Electrohydrodynamic aspects of two-fluid microfluidic systems ...

Ph.D. Thesis, PhD no. 000699Electrohydrodynamic aspects oftwo-fluid microfluidic systems:theory and simulationGoran GoranovićSupervisor: Henrik BruusMikroelektronik Centret (MIC)Technical University of Denmark15 August 2003

ivABSTRACT

ResuméMålet med dette ph.d.-studium har været at undersøge fundamentale teoretiske principperbag lab-on-a-chip systemer og at benytte computer simulationer til hjælp for udviklingenog optimeringen af sådanne systemer. Studierne har især været koncentreretom mikropumper baseret på elektroosmose, dynamikken af gasbobler i mikrokanaler samtstabilitetsundersøgelser af skillefladen mellem to dielektriske væsker under påvirkning afydre elektriske felter.Det meste af arbejdet er blevet publiceret internationalt i løbet af studiet i form af4 forskningsartikler, 7 konferenceartikler, 1 patent, og 1 kapitel i en lærebog. Især tohovedresultater kan fremhæves. Det første er den nu patenterede opdagelse af hvorledeselektroosmose i en særligt designet to-væske mikropumpe kan benyttes til at pumpe ikkepolærevæsker. Det andet er påvisningen af hvorledes selv moderate elektriske spændingsforskellei mikrofluide to-væske systemer kan føre til instabiliteter, et fænomen som ikkeiagttages i makrosystemer. Hertil kommer en række analyser og simuleringer af forskelligekonkrete mikrofluide systemer, som kolleger har fabrikeret og målt på.Endelig skal det fremhæves, at meget arbejde i løbet studiet er blevet brugt på atbidrage til etableringen af MIC’s nye gruppe, Microfluidics Theory and Simulation (MIFTS).I dag fremstår MIFTS som en velfungerende gruppe med to postdocs, fire ph.d.-studerendeog et antal ingeniørstuderende under ledelse af lektor Henrik Bruus.v

viRESUME

PrefaceThe present thesis is one of the requirements for obtaining the PhD degree from theTechnical University of Denmark (DTU). The PhD program has been completed duringthe period 1 May 2000 - 15 Aug 2003 at Mikroelektronik Centret, MIC.The first half of the PhD program was conducted within the µTAS group mainly withregard to the applications, and the second half was finalized within the MIFTS groupmainly with regard to theory.My engagement began in early 2000 as the need for simulation efforts arose in themultidisciplinary project assigned to develop lab-on-chip platform in Denmark. At thetime, there were no similar efforts done at the institute, so many different areas in boththeory and experiment needed to be initiated from the beginning. It was an exciting andon occasions tough experience but most of all nicely unpredictable. In the meantime, bothof the groups have developed significantly.Many people have contributed in supporting me on the way and I am using this occasionto thank them. I would also like to thank my first supervisor Jörg Kutter for giving thenecessary support and ”open hands” in the first part of the studies. I would then expressmy gratitude to my second and current supervisor Henrik Bruus for the tremendous help,patience and enthusiasm that he provided in the critical stages of the studies. A specialacknowledgement deserve my resourceful master students (Anders, Flemming, Lennart,and Mads - the ”four aces”) whom I co-supervised during 2001/2002.Finally, I would like to thank my family, friend Ivana and girlfriend Helle for theirnon-wavering faith and support throughout the period.Goran GoranovićMikroelektronik Centret (MIC)Technical University of Denmark15 August 2003vii

viiiPREFACE

ContentsList of figuresList of tablesxivxv1 Introduction 11.1 Publications during the project . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.1 Research papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.2 Conference proceedings . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.3 A chapter in a text book . . . . . . . . . . . . . . . . . . . . . . . . 31.1.4 Co-supervision of four M.Sc. theses . . . . . . . . . . . . . . . . . . . 32 Aspects of microfluidics 52.1 Descriptive overview of microfluidic systems . . . . . . . . . . . . . . . . . . 52.2 The Continuum Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Conservation of mass and momentum . . . . . . . . . . . . . . . . . . . . . 92.4 Conservation of energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.5 Electrostatic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.6 Reynolds and Bond number . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Electroosmotic driven microflows 133.1 The Electric Double Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Thickness λ D of the Double layer . . . . . . . . . . . . . . . . . . . . . . . . 143.3 The Electric Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.4 The Helmholtz-Smoluchowski velocity . . . . . . . . . . . . . . . . . . . . . 183.5 The Zeta Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.6 Flow in a Circular Capillary . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 Simulations of interfaces in two-fluid systems 254.1 Miscible streams in micro cell sorters . . . . . . . . . . . . . . . . . . . . . . 254.2 Bubbles in microchannels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.2.1 Formation of bubbles . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.2.2 Clogging of microchannels . . . . . . . . . . . . . . . . . . . . . . . . 284.2.3 Passive bubble trap . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.3 Two-liquid viscous pump . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30ix

xCONTENTS4.3.1 Submerged jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.3.2 EOF induced viscous drag . . . . . . . . . . . . . . . . . . . . . . . . 304.3.3 Instability problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 Fluid dielectrics and the Maxwell stress tensor 355.1 The Maxwell stress tensor in free space . . . . . . . . . . . . . . . . . . . . 355.2 Fluid dielectrics in electric field . . . . . . . . . . . . . . . . . . . . . . . . . 385.2.1 The internal and the free energy of a dielectric . . . . . . . . . . . . 385.2.2 The Maxwell stress tensor in fluid dielectrics . . . . . . . . . . . . . 405.2.3 Electric force in fluid dielectrics . . . . . . . . . . . . . . . . . . . . . 415.2.4 Alternative form of the force density . . . . . . . . . . . . . . . . . . 435.2.5 The boundary between two dielectrics . . . . . . . . . . . . . . . . . 446 Hydrodynamic stability analysis: basics 476.1 General concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486.2 Standard procedure in linear stability analysis . . . . . . . . . . . . . . . . . 506.3 Example: The Rayleigh-Taylor instability . . . . . . . . . . . . . . . . . . . 526.3.1 A note on viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547 Interfaces in electrohydrodynamics: theory 577.1 Two immiscible liquids between infinite planes . . . . . . . . . . . . . . . . 577.1.1 The governing equations . . . . . . . . . . . . . . . . . . . . . . . . . 597.1.2 General boundary conditions: rigid boundaries . . . . . . . . . . . . 597.1.3 General boundary conditions: the interface . . . . . . . . . . . . . . 607.2 Perturbation equations and boundary conditions . . . . . . . . . . . . . . . 617.2.1 The stationary solution for still liquids . . . . . . . . . . . . . . . . . 617.2.2 Linearization of the perturbed equations . . . . . . . . . . . . . . . . 627.2.3 Linearization of the perturbed boundary conditions . . . . . . . . . 637.2.4 Expansion into normal modes . . . . . . . . . . . . . . . . . . . . . . 667.2.5 Summary of solutions and linearized boundary conditions . . . . . . 678 Interfaces in electrohydrodynamics: applications in microfluidics 718.1 Zero viscosity and perpendicular E-field . . . . . . . . . . . . . . . . . . . . 718.1.1 Dispersion relation ω(k): general case . . . . . . . . . . . . . . . . . 728.1.2 The onset of instability: limiting cases for infinite planes . . . . . . . 758.1.3 The onset of instability: general case for infinite planes . . . . . . . 788.1.4 Instability voltages for infinite planes, kL ≫ 1 (macrofluidic limit) . 808.1.5 Instability voltages for infinite planes, kL ≪ 1, (microfluidic limit) . 808.1.6 Finite-size effect on instability in microfluidics . . . . . . . . . . . . 818.1.7 Instability voltages in microfluidic channels . . . . . . . . . . . . . . 838.1.8 Low aspect-ratio microchannels, L y ≫ L 0 . . . . . . . . . . . . . . . 858.2 Zero viscosity and parallel E-field . . . . . . . . . . . . . . . . . . . . . . . . 869 Conclusion 879.1 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

CONTENTSxiA Paper published in J. Micromech. Microeng. 91B Paper published in Sens. Actuators B Chem. 101C Paper published in J. Micromech. Microeng. 109D Paper published in Proc. microTAS 2002 119E Paper published in Proc. NanoTech 2003 123F Book chapter published by Wiley-VCH 2003 129Bibliography 131

xiiCONTENTS

List of Figures2.1 Integrated-chemiluminescence-detector chip . . . . . . . . . . . . . . . . . . 62.2 The race-track effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.1 The structure of an electric double layer. . . . . . . . . . . . . . . . . . . . . 143.2 Electric potentials in a circular channel and between parallel plates. . . . . 183.3 EOF and the electric field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.4 Structure of the electric potential in the double layer. . . . . . . . . . . . . 213.5 Chemical reactions for silanol groups. . . . . . . . . . . . . . . . . . . . . . 214.1 Micro cell sorter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.2 Micro cell-sorter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.3 Sudden and tapered channel constrictions . . . . . . . . . . . . . . . . . . . 284.4 Passive bubble trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.5 Viscous-drag pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.6 Two-liquid viscous pump: our new design . . . . . . . . . . . . . . . . . . . 324.7 Two-liquid viscous pump: simulation . . . . . . . . . . . . . . . . . . . . . . 335.1 Normal and tangential electric stresses . . . . . . . . . . . . . . . . . . . . . 365.2 Interface between two liquid dielectrics . . . . . . . . . . . . . . . . . . . . . 446.1 Potential energy of a conservative system . . . . . . . . . . . . . . . . . . . 486.2 Instability of a bent rectangular beam . . . . . . . . . . . . . . . . . . . . . 496.3 Two liquids between infinite planes . . . . . . . . . . . . . . . . . . . . . . . 526.4 The Rayleigh-Taylor (gravitational) instability . . . . . . . . . . . . . . . . 547.1 Instabilities in electrohydrodynamics . . . . . . . . . . . . . . . . . . . . . . 587.2 Interface perturbations in normal and tangential E-field . . . . . . . . . . . 608.1 Instability: phase velocities and frequencies . . . . . . . . . . . . . . . . . . 758.2 Instability: dependency on We el and Bo . . . . . . . . . . . . . . . . . . . . 778.3 The onset of instability as function of Bo . . . . . . . . . . . . . . . . . . . 798.4 Two liquids in finite size microfluidic channels . . . . . . . . . . . . . . . . . 828.5 Instability voltages in finite microchannels - small and large k . . . . . . . . 848.6 Instability voltage in finite size channels - general case. . . . . . . . . . . . . 858.7 Vertically superposed layers in a low-aspect-ratio microchannel. . . . . . . . 86xiii

xivLIST OF FIGURES

List of Tables4.1 Comparison between the circuit model and a CFD simulation . . . . . . . . 338.1 Parameters for the oil-water system . . . . . . . . . . . . . . . . . . . . . . 72xv

xviLIST OF TABLES

2 CHAPTER 1. INTRODUCTIONIn Chap. 6 basics of hydrodynamic stability analysis are presented with a simple exampleof gravitational instability.In Chap. 7 linear stability theory is applied to the problem of stability of the interfacebetween two still, immiscible dielectrics placed in an electric field (normal or tangential).The chapter contains full derivation of the linearized perturbed equations of motion andboundary conditions.Chap. 8 presents the solutions of the equations derived in Chap. 7 applied to the casesof normal and tangential E-fields. The interface instability is discussed with respect to twomicrofluidic cases where either one or two characteristic lengths are of micron dimensions.Chap. 9 concludes the report and outlines the future directions.Finally, the most relevant publications have been attached in the Appendix.1.1 Publications during the projectThe following list comprises all of the publications that I have contributed to during myPhD study at MIC.1.1.1 Research papers1. PIV measurements in a microfluidic 3D-sheathing structure with three-dimensionalflow behavior,Henning Klank, Goran Goranović, Jörg P. Kutter, Henrik Gjelstrup, Jess Michelsen,Carsten Westergaard,J. Micromech. Microeng. 12, 862-869 (2002).2. Integrating advanced functionality in a microfabricated high-throughput flourescentactivatedcell sorter,A. Wolff, I.R. Perch-Nielsen, U.D. Larsen, P. Friis, G. Goranović, C.R. Poulsen, J.P.Kutter, P. Telleman,Lab. Chip 3, 22-27 (2003).3. Theoretical analysis of the low-voltage cascade electroosmotic pump,Anders Brask, Goran Goranović, and Henrik Bruus,Sens. Actuators B Chem. 92, 127-132 (2003).4. The clogging pressure of bubbles in hydrophilic microchannel contractions,Mads Jakob Jensen, Goran Goranović, and Henrik Bruus,J. Micromech. Microeng. 14 (2004), (8 pages, www.mic.dtu.dk/research/MIFTS)1.1.2 Conference proceedings5. Three-dimensional single step flow sheathing in micro cell sorters,Goran Goranović, Ivan R. Perch-Nielsen, Ulrik D. Larsen, Anders Wolff, Jörg P. Kutter,Pieter Telleman,Proc. MSM 2001, Hilton Head Island (SC) USA, March 2001, p. 242-245.

1.1. PUBLICATIONS DURING THE PROJECT 36. Computer modeling as a design tool for microfluidic structures,Ivan R. Perch-Nielsen, Goran Goranović, Henning Klank, Ulrik D. Larsen, AndersWolff, Jörg P. Kutter, Pieter Telleman,Proc. MSM 2001, Hilton Head Island (SC) USA, March 2001, p. 629-630.7. Characterization of flows in laser-machined polymeric microchannels,Goran Goranović, Henning Klank, Carsten Westergaard, Oliver Geschke, PieterTelleman, and Jörg P. Kutter,Proc. µTAS 2001, Monterey (CA) USA, October 2001.8. Electroosmotically driven two-liquid viscous pump for nonconducting liquids,Anders Brask, Goran Goranović, and Henrik Bruus,Proc. µTAS 2002, Nara, Japan, November 2002, vol. 1, p. 45-47.9. The low-voltage cascade EOF pump: comparing theory with published data,Anders Brask, Goran Goranović, and Henrik Bruus,Proc. µTAS 2002, Nara, Japan, November 2002, vol. 1, p. 79-81.10. Dynamics of bubbles in microchannels,Mads Jakob Jensen, Goran Goranović, and Henrik Bruus,Proc. µTAS 2002, Nara, Japan, November 2002, vol. 2, p. 733-735.11. Electroosmotic pumping of nonconducting liquids by viscous drag from a secondaryconducting liquid,Anders Brask, Goran Goranović, and Henrik Bruus,Proc. NanoTech 2003, San Francisco (CA) USA, March 2003, vol. 1, p. 190-193.12. Quasi-static motion of bubbles in microchannel contractions,Mads Jakob Jensen, Goran Goranović, and Henrik Bruus,Proc. NanoTech 2003, San Francisco (CA) USA, March 2003, vol. 1, p. 258-261.1.1.3 A chapter in a text book13. Simulations in microfluidcsGoran Goranović and Henrik Bruus,Chap. 5 (38 pages, 21 figs., 25 refs.) in Microsystem Engineering of Lab-on-a-ChipDevices. Eds. O. Geschke, H. Klank, and P. Telleman, (Wiley-VCH, New York,autumn 2003)1.1.4 Co-supervision of four M.Sc. theses• Principles of electroosmotic pumpsAnders BraskM.Sc. thesis, 156 pages, MIC, DTU, 2002 (www.mic.dtu.dk/research/MIFTS)• Bubbles in microchannelsMads Jakob JensenM.Sc. thesis, 155 pages, MIC, DTU, 2002 (www.mic.dtu.dk/research/MIFTS)

4 CHAPTER 1. INTRODUCTION• Blood flow in microchannelsLennart BitschM.Sc. thesis, 111 pages, MIC, DTU, 2002 (www.mic.dtu.dk/research/MIFTS)• Dispersion in electrokinetically and pressure driven microflowsFlemming Rytter HansenM.Sc. thesis, 121 pages, MIC, DTU, 2002 (www.mic.dtu.dk/research/MIFTS)

Chapter 2Aspects of microfluidicsIn this chapter we give important aspects of microfluidics.2.1 Descriptive overview of microfluidic systemsThis section is adapted from my text-book chapter Simulations in Microfluidics[13]. Whendesigning new microfluidic devices, one must have a proper understanding of the physicalaspects of the problem. Some of these are given below.• DimensionsA microfluidic system is a network of fluidic channels and other components suchas valves and pumps. It typically occupies an area of one square centimeter, withthe typical channel widths on the order of 10 - 100 µm. For comparison, a humanhair has a thickness of approximately 50 µm and the diameter of red blood cells areabout 7 µm. From a practical point of view even smaller dimensions in a lab-ona-chipare not necessarily advantageous, since the requirement for easier handlingas well as measurements can impose some restrictions. However, if the goal is todevelop systems for single molecule detection, the sub-micrometer is unavoidable,and one enters the new and exciting field of nanofluidics.• GeometryThe basic component in a microfluidic network is a channel. Main features includelength, cross-section and surface properties such as roughness. After a channel ismade in a substrate it is covered with a bonded lid that can be made from a differentmaterial. Long channels are usually needed if a reaction of several mixed chemicals isto occur (in laminar flow regime diffusion is the main mixing mechanism and it takestime). Also, when a chemical compound consisting of several components needs tobe analyzed, usually by electrical separation techniques, the separation channel ismade longer in order to allow for the proper separation of the components. Sincemicrochips are confined to a small area, the extra length is achieved by meander orspiral structures, Fig. 2.1. Such channel bending, however, induces dispersion of5

6 CHAPTER 2. ASPECTS OF MICROFLUIDICSFigure 2.1: (a) A mask of an integrated microsystem for detecting chemical reactions viachemiluminescence. Before entering the mixing region, a sample is passed through thelong spiral for a prolonged reaction with immobilized enzyme. Then, a product, usuallyH 2 O 2 , is entering the mixing region and in catalyzed reaction with reagent produces light.Back-side photodiodes, used for the light detection, are not shown. The dimensions of thechip are 20 mm × 10 mm, and width of the enzyme reactor is 400 µm. Courtesy of A. M.Jørgensen.a chemical species since molecules travel different distances inside the turn (racetrackeffect, Fig. 2.2. The dispersion lowers the concentration resulting in reducedresolution, i.e. larger overlaps of the concentration peaks, and diminished detectionsignals. Another common source of the geometrical dispersion are interconnections,such as tubings, between a chip and external liquid reservoirs.Channels can have different cross-sections depending on the material in which theyare embedded. In silicon, the typical profiles are rectangular, or actually slightlytrapezoid due to underetching. In polymers, laser beams can produce rectangular,triangular and Gaussian-like shapes. Circular cross-sections as in capillaries are alsoencountered. In addition, channels can have various degrees of surface roughnessdepending on the fabrication process. Geometrical features are important since theydetermine the characteristic resistances of a channel. The hydraulic resistance, aconcept used in the uniform flow regime, relates an applied pressure to the correspondingflow rate, while the electrical resistance of a channel filled with a conductingliquid relates an applied voltage to the passing current.• SurfaceChemical groups at the surface of channel walls, such as silanol (SiOH) groups inglass, can react with the ions in an electrolyte solution and create very thin polarizedlayer. This electric double layer, or Debye layer, has a length scale of 1-10 nm andis responsible for electroosmotic flow (EOF) in an applied electric field. Surfacedouble-layers are characterized by the zeta potential. Different surfaces, such as thelid and the bottom of a channel, can have different zeta potentials. The characteristicEOF velocity depends on the zeta potential through electroosmotic mobility whichis largely an empirical value.

2.1. DESCRIPTIVE OVERVIEW OF MICROFLUIDIC SYSTEMS 7(a)(b)(c)(d)Figure 2.2: The race-track effect in a turn in an electrokinetically driven flow, with avoltage ∼10 V applied at the ends. The width of the channel is 50 µm. The motion startsat the top left figure and goes clockwise. The concentration contours of a sample aredisplayed, obtained from a simulation. The inside track has shorter path as well as higherelectrical field inducing faster movement of the inner molecules, and therefore enhancingsample dispersion.• Fluid propertiesThe channels are filled with liquids which can have different properties such asdensity, viscosity, electrical and thermal conductivity, diffusion coefficients, surfacetension etc. In many cases there are two types of liquids in the system - a buffer anda sample. The buffer is usually the transporting liquid resisting the changes in pHvaluewhich can for example be induced close to electrodes or in chemical reactions.A sample is a liquid of interest which needs to be analyzed. Both buffer and sampleare usually placed on the chip through separate reservoirs, with the buffer finallysurrounding the sample. A sample confined to a specific region is called a sampleplug. In separation science, the sample consists of several chemical species such asa positively charged, a negatively charged and a neutral. As mentioned above, thesample plug lowers its initial concentration by dispersing through the system. Inaddition to the geometrical constrictions, dispersion is caused by velocity profiles,such as the parabolic in the pressure driven flows, and by diffusion. Fluids suchas blood or gels are so-called non-Newtonian liquids, for which the viscosity is notconstant but is a function of the local shear stress. Since the shear stresses arepronounced at the walls more than in the middle of the channels, the resulting flowprofile is a blunted one rather than the usual parabolic one.• HeatingJoule heating is generated when an electrical current flows through a channel. Insome applications such heating may destroy biological samples. It may also inducechanges in viscosity and thus affect the velocity profile. Moreover, due to enhanced

8 CHAPTER 2. ASPECTS OF MICROFLUIDICSthe Brownian motion, heating may increase the dispersion of a sample plug. Heatingmay also increase the mobility of the ions in electric double layers leading to enhancedflow rates in electroosmotic flows. Finally, heating will stimulate production ofbubbles of the gases dissolved in the liquid. The bubbles are usually unwanted andcan block the flow if getting stuck in narrow contractions.2.2 The Continuum HypothesisIn this thesis all fluids discussed are Newtonian and are generally treated as continuousisotropic matter (mean molecular fields). The continuum description could however bequestioned because of the small length scales present in microfluidics.A way of defining deviations from the continuum description is through the Knudsennumber [58]Kn = λ L , (2.1)where λ is the mean-free-path of the molecules and L is a characteristic length scale. Lshould be chosen to include gradients in the velocity field. For liquids λ is approximatelythe intermolecular length L mol (bond length). The volume occupied by one molecule L 3 molis easily approximated, yieldingL mol =(MmolρN A) 1/3, (2.2)where ρ is the density, M mol the molar mass, and N A is Avogado’s number. For waterλ = L mol ≈ 0.31nm. The Knudsen number identifies the continuum regimes and thegoverning equations. 1A more intuitive way of justifying the use of the continuum approximation is through aprecision criterion. At sufficiently high precision fluids are never continuous. The conceptof a continuous velocity field does, e.g., only makes sense for small changes in the center ofmass, of a small characteristic volume, due to random fluctuations in velocities. Anotherexample could be to determine the mass density to a certain relative precision r.• ExampleDetermining the mass density ρ = mN/V to the relative precision r, with N thenumber of molecules of mass m in a small volume V . There are only fluctuationsin the relative number of molecules ∆N due to random walk which is typically∆N ≈ √ N. For r = 1% the requirement ∆N/N < r yields N > 1/r 2 = 10 4 . Thegiven volume hence needing dimensions L > 22L mol . For the case of water L > 7nmwhich is a factor 10 3 smaller than the geometric dimensions of the microfluidicsystems analyzed in this thesis.1 For 0 < Kn < 0.01 the continuum approximation is applicable and the Navier-Stokes equation usable,this regime is generally in use in microfluidic flows. For 0.01 < Kn < 0.1 the Navier-Stokes equation is stillapplicable however with use of so-called slip conditions to account for microscopic phenomenon at solidboundaries.

2.3. CONSERVATION OF MASS AND MOMENTUM 92.3 Conservation of mass and momentumLet us consider a volume V on which external forces f act. The equations of motion canbe derived by considering the flow of fluid out of the volume, Landau-Lifshitz [14]. Integralconservation laws of mass and momentum are respectively∫ ∫dρ dV + ρu·n dS = 0, (2.3)dt VS∫ ∫∫dρu dV + [ρu(u·n) − τ ′ n] dS = f dV, (2.4)dt VSVwhere u(x, y, z, t) = (u, v, w) is the velocity field, ρ is mass density and n a normal vectordirected out of the volume. For Newtonian fluids the stress tensor τ ′ isτ ′ ik = −pδ ik + τ ik, (2.5)where the normal component p is pressure and τ the viscous stress tensorτ ik = µ ( ∂ ku i+ ∂ iu k)+ (λ −23 µ)∂ l u l δ ik . (2.6)The scalars µ and λ are called coefficients of viscosity.By applying Gauss’ theorem to Eqs. (2.3) and (2.4) we obtain the equation of continuityand the Navier-Stokes equationDρDtρ DuDt= −ρ∇·u, (2.7)= −∇p + ∇·τ + f, (2.8)with D/Dt = ∂ t + u·∇ being the convective derivative.In case of incompressible fluids Dρ/Dt = 0 and Eqs. (2.7) and (2.8) simplify towhile the viscous stress tensor becomes2.4 Conservation of energy∇·u = 0, (2.9)ρ DuDt = −∇p + µ∇2 u + f. (2.10)τ ik = µ ( ∂ ku i+ ∂ iu k). (2.11)In viscous flows the energy is dissipated through irreversible processes into heat. Thetemperature of the system will increase until an equilibrium between heat generation andheat transport is obtained. The conservation of energy is given by, Landau-Lifshitz, [14],( 1) [∂ t2 ρu2 + ρe = −∇ · ρu( 1 ]2 u2 + h) − u · τ − κ∇T , (2.12)

10 CHAPTER 2. ASPECTS OF MICROFLUIDICSHere e and h are respectively internal energy and enthalpy per unit mass, κ(T ) thermalconductivity and T temperature. The left hand side of Eq. (2.12) is the rate of changein mechanical and thermal energy respectively, while the right hand side is the divergenceof the energy flux. The first term in the brackets represents the mass transfer due to thefluid motion (kinetic energy ρu 2 /2 and enthalpy ρh, per unit volume); the second termis the internal friction due to the viscosity. The last term represents the heat conductiondue to temperature gradients.The thermodynamic relations for e and h arede = T ds − pdV = T ds + p dρ, (2.13)ρ2 dh = T ds + V dp = T ds + 1 dp, (2.14)ρs being the entropy per unit mass and V = 1/ρ specific volume. If these expressions areused together with the continuity and momentum equations, Eq. (2.12) can be transformedinto the equation of heat transfer, [14]ρ DqDt= Ψ + ∇·(κ∇T ), (2.15)Left side of Eq. (2.15) represents the amount of heat gained per unit volume. The termΨΨ = τ ·∇v (2.16)is called the dissipation function and gives the amount of energy dissipated into heat byviscosity. It is always positive and for incompressible fluids equal to2.5 Electrostatic fieldsΨ = µ 1 2 (∂ k u i + ∂ i u k )2 . (2.17)In the thesis we will treat phenomena where only electrostatic fields are applied. When externalmagnetic fields are absent, magnetic effects can be completely ignored, [15]. Maxwellequations for electric fields reduce to∇·D = ρ e , (2.18)∇×E = 0, (2.19)where D = ɛE is the electric displacement vector. Thorough discussion of the electricforces in dielectrics will be made in Chap. 5.2.6 Reynolds and Bond numberA measure for the ratio between inertial and viscous forces is the Reynolds number definedasRe = L uν , (2.20)

2.6. REYNOLDS AND BOND NUMBER 11where L and u is a characteristic length and velocity, respectively. The Reynoldsnumber at the transition between laminar and turbulent flow is of the order 10 3 , dependingon the geometry and type of flow 2 . For a Poiseuille flow in a tube the critical Reynoldsnumber Re crit > 2000. In microfluidics the flow is almost always laminar due to themicro length scales L ≈ 100 µm, small velocities u ≈ 1 mm/s and kinematic viscosity1.00 × 10 −6 m 2 /s yielding a Reynolds number of Re = 0.1.If the flow is free of vorticity, the governing equations simplify considerably. Thistype of flow is called potential flow. Viscous flow is generally not potential flow, but insome special cases it may be described as such. Given the appropriate boundary conditionselectroosmotic flow can be described as potential flow independent of the Reynolds number,[16].A measure of the surface tension force to the gravitational force is given by the BondnumberBo = ρgL2(2.21)σThe characteristic length L may, for example, be the height of meniscus, the curved uppersurface of a column of liquid. When the Bond number is large the surface pressure effectsmay be neglected in the liquid at rest.A length scale for surface tension governed phenomenon is derived from the Bondnumber Eq. (2.21). HavingL ≪( σρg) 12≡ ∆c ⇒ Bo ≪ 1 (2.22)where ∆ c is the so-called capillary length. For L = ∆ c the Bond number is 1. ∆ c providesa measure (length scale) for when phenomena are either controlled by gravity (> ∆ c ) orsurface tension < ∆ c .We will return to characteristic dimensionless numbers for the problem of two dielectricfluids places in E-field in Chap. 8.2 For example, the wave boundary layer in an ocean wave may not be fully turbulent before Re = 10 6

12 CHAPTER 2. ASPECTS OF MICROFLUIDICS

Chapter 3Electroosmotic driven microflowsIn this chapter 1 we will present the phenomena behind the electroosmotic (EO) microflows[18, 19, 20, 21]. It was already mentioned that surface reaction between electrolytes andglass walls lead to the charge separation close to the walls. In an axial E-field these chargeswill move and by viscous drag set the bulk liquid in motion as well.In microsystems, surface-to-volume ratios are large. Since EOF is governed by surfacereactions, it will be more efficient than ordinary pressure driven flows, a feature exploitedin EO pumps [17]. Micropumps play a key role in the quest for fabricating versatile,cheap, and highly efficient microfluidic lab-on-a-chip devices. In particular, EO pumpsare attractive since they contain no moving parts and are relatively easy to integrate inmicrofluidic circuits during fabrication [22, 23, 24, 25, 26, 27]. Design of these pumpsrequires the understanding of the underlying physical theory outlined below. We haveused our theoretical insight to analyze the low-voltage EO pump invented by Takamuraet al. [23]. Our results are presented in a research paper [3] enclosed in Appendix B.3.1 The Electric Double LayerAs a starting point, we consider the charge transport in a liquid. The flux J of a chargedspecies is driven by the gradient of a general potential, which is a function of pressure,temperature, electric state, concentrations, composition of phases, etc. An analyticaldescription, however, is not feasible due to the nonlinear characteristics of this potential.However, if the electrolyte is dilute this is not a problem, because then the transporteffects may be linearly superposed.The involved transport processes are electromigration due to electrical fields, diffusiondue to density gradients, and convection due to velocity fields. If we neglect the effectsof external and induced magnetic fields, we obtain for dilute concentrations the followingform of the Nernst-Planck equation [18]J i = −α i z i c i ∇φ − D mass,i ∇c i + c i u, (3.1)1 The chapter is adapted from the M.Sc. thesis by A. Brask [17], supervised by Goranović and Bruus.13

14 CHAPTER 3. ELECTROOSMOTIC DRIVEN MICROFLOWSconcentrationfused silica capillary wallimmobileO +OOO-O-OInnerlayer+++-+mobile++--+-++Diffusivelayer---++++----normal to surfaceBulksolutionPositive ionNegative ionNeutral molecule++Figure 3.1: The structure of an electricdouble layer with an inner immobilelayer. The system consistingof wall and liquid is electrical neutral.The diffuse layer is mobile andcan therefore be manipulated by anexternal electric field. The positiveions are hydrated. Some of the negativeions are specifically adsorbedat the surface [60]where J i is the molar flux of the i-th species due to electromigration, mass diffusion, andconvection, respectively. The i-th mobility α i is related to the mass diffusivity D mass,i andthe thermal energy kT as α i = eD mass,i /(kT ). In the following analysis the electrolytesare assumed to be dilute.If we consider a charged surface in contact with a dilute, fully dissociated salt, thecharged surface will attract the ions of the opposite charge, the so-called counterions.Hence, a region will exist where the concentration of counterions is larger than that ofthe coions 2 . This phenomena leads to the formation of an electric double layer, or Debyelayer. In Fig. 3.1 the wall charge is assumed negative. The term ”double” refers to the factthat there are two layers: the inner layer, which is immobile due to strong electrical forces,and the outer layer (the diffusive layer) which may be affected by an external electric field.The surface charge arises from chemical reactions as discussed in Sec. 3.5.3.2 Thickness λ D of the Double layerThe thickness λ D of the electrical double layer is governed by a balance between diffusionand electrical forces. The λ D is also known in the literature as the Debye length. It isa characteristic length scale for the electrical screening. First we introduce the electrochemicalpotential µ chem,i , which is the free energy of the last arrived particle. It consistsof an electric and chemical potential. The gradient of the electrochemical potential mustbe zero in equilibrium. If this were not the case, a flow would be induced. The chemicalpotential is given asµ chem,i = µ 0 chem,i + kT ln n i, (3.2)where n i = c i /c 0 i is the concentration of ions relative to a reference concentration. µ0 chem,iis the standard chemical potential which is equal to the chemical potential when n i = 1.2 The charges of the coions have the same sign as the charges of the surface.

3.2. THICKNESS λ D OF THE DOUBLE LAYER 15The Debye length is found by balancing repulsive diffusive forces with attractive electricalforces, which is the same as requiring ∇µ i = 0.In one dimension we get∇µ i = ∇µ chem,i + z i e∇φ, (3.3)∇µ chem,i = −z i e∇φ, (3.4)dµ chem , idxInserting Eq. (3.2) into Eq. (3.5) yields= −z i e dφdx . (3.5)1 dn in i dx = − z ie dφkT dx . (3.6)Integrating this and requiring that φ = 0 when n i = 1 we obtainn i = exp(− z ieφ),kT(3.7)c i = c 0 i exp(− z ieφ).kT(3.8)The Debye layer is not electrically neutral because the neutralizing charges are at thewall. Hence, the Poisson equation,is invoked.hence∇ · (ɛ∇φ) = −ρ E , (3.9a)For simplicity the dielectric constant ɛ is assumed spatially invariant, and∇ 2 φ = − ρ Eɛ(3.9b)This approximation needs further discussion because of the strong polarization effects inthe Debye layer, see Sec. 3.4. The charge density is now inserted in the Poisson equationEq. (3.9b) along with the Boltzmann distribution, Eq. (3.8), which leads tod 2 φdx 2 = − 1 ɛ∑ci z i e = − 1 ɛ∑c0i z i e exp(− z ieφ). (3.10)kTThis is a nonlinear differential equation, due to the exponential terms on the right-handside. The general solutions will be discussed in Sec. 3.3. For now we will use the Debye-Hückel approximation z iφkT≪ 1, in which case exp( z iφkT ) ≈ 1 − z iφkTleading tod 2 φdx 2 = − 1 ∑c0ɛ i (1 − z ieφkT )(3.11a)= − 1 ( ∑ec0ɛ i z i − ∑ c 0 i zi 2 e 2 φ )zi e (3.11b)kT= e2ɛkT (∑ zi 2 c 0 i )φ(3.11c)= κ 2 φ. (3.11d)

16 CHAPTER 3. ELECTROOSMOTIC DRIVEN MICROFLOWSIn Eq. (3.11b), the first sum is zero because the bulk liquid is electrically neutral. In thelast equation, the factor κ 2 in front of φ is by definition related to the Debye length since1/κ ≡ λ D :√λ D = 1 κ = ɛkT(3.12a)=e 2 ∑ z 2 i c0 i√ɛkT2e 2 z 2 c(3.12b)In Eq. (3.12b), the salt is assumed symmetric for simplicity, i.e. −z − = z + = z andc 0 + = c 0 −.The thickness of the double layer depends on the temperature T , concentration c andvalence number z of the ions, but normally it is of the order 10 nm. At a distance ofapproximately 3λ D , the potential will be reduced to 2% of its near surface value.At high temperatures the ions move quickly which leads to a decrease in the shieldingsince they are dispersed more. For high concentrations the shielding is more effectivebecause the charge density is larger.Moreover, since λ D ∝ (z 2 c) −1/2 the double layer is compressed if the buffer concentrationor the ionic strength 3 is increased. This affects the potential distribution, and thezeta potential is reduced.The zeta potential is expected to become larger (in magnitude) for small electrolyteconcentrations.Example: The thickness of the double LayerAt a temperature of 25 ◦ C in water ) with ɛ = 78.3ɛ 0 , the above expression computesto λ D =(9.61 × 10 −9 (z 2 c) −1/2 nm, with c in units of mol/m 3 . For a monovalentconcentration of 10 2 mol/m 3 (0.1M), λ D = 0.961 nm. So in this case, theEDL is no more than approximately 10 atomic diameters wide.3.3 The Electric PotentialConsider a circular capillary filled with an aqueous solution. At the solid-liquid interfacean EDL is formed. The ion concentrations c i are connected with the electric potential φthrough the Poisson equation, Eq. (3.9b). Adopting cylindrical coordinates with x as theaxial coordinate, the ∂2 -terms may be neglected because variation is very small compared∂x 2to the radial variation. Furthermore, the solution is assumed symmetric, so the angularvariation is also disregarded, so we get(1 ∂r ∂φ )r ∂r ∂r= − ρ Eɛ= − F zɛ (c + − c − ).(3.13a)(3.13b)3 Ionic strength is a quantity representing interactions of ions with water molecules and other ions in asolution I = 1 2∑ z2i n i.

