Hostname: page-component-745bb68f8f-v2bm5 Total loading time: 0 Render date: 2025-01-09T21:43:36.390Z Has data issue: false hasContentIssue false
  • English
  • Français

Directional locking and deterministic separation in periodic arrays

Published online by Cambridge University Press:  25 May 2009

JOELLE FRECHETTE  and
GERMAN DRAZER
JOELLE FRECHETTE
Affiliation:
Chemical & Biomolecular Engineering Department, Johns Hopkins University, Baltimore, MD 21218, USA
GERMAN DRAZER*
Affiliation:
Chemical & Biomolecular Engineering Department, Johns Hopkins University, Baltimore, MD 21218, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate the dynamics of a non-Brownian sphere suspended in a quiescent fluid and moving through a periodic array of solid obstacles under the action of a constant external force by means of Stokesian dynamics simulations. We show that in the presence of non-hydrodynamic, short-range interactions between the solid obstacles and the suspended sphere, the moving particle becomes locked into periodic trajectories with an average orientation that coincides with one of the lattice directions and is, in general, different from the direction of the driving force. The locking angle depends on the details of the non-hydrodynamic interactions and could lead to vector separation of different species for certain orientations of the external force. We explicitly show the presence of separation for a mixture of suspended particles with different roughness, moving through a square lattice of spherical obstacles. We also present a dilute model based on the two-particle mobility and resistance functions for the collision between spheres of different sizes. This simple model predicts the separation of particles of different size and also suggests that microdevices that maximize the differences in interaction area between the different particles and the solid obstacles would be more sensitive for size separation based on non-hydrodynamic interactions.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 627 , 25 May 2009 , pp. 379 - 401
Copyright
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Adamczyk, Z., Adamczyk, M. & Vandeven, T. G. M. 1983 Resistance coefficient of a solid sphere approaching plane and curved boundaries. J. Colloid Interface Sci. 96 (1), 204–213.CrossRefGoogle Scholar
Allen, M. P. & Tildesley, D. J. 1989 Computer Simulation of Liquids. Oxford University PressGoogle Scholar
Arnold, V. I 1988 Geometrical Methods in the Theory of Ordinary Differential Equations, 2nd ed. Springer.Google Scholar
Austin, R. H., Darnton, N., Huang, R., Sturm, J., Bakajin, O. & Duke, T. 2002 Ratchets: the problems with boundary conditions in insulating fluids. Appl. Phys. A 75 (2), 279–284.CrossRefGoogle Scholar
Bak, P. 1986 The devil's staircase. Phys. Today 39 (12), 38–45.CrossRefGoogle Scholar
Bear, J. 1988 Dynamics of Fluids in Porous Media. Dover.Google Scholar
Blom, M. T., Chmela, E., Oosterbroek, R. E., Tijssen, R. & van den Berg, A. 2003 On-chip hydrodynamic chromatography separation and detection of nanoparticles and biomolecules. Anal. Chem. 75 (24), 6761–6768.CrossRefGoogle Scholar
Brady, J. F. & Bossis, G. 1988 Stokesian dynamics. Annu. Rev. Fluid Mech. 20, 111–157.CrossRefGoogle Scholar
Brenner, H. 1980 Dispersion resulting from flow through spatially periodic porous-media. Philos. Trans. R. Soc. Lond. A 297 (1430), 81–133.Google Scholar
Brenner, H. & Adler, P. M. 1982 Dispersion resulting from flow through spatially periodic porous-media. Part 2. Surface and intraparticle transport. Philos. Trans. R. Soc. Lond. A 307 (1498), 149–200.Google Scholar
