagen, 264 electric susceptibility, 100 diffusion, 268 electrical electrokinetic coupling, 64, 242, 338 conductivity, complex, 114 Faraday’s, 98, 201 double layer, 132, 133, 281, 355, 363 Gauss charging time, 356 electrical, 98, 105 permittivity, 98, 145, 391, 395, 475 electrical boundary conditions, 106, 110 complex, 113 magnetic, 98 permittivity of free space, 100, 406 Grahame, 232 electrochemical potential, 227, 245, Huckel,¨ 288, 291 413 Hagen–Poiseuille, 61, 73 electrode, 123, 394 Henderson–Hasselbach, 409 reaction,357 Kirchoff’s Law, 119, 122 electrodynamics, 1 Laplace, 105, 156, 178, 436, 450, 456 electrokinetic Lippman, 394 coupling equation, 64, 242, 338 modified Poisson–Boltzmann, 213, 215, 216, coupling matrix, 65,242, 337 360 mobility, 265 Navier–Stokes, 13, 133, 153, 438, 440 potential, 225 Nernst, 231 pump, 3, 140, 143 Nernst–Einstein, 252, 307 velocity, 269 Nernst–Planck, 254, 255 electrolyte, 407 Ohm’s Law, 119, 122 charge-determining, 229 Ornstein–Zernike, 479, 482 indifferent, 201, 228 Poisson, 104, 201 potential-determining, 229 Poisson–Boltzmann, 1, 202, 213, 437, 445 symmetric, 203, 214, 360, 408 Smoluchowski, 225, 287 electron charge, 201, 251, 406 Stokes, 178, 458 electroosmosis, 32, 131, 138, 209, 269, Stokes–Einstein, 191 363 transformed Nernst–Planck, 259 AC, 355, 364, 368 Young, 21 induced-charge, 355, 364, 368 Young–Laplace, 20, 22, 28

508 Index

equilibrium constant, 409, 411 Huckel¨ equation, 288, 291 equivalent circuit, 356 Hagen–Poiseuille law, 61, 73 exchange current density, 112 hard-sphere extensional strain, 11 model, 213, 477 packing length, 213 packing parameter, 215, 361 Faraday constant, 99, 406 Hele-Shaw flow, 181 Faraday’s law of induction, 98, 201 Henderson–Hasselbach equations, 409 ferromagnetism, 390 Henry’s function, 288–291 Fick’s law, 80, 253 high-pressure liquid chromatography, 274 field hydrated radius, 273 electric, 97, 102, 104 hydraulic electric force, 251 impedance, 69 field-amplified sample stacking, 276 radius, 63 flow–current similitude, 137 resistance, 61, 63 fluid hydrodynamic interaction tensor, 187 Bingham plastic, 20 hydrostatics, 6 definition of, 9 hypernetted-chain model, 479, 483 Newtonian, 17 hysteresis, magnetic, 392 particle, 7 shear-thickening, 19 shear-thinning, 19 ideal dielectric, 105 statics, 6 ideal solution (dilute solution), 226, 412 viscoelastic, 19 identity tensor, 16 flux impedance, 119 convective, 80, 254, 255 hydraulic, 69 diffusive, 80, 253, 255 indifferent electrolyte, 201, 228 scalar, 80 inductance, 62, 117 focal depth, 190 inductor, 117 force inner Helmholtz plane, 234 on a dipole, 109 insular, perfect, 105 on a monopole, 108 interfacial energy, 20 four-quadrant inverse tangent, 158 interrogation regions, 190 Fourier’s law, 80, 253 inverse tangent, 158 free charge, 97 ion free-draining polymer, 306 charge-determining or potential determining, freely jointed chain model, 312 229 end-to-end length, 313 indifferent, 201, 228 persistence length, 313 specifically adsorbed, 228 freely rotating chain model, 315 ionic strength, 408 end-to-end length, 316 irrotational field, 13, 154, 155 persistence length, 315 isoelectric focusing, 274 isoelectric point, 273, 274 isotachophoresis, 276 gas constant, 112, 406 Gauss equation(s) Joukowski transform, 171, 472 electricity, 98, 105 Joule heating, 270 electricity boundary conditions, 106, 110 magnetism, 98 Gaussian chain model, 317 kinematics, 43 Gibbs free energy, 20, 22, 324 Kirchoff’s Law, 119, 122 good solvent, 300 Kramers–Kronig¨ relation, 115 Gouy–Chapman model, 199, 213 Kratky–Porod model, 309, 313 gradient operator, 426 persistence length, 310 Cartesian, 427 Kuhn length, 312, 315, 317 cylindrical, 427 Kutta condition, 156 spherical, 427 Grahame equation, 232 laminar flow patterning, 79, 88 gravitational acceleration, 6, 22 Langevin function, 334 Green’s function, 458 Laplace equation, 105, 156, 178, 436, 450, Grotthus mechanism, 252 456

