Lecture notes second edition, fall 2005 Theoretical microfluidics Henrik Bruus + + − + − + MIC { Department of Micro and Nanotechnology Technical University of Denmark ii Preface In the fall 2003 MIC launched a new ¯fth semester course at the Technical University of Denmark (course no. 33241, 5 ECTS) to provide a general and broad introduction to theoretical aspects of the new ¯eld of lab-on-a-chip systems. In the ¯rst run of the course I tried to use existing books as basic material. However, it soon became clear that these books did not cover the material I wanted to teach. I let more and more of the teaching rely on substantial exercises, many of which were based on experimental problems from the laboratories at MIC. These exercises form the basis of the lecture notes at hand. The notes are being written during the course from September to December 2004. The ¯rst chapter is ready for the ¯rst lecture, while the rest will follow at a rate of one chapter per week. I hope that the students will bear over with the many printing mistakes and less than optimal formulations that undoubtedly will appear, and that they will participate actively in the e®orts to create new and up-to-date teaching material at the right level of di±culty. Hopefully, these lecture notes will be both inspiring and challenging. Henrik Bruus MIC { Department of Micro and Nanotechnology Technical University of Denmark 30 August 2004 This second edition of the lecture notes has bene¯ted from numerous corrections and com- ments from my students and colleagues. Moreover, three new chapters and two appendices have been added. I hope that the notes appear even more useful in their present form. Henrik Bruus MIC { Department of Micro and Nanotechnology Technical University of Denmark 28 August 2005 iii iv PREFACE Contents 1 Basic concepts in microfluidics 1 1.1 Fluids and ¯elds . 2 1.1.1 Fluids: liquids and gases . 3 1.1.2 The continuum hypothesis and fluid particles . 4 1.1.3 The velocity, pressure and density ¯eld . 5 1.2 SI units and mathematical notation . 6 1.2.1 SI units . 6 1.2.2 Vectors, derivatives and the index notation . 7 1.3 The continuity equation . 9 1.3.1 Compressible fluids . 10 1.3.2 Incompressible fluids . 11 1.4 The Navier{Stokes equation . 11 1.4.1 The material time-derivative . 12 1.4.2 Body forces . 12 1.4.3 The pressure-gradient force . 12 1.4.4 The viscous force and the viscous stress tensor . 13 1.4.5 The Navier{Stokes equation for compressible fluids . 13 1.4.6 The Navier{Stokes equation for incompressible fluids . 14 1.5 Exercises . 14 2 Analytical Navier{Stokes solutions 17 2.1 Fluids in mechanical equilibrium . 17 2.2 Liquid ¯lm flow on an inclined plane . 19 2.3 Couette flow . 20 2.4 Poiseuille flow . 21 2.4.1 Arbitrary cross-sectional shape . 22 2.4.2 Elliptic cross-section . 23 2.4.3 Circular cross-section . 25 2.4.4 Equilateral triangular cross-section . 26 2.4.5 Rectangular cross-section . 26 2.4.6 In¯nite parallel-plate channel . 30 2.5 Shape perturbation in Poiseuille flow problems . 30 2.6 Stokes drag on a sphere moving in steady-state . 32 v vi CONTENTS 2.7 Exercises . 34 3 Hydraulic resistance 37 3.1 Viscous dissipation of energy for incompressible fluids . 37 3.2 Viscous dissipation of energy in steady-state . 39 3.3 Hydraulic resistance of some straight channels . 41 3.4 Shape dependence of hydraulic resistance . 42 3.4.1 The geometrical correction factor versus compactness . 43 3.4.2 Summarizing remarks . 45 3.5 The dimensionless Reynolds number . 45 3.5.1 Reynolds number for systems with only one length scale . 