Skip to main content
SpringerLink
Log in

Three-dimensional and analytical modeling of microfluidic particle transport in magnetic fluids

  • Research Paper
  • Published:
Microfluidics and Nanofluidics Aims and scope Submit manuscript

Abstract

We present an analytical model that can predict the three-dimensional (3D) transport of non-magnetic particles in magnetic fluids inside a microfluidic channel coupled with permanent magnets. The magnets produce a spatially non-uniform magnetic field that gives rise to a magnetic buoyancy force on the particles. Resulting 3D trajectories of the particles are obtained by (1) calculating the 3D magnetic buoyancy force exerted on the particles via an analytical distribution of magnetic fields as well as their gradients, together with a nonlinear magnetization model of the magnetic fluids, (2) deriving the 3D hydrodynamic viscous drag force on the particles with an analytical velocity profile of a low Reynolds number ferrohydrodynamic flow in the channel including “wall effect” and magnetoviscous effect of the magnetic fluids, and (3) constituting and solving the governing equations of motion for the particles using the analytical expressions of magnetic buoyancy force and hydrodynamic viscous drag force. We use such a model to study the particles’ trajectories in the channel and investigate the magnitude of their deflections at different flow rates, with different properties of magnetic fluids and different geometrical parameters of the system.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  • Adams AA, Okagbare PI, Feng J, Hupert ML, Patterson D, Gottert J, McCarley RL, Nikitopoulos D, Murphy MC, Soper SA (2008) Highly efficient circulating tumor cell isolation from whole blood and label-free enumeration using polymer-based microfluidics with an integrated conductivity sensor. J Am Chem Soc 130(27):8633–8641. doi:10.1021/Ja8015022

    Article  Google Scholar 

  • Beyor N, Seo TS, Liu P, Mathies RA (2008) Immunomagnetic bead-based cell concentration microdevice for dilute pathogen detection. Biomed Microdevices 10(6):909–917. doi:10.1007/S10544-008-9206-3

    Article  Google Scholar 

  • Bonner WA, Sweet RG, Hulett HR, Herzenbe La (1972) Fluorescence activated cell sorting. Rev Sci Instrum 43(3):404

    Article  Google Scholar 

  • Brody JP, Yager P, Goldstein RE, Austin RH (1996) Biotechnology at low Reynolds numbers. Biophys J 71(6):3430–3441

    Article  Google Scholar 

  • 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. doi:10.1073/Pnas.0605967103

    Article  Google Scholar 

  • Dharmasiri U, Witek MA, Adams AA, Osiri JK, Hupert ML, Bianchi TS, Roelke DL, Soper SA (2010) Enrichment and detection of Escherichia coli O157:H7 from water samples using an antibody modified microfluidic chip. Anal Chem 82(7):2844–2849. doi:10.1021/Ac100323k

    Article  Google Scholar 

  • Furlani EP (2001) Permanent magnet and electromechanical devices. Academic Press, New York

    Google Scholar 

  • Furlani EP (2006) Analysis of particle transport in a magnetophoretic microsystem. J Appl Phys 99(2):024912

    Article  Google Scholar 

  • Furlani EP (2007) Magnetophoretic separation of blood cells at the microscale. J Phys D Appl Phys 40(5):1313

    Article  Google Scholar 

  • Furlani EP, Sahoo Y (2006) Analytical model for the magnetic field and force in a magnetophoretic microsystem. J Phys D Appl Phys 39(9):1724

    Article  Google Scholar 

  • Ganatos P, Weinbaum S, Pfeffer R (1980) A strong interaction theory for the creeping motion of a sphere between plane parallel boundaries. Part 1. Perpendicular motion. J Fluid Mech 99(04):739–753. doi:10.1017/S0022112080000870

    Article  MATH  Google Scholar 

  • Gassner A-L, Abonnenc M, Chen H-X, Morandini J, Josserand J, Rossier JS, Busnel J-M, Girault HH (2009) Magnetic forces produced by rectangular permanent magnets in static microsystems. Lab Chip 9(16):2356–2363

    Article  Google Scholar 

  • Gijs MAM, Lacharme F, Lehmann U (2010) Microfluidic applications of magnetic particles for biological analysis and catalysis. Chem Rev 110(3):1518–1563. doi:10.1021/Cr9001929

    Article  Google Scholar 

  • 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. doi:10.1007/S00216-010-3721-9

    Article  Google Scholar 

  • He L, Hu Y, Kim H, Ge J, Kwon S, Yin Y (2010) Magnetic assembly of nonmagnetic particles into photonic crystal structures. Nano Lett 10(11):4708–4714. doi:10.1021/nl103008v

    Article  Google Scholar 

  • Hoshino K, Huang YY, Lane N, Huebschman M, Uhr JW, Frenkel EP, Zhang XJ (2011) Microchip-based immunomagnetic detection of circulating tumor cells. Lab Chip 11(20):3449–3457. doi:10.1039/C1lc20270g