509 Index

Laplacian operator, 430 expansion, 108, 452 cylindrical, 430 for Laplace equation, 450 spherical, 431 mathematical or ideal, 452 Lattice-Boltzmann, 492 physical, 452 Legendre polynomials, 451, 456 Lennard–Jones potential, 477, 481 nabla, 426 Lippman equation, 394 nanochannel, 327 liquid state theory, 200, 478 nanofluidics, 336 Lyapunov exponent, 92 nanoslit, 327 Navier–Stokes equations, 13, 133, 153, 438, 440 magnetic beads, 391 Nd:YAG laser, 189 magnetic properties negative dielectrophoresis, 374 applied magnetic field, 391 Nernst equation, 231 hysteresis, 392 Nernst–Einstein relation, 252, 307 induced magnetic field, 98, 391 Nernst–Planck equation, 254, 255 permeability, 391, 393 transformed, 259 permeability of free space, 98, 391, 406 Nernstian surface, 225, 274 saturation, 392 Newtonian fluid, 17 susceptibility, 98, 391, 393 no-penetration condition, 25, 153, 155, 156 magnetic response no-slip condition, 26, 135, 138, 153, 156 diamagnetic, 390 nondimensional parameter ferromagnetic, 390 Dukhin number, 263 paramagnetic, 390 Peclet number, 79, 81, 84, 87, 88, 90, 443, magnetization, 98, 391 444 magnetophoresis, 389 Reynolds number, 44, 48, 79, 87, 440, 442 mass, conservation, 13, 62, 153, 155, 437 Strouhal number, 442 matched asymptotic, 132 nondimensionalization, 440 matching condition, 136 nondraining polymer, 304 material line, 9 normality, 407 Maxwell stress tensor, 107 northern blot, 300 Maxwell–Wagner interfacial charge, 374 Mayer f function, 480, 481 melting temperature of DNA, 300 octupole, 455, 458 micro-PIV, 189 Ohm’s Law, 119, 122 microarray, 300, 329 Onsager reciprocity, 242 microfluidic mixing, 79 Ornstein–Zernike equations, 479, 482 microscope, 189 Oseen–Burgers tensor, 187 mixing osmolarity, 408 chaotic, 329 osmotic pressure, 221 microfluidic, 79 outer Helmholtz plane, 234 mobility dielectrophoretic, 374, 379 packing parameter, 215, 361 electrokinetic, 265 pair potential(s), 476 electroosmotic, 138, 139, 241 paramagnetism, 390 electrophoretic, 252, 265, 273, 288 particle viscous, 252 electrophoresis, 281 modified Poisson–Boltzmann equation, 213, 215, image velocimetry, 189 216, 360 lag, 183, 191 molality, 408 particle-image velocimetry molar conductivity, 256, 258 interrogation regions, 190 mole fraction, 408 microscale, 189 molecular dynamics, 200, 484 partition function, 326 monopole, 391, 452, 453, 457 pathline, 7 force on, 108 PCR (polymerase chain reaction), 328 multidimensional separations, 275 Peclet number, 79, 81, 84, 87, 88, 90, 443, multipoint model, 486 444 multipole perfect insulator, 105 creating higher-order from lower-order, permeability, 391, 393 452 of free space, 98, 391, 406

