Abstract
Conventional approaches for simulating steady-state distributions of dilute particles under diffusive and advective transport involve solving the diffusion and advection equations in at least two dimensions. Here, we present an alternative computational strategy by combining a particle-based rather than a field-based approach with the initialisation of particles in proportion to their flux. This method allows accurate prediction of the steady state and is applicable even at intermediate and high Péclet numbers
Correction note
Correction added after online publication 15 April 2016: In the first sentence of the Abstract the word “numbers” was deleted from “…diffusive and advective transport numbers involve…”.
Acknowledgments
Financial support from the Biotechnology and Biological Sciences Research Council (BBSRC), the European Research Council (ERC), the Frances and Augustus Newman Foundation as well as the Swiss National Science Foundation is gratefully acknowledged.
References
[1] A. E. Kamholz, B. H. Weigl, B. A. Finlayson, and P. Yager, Quantitative analysis of molecular interaction in a microfluidic channel: the T-sensor, Anal. Chem. 71 (1999), 5340.10.1021/ac990504jSearch in Google Scholar
[2] R. F. Ismagilov, A. D. Stroock, P. J. A. Kenis, G. Whitesides, and H. A. Stone, Experimental and theoretical scaling laws for transverse diffusive broadening in two-phase laminar flows in microchannels, Appl. Phys. Lett. 76 (2000), 2376.10.1063/1.126351Search in Google Scholar
[3] C. D. Costin and R. E. Synovec, A Microscale-molecular weight sensor: probing molecular diffusion between adjacent laminar flows by refractive index gradient detection, Anal. Chem. 74 (2002), 4558.10.1021/ac020143zSearch in Google Scholar
[4] C. D. Costin, R. K. Olund, B. A. Staggemeier, A. K. Torgerson, and R. E. Synovec, Diffusion coefficient measurement in a microfluidic analyzer using dual-beam microscale-refractive index gradient detection application to on-chip molecular size determination, J. Chromatogr. A 1013 (2003), 77.10.1016/S0021-9673(03)01101-4Search in Google Scholar
[5] T. M. Squires and S. R. Quake, Microfluidics: fluid physics at the nanoliter scale, Rev. Mod. Phys. 77 (2005), 977.10.1103/RevModPhys.77.977Search in Google Scholar
[6] E. V. Yates, T. Müller, L. Rajah, E. J. De Genst, P. Arosio, S. Linse, M. Vendruscolo, C. M. Dobson, and T. P. J. Knowles, Latent analysis of unmodified biomolecules and their complexes in solution with attomole detection sensitivity, Nat. Chem. 7 (2015), 802.10.1038/nchem.2344Search in Google Scholar
[7] P. Arosio, T. Müller, L. Rajah, E. V. Yates, F. A. Aprile, Y. Zhang, S. I. A. Cohen, D. A. White, T. W. Herling, E. J. De Genst, S. Linse, M. Vendruscolo, C. M. Dobson, and T. P. J. Knowles, Microfluidic diffusion analysis of the sizes and interactions of proteins under native solution conditions, ACS Nano. 10 (2016), 333.10.1021/acsnano.5b04713Search in Google Scholar
[8] P. Arosio, K. Hu, F. A. Aprile, T. Müller, and T. P. J. Knowles, Microfluidic diffusion viscometer for rapid analysis of complex solutions, Anal. Chem. 88 (2016), 3488–3493.10.1021/acs.analchem.5b02930Search in Google Scholar
[9] M. Spiga and G. L. Morini, A Symmetric solution for velocity profile in laminar flow through rectangular ducts, Int. Commun. Heat Mass Transfer 21 (1994), 469.10.1016/0735-1933(94)90046-9Search in Google Scholar
[10] G. Taylor, Dispersion of soluble matter in solvent flowing slowly through a tube, Proc. R. Soc. A 219 (1953), 186.10.1098/rspa.1953.0139Search in Google Scholar
[11] A. E. Kamholz and P. Yager, Theoretical analysis of molecular diffusion in pressure-driven laminar flow in microfluidic channels, Biophys. J. 80 (2001), 155.10.1016/S0006-3495(01)76003-1Search in Google Scholar