Brenner, H. & Edwards, D. A. 1993 Macrotransport Processes. Butterworth-HeinemannGoogle Scholar
Cabodi, M., Chen, Y. F., Turner, S. W. P., Craighead, H. G. & Austin, R. H. 2002 Continuous separation of biomolecules by the laterally asymmetric diffusion array with out-of-plane sample injection. Electrophoresis 23 (20), 3496–3503.Google Scholar
daCunha, F. R. & Hinch, E. J. 1996 Shear-induced dispersion in a dilute suspension of rough spheres. J. Fluid Mech. 309, 211–223.Google Scholar
Davis, J. A., Inglis, D. W., Morton, K. J., Lawrence, D. A., Huang, L. R., Chou, S. Y., Sturm, J. C. & Austin, R. H. 2006 Deterministic hydrodynamics: taking blood apart. Proc. Natl. Acad. Sci. 103 (40), 14779–14784.CrossRefGoogle Scholar
Davis, R. H. 1992 Effects of surface-roughness on a sphere sedimenting through a dilute suspension of neutrally buoyant spheres. Phys. Fluids 4 (12), 2607–2619.Google Scholar
Davis, R. H., Zhao, Y., Galvin, K. P. & Wilson, H. J. 2003 Solid–solid contacts due to surface roughness and their effects on suspension behaviour. Philos. Trans. R. Soc. Lond. A 361 (1806), 871–894.Google Scholar
Dittrich, P. S., Tachikawa, K. & Manz, A. 2006 Micro total analysis systems: latest advancements and trends. Anal. Chem. 78 (12), 3887–3907.CrossRefGoogle ScholarPubMed
Dorfman, K. D. & Brenner, H. 2001 ‘Vector chromatography’: modeling micropatterned separation devices. J. Colloid Interface Sci. 238 (2), 390–413.CrossRefGoogle Scholar
Dorfman, K. D. & Brenner, H. 2002 Separation mechanisms underlying vector chromatography in microlithographic arrays. Phys. Rev. E 65 (5), 052103.CrossRefGoogle ScholarPubMed
Drazer, G., Khusid, B., Koplik, J. & Acrivos, A. 2005 a Hysteresis, force oscillations, and nonequilibrium effects in the adhesion of spherical nanoparticles to atomically smooth surfaces. Phys. Rev. Lett. 95 (1), 016102.Google Scholar
Drazer, G., Khusid, B., Koplik, J. & Acrivos, A. 2005 b Wetting and particle adsorption in nanoflows. Phys. Fluids 17 (1), 017102.Google Scholar
Drazer, G., Koplik, J., Acrivos, A. & Khusid, B. 2002 a Adsorption phenomena in the transport of a colloidal particle through a nanochannel containing a partially wetting fluid. Phys. Rev. Lett. 89 (24), 244501.Google Scholar
Drazer, G., Koplik, J., Khusid, B. & Acrivos, A. 2002 b Deterministic and stochastic behaviour of non-Brownian spheres in sheared suspensions. J. Fluid Mech. 460, 307–335.CrossRefGoogle Scholar
Drazer, G., Koplik, J., Khusid, B. & Acrivos, A. 2004 Microstructure and velocity fluctuations in sheared suspensions. J. Fluid Mech. 511, 237–263.Google Scholar
Duke, T. 1998 Separation techniques. Curr. Opin. Chem. Biol. 2 (5), 592–596.CrossRefGoogle Scholar
Duke, T. A. J. & Austin, R. H. 1998 Microfabricated sieve for the continuous sorting of macromolecules. Phys. Rev. Lett. 80 (7), 1552–1555.CrossRefGoogle Scholar
Durlofsky, L., Brady, J. F. & Bossis, G. 1987 Dynamic simulation of hydrodynamically interacting particles. J. Fluid Mech. 180, 21–49.Google Scholar
Edwards, D. A., Shapiro, M., Brenner, H. & Shapira, M. 1991 Dispersion of inert solutes in spatially periodic, 2-dimensional model porous-media. Transport Porous Med. 6 (4), 337–358.Google Scholar
Eijkel, J. C. T. & van den Berg, A. 2006 The promise of nanotechnology for separation devices: from a top-down approach to nature-inspired separation devices. Electrophoresis 27 (3), 677–685.CrossRefGoogle ScholarPubMed
Ekiel-Jezewska, M. L., Feuillebois, F., Lecoq, N., Masmoudi, K., Anthore, R., Bostel, F. & Wajnryb, E. 1999 Hydrodynamic interactions between two spheres at contact. Phys. Rev. E 59 (3), 3182–3191.CrossRefGoogle Scholar
