Title Details

Kenneth Brakke is Professor of Mathematics and Computer Science at Susquehanna University in Pennsylvania. He received his PhD in Mathematics from Princeton University, in the field of Geometric Measure Theory. Since 1988 he has written and maintained his freely-available Surface Evolver software, which shows computer models of liquid surfaces. He is the second author of the book The Physics of Microdroplets (Wiley-Scrivener 2012).

Erwin Berthier is the VP of R&D of Tasso, Inc, a biotechnology startup that is developing blood sample collection and analysis technologies based on open microfluidic concepts as well as an affiliate professor in the Department of Chemistry at the University of Washington. He has received a PhD in Biomedical Engineering at the University of Wisconsin-Madison where he became an expert in user-centered microfluidic technologies. His believes that technologies must be made simpler to disseminate, be widely adopted, and find killer applications.

Acknowledgements xi

Preface xiii

Online Materials xv

Introduction 1

1 Theory of Spontaneous Capillary Flows 13

1.1 Introduction 13

1.2 Quasi-static Approach to SCF 16

1.2.1 Open and Confined Systems 17

1.2.2 Theoretical Approach 17

1.2.3 Numerical Approach 21

1.2.3.1 Numerical Verification of the Capillary Force 22

1.2.3.2 Composite Confined Channel 22

1.2.3.3 Composite Open Channel 23

1.2.3.4 Fiber Bundle 24

1.2.3.5 Usual Geometries 27

1.2.3.6 Conclusion 27

1.2.4 Dynamic Aspects 27

1.2.4.1 Generalization of the Lucas-Washburn-Rideal Law to Composite, Confined Microchannels of Arbitrary Cross-section 30