3.3. THE ELECTRIC POTENTIAL 17It is convenient to split up the potential in two parts: a radial component ψ(x, r) and anaxial component Φ(x),φ(x, r) = Φ(x) + ψ(x, r). (3.14)In order to find the concentrations, the ion flux is considered. In equilibrium the radialflux of ions must be zero. This requirement is inserted into the Nernst-Planck equation,Eq. (3.1),0 = ∓zRT D ∂ψmassF c ±∂r − D ∂c ±mass∂r . (3.15)By integration of this equation and using that the diffusivity is D mass = RT α, the Boltzmanndistribution is obtained for c + (z) and c − (z),(c ± = c 0 exp ∓ zF ψ ). (3.16)RTBy inserting this result into Eq. (3.13b) and making the variable dimensionless accordingtor ∗ = r a , λ∗ = λ Da , and ψ∗ = zF ψRT , (3.17)the following nonlinear second order ordinary differential equation (ODE) arises:λ ∗ 2 1 ( )∂r ∗ ∂r ∗ r ∗ ∂ψ∗∂r ∗ = sinh(ψ ∗ ). (3.18)In combination with the boundary conditions this forms a nonlinear boundary value problem(BVP).∂ψ ∗∂r ∗ = 0, at r∗ = 0, (3.19)ψ ∗ = ζ ∗ , at r ∗ = 1. (3.20)The zeta potential will be explained in Sec. 3.5 for now it is simply the potential at the wallinterface. In Fig. 3.2 the potentials have been calculated numerically for different valuesof λ ∗ using a shooting algoritm in MatLab. Two geometries are considered. A circularcapillary in Fig. 3.2(a), and a one dimensional parallel plate geometry in Fig. 3.2(b).In the case of large λ, the potential equals the zeta potential across the entire channelwidth. For small values of λ the potential is zero everywhere except near the boundaries.This illustrates the different shielding effects. For small ratios the bulk flow is completelyshielded, refer to Sec. 3.2. This will often be the case, since the Debye length being lessthan 10 nm is very small compared to most capillaries.An analytical solution for the potential may be obtained by linearizing the right-handside of Eq. (3.18), and the double layer thickness λ D ≪ a must be small compared tothe radius of the capillary. Hence curvature effects can be neglected. This is termed theDebye-Hückel approximation. The result is a simple exponential function ψ(r),(ψ(r) = ζ exp − a − r ), (3.21)λ D

18 CHAPTER 3. ELECTROOSMOTIC DRIVEN MICROFLOWS3ζ=2.793ζ=2.792.5λ*=22.5λ*=22λ*=12λ*=1ψ‘1.5ψ‘1.51λ*=0.41λ*=0.40.5λ*=0.2λ*=0.1λ*=0.0100 0.2 0.4r/a0.6 0.8 10.5λ*=0.2λ*=0.1λ*=0.0100 0.2 0.4 0.6 0.8 1x/WFigure 3.2: Nondimensional screening potential ψ ∗ in two different geometries: (a) In aradial channel where the length scale is the radius of the channel. (b) Potentials betweentwo planes where the length scale is half the spacing of the planes.a result that also emerges from the linearized parallel planes case, since the curvatureeffects here are manifestly zero.3.4 The Helmholtz-Smoluchowski velocityThe purpose of this section is to find the velocity as a function of an external electricfield. The concepts introduced here are essential for a description of electroosmotic flow.Electroosmosis is the movement of liquid relative to a stationary charged surface (e.g., acapillary or porous plug) by an electric field. The pressure necessary to counterbalanceelectroosmotic flow is termed the electroosmotic pressure. The equation governing theflow is the Navier-Stokes equation (see Sec. 2.3).Since the length scale of the Debye layer is very small λ D ≈ 1−10 nm the correspondingReynolds number is very small for normal velocities and viscosities. Hence the inertialforces may be neglected here. For a further discussion of inertial forces and Reynoldsnumbers see Sec. 2.6. So the problem is reduced to a balance between electric forces andviscous forces. The maximum velocity in the boundary layer is then used as a boundarycondition for the bulk flow. Introducing Poisons equation to eliminate the charge densityρ E we obtain a relation between the velocity u = (u, v, 0) and the applied axial potentialφ a (x):µ∇ 2 u = ρ E ∇φ a (3.22)= −ɛ ( ∇ 2 φ ) ∇φ a (3.23)To identify nonessential terms the x and y coordinates are scaled to ξ and η by the radius

3.4. THE HELMHOLTZ-SMOLUCHOWSKI VELOCITY 19a and the Debye length λ D , respectively,ξ = x/a, (3.24)η = y/λ D . (3.25)The Laplace operator of the intrinsic potential φ i and the gradient of the applied potentialφ a in Eq. (3.23) are computed separately in Eqs. (3.26b) and (3.26c) below. Note thatthe tangential term in the gradient of the intrinsic potential is expressed by means of theapplied potential. Its magnitude is proportional to the applied field by the factor χ. Thisis a correction factor due to the polarization of the dielectric liquid within the Debye layer.The electric field in the Debye layer E ≈ 10 7 V/m is much larger than the applied fieldE ≈ 10 4 V/m. The terms computes to( )χ ∂φ a∇φ i =a ∂ξ , 1 ∂φ iλ ∂η , 0 , (3.26a)∇ 2 φ i = χ ∂ 2 φ aa 2 ∂ξ 2 + 1 ∂ 2 φ iλ 2 ∂η 2 ,(3.26b)( 1∇ 2 ∂ 2 uu =a 2 ∂ξ 2 + 1 ∂ 2 uλ 2 ∂η 2 , 1 ∂ 2 va 2 ∂ξ 2 + 1 ∂ 2 )vλ 2 ∂η 2 , 0 . (3.26c)A value of the χ-factor is difficult to obtain. A more detailed physical description of theDebye layer is required. This is beyond the scope of the this analysis, but the χ-factorcan be incorporated in an suitable choice of the dielectric constant for the Debye layer, inthe following just denoted ɛ.Collecting terms in the axial direction x, and neglecting terms of order λ/a or higherEq. (3.23) becomesµ ∂2 u∂η 2 = ɛE ∂ 2 φ ix∂η 2 , (3.27)where E x = −dφ a /dx is the applied electric field. Note that the gradient of the appliedfield is given in the x-y coordinates. By double integration with respect to η and use ofthe conditions ∂u∂η = ∂φ i∂η= 0 as y → ∞ and that φ i = ζ at u = 0, the integrated equationyields the very important Helmholtz-Smoluchowski relation (see also Fig. 3.3),u eo ≡ u(∞) = ɛζ µdφ adx = α eoE x , (3.28)α eo = − ɛζ µ , (3.29)where u eo is the so-called electroosmotic velocity, and where α eo is the EO mobility. Recentresearch has shown that the zeta potential is ambiguous in that sense that for one mobilitythere may be two corresponding zeta potentials [31]. It is therefore important to emphasizethat it is the EO mobility α eo that is important with respect to EO pumps.The electric field is given by the potential and the electrode separation. If a dielectric ispresent the field is unchanged, but the charges at the electrodes are increased in magnitude.

20 CHAPTER 3. ELECTROOSMOTIC DRIVEN MICROFLOWSueoEElectrolytesolutionuCharged surface(non-conducting)Diffuse layerFigure 3.3: Flow produced by anelectric E field acting on a chargeddiffuse double layer. The electroosmoticvelocity u eo is given byu eo = α eo E. There is overall chargeneutrality.One should be careful not to confuse this setup with an isolated capacitor, with fixedcharges, in which the field is reduced because of the dielectric.Example: EO velocityIn an experiment conducted by Cummings et al. [16], the applied electric field is50 V over 20 mm, thus E x = 2.5 × 10 3 V/m. The zeta potential, ζ = −100 mV,the dielectric constant is ɛ = 78.3 ɛ 0 , and the dynamic viscosity µ = 1.0 ×10 −3 kg (m s) −1 are those of water at room temperature. This yields u eo =200 µm/s3.5 The Zeta PotentialThe thickness of the double layer is given by the Debye length λ D . The derivation of theDebye length was based on the assumption that the ions are points. In real life the ionsare not that small compared with the Debye length. So when considering the boundarycondition for the potential, we must have a more accurate description. The center of theinner most ions are in a finite distance from the surface. This inner layer is called the Sternlayer, Fig. 3.4. This layer is bounded by strong electrostatic forces E ≈ 10 7 −10 8 V/m andhence is immobile. The counterions are attracted to the surface and in this way screeningthe surface charge. The zeta potential is defined as the potential difference across theDebye layer from the shear surface between the charged surface and the electrolyte to thefar field (zero) [18]. The zeta potential is typically of the order ζ ≈ 1 − 200 mV. Thesystem, liquid and surface, is electrically neutral. So the wall charge must equal that of thefluid charge with opposite sign. Far away from the wall, the fluid is electrically neutral [32].We are considering glass as the main material that the capillaries are made of. Othermaterials such as polymers (e.g., PMMA) can also be used. Such materials are interestingwith respect to mass production and fast prototyping, but glass/siliconoxide offers thehighest level of refinement available, i.e., the most accurate and reproducible structures.

3.5. THE ZETA POTENTIAL 21Figure 3.4: Structure of electricpotential in the double layer withinner Stern layer. The thicknessof the Stern layer is approximatelythe width of an atom δ ≈ 0.1 nm.The zeta potential is defined at thesurface of shear, i.e. where the ionsstart to move [18].Figure 3.5: Chemical reactions for silicon in aqueousmediums with different pH. Silanol groups react withthe ions in the aqueous solution, creating an excess ofcharge at the surface.If siliconoxide (silica or quartz) is immersed in water, the potential is generated by achemical reaction at the surface. The silanol groups SiOH are changed to either SiOH + 2or SiO − depending on the buffers pH value, Fig. 3.5.In this way the zeta potential may be varied by the pH of the buffer solution. At thesome pH-level, point of zero charge, the wall potential is zero. For high pH levels thechange in EO mobility is marginal. A suitable choice of the buffers pH-value must beconsidered in order to have effective electroosmotic pumping.SiOH + H + → SiOH + 2 (3.30)SiOH + OH − → SiO − + H 2 O (3.31)pK a (−SiOH) ≈ 6 (3.32)

22 CHAPTER 3. ELECTROOSMOTIC DRIVEN MICROFLOWS3.6 Flow in a Circular CapillaryIn this section we shall find the velocity distribution u(x, y) without the infinitely thinDebye layer assumption. An analytical solution is possible due to the simple geometry.The electric potential and the corresponding charge distribution were found in Sec. 3.3.The Navier-Stokes equation is applied to derive the velocity distribution in a circularcapillary. The inertia terms vanish identically due to the high symmetry of the problem:only u x ≠ 0, but u x = u x (y) so u x ∂ x u x ≡ 0. Pressure on the other hand needs to beincluded, because the entire capillary flow may not be driven solely by electroosmosis. Forstationary flow we therefore end up with0 = −∇p + µ∇ 2 u − F (z + c + + z − c − )∇φ a , (3.33)where the electric field has been replaced by minus the gradient of the applied potentialφ a . Using cylindrical coordinates for the axisymmetric capillary, we adopt (r, θ, x) with xbeing the axial direction, i.e. the direction of the flow. The azimuthal and radial directionsare described by the coodinates θ and r, respectively. We are solving for the velocitycomponent u in x-direction, u = (0, 0, u).Eq. (3.33) is a sexond order partial differential equation. Due to the rotational andtranslation invariance of both the differential equation and the boundary conditions, thesolution u depends only on r, thus u = u(r) and Eq. (3.33) becomesµrddr(r du )= dpdr dx + F z(c + − c − ) dφ adx . (3.34)Using Eq. (3.13b) to eliminate the charge densities, one obtains the modified Navier-Stokesequation, which is then integrated with the appropriate boundary conditions.µr(dr dudr dr∣)= dpdx − ɛ rdu∣ = 0, anddr r=0(dr dψdr dr∣dψdr) dφadx ,∣ = 0,r=0(3.35a)(3.35b)u(a) = 0, and ψ(a) = ζ. (3.35c)The solution isu(r) =[ ]ψ(r) − ζ− ɛ µdφ adx + r2 − a 24µdpdx . (3.36)Here the Debye-Hückel approximation Eq. (3.21) may be utilized for ψ(r), or more correctlythe potentials given in Fig. 3.4. If the Debye-Hückel approximation is used, thesolution becomesu(r) = ɛζ µ[ (1 − exp− a − r )] dφaλ D dx + r2 − a 2 dp4µ dx . (3.37)

3.6. FLOW IN A CIRCULAR CAPILLARY 23From this the flow rate Q is easily obtained by integrationQ = ɛζ µdφ adx πa2( 1 − 2 λ )D− πa4 dpa 8µ dx . (3.38)Two important conclusions can be derived from Eq. (3.38):(1) The flow rate, Q is reduced in the case of channel dimensions comparable with theDebye layer thickness. However Eq. (3.38) is not valid for a ≈ λ D .(2) The flow rate induced by EO is proportional to Q ∝ a 2 , whereas the pressuredriven flow is proportional to Q ∝ a 4 . Scaling of a capillary will therefore make EOFmore effective in small capillaries. In the extreme case of uniform charge distribution(total Debye layer overlap), a parabolic flow profile will be induced, identical to that ofpressure driven flow and Q ∝ a 4 .

24 CHAPTER 3. ELECTROOSMOTIC DRIVEN MICROFLOWS

Chapter 4Simulations of interfaces intwo-fluid systemsIn this chapter we give an account on several microfluidic systems which make use of twofluidinterfaces in their operating principles. The applications are interesting since eachcomprises a number of phenomena characteristic for microfluidics - an advantage withrespect to one aspect can sometimes mean a disadvantage with respect to another. All ofthese systems have been simulated on a CFD software. One device has been microfabricatedand measurements performed in order to test the simulations. This chapter is anoverview of principles described in my four publications [1, 4, 10, 11] in Appendices A, C,D, and E, respectively.4.1 Miscible streams in micro cell sortersThe first example that we are going to consider is a type of device called micro-fabricatedcell sorter (µFACS), Fig. 4.1 (a), which is used to focus a stream of cells inside a microfluidicsystem, [2, 33]. When three separate adjacent streams enter a long microchannel, thewidth of the middle stream can be regulated (focused) by adjusting the flow rates of twoouter streams. The idea is to use the focusing effect so that cells, carried by the middlestream, can be better manipulated, e.g. counted and sorted in an easier manner.In a traditional cell sorting system, the focusing is done only sidewise, while usinga so-called chimney structure it is done from all sides i.e. coaxially, the middle streambeing now completely surrounded by the outer liquid, Fig. 4.1 (b). The flows thus representsubmerged jets of one viscous liquid within another, producing a viscous drag schematicallyshown in Fig. 4.2 (a). This issue will be discussed further below when viscous pumping isconsidered.Since the two streams are miscible liquids, the boundary between them will be adiffused one. The focusing effect will be more pronounced as the difference of the flowrates between the outer and the middle streams is increased [5]. This is because theconvective term, depending on the higher velocity of the outer stream, will dominate thelateral diffusion. This can be seen in Fig. 4.2 (b) where the cross-sections of the focused25

26 CHAPTER 4. SIMULATIONS OF INTERFACES IN TWO-FLUID SYSTEMS(a)(b)Figure 4.1: (a) A schematic view of a micro cell sorter, [2]. (b) Schematic differences infocusing of the middle stream. The sheathing from the sides (left) as compared with thecoaxial sheathing using chimney (right). In a chimney set-up, only one sheathing streamis needed. Courtesy of I.R. Perch-Nielsen.middle stream are shown for two sheath-to-samplestream flow rate ratios.The middle stream coming out of chimney is peculiar in that it is not clear whether itwill remain positioned vertically in the middle for all times. Issues such as stability andthe Coanda effect deflecting the jet to nearby geometry, play important role. In Ref. [1],the chimney is embedded in a basin lowered from the main channel, see Appendix A. Apressure drop will occur inside the basin below the middle stream, causing the stream tobe deflected downwards. Concentration contours extracted from simulations predict thatthe stream will tend to stick to the bottom wall Fig. 4.2 (b).The measurements in microfluidic systems are hard to obtain due to their small size.In a set of experiments using microPIV (Particle-Image-Velocimetry) measurements, thevelocity profiles obtained by simulations were compared with experiments. The resultsdiffer mainly due to numerical diffusion.From a practical point of view, a chimney embedded in cavity-like basin will tend tocollect bubbles which are unwanted. The reasons for this we address next.4.2 Bubbles in microchannelsAnother example of two-fluid interfaces in microfluidic systems are gas bubbles. Suchbubbles present a significant problem: they can block the microchannels, disrupt the flow,and disturb measurements in an uncontrolled manner.4.2.1 Formation of bubblesThe following are the most common reasons for bubble formation:• Cavitation (or, nucleation). When the local pressure p(x) drops below the vaporpressure p v of a fluid, a bubble containing mainly the vapor molecules spontaneouslydevelop. The liquid is said to ”break” at the point of cavitation. The effect can be

! "#¤$ %&"'4.2. BUBBLES IN MICROCHANNELS 27¢¡¤£¦¥¨§©(a) ¢ ¤ (b)Figure 4.2: (a) Schematic view of the viscous drag by two outer streams. (b) Simulationsof concentration contours for different focusing ratios R (R=sheath-flow rate/sample-flowrate). On the left, the cross-sections of the outer channel are shown with the middlestream being more (top) and less (bottom) focused. On the right, the corresponding topview of a part of the chimney structure, see Appendix A. The middle stream contains asample species that can diffuse into the sheathing liquid. The focusing is better for higherR, and confined to the bottom of the channel.enhanced either by raising the temperature and thus increasing the vapor pressure 1 ,or by decreasing the local pressure. The latter effect is particularly pronouncedat junctions of microfluidic systems when changes in pressure occur due to suddencontractions/expansions.• Solubility. The solubility of a gas in a liquid is governed by Henry’s law, whichstates that the partial pressure of a gas above a solution is directly proportionalto the concentration of that gas in the solution, [32]. The solubility decreases withincreasing temperature. A common practice with respect to lab-on-chip systems is topush helium (which itself does not nucleate under ordinary conditions) through thesystem to lower the partial pressure of other gases and therefore their concentrationinside the liquids. Such a degassing procedure is usually good for several hours.• Electrolysis. In microfluidic systems containing electrodes and operating under DCvoltages, e.g. some types of EO micropumps, gases will form next to the electrodesdue to electrolysis. The buffers are usually water solutions, and so hydrogen andoxygen will develop. The problem will be circumvented if AC fields are applied (anddesign allows it).• Heating. As mentioned, heating of devices will cause bubbles to form: liquid mightevaporate or dissolved gases expand.Apart from the degassing method, the removal of unwanted bubbles might involve highfrequencysound vibrations or control of surface tension. However, in spite of such effortsgas bubbles may still be present, and once there it often requires large driving pressuresto push them out of the system, [35].1 At 20 ◦ C the vapor pressure for water is 0.023 bar, but at 100 ◦ C it is 1 bar and the water boils, [34].

H28 CHAPTER 4. SIMULATIONS OF INTERFACES IN TWO-FLUID SYSTEMS¢¡¤£ ¦¥§£tttDMRrmhdLFigure 4.3: (a) A bubble passing through a sudden contraction. The curvatures at bothends are different and constant while the bubble is in the contraction. (b) A bubble in atapered channel. Appropriate names have been given to different lengths and angles inuse. The curvatures depend on the position of the bubble because of the tapering. SeeAppendix D.4.2.2 Clogging of microchannelsIf a slowly moving bubble is present in a circular capillary of diameter d, the gas-liquidinterface has a spherical shape, [36], as sketched in Fig. 4.3. This is true as the sphericalshapes provide a minimal pressure drop and thus a minimum surface energy. The pressuredrop over each of the interfaces is given by Young-Laplace equation( 1∆P i = σ + 1 )R 1 R 1= 4σ cos θd , (4.1)where θ is the contact angle, and the two main curvatures are identical due to symmetry,R 1 = R 2 = R. In a straight channel the total pressure drop, due to curvature, over thebubble is obviously zero. ∆P i is identical with different sign at each end of the bubble. Tomove the bubble through a straight channel only the friction needs to be overcome. Therequired pressure is∆P f = 4σ fd , (4.2)where σ f is a frictional surface tension parameter. However, if there is a sudden changein the capillary radius the pressures ∆P i on two sides of the bubble will not balance eachother. The pressure drop over the bubble is∆P suddenb= 4σ( 1d − 1 D), (4.3)and acts as a blocking pressure, in addition to ∆P f , making it harder to push the bubbleinto the part of a smaller radius. A bubble in a sudden contraction is shown in Fig. 4.3 (a).If on the other hand the contraction is tapered, the pressure drop is given by∆P taperedb( cos(θ + θt )= 4σd− cos(θ − θ )t), (4.4)D

4.2. BUBBLES IN MICROCHANNELS 29¢¡¤£ ¦¥§£ ©¨£©£ ¦£Figure 4.4: (a) A sketch of the trap geometry where the central length scales are marked:the width d of the main channel, the size l 0 of the offset between the upper and lower wall,and the width l t of the narrow trapping channels. (b)-(e) Four snapshots of the simulatedtrapping procedure near the offset and trap regions. (b) A bubble moving towards theoffset region. (c) Detaching of one side of the bubble by the offset effect. (d) The bubble ispushed towards the trapping region. (e) The bubble gets trapped. See also Appendix D.where θ t is tapering angle, Fig. 4.3 (b). The pressure contribution ∆P b due to the curvatureeffects will be smaller in case of the tapered channels than in the case of sudden contraction.Therefore, it may be beneficial to use tapered constrictions in microfluidic networks wherethe presence of bubbles poses a problem.However, some interesting and non-intuitive behavior appears when the full analysisis made about the dependence of ∆Pb sudden on arbitrary tapering angle. A systematicstudy of a bubble passing through tapered constriction in a quasi-static motion has beenconducted and presented in the paper enclosed as Appendix C.4.2.3 Passive bubble trapThe curvature effects described above can be used to manipulate the bubbles by directingthem away from the main flow into specific locations from which they could be removed.For this purpose a special passive bubble trap is developed, using a combination of geometryand surface tension effects. Fig. 4.4 (b)-(e) show a 2D CFD simulation of abubble-trapping process allowing unhindered flow. In a real system, several bubble trapscould be placed at locations where there is a tendency of bubble creation. Using specialsemi-permeable materials (e.g. Gore-Tex), the bubbles could then be extracted from theregion and out of the system.

30 CHAPTER 4. SIMULATIONS OF INTERFACES IN TWO-FLUID SYSTEMS4.3 Two-liquid viscous pumpAs a final example of systems with two fluid interfaces, we will present a device operatingwith two immiscible liquids such as water and oil; namely, a unique EO pump able topump non-conducting liquids. Non-conducting liquids do not form the electric doublelayer at channel walls since they have very low conductivity (< 10 −6 S/m) i.e. very lowamount of dissociated ions. Therefore, they cannot be pumped by ordinary EO pumps.The device has been developed during the PhD project together with A. Brask, andis even patented (provisional US Patent application; see the conference paper enclosed inAppendix E.).4.3.1 Submerged jetsBefore the attempt to explain the mechanism of this two-liquid viscous pump, we willexpose a known phenomenon called entrainment of one fluid by another, [37]. If a fluidjet, say water, is submerged in another fluid, say air, such as the stream from a water tap,the jet will drag with itself the air not only from the border where the fluids are in directcontact, but also the air within the boundary layer of a finite thickness. This phenomenonis exploited in so-called filter pump used in chemical laboratories, Fig. 4.5 (a). The moreair is entrained with the streaming water, the more air will flow from the side channel toreplace it, creating a significant suction. The emphasis here is on the viscous drag withinthe boundary layer. If a water jet is submerged within water the same phenomenon willappear; this is precisely what happens in µFACS using the chimney except that the outer,faster jets do the drag.In microfluidics the boundary layer can occupy large portions of channels, so the sametype of filter pump could, in principle, be possible to build in microsystems. The problemis, however, to pump the driving jet in the first place.4.3.2 EOF induced viscous dragAn idea was proposed by O. Geschke, MIC, to possibly use an EOF as a driving mechanismto drag other liquids. The intention was to direct a conducting, so-called pumping liquid,through two side-channels, and then try to drag any, in particular non-conducting, workingliquid into the main channel, Fig. 4.5 (b). The original design was not successful since itbasically did not enable the pressure build-up, due to the equal flow resistances of theouter channels. Three features have been introduced in the improved design, Fig. 4.6.• The EOF is more efficient than pressure in microflows, Eq. (3.38). By increasing theflow resistance of the side channels, the conducting pumping liquid is able to flowthrough them, while the working liquid is largely prevented from entering - a builtpressure will not ”leak” (as much) through these side pressure valves.• The conducting liquid needs to be confined close to the walls to provide a freeboundary to enable the entrainment effect i.e. to enhance the amount of workingliquid through the main channel. This is achieved by lowering EO mobility in theside channels as compared to the main channel.

4.3. TWO-LIQUID VISCOUS PUMP 31Top viewInlet B aInterfaceOutlet BInlet ASymmetryOutlet AFigure 4.5: (a) The water-jet pump creates suction effect by entrainment of the air withinthe boundary layer at the free air-water interface, [37] (b) The original but not usable ideaof a two-liquid viscous pump. The pumping liquid driven by EOF enters through inlet Band the working liquid at inlet A. Inlet/outlet B are connected to external reservoirs, [17].• In order to achieve the largest possible E-field in the main channel, and thus toincrease the driving velocity, Eq. (3.28), the pressure valves are made short (smallelectric resistances). This will enable a desirable low voltage operation.The performance of the pump has been analyzed using two methods, [11]. The pressureand flow rates have been calculated by equivalent circuit theory, and in addition, a setof CFD simulations was carried yielding more detailed information on the distribution ofpressures and velocities. The two methods were applied to the cases of both immiscibleand miscible liquids. Pumping of an immiscible liquid is shown in Fig. 4.7. Three setsof resulting backpressures and corresponding flow rates are shown in Table 4.3.3. Thegenerated pressures are not very large, but the pump Q-p characteristic largely depends onthe geometrical factors and can be significantly enhanced by advanced etching techniques.The pump still works for miscible liquids, but here the working liquid gets mixed withthe pumping liquid. Because of the possibility to pump all types of liquids, the describedconcept seems very promising.4.3.3 Instability problemOur analysis of the pump did not, however, include a stability analysis. The interfacebetween two immiscible liquids will be subject to flow disturbances which are alwayspresent in the flow. It is therefore important to estimate the flow regimes under which theinstabilities, that lead eventually to flow disruption, can develop in course of time.The analysis, however, is a comprehensive task. It requires the understanding of electrohydrodynamics(EHD) i.e. the way how electric fields couple with hydrodynamics. Wewill dedicate the next four chapters to address to some extent the aspects of electrohydrodynamicinstability. This is then applied to two-liquid microfluidic systems which bring

32 CHAPTER 4. SIMULATIONS OF INTERFACES IN TWO-FLUID SYSTEMSFigure 4.6: A schematic view of our two-liquid viscous pump, Appendix E. The pumpingliquid and the working liquid are shown in light gray and dark gray, respectively. Theexpected velocity profiles are shown as inserts to give an idea of the overall flow. Internalfriction gives rise to a counter pressure, which induces the parabolic flow profiles. Thethickness of the pumping liquid layer is D layer .into light some interesting features.

4.3. TWO-LIQUID VISCOUS PUMP 33Figure 4.7: A CFD simulation of an immiscible two-phase flow. The pumping liquid ofmass fraction 1 is introduced through inlet B. At outlet B the mass fraction drops to 0.87and lower as shown, and a part of the pumping liquid continues in the main channel.EO mobility Max flow rate Backpressureα eo /µm 2 (V s) −1 Q / nL/s PaModel 0 0.296 31.6Simulation 0 0.290 30.6Deviation 2% 3%Model 5000 0.267 28.5Simulation 5000 0.260 27.4Deviation 3% 4%Model 10000 0.237 25.3Simulation 10000 0.230 24.3Deviation 3% 4%Table 4.1: Comparisons between the circuit model and a CFD simulation. The results forthree different EO mobilities in the valve channels are shown. The agreements are within4%. Parameters: φ a = 10 V and p in = p out = 0.