Ekiel-Jezewska, M. L., Lecoq, N., Anthore, R., Bostel, F. & Feuillebois, F. 2002 Rotation due to hydrodynamic interactions between two spheres in contact. Phys. Rev. E 66 (5), 051504.CrossRefGoogle Scholar
Ertas, D. 1998 Lateral separation of macromolecules and polyelectrolytes in microlithographic arrays. Phys. Rev. Lett. 80 (7), 1548–1551.CrossRefGoogle Scholar
Fannjiang, A. & Papanicolaou, G. 1994 Convection enhanced diffusion for periodic flows. SIAM J. Appl. Math. 54 (2), 333–408.Google Scholar
Gadala-Maria, F. & Acrivos, A. 1980 Shear-induced structure in a concentrated suspension of solid spheres. J. Rheol. 24 (6), 799–814.Google Scholar
Giddings, J. C. 1991 Unified Separation Science. Wiley Interscience.Google Scholar
Goldstein, H. 1980 Classical Mechanics, 2nd ed. Addison-Wesley.Google Scholar
Gopinathan, A. & Grier, D. G. 2004 Statistically locked-in transport through periodic potential landscapes. Phys. Rev. Lett. 92 (13), 130602.CrossRefGoogle Scholar
Grier, D. G. 2003 A revolution in optical manipulation. Nature 424 (6950), 810–816.CrossRefGoogle Scholar
Hasimoto, H. 1959 On the periodic fundamental solutions of the Stokes equations and their application to viscous flow past a cubic array of spheres. J. Fluid Mech. 5 (2), 317–328.Google Scholar
Heller, M. & Bruus, H. 2008 A theoretical analysis of the resolution due to diffusion and size dispersion of particles in deterministic lateral displacement devices. J. Micromech. Microengng 18 (7), 075030.CrossRefGoogle Scholar
Huang, L. R., Cox, E. C., Austin, R. H. & Sturm, J. C. 2004 Continuous particle separation through deterministic lateral displacement. Science 304 (5673), 987–990.Google Scholar
Huang, L. R., Silberzan, P., Tegenfeldt, J. O., Cox, E. C., Sturm, J. C., Austin, R. H. & Craighead, H. 2002 Role of molecular size in ratchet fractionation. Phys. Rev. Lett. 89 (17), 178301.CrossRefGoogle Scholar
Inglis, D. W., Davis, J. A., Austin, R. H. & Sturm, J. C. 2006 Critical particle size for fractionation by deterministic lateral displacement. Lab Chip 6 (5), 655–658.Google Scholar
Jeffrey, D. J. & Onishi, Y. 1984 Calculation of the resistance and mobility functions for two unequal rigid spheres in low-Reynolds-number flow. J. Fluid Mech. 139, 261–290.CrossRefGoogle Scholar
Keller, C., Marquardt, F. & Bruder, C. 2002 Separation quality of a geometric ratchet. Phys. Rev. E 65 (4), 041927.CrossRefGoogle Scholar
Khandurina, J. & Guttman, A. 2003 Microscale separation and analysis. Curr. Opin. Chem. Biol. 7 (5), 595–602.Google Scholar
Koch, D. L., Cox, R. G., Brenner, H. & Brady, J. F. 1989 The effect of order on dispersion in porous media. J. Fluid Mech. 200, 173–188.CrossRefGoogle Scholar
Korda, P. T., Taylor, M. B. & Grier, D. G. 2002 Kinetically locked-in colloidal transport in an array of optical tweezers. Phys. Rev. Lett. 89 (12), 128301.CrossRefGoogle Scholar
Lacasta, A. M., Khoury, M., Sancho, J. M. & Lindenberg, K. 2006 Sorting of mesoscopic particles driven through periodic potential landscapes. Mod. Phys. Lett. 20 (23), 1427–1442.Google Scholar
Li, Z. & Drazer, G. 2007 Separation of suspended particles by arrays of obstacles in microfluidic devices. Phys. Rev. Lett. 98 (5), 050602.Google Scholar
Lin, Y. W., Huang, M. F. & Chang, H. T. 2005 Nanomaterials and chip-based nanostructures for capillary electrophoretic separations of dna. Electrophoresis 26, 320–330.CrossRefGoogle ScholarPubMed
Malysa, K., Dabros, T. & Van de Ven, T. G. M. 1986 The sedimentation of one sphere past a second attached to a wall. J. Fluid Mech. 162, 157–170.CrossRefGoogle Scholar
