Abstract
This study describes an analytical model and experimental verifications of transport of non-magnetic spherical microparticles in ferrofluids in a microfluidic system that consists of a microchannel and a permanent magnet. The permanent magnet produces a spatially non-uniform magnetic field that gives rise to a magnetic buoyancy force on particles within ferrofluid-filled microchannel. We obtained trajectories of particles in the microchannel by (1) calculating magnetic buoyancy force through the use of an analytical expression of magnetic field distributions and a nonlinear magnetization model of ferrofluids, (2) deriving governing equations of motion for particles through the use of analytical expressions of dominant magnetic buoyancy and hydrodynamic viscous drag forces, (3) solving equations of motion for particles in laminar flow conditions. We studied effects of particle size and flow rate in the microchannel on the trajectories of particles. The analysis indicated that particles were increasingly deflected in the direction that was perpendicular to the flow when size of particles increased, or when flow rate in the microchannel decreased. We also studied “wall effect” on the trajectories of particles in the microchannel when surfaces of particles were in contact with channel wall. Experimentally obtained trajectories of particles were used to confirm the validity of our analytical results. We believe this study forms the theoretical foundation for size-based particle (both synthetic and biological) separation in ferrofluids in a microfluidic device. The simplicity and versatility of our analytical model make it useful for quick optimizations of future separation devices as the model takes into account important design parameters including particle size, property of ferrofluids, magnetic field distribution, dimension of microchannel, and fluid flow rate.
Similar content being viewed by others
References
Berkovsky BM, Medvedev VF, Krakov MS (1993) Magnetic fluids: engineering applications. Oxford University Press, New York
Bonner WA, Hulett HR, Sweet RG, Herzenberg LA (1972) Fluorescence activated cell sorting. Rev Sci Instrum 43(3):404–409
Brody JP, Yager P, Goldstein RE, Austin RH (1996) Biotechnology at low Reynolds numbers. Biophys J 71(6):3430–3441
Cho YK, Lee JG, Park JM, Lee BS, Lee Y, Ko C (2007) One-step pathogen specific DNA extraction from whole blood on a centrifugal microfluidic device. Lab Chip 7(5):565–573
Davis JA, Inglis DW, Morton KJ, Lawrence DA, Huang LR, Chou SY, Sturm JC, Austin RH (2006) Deterministic hydrodynamics: taking blood apart. Proc Natl Acad Sci USA 103(40):14779–14784
Deen WM (1998) Analysis of transport phenomena. Oxford University Press, New York
Di Carlo D (2009) Inertial microfluidics. Lab Chip 9(21):3038–3046
Easley CJ, Karlinsey JM, Bienvenue JM, Legendre LA, Roper MG, Feldman SH, Hughes MA, Hewlett EL, Merkel TJ, Ferrance JP, Landers JP (2006) A fully integrated microfluidic genetic analysis system with sample-in-answer-out capability. Proc Natl Acad Sci USA 103(51):19272–19277
Einstein A (1956) Investigations on the theory of Brownian movement. Dover, New York
Furlani EP (2006) Analysis of particle transport in a magnetophoretic microsystem. J Appl Phys 99(2):024912
Furlani EP, Sahoo Y (2006) Analytical model for the magnetic field and force in a magnetophoretic microsystem. J Phys D 39(9):1724–1732
Ganatos P, Pfeffer R, Weinbaum S (1980) A strong interaction theory for the creeping motion of a sphere between plane parallel boundaries.2. Parallel motion. J Fluid Mech 99:755–783
Gassner AL, Abonnenc M, Chen HX, Morandini J, Josserand J, Rossier JS, Busnel JM, Girault HH (2009) Magnetic forces produced by rectangular permanent magnets in static microsystems. Lab Chip 9:2356–2363
Gijs MAM, Lacharme F, Lehmann U (2010) Microfluidic applications of magnetic particles for biological analysis and catalysis. Chem Rev 110(3):1518–1563
Gossett DR, Weaver WM, Mach AJ, Hur SC, Tse HTK, Lee W, Amini H, Di Carlo D (2010) Label-free cell separation and sorting in microfluidic systems. Anal Bioanal Chem 397(8):3249–3267