34 CHAPTER 4. SIMULATIONS OF INTERFACES IN TWO-FLUID SYSTEMS

Chapter 5Fluid dielectrics and the Maxwellstress tensorIn the following chapter we investigate the effects of the electric fields on fluid dielectrics.In microfluidic systems large E-fields can be achieved by applying relatively small voltages,due to the smaller length-scales involved. The effect of E-fields, therefore, will be quitesignificant.We first introduce the electrical, Maxwell, stress tensor in vacuum, a useful concept inexpressing the electrical body forces, [15].We then define thermodynamic properties of dielectrics placed in an electric field tounderstand how does the presence of the field affect these properties, Landau and Lifshitz[38].The thermodynamical approach leads to important general expressions for the stresstensor and the resulting electrical force on a dielectric placed in an electric field. Thevarious terms in the force equation are then explained.Finally, the stress conditions on the boundary between two fluid dielectrics are discussedand the non-trivial condition on the pressure difference derived.5.1 The Maxwell stress tensor in free spaceConsider a system of stationary charges inside a volume V placed in an electrostatic fieldin vacuum. We assume that the volume is free of any neutral dielectric material. If ρ isthe charge density, the force acting on it is∫F e =V∫ρE dV =V[∇·(ɛ 0 E)] E dV. (5.1)By the following transformation, the integrand can be written as the divergence of a tensor.Since ∇×E = 0 we can subtract ɛ 0 E×(∇×E) and obtain the i-th component of the force35

36 CHAPTER 5. FLUID DIELECTRICS AND THE MAXWELL STRESS TENSORS( )E( )S( )nFigure 5.1: The electric stress S e acting on a surface enclosing charges placed in an electricfield. For an arbitrary angle θ, the E-field bisects the angle between the stress and thenormal n. In a perpendicular E-field, the stress is normal to the surface, pointing outwards.In a tangential E-field, the stress is directed inwards.per unit volume in the following form [15]1ɛ 0ρE i= E i(∇·E)−[E×(∇×E)] i(5.2a)= E i∂ kE k−[E j(∂ iE j− ∂ jE i)−E k(∂ kE i− ∂ iE k)] (5.2b)= ∂ i(E 2 i − 1 2 E2) + ∂ j(E iE j) + ∂ k(E iE k) (5.2c)= 1 ɛ 0∂ kT Mik .(5.2d)This is divergence of a tensor. Applying Gauss theorem, the i-th component of the totalforce can therefore be written as a surface integral∫∫∫F ei= ρE i dV = Tik M df k = Tik M n kdf, (5.3)VSwhere the normal vector n = (n x , n y , n z ) is directed out of the volume. The tensor TikM iscalled the Maxwell stress tensor,ST Mik = −1 2 ɛ 0 E2 δ ik+ ɛ 0 E iE k. (5.4)Note that the force on an empty volume placed in a static E-field vanishes while thestress tensor TikM does not. From Eq. (5.3) we see that the force on a stationary chargedistribution can be determined by calculating the electric stress over a surface surroundingthe charges. The force acting on a unit area of the volume in vector notation isS e = − 1 2 ɛ 0 E2 n + ɛ 0 (n·E)E. (5.5)It is useful to express the applied E-field in terms of the normal and tangential componentswith respect to the surfaceE = E cos θ n + E sin θ t, (5.6)

5.1. THE MAXWELL STRESS TENSOR IN FREE SPACE 37where θ is the angle between the E-field and the normal, Fig. 5.1 (a). Inserting the abovefield into Eq. (5.5) we getS e = − 1 2 ɛ 0 E2 n + ɛ 0 (E cos θ)(E cos θ n + E sin θ t)= − 1 2 ɛ 0 E2 (1 − 2 cos 2 θ) n + 1 2 ɛ 0 E2 sin 2θ t(5.7)= 1 2 ɛ 0 E2 (cos 2θ n + sin 2θ t),so that E bisects the angle between the stress and the normal. The following cases are ofspecial interest:S e = 1 2 ɛ 0 E2 n, (θ = 0, π), (5.8a)S e = − 1 2 ɛ 0 E2 n, (θ = π/2), (5.8b)|S| = 1 2 ɛ 0 E2 , (all θ). (5.8c)If the E-field is perpendicular to the surface (θ = 0, π), the stress is in the direction of thenormal (and the field) pulling the element; if the field is parallel to the surface (θ = π/2),the stress is directed inwards i.e., pushes the surface in the opposite direction of n. In allcases the total magnitude of the stress is constant and equal to ɛ 0 E 2 /2.To understand the true meaning of the Maxwell stress tensor, one needs to considerthe total force on a system of charges moving within variable EM-fields i.e. the Lorentzforce∫F e + F m = ρ(E + v×B) dV = d dt P mech. (5.9)VF e and F m are electric and magnetic forces, respectively, and we have expressed the totalforce as the rate of change of linear mechanical momentum P mech. After somewhatinvolved transformations, similar to the one giving Eqs. (5.2a) - (5.2d), we arrive at [15]∫ddt P mech + ∂ t ɛ 0 E×B dv = Tik ∫VM df k, (5.10)Vwhere now T Mikincludes also the magnetic contributionTik M = −1 [ɛ0 E 2 + µ20 B 2] δ ik+ ɛ 0 E iE k+ 1 Bµ iB k. (5.11)0From Eq. (5.10) we see that, unlike for the static fields, the time-varying fields will exerta force ∂ t∫V ɛ 0 E×B dv over an empty volume, that is, when Fe = F m = 0. The quantityg = ɛ 0 E×B, (5.12)has dimension of momentum density and is interpreted as the momentum density of electromagneticfield. Eq. (5.10) now can be written∫ddt (P mech + P field ) i = Tik M n kdf, (5.13)V

38 CHAPTER 5. FLUID DIELECTRICS AND THE MAXWELL STRESS TENSORexpressing the conservation of linear momentum of particles and fields. The term Tik Mnkmeasures therefore the momentum flux per unit area into the volume.We have seen that Maxwell stress tensor, Tik M , can be used to calculate the electric forceon the system of particles and fields enclosed inside a volume V . However, an importantaspect discussed in Ref. [39] is worth of attention. In the early days of electromagnetictheory, Maxwell and Faraday have considered these stresses primary over the fields in determiningthe electrical forces. In particular, propagation of electromagnetic fields throughan empty space could not be conceived and an elastic medium, the ether, was postulated.The stresses in Eqs. (5.3) and (5.10), were then associated with deformations of the etherand the electric force propagated via the contact with it. It was then confirmed thatelectric fields are primary quantities and that the electric force in a point in space solelydepends on the E-field at that point, without any underlying support. In that sense, thestress formulation is rather a convenience than (essential) physical reality. What Eqs. (5.3)and (5.10) really mean is that the bulk force on the system of charges can be correctlycalculated assuming the existence of a fictitious state of stress at the surface, [39].5.2 Fluid dielectrics in electric fieldWe will now investigate some of the interesting phenomena arising when a dielectric isplaced in an electric field. Unlike with conductors, external E-fields do not vanish insidethe dielectrics and that greatly affects their thermodynamic properties. Of a particularinterest to us are electric forces acting on a dielectric material as well as stresses onboundaries between two dielectrics.Because of its generality we will follow the free-energy approach developed in Landauand Lifshitz [38]. The formalism is rather comprehensive and instead of deriving it all,we will borrow some of the main results. When necessary, the terms will be explained inmore detail.5.2.1 The internal and the free energy of a dielectricLet a thermally insulated dielectric be placed in an external electric field produced by aset of charged conductors. If now the field undergoes an infinitesimal change, due to aslight change in the charges on the conductors, a work done in the process is∫δW =∫=∫=∫=VVVVφ δρ dVφ∇·δD dV∫−∇φ·δD dV +E·δD dV.Sφ δD·n da(5.14a)(5.14b)(5.14c)(5.14d)

5.2. FLUID DIELECTRICS IN ELECTRIC FIELD 39where we made use of ∇·D = ρ and the fact that the surface integral in Eq. (5.14c) vanishesfor a localized charge density ρ when S is taken to infinity. The volume integration is overthe whole volume outside the conductor including the vacuum if the dielectric does notoccupy all the space.The well-known integral Eq. (5.14d) represents the increment of energy stored in theelectrostatic field. Now both the total energy and the total free energy of the dielectric(each including also the energy of the electric field), change by this amount. The changeof the internal energy U and the free energy F per unit volume are thendU = T dS − pdV + µ c dρ + E·dD, (5.15a)dF =−SdT − pdV + µ c dρ + E·dD,(5.15b)where µ c is the chemical potential with respect to unit mass 1 . A Legendre transforamtionfrom variable D to variable E is performed to get the new thermodynamic potentials Ũ(E)and ˜F (E)Ũ = U − E·D,˜F = F − E·D,(5.16a)(5.16b)for which the thermodynamic relations becomedŨ = T dS − pdV + µ cdρ − D·dE, (5.17a)d ˜F =−SdT − pdV + µ c dρ − D·dE.(5.17b)To find the internal and free energies per unit volume stored in a field, Eqs. (5.15a)and (5.15b) have to be integrated from the initial state D = 0 till the final D. If themedium is linear, D = ɛE, we getU = U 0 (S, V, ρ) + 1 2ɛ D2 ,F = F 0 (T, V, ρ) + 1 2ɛ D2 ,(5.18a)(5.18b)where U 0 and F 0 are zero-field quantities. In similar wayŨ = U 0 (S, V, ρ) − 1 2 ɛE2 ,˜F = F 0 (T, V, ρ) − 1 2 ɛE2 .(5.19a)(5.19b)The total internal and free energies, U tot and F tot , are obtained by integration of Eqs. (5.18a)and (5.18b) over all space.Various thermodynamic potentials describing a system acquire minimal values in astate of thermal equilibrium, with respect to infinitesimal changes in the state of the1 In Chap. 3 chemical potential µ chem was defined with respect to concentration. These two differ by aconstant factor since ρ = c m, where concentration c = n/V is the number of particles per unit volumeand m the mass of one particle

40 CHAPTER 5. FLUID DIELECTRICS AND THE MAXWELL STRESS TENSORsystem, Landau and Lifshitz [40]. The thermal equilibrium in an electric field will dependwhether the changes of the state are happening at the constant charges or the constantpotentials. When the charges (and temperature) are held constant, the state of thermalequilibrium is characterized by the minima of U and F ; when the potentials are constant,the state of equilibrium is described by the minima of Ũ and ˜F . By inspecting Eqs. (5.18a)and (5.19a) we note that for linear dielectricsδŨ= −δU. (5.20)The usefulness of the formalism can be illustrated by considering the change in theinternal energy if the dielectric constant ɛ is changed. From Eq. (5.18a) we get∫1δU tot =V ɛ∫VD·δD dV − 12ɛ 2 δɛD2 dV(5.21a)∫= E·δD dV − 1 ∫δɛE 2 dV.(5.21b)2VIf the charges are held constant, the first term in Eq. (5.21b) vanishes, being the chargecontribution from Eq. (5.14a), and we getδU tot = − 1 ∫δɛE 2 dV. (5.22)2Eq. (5.22) says that an increase in the dielectric constant somewhere inside the systemreduces the electrostatic energy (with charges being fixed). On the other hand, when thepotentials are kept constant, the energy increases by the same amount, Eq. (5.20).5.2.2 The Maxwell stress tensor in fluid dielectricsWe have seen in Sec. 5.1 that a force acting on a finite volume can be represented as a stressapplied to the surface of the volume. This general fact stems from the conservation of linearmomentum, since the rate of change of momentum inside a volume can be represented asthe momentum flux through its surface [41]. To find the force on a fluid dielectric placedinside an electric field we will have identical approach; that is, we want to find the stresstensor σik M satisfying ∫ ∫f idV = σik M n kdf. (5.23)VAs before, σik Mdf k is the i-th component of the force acting on a surface element df, withn being directed out of the volume. Once σikM is determined the force f i can be found as,Eq. (5.2d)f i = ∂ kσik M . (5.24)There are several approaches to determine the stress tensor σik M in dielectrics, [39, 42,38]. The most general method is described in Landau and Lifshitz [38]. Here we brieflydescribe the procedure 2 . A fluid dielectric is placed between two conducting planes held at2 The terms appearing in the stress tensor and in the resulting force are discussed below.VSV

5.2. FLUID DIELECTRICS IN ELECTRIC FIELD 41a constant potential. The planes are then subjected to an isothermal virtual displacementwhich causes the homogeneous deformations within the dielectric. The work done on theconducting planes is equal to the decrease in the free energy ˜F of the dielectric. Theresult of the procedure is the stress tensor σik M , which in case of isotropic and linear media,becomes the symmetric tensorσik M = − ˜P 0 δ ik− 1 [2 ɛ rɛ 0 1 − ρ ]( )∂ρ ɛɛ r E 2 δ ik+ ɛ r ɛ 0 E iE k. (5.25)rHere ˜P 0 (ρ, T ) is the pressure present in the dielectric when E = 0, while ∂ ρ is the partialderivative with respect to mass density. If we consider vacuum, ɛ r = 1, and neglect thepressure we recover Eq. (5.4). The termTc e = ρ ɛ r(∂ρ ɛ r)T , (5.26)is called the electrostriction term and is in connection with the deformations imposed bythe electric field.5.2.3 Electric force in fluid dielectricsDifferentiating Eq. (5.25) according to Eq. (5.24) we get for the force per unit volumef e = −∇ ˜P 0 (ρ, T ) + 1 2 ɛ 0 ∇ [E 2 ρ ( ∂ ρ ɛ r)T]− 1 2 ɛ 0 E2 ∇ɛ r + ρ e E, (5.27)where ρ e is the density of free (external) charge. Eq. (5.27) is a very important resultand represents the most general expression for the force on dielectric media placed in anelectric field. Therefore, it is worth now to investigate how the terms came about.PressureGenerally, the pressure in compressible fluids is a function of mass density and temperatureand a separate equation of state is needed to fully characterize the fluid systems. Bythe additional equation the pressure value is completely determined. We will investigateonly incompressible liquids and therefore the pressure will appear only in the pressuregradient term in the force density of Eq. (5.27). This means that any constant pressureadded to ˜P 0 would yield the same motion. If we look at Eq. (5.25) we realize that thepressure term does play a role in the determination of stresses on a boundary between twomedia (to be considered later); however, there again it will enter as a pressure difference.In passing we remark that non-unique pressure solutions in incompressible liquids isa known problem from computational fluid dynamics. Numerical algorithms are thus devisedto deduce the pressure field in an iterative process, starting with a guessed initialvalue until a converged solution is reached.ElectrostrictionThe electrostriction term of Eq. (5.26) gives rise to the second term in Eq. (5.27). Aspointed, it arises due to the deformations of a dielectric in an electrostatic field.

42 CHAPTER 5. FLUID DIELECTRICS AND THE MAXWELL STRESS TENSORLet us assume that the dielectric constant depends on position as well as density,ɛ = ɛ(r, ρ). When a dielectric is placed in an electrostatic field it undergoes a deformationsince material points are displaced due to dipole formation. As a result, the mass densitychanges.We introduce the displacement field s(r); a point inside the dielectric at a positionr before the deformation, finds itself at the new position r + s(r) afterwards. An initialsmall volume dV becomes a deformed dV ′dV ′ = (1 + ∇·s) dV, (5.28)where ∇·s is the volume dilatation up to the first order in s, [Stratton]. Since mass isconserved we haveρdV = ρ ′ (1 + ∇·s) dV, (5.29)where ρ and ρ ′ are the densities before and after the deformation, respectively. For aninfinitesimal changeδρ = ρ ′ − ρ = −ρ(∇·s), (5.30)which results in the corresponding change of the dielectric constantδ ρ ɛ = (∂ ρ ɛ)δρ = −ρ(∂ ρ ɛ)∇·s. (5.31)If the above variation is inserted into the change of the electrostatic energy, Eq. (5.22),we eventually obtain the second term in Eq. (5.27).The electrostriction term c for liquids could be estimated from Clausius-Mossotti equation3 Nαɛ r − 1 =1 − 1 (5.32)3Nα, where N is number of atoms per unit volume and α molecular polarizability. Since N = βρis proportional to the density, after a quick manipulation we getc e = 13ɛ r(ɛ r − 1)(ɛ r + 2). (5.33)The relative dielectric constant ɛ r is always bigger than 1 resulting in a positive electrostrictionterm c. It is worth mentioning that in Ref. [43] performed a set of experimentswith perpendicular and tangential electric fields imposed on the boundary between twoliquid dielectrics. The electrostriction effects, although present, were much less than suggestedby Eq. (5.33).Generally, the electrostriction involves change in shape of the dielectric. For a slightdeformation, that is for small E-fields, the first order effect include a volume expansionwithout the change in shape. Therefore, the effect can be described as an effective pressureon the dielectric. Due to the large E-fields present in microfluidic systems, order of10 5 V/m, the effects of the electrostriction could be quite significant. In case of incompressibleliquids, however, we can in first approximation treat the deformation as small3 For water, or polar molecules in general, the equation does not apply and the approach developed byOnsager must be used, [44].

5.2. FLUID DIELECTRICS IN ELECTRIC FIELD 43(ρ = const.) and so group the electrostriction term with ˜P 0 into a redefined pressure.As already mentioned, such a procedure will not affect the equations of motion. We canrewrite Eq. (5.27)[f = −∇ ˜P 0 − 1 2 ɛ 0 E2 ρ ( ])∂ ρ ɛ r − 1 T 2 ɛ 0 E2 ∇ɛ r + ρ e E. (5.34)InhomogeneityThe term proportional to ∇ɛ gives the force contribution due to the non-homogeneousdielectric constant. Since ɛ = ɛ(r, ρ), the small element of the dielectric positioned at rafter the deformation was at the point r − s before; the contribution to δɛ due to theinhomogeneity isδ r ɛ = −s·∇ɛ, (5.35)which together with Eq. (5.22) leads to the third term in Eq. (5.27).In general ɛ is a function of density, temperature and (mixture) concentration c whichgenerally depends on position. We have∇ɛ = (∂ ρ ɛ) T,c ∇ρ + (∂ Tɛ) ρ,c ∇T + (∂ c ɛ) ρ,T ∇c. (5.36)For incompressible, non-homogeneous liquid at constant temperature the force will dependon the concentration gradient. In microfluidics, the systems used for analytical methods,for example separations, consist of usually two miscible liquids - a buffer and a sample.Due to diffusion, their interface will eventually acquire transitional dielectric constant (andconductivity), which will contribute to the force in proportion to Eq. (5.36).5.2.4 Alternative form of the force densityThe equation Eq. (5.27) has been derived from the energy principle. The derivation wasfirst proposed by Korteweg and later developed by Helmholtz in 1881.An alternative way to obtain the force on a dielectric placed in an E-field is to supposethat the electric forces exerted on free charge and polarization charge (dipoles) aretransferred directly to the dielectric, [45, 46]. Since the force on a single dipole is p·∇E,then for single-species material with N d dipoles per unit volume we havef = N dp·∇E = P·∇E, (5.37)where P = N dp is the polarization density. Now, P = (ɛ − ɛ 0 )E and ∇×E = 0. With theuse of known vector identitiesV·∇V = (∇×V)×V + 1 2 ∇(V·V),∇(φψ) = φ∇ψ + ψ∇φ,(5.38a)(5.38b)we can transform Eq. (5.37) intof = 1 2 ɛ 0 ∇ [ (ɛ r − 1)E 2] − 1 2 ɛ 0 E2 ∇ɛ r . (5.39)

44 CHAPTER 5. FLUID DIELECTRICS AND THE MAXWELL STRESS TENSOR 1 2E=(E n,E t)D n1n(1)P0D n2Et1Et 2P0(2)Figure 5.2: A disturbed interface between two liquid dielectrics. In case of perfect dielectrics,the stresses are normal to the interface causing a change in pressure,‖P 0 ‖=‖ 1 2 ɛ ( En 2 − Et2 )‖.Comparing Eq. (5.27) (without the free-charge term) and Eq. (5.39) we see that theyare not identical. It is not surprising since Eq. (5.27) is not restricted to dipoles - multipolescould in principle be present 4 . The two expressions will be equal when(∂ρ ɛ r)T = c 1 = ɛ r − 1, (5.40)ρor, whenɛ r = 1 + c 1 ρ. (5.41)But Eq. (5.41) is just the first term in the expansion of ɛ r in terms of ρ. So if the compressionsare assumed small, the two equations could be made identical. This is in supportof the previous statement that in the case of incompressible liquids, the deformations,manifesting themselves as a pressure, do not affect the motion.However, if the compressibility is significant the approach leading to Eq. (5.39) will notgive correct results. Once calculated, the polarization force could indeed be used to predictthe corresponding deformation; but in turn the deformation affects the polarization, andif this change is not included the prediction will be only approximative. Such an erroneousattempt is discussed in Ref. [39]. If a force description is used it is important to accountfor all possible contributions which is usually difficult. The difficulties are circumventedin the (free) energy approach, used to obtain Eq. (5.27).5.2.5 The boundary between two dielectricsWe will now consider the boundary conditions at the interface between two incompressibleliquid dielectrics placed next to each other, Fig. 5.2. Let us first introduce a notationwhich will be of extensive use in later chapters. The ”jump” in a function f(x, y, z) at theboundary z = z 0 between two media represents the change in the function f across the4 The relationship between vectors D and E can be written as D i = ɛ 0 E i + (P i − ∑ j ∂ j Q′ ij + ...),where Q ′ ij is the macroscopically averaged electric quadrupole moment, [15]. The higher terms containingderivatives, however, are vanishingly small in comparison with the terms that do not contain them.

5.2. FLUID DIELECTRICS IN ELECTRIC FIELD 45boundary‖f ‖≡ f(x, y, z = z + 0 ) − f(x, y, z = z− 0 ) = (f 2 − f 1 ) z=z 0, (5.42)where + and − refer to the different sides, 2 and 1 respectively, of the boundary.If the normal vector n is directed from the medium 2 into the medium 1, we can writethe known conditions imposed on E and D fields at the boundary between two media asn× ‖E‖ = ‖E t ‖ = 0, (5.43)n · ‖D‖ =‖D n ‖ =−q s , (5.44)expressing the continuity of the tangential component E t across the interface, and thejump in the normal component D n due to a surface charge density.The forces at each side of the interface are equal and opposite. In terms of the stresstensor σik M we have ‖σik M ‖ n k= 0. (5.45)In Eq. (5.25) we can put the electrostriction term into a redefined pressure P 0 , and expressthe E-field in terms of normal and tangential components. We get for the normal stressesand for the tangential stresses‖P 0 ‖=‖ 1 2 ɛ ( E 2 n − E 2 t)‖=12 ‖ɛE2 n ‖ − 1 2 E2 t‖ɛ‖, (5.46)‖E t D n ‖= E t ‖D n ‖= −E t q s , (5.47)where the use has been made of Eqs. (5.43) and (5.44).Several important remarks need to be underlined. We have already seen in Sec. 5.1that an E-field acts as an effective pressure with the normal and tangential componentssuppressing each other’s contribution, and that is what the balance of normal stresses inEq. (5.46) reveals.In case of perfect dielectrics, q s = 0, or in case when the interface is perfectly conducting,E t = 0, the equality of tangential stresses in Eq. (5.47) is satisfied identically.This means, that for these cases the electric stresses are always perpendicular to the interface.Alterations of interface shape together with the surface tension serve to balancethe electric stress.If there are free charges accumulated at the interface, a tangential E-field acting onthe charge will produce a net tangential stress. In case of liquids, a viscous flow will resultto produce the stresses to balance the action. This will be shown in Chap. 7. In otherwords, an inviscid approximation in the case of free charge at the interface will not be avalid one. This is the mechanism behind realistic, leaky dielectrics with small but finiteconductivities, Saville [46].

46 CHAPTER 5. FLUID DIELECTRICS AND THE MAXWELL STRESS TENSOR

Chapter 6Hydrodynamic stability analysis:basicsBy a dynamical system we mean any physical system that evolves in time; it could bean atom, a swinging pendulum or a general hydrodynamic flow to which we will focusour attention. A dynamical system is said to be unstable if a small perturbation of itsequilibrium state increases exponentially in time [37].For any viscous flow that is subject to steady conditions the equations of motion,Eq. (2.8), will yield in principle the exact steady solutions, Landau-Lifshitz [14]. Simpleexamples of the exact (analytical) steady solutions of the flow equations are Poiseuille andCouette flows. However, these stationary solutions exist only within certain range of therelevant flow parameters defined by Reynolds’ or other dimensionless numbers. Withinthis stable range any fluctuation imposed on the stationary flow will decay in time leavingthe steady solution unaltered. Outside the range, the small fluctuations which are alwayspresent within the flow will destabilize the flow and the stationary solution will seize toexist. The outside regime is termed unstable. Such transitions from stable to unstableregimes are the topic of hydrodynamic stability analysis and have been the centerpiece ofenormous number of research studies dating all the way from 19-th century, [47, 48, 49, 50],to name a few.The flow instabilities have a big practical impact. In large systems they result inturbulence and big energy losses associated with fluid transport, and are thus undesirable.In microfluidic systems, devices such as EOF pumps require operational stability whilesome of them, such as mixers or microdispensers, would benefit from the instability.In this chapter we will try to introduce general concepts of the linear stability analysis.We would like to find a systematic way to quantitatively separate stable from unstable flowregimes. In later chapters, the analysis will be applied to a specific problem of interfacestability between two immiscible liquids in microfluidic systems.47

48 CHAPTER 6. HYDRODYNAMIC STABILITY ANALYSIS: BASICSSTRXRX SXTXFigure 6.1: Typical potential energy φ of a conservative system. The points R, S, and Tcharacterize the stable, neutral and unstable equilibrium, respectively.6.1 General conceptsConservative systemsIf we consider a one-dimensional conservative system, its equation of motion is describedbym d2 x(t)dt 2 = f(x), (6.1)where m is a constant parameter having different meaning for different conservative systemsand f(x) is an analytical function.In a mechanical system m represents a point mass and f(x) force which can be expressedas negative gradient of the potential energy φ(x). Generally, the potential energyconsists of local minima and maxima and can be depicted as in Fig. 6.1. The force appearingwhen a system is moved the distance δx P away from an arbitrary but fixed pointcan be found by a Taylor expansion(∂ x φ) xP +δx P= (∂ x φ) xP+ (∂ xx φ) xPδx P+ ... (6.2)Discarding the terms higher than linear and expanding around the points where ∂ x φ = 0(any of the three marked points in Fig. 6.1), we have(∂ x φ) xP +δx P≈ (∂ xx φ) xPδx P, (6.3)which gives the equation of motion of a point mass in terms of small displacement δx P:The solutions of the above equation can be expressed aswhere ω Pis the frequency of oscillations given bym∂ tt δx P= −(∂ xx φ) xPx P. (6.4)δx P(t) = ±δx P 0exp(−iω Pt), (6.5)ω 2 P = 1 m (∂ xxφ) xP. (6.6)

6.1. GENERAL CONCEPTS 49FF>F cuuF cF(a)(b)Figure 6.2: (a) Buckling of a solid rectangular beam upon exerting a vertical force. Thedisplacement u increases with F . (b) ”Pitchfork” bifurcation diagram: maximal displacementu(F ) as function of the force F . After critical F c , two stable states appear. Theformer stable (undistorted) state exists but has become unstable (dashed line).From Eq. (6.6) we notice that the frequency ω Pdepends on the curvature of thepotential energy, given by the second derivative or generally the Laplacian. The point x Rin Fig. 6.1 is the local minimum, and so the curvature is positive resulting in real ω R.The point mass m experiences an oscillatory behavior, since the restoring force is directedtowards the stable equilibrium point x R. For the point x T, however, the curvature isnegative and ω Tis a complex number. In this case (or whenever the imaginary part of thefrequency is positive) Eq. (6.5) shows that the displacement δx Twill exponentially growin time. At the point T the system is in an unstable state of higher energy, the restoringforce now directed away from it.A system can undergo a change from a stable to an unstable state, passing throughthe point of marginal or neutral stability where the frequency is equal to zero i.e., thecurvature changes the sign. In our case this is valid for the point x S. In order to find theforce on the mass m at that point, we need to include the higher terms in the expansionof Eq. (6.2).The instability transitions can be visualized in parameter space by means of bifurcationdiagrams. For that we use another example of a solid beam on which a vertical force isapplied. After a certain critical force (Euler force), the beam will buckle and will deformas depicted in Fig. 6.2 (a).If we now plot the maximal displacement u(F ) as a function of the force F , then thequalitative graph looks like the one in Fig. 6.2 (b). After a critical force F c is applied twonew stationary states appear; the former equilibrium state, although still exists, becomesunstable. This outlines the principle of exchange of stabilities manifested when at theonset of instability stationary solutions prevail [49].

50 CHAPTER 6. HYDRODYNAMIC STABILITY ANALYSIS: BASICSDissipative systemsUnlike the conservative systems, dissipative systems, such as viscous fluids, exchange energy,volumes, particles, etc. with their surroundings, resulting in fluctuations of thesevariables. The concept of stable and unstable equilibria is still valid, [37], with mass particlesnow being subjected to a dissipative retarding force; oscillations are damped, andthe system, driven by fluctuations, cannot infinitely stay in an unstable state of highestenergy.In order for a flow to be stable, it has to be stable with respect to every possibledisturbance to which it can be subject; contrary, it will be unstable if only one particulardisturbance grows in time. The conditions under which instability starts are usuallyexpressed in terms of some critical parameters governing the flow.Of a special interest are marginal stability states marking the onset of instability andthus dividing stable from unstable regimes. Unlike the conservative systems, dissipativesystems have two different types of the marginal states, depending on the way the flowperturbations grow/decay, [49]. In stationary marginal states perturbations grow/decayaperiodically and the principle of the exchange of stabilities is applicable - the instabilityis manifested as a secondary flow e.g. Bénard cells appearing in the case of thermalinstability. In oscillatory marginal states perturbations oscillate with growing/decayingamplitudes. An oscillatory growth transition is termed overstability. A tangential electricfield imposed on the boundary between two viscous, conducting liquids can cause theoscillatory marginal states.6.2 Standard procedure in linear stability analysisWe will now summarize the usual procedure in hydrodynamic linear stability analysis interms of four basic steps.1. The equations are set for the initial, stationary (unperturbed) flow resulting in thevelocity field v 0 . The hydrodynamic equations include the continuity equation, themomentum and energy equations. In case of electrohydrodynamics the Maxwellequations for electroquasistatic fields need to be included as well. The equations aresimplified if incompressible liquids are considered.2. A small (infinitesimal) perturbation v ′ is superposed to the main velocity v 0 and theequations governing the perturbation are obtained. The equations and the boundaryconditions are then linearized neglecting the terms with quadratic and higher termsin v ′ 1 .3. The small disturbance v ′ is expanded into a complete set of normal modes characteristicfor the system with given boundary conditions. For example, if a system isconfined between two infinitely long horizontal planes with normals in z direction,and the flow parameters (density, viscosity etc.) depend only on the z coordinate,1 The general case of finite disturbances would require inclusion of non-linear (higher order) terms.

6.2. STANDARD PROCEDURE IN LINEAR STABILITY ANALYSIS 51the expansion in term of two-dimensional normal modes is∫v ′ (r, t) = dk v k (z) exp(−iω kt) exp[i(k·r)]. (6.7)where we have separated the space and the time dependence of normal modes.4. Finally, a normal modev ′ k (r, t) = v k(z) exp(−iω kt) exp[i(k·r)], (6.8)where k serves as a parameter, is put into the system of linearized equations. Bychoosing the solution in the form of Eq. (6.8), the linear PDE system of equationsand boundary conditions describing v ′ becomes a simpler linear ODE system 2 . Theproblem is then transformed to an eigenvalue problem for ω kwhich is generally acomplex number.We have mentioned that in unstable systems the disturbances exponentially grow intime, whereas in stable systems they decay. Different stability regimes impose differentconditions on ω k. These can be found by considering the time dependence exp(−iωt)given by Eq. (6.8). If ω k= Re ω k+ i Im ω kis inserted into Eq. (6.8), the stability regimescan be distinguished as follows:⎧⎪⎨ > 0 → instability,Im ω k = 0 → marginal (neutral) stability for k⎪ 1c ,⎩< 0 → stability.(6.9)The critical wave number k c marks the border between the stable and unstable states, seeFig. 6.2 (b). We can also make the distinction between the oscillatory and the stationarymarginal states. Let Re ω k ≠ 0. Then in accordance with Eq. (6.9)⎧⎪⎨ > 0 → overstability,Re ω k ≠ 0 ∧ Im ω k = 0 → stability or marginal overstability for k⎪ 2c ,⎩< 0 → stability.(6.10)An oscillatory marginal state exists when the real part of the frequency ω k, which determinesoscillatory behavior, is different from zero when the imaginary part is zero. IfIm ω k= 0 for all k-s, we are in a stable regime; when Im ω k> 0 at some value k 2c , thenoverstability sets in. On the other hand, if Im ω k= 0 implies Re ω k= 0 for every k, themarginal state is stationary and the principle of exchange of stabilities is valid [49].Our main goal in the next chapters is to determine the characteristic frequency ω kaswell as the critical wave number k c manifested at the onset of instability. In electrohydrodynamics,the k c will depend on the applied voltage.2 The formal analysis is presented in Chap. 7.

52 CHAPTER 6. HYDRODYNAMIC STABILITY ANALYSIS: BASICSwall12zxasmall perturbationinterfacebFigure 6.3: Two liquids of different densities confined between two infinite planes. Thedistances from the interface are a and b for the liquids 1 and 2, respectively. The verticaldisplacement of the interface during perturbations is ζ(x, t).6.3 Example: The Rayleigh-Taylor instabilityTo illustrate some of the outlined ideas we will now give a simple example of gravitationalinstability. A note to the reader is that the boundary conditions on interface and wavenumbers used here will be borrowed from the chapter ahead, where the general treatmentis made.Consider two ideal incompressible liquids on top of each other and confined betweentwo infinitely long, horizontal planes, Fig. 6.3. The densities of the upper and lower liquidsare ρ (1) and ρ (2) , respectively, and the surface tension is σ. The liquids are assumed tobe motionless in the beginning. We want to find the characteristic frequencies when theinterface is slightly perturbed i.e. z = ζ(x, t).The ideal liquids which are vorticity-free in the beginning, are vorticity-free in subsequenttimes by the conservation of circulation i.e. Kelvin’s theorem, [14]. The flow istherefore potential and the velocity in each of the liquids given byv (i) = ∇φ (i) , (6.11)where i = 1, 2 and φ = φ(t). Inserting the above velocity into the Euler’s equationfor incompressible flow, Eq. (2.8) with zero viscosity, we get generalization of Bernoulli’stheorem for time dependent flowρ (i) ∂ t φ (i) + 1 2 ρ(i) v (i)2 + ρ (i) gz + p(i) = f(t). (6.12)Since the potential φ(t) is not uniquely determined, it can be redefined by a time transformationφ(t) = φ(t) + tf(t)/ρ without loss of generality 3 . We can thus eliminate f(t)from Eq. (6.12). If the velocities are small the quadratic term can also be neglected andwe get for the pressures anywhere in the liquidsp (i) = −ρ (i) gz − ρ (i) ∂ t φ (i) . (6.13)3 Adding an arbitrary function of time to φ in Eq. (6.11) does not change the velocity.

6.3. EXAMPLE: THE RAYLEIGH-TAYLOR INSTABILITY 53At the rigid boundaries a and b, the vertical velocities w (1) and w (2) are zero; fromEq. (6.11) we then findφ (1) = A cosh k(z − a) exp[i(kx − wt)],φ (2) = B cosh k(z + b) exp[i(kx − wt)],(6.14)for the waves propagating in the x-direction.The conditions at the interface z = ζ, Eqs. (7.13a) and (7.13b), and Eq. (7.13e),become‖w(0)‖=‖∂ z φ(0)‖ = 0,∂ z φ (1) (0) = ∂ z φ (2) (0) = ∂ t ζ,‖−p‖ = σ∇ 2 sζ,(6.15a)(6.15b)(6.15c)where we evaluated the derivatives at z = 0 since the oscillations are small. Inserting thepressure from Eq. (6.13) into Eq. (6.15c) we get‖ρgζ + ρ ∂ t φ(ζ)‖= σ∇ 2 sζ, (6.16)which after differentiation with respect to time and using Eq. (6.15b), gives the conditionon the potential φ‖ρg∂ z φ(0) + ρ∂ tt φ(0)‖= σ∂ z[∂xx φ(0) + ∂ yy φ(0) ] . (6.17)Inserting the solutions from Eq. (6.14) into the conditions Eqs. (6.15a) and (6.17)we arrive at two algebraic equations for two unknowns A and B. The frequency ω kisdetermined when the determinant of the system vanishes:ω 2 k = (ρ (2) − ρ (1) )gk + σk 3ρ (1) coth(ka) + ρ (2) coth(kb) . (6.18)If ρ (2) > ρ (1) , ω is always positive and the oscillations are stable, Eq. (6.10). However,if ρ (2) < ρ (1) , the first term in Eq. (6.18) changes the sign and the system becomes unstablefor a range of k-s. From the condition ω 2 = 0 we find for the critical wave number andwavelength,√(ρk c =(1) − ρ (2) )g, (6.19)σλ c = 2πk c= 2π√σ(ρ (1) − ρ (2) )g . (6.20)For perturbations with k < k c , or λ > λ c , Imω k < 0 and the interface is unstable. From theenergy point of view, the decrease in the gravitational potential energy is larger than the

54 CHAPTER 6. HYDRODYNAMIC STABILITY ANALYSIS: BASICSz=012x=0 x=LFigure 6.4: The onset of instability of the system with a heavier liquid on top of a lighterone. In a container of length L, the smallest non-zero wave number consistent withthe boundary conditions is k L= π/L corresponding to λ L = 2L. Instability starts whenλ L = λ c , and is present when λ L > λ c i.e. when L exceeds the critical length L c , Eq. (6.21).increase in the surface energy, and so the system goes into the more stable configuration.The elegant energy argument leading to the same Eqs. (6.19) and (6.20) is outlined in [37].In finite-size systems not all of the wavelengths can be realized. The boundary conditionson possible wave numbers imposed in a rectangular channel are given by Eq. (8.27).We imagine now that the vertical boundaries are far apart, ka, kb ≫ 1 in Eq. (6.18), andthat the liquids are confined between walls at x = 0 and x = L, as in Fig. 6.4.The wavelength associated with the smallest allowable wave number k L= π/L isλ L= 2L. The instability will happen if this wavelength becomes greater than the criticalλ c i.e. if the length L of the system exceeds the smallest length enabling the onset ofinstability√σL > L c = π(ρ (1) − ρ (2) )g . (6.21)For a water-oil system L c = 15 mm, and for water-air L c = 8.9 mm. In microfluidicchannels typical widths are 100 µm while lengths can vary. For the systems shorter thanL c the Rayleigh-Taylor instability will not appear.6.3.1 A note on viscosityThe stability analysis involving two superposed viscous fluids is substantially more involved.Here we only mention an important conclusion of the analysis made in Ref. [49]which says that the critical wavelength λ c , Eq. (6.20), at which the gravitational instabilitysets in, is not affected by the presence of viscosity. In other words, ”the wave numbersstabilized by the surface tension are independent of viscosity”, [49].In microfluidic regime gravity effects can be neglected since the surface tension is thedominant force. From Eq. (6.18) we see that the onset of instability in such case startsat k = 0 4 , with the other wave numbers being stabilized by the surface tension. It wasjust mentioned that these wave numbers are not affected by viscosity. In this limit also4 It will be more evident in Chap. 8.

6.3. EXAMPLE: THE RAYLEIGH-TAYLOR INSTABILITY 55the beginning of instability is independent of viscosity. Even though the viscous stressesare important in microfluidics, Re ≪ 1, as far as the onset of instability is concerned, theinviscid approximation will suffice. For any realistic and thorough analysis, however, theviscosity must be included.