Marconi, V. I., Candia, S., Balenzuela, P., Pastoriza, H., Dominguez, D. & Martinoli, P. 2000 Orientational pinning and transverse voltage: Simulations and experiments in square Josephson junction arrays. Phys. Rev. B 62 (6), 4096–4104.Google Scholar
Masliyah, J. H. & Bhattacharjee, S. 2006 Electrokinetic and Colloid Transport Phenomena. Wiley InterscienceGoogle Scholar
Morris, J. F. & Brady, J. F. 1998 Pressure-driven flow of a suspension: Buoyancy effects. Intl J. Multiphase Flow 24 (1), 105–130.Google Scholar
Nitsche, J. M. & Brenner, H. 1988 Sedimentation and dispersion of Brownian particles in spatially periodic potential fields. J. Chem. Phys. 89 (12), 7510–7520.Google Scholar
Pamme, N. 2007 Continuous flow separations in microfluidic devices. Lab Chip 7, 1644–1659.CrossRefGoogle Scholar
Phillips, R. J., Deen, W. M. & Brady, J. F. 1989 Hindered transport of spherical macro-molecules in fibrous membranes and gels. AIChE J. 35 (11), 1761–1769.CrossRefGoogle Scholar
Phillips, R. J., Deen, W. M. & Brady, J. F. 1990 Hindered transport in fibrous membranes and gels: Effect of solute size and fiber configuration. J. Colloid Interface Sci. 139 (2), 363–373.Google Scholar
Rampall, I., Smart, J. R. & Leighton, D. T. 1997 The influence of surface roughness on the particle-pair distribution function of dilute suspensions of non-colloidal spheres in simple shear flow. J. Fluid Mech. 339, 1–24.Google Scholar
Reichhardt, C. & Reichhardt, C. J. O. 2004 Directional locking effects and dynamics for particles driven through a colloidal lattice. Phys. Rev. E 69 (4), 041405.CrossRefGoogle Scholar
Reimann, P. 2002 Brownian motors: noisy transport far from equilibrium. Phys. Rep. 361 (2-4), 57–265.CrossRefGoogle Scholar
Risken, H. 1989 The Fokker–Planck Equation, Methods of Solution and Application, 2nd ed. Springer.Google Scholar
Slater, G. W., Desrulsseaux, C., Hubert, S. J., Mercier, J. F., Labrie, J., Boileau, J., Tessier, F. & Pepin, M. P. 2000 Theory of DNA electrophoresis: a look at some current challenges. Electrophoresis 21 (18), 3873–3887.3.0.CO;2-8>CrossRefGoogle Scholar
Smart, J. R. & Leighton, D. T. 1989 Measurement of the hydrodynamic surface-roughness of noncolloidal spheres. Phys. Fluids 1 (1), 52–60.CrossRefGoogle Scholar
Squires, T. M. & Quake, S. R. 2005 Microfluidics: fluid physics at the nanoliter scale. Rev. Mod. Phys. 77 (3), 977–1026.Google Scholar
Szantai, E. & Guttman, A. 2006 Genotyping with microfluidic devices. Electrophoresis 27 (24), 4896–4903.CrossRefGoogle Scholar
Szekely, L. & Guttman, A. 2005 New advances in microchip fabrication for electrochromatography. Electrophoresis 26 (24), 4590–4604.CrossRefGoogle Scholar
Tegenfeldt, J. O., Prinz, C., Cao, H., Huang, R. L., Austin, R. H., Chou, S. Y., Cox, E. C. & Sturm, J. C. 2004 Micro- and nanofluidics for DNA analysis. Anal. Bioanal. Chem. 378 (7), 1678–1692.CrossRefGoogle ScholarPubMed
Vilkner, T., Janasek, D. & Manz, A. 2004 Micro total analysis systems: recent developments. Anal. Chem. 76 (12), 3373–3385.CrossRefGoogle Scholar
Weinan, E. 1992 Homogenization of linear and nonlinear transport-equations. Comm. Pure Appl. Math. 45 (3), 301–326.Google Scholar
Zeng, S. L., Kerns, E. T. & Davis, R. H. 1996 The nature of particle contacts in sedimentation. Phys. Fluids 8 (6), 1389–1396.CrossRefGoogle Scholar
Zhao, Y. & Davis, R. H. 2002 Interaction of two touching spheres in a viscous fluid. Chem. Engng Sci 57 (11), 1997–2006.Google Scholar
Zhao, Y. & Davis, R. H. 2003 Interaction of sedimenting spheres with multiple surface roughness scales. J. Fluid Mech. 492, 101–129.CrossRefGoogle Scholar