    Article  Google Scholar 

  • Huang LR, Cox EC, Austin RH, Sturm JC (2004) Continuous particle separation through deterministic lateral displacement. Science 304(5673):987–990

    Article  Google Scholar 

  • Huang P, Guasto JS, Breuer KS (2006) Direct measurement of slip velocities using three-dimensional total internal reflection velocimetry. J Fluid Mech 566:447–464. doi:10.1017/S0022112006002229

    Article  MATH  Google Scholar 

  • Ichikawa N, Hosokawa K, Maeda R (2004) Interface motion of capillary-driven flow in rectangular microchannel. J Colloid Interface Sci 280(1):155–164. doi:10.1016/j.jcis.2004.07.017

    Article  Google Scholar 

  • Kose AR, Fischer B, Mao L, Koser H (2009) Label-free cellular manipulation and sorting via biocompatible ferrofluids. Proc Natl Acad Sci. doi:10.1073/pnas.0912138106

    MATH  Google Scholar 

  • Krishnan GP, David T, Leighton J (1995) Inertial lift on a moving sphere in contact with a plane wall in a shear flow. Phys Fluids 7(11):2538–2545

    Article  MATH  Google Scholar 

  • 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. doi:10.1039/B601326k

    Article  Google Scholar 

  • 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. doi:10.1021/Nl049562x

    Article  Google Scholar 

  • Lenshof A, Laurell T (2010) Continuous separation of cells and particles in microfluidic systems. Chem Soc Rev 39(3):1203–1217. doi:10.1039/B915999c

    Article  Google Scholar 

  • Liang L, Zhu J, Xuan X (2011) Three-dimensional diamagnetic particle deflection in ferrofluid microchannel flows. Biomicrofluidics 5(3):034110

    Article  Google Scholar 

  • Liu RH, Yang JN, Lenigk R, Bonanno J, Grodzinski P (2004) Self-contained, fully integrated biochip for sample preparation, polymerase chain reaction amplification, and DNA microarray detection. Anal Chem 76(7):1824–1831. doi:10.1021/Ac0353029

    Article  Google Scholar 

  • Liu CX, Stakenborg T, Peeters S, Lagae L (2009), Cell manipulation with magnetic particles toward microfluidic cytometry. J Appl Phys 105(10):102014-1–102014-11

    Google Scholar 

  • Mao X, Huang TJ (2012a) Exploiting mechanical biomarkers in microfluidics. Lab Chip 12(20):4006–4009. doi:10.1039/c2lc90100e

    Article  Google Scholar 

  • Mao X, Huang TJ (2012b) Microfluidic diagnostics for the developing world. Lab Chip 12(8):1412–1416. doi:10.1039/c2lc90022j

    Article  Google Scholar 

  • McFaul SM, Lin BK, Ma H (2012) Cell separation based on size and deformability using microfluidic funnel ratchets. Lab Chip 12(13):2369–2376. doi:10.1039/c2lc21045b

    Article  Google Scholar 

  • 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. doi:10.1038/Nature06385

    Article  Google Scholar 

  • Odenbach S (2002) Magnetoviscous effects in ferrofluids. Springer, New York

    MATH  Google Scholar 

  • Pamme N (2006) Magnetism and microfluidics. Lab Chip 6(1):24–38. doi:10.1039/B513005k

    Article  Google Scholar 

  • Pamme N (2007) Continuous flow separations in microfluidic devices. Lab Chip 7(12):1644–1659. doi:10.1039/B712784g

    Article  Google Scholar 

  • Pethig R (2010) Review article—dielectrophoresis: status of the theory, technology, and applications. Biomicrofluidics 4(2):022811

    Article  Google Scholar 

  • Robert D, Pamme N, Conjeaud H, Gazeau F, Iles A, Wilhelm C (2011) Cell sorting by endocytotic capacity in a microfluidic magnetophoresis device. Lab Chip 11(11):1902–1910. doi:10.1039/c0lc00656d

    Article  Google Scholar 

  • Rosensweig RE (1985) Ferrohydrodynamics. Cambridge University Press, Cambridge

    Google Scholar 

  • Shen F, Hwang H, Hahn YK, Park J-K (2012) Label-free cell separation using a tunable magnetophoretic repulsion force. Anal Chem 84(7):3075–3081. doi:10.1021/ac201505j

    Article  Google Scholar 

  • 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. doi:10.1039/B915113c

    Article  Google Scholar 

  • Staben ME, Zinchenko AZ, Davis RH (2003) Motion of a particle between two parallel plane walls in low-Reynolds-number Poiseuille flow. Phys Fluids 15(6):1711–1733

    Article  Google Scholar 

  • Tarn MD, Peyman SA, Pamme N (2013) Simultaneous trapping of magnetic and diamagnetic particle plugs for separations and bioassays. RSC Advances 3(20):7209–7214. doi:10.1039/c3ra40237a

    Article  Google Scholar 

  • Toner M, Irimia D (2005) Blood-on-a-chip. Annu Rev Biomed Eng 7:77–103. doi:10.1146/Annurev.Bioeng.7.011205.135108