56 CHAPTER 6. HYDRODYNAMIC STABILITY ANALYSIS: BASICS

Chapter 7Interfaces inelectrohydrodynamics: theoryIn this chapter we will apply the theory and concepts of the previous two chapters to theproblem of hydrodynamic stability of the interface between two dielectric liquids. Theresults of the electrohydrodynamic (EHD) instability analysis will later be applied to twoimmiscible liquids placed in an E-field in a microfluidic devices. As already mentioned theinfluence of the electric fields is very important in microsystems since very large electricfields can be achieved with applying small voltages. To give an idea, a voltage of 10 Vapplied across the typical 100 µm will give 10 5 V/m. As far as the stability is concerned,the different situations will occur depending whether the E-field is applied perpendicularor tangential to the interface.The problem of electrohydrodynamic (EHD) instability is not new. Many studieshave been conducted over the years. The pioneering theoretical work started with LordRayleigh in 1882, [51], but the foundations of the EHD could be credited to Taylor andMelcher, namely for establishing the leaky dielectric model and extensively developing thetheory, [52]. Many later contributions are listed in Saville [46]. With the development ofmicrofluidic systems through 1990s, the EHD aspects are reappearing in the light of newapplications.7.1 Two immiscible liquids between infinite planesIn Chap. 4 we introduced a pump where a conducting liquid driven by EOF is used topump a non-conducting liquid by viscous drag.Although the actual design and characteristics of the pump have been investigated,no stability analysis has been performed yet. The full stability analysis of the system,however, is a rather involved procedure: the liquids differ in densities, viscosities, dielectricconstants, and conductivities; they are moving and are subject to both normal andtangential E-fields in a finite-size system. In addition, the electrokinetic effects generallyplay the role since the pump is driven by EOF. Actually, the pump represents a jet of oil(i.e. the two interfaces) entering the surrounding water, Fig. 7.1 (a).57

58 CHAPTER 7. INTERFACES IN ELECTROHYDRODYNAMICS: THEORYFigure 7.1: Two-liquid instabilities in electrohydrodynamics (a) Schematic view of the jetinstability present in the two-liquid viscous pump. (b) Two moving or still liquids placedin an E-field between infinitely long planes spaced by L 0 = a + b.In order to get the insight how various physical parameters affect the stability of theinterface, we will restrict our initial analysis. This will give us a good starting point for alater, more realistic one.As sketched in Fig. 7.1 (b) we will consider two immiscible, non-moving liquids oneabove the other, which are confined between two horizontal planes. Liquids are assumedincompressible and homogeneous perfect dielectrics separated by a plane interface. Theeffects of gravity, surface tension, the electrical and viscous forces will then successively beadded. Each of these forces will eventually contribute a term in the characteristic equationfor the normal mode frequencies. In this chapter we want to find general solutions andboundary conditions resulting when the interface is slightly perturbed.The following notation will be in place: superscripts (1) and (2) are used for thetwo regions, while subscripts are used for vector components. The exceptions are thesuperscript ′ indicating a perturbed variable, and the subscript 0 relating exclusively tothe unperturbed variables.Our analysis will distinguish two cases: the case in which the imposed E-field is normalto the (still) interface, and the case in which the E-field is tangential. To shorten thenotation, the variables pertaining to the two cases will be presented in the curly bracketsas { } N(7.1)Twhere the upper values correspond to the case of normal E-field and the lower values tothe case of tangential E-field.Finally, in this chapter the symbol TikM will be used for the Maxwell stress tensorcontaining only the electrical terms. We will lump together the electrostriction term intothe zero-field pressure p which will be pulled out of Eq. (5.25) for the sake of transparency.

7.1. TWO IMMISCIBLE LIQUIDS BETWEEN INFINITE PLANES 597.1.1 The governing equationsFor each of the liquids in Fig. 7.1 (b), we summarize the equations of motion and of theelectric fields(i) Du(i)ρ= −∇p (i) + µ (i) ∇ 2 u (i) + ρ (i) g + ∇·T M (i) , (7.2)Dt∇·u (i) = 0, (7.3)∇·(ɛ (i) E (i) ) = 0, (7.4)∇×E (i) = 0, (7.5)where u (i) (x, y, z) = (u (i) , v (i) , w (i) ). In homogeneous perfect dielectrics the electric forceper unit volume (inside the bulk) in Eq. (7.2) is zero. This can be seen from Eq. (5.27)as the terms including ∇ɛ and free charge ρ e both vanish. So the coupling with theelectric fields happens at the interface, and for that we will need to specify the boundaryconditions. The only remaining body force in Eq. (7.2) is gravity.7.1.2 General boundary conditions: rigid boundariesOn the rigid horizontal boundary planes at z = a and z = −b the no-slip condition isassumed for the velocitiesu (1) = 0 (z = a), (7.6a)u (2) = 0 (z = −b). (7.6b)The boundary conditions on E and D-fields are given by Eqs. (5.43) and (5.44), respectively.In case of a normal E-field, Fig. 7.2 (a), it is assumed that both rigid planesare perfectly conducting electrodes kept at a constant potential V , and in direct contactwith the liquids. The two electrodes are large enough so that the fringing fields close tothe edges are not important. For small microfluidic systems the electrodes could alwaysbe made in such a way as to ensure a uniform field. From Eq. (5.43),n×E (1) = 0 (z = a), (7.7a)n×E (2) = 0 (z = −b). (7.7b)In case of imposed tangential fields, Fig. 7.2 (b), the situation is different. We assumethat the liquids are placed in a uniform horizontal field produced from two large electrodesspaced far apart. This is particularly valid for microfluidic case where the distance betweenthe confining planes is very small. The plane boundaries are assumed insulating. Thus.the tangential component is continuous across all the boundaries and it will suffice toconsider the normal E-field componentn·E (1) = 0 (z = ∞), (7.8)n·E (2) = 0 (z = −∞). (7.9)

60 CHAPTER 7. INTERFACES IN ELECTROHYDRODYNAMICS: THEORYwallzxa1 12 212wallsmall perturbationE1interfaceVa-b1 12 212small perturbationE1E2interfacezx-bE2wallwallVFigure 7.2: Two-liquid interface perturbations (a) in a normal E-field (b) and in a tangentialE-field.7.1.3 General boundary conditions: the interfaceWe will now describe main features of deformable interface between two fluids, Fig. 7.1 (b).General expressions provided below will be linearized later since we are interested in smalldeformations.When unperturbed, the interface is described by the equation z = 0. Upon deformationit is given byz = ζ(x, y, t), (7.10)where ζ is the vertical displacement from the equilibrium position z = 0.To determine the velocity of a particle which moves with the surface we evaluate therate of change of ζ with respect to a fixed coordinate systemdzdt= w(ζ) =dζdt = (∂ t ζ + u∂ xζ + v∂ y ζ) z=ζ . (7.11)At fixed coordinates x and y the surface (and particles on it) move up and down withvelocity ∂ t ζ. An important variable characterizing a surface deformation is vector normalto the surface. From differential geometry the unit vector n normal to the surface definedin Eq. (7.10) isn =1√(∂x ζ) 2 + (∂ y ζ) 2 + 1 (−∂ xζ, −∂ y ζ, 1). (7.12)We can now formulate the boundary conditions for velocities, electric fields and stressesat the interface between two dielectric viscous fluids. We will recall the notation and

7.2. PERTURBATION EQUATIONS AND BOUNDARY CONDITIONS 61discussion at the end of Chap. 5. At z = ζ we have,velocity BCs : ‖u‖=‖v ‖=‖w ‖ = 0, (7.13a)w 1 = w 2 = dζdt ,(7.13b)electric fields BCs : n×‖E‖ = 0, (7.13c)n·‖D‖= n·‖ɛE‖ = 0,(7.13d)stress BCs :‖−p‖ n i + ‖τ ik + T Mik ‖ n k = σ∇ 2 sζ n i ,(7.13e)where ∇ 2 s is two-dimensional (surface) Laplacian expressing the surface curvature for smalldeformations.The interface is characterized by a surface tension introduced through Young-Laplaceequation. There is an abrupt change (jump) in liquid properties at z = ζ. Formally,the surface tension is represented by a delta-function and the above boundary conditionsobtained by integration of the bulk Eqs. (7.2) - (7.5) across the interface, Chandrasekhar,[49].Eq. (7.13a) expresses the continuity of the velocities while Eq. (7.13b) follows fromEq. (7.11). The boundary conditions on the electric fields are Eqs. (5.43) and (5.44) withthe surface charge density q s equal to zero (no free charge present). Eq. (7.13e) representsthe balance of the surface stresses. Pressure and surface tension contribute only to normalstresses while viscous and Maxwell stress tensors contribute also to tangential stresses.7.2 Perturbation equations and boundary conditionsThe procedure outlined in Sec. 6.2 will now be used to linearize equations of motion,Eqs. (7.2) - (7.5), and the boundary conditions Eqs. (7.6a) and (7.6b), Eqs. (7.7a) and (7.7b),and Eqs. (7.13a) - (7.13e).7.2.1 The stationary solution for still liquidsWe are considering the situation depicted in Fig. 7.2, i.e. two immiscible still liquids ontop of each other with an electric field present. The vector n normal to the unperturbedinterface is given byn 0 = (0, 0, 1), (7.14)while the stationary solutions in each region i = 1, 2 arep (i)0 = −ρ (i) gx + P (i)0 , (7.15)u (i)0 = 0, (7.16){ } ⎧ ⎫EN ⎨0 (0, 0, E (i)z0=) ⎬E T ⎩0 (E (i)x0 , 0, 0) ⎭ . (7.17)

62 CHAPTER 7. INTERFACES IN ELECTROHYDRODYNAMICS: THEORYE V 0 and EH 0 are the unperturbed vertical and horizontal fields shown in the two cases inFig. 7.2, respectively. The curly brackets will be used throughout the chapter to representthe quantities corresponding to the two cases: the upper value always representing aquantity in the case of imposed vertical field (i.e. normal to the unperturbed interface)and the lower value always representing a quantity in the case of imposed horizontal field(i.e. tangential to the unperturbed interface).As expected, the equilibrium pressure is the hydrostatic pressure resulting after theintegration of Eq. (7.2). In motionless liquids there are only normal contributions to thestress, coming from pressure and electric fields. We can use results for electric stressobtained for a plane interface in Chap. 5. The stress balance at the interface is‖P 0 ‖ n 0x = ‖P 0 ‖ n 0y = 0,(7.18a){ 12‖P 0 ‖ n 0z =z0 ‖},− 1 2 ‖ɛ‖E2 x0(7.18b)where we have used the fact that in the case of imposed horizontal field, E x0 is the samein the two dielectrics.7.2.2 Linearization of the perturbed equationsA slight disturbance of the interface corresponds to a small perturbation of the stationarysolutions. Expanding the perturbed fields u, p and φ we getu = u 0 + u ′ + ... ,p = p 0 + p ′ + ... ,φ = φ 0 + φ ′ + ... ,(7.19a)(7.19b)(7.19c)where we kept only linear terms in the increments u ′ , p ′ and φ ′ . Putting the perturbedvariables into the equations of motion, Eqs. (7.2) and (7.3), and subtracting the stationarysolutions we get the equations for the perturbations:ρ∂ t u ′ = −∇p ′ + µ∇ 2 u ′ , (7.20)∇·u ′ = 0, (7.21)∇ 2 φ ′ = 0, (7.22)where u ′ = (u ′ , v ′ , w ′ ). By applying ∇· to Eq. (7.20) and using Eq. (7.21) we see that p ′satisfies the Laplace equation∇ 2 p ′ = 0. (7.23)To get the equations for electric fields we apply ∂ x to Eq. (7.22),∇ 2 E ′ x = 0. (7.24)

7.2. PERTURBATION EQUATIONS AND BOUNDARY CONDITIONS 63The other components of E ′are determined from∇×E ′ = 0, (7.25)which is a consequence of Eq. (7.5).As can be noticed we have not considered the changes in the density or viscosity sincein each region they are assumed constant.7.2.3 Linearization of the perturbed boundary conditionsRigid boundariesAt the rigid boundaries the normal vector is n 0 from Eq. (7.14). When perturbed variablesfrom Eqs. (7.19a) - (7.19c) are put into the boundary conditions, we get for the velocityperturbationsw ′ = u ′ = v ′ = 0 (z = a, −b), (7.26)and for the E-field perturbations{ }n0 ×E ′ = 0 (z = a, −b), (7.27)n 0 · E ′ = 0 (z = ± ∞)where in the case of imposed tangential field we evaluate the perturbations at the infinity,in accordance with Eqs. (7.8) and (7.9).The perturbed interfaceLinearization of the interface BCs is a more complicated procedure. The perturbed valuesgiven by Eqs. (7.19a) - (7.19c) have to be evaluated, strictly speaking, at z = ζ. However,since ζ is assumed small, for the first order approximation it will suffice to consider theconditions at z = 0. For example, the approximationE ′ y(ζ) = E ′ y(0) + ∂ z E ′ y(0)ζ + ... ≈ E ′ y(0), (7.28)is valid if both E ′ y and ζ remain small. In addition, the normal vector is nown = n 0 + n ′ ,(7.29a)where n ′ is according to linearized Eq. (7.12)n ′ = (−∂ x ζ, −∂ y ζ, 0). (7.29b)With the above remarks in mind the linearized BCs at the interface, z = 0, come asfollows. The conditions on the velocities, Eqs. (7.13a) and (7.13b), become‖u ′ ‖=‖v ′ ‖=‖w ′ ‖ = 0,w 1 (0) = w 2 (0) = ∂ t ζ.(7.30a)(7.30b)The perturbed electric fields for the normal and tangential case in each of the regions are{ } {EN (E′x , E y, ′ E z0 + E ′ }z)=, (7.31)E T (E x0 + E x, ′ E y, ′ E z)′

64 CHAPTER 7. INTERFACES IN ELECTROHYDRODYNAMICS: THEORYwhich together with Eqs. (7.29a) and (7.29b) yields{ }‖E′x + E z0 ∂ x ζ ‖ = 0, (7.32a)‖E x ′ ‖ = 0{}‖ɛE z ′ ‖ = 0, (7.32b)‖−ɛE x0 ∂ x ζ + ɛE z ′ ‖ = 0instead of Eqs. (7.13c) and (7.13d), respectively.Linearized stressesThe perturbed stress condition is given by Eq. (7.13e). Expanding the pressure accordingto Eq. (7.19b), with p 0 defined in Eq. (7.15), we arrive at− ‖p ′ ‖ n i − ‖P 0 ‖ n i + ‖ρ‖ gζ n i + ‖τ ik + T Mik ‖ n k = σ∇ 2 sζ n i , (7.33)where, of course, n is given by Eq. (7.29a) and therefore has components in all threedirections. A useful thing is now to consider the stress components from the above equationseparately.We have mentioned earlier that the variables at the interface z = ζ will be approximatedwith the values at z = 0. However, in the above term ‖ρ‖ gζ we cannot immediatelyput ζ = 0 since the time derivative of the displacement ζ equals the vertical velocity w,Eq. (7.30b). In other words, to account for the gravity effects, the above equation wouldfirst need to be differentiated with respect to time, after which we can put ζ = 0. In termsof the components, both the gravity and the surface tension term have only z componentsup to the first order.We saw in Eq. (7.18b) that ‖P 0 ‖ n i has only the z component in the stationary case.It will however have the first-order components in x and y directions after the interface isperturbed{ 12‖P 0 ‖ n x =‖ɛE2 z0 ‖}(−∂− 1 x ζ),(7.34a)2 ‖ɛ‖E2 x0{ 12‖P 0 ‖ n y =‖ɛE2 z0 ‖}(−∂− 1 y ζ),(7.34b)2 ‖ɛ‖E2 x0{ 12‖P 0 ‖ n z =‖ɛE2 z0 ‖}. (7.34c)− 1 2 ‖ɛ‖E2 x0Comparing the above equations with the Eqs. (7.18a) and (7.18b) we see that after thestationary solution is subtracted from the perturbed pressure, the only contributions thatare left from the pressure P 0 will be in x and y direction! It is crucial to realize theimportance of the P 0 contribution, if the stress balance is to match consistently.Since the stationary velocity u 0 is zero, the perturbed velocity u from Eq. (7.19a) isjust u ′ . The perturbed viscous stresses reduce to the stresses for perturbationsτ ik = τ ik ′ = µ(∂ k u′ i + ∂ i u′ k). (7.35)

7.2. PERTURBATION EQUATIONS AND BOUNDARY CONDITIONS 65We now wish to linearize the components of the stress in the three directions. Up to thefirst order we haveτ xk n k = µ(∂ z u ′ + ∂ x w ′ ),τ yk n k = µ ( ∂ z v ′ + ∂ y w ′) ,τ zk n k = 2µ∂ z w ′ ,(7.36a)(7.36b)(7.36c)in x, y and z direction, respectively.In similar way the perturbed components of the Maxwell stress tensor areT Mik = − ɛ 2 E2 n i + E iD kn k . (7.37)In this case the stationary stresses have non-zero values, Eqs. (7.18b) and (7.18a). Afterthe linearization of Eq. (7.37) the perturbation components areT M ′xk n k =T M ′yk n k =T M ′zk n k ={ 12 ɛE2 z0 ∂ }xζ + ɛE 0 E x′− 1 2 ɛE2 x0 ∂ , (7.38a)xζ + ɛE 0 E z′{ 12 ɛE2 z0 ∂ yζ + ɛE 0 E y′ }12 ɛE2 x0 ∂ , (7.38b)yζ{ }ɛEz0 E z′ . (7.38c)−ɛE x0 E x′All of the above stresses are to be evaluated at z = 0.Grouping the above perturbation stress components together, we have in x, y and zdirection, respectively,{ }‖ɛE2z0 ‖(∂−‖ɛ‖Ex02 x ζ) + ‖µ ( { }∂ z u ′ + ∂ x w ′) ‖ɛEz0 E x ′ ‖‖ += 0,‖ɛE x0 E z ′ ‖(7.39a){ }‖ɛE2z0 ‖(∂ y ζ) + ‖µ ( { }∂ z v ′ + ∂ y w ′) ‖ɛEz0 E y ′ ‖‖ += 0,00(7.39b){ }‖ɛEz0 E− ‖p ′ ‖ + ‖2µ∂ z w ′ z ′ ‖‖ ++ (‖ρ‖ g − σ∇ 2 s)ζ = 0.‖−ɛE x0 E x ′ ‖(7.39c)

66 CHAPTER 7. INTERFACES IN ELECTROHYDRODYNAMICS: THEORY7.2.4 Expansion into normal modesWe will now expand the perturbations into the normal modes, assuming the two liquidsare confined between two infinite planes having the normal in z direction (see Chap. 6):ζ = ˆζ exp[i(k x x + k y y) − iωt],u ′ = û(z) exp[i(k x x + k y y) − iωt],p ′ = ˆp(z) exp[i(k x x + k y y) − iωt],E x ′ = Êx(z) exp[i(k x x + k y y) − iωt].(7.40a)(7.40b)(7.40c)(7.40d)The above solutions are put into the governing Eqs. (7.2) and (7.3)−iωρû = −ik x ˆp + µ(−k 2 + ∂z 2 )û,−iωρˆv = −ik y ˆp + µ(−k 2 + ∂z 2 )ˆv,−iωρŵ = −∂ z ˆp + µ(−k 2 + ∂z 2 )ŵ,ik x û + ik yˆv + ∂ z ŵ = 0,(7.41a)(7.41b)(7.41c)(7.41d)and into the stress boundary conditions Eqs. (7.39a) - (7.39c)where k ={ }{ }‖ɛE2z0 ‖‖ɛEz0 Ê x ‖(ik−‖ɛ‖Ex02 x ˆζ) + ‖µ (∂z û + ik x ŵ)‖ += 0,‖ɛE x0 Ê z ‖(7.41e){ }{ }‖ɛE2z0 ‖‖ɛEz0 Ê y ‖(ik y ˆζ) + ‖µ (∂zˆv + ik y ŵ)‖ += 0,00(7.41f){ }‖ɛEz0 Ê z ‖− ‖ ˆp‖ + ‖2µ∂ z ŵ ‖ ++ (‖ρ‖ g + σk 2 )ˆζ = 0,‖−ɛE x0 Ê x ‖(7.41g)√k 2 x + k 2 y. Note that the above equations do not have the time dependence.In addition, from Eqs. (7.40c) and (7.23) we get−k 2 ˆp + ∂ 2 z ˆp = 0. (7.42)Let us now obtain the form of solutions. Eqs. (7.40c) and (7.42) give for pressure ˆpwhich together with Eq. (7.41c) gives for ŵˆp(z) = exp(±kz), (7.43)ŵ = exp ±(kz) and exp ±(qz), (7.44)

7.2. PERTURBATION EQUATIONS AND BOUNDARY CONDITIONS 67where q =√k 2 − iω ν. In case of inviscid liquids the terms with q vanish. Finally,Eqs. (7.24) and (7.25) give for ÊÊ x = exp(±kz),Ê y = k yk xE x ,Ê z = 1ik x∂ z E x .(7.45a)(7.45b)(7.45c)Elimination of û, ˆv, ˆpWe will now simplify Eqs. (7.41a) - (7.41g) by eliminating dependence on û, ˆv and ˆp.From Eq. (7.41d) the conditions imposed on û and ˆv can be expressed by conditionsimposed on ∂ z ŵ, since k is a continuous variable. The no-slip conditions at rigid boundaries,Eqs. (??) and (??), and the continuity of velocities at the interface, Eq. (7.30a),now becomeŵ = 0 = ∂ z ŵ (z = a, −b), (7.46a)‖ŵ ‖ = 0 =‖∂ z ŵ ‖ (z = 0). (7.46b)Multiplying Eqs. (7.41c) and (7.41g) by k 2 and ∂ z , respectively, adding them and makinguse of Eq. (7.41d) we get for the stress in z direction‖[iωρ + µ(∂ 2 z − 3k 2 )]∂ z ŵ ‖ −k 2 [(‖ρ‖ g + σk 2 )ˆζ +{‖ɛEz0 Ê z ‖‖−ɛE x0 Ê x ‖}]= 0. (7.47)Finally, we multiply Eqs. (7.41e) and (7.41f) by ik x and ik y , respectively, and add themtogether. Using Eqs. (7.41d) and (7.22) we arrive at{ }‖−ɛE2− ‖µ(k 2 + ∂z 2 z0 k 2 ˆζ − ɛEz0 ∂ z Ê z ‖)ŵ ‖ +‖ɛEx0 2 k2 ˆζ= 0, (7.48)x + ik x ɛE x0 Ê z ‖which comprises the continuity of x and y stresses across the interface.7.2.5 Summary of solutions and linearized boundary conditionsFor the sake of easier reading we will now finally summarize relevant solutions and all ofthe linearized boundary conditions.Solutionsvelocities:ŵ 1 = A 1 exp(kz) + A 2 exp(−kz) + A 3 exp(qz) + A 4 exp(−qz),ŵ 2 = B 1 exp(kz) + B 2 exp(−kz) + B 3 exp(qz) + B 4 exp(−qz),(7.49a)(7.49b)

68 CHAPTER 7. INTERFACES IN ELECTROHYDRODYNAMICS: THEORYE fields:Ê x1 = C 1 exp(kz) + C 2 exp(−kz),Ê z1 =k [C 1 exp(kz) − C 2 exp(−kz)] ,ik xÊ x2 = D 1 exp(kz) + D 2 exp(−kz),Ê z2 =k [D 1 exp(kz) − D 2 exp(−kz)] .ik x(7.49c)(7.49d)(7.49e)(7.49f)BCs at rigid boundariesvelocities:ŵ 1 (a) = 0 = ∂ z ŵ 1 (a), (7.50a)ŵ 2 (−b) = 0 = ∂ z ŵ 2 (−b),(7.50b)E fields:}{Êx1 (a) = Êy1 (a) = 0,Ê z1 (∞) = 0(7.50c)}{Êx2 (−b) = Êy2 (−b) = 0.Ê z2 (−∞) = 0(7.50d)BCs at interface z=0velocities:‖ŵ(0)‖= 0 =‖∂ z ŵ(0)‖,(7.50e)surface displacement:E and D fields:ˆζ =iŵ1(0)ω , (7.50f){‖ik 2 ŵE 1 (0)}z0 ω+ ∂ z Ê z (0)‖= 0, (7.50g)‖Êx(0)‖= 0{‖ɛÊz(0)‖= 0‖k x ɛE x0ŵ 1 (0)ω+ ɛÊz(0)‖= 0}, (7.50h)(7.50i)

7.2. PERTURBATION EQUATIONS AND BOUNDARY CONDITIONS 69stresses:[{ }]‖[iωρ + µ(∂z 2 − 3k 2 )]∂ z ŵ(0)‖ −k 2 2 (0) ‖ɛEz0 Ê z (0)‖i(‖ρ‖ g + σk )ŵ1 += 0,ω ‖−ɛE x0 Ê x (0)‖⎧⎫⎨‖−iɛE − ‖µ(k 2 + ∂z 2 z0 2)ŵ(0)‖ +k2 ŵ 1 (0)ω− ɛE z0 ∂ z Ê z (0)‖ ⎬⎩‖iɛEx0 2 k2 ŵ 1 (0)x ω+ ik x ɛE x0 Ê z (0)‖⎭ = 0.(7.50j)(7.50k)

70 CHAPTER 7. INTERFACES IN ELECTROHYDRODYNAMICS: THEORY

Chapter 8Interfaces inelectrohydrodynamics:applications in microfluidicsHaving summarized the solutions and the boundary conditions we can now start solvingsome special cases. It is important to say that many aspects of the electrohydrodynamicinstability of two-inviscid-dielectric systems have already been discussed, [42]. The equationsEqs. (7.49a) - (7.50k) include the viscous effects and for zero viscosity recover theresults stated in [42].We will consider three types of regimes in our analysis: re-derived macro regime consideredin [42], as a reference point for comparison, with no micro length-scales involved;microfluidic-plane regime with only one micro dimension L 0 (the case of two infinitelylong planes spaced by L 0 ); and microfluidic-channel regime, with two or even three microdimensions involved - L 0 , L x and L y - being the height, length and width of a channel,respectively. The analysis will provide reasonable estimates with respect to the onset ofinstability in case of two-liquid viscous pump previously considered.All numerical calculations will include the oil-water system from Table 8.1. Notationwill follow Chap. 7.We will treat only the case which is stable when E = 0. This means that oil will bealways upper liquid 1 of thickness a, and water the lower liquid 2 with thickness b.8.1 Zero viscosity and perpendicular E-fieldIn the inviscid case the viscous stresses are neglected. The E-field is perpendicular tothe interface and therefore the upper values in the curly brackets in Chap. 7 need to be71

72 CHAPTER 8. INTERFACES IN EHD: APPLICATIONSρ ɛ r λ c σkg/m 3 S/m J/m 2corn oil 920 3.1 5 ×10 −11 −water, distilled 1000 80.1 2 ×10 −4 −copper − ∞ 5.8 ×10 7 −oil-water interface − − − 18 ×10 −3Table 8.1: Parameters for the oil-water system. ρ is mass density, ɛ r is the relativedielectric constant, λ c is the conductivity, and σ is the surface tension, [45].considered. From Eqs. (7.50a) - (7.50d) velocities and fields are of the formŵ (1) = A sinh k(z − a),ŵ (2) = B sinh k(z + b),(8.1a)(8.1b)Ê (1)x = C sinh k(z − a), (8.1c)Ê (2)x = D sinh k(z + b). (8.1d)The four constants A, B, C and D can be determined from the interface BCs, Eqs. (7.50e) -(7.50j), which reduce to‖ŵ(0)‖= 0,(8.2a)‖k 2 E z0ŵ (1) (0)n‖ɛÊz(0)‖= 0,‖−ρnDŵ(0)‖ −k 2 [+ DÊz(0)‖= 0, (8.2b)](‖ρ‖ g + σk 2 )ŵ(1) (0)+ ‖ɛEnz0 Ê z (0)‖(8.2c)= 0. (8.2d)8.1.1 Dispersion relation ω(k): general caseInserting the solutions Eqs. (8.1a) - (8.1d) into Eqs. (8.2a) - (8.2d) defines an eigenvalueproblem from which we can determine the eigenfrequency. From the condition that thedeterminant of the system vanishes we obtainω 2 = k 2 [(ρ (2) − ρ (1) )ρgk + σkρ − 1 (ɛ (2) − ɛ (1) ) 2ρ ɛ (1) tanh(kb) + ɛ (2) tanh(ka) E(1)z0 E(2) z0], (8.3)where ρ = ρ (1) coth(ka) + ρ (2) coth(kb). As a reminder, the electrostriction effect does notappear in the above equation since we included it into the pressure term, see Sec. 5.2.

ZERO VISCOSITY AND PERPENDICULAR E-FIELD 73Eq. (8.3) can be cast into the non-dimensional form by introducing the following setof variables marked with a primea ′ = a L 0,b ′= bL 0,k ′ = kL 0 ,ρ ′ (i)E ′ (i)ω ′= ρ(i)ρ 0,= E(i) z0 L 0V,0√= ωρ 0 L 3 0σ , (8.4)which lead toIn Eq. (8.5) we haveω ′2 = k′ρ ′ (k ′2 − ε ′ (k ′ )W elk ′ + Bo ) = k′ρ ′ Y (k′ ). (8.5)ρ ′ = ρ ′ (1) coth(k ′ a ′ ) + ρ ′ (2) coth(k ′ b ′ ),(8.6a)ε ′ (k ′ ) =(ɛ (2)r − ɛ (1)r ) 2 E ′ (1) E ′ (2)ɛ (1)r tanh(k ′ b ′ ) + ɛ (2)r tanh(k ′ a ′ ), (8.6b)with the non-dimensional (measures of) electric fieldsE ′ (1) = −E ′ (2) = −ɛ (2)rɛ (1)r b ′ + ɛ (2)r a ′ , (8.7a)ɛ (1)rɛ (1)r b ′ + ɛ (2)r a ′ . (8.7b)The two characteristic non-dimensional numbers which appear areWe el = ɛ 0 E2 0 L 0σ= ɛ 0 V 02,σL 0(8.8a)Bo = (ρ(2) − ρ (1) )gL 2 0. (8.8b)σThe ”electrical” Weber number 1 We el and the BOnd number Bo represent the ratio betweenthe corresponding energy density, electric and gravitational respectively, and thesurface energy density. For a typical voltage V 0 = 1 kV and a length L 0 = 100 µm, wefind for the oil-water system We el ∼ 5 and Bo ∼ 4 × 10 −4 . The effect of gravity canbe completely ignored in microfluidics. For the (capillary) length L 0 = 4.8 mm the Bondnumber Bo = 1.It is worth to highlight four main aspects of Eq. (8.3).1 The electrical Weber number is obtained from the ordinary Weber number by substituting the kineticenergy density 1 2 ρv2 with the electric energy density 1 2 ɛ 0 ɛ rE 2 . The dependence on ɛ r we have included inEqs. (8.7a) and (8.7b).

74 CHAPTER 8. INTERFACES IN EHD: APPLICATIONS1. The dimension of each of the terms in brackets is [(l/t) 2 ]. Each of the factorsrepresents the square of the corresponding phase velocity for the gravity waves,Vg 2 , the surface-tension or capillary waves, Vc 2 , and the electrohydrodynamic (EHD)waves, Ve 2 , respectively. If the field is zero we have only the gravity and the capillarywaves, the situation described in Chap. 6. A non-zero normal E-field producesstresses which result in the third term in Eq. (8.3). There will be a contributiononly if two fluids with different dielectric constants are present.2. For a given wave vector k the E-field can be made large enough so that its phasevelocity contribution, Ve 2 , prevails over those of gravity and capillary waves. Conversely,for a given field strength (before the instability sets in), there is always ak value for which either gravity or capillary waves prevail. The effects of electricfields on long or short waves, therefore, cannot be considered without the inclusionof gravity or capillary forces in one of the two important limits, k → 0 and k → ∞.The phase velocities for Bo = 39 are shown in Fig. 8.1 (a). The increase in voltageincreases Ve 2 ; at V = 5.83 kV, Ve 2 is just slightly smaller than (Vg 2 + Vc 2 ), meaningthat the frequency ω is just slightly positive, see Fig. 8.1 (c). For Bo = 4 × 10 −5 ,both Vg 2 and Ve2 are constant for the k-range shown in Fig. 8.1 (b). If the voltage isincreased in small steps from the value V = 0.1 V, there will come a point when thetwo phase velocities exactly cancel each other. This happens whenV 2 =≈ɛ (1)r b + ɛ (2)r a(ɛ (1)r − ɛ (2)r ) 2 ɛ (1)r ɛ (2) (ρ (2) − ρ (1) )g (8.9)r ɛ 0a3ɛ (1)r ɛ 0(ρ (2) − ρ (1) )g, (8.10)where the approximation holds if ɛ (2)r ≫ ɛ (1)r , and yields V = 0.17 V.3. Each of the three terms in the brackets, V 2g , V 2c and V 2e , have dependence on thewave vector k i.e. the waves are dispersive. We have just seen in Fig. 8.1 (b)that for ka, kb ≪ 1 (and small Bo values), the gravity and the EHD waves bothbecome dispersionless. But also, the EHD waves become dispersionless in the shortwavelengthlimit when ka, kb ≫ 1. This effect can be seen in Fig. 8.1 (a) or directlyfrom Eq. (8.3). An important conclusion is that in the limits when a phase velocitybecomes constant, the minimum of the total phase velocity does not depend onthat particular contribution. Furthermore, the wave vector k min for which the totalphase velocity is both minimal and zero also does not depend on the constant phasevelocitycontributions.4. The point when instability sets in is when the total phase velocity, and thereforeω, becomes zero. If the E-field is not present there can only be the gravitationalinstability if a heavier liquid is on top of a lighter one, Sec. 6.3. However, Eq. (8.3)shows that the normal field always destabilizes the interface by reducing the phasevelocity; this happens even in the absence of the ordinary gravitational instability.

