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  • Ph.D. Thesis, s010132

    Magnetic Separation andHydrodynamic Interactions in

    Microfluidic Systems

    Christian Ingemann Mikkelsen

    Main Supervisor: Henrik Bruus, Co-supervisor: Mikkel Fougt Hansen

    MICDepartment of Micro and NanotechnologyTechnical University of Denmark

    May 18, 2005

  • ii

  • Abstract

    Magnetic bead manipulation, in particular separation, in microfluidic systems is a tech-nique which offers to simplify and integrate separation and rinsing procedures for minutesamples of biological material. We study the physics of magnetic bead motion in suchsystems.

    The force on a body in a magnetic field is determined from principles of thermody-namics. From that result, the force between two spherical linearly paramagnetic particlesimmersed in an external magnetic field is derived provided that they are well separated.Furthermore, we find the magnetic dipole moments that two linearly paramagnetic spher-ical particles induce in one another; both being immersed in an external magnetic field.The leading magnetic interaction force decays as particle separation to the power 3.

    The hydrodynamic interaction, which stems from the fluid motion set about by par-ticles moving in a viscous fluid, is shown to decay with separation to the power 1 bymeans of hydrodynamic Greens functions. The magnetic interaction decays much fasterwith separation. This significantly influences the dynamics of magnetic bead motion whichis illustrated through numerical simulations that study individual beads.

    Instead of adding more and more beads on an individual basis, we go on to treat thebeads as a continuum described by a distribution that is coupled to the problem of fluidflow in a model microfluidic channel. This shows that hydrodynamic interactions help thecapturing of magnetic beads and that this depends on the concentration of beads. Aneffort is underway to test this prediction in experiments.

    Finally, we have derived an analytic framework for the description of the slow motionof spherical particles in a viscous fluid in confined geometries. This enables us to derivea first approximation to the mobility of a spherical particle at the centre of a cube filledwith viscous fluid.

    iii

  • iv ABSTRACT

  • Resume

    Manipulation af magnetiske partikler, i srdeleshed separation, i mikrovskesystemer eren teknik, som tillader simplifikation og integration af separations- og oprensningsproce-durer for sma mngder af biologisk materiale. Vi studerer her fysikken bag magnetiskekuglers bevgelse i sadanne systemer.

    Vi udleder kraften pa et legeme i et magnetfelt udfra termodynamiske principper ogfra dette udledt kraften mellem to sfriske, linert paramagnetiske partikler i et eksterntmagnetfelt under antagelse af, at de ikke er for tt sammen. Derudover finder vi demagnetiske dipolemomenter, som to sfriske, linert paramagnetiske partikler inducerer ihinanden, nar de er patrykt et eksternt magnetfelt. Den frende magnetiske vekselvirkingaftager med partikelafstanden i tredie potens.

    Den hydrodynamiske vekselvirkning, der skyldes vskebevgelse, der opstar nar par-tiklerne bevger sig gennem vsken, vises ved hjlp af hydrodynamiske Greensfunktionerat aftage med partikelafstanden i frste potense. Den magnetiske vekselvirkning aftageraltsa meget hurtigere med afstanden. Dette pavirker afgrende de magnetiske kuglersbevgelse, hvilket vi illustrerer ved numeriske simulationer, hvor vi behandler kuglernesrskilt.

    I stedet for at tilfre flere kugler en ad gangen gar vi videre og behandler kuglerne somet kontinuum, som beskrives ved en fordeling, som er koblet sammen med vskestrmningsproblemeti en modelmikrovskekanal. Dette viser, at hydrodynamiske vekselvirkninger hjlper ind-fangningen af magnetiske kugler, og at dette afhnger af koncentrationen af kugler. Derbliver arbejdet pa at undersge denne forudsigelse eksperimentelt.