510 Index

permittivity, 98, 145, 391, 395, 475 Rankine solids, 172 complex, 113 reaction, Faradaic, 357 dissipative, 115 reaction rate, 409 of free space, 100, 406 reciprocity, Onsager, 242 reactive, 115 rectangular coordinate system, 419 relative, 101 relative permittivity, 101 of Stern layer, 360 resistance persistence length, 299, 302 electrical, 117 of bead–spring chain polymer, 317 hydraulic, 61, 63 of DNA, 303 resistor, 117, 122 of freely jointed chain polymer, 313 resolution of chemical separation, 267 of freely rotating chain polymer, 315 Reynolds number, 44, 48, 79, 87, 440, of Kratky–Porod polymer, 310 442 photobleaching, 259 RNA, 300 photodissociation, 259 rotation rate tensor, 11 Poincare map, 92 rotlet (couplet), 461 point charge approximation, 200, 202, 214, Rouse dynamics, 304, 306 281 point of zero charge, 230 Poiseuille flow, 46, 90, 140 Sanger sequencing, 328, 331 Poisson equation, 104, 201 saturation, magnetic, 392 Poisson–Boltzmann equation, 1, 202, 213, 437, scalar, 420 445 flux, 80 polarization, orientation, 102 image velocimetry, 259 polymerase chain reaction, 328 product, 423 polystyrene beads, 189 Schwarz–Christoffel transform, 174, 473 positive dielectrophoresis, 373 SDS-PAGE (sodium dodecyl potential sulfate-polyacrylamide gel double layer, 133 electrophoresis), 276 electric, 97, 154 second moment of area, 309 electrokinetic, 225 self-avoidance, 321 Lennard–Jones, 477, 481 separation of variables, 32 of mean force, 227, 480, 481 for Laplace equation, 450 pair, effective, 476 separation(s), 265 singlet, 475 band broadening, 268, 271 surface, 225 capillary zone electrophoresis, 241, 273 triplet, 475, 476 field-amplified sample stacking, 276 velocity, 153, 155 high-pressure liquid chromatography, well depth, 477 274 potential flow, 153, 156 isoelectric focusing, 274 doublet, 165 isotachophoresis, 276 over circular cylinder, 171 low-dispersion turns, 270 source, 160 multidimensional, 275 uniform flow, 168 protein, 273 vortex, 163 resolution, 267 potential-determining ion, 229 sample injection, 266, 274 pressure, osmotic, 221 SDS-PAGE, 276 pressure interaction tensor, 187 shear strain, 12 probability density function, 313 shear-thickening fluid, 19 pseuodovector, 420 shear-thinning fluid, 19 sign function, 232 similarity solution, 32 quadrupole, 455, 457 similitude dynamic, 443 racetrack effect, 271 flow-current, in electroosmotic systems, radial velocity, 9, 159 137 radius singlet potential(s), 475 of gyration, 299, 303 slip hydrated, 273 effective, 136, 199 hydraulic, 63 length, 31

511 Index

Smoluchowski surface tension, 20, 24, 394 equation, 225, 287 symmetric electrolyte, 203, 214, 360, 408 velocity, 287 solubility product, 412 tangent, two-argument inverse, 158 solvent Taylor–Aris dispersion, 79, 89, 91, 271 bad, 300, 304 Teflon AF, 394 good, 300 tensor, 423 theta, 305 hydrodynamic interaction, 187 source Oseen–Burgers, 187 in potential flow, 160 rotation rate, 11 strength, 160 strain rate, 10 Southern blot, 300 velocity gradient, 11 SPC model, 488 thermal diffusivity, 80 SPC/E model, 488 thermal voltage, 445 species diffusivity, 252, 255 thermodiffusion, 261 specific heat, 80 thermodynamic efficiency, 143 specifically adsorbed ion, 228 theta solvent, 305 spherical coordinates, 418 thick-EDL approximation, 336 curl, 430 thin-EDL approximation, 136, 269 divergence, 428 TIP3P model, 489 gradient, 427 TIP4P model, 489 Laplacian, 431 TIP5P model, 489 spring constant torque on a dipole, 109 entropic, 323, 325 total correlation function, 482, 483 of bead–spring model, 317 transport properties startup, 49 dynamic viscosity, 17 Stern layer, 217, 234 species diffusivity, 252, 255 thickness, 360 triple point, 21, 22 Stokes triplet potential(s), 475, 476 dipole, 459 twist map, 84 equations, 178, 458 two-argument inverse tangent, 158 numerical solution, 180 flow uniform flow, 168 instantaneity, 179 over sphere, 182 point force solution, 187 valence, 99, 201, 251, 408 time-reversibility, 179 magnitude, 203 number, 186 Van Dyke condition, 136 stream function, 9 vector, 420 theorem, 429 basis, 418 Stokes–Einstein relation, 191 vector operation Stokeslet, 185, 458 cross product, 423, 425 strain curl, 426 extensional, 11 divergence, 426 shear, 12 dot product, 423 strain rate tensor, 10 gradient, 426 stream function, 8 vector operator, 426 streaming current, 242 velocity streaming potential, 212, 242 circumferential, 9, 159 streamline, 7 electrokinetic, 269 stress tensor, 15 electroosmotic, 140 pressure, 16 electrophoretic, 255 viscous, 16 radial, 9, 159 stresses, surface, 15 Smoluchowski, 287 stresslet, 185, 459 velocity gradient tensor, 11 strong acids and bases, 410 velocity potential, 153, 155 Strouhal number, 442 viscoelectric coefficient, 235 superparamagnetism, 392 viscosity, dynamic, 17 surface stresses, 15 viscous mobility, 252 visoelastic fluid, 19

512 Index

vortex TIP5P model, 489 in potential ﬂow, 163 weight percent, 408 strength, 163 wormlike chain model, 309, 313 vorticity, 12, 138, 154 in electrical double layer, 139 Young’s equation, 21 Young’s modulus, 309 water model Young–Laplace equation, 20, 22, 28 multipoint model, 486 SPC model, 488 zeta potential, 139, 225, 238, 271, 291 SPC/E model, 488 effective, 291 TIP3P model, 489 Zimm dynamics, 304 TIP4P model, 489 zwitterion, 414