ZERO VISCOSITY AND PERPENDICULAR E-FIELD 75v ms Ω 1s 0.10.080.060.040.021412108642v cv e7.07kV5.83kV4.69kV100 200 300 400 500k 1m a⩵ 10 2 mb⩵2⨯10 2 mBo⩵394.69kV5.83kV5.92kVv gac100 200 300 400 500k 1m v ms Ω 1s 0.010.0080.0060.0040.00232.521.510.5↙v cv gbv e 0.1V100 200 300 400 500k 1m a⩵ 10 5 mb⩵2⨯10 5 mBo⩵4⨯10 5 k minV⩵0.1Vd100 300 500k 1m Figure 8.1: Phase velocities and eigenfrequencies as function of wave number for the twolquidoil-water system with parameters given in Table 8.1 and layer thickness denoteda and b, respectively. (a) Phase velocities for large Bo = 39 corresponding to the limitka, kb ≫ 1. The velocity V e is dispersionless for large k. For V = 5.83 kV, Ve 2 < Vg 2 + Vc2just before the instability sets in, see panel (c). (b) Phase velocities for microfluidicBo = 4 × 10 −5 corresponding to the limit ka, kb ≪ 1. Both V e and V g are dispersionless.Instability starts for V = 0.17 V when V e = V g . (c) The frequency ω for Bo = 39. ForV = 5.92 V there is a range of k-values yielding instability. Note the region of negativegroup velocity dω/dk. (d) The frequency ω for Bo = 4 × 10 −5 . The instability sets in atk = 0, a characteristic of microfluidic systems.If the E-field is increased beyond the onset of instability, the frequency becomesnegative for a range of k values. In Fig. 8.1 (c) this happens for V = 5.92 V.Comparing Fig. 8.1 (c) with Fig. 8.1 (d) we notice that in the case of very small Bondnumber the position of the k min is shifted towards very long wavelengths (smallerk’s). This is one of the characteristics of the microfluidic regime - instability startsfrom the limit k = 0. This, of course, is to be expected since the surface tension,which is the dominant force, has negligible effect on the long wavelengths.8.1.2 The onset of instability: limiting cases for infinite planesWe will now quantify the conditions for the onset of instability by considering Eq. (8.5).To observe the change in the sign of the frequency ω ′ it is sufficient to consider the behavior

76 CHAPTER 8. INTERFACES IN EHD: APPLICATIONSof Y (k ′ )Y (k ′ ) = k ′2 − ε ′ (k ′ )We el k ′ + Bo. (8.11)If there is no E-field, We el = 0 and Y (k ′ ) becomes a parabola displaced vertically by Bo,Fig. 8.2 (a). Y (k ′ ) acquires the negative values only if Bo < 0. The instability is presentfor the k ′ values smaller than k ′ c = √ −Bo which is just the critical wave number for theRayleigh-Taylor instability, Eq. (6.19).The effect of the field is described by the EHD term −ε ′ (k ′ )We el k ′ which acquiresdifferent values in the two limits of short and long wavelengths. If a ′ /b ′ ∼ 1, we have⎧⎨ ɛ (1)rε corrε ′ (k ′ a) =′2 ∞ k ′ a ′ , k ′ b ′ ≫ 1,⎩ 1 r0 k ′ a ′ , k ′ b ′ ≪ 1.ɛ (1)kε corr′ a ′3(8.12)The factors ε corr∞ and ε corr0 are functions of ɛ (1)r /ɛ (2)r and b ′ /a ′ , and they are close to 1for ɛ (1)r /ɛ (2)r ≪ 1. When the thickness-ratio a ′ /b ′ differs significantly from 1 we need toevaluate Eq. (8.5) numerically.Using Eq. (8.12) we can easily determine the function Y (k ′ ) in the limits:{Y (k ′ k ′2 − α) =∞ k ′ + Bo k ′ a ′ , k ′ b ′ ≫ 1,k ′2 − α 0 + Bo k ′ a ′ , k ′ b ′ (8.13)≪ 1,where α = α(We el ) is independent of k ′ . The two parabolas will have the extremum(minimum) at the point⎧( )⎨ α∞2, Bo − α2 ∞T (k min)) ′ 4k ′ a ′ , k ′ b ′ ≫ 1,=(8.14)⎩(0, 0) k ′ a ′ , k ′ b ′ ≪ 1.The onset of instability happens when Y (k min ′ ) in Eq. (8.13) first becomes zero, i.e.α ∞ = 2 √ Bo k ′ a ′ , k ′ b ′ ≫ 1,α 0 = Bo k ′ a ′ , k ′ b ′ ≪ 1.(8.15)In terms of We el Eqs. (8.14) and (8.15) givek ′ min = √ Bo,k ′ min = 0,We el = 2√ Boε ′ ∞We el = Boε ′ 0k ′ a, k ′ b ≫ 1, (8.16)k ′ a, k ′ b ≪ 1, (8.17)where ε ′ 0 and ε′ ∞ are constants given in Eq. (8.12).In Fig. 8.2 (b), Y (k ′ ) is plotted for a large Bond number, Bo = 25, and different valuesof We el . Eq. (8.16) predicts k min ′ = 5 for We el = 0.97. The actual k min ′ = 4.535 is obtainednumerically, see Fig. 8.3 (a). Further increase in We el leads to negative Y (k ′ ) values.

ZERO VISCOSITY AND PERPENDICULAR E-FIELD 7730100yk’252015105y⩵k' 2 Boy⩵k' 2 Boyk’80604020Bo⩵25 We⩵.33.51.74.971.310-5k' c ⩵ Bo k' La0-20k' min k' L k' cbyk’50403020Bo⩵62.850.225.12 4 6 8 10k ′ΣV 0 ⩵5kVΣ⩵18mJm 2yk’10080604020Bo⩵4⨯10 44.54 10 15k ′V 0 ⩵10V3004001008.42.5 10Σ c0 2 4 6 8 10 12 14k ′0-20k' c ⩵k' Ld2.5 5 7.5 10 12.5 15 17.5 20k ′Figure 8.2: Dependence of Y (k ′ ) on We el and Bo. (a) Without E-field present, instabilityis possible when the heavier liquid is on top of the lighter one (negative Bo). The unstablewave numbers are k ′ < k c ′ = √ Bo (dashed region). If a finite-size system is characterizedby a kL ′ , the system will be stable as long as k′ L > k′ c (bold region). (b) For a large Bondnumber the onset of instability happens at k min ′ ∼ √ Bo. Further increase in the voltagecauses a range of unstable k ′ -s. For We el = 1.31, the kL ′ shown is within the unstableregion. (c) Shifting of k min ′ towards zero as the Bond number is lowered, Eq. (8.17). Tomake the effect more apparent, the surface tension has been varied (not the length scale) ata constant voltage. (d) The effect of the voltage increase in microfluidic regime. Instabilitystarts at longer wavelengths. In order to reach the size-dependent kL ′ , the applied voltagehas to be increased substantially .

78 CHAPTER 8. INTERFACES IN EHD: APPLICATIONSSmall Bond numbers correspond to the the large values of surface tension. If the Bondnumber is lowered the minimum is shifting towards smaller k ′ -values. As mentioned earlier,in terms of Y (k ′ ) this is shown in Fig. 8.2 (c) 2 . The larger surface tension stabilizes smallwavelengths pushing the instability to the longer ones. We can apply Eq. (8.17) whichshows that the instability begins at k min ′ = 0 when the Bond number exactly cancels theelectric term. This corresponds to the cancellation of the V e and V g phase velocities inthis limit as already discussed earlier. If the Bond number is lowered beyond that point,the electric term will prevail over Bo and there will be a range of k ′ -values for which Y (k ′ )is negative.So if we have a small Bond number to begin with, as in microfluidic systems, raising thevoltage from zero to the value giving ε ′ 0 We el, the instability is first induced at the infinitelylong wavelength; raising the voltage further, more and more of the longer wavelengthsbecome unstable - a situation completely analogous to the negative Bond number (heavierliquid on top of the lighter one) without E-field, as shown in Fig. 8.2 (d) and (a).We have seen in Chap. 6 that finite size microfluidic systems impose constraints onthe possible wavelengths. In order to generate the instability, the voltage will need tobe raised so that the unstable wave numbers fall in the range below the critical k c . Wewill return to this important point later where the finite-size effects in microfluidics areconsidered in detail.8.1.3 The onset of instability: general case for infinite planesWe will now numerically determine the value k min ′The wave number satisfies the requirementsfor which the instability first occurs.Y (k ′ min) = 0,dY (k ′ min )dk ′ = 0;(8.18a)d 2 Y (k ′ min )dk ′2 < 0. (8.18b)As an illustration consider two boundaries equally spaced from the interface, i.e. a ′ = b ′ =0.5, then Eq. (8.11) can be written in the formY (k ′ ) = k ′2 − αk ′ coth(0.5k ′ ) + Bo,(8.19a)where α = α 0 /2 and α = α ∞ in the respective limits. Applying the conditions Eqs. (8.18a)and (8.18b) on Eq. (8.19a) yieldsα =k ′2 min + Bok ′ min coth(0.5 k′ min ),sinh(k ′ min) ( k ′2 min − Bo) + k ′ min(k′2min + Bo) = 0,(8.19b)(8.19c)which in the limits of short and long wavelengths, k ′ a ′ ≫ 1 and k ′ a ′ ≪ 1, recover Eq. (8.15).2 To emphasize the effect the voltage is kept constant and the Bond number was lowered by increasingthe surface tension.

ZERO VISCOSITY AND PERPENDICULAR E-FIELD 791616141412121010k'min86k'⩵ Bok'min86ab⩵1.542ab⩵1a420.430.25b12 50 100 150Bo12 50 100 150BoFigure 8.3: Wave number k min ′ of marginal stability as function of Bo, Eq. (8.19a). Stableand unstable wave numbers cannot be directly deduced. (a) Equal layer thicknesses, a = b.For Bo < 12 the onset of instability starts at k ′ = 0 (b) Unequal layer thicknesses, a ≠ b.The case a = b shown as a dashed line. Due to the stronger field thin-oil configurationsare less stable, see the discussion in the text.The dependence of k min ′ on Bo obtained from Eqs. (8.19b) and (8.19c) is shown inFig. 8.3 (a). The solid curve approaches √ Bo (zero) for large (small) Bo as it should. Thenumerical analysis shows that at a particular finite value of the Bond number, Bo = 12,the Bond number exactly cancels the EHD term, and the onset of instability is shifted tok ′ = 0. To emphasize this point we write:for a b = 1 and Bo = 12 ⇒ −ε′ (k ′ )We el k ′ = Bo ⇒ Y (k ′ ) = k ′2 . (8.20)The graph in Fig. 8.3 (a) (solid line) represents the wave numbers of the states withmarginal stability, and yet it does not divide stable from unstable k ′ -values.It is important to note that the stable and unstable k ′ -values cannot be deduceddirectly from Fig. 8.3 (a). For example, for Bo > 12, the onset of instability happens ata k min ′ (a value on the solid line) while the values of k′ both above and below k min ′ arestable (analogous to the k ′ values left and right from k min ′ in Fig. 8.2 (b) for We el = 0.97).Unstable wave numbers exist after the voltage is raised further, and need to be found bysolving the condition Y (k ′ ) < 0, where Y (k ′ ) is given by Eq. (8.11). In case of Bo < 12,unstable wave numbers are present when the EHD term exceeds the Bond number, similarto Fig. 8.2 (b) for V = 300 or 400 V.The governing factor in the EHD term from Eq. (8.11) is the E-field inside the dielectricwith smaller ɛ r . In Fig. 8.3 (b), the dependence of k min ′ on Bo is shown for four differentlayer-thickness ratios. If the oil layer is thicker than the water layer, a/b = 1.5, the E-field within the oil layer is smaller compared to the case of equal thicknesses a/b = 1 3 .This yields a smaller EHD term - a smaller Bond number is required to cancel the EHDcontribution. If the oil layer is slightly thinner than the water layer, the Bond numberneeded to cancel now larger EHD term is larger. The graphs will be shifted left or right,respectively, from the case a/b = 1.3 Keeping the voltage constant.

80 CHAPTER 8. INTERFACES IN EHD: APPLICATIONSTo see (qualitatively) which of the two configurations (represented by the curves left orright from the a/b = 1 curve) is more stable, we now apply a voltage to bring the thickeroil(”left”) configuration to the onset of instability for e.g. Bo = 12. The same voltage willproduce a range of unstable wave numbers for the thinner-oil (”right”) configurations, sincein this case the EHD contribution in Eq. (8.11) prevails over the Bond number Bo = 12,making Y (k ′ ) < 0. The discussion will be illuminated further when the instability voltagesare found.8.1.4 Instability voltages for infinite planes, kL ≫ 1 (macrofluidic limit)The limits ka, kb ≫ 1 correspond to the cases where the wavelength is much shorter thanthe distances a and b between the interface and the boundaries. In other words, thereis a small interaction between the short-wavelength disturbances at the interface and theelectrodes. In this limit surface tension is the dominant force.Large values of the Bond number correspond to the small values of surface tension (orlarge distance between the electrodes). As the Bond number is increased we expect thatstability border is shifted further towards larger k ′ -s, i.e. smaller wavelengths, where thestabilizing effect of the surface tension is still significant (or where the more pronouncedgravity effect did not yet take over). This is revealed in Eq. (8.16).From Eq. (8.16) we further observe that the wavelength λ min = 2π/k min at which theinstability sets in is the same as in the ordinary gravitational instability i.e. given bythe Rayleigh-Taylor condition, Eq. (6.20). This is a consequence of the fact that Ve2 isdispersionless in this limit, as already mentioned earlier, see Fig. 8.1 (a). From Eq. (8.16)we also obtain the voltage at the onset of instability:[]4σ(ρ (2) − ρ (1) 14)V 0 = L 0ɛ 2 . (8.21)20 ε′ ∞Since microfluidics is characterized with extremely low Bond numbers this voltageis irrelevant for microfluidic regime. For the sake of comparison we evaluate the abovevoltage for the symmetric, oil-water system: for L 0 = 2a = 2 cm, V 0 = 5.76 kV.In [43], a set of the instability experiments was performed in a large apparatus in theregime of Bo ∼ 25. The dimensions were confined basically in one direction by electrodeswhose distance (∼ cm) could be adjusted to remove the boundaries from the interface.The instability in the short-wavelength limit could then be well tested; both λ min givenby Eq. (6.20) and the linear dependence on L 0 in Eq. (8.21) were confirmed.8.1.5 Instability voltages for infinite planes, kL ≪ 1, (microfluidic limit)We now turn to the limit relevant for microfluidics. When ka, kb ≪ 1, the wavelengths aremuch longer than the boundary distances and there will be a strong interaction betweenthe waves and the electrodes.We can investigate this limit by lowering the characteristic distance L 0 = a + b whichcorresponds to the regime of small Bond numbers. We have seen in Fig. 8.3 that for small

ZERO VISCOSITY AND PERPENDICULAR E-FIELD 81Bond numbers instability is shifted towards longer wavelengths (since destabilizing effectsof gravity become less and less pronounced).When a ′ = b ′ , Fig. 8.3 (a) shows that for Bo = 12 (L 0 = 1.66 cm), instability starts atthe infinitely long wavelength. For the oil-water system this happens when V 0 = 4.45 kV.We wish to find the condition on the voltage causing the onset of instability for any Bondnumber smaller than 12. For all these Bond numbers we are in the limit where the resultsof Eq. (8.17) hold, and so the voltage condition is obtained by equalizing the EHD termwith the given Bond number. The result is Eq. (8.9), or slightly rewrittenV 0 = L 3 20√(ρ (2) − ρ (1) )gɛ 0 ε ′ . (8.22)0For Bo = 4 × 10 −4 (L 0 ∼ 100 µm), Eq. (8.22) gives the voltage on the order of 2 V.Inducing the instability in microfluidic systems with a small voltage can be used toenhance the mixing of fluids. Although the above analysis is based on still, perfectlyinsulating (immiscible) liquids, the order of magnitude cannot be far off. An active micromachinedmixer applying the electrical fields has already been realized in literature,[53]. The electrodes were arranged so that a normal field could act on the surface of twoconducting, streaming fluids. The characteristic distance was L 0 = 200 µm. At a flow rateof 10µl/min, two liquids, one with a low and the other one with much higher conductivity,were fully mixed with a low voltage of 7 V (over a distance of 500 µm).The instability experiments in microfluidic regime have not yet been performed systematically.To illustrate how could the instability manifest itself for dielectric liquidswe quote Melcher, [42], who performed the experiments with rounded electrodes downto a = 0.4 cm: ”For close plate-interface spacings, the surface remains rounded until theinterface arcs over to the electrode.”8.1.6 Finite-size effect on instability in microfluidicsSo far we have been concerned with the instability between two infinitely long plates. Butmicrofluidic systems, or actually the channels, are always finite in size, usually long andnarrow, and behave more like resonators, Fig. 8.4. We have seen already, Sec. 6.3, thatthe finite size will impose restriction on the possible wavelengths and therefore on theinstability requirements.In some microfluidic systems, such as micro cell-sorters, liquid streams are parallel toeach other, and gravity does not play any role. The Bond number does not enter theequations. However, the shape of the interface between such immiscible streams (in aclosed microchannel) will be curved due to the strong capillary forces, Fig. 8.4 (b). Thesame reasoning goes if a gas bubble is stuck inside a channel in which an E-field is present.The meniscus will be especially pronounced in the smaller dimensions. The assumptionof a field normal to the interface in such case will therefore not be a valid one, to say atleast. Also, small parasitic capillary waves will develop on the surface of the meniscus.A more thorough analysis is needed to address these issues. For now, in order to getsome estimates, we will proceed as though the field is normal to the interface and the

82 CHAPTER 8. INTERFACES IN EHD: APPLICATIONS+V1+VL 02ZXYLXLYZXY- V(a)-VFigure 8.4: Two liquids in finite size microfluidic channels. The liquids will align themselves tominimize the surface free energy. (a) High aspect ratio microfluidic channel, ̷L 0 > L y . Capillaryeffects will be pronounced in the smaller dimension i.e. y direction. E-field produced by theelectrode configuration is not normal to the meniscus shown. The voltages inducing instabilityin y direction are substantially larger, since the allowed smaller wavelengths are stabilized bysurface tension. (b) Low aspect ratio microfluidic channel, ̷L 0 < L y . The two liquid configurationis similar to the two-infinite-planes case. However, the configuration shown is unstable due toincreased interface area, and the liquids will spontaneously position themselves next to each other.contact angle does not affect the unstable wavelengths. The latter approximation wouldbe of some validity at least for the waves propagating in x direction, Fig. 8.4 (a).Our analysis from Chap. 7 included the disturbances in both x and y-direction, butdid not impose any restrictions on the wave numbers. To establish the conditions on k xand k y we consider the normal modes of Eqs. (7.40b) and (7.40c) in a rectangular cavityof length L x , width L y and height L 0 .Consider the perturbation pressure p ′ in the top 4 layer filling the cavity. The pressuresatisfies the Laplace equation Eq. (7.23) with the solution given by Eq. (7.40c). Because ofthe inviscid approximation, the perpendicular velocities at each of the walls vanish; thatis, u ′ and v ′ vanish at the boundaries x = 0, L x and y = 0, L y , respectively, and w ′ vanishat z = a. By Eqs. (7.41a) - (7.41c) these conditions are equivalent toWe therefore choose the solution for p ′ in the form∂ x p ′ = 0 (x = 0, L x ), (8.23)∂ y p ′ = 0 (y = 0, L y ), (8.24)∂ z p ′ = 0 (z = a). (8.25)p ′ = cos x cos y cosh k(z − a). (8.26)Inserting the above solution into the Laplace equation Eq. (7.23) with the boundary conditionsEqs. (8.23) and (8.24) lead to the familiar relationsk x = ± nπL x, n = 0, 1, 2, 3...k y = ± mπL y, m = 0, 1, 2, 3...(8.27)4 The layer with thickness a in z-direction.

ZERO VISCOSITY AND PERPENDICULAR E-FIELD 83The magnitude of the total wave number k is given by[ ( ) n 2 ( ) ]m2k 2 = kx 2 + ky 2 = π 2 + , (8.28)L x L ywhere integers n, m differ from zero at the same time. For any given mode defined by nand m, the wave numbers k x and k y , and therefore k are fixed. The dispersion relationfor ω(k), Eq. (8.3), shows that for a given (n, m) mode ω 2 is linearly proportional to V 20 .8.1.7 Instability voltages in microfluidic channelsWe will identify two different instability voltages for L x ≫ L 0 > L y : V x0 and V y0 in the xand y directions, respectively.The instability will occur when the voltage is raised so that the size dependent k L fallsin the range of unstable k-values, see Fig. 8.2 (b). For a given k L from Eq. (8.28), theonset of instability starts whenY (k ′ L) = k ′2 L − ε′ (k ′ L)We el k ′ L + Bo = 0, (8.29)where kL ′ = k LL 0 . Inserting We el from Eq. (8.8a) into Eq. (8.29) we obtain for theimpending voltage√ (σLV 0 = 0 k′2L + Bo)ɛ 0 ε ′ (kL ′ ) . (8.30)k′ LTo evaluate Eq. (8.30) in the long- and short-wavelength limits, we consider disturbancesin a long and narrow rectangular microchannel with L x ≫ L 0 > L y , Fig. 8.4 (a).As before, it is filled with an oil (liquid 1) of thickness a in the z-direction, and the waterof thickness b.The k-mode (1, 0) represents the wave travelling down the channel in x-direction i.e.k = k x . From Eqs. (8.4) and (8.27) we get the value of k ′ for which the onset of instabilityis reachedk ′ L = π L xL 0 . (8.31)To catch the essential features, we will assume that a = b and ɛ (1)r ≪ ɛ (2)r . For k ′ a ′ , k ′ b ′ ≪1, ε ′ (kL ′ ) is found from Eq. (8.12) to beε ′ (k ′ L) = 1k ′ Lwhich together with Eq. (8.30) gives√V x0 = a 3 σπ 22ɛ (1)r ɛ 0 L 2 xɛ (1)rWhen L x → ∞, the above result goes into the familiar Eq. (8.9)., (8.32)a ′3+ (ρ(2) − ρ (1) )gɛ (1)r ɛ 0. (8.33)

84 CHAPTER 8. INTERFACES IN EHD: APPLICATIONS10080a14001200bL y ⩵10ΜmV0 xV604020L x ⩵1mm3mm9mmV0 yV100080060040020020Μm50Μm0 20 40 60 80 100aΜm20 40 60 80 100aΜmFigure 8.5: Instability voltages as function of the oil-thickness a. Instability is easier toinduce in a longer dimension. The full solution, Eq. (8.30), is plotted with dashed line.(a) The voltage V ∼ a 3/2 in the long-wavelength limit. (b) The voltage V ∼ a in the shortwavelength limit.The voltage dependence on the oil-thickness a is shown in Fig. 8.5 (a). For L x = 1 mmand L 0 = 2a = 100 µm, V x0 = 28.5 V. This is two orders of magnitude higher than thepure Bond number contribution of 0.17 V for the instability when L x → ∞ (Eq. (8.9) orsecond term in Eq. (8.33)). We now see if V x0 = 10 V is applied across L 0 = 2a = 100 µm,the instability of the (1, 0) mode will happen for L x > L c = 2.9 mm.To find V y0 we proceed as follows. The (0, 1) mode represents a disturbance in y-direction. If the channels are such that L y < a we are in the limit k ′ a ′ , k ′ b ′ ≫ 1 andε ′ = ɛ (1)r /a ′2 . We finally get√ σπV y0 = aɛ (1) , (8.34)r ɛ 0 L ywhere we have neglected the Bond number. For a choice, say L y = a = 50 µm, Eq. (8.34)gives V y0 = 321 V . Even smaller L y lengths will lead to higher voltages. This conclusion,as well as the linear dependence of V y0 on a is shown in Fig. 8.5 (b). As expected, thedisturbances in the smaller dimension are more stable (higher instability voltages needed)since the shorter wavelengths are stabilized by the surface tension.When ɛ (1)r ≪ ɛ (2)r we have seen that the voltage primarily depends on the thickness a ofthe ”weaker” dielectric 1 (the characteristic length in such systems is really a). Therefore,Eqs. (8.33) and (8.34) can be used to predict the voltages also in the asymmetric cases,a ≠ b. To confirm that, the full voltage solution from Eq. (8.30) is plotted next to theapproximate solutions in Fig. 8.5 (dashed line). If L 0 is held fixed, e.g. 100 µm, theimpending voltages for 0 < a < L 0 can be read directly from the figure 5 . The graphs fromFig. 8.5 complement Fig. 8.3 (b).One final remark about the voltages. Eq. (8.30) is an important result. In deriving it,we did not put any restrictions on the Bond number. Indeed, if kL ′ in Eq. (8.30) is replaced5 The results obviously do not hold at a = 0 or b = 0 since this corresponds to one dielectric beingpresent i.e. when the EHD term vanishes.

ZERO VISCOSITY AND PERPENDICULAR E-FIELD 850.1mmV0V140012001000800600400200L c ⩵1mm0 200 400 600 800 1000aΜmFigure 8.6: Instability voltage from Eq. (8.30) as function of the oil-thickness a. Thelength of the system (channel) is L c = 1mm. The general expression (dashed line) goesinto the limiting cases of Eqs. (8.33) and (8.34) for small and large a respectively.by one of the limiting values, 0 or √ Bo from Eqs. (8.16) and (8.17), we recover the voltagesfor the onset of instability for large and small Bo, Eqs. (8.21) and (8.22), respectively. FromEq. (8.30) therefore, we can find the impending voltages for any Bond number and anyfinite size system having an arbitrary liquid-thickness ratios. To emphasize this, a graphis shown for a larger range of thicknesses a in a separate Fig. 8.6. As mentioned earlier,Eq. (8.30) does not account for the capillary (meniscus) effects.8.1.8 Low aspect-ratio microchannels, L y ≫ L 0For many applications, microfluidic channels are of small height-to-width aspect ratios,Fig. 8.4 (b). When two streams of liquid go into such a channel they will adjust to flownext to each other rather than on top of each other. By energy argument, surface willposition itself to minimize the free energy; by instability argument, the smallest dimensionallows (relatively) the most stable disturbances.The question is whether the two liquids could at all be forced on top of each other,so that the meniscus effect is less pronounced. This could in fact be achieved in the twoliquidviscous pump described in Chap. 4. The conducting liquid introduced at the top(or bottom) is driven by EOF producing a sheet of width L y as sketched in Fig. 8.7. Bya connection to a reservoir with a non-conducting liquid, the non-conducting liquid couldthen be dragged by the boundary layer as explained before. Such system will be moreprone to the instabilities not only in the x but also in the y direction. If there is a normalcomponent of the E-field inside the non-conducting liquid, the voltage in Eq. (8.30) couldthen be used as an estimate for the impending instability in the x or y direction.However, in the configuration of Fig. 8.7 the more significant contribution will be fromthe tangential component of the E-field, 6 and this is what we investigate next.6 On the boundary between two conductors with conductivities σ 1 and σ 2, the current normal to theboundary is conserved i.e. σ 1E 1n = σ 2E 2n. Therefore, on the interface between a conductor and a perfect

86 CHAPTER 8. INTERFACES IN EHD: APPLICATIONS+V- VEOFL 0ZXYL XL YFigure 8.7: A modification of two viscous-liquid pump, Chap. 4, enabling two liquid layersto slide on top of each other in a low-aspect-ratio microfluidic channel. With L y ≫ L 0 ,the capillary effects at the boundaries in the y-direction are less pronounced, and theinstability-voltage estimate from Eq. (8.30) can be applied.8.2 Zero viscosity and parallel E-fieldThe influence of a parallel E-field on a two-dielectric system can be found by consideringthe lower values in the curly brackets of Eqs. (7.50a) - (7.50d). For a tangential E-field inthe x-direction, we arrive atω 2 = k 2 [ Vg 2 + Vc 2 + cos 2 θVe2], (8.35)where V g and V c are the gravitational and capillary phase velocity from Eq. (8.3), θ = k x /kis the angle between the direction of the wave propagation and the E-field, while Ve2 isnowVe2 = 1 (ɛ 2 − ɛ 1 ) 2Ex0 2 ρ ɛ 1 + ɛ . (8.36)2From Eq. (8.35) we see that the influence of a tangential E-field on the waves dependson the direction of the waves. The waves travelling perpendicular to the field, θ = 0,are not affected by it; the waves travelling in the direction of the field, θ = 1, will havethe phase velocity increased by Ve 2 . In addition, V e is dispersionless due to the fact thatboundaries where the field perturbations vanish are at infinity, see Eq. (7.27). A typicalwave front for a radial disturbance will be an ellipsoid elongated in the direction of theE-field. Such wave fronts were confirmed experimentally in Ref. [43].Because of the positive EHD term in Eq. (8.35) the frequency will not acquire anynegative values (within the linear approximation). Unlike a normal E-field, a tangentialfield acts to enhance the interface stability.With regard to the two-liquid viscous pump, the above analysis gives an idea that atleast in a (fictitious) ”stationary” pump-mode, instability of the waves travelling down themain channel will be suppressed. Instability of the waves propagating in the y-directionwill not be influenced by the tangential field. However, for these waves L c = L y ≪ L xand substantially larger normal fields are needed to induce the flow disruption.A valid criticism of the above analysis is that the conductivities of the liquids (particularlywater) were neglected. Such treatment is much more involved, especially in thecase of an applied tangential field, as it has to properly account for the charge movement.It is planned for the future work.dielectric, the normal E-field inside the conductor vanishes.

Chapter 9ConclusionThe following report serves to highlight a variety of relevant topics in the field of microfluidics,from a theoretical and a simulation perspective. I hope that my investigations, incollaboration with my colleagues, helped to resolve some of the multidisciplinary ”mysteries”behind lab-on-a-chip systems.The thesis consists of essentially two parts. Chapters 1-4 give a general introduction tomicrofluidic systems and a brief description of microfluidic applications further developedin the research papers (Appendices A,B,C,D,E). In the second half, chapters 4-8, a detaileddescription of electrohydrodynamic (EHD) effects is presented with the accent on theinstability between two immiscible dielectric liquids.Chapter 1 is an outline of the manuscript. Chapter 2 underlines the important physicsat work in microfluidic systems, and a section of it is adapted from my chapter publishedin a text-book (under references). Chapter 3 gives a thorough introduction to steady-stateelectroosmosis, an important phenomenon of driving liquids at the microscale. In chapter4, the three different micro applications further analyzed in appendices A, C+D and E,are briefly reviewed.Appendix A reports a study of the focusing effect of a chimney-like structure in amicrochannel geometry, a system used in micro cell-sorters. Experimental PIV measurementsare compared to the CFD simulations for a similar geometry, showing reasonableagreement. The small discrepancies are mainly due to numerical diffusion.In Appendix B, an EOF micropump, recently built by Takamura et al., has beenanalyzed by means of equivalent electric and hydraulic circuits, backed by CFD simulations.The results of the combined theory/simulation technique are confronted to theexperimental findings and suggest a need for stronger theoretical efforts in microfluidics.Appendix C investigates a quasi-statical motion of a bubble in microfluidic axisymmetriccontractions, based on the interplay between the surface energies and geometry.Capillary forces and partial-wetting concept were further used in a CFD simulation inAppendix D, where a passive bubble-trap is designed to ease the bubble removal frommicrochannels.In Appendix E a preliminary study is presented on performance of patented two-liquidviscous pump. The pump uses a conducting liquid driven by the electroosmotic flow,87

88 CHAPTER 9. CONCLUSIONto drag a nonconducting liquid. A pressure/flow-rate characteristics are calculated usingCFD and equivalent circuit theory.The second part of the thesis starts with a general theoretical description of electricfields in dielectrics and how they couple to hydrodynamics. Next, a general discussion ofhydrodynamic stability is presented.The equations are then applied to study EHD instability of a flat interface betweentwo perfect dielectric liquids in an electric field (perpendicular and parallel to it). Theliquids are described by Stokes’ equation with the electrical forces acting on the interface.The expansion of linearized perturbations into normal Fourier modes allows a descriptionin terms of dispersion relations, which easily discriminate the unstable modes. The maingeneralization from the previous work lies in the inclusion of viscous terms, finite-sizeeffects and the long-wavelength limit relevant for instabilities in microfluidics.It was found that gravity is completely irrelevant in microsystems and that finite-sizeof microchannels can significantly suppress a number of unstable modes. In addition,normal electric fields always act to destabilize, while tangential fields act to stabilize theinterface between two dielectrics (within linear stability approximation). Because of smalldimensions of microfluidic systems, relatively small applied voltages (∼ 10 V) can induceEHD instability and cause flow disruption.Finally, all of my research details and more, including the thesis in PDF format, canbe obtained from the group’s web-page: http ://www.mic.dtu.dk/research/MIF T S.9.1 OutlookThe work in the immediate future will be focused on several topics that were not treatedin the thesis. In brief, they are:(1) Viscous streaming liquids.The convective terms in Navier-Stokes equations will give first-order contributions in thelinear stability analysis of two streaming viscous liquids. For parallel flows, the analysisinvolves solving of the Orr-Sommerfeld equation[∂z 4 ψ − 2k 2 ∂z 2 ψ + k 4 ψ = ikRe (U − ω ]k )(∂2 z ψ − k 2 ψ) − ∂z 2 Uψ , (9.1)where ψ(x, z) is the stream function and U(z) the velocity of a primary flow. My attemptwill be to find the solutions when electric fields are included. The work in this directionapplied to the microfluidics will extend the existing literature, e.g. [54], and will give adefinite answer on the stability of the two-liquid viscous pump.(2) Finite conductivities.We notice from Table 8.1 that the liquid conductivities, although small in comparisonwith metals, are finite i.e. the liquids are leaky dielectrics. As stated before, this willhave an impact in balancing the shear stresses. Inclusion of the conductivities requiresa proper treatment of the charge relaxation. In the case of a perpendicular field appliedto a system where charges reside on the interface effectively all the time (short chargerelaxationtimes), the stress conditions Eqs. (7.50j) and (7.50k) can still be used if the

9.1. OUTLOOK 89condition on D-field is modified to include the surface charge density, as in Eq. (5.44).The full treatment, however, involving the Nernst-Planck equation for charged species,is needed in the case of tangential fields acting on the streaming fluids. Such study willanswer the problems of the so-called electrokinetic instability.(3) Electrowetting effects.The effects of a finite contact angle on the electric stresses need to be examined.(4) Electrostriction term.The electrostriction, Eq. (5.26), which has been put as a redefined pressure, plays a moreimportant role in case of tangential fields, [43]. In combination with the Clausius-Mossottidependence, Eq. (5.32), the contribution actually makes Ve2 negative. However, in theexperiments made by Melcher [43], large electrostriction causing negative phase velocityVe2 has not been observed. Whether such a conclusion is valid in microfluidics, remainsan open question.