    Endeligt har vi udledt et analytisk redskab til beskrivelse af langsomtbevgende sfriskepartikler i visks vske indesluttet i en beholder. Dette gr det muligt at udlede enfrstetilnrmelse til mobiliteten for en kugleformet partikel placeret i centrum af en tern-ingformet kasse fyldt med visks vske.

    v

  • vi RESUME

  • Preface

    The present dissertation is the result of the work I have undertaken over the last 18months under the supervision of professor Henrik Bruus and associate professor MikkelFougt Hansen both at MIC who both kindly took me under their scientific wings after Ihad to terminate the PhD-project that I had hitherto pursued.

    I happily seized the opportunity to work in a, to me, completely new field and goingthrough the quandary phase. A year and a half is not a long time to contribute tothe advancement of knowledge and a comparably large part of my effort has gone intocomputer simulations which are forgiving in the sense that one can be rather certain ofobtaining something for the effort. To my own taste, simulation may lack some of thetechnical appeal of analytical arguments so it is a great joy to be able to present someoriginal theoretical work reaching something quintessential.

    Over the course of my PhD-work, I have published two papers in international scientificjournals [9, 36], submitted one [37] and contributed three times to research conferences[10, 38, 36].

    This thesis submitted for my PhD degree at the DTU is wholly my work but I haverelied on the guidance and help of many. Firstly, my thanks go to my two supervisorsfor their guidance and to the Dean of Research professor Kristian Stubkjr for letting mechange projects however implausible it may have seemed so that I would be able to handin a thesis a year and a half later. Secondly, my thanks are extended to my very goodfriends Thoa Nguyen, Karin Nordstrom Andersen, Haiyan Ou, Jacob Sparre Andersenand Rasmus Kousholt Sandberg for their never failing encouragements. Especially, Iacknowledge the care and patience that Thoa and Jacob have had to mount after hours ofcareful proof-reading of my ever changing manuscript just having me vehemently defendevery single convoluted and awkward sentence or attack the use of the gerund. In honesty,I can say that I meant every single misprint at the time I wrote them.

    Lastly, I wish to thank my family for extending patience to cover everything from foodand lodging to the pieces of graphics that were beyond my own skills to produce.

    Christian Ingemann MikkelsenMICDepartment of Micro and Nanotechnology

    Technical University of Denmark15 May 2005

    vii

  • viii PREFACE

  • Contents

    List of figures xiii

    List of symbols xv

    1 Introduction 11.1 Dielectrophoresis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Magnetophoresis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.5 Finished and published papers . . . . . . . . . . . . . . . . . . . . . . . . . 6

    2 Force on a magnetized object 92.1 Magnetic force on a body . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

    2.1.1 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.1.2 The magnetic force from the free energy . . . . . . . . . . . . . . . . 13

    2.2 Magnetic forces between two particles . . . . . . . . . . . . . . . . . . . . . 142.3 Magnetic particle in pure dipole field . . . . . . . . . . . . . . . . . . . . . . 152.4 Two magnetic spheres in an external magnetic field . . . . . . . . . . . . . . 192.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

    3 Motion of few beads 213.1 Hydrodynamic interactions between beads . . . . . . . . . . . . . . . . . . . 21

    3.1.1 Motion in a viscous fluid . . . . . . . . . . . . . . . . . . . . . . . . . 223.1.2 Motion in fluid flow and external magnetic field . . . . . . . . . . . . 233.1.3 Hydrodynamic Greens functions . . . . . . . . . . . . . . . . . . . . 24

    3.2 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2.1 Two-bead simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2.2 Few-bead simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 29

    3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

    4 Motion of many beads 354.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.2 The physical problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.3 Continuum model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

    ix

  • x CONTENTS

    4.3.1 Fluid flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.3.2 Bead motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.3.3 Magnetic force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.3.4 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.3.5 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

    4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.4.1 Qualitative picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.4.2 Quantitative measures . . . . . . . . . . . . . . . . . . . . . . . . . . 424.4.3 Concentration dependence . . . . . . . . . . . . . . . . . . . . . . . . 444.4.4 Influence of diffusion constant . . . . . . . . . . . . . . . . . . . . . . 464.4.5 Other force laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

    4.5 Bead current for finite size beads . . . . . . . . . . . . . . . . . . . . . . . . 484.6 Simplified view . . . . . . . . .

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