[12] Z. Wu and N. T. Nguyen, Hydrodynamic focusing in microchannels under consideration of diffusive dispersion: theories and experiments, Sens. Actuators B 107 (2005), 965.10.1016/j.snb.2004.11.014Search in Google Scholar
[13] M. J. Kennedy, H. D. Ladouceur, T. Moeller, D. Kirui, and C. A. Batt, Analysis of a laminar-flow diffusional mixer for directed self-assembly of liposomes, Biomicrofluidics 6 (2012), 044119.10.1063/1.4772602Search in Google Scholar PubMed PubMed Central
[14] M. R. Schure, Digital simulation of sedimentation field-flow fractionation, Anal. Chem. 60 (1988), 1109.10.1021/ac00162a006Search in Google Scholar
[15] Z. Li and G. Drazer, Separation of suspended particles by arrays of obstacles in microfluidic devices, Phys. Rev. Lett. 98 (2007), 050602.10.1103/PhysRevLett.98.050602Search in Google Scholar PubMed
[16] A. Fokker, Die mittlere energie rotierender elektrischer dipole im strahlungsfeld, Ann. Phys. 348 (1914), 810.10.1002/andp.19143480507Search in Google Scholar
[17] M. von Smoluchowski, Über Brownsche Molekularbewegung unter Einwirkung äußerer Kräfte und deren Zusammenhang mit der verallgemeinerten Diffusionsgleichung, Ann. Phys. 48 (1915), 1103.10.1002/andp.19163532408Search in Google Scholar
[18] M. Planck, Über einen satz der statistischen dynamik und seine erweiterung in der quantentheorie, Sitzber. Preuss. Akad. Wiss. (1917), 324.Search in Google Scholar
[19] H. Risken, The Fokker-Planck Equation, Springer, Berlin, 1989.10.1007/978-3-642-61544-3Search in Google Scholar
[20] P. Langevin, Sur la théorie du mouvement brownien, C. R. Acad Sci. 146 (1908), 530.Search in Google Scholar
[21] G. Segré and A. Silberberg, Behaviour of macroscopic rigid spheres in poiseuille flow part 2. experimental results and interpretation, J. Fluid Mech. 14 (1962), 136.10.1017/S0022112062001111Search in Google Scholar
[22] J. Rotne and S. Prager, Variational treatment of hydrodynamic interaction in polymers, J. Chem. Phys. 50 (1969), 4831.10.1063/1.1670977Search in Google Scholar
[23] E. C. Eckstein, D. G. Bailey, and A. H. Shapiro, Self-diffusion of particles in shear flow of a suspension, J. Fluid Mech. 79 (1977), 191.10.1017/S0022112077000111Search in Google Scholar
[24] C. W. J. Beenakker, Ewald sum of the rotne–prager tensor, J. Chem. Phys. 85 (1986), 1581.10.1063/1.451199Search in Google Scholar
[25] D. Leighton and A. Acrivos, Measurement of shear-induced self-diffusion in concentrated suspensions of spheres, J. Fluid Mech. 177 (1987), 109.10.1017/S0022112087000880Search in Google Scholar
[26] F. R. da Cunha and E. J. Hinch, Shear-induced dispersion in a dilute suspension of rough spheres, J. Fluid Mech. 309 (1996), 21110.1017/S0022112096001619Search in Google Scholar
[27] Y. Wang, R. Mauri, and A. Acrivos, Transverse shear-induced gradient diffusion in a dilute suspension of spheres, J. Fluid Mech. 327 (1996), 255.10.1017/S0022112097008148Search in Google Scholar
[28] A. Hatch, A. E. Kamholz, K. R. Hawkins, M. S. Munson, E. A. Schilling, B. H. Weigl, and P. Yager, A rapid diffusion immunoassay in a T-sensor, Nat. Biotechnol. 19 (2001), 461.10.1038/88135Search in Google Scholar
[29] D. Bedeaux and P. Mazur, Brownian motion and fluctuating hydrodynamics, Physica 76 (1974), 247.10.1016/0031-8914(74)90198-0Search in Google Scholar
[30] D. C. Duffy, J. C. McDonald, O. J. A. Schueller, and G. M. Whitesides, Rapid prototyping of microfluidic systems in poly(dimethylsiloxane), Anal. Chem. 70 (1998), 4974.10.1021/ac980656zSearch in Google Scholar PubMed
Correction note
Correction added after online publication 15 April 2016: For Reference 8 the authors were changed from “P. Arosio, K. Hu, T. Müller, and T. P. J. Knowles” to “P. Arosio, K. Hu, F. A. Aprile, T. Müller, T. P. J. Knowles”.
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