90 CHAPTER 9. CONCLUSION

Appendix APaper published in J. Micromech.Microeng.TitlePIV measurements in a microfluidic 3D-sheathing structure with three-dimensional flowbehaviorAuthorsHenning Klank, Goran Goranović, Jörg P. Kutter, Henrik Gjelstrup, Jess Michelsen,Carsten WestergaardReferenceJ. Micromech. Microeng. 12, 862-869 (2002)91

INSTITUTE OF PHYSICS PUBLISHINGJOURNAL OF MICROMECHANICS AND MICROENGINEERINGJ. Micromech. Microeng. 12 (2002) 862–869 PII: S0960-1317(02)37632-0PIV measurements in a microfluidic3D-sheathing structure withthree-dimensional flow behaviourHKlank 1 ,GGoranović 1 ,JPKutter 1 ,HGjelstrup 2 ,JMichelsen 2and C H Westergaard 31 MIC-Mikroelektronik Centret, µTAS Project, Technical University of Denmark,2800 Lyngby, Denmark2 Department of Mechanical Engineering, Fluid Mechanics Section, Technical University ofDenmark, 2800 Lyngby, Denmark3 Dantec Dynamics A/S, Tonsbakken 16-18, 2740 Skovlunde, DenmarkE-mail: hk@mic.dtu.dk and carsten.westergaard@dantecdynamics.comReceived 31 May 2002, in final form 2 September 2002Published 3 October 2002Online at stacks.iop.org/JMM/12/862AbstractThe design and production time for complex microfluidic systems isconsiderable, often up to several months. It is therefore important to be ableto understand and predict the flow phenomena prior to design andfabrication of the microdevice in order to save costly fabrication resources.The structures are often of complex geometry and include stronglythree-dimensional flow behaviour, which poses a challenge for the microparticle image velocimetry (micro-PIV) technique. The flow in amicrofluidic 3D-sheathing structure has been measured throughout thevolume using micro-PIV. In addition, a stereoscopic principle was applied toobtain all three velocity components, showing the feasibility of obtaining fullvolume mapping (x, y,z,U,V,W)from micro-PIV measurements. Theresults are compared with computational fluid dynamics (CFD) simulations.(Some figures in this article are in colour only in the electronic version)1. IntroductionAlong with the requirements for more advanced liquidhandling capabilities, the layout of microstructures becomesmore and more complex. Computer modelling can accelerateand help inthe labour intensive and time consuming processof intuitive design, fabrication and testing. However, in orderto use simulations reliably as a design tool, the software usedfor modelling must initially be validated experimentally, e.g.with particle image velocimetry (PIV).Amicrostructure that allows coaxial sheathing of a samplestream with a buffer stream was chosen as a demonstratorand is shown in figures 1 and 2. This so-called chimneystructure is a crucial part of a micro cell sorter (Wolff et al2000). The purpose of the cell sorter is to select cells accordingto specific biochemical and physical properties, e.g. whetherthey are leukaemia cells, or according to their size. For thecells to be recognized and selected, they have to be markedwith fluorescent labels before they can be detected by thesystem. The detection system is typically organized like anepi-fluorescent microscope.In a conventional cell sorter the sample fluid containingthe labelled cells enters the sorter centrally in such a way thatabuffer stream surrounds the sample. This sheathing of thesample ensures that cells are only travelling within the verynarrow field of view of the detection system. Sheathing canonly work under laminar flow conditions, which prevents thepremature mixing of buffer and sample.The type of sample sheathing as described above isdifficult to achieve within micro cell sorters due to thetwo-dimensional, planar character of the fabricated devices.First-generation micro cell sorters therefore reverted to usingtwo-dimensional sample sheathing. Such sample sheathing,however, has the disadvantage of signal and speed variations.Since the sample is neither sheathed on topnor sheathed onthe bottom, cells can travel at any height within the fluidic0960-1317/02/060862+08$30.00 © 2002 IOP Publishing Ltd Printed in the UK 862

PIV measurements in a microfluidic structure(i)(b)1475 µm(c)2970 µm600 µmFigure 1. Overview of the chimney structure. The laminating bufferflowisentering the basin (b) from the left through the inlet (i), whilethe sample flow is entering the basin through the centre of thechimney (c). The total flow is leaving the basin over a step into theexit channel (e), where the lamination of the samples is completed.PIVmeasurements were performed in the outlined dashed areas.Stereoscopic PIV recombination was performed in an overlappingarea from two camera positions over the chimney entrance, asindicated by shading.channel of the micro cell sorter. Cells travelling outside thefocal plane of the detection system will generate a lowersignal than those travelling within the focal plane. Cellswill also travel at different speeds due to the parabolic flowprofile that develops in the channel. These speed differencesdiminish the accuracy of the cell sorter. To overcomethesetwo disadvantages, second-generation micro cell sorterswere developed that were able to sheath the sample threedimensionally, for example by Larsen (2000)andTashiroet al(2000). The chimney structure, which has been developedat MIC (Larsen 2000) andisdiscussed in this paper, usedonly one input for the buffer to achieve coaxial sheathing ofthe sample. The chimney structure was described in moredetail by Goranović et al (2001). It should be noted that theflow within amicrofluidic system is always laminar under allpractical conditions due to the extraordinary low Reynoldsnumbers. The above mentioned condition for a cell sorterthat the sheathing flow should always be laminar is thereforefulfilled in all microfluidic cell sorters.As shown in figures 1 and 2, the chimney structureconsists of an inlet channel, the chimney itself, a basin andan exit channel. The sheathing buffer flow enters the basinthrough the inlet channel, where it merges with the sample flowentering through the chimney. The sheathing of the sampleis gradually completed as the fluids exit through the outlet(e)600 µmchannel. The chimney basin was designed to take care ofthelaminatingpartofthe buffer stream underneath the samplestream. At the end of the basin, the reduction in cross sectionalarea leads to an increase in flow velocity, which is especiallyhigh just at the basin corner. This layer of high-velocity bufferstream ensures sheathing at the bottom of the sample stream.Sheathing at the topof the sample stream is ensured as longas the buffer stream is fast enough to avoid being pushedbackwards by the incoming sample stream.Particle image velocimetry (PIV) has been applied in anumber of microscopic devices with nozzle- or channel-likeshapes, for instance by Meinhart et al (1999). In micro-PIV,where microscopes are used for imaging, all three importantvariables, spatial resolution, the equivalent of the lightsheet thickness and field of view, are intertwined. Themicro-PIV technique therefore has to balance these variables.Especially, the chimney structure provides additionalchallenges, since the structure is characterized by a complexgeometry with a strongly three-dimensional flow.The flow is visualized in figure 2 by introducing a tracerinto the sample flow. The strongest out-of-plane movementis found near the outlet of the chimney and at the exit of thebasin.2. Experimental set-upThe chimney structure was fabricated in silicon by reactive ionetching. The top of the structure was closed with a glass cover,which was bonded directly to the silicon. The thickness of thecover glass was 0.75 mm. The fluid was purified water, whichfor the purpose of the PIV measurement was seeded with1.0 µm ‘orange fluorescent’ polystyrene particles(Fluospheres, Molecular Probes). The excitation maximumof these particles was at around 540 nm, while the emissionmaximum was at around 560 nm. Sample and buffer wereinfused with one syringe pump (Pump 22, Harvard Apparatus)each. The flow rate for the sample flow through the chimneywas kept constant at 0.02 ml min −1 , while the flow rateof the laminating flow was varied. Three flow cases wereinvestigated giving a volume flow ratio of R = 0.5, 5 and 25,where R is the ratio of the laminating flow over the sampleflow, also shown in table 1. The coordinates were placedwith their origin in the centre of the chimney and the z-axispointing out of the structure. The y-axis was positive in themain flow direction.ADantec HiSense camera with a 1280 by 1024 pixelarray was used in conjunction with a NewWave Solo Nd:YAGFigure 2. Outline of the chimney structure as modelled with the buffer fluid coming from the left. The shaded contours representconcentration of tracer fluid injected through the chimney illustrating how the buffer fluid laminates the sample fluid in the middle of the exitchannel. The buffer to sample volume flow ratio was R = 25.863

HKlanket alTable 1. Flow setting and key parameters for the cases investigated.Inlet Flow in Flow in exit Maximum Time betweenFlow flow chimney (total flow) velocity in exit light pulsesratio (ml min −1 ) (mlmin −1 ) (mlmin −1 ) (mms −1 ) (µs)R = 0.5 0.01 0.02 0.03 ≈24 250R = 5 0.1 0.02 0.12 ≈90 60R = 25 0.5 0.02 0.52 ≈400 12laser emitting at λ = 532 nm. The maximum energy usedwas 10 mJ. The light from the laser was delivered into themicroscope illumination path with an optical fibre fitted to themicroscope. The seeding was sufficiently efficient to applyan adaptive (multi-pass) PIV algorithm with 32 by 32 pixelinterrogation areas. Only average velocities are presented inthe paper. Ensemble correlation, as presented by Meinhartet al (1999), was also tested. Although the ensemblecorrelation is known to provide better results in a sparselyseeded flow, this technique did not provide additionalinformation in our case, apparently because the flow wasefficiently seeded.The imaging was done using an epi-fluorescentmicroscope (DMLB, Leica) with a 20× /0.40 planachromaticobjective and a 0.5× relay lens, resulting in a tenfoldmagnification and a field of view of 857 by 686 µm.Complementary measurements were also made with a 63 ×/0.7 semi-apochromat objective. Both objectives weredesigned to have a relatively long working distance of upto 3.2 mm and 2.6 mm, respectively, which allowed practicalroom for the clamping of the holder of the chimney structure.The comparatively thick cover glass of 0.75 mm could becompensated for with adjustable optics integrated in bothobjectives.Although oil-immersion objectives are beneficial to thelight budget and to the quality of the imaging, these were notapplicable in this case due to the fact that the working distancesof the oil-immersion objectives were limited to a maximum of0.22 mm.With a magnification of 10, the 32 by 32 pixelinterrogation area corresponded to 21 by 21 µm. This wasasatisfactory size compared to the width of the channel. Thefocal depth of the microscope objective defines the thicknessof the measurement plane. This was calculated to be 16 µmaccording to Meinhart (2000), which then gives a measurementvolume of 21 × 21 × 16 µm 3 .Although the 16 µm presentarelatively large fraction of the 60 µm channel depth, this waspart of a practical trade-off.The uncertainty of the measurement is estimated to be0.45 mm s −1 for the flow case R = 5. The value is justified byobserving the rms velocity of the U-velocity component of 50vector maps in an area of the flow, which can be expected to beundisturbed by actual flow phenomena, i.e. in the vicinity of thewalls in the exit channel, where disturbance fromthesheddingprocess is small. The value of 0.45 mm s −1 corresponds toan absolute uncertainty of 0.04 pixel displacement based on50 samples. For one single measurement realization, thiscorresponds to 0.28 pixel (0.04 × √ 50). This is slightlyhigher than ordinary PIV experiments, where a typical absoluteuncertainty between 0.1 and 0.2 pixel displacement on asingle measurement realization can be found. The uncertaintyTable 2. Volume flow in the exit channel from PIV data andnominal volume flow of the syringe pumps. The PIV based volumeflowwas derived by integration of the velocities over the area of theexit channel.Nominal flow rate Flow rate from PIV data(ml min −1 ) (ml min −1 )0.030 0.0280.12 0.120.52 0.48for the two other flow cases in table 1 is estimated tobe 0.11 mm s −1 and 2.3mm s −1 for the cases R = 0.5and R = 25, respectively. The absolute accuracy of themeasurement depends on a number of scaling factors, i.e.the optical magnification. A direct indication of the absoluteaccuracy is seen in table 2 by the fine agreement between theflow rates shown.The PIV measurements were done in two regions asoutlined in figure 1. In the exit channel, the measurementswere performed at five planes spaced 10 µm apartwith the857 by 686 µm fieldofview, as shown infigure3. Thefirst plane was estimated to be located approximately 10 µmbelow the glass surface, placing the volume that was mappedin the vertical centre of the channel. The 10 µm spacingprovides a vertical overlap of 40%, given the focal depth ofthe microscope of 16 µm. Hence, the vertical overlap wassimilar to the overlap used for the in-plane PIV processing.Using the velocities from each measurement plane, velocitieswere extracted in the x–z-plane, as illustrated in figure 3.The measurement area outlined in figure 1 in the centralregion around the chimney and upstream towards the stepinto the exit channel is composed of images originating fromseveral camera positions. From each camera position, the fieldof view was 857 by 686 µm, resulting in a combined area of857 by 1646 µm.3. Stereoscopic micro-PIV measurementsAtranslation stage was used to map a larger area of themicrostructure outlined in figure 1. The stage could also beused to obtain stereoscopic information from the flow in thesame manner as cartographic information is gathered by flyingover a terrain. In the shaded area of figure 1, twosuccessivecamera positions have significant overlap. The translationbetween these two positions was 320 µm orapproximatelyhalf of the field of view. Inside the overlapping area, it waspossible to perform stereoscopic recombination of the results,assuming the flow was stationary or more precisely that theaverage of several realizations was stationary. In this case, theflow measured at each camera position was an average of 50samples, each sample recorded at statistical unrelated times.864

PIV measurements in a microfluidic structureFigure 3. PIV measurements in the exit channel were performed in five x–y-planes spaced 10 µmindepth(z-direction). The measurementarea is indicated in the inset. There were 63 by 79 vectors in each plane originating from an interrogation area of 21 by 21 µm. Thethickness of the measurement plane was defined by the focal depth of the microscope objective to be 16 µm. Stream traces on the top planeindicate the end of the contraction and the beginning of the exit channel.The most common technique used to computestereoscopic PIV results is to pre-calibrate the two cameraviews with a known calibration target placed at differentz-positions. From the calibration a transfer function isgenerated, which is often a direct linear transform. Thisapproach would also have worked satisfactorily for thisapplication, except for the obvious access problems, making itimpossible to position a calibration target in situ. Alternatively,an analytical approach could be chosen as described byPrassad and Adrian (1993). The two main parameters arethe translation between the camera positions and the opticalpath from the measurement plane to the principal plane ofthe optical imaging system. Although the translation wasaccurately known, the optical path was only known in coarsedetails.Mapping the entire parameter set needed for thestereoscopiccombination would have been a complicated task.It requires that refraction of light in the air/glass interface ofthecoverglass on the microstructure be taken into account. Inparticular, the orientation of the microstructure surface mustbe well known in order to determine the refraction of light inthe cover glass. A scheme to map the various parameters failedin the measurement campaign presented here and therefore amore pragmatic approach was chosen. Ignoring the air/glassinterface and pretending the set-up to appear in a medium witha uniform refractive index, the equations for recombinationwere significantly reduced. Ignoring the interface and coverglass (750 µm thick)seemed reasonable, since it was lessthan one-third of the optical path from the objective to themeasurement plane. Further, the stereo view angle was lessthan 6 ◦ and the imaging was sharp all over the field of view ofthe camera. Prassad and Adrian (1993)presentthree equationsfor fluid displacement in air (equations (11)–(13)). Of thesewe work with the equation for the z-velocity component:W = zt = 1 −d 0 (X 1 − X 2 )t M 0 S − (X 1 − X 2 )(1)where z is the displacement, t is the time between thetwo laser pulses, S is the displacement between the twocamera positions, d 0 is the object to lens distance, M 0 isthe magnification and X 1 and X 2 are the displacementsmeasured by the camera in the two view positions.Observing that M 0 S ≫ (X 1 − X 2 ),equation (1)maybe simplified toW = d 0 (X 2 − X 1 )= d 0S M 0 t S (U 2 − U 1 ). (2)Hence theW -velocity component may be expressed asthe difference between the velocities measured from eachcamera in the translation direction (x-direction) scaled withafactor (d 0 /S). This means the W -component may alwaysbe visualized qualitatively, and additionally be quantitativelyestimated, except for the scaling factor. For this case using aqualified guess, an absolute velocity magnitude was obtained.Compared to the result from computational fluid dynamicsshown in figure 7, the factor was underestimated by at least20%. The correction was included in the PIV plot shown infigure 7.4. SimulationsFlow in the chimney structure was simulated using thecommercial software package CFD-ACE+ 6.4 from CFDResearch Corporation, Huntsville, AL, for computational fluiddynamics (CFD). This piece of software is a multi-physicspackage based on the finite-volume method. The programwas run on an 800 MHz Pentium III processor with 512 MBof RAM. Although the experiments were carried out inthe laminar flow regime, simulations were done on a finemesh of about 100 000 cells to account for the complexgeometry of the fabricated design. The geometry used in thesimulations included some simplifications compared to theactual microstructure usedfortheexperiments. The sidewalls865

HKlanket al600 µm65 µmy = 1500 µmy = 1560 µmy = 1845 µm y = 2010 µmFigure 4. V -velocity component shown in x–z-planes. The data were extracted from the PIV results shown in figure 3 at four y locations.Note that channel height and width are scaled differently. Colour coding of V -component was the same in figures 4 and 6.65 µm600 µmFigure 5. Realistic dimensions of the x–z-planes of figure 4.Figure 6. V -velocity component shown in x–z-planes. The data were extracted from the CFD results shown in figure 3 at two y locations.Colour coding of V -component was the same in figures 4 and 6.of the structure have a slope originating from the underetchingduring the manufacturing process. This slope was not takeninto account in the simulations; i.e. the exit channel wasmodelled as being rectangular. Furthermore, the width todepth ratio in the exit channel was 7.8, whereas the actualratio was 9.5. The actual channel dimensions are depicted infigure 5.The CFD-ACE+ 6.4 software offers several differentboundary conditions for the buffer and sample streams to beset, the initial flow rate, the average velocity or the velocityprofile. The small dimensions of microsystems usuallymean that velocity profiles do not develop fully within themicrosystem. Therefore, we chose to run the simulations withboththe set flow rates and the average velocities instead of thefully developed profile. Since the results using a set averagevelocity boundary condition were far more stable, only theseare presented in this paper. The known flow rates for bufferand sample stream listed in table 1 were used to derive one setof boundary conditions. The initial velocity of the sheathingstream was set to 44 mm s −1 and the velocity of the samplestream was set to 11 mm s −1 . Results from the simulationscan be seen in figures 6 and 7.5. Results and discussionAs mentioned above, the measurement volume was about21 × 21 × 16 µm 3 (H × W × D). Thehorizontal dimensions,i.e. the dimensions of the interrogation area, are given bythe number of pixels chosen, their size and the magnificationof the microscope. The vertical dimension was given bythe depth of field of the microscope and depends on the866

PIV measurements in a microfluidic structure50403020100w(m m/s)5.04.64.23.83.32.92.52.11.71.30.80.40.0Figure 7. All three velocity components are shown in a plane 30 µmabove the chimney. CFD results are shown to the left and PIV to theright. The colour indicates the magnitude of the W-velocity component. In the PIV results, the U, V -velocity vectors were obtained fromseveral camera positions. Stream traces of the U, V -component were shown. The W-component was obtained by combining the resultsfrom two overlapping camera positions with a stereoscopic technique.magnification and the numerical aperture. The higher themagnification of the system the smaller both the interrogationarea as well as the depth of field. This means that an increasein magnification produces in principle an overall decreasedmeasurement volume, which would increase the resolution ofthe PIV measurements.Nevertheless, opting for an objective with a lowmagnification has several practical advantages over choosingahigh-magnification objective. Low magnification objectiveshave larger working distances and provide a larger field ofview as well as a stronger irradiance onto the light detector.In our set-up it was not easily possible to use a highmagnificationobjective due to the thick cover glass of thechimney. Apart from this, a second reason for not choosingahighmagnification was that the field of view of the systemis decreasing with increasing magnification. In our case, thefield of view was chosen to cover the entire exit channel widthof 600 µm conveniently within one measurement. Therefore,the system magnification for the PIV measurements was set tobe tenfold.This magnification was achieved using a 20 × /0.40objective in combination with a 0.5× relay lens, which isbetter than using the 10× /0.25 objective of the microscope,since the higher numerical aperture of the 20× objectiveprovides a smaller depth of field and an image of highercontrast. The signal strength measured by the detector isgiven by the irradiance of an imaged particle within the imageplane. This irradiance is proportional to the square of the ratioof the numerical aperture to the linear magnification of thechosen objective (Schröder 1998). At a given magnification, ahigher numerical aperture value results, therefore, in increasedsignal strength, which makes the 20× objective/0.5× relaylens combination superior to the 10× objective.The reliability of the PIV results was examined bycomparing the volume flow of the PIV data in the exit channelto the nominal volume flow given by the syringe pumps.The flow rate was calculated by integration of thevelocities over an exit channel cross section, where a standardnumerical method (Simpson’s rule) was used. A systematicalerror of about 10% was found, as presented in table 2. Theerror probably originates from uncertainty in the magnificationof the system, since it was not calibrated, and the limitednumber of planes used for the integration. During all PIVmeasurements, the flow in the chimney structure was observedto be laminar and without separations, i.e. no convection dueto flow separation inthe microsystem could be found.Figures 4 and 6 show the axial velocity in the exit channelat different y locations for R = 5. Note that the z-axis inthe images is stretched in comparison to the x-axis. A morerealistic view of the dimensions is depicted in figure 5.The PIV data presented in figure 4 show a pair of higheraxial velocity streams at all four y locations. The streampair moves from the bottom of the channel upward passingthrough the channel. At the same time the two streams wereapproaching each other slowly up to the point where they beginto join in the last x–z-slice.The high-velocity stream pair was a result of the five timeshigher volume flow rate of the sheathing stream interrupted bythe sample stream. The highest measured speed of the pair isabout 84 mm s −1 as indicated by the colour code in figure 4.At first the stream pair flows low in the exit channel, since thesheathing stream acceleration was a result of the reduced crosssection at the end of the chimney basin. Within the basin themaximal V -component of the stream velocity was measuredto be 78 mm s −1 ,whichis8% less than the highest speedin the exit channel. Further along the channel, the sheathingstream pair accelerated the sample stream until an equilibriumvelocity distribution was reached. This equilibrium profilewas not yet fully developed within the measurement range.The simulated data show similar axial velocity profiles.The cross section at y = 1545 µm, in figure 6, contains thehigh-velocity stream pair, while the downstream cross section867

HKlanket alshows the already merged pair. The colour codes that wereused in figures 4 and 6 are equal and it can be seen that inboth cases the maximum V -component of the stream velocityis the same, about 84 mm s −1 . In contrast to the measuredcross sections, the high-velocity areas in the simulated crosssections are closer to the sidewalls. Measuring between thehighest speed and a speed of about 20 mm s −1 ,thedistanceto the sidewall is 40 and 70 µm inthe simulations and about90 µminthemeasurements.There are two main differences between the PIV andsimulation results, the facts that the PIV data show a certainasymmetry and that the stream pair merges earlier in thesimulations than in the PIV results. The simulated data arenecessarily symmetric, since they have been computed in asymmetrical way to save computing power. The measureddata on the other hand reflect thefact that the sheathing bufferstream is pushed up at the end of the basin.The second main difference between the PIV andsimulation results was that the stream pair merged about300 µm earlier in the simulations. While the simulated crosssection at y = 1545 µm can best be compared to the measuredcross section at y = 1560 µm, the simulation results show anequilibrium already at y = 1845 µm, where the measurementsstill show two distinct high-speed streams. The reason for theearly stream merger in the simulations was due to numericaldiffusion.The speeds of the two sheathing streams and the samplestream differ where the two streams meet. The slower samplestream exerts a viscous drag on the sheathing streams, whichflow next to it on both sides. The resulting accelerationof the sample stream can be described as a diffusion ofmomentum from the high-speed sheathing streams to the slowsample stream. The numerical computing method used inthe presented simulations adds an additional equalizing term,which can also be described as adiffusion. This numericaldiffusion is an intrinsic part of all numerical calculations. Itcan only be alleviated by defining a finer mesh or by choosingadifferent, more complex, method. In both cases a smallernumerical diffusion is bought at the cost of either longercomputing time or increased instability. We therefore decidedto accept a certain numerical diffusion, which can be found inour results.The velocity cross sections shown in figure 4 provideinformation that would otherwise not be easily obtainablewithout PIV. Although the distribution of the sample flowconcentration can be visualized by injecting a dye(Perch-Nielsen et al 2001), it is not straightforward to measurethe concentration quantitatively. It is even more difficultto make a statement about the flow velocities in themicrostructure. PIV on the other hand is designed to measurethe flow velocities and with a slight modification of theexperiment, the concentration information can be extracted.To achieve this, only the sample flow is seeded with tracerparticles, after which it is possible to observe the samplestream qualitatively and quantitatively using the PIV systemcomponents.Observing the chimney structure at different locationswith overlapping field of view allowed the construction of athree-dimensional vector field in the area of the actual chimneyas shown in figure 7. Both CFD simulation results and PIVmeasurement results are shown in the figure, where the U-and V -velocity components are rendered as streamlines whilethe W -velocity components, normal to the plane shown, areencoded in colour. The in-plane velocities were almost zeroinside the area of the chimney. In the vicinity of the chimney,the in-plane velocities were in the range of 10 – 20 mm s −1 andtheflow through the plane (W -component) had a maximum ofabout 5 mm s −1 .Theforward flow sped up towards the exit ofthe basin and reached about 70 mm s −1 within the area shownin figure 7.Since the absolute value of the W -velocity componentswas adjusted following the CFD results, only the relativePIV measurement results for the W -velocity components canbe compared. These results compare reasonably, with themedium W -velocity components between approximately 13and 30 mm s −1 coming up from the chimney in a roughly530 µm broad and 190 µm long area in the simulationimage, while being 480 µm broadand 120 µm long in themeasurement image.Outside the stereoscopic area, the two-dimensional datawere also in agreement. From thechimneystructure one wouldexpect the average flow velocity to rise somewhere above theactual chimney due to the increase in volume flow and alsoat the end of the chimney basin, where the cross sectionalarea decreases. Both speed increases are observed in boththe simulation and PIV results. An interesting result of bothsimulation and PIV measurement was that the highest sampleflow velocity was concentrated at the exit-facing rim of thechimney.In section 3, stereoscopic recombination was discussedand a method was presented to find the scaling factorbetween the difference in U-component velocities and theW -component velocity. Another approach to find the scalingfactor would be to integrate the volume flow over the entirechimney, since the volume flow is known from the pumpsettings given in table 1. Within the area presented in figure 7,the volume flow is computed to be 0.011 ml min −1 .However,the field of view does not cover the entire exit. Using the CFDresult, it is seen that the majority (about 80%) of the volumeflow exits the chimney within the field of view from themeasurement area. Using this fact confirms that the assumedscalingfactor was underestimated by more than 20%.Observing figure 7, itisfairly obvious that the accuracyof the W -velocity component was only limited, given that itwas based on the average of 50 samples. The main sources forthis large uncertainty are the very small stereoscopic viewangles used. The translation of 320 µm corresponds toastereoscopic view angle of approximately 2 × 6degrees,which challenges the accuracy of the measurement. Themeasurement could be significantly improved by using ahigher magnification with shorter working distance, includinglarger stereo view angles into the measurement. Additionally,alarge-format CCD camera could be used to increase thestereoscopic viewing angle, since it allows exploitation ofthe full viewing capability of the objectives. Finally, moreadvanced recombination schemes, which take into accountthe image distortion introduced by the thick cover glass,could be included in the computation of the W -velocitycomponents.The most desirable measure for improvement of theexperimental results would be to equip the examined868

PIV measurements in a microfluidic structuremicrofluidic structure with a thinner cover glass. This woulddecrease the distortions and make it possible to use objectiveswith higher numerical aperture. A cover glass thickness of170 µmwould be desirable, since most objectives are designedto be used with such cover glasses.6. ConclusionThe flow velocities of a microstructure used in cell sorting weremapped in several planes using PIV. The PIV measurementshad an adequate resolution and a good field of view.The quantitative results from measurements and simulationswere in reasonable agreement, considering the differencein geometry. Furthermore, it was possible to expand themicro-PIV technique to stereoscopic viewing. Combiningthe stereoscopic technique with mapping of several planes, thepossibility of full volume mapping (x, y,z,U,V,W) micro-PIV was demonstrated.Micro-PIV results can be used to evaluate and validatethe simulation results of micro-fluidic flows. In case of thechimney basin, the PIV results support the simulation results.While comparing simulation and PIV data of the exit channel,a certain difference was found, which was helpful as it pointsout the care that has to be exercised when simulating flow. Toimprove the agreement between simulation and measurement,the simulation can in principle be improved by using morecomplex algorithms or by defining a denser mesh at the cost ofincreased instability or computation time. The measurementcan be improved by employing a higher aperture objective,which would increase the viewing angle as well as collectmore light. One crucial step towards this end would be the useof a thinner cover glass.In general, the good agreement between simulation andPIV will allow the simulation software to be used as an efficienttool to speed up the microstructure design cycle.AcknowledgmentWe acknowledge Ulrik Darling Larsen, whose chimneystructure we used in our PIV measurements.ReferencesGoranovićG,Perch-Nielsen I P, Larsen U D, Wolff A, Kutter J Pand Telleman P 2001 Proc. 4th Int. Conf. on Modeling andSimulation of Microsystems (March, Hilton Head Island, USA)pp 242–5Larsen U D 2000 Micro liquid handling—passive microfluidics PhDThesis Technical University of Denmark, Lyngby, Denmarkpp 93–104Meinhart C D, Wereley S T and Gray M H B 2000 J. Fluids Eng.122 285–9Meinhart C D, Wereley S T and Santiago J G 1999 Exp. Fluids 27414–9Perch-NielsenIP,GoranovićG,KlankH,Larsen U D, Wolff A,Kutter J P and Telleman P 2001 Proc. 5th Int. Conf. onMiniaturized Chemical and Biochemical Analysis Systems(October, Monterey, USA) pp 629–30Prassad A K and Adrian R J 1993 Exp. Fluids 15 49–60Schröder G 1998 Technische Optik 8th ed (Würzburg: Vogel)p77Tashiro K, Sekiguchi T, Shoji S, Funatsu T, Masumoto W andSato H 2000 Proc. 4th Int. Symp. on Micro Total AnalysisSystems (May, Enschede, The Netherlands) (The Hague:Kluwer) pp 209–12Wolff A, Larsen U D, Friis P and Telleman P 2000 Proc. 14thEuropean Conf. on Solid-State Transducers (August,Copenhagen, Denmark) pp 235–8869

100 APPENDIX A. PAPER PUBLISHED IN J. MICROMECH. MICROENG.

Appendix BPaper published in Sens.Actuators B Chem.TitleTheoretical analysis of the low-voltage cascade electroosmotic pumpAuthorsAnders Brask, Goran Goranović, and Henrik BruusReferenceSens. Actuators B Chem. 92, 127-132 (2003)101

Sensors and Actuators B 92 (2003) 127–132Theoretical analysis of the low-voltage cascade electro-osmotic pumpAnders Brask, Goran Goranović, Henrik Bruus *Mikroelektronik Centret (MIC), Technical University of Denmark, DK-2800 Kongens Lyngby, DenmarkReceived 11 July 2002; received in revised form 19 December 2002; accepted 17 January 2003AbstractThe recently published experimental results obtained by Takamura et al. [Y. Takamura, H. Onoda, H. Inokuchi, S. Adachi, A. Oki, Y.Horiike, in: J.M. Ramsey, A. van den Berg (Eds.), Proceedings of the mTAS 2001, Monterey, CA, USA, Kluwer Academic Publishers,Drodrecht, 2001, p. 230], on their low-voltage cascade electro-osmotic pump are analyzed using two different theoretical approaches. One isthe semi-analytical equivalent circuit theory involving hydraulic resistances, pressures, and flow rates. The other is a full numerical simulationusing computational fluid dynamics. These two approaches give the same results, and they are in good qualitative agreement with thepublished data. However, our theoretical results deviate quantitatively from the experiments. The reason for this discrepancy is discussed.# 2003 Elsevier Science B.V. All rights reserved.PACS: 47.11.þj; 47.60.þi; 47.85.Dh; 82.45Keywords: Electro-osmotic pump; Equivalent circuit model; CFD simulation1. IntroductionMicropumps play a key role in the quest for fabricatingversatile, cheap, and highly efficient microfluidic lab-on-achipdevices. In this growing field especially micropumpsbased on electroosmotic flow (EOF) [1–3] are becomingimportant [4–9]. They contain no moving parts and arecompact. Moreover, they are relatively easy to integratein microfluidic circuits during fabrication.One major drawback in the conventional design of EOFmicropumps is the use of high voltage to drive the pump. Theinvention in 2001of the low-voltage cascade EOF pump byTakamura et al. [5] therefore marks an interesting developmentin the field. In the future, EOF pumps may be poweredby battery; and hence, portable.The aim of our work is two-fold. (1) We want to providethe first theoretical analysis of the experimental resultsobtained by Takamura et al. [5,10]. (2) Using computationalfluid dynamics (CFD) we want to demonstrate that thecomplex EOF pump is adequately described by the semianalyticalequivalent circuit theory involving hydrodynamicresistances, pressures, and flow rates. Both calculationalmethods are approximate, but since they are independent* Corresponding author. Tel.: þ45-4525-6399; fax: þ45-4588-7762.E-mail address: bruus@mic.dtu.dk (H. Bruus).and yield comparable results, we have gained confidence inour theoretical results.2. Principles of the EOF pumpTo set the stage for our analysis we briefly recapitulate theworking principles of the low-voltage cascade EOF pump[5]. The pump is designed to work as an effective pressuresource for low applied voltages. The layout of the pump isshown in Fig. 1.The main principle is to connect multiple EOF pumps inseries in order to accumulate pressure. Each elementary EOFpump (denoted a step) consists essentially of a narrowchannel section, marked B in Fig. 1, containing 10 parallelchannels, followed by a wide channel section, marked C inFig. 1, containing a single channel. The EOF in the narrowchannel section acts as a high pressure pump with forwardelectric field. In the wide channel section the electric field isreversed, but here the channel is so wide that the inducedback pressure is small compared to the previous pressure.After flowing through one such pump step the accumulatedvoltage is thus zero, while an appreciable pressure is maintaininga net flow. This ensures EOF pumping using alow operating voltage, indeed an attractive feature allowingthe pump to be operated with a battery and thus to beportable. Furthermore, it is more safe to use low voltages.0925-4005/03/$ – see front matter # 2003 Elsevier Science B.V. All rights reserved.doi:10.1016/S0925-4005(03)00130-8

128 A. Brask et al. / Sensors and Actuators B 92 (2003) 127–132where R hyd and the EOF pressure Dp eo is introduced in thesecond and third equality, respectively, so that Q eo appears asQ hyd in Eq. (2). In accordance with [6], Dp eo is thus defined by:ADp eo ¼ m eo Df eo R hydL ¼ m reo Df eo R elhydR ; (5)where the geometry ratio A=L equals the ratio between theelectric resistivity r el of the liquid phase and the electricresistance R of the channel.Equivalently, Dp eo can be defined as the hydraulic backpressure needed to balance an EOF, i.e.jQ eo j¼jQ hyd j. Thetotal flow rate Q of a channel with both a pressure-drivenflow and an EOF is simplyFig. 1. Top view of the low-voltage cascade EOF pump with three steps(adapted from Fig. 3 in [5]). The total pressure is proportional to the numberof steps. The EOF is driven through many parallel narrow channels by aforward electric field and through one wide channel with a reversed electricfield. A net flow is thus generated without accumulation of voltage.The disadvantage of a cascade pump is the extra complexityof the many electrodes.We begin our analysis by noting that water at roomtemperature has the kinematic viscosity n ¼ 10 6 m 2 /s,and when, as in the experiment, it flows with the velocityu 10 3 m/s through a channel of width D ¼ 50 mm, theReynolds number, Re, is minute:Re ¼ uD ¼ 0:05: (1)nFor such a low Reynolds number the flow is laminar and theviscous forces are dominant. In the case of uniform flow, i.e.zero spatial derivatives in the flow direction, the Navier-Stokes equation becomes linear.In this limit, for a steady-state flow through a channel witha hydraulic resistance R hyd , the flow rate Q hyd induced by thepressure drop Dp hyd isQ ¼ Q eo þ Q hyd ¼ Dp eo þ Dp hyd: (6)R hydLikewise, the resulting velocity profile is given by a superpositionof the velocity profiles of the EOF and the pressuredrivenflow, respectively [2].Consider a pump in a fluidic network. If the back pressureDp hyd is zero, the flow is denoted a ’’free run flow’’ and thecorresponding flow rate is termed Q max . The counter pressureneeded to stop the flow (i.e.Q ¼ 0) through the entirepump is denoted the maximum back pressure p max .3. Analysis of the EOF pumpFrom Eq. (5), it follows that the EOF-induced pressurebuildup is very large in the narrow channel section, and thusthe flow profile in this section becomes only slightly deformedunder the influence of the actual back pressure. The resultingvelocity profile u 1 , shown in Fig. 2, is a sum of a large, positiveQ hyd ¼ Dp hyd: (2)R hydPolar liquids offer another possibility beyond pure hydrodynamicsto generate a flow: namely the electrically drivenEOF [1–3]. Ions in the liquid form a thin (