Han KH, Frazier AB (2004) Continuous magnetophoretic separation of blood cells in microdevice format. J Appl Phys 96(10):5797–5802
Huang LR, Cox EC, Austin RH, Sturm JC (2004) Continuous particle separation through deterministic lateral displacement. Science 304(5673):987–990
Ichikawa N, Hosokawa K, Maeda R (2004) Interface motion of capillary-driven flow in rectangular microchannel. J Colloid Interface Sci 280(1):155–164
Jones TB (1995) Electromechanics of particles. Cambridge University Press, Cambridge
Kose AR, Fischer B, Mao L, Koser H (2009) Label-free cellular manipulation and sorting via biocompatible ferrofluids. Proc Natl Acad Sci USA 106(51):21478–21483
Krebs MD, Erb RM, Yellen BB, Samanta B, Bajaj A, Rotello VM, Alsberg E (2009) Formation of ordered cellular structures in suspension via label-free negative magnetophoresis. Nano Lett 9(5):1812–1817
Krishnan GP, Leighton DT (1995) Inertial lift on a moving sphere in contact with a plane wall in a shear-flow. Phys Fluids 7(11):2538–2545
Kumar A, Bhardwaj A (2008) Methods in cell separation for biomedical application: cryogels as a new tool. Biomed Mater 3(3):034008
Laurell T, Petersson F, Nilsson A (2007) Chip integrated strategies for acoustic separation and manipulation of cells and particles. Chem Soc Rev 36(3):492–506
Lee H, Purdon AM, Chu V, Westervelt RM (2004) Controlled assembly of magnetic nanoparticles from magnetotactic bacteria using microelectromagnets arrays. Nano Lett 4(5):995–998
Leighton D, Acrivos A (1985) The lift on a small sphere touching a plane in the presence of a simple shear flow. Z Angew Math Phys 36:174–178
Li J, Zhang Z, Rosenzweig J, Wang YY, Chan DW (2002) Proteomics and bioinformatics approaches for identification of serum biomarkers to detect breast cancer. Clin Chem 48(8):1296–1304
Liu C, Stakenborg T, Peeters S, Lagae L (2009) Cell manipulation with magnetic particles toward microfluidic cytometry. J Appl Phys 105(10):102011–102014
Mao L, Koser H (2005) Ferrohydrodynamic pumping in spatially traveling sinusoidally time-varying magnetic fields. J Magn Magn Mater 289:199–202
Mao LD, Koser H (2006) Towards ferrofluidics for mu-TAS and lab on-a-chip applications. Nanotechnology 17(4):S34–S47
Miltenyi S, Muller W, Weichel W, Radbruch A (1990) High gradient magnetic cell separation with MACS. Cytometry 11(2):231–238
Mirica KA, Shevkoplyas SS, Phillips ST, Gupta M, Whitesides GM (2009) Measuring densities of solids and liquids using magnetic levitation: fundamentals. J Am Chem Soc 131(29):10049–10058
Mirica KA, Phillips ST, Mace CR, Whitesides GM (2010) Magnetic levitation in the analysis of foods and water. J Agric Food Chem 58(11):6565–6569
Muldoon LL, Sandor M, Pinkston KE, Neuwelt EA (2005) Imaging, distribution, and toxicity of superparamagnetic iron oxide magnetic resonance nanoparticles in the rat brain and intracerebral tumor. Neurosurgery 57(4):785–796
Nagrath S, Sequist LV, Maheswaran S, Bell DW, Irimia D, Ulkus L, Smith MR, Kwak EL, Digumarthy S, Muzikansky A, Ryan P, Balis UJ, Tompkins RG, Haber DA, Toner M (2007) Isolation of rare circulating tumour cells in cancer patients by microchip technology. Nature 450(7173):1235–1239
Odenbach S (2002) Ferrofluids: magnetically controllable fluids and their applications. Springer, London
Pamme N (2006) Magnetism and microfluidics. Lab Chip 6(1):24–38
Pamme N (2007) Continuous flow separations in microfluidic devices. Lab Chip 7(12):1644–1659
Pamme N, Manz A (2004) On-chip free-flow magnetophoresis: continuous flow separation of magnetic particles and agglomerates. Anal Chem 76(24):7250–7256
Pamme N, Wilhelm C (2006) Continuous sorting of magnetic cells via on-chip free-flow magnetophoresis. Lab Chip 6(8):974–980
Pankhurst QA, Connolly J, Jones SK, Dobson J (2003) Applications of magnetic nanoparticles in biomedicine. J Phys D 36(13):R167–R181
Petersson F, Aberg L, Sward-Nilsson AM, Laurell T (2007) Free flow acoustophoresis: microfluidic-based mode of particle and cell separation. Anal Chem 79(14):5117–5123