46 3.5.2 Reynolds number for systems with two length scales . 47 3.6 Hydraulic resistance, two connected straight channels . 49 3.6.1 Two straight channels in series . 49 3.6.2 Two straight channels in parallel . 50 3.7 Equivalent circuit theory . 50 3.8 Exercises . 52 4 Time-dependent phenomena 55 4.1 A random walk model of di®usion . 55 4.2 The convection-di®usion equation for solutions . 57 4.3 The di®usion equation . 59 4.3.1 Limited point-source di®usion . 59 4.3.2 Limited planar-source di®usion . 60 4.3.3 Constant planar-source di®usion . 60 4.3.4 Di®usion of momentum and the Navier{Stokes equation . 61 4.4 Taylor dispersion; a convection-di®usion example . 62 4.4.1 Dimensional analysis and the P¶ecletnumber . 62 4.4.2 Taylor's model for dispersion in microfluidic channels . 63 4.4.3 The solution to the Taylor dispersion problem . 65 4.5 Initiating a Poiseuille flow; circular cross section . 65 4.6 Accelerated motion of a spherical body in a liquid . 68 4.7 Exercises . 69 5 Capillary e®ects 73 5.1 Surface tension . 73 5.1.1 De¯nition of surface tension . 74 5.1.2 The Young{Laplace pressure across curved interfaces . 75 5.2 Contact angle . 76 5.2.1 De¯nition of the contact angle . 76 5.2.2 Young's equation; surface tensions and contact angle . 77 5.3 Capillary rise . 78 5.3.1 Capillary rise height . 79 5.3.2 Capillary rise time . 80 CONTENTS vii 5.3.3 Capillary rise and dimensionless numbers . 81 5.4 Capillary pumps . 81 5.4.1 Capillary pump advancement times . 82 5.4.2 A bio-sensor chip with a capillary-force pump . 83 5.5 Marangoni e®ect; surface tension gradients . 84 5.6 Exercises . 84 6 Numerical simulations 87 6.1 The ¯nite element method (FEM) . 88 6.1.1 Discretization using ¯nite elements . 88 6.1.2 Weak solutions . 89 6.1.3 The Galerkin method . 90 6.1.4 The Navier{Stokes equation in FEM . 90 6.2 A short introduction to FemLab . 92 6.2.1 The structure of problem-solving in FemLab . 92 6.2.2 Solving a problem using the FemLab graphical user interface . 92 6.3 Some FemLab scripts for microfluidics . 93 6.3.1 Incompressible flow in a backstep geometry . 94 6.3.2 Multipolar deformations and Poiseuille flows . 95 6.4 Exercises . 97 7 Electrohydrodynamics 101 7.1 Polarization and dipole moments . 102 7.2 Electrokinetic e®ects . 103 7.2.1 Electrophoresis . 104 7.2.2 Ionic mobility and conductivity . 105 7.3 The Debye layer near charged surfaces . 105 7.3.1 The continuum model of the Debye layer . 106 7.3.2 The Debye{HÄuckel approximation for the Debye layer . 108 7.3.3 Surface charge and the Debye layer capacitance . 109 7.3.4 Electrophoresis and Debye layer screening . 111 7.4 Exercises . 112 8 Electroosmosis 115 8.1 Electrohydrodynamic transport theory . 115 8.2 Ideal electroosmotic flow . 116 8.3 Debye layer overlap . 118 8.4 Ideal EO flow with back-pressure . 119 8.5 The many-channel EO pump . 122 8.6 The cascade EO pump . 123 8.7 Exercises . 126 viii CONTENTS 9 Dielectrophoresis 129 9.1 Induced polarization and dielectric forces; heuristically . 129 9.2 A point dipole in a dielectric fluid . 130 9.3 A dielectric sphere in a dielectric fluid; induced dipole . 131 9.4 The dielectrophoretic force on a dielectric sphere . 134 9.5 Dielectrophoretic trapping of particles in microfluidics . 134 9.6 The AC dielectrophoretic force on a dielectric sphere . 137 9.7 Exercises . 139 10 Magnetophoresis 143 10.1 Magnetophoresis and bioanalysis . 143 10.2 Magnetostatics . 145 10.3 Basic equations for magnetophoresis . 147 10.4 Magnetophoretic lab-on-a-chip systems . 148 10.5 Exercises . ..