A. Brask et al. / Sensors and Actuators B 92 (2003) 127–132 129and flat EOF velocity profile and a small, negative andparabolic back pressure velocity profile. The pressure buildupin the narrow channel section is denoted Dp N eo .In the wide channel the electric field is reversed. A part ofthe pressure generated in the narrow section is therefore usedboth for overcoming the reversed, small, and flat EOF profileand for driving the flow by pressure. The resulting velocityprofile is the large, forward parabolic-like velocity profile u 2shown in Fig. 2. It is evidently imperative to make thechannel width considerably larger in this section to keep itsback pressure to a minimum. The pressure drop in the widechannel section is denoted Dp W eo .The final pressure buildup Dp maxstep along the whole step isthe EOF pressure buildup in the narrow channel sectionminus the back pressure drop in the wide channel section,Dp maxstep ¼ DpN eo Dp W eo : (7)3.1. Equivalent circuit theoryThe concepts from the previous section can be used toanalyze the pump using the so-called equivalent circuittheory. In this theory, the hydraulic resistance of a fluidicnetwork is calculated by representing individual sections withequivalent hydraulic resistors using the usual rules for seriesand parallel resistors. The flow rate and the pressure drop for ahydraulic resistor are related by Eq. (2). For details see [6].The equivalent circuit theory is only exact for a uniformand laminar flow. Hence, it is not possible to analyze the flownear a bend. In the following analysis the bends are neglected,an approximation justified in Section 3.2 by CFD simulation.The first step is to find the equivalent diagram for thepump. It consists of a parallel coupling of N ¼ 10 identicalresistors R hyd;1 followed by R hyd;3 and R hyd;2 in series asshown in Fig. 3b. For a single rectangular channel of widthW and height H, R hyd is given by [6],since R hyd;3 R hyd;1 ; R hyd;2 . The total hydraulic resistanceis therefore given asR totalhydR totalhyd ¼ N 1 R hyd;1 þ R hyd;2 : (9)To drive the EOF an external electrical potential, Df ch ,isapplied to the resistor network (Fig. 3a). Only a fraction a ofDf ch is dropped over the EOF channels. This effectivepotential drop is denoted Df eo ,Df eo ¼ a Df ch : (10)Using the circuit diagram we determine a. The electricalresistance of a single rectangular channel with length L,width W, and height H is given byLR ¼ r elHW : (11)Combining this with the actual geometry of Fig. 1, it followsthata ¼ ðN=R 1 þ 1=R 2 Þ 1ðN=R 1 þ 1=R 2 Þ 1 ¼ 0:885: (12)þ R 3The actual applied potential Df app does not equal Df chbecause further potential drops occur in the gel electrodes.Hence, Df eo is unknown but could have been estimated ifthe electrical current and conductivity of the liquid had beenmeasured experimentally.At this point the EOF pressures Dp W eo , DpN eo , and Dp step canbe found for a given potential. However, it remains to verifythe validity of the approximate equivalent circuit theory.This is done by numerical simulations in the following.3.2. Computational fluid dynamicsR hyd ¼ðW=HÞ12ZLH 4P 1m¼0 ð192=ðp5 ð2m þ 1Þ 5 ÞÞ tanh ½ðð2m þ 1ÞpWÞ=2HŠ ; (8)where Z is the dynamic viscosity. R hyd;1 and R hyd;2 are thencomputed using Eq. (8). The short channel connecting thenarrow and wide channel sections can safely be neglected,Fig. 3. (a) The equivalent electrical circuit used in the calculation ofpotential drop across the EO section. (b) The slightly different equivalentcircuit for the fluidic network used in the calculation of flow rates. Thereare N ¼ 10 narrow channels each with the hydraulic resistance R hyd;1 .Inour calculations R hyd;3 is neglected.A full numerical simulation was performed using thecommercial computational fluid dynamics program Coventor2001.3. The program can simulate EOF in the limit ofinfinitely thin Debye layers. The electric potential is calculatedfirst, and by the Smoluchowski relation Eq. (3) it isused to establish the boundary conditions for the velocityfield at the walls. The accuracy of the CFD could bemeasured for straight channels with rectangular cross-section,since there the equivalent circuit model gives analyticalanswers.In order to save calculation time, it is important to identifythe minimal computational domain. Only one pump stepneeds to be analyzed. Further simplifications can beachieved by symmetry considerations. Clearly, the pumpis symmetric about the horizontal plane at half the channeldepth. Due to the very low Reynolds number, Re < 0:05, theflow is said to be creeping. Numerical investigation by Yang

130 A. Brask et al. / Sensors and Actuators B 92 (2003) 127–132et al. [11] showed that for a 90 bend, inertial effect wasnegligible for Re < 5. Consequently, it does not matterwhich way the liquid flows. The flow pattern in the firsthalf of a pump step (e.g. between A and B in Fig. 1) thereforeequals that in the second half (e.g. between B and C inFig. 1). Hence, only half of the computational domain needsto be considered. This was verified by a full-geometrysimulation. The flow rate and pressure buildup for thecomplete pump step are twice the values obtained usingthe symmetry-reduced geometry.The CFD simulation was done with various grids to findthose yielding grid-independent results. We focused especiallyon problems at the corners, since they are known to beproblematic [12]. We ended using a grid containing 10 5rectangular cells with 14 7 cells in each cross-section. TheCFD results for this grid matched the analytical circuitmodel within 5%.4. Results and discussion4.1. Velocity profilesA simulation of the low-voltage cascade EOF pump gavethe velocity profiles depicted in Fig. 4. There is one profilefor the wide channel, and one for a single narrow channel.The profiles have been extracted from the symmetry plane.The CFD program solves the Poisson equation for a givenpotential drop Df ch and finds that the electric fields areslightly different in the wide and narrow channels. In thecircuit model they are assumed to be the same. Thecalculated velocity profiles at two positions are sketchedin Fig. 2.4.2. Comparison with experimentsThe circuit model is now used to calculate the back pressurep max and the EOF velocity u eo as a function of the potentialdrop. Unfortunately, neither the actual potential drop Df ch northe EOF mobility m eo was measured in the experiment.However, both back pressure and flow velocity depend linearlyon the product m eo Df ch , and, as we shall see, this makesa comparison between experiments and theory possible.In the case of a 15-step cascade pump, Takamura et al. [5]measured a max back pressure of 800 Pa and for the samevoltage and under free run conditions, a max velocityu max ¼ 0:50 mm/s. The velocity measurements were conductedin the last wide section, which had no electric field.Hence, the flow is solely pressure driven.The relation between the back pressure and the productm eo Df ch is found using Eqs. (5) and (7), and it is plotted inFig. 5. From the figure, it is found that the measured 800 Pacorresponds to m eo Df ch ¼ 0:114 mm 2 /s. Using this valuethe model predicts the flow rate Q max ¼ 0:110 nl/s. Fromthis flow rate, the maximum velocity u max may be calculatedin the wide channel section.Using the aspect ratio as a variable parameter, we performeda general analysis of the relation between flow ratesand maximal flow velocities in rectangular channels withcross-section area A. For the actual aspect ratio W=H ¼ 2:5we found a numerical factor 0.52:u max ¼ Q max¼ 0:21 mm=s; (13)0:52 Awhere A ¼ 50 20 mm 2 . This velocity, found both by simulationand by the circuit model, deviates by a factor 2.4 fromthe measured 0.50 mm/s.In [5], results from a 6-step and 15-step pump are presented.The back pressure measurements give 380 and 860 Pafor the 6-step and 15-step pump, respectively, at f app ¼ 25 V.The corresponding ratio is then 860=380 ¼ 2:26 whichshould be compared with 15=6 ¼ 2:5. Hence, it can beconcluded that the pressure is not accumulated linearly withthe number of steps in contrast to theoretical expectations.The deviation may be within the range of uncertainty for thepressure measurements.To obtain realistic estimates for the absolute values of theEOF pump parameters, we use some typical values for m eoand Df ch . The results for Q max , p max , and R hyd obtained bymodel calculations and simulations are compared in Table 1Table 1Comparisons between equivalent circuit model and numerical simulationfor a single pump stepQ max (nl/s) p max (Pa) R hyd (kg/(m 4 s))Fig. 4. CFD simulated velocity profiles at the symmetry plane in a wideand a narrow channel. The velocity at the walls is the EOF velocity. Theparameters are: Z ¼ 1:00 10 3 Pa s, m eo ¼ 0:06 mm 2 /(V s), and Df ch ¼10 V.Model 0.577 281.0 4:88 10 14Simulation 0.560 274.0 4:89 10 14Deviation 3% 2% 0%Parameters: m eo ¼ 0.06 mm 2 /(V s), Z ¼ 1.0 10 3 Pa s, Df ch ¼ 10 V.

A. Brask et al. / Sensors and Actuators B 92 (2003) 127–132 131Fig. 5. Maximum back pressure as a function of m eo Df ch . The pressures are calculated using the equivalent circuit theory. The one step slope is 468 Pa s /mm 2 for Z ¼ 1:00 10 3 Pa s, a ¼ 0:885. The measured pressure of 800 Pa leads to a value m eo Df ch ¼ 0:114 mm 2 /s.for a given set of parameters. The two sets of results agreewell. Calculations with m eo ¼ 0:06 mm 2 /(V s) give considerablylarger pressures than those measured, e.g. 4200 Pa forthe 15-step pump with Df ch ¼ 10 V. This indicates thepossibility for improving the performance of the pump.5. Discussions and conclusionsDiscussions with Takamura [10] have led to the followingconclusions regarding the discrepancies between experimentand theory.(1) The velocity mismatch of a factor of 2.4 is probablydue to fabrication difficulties. The actual width of thenarrow channels may have been as large as 8 mminstead of the design value of 5 mm [10]. SinceDp eo / W 2 for W H, the pressure for a 8 mm widechannel is ð8=5Þ 2 ¼ 2:6 times smaller than for a 5 mmchannel. This would lower the slopes of the lines inFig. 5 and instead of reading off 0.114 mm 2 /s on theabscissa, we would get 0.296 mm 2 /s. Using thisnumber for m eo Df ch , we obtain u max ¼ 0:55 mm/s.This result agrees better with the observed 0.5 mm/s. Asource for minor inaccuracies is the actual shape of thechannel cross-section.(2) The pressure does not increase linearly with thenumber of stages. Two possible explanations of thisobservation are: (a) due to the design of the pump theindividual gel electrodes do not have the samepotential; (b) pressure-dependent hydraulic resistancesof the gel electrodes, i.e. leaks in the gel electrodesincreases with increasing pressure.(3) The experimental accuracy of the velocity and pressuremeasurements is 15–30%. Part of the discrepanciescould therefore simply arise from inaccurate measurements.The above explanations can account for the large quantitativedeviations between theory and experiment, and theproblems will be addressed in coming improved version ofthe low-voltage cascade EOF pump [10].In summary, we have presented a simple analytical modelfor the low-voltage EOF pump obtained by using equivalentcircuit theory. This approximate model has proved sufficientfor making good estimates of the performance of thepump. Full CFD numerical simulations were made to verifythe model and to provide more detailed information aboutthe flow. The model and the simulations agree within 3%.Our theoretical analysis was compared to experimentalresults obtained by Takamura et al. [5], and good qualitativeagreement was observed. However, a considerable quantitativedeviation (a factor of 2.4) was found for values ofcalculated versus measured maximum velocity. The possiblesources for this discrepancy were identified as inaccuraciesin the measurement of various parameters (channel widths,velocities, and pressures) and pressure leakages.Our work shows the advantage of theoretical analysis as asupplement to the experimental approach in the study ofEOF pumps. Our pressure calculations indicate, for example,that there is room for improving the performance of thepump. We have also shown that, although approximate, theequivalent circuit model is applicable to the cascade EOFpump. This is mainly because the corrections from electricaland hydraulic corner effects are negligible. Use of the circuitmodel facilitates the analysis of the pump enormously.

132 A. Brask et al. / Sensors and Actuators B 92 (2003) 127–132AcknowledgementsWe are grateful to Yuzuru Takamura and his colleagues atthe University of Tokyo for their openness and great help inour discussions by email during this work and for providingus with experimental details not presented in [5]. We alsothank our colleagues from the mTAS project at MikroelektronikCentret, especially Jörg Kutter, for support and stimulatingdiscussions. This work was partly supported by theDanish Technical Research Council, mTAS Frame ProgramGrant No. 9901288.References[1] J.F. Osterle, J. Appl. Mech. 31 (1964) 161.[2] C.L. Rice, R. Whitehead, J. Phys. Chem. 69 (1965) 4017.[3] A. Manz, C.S. Effenhauser, N. Burggraf, D.J. Harrisson, K. Seiler, K.Fluri, J. Micromech. Microeng. 4 (1994) 257.[4] P.H. Paul, D.W. Arnold, D.J. Rakestraw, Proceedings of the mTAS1998, Banff, Canada, 1998, p. 49.[5] Y. Takamura, H. Onoda, H. Inokuchi, S. Adachi, A. Oki, Y. Horiike,in: J.M. Ramsey, A. van den Berg (Eds.), Proceedings of the mTAS2001, Monterey, CA, USA, Kluwer Academic Publishers, Drodrecht,2001, p. 230.[6] W.E. Morf, O.T. Guenat, N.F. de Rooij, Sens. Actuators B-Chem. 72(2001) 266.[7] O.T. Guenat, D. Ghiglione, W.E. Morf, N.F. d Rooij, Sens. ActuatorsB-Chem. 72 (2001) 273.[8] S. Zeng, C.-H. Chen, J.C. Mikkelsen Jr., J.G. Santiago, Sens.Actuators B-Chem. 79 (2001) 107.[9] D.R. Reyes, D. Lossifidis, P.-A. Auroux, A. Manz, Anal. Chem. 74(2002) 2623.[10] Experimental parameters not quoted in the original experimentalpaper were kindly provided by Dr. Y. Takamura, Univ. Tokyo.[11] R.-J. Yang, L.-M. Fu, Y.-C. Lin, J. Colloid Interf. Sci. 239 (2001) 98.[12] N.A. Patankar, H.H. Hu, Anal. Chem. 70 (1998) 1870.

108 APPENDIX B. PAPER PUBLISHED IN SENS. ACTUATORS B CHEM.

Appendix CPaper published in J. Micromech.Microeng.TitleThe clogging pressure of bubbles in hydrophilic microchannel contractionsAuthorsMads Jakob Jensen, Goran Goranović, and Henrik BruusReferenceJ. Micromech. Microeng. 14 (2004)109

CE: RVINSTITUTE OF PHYSICS PUBLISHINGJOURNAL OF MICROMECHANICS AND MICROENGINEERINGJ. Micromech. Microeng. 14 (2004) 1–8 PII: S0960-1317(04)72206-8The clogging pressure of bubbles inhydrophilic microchannel contractionsMads Jakob Jensen, Goran Goranović and Henrik BruusMIC - Department of Micro and Nanotechnology, Technical University of Denmark,Oersteds Plads, Bldg. 345 East, DK-2800 Kgs. Lyngby, DenmarkE-mail: mjj@mic.dtu.dkReceived 19 November 2003Published DD MMM 2004Online at stacks.iop.org/JMM/14/1 (DOI: 10.1088/0960-1317/14/0/000)AbstractWe present a theoretical and numerical study of the quasi-static motion oflarge wetting bubbles in microfluidic channels with contractions. In mostcases the energy of a bubble increases when it is moved from a wide channelto a narrow one, and the bubble thus tends to clog the flow of the fluid. Acertain pressure, the so-called clogging pressure, is needed to push thebubbles out of the contraction. However, we show that in the case of ahydrophilic channel contraction there exists a range of parameter valueswhere the bubble actually gains energy by moving into the narrow part. Forthese specific cases we analyze how the clogging pressure depends onchannel geometry, surface tension and contact angle. Based on our analysiswe establish design rules for minimizing the clogging pressure ofmicrochannel contractions.1. IntroductionMany microfluidic networks on modern lab-on-a-chip devicescontain channel contractions. These tend to becomeproblematic if, as is often the case, gas bubbles are introducedinto the liquid at the inlets or by electrochemical processes.Due to the small channel dimensions gas bubbles can easilybe large enough to span the entire channel cross-section.Such ‘large’ bubbles are prone to get stuck at the channelcontraction, whereby they can clog the flow and disturbmeasurements or functionality of the system in an uncontrolledmanner. To clear the clogged channel an external pressure,the so-called clogging pressure, has to be applied to pushthe clogging bubble out of the system. Although alreadyidentified nearly a decade ago [1, 2], this important problemin microfluidic systems has not been studied theoreticallyto a wide extent, a situation we would like to amend withthis paper. The present work is a substantial extension of apreliminary and specialized study presented at the NanoTech2003 conference, [3] now including an analysis of the bubbleenergies in general cases, inclusion of compressibility effectsand the use of different parameter values.A complete analysis of the motion of a large bubblethrough a microchannel contraction involves many differentphysical effects, some which are not completely understood.Any comprehensive analysis would at least require detailedmodeling of the liquid–gas, liquid–solid and solid–gasinterfaces as well as the dynamics in the bulk fluids. Butalso more complicated processes near the contact lines needto be addressed, e.g. wetting [4–6], contact line pinning andhysteresis [4, 7], dynamic contact angles and contact lines[8–10] and static and dynamic friction [11–13]. It should bestressed that many of these surface effects are hard to controlprecisely, therefore dynamical systems where the lubricationassumption is used are also widely analyzed [14, 15].In this work, however, we will restrict our analysis toquasi-static motion of bubbles. By this we mean that thevelocity of the bubble is nearly zero and that the entiremodel system remains arbitrarily close to equilibrium for allbubble positions. All dynamic aspects are thus neglected,and basically the model involves only the free energy of theinternal interfaces of the system and external pressures. Thisis motivated by the fact that it is difficult to experimentallycontrol surface related properties. We thus only studygeometry related effects. We also choose to work only withaxisymmetric channels of smooth (but otherwise arbitrary)contraction geometries free from any sharp corners andother singularities. With these simplifications the forces orpressures needed to push a bubble through the system can becalculated accurately without losing the essential physics ofthe problem. This in turn enables us to formulate design rulesfor microchannel contractions to prevent or reduce clogging.0960-1317/04/000001+08$30.00 © 2004 IOP Publishing Ltd Printed in the UK 1

MJJensenet alRP Lθliquid (l)solid (s)r(x)axis ofx L x cm symmetry x RP igas (g)θ trxP R =P L +∆P bFigure 1. A bubble with internal pressure P i and center of mass x cmin a hydrophilic axisymmetric channel. The left (right) contact linehas the coordinate x L (x R ) and contact angle θ. The channel iscontracting from a straight part of radius R to one of radius r. Thespecific channel profile is defined by some function r(x).Throughout this paper we have chosen r(x) to be a sloped straightline joined to the straight parts by two circle arcs. The taperingangle θ t is given by tan θ t =−r ′ (x). The pressure left (right) of thebubble is denoted as P L (P R ) and the pressure difference across thebubble is P b .To our knowledge similar analyses have only been made onchannels of constant cross-sections [16] and for the specialcase of sudden contractions [17].2. The model and basic physical assumptionsConsider a hydrophilic microfluidic channel or capillary, suchas the one depicted in figure 1, which is axisymmetric aboutthe x axis with a position dependent channel radius r(x).Thechannel is filled with a liquid. A large bubble of some otherfluid (we think mainly of a gas such as air) is present in theliquid. By large we mean that the volume of the bubble islarger than the volume Vsphmax of the largest inscribed spherethat can be placed anywhere in the microchannel. A largebubble divides the liquid into two disconnected parts, left andright of the bubble. The bubble itself consists of a bulk part indirect contact with the walls of the channel and of two menisci,in contact with the liquid, capping the ends of the bubble.The bubble is assumed to be in quasi-static equilibrium. Inthat case it is relatively simple to combine mass conservationwith geometric constraints to determine, as a function of thebubble position, the pressure drops over the two meniscineeded to maintain this equilibrium. We define our centralconcept, the clogging pressure, as the maximum of the positiondependent pressure drop across the bubble, i.e. the minimalexternal pressure that must be supplied to push the bubblethrough the microchannel.2.1. The Young and Young–Laplace equationsOur model system consists of a solid channel containing aliquid and one large gas bubble. Therefore, the essentialphysical parameters are the three surface tensions (surfacefree energy per area) σ lg ,σ sl and σ sg for the liquid–gas, solid–liquid and solid–gas interfaces, respectively. In equilibriumthe contact angle θ is determined by the surface tensionsthrough the Young equation [18, 19]σ sg − σ sl = σ lg cos θ. (1)In the following the contact angle is taken as the equilibriumangle or rather as an average contact angle. Because contactangle hysteresis is very sensible to surface effects, we do notaddress these questions in this work.To sustain a curved interface with the main radii ofcurvature R1 c and Rc 2 between a gas of pressure P g and a liquidof pressure P l , the pressure difference P = P g − P l mustobey the Young–Laplace equation [20]( 1P = σ lgR1c + 1 )cos θR2c = 2σ lg , (2)rwhere the last equation is applicable for a constant circularcross-section of radius r. We use the standard convention thatthese radii are taken as positive if the interface is concave whenseen from the gas.2.2. Isothermal motion and compressibilityIn the rest of the paper we consider a ‘large’ bubble having theinitial position ‘1’ in the widest part of the channel. The initial /volume is V 1 = γVsph maxmax,whereγ > 1andVsph = 4πr1 3 3,and the corresponding internal pressure is P i,1 . At a later stagethe bubble is moved to a position ‘2’, where the volume isV 2 and the internal pressure is P i,2 . In the quasi-static casethe bubble motion is isothermal and hence the compressibilitycondition applies,P i,1 V 1 = P i,2 V 2 . (3)The pressure P i within the bubble is given as the externalpressure P 0 plus the pressure change P across the curvedinterface, given by equation (2).The most extreme compression is obtained by pressing alarge bubble, which floats without geometrical constraints ina bulk liquid of pressure P 0 , into a narrow circular channel ofradius r. Combining equations (2) and (3) yieldsV 1= P i,2≈ P i,2= 1+ 2σ lg cos θ. (4)V 2 P i,1 P 0 rP 0For example, moving a large spherical air bubble in water(σ lg = 0.0725 J m −2 ) at the ambient pressure P 0 = 10 5 Painto a channel of radius r = 25 µm leads to V 1 /V 2 ≈ 1.06,i.e. a volume compression of 6%. Moving, as in section 6,a bubble from a 300 µm to a 190 µm wide channel yields acompression of about 0.2%.In the case of laser ablated microchannels in plasticchips, compressibility effects are negligible as the smallestdimensions typically are greater than 100 µm. However, forsilicon based micro- or nanofluidic devices, compressibilitymay play a significant role.2.3. Quasi-static motion and geometryFor a bubble positioned in a microchannel contraction, thetotal internal energy E tot is the sum of the surface free energy,gravitational energy, kinetic energy and frictional energy. Weregard the surrounding pressures as external energy. By ourdefinition quasi-static motion of an incompressible bubbleimplies that the kinetic energy is zero and friction is also2

zero because of hydrostatic and thermodynamic equilibrium.Finally, we treat channels of characteristic dimensions 2r lessthan 300 µm, which is significantly smaller than the capillarylength of water, c = √ σ lg /ρ lg ≈ 2700 µm, where ρ l=10 3 kg m −3 and g = 9.82 m/s 2 . So the gravitational energycan also be neglected, which ensures that the menisci may beapproximated by spherical caps.The total internal energy E tot of the microchannelcontaining a quasi-statically moving bubble is given only bythe surface free energy, i.e. the sum of interfacial energies σ itimes interfacial areas A i ,E tot = ∑ σ i A i = σ lg A lg + σ sg A sg + σ sl A sl . (5)iThe pressure-related applied external force F needed tobalance the bubble is given by the gradient of the total internalenergy with respect to the center of mass coordinate of thebubble x cm . HenceF = dE tot, (6)dx cmwhich thus depends on the bubble position x cm and, throughthe areas A i , on the geometry of the channel.The clogging pressure of bubbles in hydrophilic microchannel contractionscos θP i,1 = P R +2σ lgR . (10) In figure 2 the energy E tot (equation (13)) is plottedas a function of the ratio r/R for given values of γ . The0 0.2 0.4 0.6 0.8 1.0r/RFigure 2. Plot of the energy change E tot as a function of the ratior/R. The bubble is moved from a wide channel of radius R =150 µm to a narrow channel of radius r. Five curves are showncorresponding to the volume ratio γ = 1, 2, 3,γ c and 8, respectively,γ c ≈ 4.75. For ‘small’ volumes 1 γγ c the movement requires energyin all cases.The volume of the bubble is the sum of two spherical capvolumes and the volume of a cylinder of initial length L. Once2.4. The clogging pressurethe length L is known, the relevant interfacial areas A lg andA sg may be found.The Young–Laplace pressure drops (cf equation (2)) at themenisci are given by,The gas bubble is now moved to the cylindrical channelof radius r, and according to equations (2), (3) and (7b) theP L = P i − P L ,(7a)pressure P i,2 and volume V 2 areP R = P i − P R .(7b)cos θP i,2 = P R +2σ lg ,r(11)The total pressure drop P b (x cm ) over the bubble as a functionof its center of mass x cm is given byV 2 = P i,1V 1 .P i,2(12)P b (x cm ) = P R − P L = P L (x cm ) − P R (x cm ). (8)By solving equation (12) it is straightforward to find the changeThe clogging pressure P clog is defined as the maximal position in total free surface energy,dependent pressure drop across the bubble,E tot = E tot,2 − E tot,1 = σ lg (A lg,2 − A lg,1 )P clog = max {−P b (x cm )}. (9)+ σ lg 2π cos θ(rl − RL), (13)The clogging pressure expresses the minimal amount by whichwhere l is the length of the bubble in the channel of radiusthe left-hand-side pressure P L must exceed the right-hand-sider < R (situation 2). In equation (13) the Young relationpressure P R to push the bubble through the contraction quasistaticallyfrom left to right.(1) has been used to eliminate the solid–liquid and solid–gasinterfacial energies.Basedonequation(13) we can analyze the energy change3. General energy considerations for axisymmetric when moving the bubble from the wide channel of radius R tomicrochannelsthe narrow channel of radius r. First we give the limiting valuesof E tot . In the limit r/R → 1 we obviously get E tot → 0.Consider a bubble placed in a cylindrical channel of radius R.In the opposite limit, r/R → 0, the compressibility of theWe want to determine the change in energy resulting frombubble results in convergence of E tot ,moving it into a smaller channel of radius r < R, e.g.()by moving it from left to right in the channel depicted infigure 1. Intuitively, we would expect the energy to increase lim rRas a result of the movement. In most cases this intuition istot = πR34+sin(3θ)− 3tanθ4γRP R − σ lg .3cos 2 θ(14)correct; however, we shall see that in some cases the systemgains energy by the move, solely due to geometric conditions.The bubble has the initial volume V 1 = γVsph max,whereTo discuss E tot for general values of r/R we use a numericalexample: an air bubble in a water filled PMMA channel forγ>1andVsphmax = 4πR 3 /3. With this constraint the bubble is which we have the parameter values P R = 10 5 Pa, σ lg =forced to touch the walls regardless of its position. According 72.5 mJandθ = 72 ◦ . The radius ratio r/R and the volumeto equations (2) and (7b) the internal pressure of the bubble is parameter γ are then varied.∆Etot (nJ)3020100γ = γ cγ =3γ =2γ =1γ =83

MJJensenet al∆Etot (nJ)3020100-10γ =1R = 100 µmR = 150 µmR = 200 µmR = 250 µmR = 300 µmγ =3R = 100 µmR = 150 µmR = 200 µmR = 250 µmR = 300 µm-200 0.2 0.4 0.6 0.8 1.0r/RFigure 3. The energy E tot as a function of the ratio r/R fordifferent values of the wide channel radius, R = 100, 150, 200, 250and 300 µm. The plain curves correspond to the smallest bubble forγ = 1 and the dotted curves correspond to a larger bubble withγ = 3.figure shows that for large values of γ , i.e., large bubbles,it requires energy (E tot > 0) to move the bubble fromthe wide to the narrow channel. However, there exists acritical value γ c ≈ 4.75 below which the system can gainenergy by moving the bubble, if the radius ratio r/R is not toosmall. This behavior is generic for a bubble in a contractingchannel, but the specific shape of the curve and the optimalminimum depend on the material parameters and the externalpressure P R .The critical value γ c , above which energy gain isimpossible, is given by ∂E tot /∂(r/R) = 0atr/R = 1,γ c = (3 − cos(3θ)+2sinθ)(2σ lg cos θ + RP 0 ). (15)2RP R cos θ(1+sinθ)Figure 3 depicts the energy E tot as a function of theratio r/R for γ = 1andγ = 3, and for five values of thewide channel radius, R = 100, 150, 200, 250 and 300 µm.From equation (13) it may be seen that min{E tot }∝R 2 asthe area is proportional to R 2 and L is proportional to R.Deviations from this proportionality arise for small valuesof R because of compressibility. For γ = 1 in figure 3we find max{−E tot }=kR 2 with k = 0.159 J m −2 . Thisproportionality is illustrated as the energy at a given r/Rpoint is increased by a factor 4 when R is doubled, e.g. fromR = 150 µm toR = 300 µm.The previous calculations clearly show that for somegeometries it is favorable to place the bubble in the narrowrather than in the wide part of the channel. In the followingwe shall address the question of whether for such geometriesthe bubble will move spontaneously or it must cross an energybarrier to arrive at the low-energy state in the narrow channel.4. Analytical results for contractions withenergy gainCombining the geometry defined in figure 1 with equations (2)and (8), the central expression of our analysis is easily derived,). (16)P b = 2σ lg( cos[θ − θt (x L )]r(x L )− cos[θ + θ t(x R )]r(x R )From the discussion in section 2.4 it follows that if P b < 0then the contraction causes bubble clogging, whereas for(a)LdD(b)x 1 x 2 x 1 x 2Figure 4. Two generic situations for a bubble of length L = x R − x Lnear a microchannel contraction of length x 2 − x 1 .(a) Thecontraction is long enough to contain the entire bubble, i.e.x R − x L x 2 − x 1 , which is a class β 4bubble.P b > 0 the bubble tends to move spontaneously throughthe contraction toward the narrow part.Based on equation (16) a number of design rules may beestablished specifying the geometric features that may preventor decrease clogging. Consider a bubble that starts out in thewide straight section left of the contraction, where it has alength L 0 = x R − x L . The pressure drop P b is zero to beginwith, but depending on the shape of the contraction, such as thetwo examples shown in figure 4, P b changes as the bubbleadvances quasi-statically through the contraction.The first part of any contraction can always beapproximated by a circle with an arc angle which is the localtapering angle θ t . As the right contact line x R just enters thecontraction, equation (16) can be expanded to first order in θ tyieldingP b ≈ 2σ lg sin θθ t > 0. (17)RThus initially the bubble tends to move spontaneously intothe contraction. The physical reason for this is that the localtapering angle allows the meniscus to flatten a little, whichreduces the costly gas–liquid interface energy.Once the bubble moves inside the contraction defined infigure 1, a complicated interplay between the initial bubblelength L 0 , the contact angle θ, the channel radii r(x L ) andr(x R ) at the contact lines and the local tapering angle θ t (x)decides whether bubble clogging occurs or not. We classifyour systems into two main classes:Class α comprises all cases where no clogging occurs,i.e. where the bubble can move spontaneously through thecontraction without applying an external pressure.Class β contains all cases with clogging, i.e. whereP b < 0 at some point or, equivalently, where P clog > 0.For class β four sub-classes can be identified depending onwhere the bubble is when P b becomes negative and cloggingoccurs. This bubble position is classified by the position ofthe contact lines x L and x R relative to the beginning x 1 and theend x 2 of the contraction region (see figures 1 and 4):class β 1 : x L

The clogging pressure of bubbles in hydrophilic microchannel contractionsTable 1. Physical parameters for air bubbles in water flowingthrough PMMA microchannels.Parameter valuesσ lg = 72.5 × 10 −3 Jm −2 [21]σ sg = 38.9 × 10 −3 Jm −2 [21]σ sl = 16.5 × 10 −3 Jm −2 [21]θ = 72 ◦ [18]Reference100 x =120.59CM(a) 0−100100 x CM=337.02(b) 0−100Q1Q2A β 2 clogging only occurs if the bubble can move entirelywithin the tapered region as shown in figure 4(a), and if atsome point it has a length L = x L − x R such thatL> r(x [L)1 − cos(θ − θ t)tan θ t cos(θ + θ t )]. (19)In β 4 where the bubble in fact spans the entire contractionas sketched in figure 4(b), there is always clogging and theclogging pressure is maximal. The value for P b is negativeand independent of the shape of the contraction. Fromequation (16)weget( 1P b = 2σ lg cos θR − 1 )< 0. (20)rThe nonclogging class α will in general occur if the bubble issmall enough. According to the class β 4 analysis a necessary(but not sufficient) condition for avoiding clogging is that thebubble is small enough to be completely contained in thecontraction region. An analysis of the β 2 and β 3 classes showsthat it should also be short enough to avoid clogging whilethe left meniscus is still in the tapered region. The β 1 classfurthermore puts upper limits on tapering angles that allowfor clog-free flow. Examples from class α and β 4 are treatedfurther by detailed numerical analysis in sections 5.2 and 5.3.5. Numerical simulationsTo illustrate the analysis given above a detailed simulation ismade in the following. The aim is to minimize the cloggingpressure P b with respect to a given parameter. We arelimiting our analysis so that the variation comprises only oneparameter: the tapering angle θ t .5.1. The numerical algorithmIn order to find the force and clogging pressure acting on alarge bubble for a given geometry, a semianalytical modelof the contracting channel is implemented in MatLab. Anumerical Romberg integration scheme is used together witha Newton solver to determine the location of the right and leftcontacts line (x R and x L ) for a given position of the center ofmass coordinate x cm . The respective interface areas A i arethen found. For a specific geometry defined through r(x),the maximal force is found through equations (5) and (6)and the pressure drop P b is found through equation (16).The heaviest calculation ran for approximately 4 h on astandard PC.To be specific we use the geometry defined in figure 1 andtake PMMA as the solid material, water as the liquid and air asthe gas. This configuration has the physical parameters givenin table 1.(c)(d)(e)100 x CM=371.150−100100 x CM=600.210−100100 x CM=795.160−1000 200 400 600 800 1000Figure 5. Five positions (a)–(e) of a large bubble with γ = 1.02inside a 1000 µm long hydrophilic channel with a tapering angleθ t = 10 ◦ . The black dots indicate x cm . The contact angle is θ = 72 ◦ .5.2. A specific system without clogging, class αThe first example is the system with a bubble placed in arelatively gentle contraction depicted in figure 5. The totallength of the channel is 1000 µm. The wide straight channelto the left has a radius R = 150 µm and length 200 µm. Thecontraction has a length x 2 − x 1 = 350 µm and circle arclengths of 30 µm, which results in a tapering angle θ t = 10 ◦ .The narrow straight channel to the left has a radius r = 95 µmand length 500 µm. The bubble starts out in the wide channelto the left. It has a relative volume of γ = 1.02 (cf section 3)and an initial length L 0 = 180 µm.Figure 5 shows the bubble at five different positions (a)–(e). As the bubble advances through the channel it is seenhow its length x R − x L changes and how the curvatures of themenisci vary. The black dots inside the bubble indicate thecenter of mass x cm .In figure 6(a) the total internal energy of the system isplotted as a function of center of mass position x cm . The zeropoint of the energy is chosen as the energy of the system whenthe entire bubble is positioned completely within the narrowpart of the channel. The five positions (a)–(e) in figure 5 arealso marked here.It is seen that the energy decreases monotonically. Thismeans that without a negative external pressure holding it back,it would move spontaneously through the channel from the leftto the right. As long as the bubble moves completely within thewide part of the channel, the energy is constant (about 4 nJ).Then as the right edge enters the contraction, position (a),the energy drops rapidly in accordance with the pressure dropequation (17). This trend continues as the entire bubble movesinside the contraction, as is the case in position (b). The energycontinues to drop, but now less rapidly, as the right edge of the5