    Article  Google Scholar 

  • Tsutsui H, Ho CM (2009) Cell separation by non-inertial force fields in microfluidic systems. Mech Res Commun 36(1):92–103. doi:10.1016/J.Mechrescom.2008.08.006

    Article  MATH  Google Scholar 

  • Vojtíšek M, Tarn M, Hirota N, Pamme N (2012) Microfluidic devices in superconducting magnets: on-chip free-flow diamagnetophoresis of polymer particles and bubbles. Microfluid Nanofluid 13(4):625–635. doi:10.1007/s10404-012-0979-6

    Article  Google Scholar 

  • Voldman J (2006) Electrical forces for microscale cell manipulation. Annu Rev Biomed Eng 8:425–454. doi:10.1146/Annurev.Bioeng.8.061505.095739

    Article  Google Scholar 

  • Wang FC, Zhao YP (2011) Slip boundary conditions based on molecular kinetic theory: the critical shear stress and the energy dissipation at the liquid-solid interface. Soft Matter 7(18):8628–8634. doi:10.1039/C1sm05543g

    Article  Google Scholar 

  • Wang ZC, Zhe JA (2011) Recent advances in particle and droplet manipulation for lab-on-a-chip devices based on surface acoustic waves. Lab Chip 11(7):1280–1285. doi:10.1039/C0lc00527d

    Article  Google Scholar 

  • Winkleman A, Gudiksen KL, Ryan D, Whitesides GM, Greenfield D, Prentiss M (2004) A magnetic trap for living cells suspended in a paramagnetic buffer. Appl Phys Lett 85(12):2411–2413

    Article  Google Scholar 

  • 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. doi:10.1021/Ac049863r

    Article  Google Scholar 

  • Yellen BB, Hovorka O, Friedman G (2005) Arranging matter by magnetic nanoparticle assemblers. Proc Natl Acad Sci USA 102(25):8860–8864

    Article  Google Scholar 

  • Yung CW, Fiering J, Mueller AJ, Ingber DE (2009) Micromagnetic-microfluidic blood cleansing device. Lab Chip 9(9):1171–1177. doi:10.1039/B816986a

    Article  Google Scholar 

  • Zborowski M, Ostera GR, Moore LR, Milliron S, Chalmers JJ, Schechter AN (2003) Red blood cell magnetophoresis. Biophys J 84(4):2638–2645. doi:10.1016/S0006-3495(03)75069-3

    Article  Google Scholar 

  • Zeng J, Chen C, Vedantam P, Brown V, Tzeng T-RJ, Xuan X (2012) Three-dimensional magnetic focusing of particles and cells in ferrofluid flow through a straight microchannel. J Micromech Microeng 22(10):105018

    Article  Google Scholar 

  • Zhang K, Liang Q, Ai X, Hu P, Wang Y, Luo G (2011) On-demand microfluidic droplet manipulation using hydrophobic ferrofluid as a continuous-phase. Lab Chip 11(7):1271–1275. doi:10.1039/c0lc00484g

    Article  Google Scholar 

  • Zhu TT, Marrero F, Mao LD (2010) Continuous separation of non-magnetic particles inside ferrofluids. Microfluid Nanofluid 9(4–5):1003–1009. doi:10.1007/S10404-010-0616-1

    Article  Google Scholar 

  • Zhu T, Cheng R, Mao L (2011a) Focusing microparticles in a microfluidic channel with ferrofluids. Microfluid Nanofluid 11(6):695–701. doi:10.1007/s10404-011-0835-0

    Article  Google Scholar 

  • Zhu T, Lichlyter D, Haidekker M, Mao L (2011b) Analytical model of microfluidic transport of non-magnetic particles in ferrofluids under the influence of a permanent magnet. Microfluid Nanofluid 10(6):1233–1245. doi:10.1007/s10404-010-0754-5

    Article  Google Scholar 

  • Zhu TT, Cheng R, Lee SA, Rajaraman E, Eiteman MA, Querec TD, Unger ER, Mao LD (2012) Continuous-flow ferrohydrodynamic sorting of particles and cells in microfluidic devices. Microfluid Nanofluid 13(4):645–654. doi:10.1007/S10404-012-1004-9

    Article  Google Scholar 

Download references

Acknowledgments

This work is supported by the National Science Foundation and the Office of Vice President for Research at the University of Georgia.

Author information

Authors and Affiliations

  1. College of Engineering, Nanoscale Science and Engineering Center, University of Georgia, Athens, GA, 30602, USA

    Rui Cheng & Leidong Mao

  2. Department of Chemistry, Nanoscale Science and Engineering Center, University of Georgia, Athens, GA, 30602, USA

    Taotao Zhu

Authors
  1. Rui Cheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Taotao Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Leidong Mao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Leidong Mao.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cheng, R., Zhu, T. & Mao, L. Three-dimensional and analytical modeling of microfluidic particle transport in magnetic fluids. Microfluid Nanofluid 16, 1143–1154 (2014). https://doi.org/10.1007/s10404-013-1280-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10404-013-1280-z

Keywords

Search

Navigation