Peyman SA, Kwan EY, Margarson O, Iles A, Pamme N (2009) Diamagnetic repulsion—a versatile tool for label-free particle handling in microfluidic devices. J Chromatogr A 1216(52):9055–9062
Pham P, Masse P, Berthier J (2000) Numerical modeling of superparamagnetic sub-micronic particles trajectories under the coupled action of 3D force fields. Eur Phys J Appl Phys 12(3):211–216
Pieranski P, Clausen S, Helgesen G, Skjeltorp AT (1996) Braids plaited by magnetic holes. Phys Rev Lett 77(8):1620–1623
Rosensweig RE (1966) Fluidmagnetic buoyancy. AIAA J 4:1751–1758
Rosensweig RE (1985) Ferrohydrodynamics. Cambridge University Press, Cambridge
Rosensweig RE, Lee WK, Siegell JH (1987) Magnetically stabilized fluidized-beds for solids separation by density. Sep Sci Technol 22(1):25–45
Schuler D, Frankel RB (1999) Bacterial magnetosomes: microbiology, biomineralization and biotechnological applications. Appl Microbiol Biol 52(4):464–473
Shi JJ, Huang H, Stratton Z, Huang YP, Huang TJ (2009) Continuous particle separation in a microfluidic channel via standing surface acoustic waves (SSAW). Lab Chip 9(23):3354–3359
Skjeltorp AT (1983) One- and two-dimensional crystallization of magnetic holes. Phys Rev Lett 51(25):2306–2309
Smistrup K, Hansen O, Bruus H, Hansen MF (2005) Magnetic separation in microfluidic systems using microfabricated electromagnets—experiments and simulations. J Magn Magn Mater 293:597–604
Toner M, Irimia D (2005) Blood-on-a-chip. Annu Rev Biomed Eng 7:77–103
Tsutsui H, Ho CM (2009) Cell separation by non-inertial force fields in microfluidic systems. Mech Res Commun 36(1):92–103
Voldman J (2006) Electrical forces for microscale cell manipulation. Annu Rev Biomed Eng 8:425–454
Weissleder R, Stark DD, Engelstad BL, Bacon BR, Compton CC, White DL, Jacobs P, Lewis J (1989) Superparamagnetic iron-oxide—pharmacokinetics and toxicity. Am J Roentgenol 152(1):167–173
Willard MA, Kurihara LK, Carpenter EE, Calvin S, Harris VG (2004) Chemically prepared magnetic nanoparticles. Int Mater Rev 49(3-4):125–170
Winkleman A, Perez-Castillejos R, Gudiksen KL, Phillips ST, Prentiss M, Whitesides GM (2007) Density-based diamagnetic separation: devices for detecting binding events and for collecting unlabeled diamagnetic particles in paramagnetic solutions. Anal Chem 79(17):6542–6550
Wirix-Speetjens R, Fyen W, Xu KD, De Boeck J, Borghs G (2005) A force study of on-chip magnetic particle transport based on tapered conductors. IEEE Trans Magn 41(10):4128–4133
Xia Y, Whitesides GM (1998) Soft lithography. Annu Rev Mater Res 28(1):153–184
Yamada M, Seki M (2005) Hydrodynamic filtration for on-chip particle concentration and classification utilizing microfluidics. Lab Chip 5(11):1233–1239
Yamada M, Nakashima M, Seki M (2004) Pinched flow fractionation: continuous size separation of particles utilizing a laminar flow profile in a pinched microchannel. Anal Chem 76(18):5465–5471
Yellen BB, Hovorka O, Friedman G (2005) Arranging matter by magnetic nanoparticle assemblers. Proc Natl Acad Sci USA 102(25):8860–8864
Yung CW, Fiering J, Mueller AJ, Ingber DE (2009) Micromagnetic-microfluidic blood cleansing device. Lab Chip 9(9):1171–1177
Zborowski M, Ostera GR, Moore LR, Milliron S, Chalmers JJ, Schechter AN (2003) Red blood cell magnetophoresis. Biophys J 84(4):2638–2645
Zhu TT, Marrero F, Mao LD (2010) Continuous separation of non-magnetic particles inside ferrofluids. Microfluid Nanofluid 9(4–5):1003–1009
Acknowledgments
This research was financially supported by the Office of the Vice President for Research at the University of Georgia, and by the B3I Seed Grant Program with funds from the University of Georgia Research Foundation, Inc.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhu, T., Lichlyter, D.J., Haidekker, M.A. et al. Analytical model of microfluidic transport of non-magnetic particles in ferrofluids under the influence of a permanent magnet. Microfluid Nanofluid 10, 1233–1245 (2011). https://doi.org/10.1007/s10404-010-0754-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10404-010-0754-5