MJJensenet al(a)3•a(a)100 x CM=127.380−100Etot (nJ)21• b •c•d• e(b)100 x CM=300.570−100(b)0 • ac •• e 0-2• d-50-4←−◦ -100-6◦−→ -150-8-10-12• b100 200 300 400 500 600 700 800x cm (µm)F (µN)0100 200 300 400 500 600 700 800x cm (µm)Figure 6. (a) A plot of the total internal energy E tot versus thecenter of mass coordinate x cm for the same bubble as in figure 5.(b) The balancing external force F and the pressure drop across thebubble P b versus x cm . The five dots correspond to the five bubblepositions in figure 5. Note that −P b < 0 for all positions, i.e. noclogging occurs (a class α system).bubble enters the narrow channel, see position (c). However,as the left bubble edge approaches the narrow channel, theenergy drop picks up again, see position (d). Finally,the bubble moves completely inside the narrow section andthe energy becomes zero (per definition), see position (e).In figure 6(b) the corresponding balancing external force Ffrom equation (6), and the clogging pressure across the bubble−P b , equation (16), are plotted as functions of x cm . Thebalancing external force is seen to be negative, which meansthat to maintain the bubble at quasi-static equilibrium, it isnecessary to hold it back. Without this force, the bubble wouldof course, as mentioned above, move spontaneously towardthe narrow segment. At position (c) where the right edge ofthe bubble enters the narrow channel, both force and pressurereach local maxima, but even here they are both negative. Noclogging occurs in this system, and it therefore belongs to classα as defined in section 4.5.3. A specific channel with clogging, class β 4The second example is nearly the same as the first. Onlythe length of the contraction region has been reduced from350 µm to 180 µm. This leads to an increase of the taperingangle from 10 ◦ to θ t = 20 ◦ . In figure 7 four positions (a)–(d)of the large bubble are depicted. Note that since θ + θ t = 92 ◦ ,6−∆Pb (Pa)(c)(d)100 x CM=380.830−100100 x CM=500.260−1000 200 400 600 800Figure 7. Four positions (a)–(d) of a bubble with γ = 1.02 inside a1000 µm long hydrophilic channel with the tapering angle θ t = 20 ◦ .The black dots indicate x cm . The contact angle is θ = 72 ◦ .the right meniscus in the tapered section of the channel inpanel (a) is nearly flat. In fact it has a slight inward bend.In figure 8(a) the energy is shown as a function of x cm .The four positions (a)–(d) are also marked. We immediatelynote a qualitative difference between this graph and that infigure 6(a). The energy no longer drops monotonically butexhibits a marked increase between positions (b) and (c). Thiscorresponds to the case where the bubble spans the entirecontraction, i.e. the left bubble edge is still in the wide channelsegment when the right edge enters the narrow segment.This effect is of course also visible in figure 8(b) where thebalancing external force F and the clogging pressure −P bare plotted as functions of x cm . Around position (b) both F and−P b become positive, which means that external pressureforces need to be applied to move the bubble through thesystem. Using equation (20) the clogging pressure is foundto be 173 Pa. Without this external force the bubble wouldtend to move backwards out of the channel, i.e. the systemis clogging, and in fact it is an example of class β 4 clogging.5.4. Clogging pressure versus tapering angle θ tThe previous two examples showed the behavior for aparticular channel contraction from R = 150 µm tor =95 µm with tapering angles θ t = 10 ◦ and 20 ◦ , respectively. Wenow extend this analysis to the entire interval 0 ◦

The clogging pressure of bubbles in hydrophilic microchannel contractions(a)Etot (nJ)321a •b •• c• dconfiguration where the bubble can span the entire contractionregion, i.e. class β 4 . We clearly see that once the bubble isable to span the entire contraction, the specific geometry ofthe contraction (in this case the tapering angle) plays no role.We get the same clogging pressure, P clog = 173 Pa.Finally, we note, that class β 3 behavior sets in for smalltapering angles below 9.5 ◦ . A small clogging pressure isobserved, less than 30 Pa.6. Conclusion(b)F (µN)0100 200 300 400 500 600x cm (µm)50-5•b• c • d-10←−◦◦−→-15-20• a 100 200 300 400 500 600x cm (µm)1000-100Figure 8. (a) A plot of the total internal energy E tot versus thecenter of mass coordinate x cm for the same bubble as in figure 7.(b) The balancing external force F and the pressure drop across thebubble P b versus x cm . The four dots correspond to the four bubblepositions in figure 7. Note that −P b > 0forx cm around 300 µm,i.e. clogging occurs (a class β system).Fmax (µN)1086420• B◦−→ ←−◦β 3 β 40.20.103α• A 19 10 1110 20 30 40 50 60θ t (deg.)516014012010080604020Figure 9. The maximal balancing force F max and clogging pressureP clog =−P b,max plotted as functions of the tapering angle θ t .Detailed figures illustrating the situation for θ t = 10 ◦ (marked A)are given in figures 6(a) and (b), and for θ t = 20 ◦ (marked B) infigures 8(a) and (b). The β 3 ,β 4 and α classes are described insection 4. The maximal clogging pressure occurs in class β 4 .Itisfound by equation (20) tobeP clog = 173 Pa.be identified. In this window P clog < 0 corresponding to theclogging-free class α behavior.For angles greater than about 11 ◦ the maximal force isseen to increase dramatically. This transition corresponds to a−∆Pb (Pa)−∆Pb,max (Pa)The effects of geometry on the quasi-static motion oflarge bubbles through a hydrophilic microchannel (capillary)contraction are modeled. The simplicity of the model leads toa good physical understanding of bubble clogging. We haveshown that in most cases it requires energy to move a bubblefrom a wide to a narrow channel. However, we have also foundthat certain bubble sizes and specific channel geometries leadto a gain in energy.We have specifically studied the contractions where suchan energy gain is achieved. Using the central equation forthe pressure drop P b across the bubble, equation (16),we analyzed a specific contracting axisymmetric hydrophilicchannel, and we identified four different classes, denoted as β 1to β 4 , leading to bubble clogging, and one clogging-free classdenoted as α. The details of the analysis are quite complicateddue to the large number of parameters: the tapering angle θ t ,the contact angle θ, the initial bubble length L 0 , the radii R andr of the wide and narrow channel segments, etc. However, onegeneral trend is clear. The tendency for clogging increases asthe bubbles become larger.Based on our analysis, some important design rules canbe established for making microchannel contractions withminimal or even vanishing clogging pressures. These rulesonly apply for channel contractions where the energy is lowestin the narrow part.First, if the typical size L 0 of the bubbles present inthe microfluidic system is known, it is important to designcontractions which are larger than L 0 . The highest cloggingpressures occur namely for bubbles spanning the entirecontraction, the so-called β 4 class.Second, the combined effect of the tapering angle andcontact angle has to be taken into account to make sure that L 0is shorter than the critical length leading to clogging of classβ 2 being entirely within the tapered region.Third (not presented here), to smoothen out and lowerany unavoidable positive clogging pressure, it helps to makethe curved parts of the contraction as large as possible, thusdecreasing their curvature.The method of analyzing the bubble clogging problemin microchannels presented in this paper is very general.It is straightforward to extend it to other geometries(such as nonmonotonic contractions) and to hydrophobicmicrochannels. With the presented design rules at hand itis possible to design a system that may filter or sort bubblesof different volumes—one simply places contractions withdifferent tapering angles in properly arranged series. Acomparable system designed to sort bubbles is presented in[17]. The model may also be extended to include wettinglayers as used in [16], and it may be used to model two phase7

MJJensenet alflows in porous media as in [22, 23]. Some of the dynamicaleffects such as those briefly mentioned in the introduction maybe included as well.AcknowledgmentsThis work is partly supported by the Danish TechnicalResearch Council, µTAS Frame Program Grant no 26-00-0220.References[1] Gravesen P, Branebjerg J and Søndergård Jensen O 1993 J.Micromech. Microeng. 3 168[2] Elwenspoek M, Kannerubj T S, Miyake R and Fluitman J H J1994 J. Micromech. Microeng. 4 227[3] Jensen M J, Goranović G and Bruus H 2003 Proc. NanoTech2003 (San Francisco, USA, Feb. 2003) vol 1 258–61[4] de Gennes P G 1985 Rev. Mod. Phys. 57 827[5] Brochard-Wyart F and de Gennes P G 1992 Adv. ColloidInterface Sci. 39 1[6] Stoev K, Ramé E, Leonhardt T and Garoff S 1998 Phys. Fluids10 1793[7] Raphaël E and de Gennes P G 1989 J. Chem. Phys. 90 7577[8] Dussan E B, Ramé V E and Garoff S 1991 J. Fluid Mech. 23097[9] de Gennes P G, Hua X and Levinson P 1990 J. Fluid Mech.212 55[10] Pismen L M and Rubinstein B Y 2001 Langmuir 17 5265[11] Tenan M A, Hackwood S and Beni G 1982 J. Appl. Phys. 536687[12] Parlange J-Y 1983 J. Appl. Phys. 54 6744[13] Schwartz A M, Rader C A and Huey E 1964 Adv. Chem. Ser.vol 43 ed R F Gould (Washington, DC: American ChemicalSociety) p 250[14] Bretherton F P 1961 J. Fluid Mech. 10 97[15] Ratulowski J and Chang H -C 1989 Phys. Fluids A 1 97[16] Bico J and Quéré D 2002 J. Fluid Mech. 467 101[17] Tordeux C and Fournier J -B 2002 Europhys. Lett. 60 875[18] Adamson A W and Gast A P 1997 Physical Chemistry ofSurfaces 6th edn (New York: Wiley)[19] Probstein R F 1994 Physiochemical Hydrodynamics: AnIntroduction 2nd edn (New York: Wiley)[20] Batchelor G K 2000 An Introduction to Fluid Dynamics(Cambridge: Cambridge University Press)[21] Jańczuk B, Bialopiotrowicz T and Zdziennicka A 1999 J.Colloid Interface Sci. 211 96[22] Zha F F, Fane A G, Fall C J D and Schofield R W 1992 J.Membr. Sci. 75 69[23] Bear J 1998 Dynamics of Fluids in Porous Media (New York:Dover)8

118 APPENDIX C. PAPER PUBLISHED IN J. MICROMECH. MICROENG.

Appendix DPaper published in Proc.microTAS 2002TitleDynamics of bubbles in microchannelsAuthorsMads Jakob Jensen, Goran Goranović, and Henrik BruusReferencemicroTAS 2002, Nara, Japan, November 2002, proc. vol. 2, p. 733-735119

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Appendix EPaper published in Proc.NanoTech 2003TitleElectroosmotic pumping of nonconducting liquids by viscous drag from a secondary conductingliquidAuthorsAnders Brask, Goran Goranović, and Henrik BruusReferenceNanoTech 2003, San Francisco, USA, February 2003, proc. vol. 1, p. 190-193123

Electroosmotic Pumping of Nonconducting Liquidsby Viscous Drag from a Secondary Conducting LiquidAnders Brask, Goran Goranović, and Henrik BruusMikroelektronik Centret (MIC), Technical University of Denmark (DTU)DK-2800 Lyngby, Denmarkabr@mic.dtu.dkABSTRACTA novel electroosmotic pump design relying on twoliquidviscous drag to pump nonconducting liquids ispresented and analyzed theoretically. We denoted thepump a two-liquid viscous EOF pump. A conductingpumping liquid driven by electroosmotic flow (EOF)drags a nonconducting working liquid by viscous forces.In particular liquids such as oil, which normally cannotbe moved by EOF, may be pumped. This allows fornew types of analysis in the field of micro Total AnalysisSystems (µTAS) which may prove important in the drugindustry and for environmental monitoring. The characteristicflow rate Q and pressure p of the pump is in therange nL/s and kPa, respectively, but depends largelyon achievable geometrical dimensions and applied voltage.This paper presents a theoretical and modelingstudy of the novel design.Keywords: electroosmotic pump, viscous drag, twophaseflow, circuit modeling, CFD simulation1 INTRODUCTIONElectroosmotic pumps are suitable for microfluidicapplications since (1) they do not have moving parts,(2) they do not produce pulsating flows, and (3) theyare easier to integrate. The liquids are pumped by applyingan electric field to the electric double layer whichforms in the liquid by Debye screening of the immobilecharges on the pump walls. In order for such a doublelayer to form, the liquid needs to have significant electricalconductivity i.e. a sufficiently high concentrationof dissociated ions. Nonpolar liquids with very low conductivity(< 10 −6 S/m), such as oil, cannot form thenecessary double layer and therefore cannot be pumpedin this way [1]. However, this problem is circumventedin our design by introducing a secondary conducting liquid.2 DESCRIPTIONThe aim is a to have a thin layer of conducting liquidsurrounding and dragging the nonconductive liquid. Aschematic view of the novel viscous pump is shown inFig. 1. To realize this we have developed three new featuresthat are essential for the design: pressure valves,reduced flow, and optimized potential drop.2.1 Pressure ValvesTwo pressure valves, the two sets of four narrowchannels shown in Fig. 1, prevent the working liquidfrom entering the electrode reservoirs. The electrodereservoirs are open and thus bubble formation from electrolysisis not a problem. Furthermore the pressurevalves allow for placing the pump anywhere in a fluidicnetwork, which is usually not possible. The pressurevalves have a width a, and the design utilizes thefact that the electroosmotic flow rate Q eo ∝ a 2 , whereasthe hydraulic flow rate Q hyd ∝ a 4 . Hence the pressuredriven flow will be effectively reduced for small valuesof a. Reactive ion etching systems can deliver narrowand deep channels with aspect ratio as high as 40. So ifa valve channel is 1 µm wide it can be 40 µm deep.2.2 Reduced FlowThe flow from the side channels must be reducedcompared to that of the main channel in order to enablepumping between Inlet A and Outlet B in Fig. 2. Thereduced flow may be obtained in two different ways.(1) A lower EOF mobility can be present in the pressurevalves in comparison to the main channel to optimizethe amount of the working liquid going throughthe pump, see Fig. 1.(2) A flow reduction can also be achieved by makingthe channel dimensions so small that the Debye layersoverlap. Typically the Debye layers are 1-100 nm wide.2.3 Electric Potential DropOne narrow valve channel have a large flow resistancebut also a large electrical resistance. This meansthat the main potential drop would occur in the valvechannels and thus not contribute to any pressure buildup. The electrical resistance is inversely proportional tothe area of the cross section. So by making many short,and narrow channels a low electrical resistance and highhydraulic resistance is obtained.

Top viewInlet BApplied potential VOutlet BValveDL2Not etchedLow EO mobilityi i i iHigh EO mobilityL o o o o1 2 3 4 4 1 2 3 4W3Inlet AL3WL 11 2Symmetry lineWOutlet AFigure 1: Layout of one half of a two-liquid viscous EOF pump. The pump is mirror-symmetric around the centralvertical plane, and only one half is shown. The working liquid A is driven by the pumping liquid, B. The pumpingliquid enters at inlet B, goes through the narrow valves, moves along the wall, and exits through the valve and finallythrough outlet B. The two electrode compartments each have four narrow channels of width W 3 . The channels/valvesensure that the working liquid does not enter inlet or outlet B. Two regions with different EO mobilities are identified.The ratio between the mobilities roughly governs the layer thickness of the pumping liquid. The high EO mobilityarea between the valves is the EO section. The valve regions should be given a coating with low EO mobility.Figure 2: The working liquid is shown in dark gray. It is being dragged by the EOF driven pumping liquid (shown inlight gray) that flows along the edge of the main channel. The pump is mirror-symmetric around the central verticalplane, and only one half is shown. The following specific parameters are chosen to predict the performance of thepump. The displayed micro-channels are all etched 40 µm down into the substrate (either siliconoxide or plastic), andafter etching the entire structure is closed with a lid. The main channel were the working liquid flows is 150 µm longand 10 µm wide. The narrow valve channels are 1 µm wide and 42 µm long. The uncoated walls are marked as thethick edges of the main channel. Coated walls are marked with thin edges (see also Fig. 1).

Inlet BR 1R 1Outlet BR 2R 2 R 2 R 2 R 2 R 2 R 2 R 2Inlet AR 3 R 3R 3R 4R 3R 3R 3Outlet AR 2 R 2 R 2 R 2 R 2 R 2 R 2 R 2Inlet BR 1R 1Outlet BFigure 3: The equivalent electric circuit of the two-liquidviscous EOF pump. Note that the whole pump/circuitis depicted. The overall resistance is calculated asR total =2R αβ + R 4 , where R αβ is the electrical resistancebetween node point α and β. R 4 is the resistanceof the EO section.2.4 Flow ProfilesThe resulting flow will be a superposition of an EOFand a pressure driven flow. Since the Debye layer isroughly 10 4 times smaller than the total width of thechannel, we do not resolve it in the following modelingof the pump. The EO velocity appears simply as a nonzerovelocity u eo at the walls. In the absence of anybackpressure the velocity would simply be constant u eoacross the channel. However the presence of pressuregradients induces the characteristic parabolic velocityprofiles in the laminar regime, Re ∼ 0.01. For suchlow Reynolds numbers the flow is said to be creeping,i.e., inertia can be neglected. Some velocity profiles areshown in Fig. 2. A characteristic feature for creepingflow is that it is free of vorticity. This means that thevalve channels may be positioned perpendicular to themain channel without generating any eddies. For a moredetailed discussion see Ref. [2].3 RESULTSWe have analyzed the performance of the pump usingtwo methods. The pressure and flow rates have been calculatedby equivalent circuit theory, and in addition, aset of computational fluid dynamics (CFD) simulationswas carried yielding more detailed information aboutthe distribution of pressures and velocities. The twomethods, which yield the same results within a few percent,were applied to the cases of both immiscible andmiscible liquids.3.1 Equivalent Circuit ModelThe aim is to establish a model that can predict theQ-p characteristic. In this respect, the pumping layerthickness, D layer , is important. The EOF depends onthe electric field, which depends on the flow in the caseof different conductivities. A rigorous model of the two-Figure 4: The velocity field in the pump near inlet Bcalculated by use of CFD. The velocity vectors are positionedin the symmetry plane. The transition frompressure driven flow to EOF is clear: in the regions withlow EO mobility, the flow profile is a positive parabolic,but after the last junction, inlet i 4 , the EOF drags theliquid, which results in a negative parabolic profile.liquid pump is therefore a complex matter. In this contextonly an outline will be presented.The first step is to find the effective potential dropacross the EO section. The procedure is to analyze theequivalent electric circuit shown in Fig. 3. The result isthat 52% of the applied voltage is dropped over the EOsection, R 4 in Fig. 3.The next step is to find the hydraulic resistance R hydof each of the channel segments. The channel cross sectionsare all rectangular with width W and height H.The channel length is denoted L.{R hyd =12µL H 3 W − 192π 5 H4 ×∞∑[ (2m +1)πW] } −1(2m +1) −5 tanh, (1)2Hm=0where µ is the dynamic viscosity. R hyd is calculated usingEq. (1), remembering that there are four channelsin the valves. For the present geometry, the overall hydraulicresistance of the valves is 26 times larger thanthat of the EO section, implying that the unintendedcross flow is small.The electroosmotic pressure buildup ∆p eo is given by∆p eo = α eo V eff Rhydeo HWL , (2)where α eo is the electroosmotic mobility, V eff the effectivepotential drop, and Rhyd eo the hydraulic resistance ofthe EO section. In Eq. (2) the Debye layer is assumedinfinity thin.3.2 Computational Fluid DynamicsTo obtain more detailed information a more advancedtool such as CFD is advantageous. The presented simulationsare made with Coventor 2001.3. The program

Figure 5: Visualization of the pressure distribution.The floating figure above the pump also displays thepressure. The float shows that the pressure varieslinearly from junction to junction, implying uniformflow. There are no external pressure difference. Sothe pressure gradients originate from internal frictionin the pump. Parameters: dimension as in Fig. 2,αeolow = 5000 µm 2 (V s) −1 , αeo high = 50000 µm 2 (V s) −1 ,p in = p out =0,V =10V.solves the Laplace equation for the potential and theNavier-Stokes equation for the velocity field.In Fig. 4 the simulated velocity vectors are shownnear the inlets to the main channel i 3 and i 4 , see Fig. 1.If the pumping liquid is chosen to be water the EOFmobility along uncoated walls is typically 0.05 mm 2 /(V s).In the valve channels the walls are coated to lower theEOF mobility by a factor 10. With these parameters numericalsimulations yield a maximal flow rate per volt of0.03 nL/(s V) and a backpressure capacity per volt of 3Pa/V. The value for the flow rate is specific for the givengeometry whereas the backpressure is independent ofthe length of the pump, refer Eq. (2). Visualizations ofthe pressure distribution and the streamlines are shownin Figs. 5 and 6 respectively.4 CONCLUSIONThe Q-p characteristic of the two-liquid viscous EOFpump largely depends on the geometrical factors andcan be significantly enhanced by advanced etching techniques.The pump still works for miscible liquids, buthere the working liquid gets mixed with the pumpingliquid.With these parameters numerical simulations yield amaximal flow rate per volt of 0.03 nL/(s V) and a backpressurecapacity per volt of 3 Pa/V. These values arequite small and the pump is therefore suited for preciseflow manipulation rather than pumping bulk volumes.Future work involves time dependent two-phase simulations.Such work could perhaps give valuable in-Figure 6: Streamlines at the anode junctions. Eventhough the channels are orthogonal, the lamination isclear. The Reynolds number is very small Re ∼ 0.01,thus indicating creeping flow. In the case of immiscibletwo-liquid flow, the expected pumping layer thicknesswould be about 20% of the half-width. Please note thatthe figure is composed of three figures, so the density ofthe streamlines is not meaningful. The parameters areas in Fig. 5.formation about stability and position of the pumpinglayer, and how to make the initial filling of the pump.Finally a prototype should be manufactured. Becauseof the possibility to pump all types of liquids in a preciseand controlled manner, the described concept seemsvery promising.5 ACKNOWLEDGEMENTSThis work is partly supported by the Danish TechnicalResearch Council, µTAS Frame Program Grant No.9901288.REFERENCES[1] C.H. Chen, S. Zeng, J. C. Mikkelsen, and J. G. Santiago,”Development of planar electrokinetic micropump”,ASME International Mechanical EngineeringCongress and Exposition, Orlando, FL,Nov. 7, 2000.[2] N. A. Patankar and H. H. Hu, Analytical Chemistry,70, pp. 1870-1881, 1998.

128 APPENDIX E. PAPER PUBLISHED IN PROC. NANOTECH 2003

Chapter 1IntroductionMicrofluidics is fluid dynamics on the micrometer (10 −6 m) scale. A microfluidic systemhas at least one characteristic length of that order.The revival of microfluidics began in early ’90s with the attempts to miniaturizesystems for use in analytical chemistry. The terms lab-on-chip or µTAS (micro-Total-Analysis-Systems) emerged indicating the trend of shrinkage in bio/chemical sciences, aswell as the intention to make integrated systems capable of performing desirable analyses.The integration has, however, brought forward a whole spectrum of problems and theneed for a true multidisciplinary research involving process specialists, chemists, biologists,material scientists and physicists.In such a colorful community, it is not surprising that microfluidics is many timesregarded, erroneously, as a new field. The fundamentals of microfluidics have been laidin 19th century with works of Stokes, Rayleigh and others. The true novelty, however,lies in application prospects which are indeed wide, in particular within the fields ofenvironmental and medical monitoring.The accent in this report will be put on understanding how do electric fields coupleto hydrodynamics in microfluidic regime. Also, various aspects of two-fluid interfaces inmicrofluidic devices will be examined. Both fundamental (theory) and applied (simulation)angles will be shown. In combination, the two approaches lead to a good understandingof microfluidic systems.In Chap. 2 a descriptive overview of the features of microfluidic systems is given, afterwhich the fundamental flow equations and relevant dimensionless numbers follow.In Chap. 3 the microscopic theory of the electroosmotic flow is presented on a thorough,but introductory level. EOF is the driving principle behind electroosmotic pumps. Theadvanced topics of non-equilibrium thermodynamics are out of the scope of this report.In Chap. 4 examples are described of how two-fluid systems behave in microsystemsand can be advantageously applied: miscible streams in case of micro cell-sorters, bubblebehavior in microchannels, and particularly interesting viscous-drag between two immiscibleliquids.Chap. 5 gives a general, in depth analysis on the interaction between electric fields anddielectrics. The electric force is described in terms of the Maxwell stress tensor.1

Appendix FBook chapter published byWiley-VCH 2003TitleSimulations in microfluidcsAuthorsGoran Goranović, and Henrik BruusReferenceChap. 5 (38 pages, 21 figs., 25 refs.) inMicrosystem Engineering of Lab-on-a-Chip Devices.Eds. O. Geschke, H. Klank, and P. Telleman, (Wiley-VCH, New York, 2003)129

Microsystem Engineeringof Lab-on-a-Chip Devices

Bibliography[1] PIV measurements in a microfluidic 3D-sheathing structure with three-dimensionalflow behavior,Henning Klank, Goran Goranović, Jörg P. Kutter, Henrik Gjelstrup, Jess Michelsen,Carsten Westergaard,J. Micromech. Microeng. 12, 862-869 (2002).[2] Integrating advanced functionality in a microfabricated high-throughput flourescentactivatedcell sorter,A. Wolff, I.R. Perch-Nielsen, U.D. Larsen, P. Friis, G. Goranović, C.R. Poulsen, J.P.Kutter, P. Telleman,Lab. Chip 3, 22-27 (2003).[3] Theoretical analysis of the low-voltage cascade electroosmotic pump,Anders Brask, Goran Goranović, and Henrik Bruus,Sens. Actuators B Chem. 92, 127-132 (2003).[4] The clogging pressure of bubbles in hydrophilic microchannel contractions,Mads Jakob Jensen, Goran Goranović, and Henrik Bruus,J. Micromech. Microeng. 14 (2004).[5] Three-dimensional single step flow sheathing in micro cell sorters,Goran Goranović, Ivan R. Perch-Nielsen, Ulrik D. Larsen, Anders Wolff, Jörg P. Kutter,Pieter Telleman,Proc. MSM 2001, Hilton Head Island (SC) USA, March 2001, p. 242-245.[6] Computer modeling as a design tool for microfluidic structures,Ivan R. Perch-Nielsen, Goran Goranović, Henning Klank, Ulrik D. Larsen, AndersWolff, Jörg P. Kutter, Pieter Telleman,Proc. µTAS 2001, Monterey (CA) USA, October 2001, p. 629-630.[7] Characterization of flows in laser-machined polymeric microchannels,Goran Goranović, Henning Klank, Carsten Westergaard, Oliver Geschke, PieterTelleman, and Jörg P. Kutter,Proc. µTAS 2001, Monterey (CA) USA, October 2001.131

132 BIBLIOGRAPHY[8] Electroosmotically driven two-liquid viscous pump for nonconducting liquids,Anders Brask, Goran Goranović, and Henrik Bruus,Proc. µTAS 2002, Nara, Japan, November 2002, vol. 1, p. 45-47.[9] The low-voltage cascade EOF pump: comparing theory with published data,Anders Brask, Goran Goranović, and Henrik Bruus,Proc. µTAS 2002, Nara, Japan, November 2002, vol. 1, p. 79-81.[10] Dynamics of bubbles in microchannels,Mads Jakob Jensen, Goran Goranović, and Henrik Bruus,Proc. µTAS 2002, Nara, Japan, November 2002, vol. 2, p. 733-735.[11] Electroosmotic pumping of nonconducting liquids by viscous drag from a secondaryconducting liquid,Anders Brask, Goran Goranović, and Henrik Bruus,Proc. NanoTech 2003, San Francisco (CA) USA, March 2003, vol. 1, p. 190-193.[12] Quasi-static motion of bubbles in microchannel contractions,Mads Jakob Jensen, Goran Goranović, and Henrik Bruus,Proc. NanoTech 2003, San Francisco (CA) USA, March 2003, vol. 1, p. 258-261.[13] Simulations in microfluidcsGoran Goranović and Henrik Bruus,Chap. 5 (38 pages, 21 figs., 25 refs.) in Microsystem Engineering of Lab-on-a-ChipDevices. Eds. O. Geschke, H. Klank, and P. Telleman, (Wiley-VCH, New York,autumn 2003)[14] Fluid Mechanics 2nd edition Course in Theoretical Physics, Volume 6L. D. Landau and E. M. LifshitzInstitue of Physical Problems, USSR Academy of Sciences, Butterworth-Heinemann,2000[15] Classical Electrodynamics Second EditionJ. D. JacksonJohn Wiley and Sons, USA, 1975[16] Conditions for Similitude between the Fluid Velocity and Electric Field in ElectroosmoticFlowE.B. Cummings, S. K. Griffiths, R. H. Nilson, P. H. PaulAnalytical Chemistry, Vol. 72 No. 11, June 1, (2000)[17] Principles of electroosmotic pumpsA. BraskM.Sc. thesis, MIC, DTU, 2002[18] Physicochemical Hydrodynamics, An Introduction 2nd. EditionR. F. ProbsteinJohn Wiley and Sons, Massachusetts Institute of Technology, 1994

BIBLIOGRAPHY 133[19] J. F. Osterle, J. Appl. Mech. 31, 161 (1964).[20] C.L. Rice and R. Whitehead, J. Phys. Chem. 69, 4017 (1965).[21] A. Manz, C. S. Effenhauser, N. Burggraf, D. J. Harrisson, K. Seiler, and K. Fluri,J. Micromech. Microeng. 4, 257 (1994).[22] P. H. Paul, D.W. Arnold, D. J. Rakestraw, proc. µTAS 1998, Banff, Canada, p. 49(1998).[23] Y. Takamura, H. Onoda, H. Inokuchi, S. Adachi, A. Oki, and Y. Horiike, proc. µTAS2001, Monterey CA, USA, p. 230, J. M. Ramsey and A. van den Berg (eds.), KluwerAcademic Publishers (2001).[24] W.E. Morf, O.T. Guenat and N.F. de Rooij, Sensor. Actuat. B-Chem. 72, 266 (2001).[25] O.T. Guenat, D. Ghiglione, W.E. Morf, and N.F. de Rooij, Sensor. Actuat. B-Chem.72, 273 (2001).[26] S. Zeng, C.-H. Chen, J. C. Mikkelsen Jr., and J. G. Santiago, Sensor. Actuat. B-Chem. 79, 107 (2001).[27] D. R. Reyes, D. Lossifidis, P.-A. Auroux, and A. Manz, Anal. Chem. 74, 2623 (2002).[28] Experimental parameters not quoted in the original experimental paper were kindlyprovided by dr. Y. Takamura, Univ. Tokyo.[29] R.-J. Yang, L.-M. Fu, and Y.-C. Lin J. Colloid Interf. Sci. 239, 98 (2001).[30] N. A. Patankar and H. H. Hu, Anal. Chem. 70, 1870 (1998).[31] Zeta Potential of Highly Charged Thin Double-Layer Systems R. J. Hunter., B. R.Midmore and H. Zhang Journal of Colloid and Interface Science 237, p. 147-149,2001.[32] Physical Chemistry of SurfacesA. W. Adamson and A. P. GastJohn Wiley and Sons, Sixth Edition, 1997[33] Micro tools for cell handlingP. Telleman, et al.SPIE conference on microfluidic devices and systems, Santa Clara, USA, Sep. 2000,[34] Bubble PuzzlesD. LohsePhysics Today, February 2003[35] Microfluidcs - a reviewP. Gravesen, J. Branebjerg and O. S. Jensen J. Micromech. Microeng. 3, 1993.

134 BIBLIOGRAPHY[36] Transport of Gas Bubbles in Capillaries J. Ratulowski and H.-C. ChangPhysics of Fluids A 1 (10), American Institute of Physics, October 1989.[37] Fluid dynamics for physicistsT. E. FaberCambridge University Press, Cambridge, UK, 2001[38] Electrodynamics of continuous media Course in Theoretical Physics, Volume 8L. D. Landau and E. M. LifshitzInstitue of Physical Problems, USSR Academy of Sciences, Pergamon Press, 1960[39] Electromagnetic TheoryJ. A. StrattonMcGraw-Hill, USA, 1941[40] Statistical Physics 3rd Edition Part 1L. D. Landau and E. M. LifshitzInstitue of Physical Problems, USSR Academy of Sciences, Butterworth-Heinemann,2001[41] Vectors, Tensors and the Basic Equations of Fluid MechanicsR. ArisPrentice-Hall International, London, 1962[42] Field-coupled surface waves: a comparative study of surface-coupled electrohydrodynamicand magnetohydrodynamic wavesJ. R. MelcherM.I.T. Press, Cambridge, Massachusetts, 1963[43] Electrohydrodynamic and Magnetohydrodynamic Surface Waves and InstabilitiesJ. R. MelcherPhys. of Fluids, 4, 11 p. 1348-1354 (1961)[44] Introduction to Solid State PhysicsC. KittelJohn Willey and Sons, 6ed, 1986[45] Electromagnetic fields and energyH. A. Haus, J. R. MelcherPrentice Hall, Englewood Cliffs, New Jersey, 1989[46] Electrohydrodynamics: The Taylor-Melcher Leaky Dielectric ModelD. A. SavilleAnnu. Rev. Fluid Mech., 29 p. 27-64 (1997)[47] The Theory of Sound Vol.2Lord RayleighDover New York, 1945

BIBLIOGRAPHY 135[48] Disintegration of water drops in an electric fieldG. I. TaylorProc. Roy. Soc. A 331 p.263-72 1964[49] Hydrodynamic and Hydromagnetic StabilityS. ChandasekharOxford University Press, 1961[50] Hydrodynamic stabilityP.G. Drazin and W. H. ReidCambridge University Press, 1991[51] On the equilibrium of liquid conducting masses charged with electricityLord RayleighPhil. Mag. Ser. 5 14 p. 184-86, 1882[52] Electrohydrodynamics: a review of the role of interfacial shear stressesJ. R. Melcher and G. I. TaylorAnnu. rev. Fluid Mech. 1 p.111-46 1969[53] An active micro mixer using electrohydrodynamic (EHD) convectionJ. W. Choi, Chong H. AhnSolid State Sensor and Actuator Workshop, Hilton Head Island, S. Carolina, 2000[54] Electrohydrodynamic Stability of Two Superposed Viscous FluidsA. A. Mohammed, E. F. Elshehawey and M. F. El-SayedJ. Coll. Interf. Sci., 169 p. 65-78 (1995)[55] Bubbles in microchannelsM. J. JensenM.Sc. thesis, MIC, DTU, 2002[56] Blood flow in microchannelsL. BitschM.Sc. thesis, MIC, DTU, 2002[57] Dispersion in electrokinetically and pressure driven microflowsF. R. HansenM.Sc. thesis, MIC, DTU, 2002[58] Micro flows Fundamentals and SimulationG. E. Karniadakis and A. BeskokSpringer-Verlag, New York, 2002[59] Viscous fluid flow 2nd editionF. M. WhiteMcGraw-Hill, Singapore, 1991

136 BIBLIOGRAPHY[60] An Introduction to Capillary Electrochromatography D. B. Keith, M. G. Cikalo andM. M. Robson