1.2.4.2 Theory 30

1.2.4.3 Magnitude of Capillary Velocities 36

1.2.4.4 Experimental Results for Confined Channels 38

1.2.4.5 Conclusion 39

1.3 The Dynamics of Spontaneous Capillary Flows in Open-surface Channels 40

1.3.1 The Dynamics of SCF 40

1.3.2 Confined Rectangular Channels 42

1.3.3 Open Rectangular U-grooves 44

1.3.4 Suspended Rectangular Channels 45

1.3.5 Experiments 46

1.3.6 Comparison 46

1.4 Dynamic Contact Angle 49

1.5 Conclusion 53

1.6 References 53

2 Capillary Filaments 57

2.1 Introduction 57

2.2 Concus-Finn Theory 57

2.2.1 Numerical Approach 60

2.2.2 Example of Capillary Filaments in a Micro-beaker 60

2.2.3 Example of a Capillary Filament in a Micro Petri Dish 60

2.2.4 Extended Concus-Finn Relation 62

2.2.5 Capillary Filaments in a Non-ideal Corner 63

2.3 Capillary Filaments in Rectangular U-grooves 65

2.3.1 Capillary Flow Regimes with No Capillary Filaments (> 45°) 66

2.3.2 Capillary Flow Regimes with Capillary Filaments (<45°) 66

2.3.2.1 SCF Self-dividing into Filaments 67

2.3.2.2 Initially Separated Concus-Finn Filaments 69

2.3.2.3 Metastability of CF Filaments 70

2.3.2.4 Discussion 72

2.3.2.5 Imperfect Grooves 73

2.3.3 Example of a Varying Cross-sectional Area Channel 73

2.4 Capillary Filaments in V-grooves 74

2.4.1 Perfect V-grooves 74

2.4.2 Imperfect V-grooves 75

2.4.3 Parallel V-grooves 77

2.4.4 Imperfect Groovy Surface 79

2.5 Examples of Capillary Filaments 81

2.5.1 Capillary Filling of PCR Devices 82

2.5.2 Whole Blood Capillary Flow in V-grooves 82

2.6 Conclusions 85

2.7 References 86

Appendix 2.1 Capillary Flow in a Cylindrical Cavity 88

3 Spontaneous Capillary Flows in Open U-grooves 91

3.1 Introduction: SCF in Open “U-grooves” 91

3.2 Quasi-static Approach 92

3.3 Bulk SCF in Uniform Cross-section U-grooves 93

3.3.1 Single Wall Wettability 93

3.3.1.1 Theoretical Approach 93

3.3.1.2 Evolver Numerical Approach 97

3.3.2 Composite Walls 97

3.3.2.1 Rectangular Open Channel 98

3.3.2.2 Trapezoidal Open Channel 99

3.3.2.3 Roll-embossed Channel 100

3.4 Slightly Pressurized Open-surface Capillary Flow 100

3.5 SCF in Winding Channels 102

3.5.1 SCF in Winding, Open Channels, > 45° 103

3.5.2 Concus-Finn Filaments in Sharp Curves, > 45° 103

3.6 Extrapolation to the Coiling of the Flow Around a Curved Corner 104

3.7 Converging U-channels 105

3.8 Diverging U-channels 105

3.8.1 No CF Filaments 106

3.8.2 CF Filaments 108

3.9 U-groove with a Sudden Enlargement 108

3.9.1 Smooth Enlargement 109

3.9.2 Enlargement with Sharp Edges 110

3.9.3 U-groove Exiting into a Cylinder 112

3.9.4 U-groove Crossing a Polygonal Cavity 113

3.10 Open Capillary Valves 114

3.10.1 Capillary Stop Valves 114

3.10.2 Trigger Valves 115

3.11 Bifurcation 116

3.12 Capillary Filtration 118

3.13 Capillary Flow Mixing 119

3.14 Generalization: Substrate Patterned with Parallel Rectangular U-grooves 119

3.14.1 Substrate Patterned with U-grooves 119

3.14.2 Open, Rectangular U-groove with Sub-grooves in the Bottom Plate 120

3.14.3 Applications 121

3.15 Conclusion 121

3.16 References 122

4 Dynamics of Capillary Flow in a Channel with Constrictions and Enlargements 125

4.1 Introduction 125

4.2 Channel Constriction and Enlargement 126

4.2.1 Theory 126

4.2.2 Numerical Results and Discussion 130

4.2.2.1 Straight Channel 131

4.2.2.2 Channel with a Constricted Section 131

4.2.2.3 Channel with an Enlarged Section 132

4.2.3 Experimental Results 134

4.2.3.1 Constriction 135

4.2.3.2 Enlargement 136

4.2.4 Conclusion 137

4.3 SCF in a U-groove with Multiple Change of Cross-section 137

4.3.1 Theoretical Approach 138

4.3.2 Experimental Approach 140

4.3.2.1 Winding Open Rectangular U-groove 140

4.3.2.2 Open Rectangular U-groove with Constricted Sections 141

4.3.2.3 Open Rectangular U-groove with Cylindrical Chambers 144

4.3.3 Comparison with the Numerical Approach 145

4.4 Conclusion 146

4.5 References 149

Appendix 4.1 Velocity Model for Open Rectangular Channels 150

Appendix 4.2 Velocity Model for Cylindrical Tubes 152

Appendix 4.3 Friction in a Rectangular Open Channel 155

5 Suspended Capillary Flows 157

5.1 Introduction 157

5.2 Theory 158

5.3 Quasi-static Numerical Approach 159

5.3.1 Effect of Gravity 162

5.4 Dynamic Approach 162

5.4.1 Closed-form Expression of the Velocity for Newtonian Fluids 162

5.4.2 Channel Characteristics Corresponding to Maximum Velocities 164

5.4.3 Examples from Experiments 166

5.4.3.1 Suspended Channel Fabrication 167

5.4.3.2 Preparation of the Solutions and Liquid Characterization 168

5.4.3.3 Tinted Water 168

5.4.3.4 IPA Solutions 169

5.4.3.5 Whole Blood 169

5.4.3.6 Alginate Solutions 171

5.5 Comparison of a U-channel and a Suspended Channel 174

5.6 Suspended Microfluidics in Channels of Varying Section 175

5.6.1 Diverging Straight Walls 175

5.6.2 Sudden Enlargement of Suspended Channels 179

5.6.2.1 Quasi-static Approach 179

5.6.2.2 Dynamic Approach 183

5.6.3 Converging Suspended Channels 183

5.6.4 X-shape Suspended Channels 184

5.7 Capillary Flow in a Suspended Tapering Channel 186

5.8 Suspended Microfluidics in Suspended V-shaped Channels 188

5.9 Capillary Flow Over a Hole 189

5.10 Introduction to Two-phase Suspended Microflows 191

5.10.1 Parallel Walls 194

5.10.2 Tapered Walls 197

5.10.2.1 Converging Channel 197

5.10.2.2 Diverging Channel 198

5.10.3 Examples and Applications of Suspended Microfluidics 199

5.10.3.1 Formation of μDots 199

5.10.3.2 Towards a Giant Polymeric Micromembrane 201

5.10.3.3 Suspended Microfluidics for Measurement of Contact Angles 201

5.11 Conclusion 203

5.12 References 203

6 Spontaneous Capillary Flow Between Horizontal Rails 207

6.1 Introduction 207

6.2 Spontaneous Capillary Flows Between Rails 209

6.3 Winding Channels 210

6.4 Diverging Rails 211

6.5 Rails with Lateral Enlargement 212

6.6 Converging Rails 212

6.7 Rails with Constriction 212

6.8 Stopping a Capillary Flow at a Neck 213

6.9 SCF in Sinusoidal Railed Channels 215

6.10 Divisions and Bifurcations 217

6.10.1 Flow Separation 217

6.10.2 Flow Around a Hole 217

6.10.2.1 Two Plates Pierced by a Hole 218

6.10.2.2 Bottom Plate Pierced by a Hole 221

6.10.2.3 Rails Around a Hole 221

6.10.3 Capillary Flow Around Pillars 224

6.10.3.1 Single Pillar 224

6.10.3.2 Multiple Pillars 225

6.11 Conclusion 227

6.12 References 227

7 Paper-based Microfluidics 229

7.1 Introduction 229

7.2 Principles of Labs-on-Paper and Paper-based Devices 230

7.3 Paper-based Microfluidics 231

7.3.1 Spontaneous Imbibition-wicking 231

7.3.2 Fully Wetted Medium – Darcy’s law 234

7.3.3 Velocity in Paper Strips of Piecewise Varying Width 236

7.3.4 Filtration and Separation 237

7.3.5 Mixing 238

7.3.6 Y-junctions 240

7.3.7 Hydrodynamic Focusing 241

7.3.8 H-filters: Separation and Extraction 242

7.3.9 Valves 243

7.3.10 Architecture for Time Sequencing 244

7.3.11 3D paths – Fluidic Origamis 244

7.3.12 Electrokinetics on Paper 244

7.4 Paper-based Systems Fabrication and Detection 245

7.4.1 Fabrication Techniques of Paper Strips 246

7.4.2 Fabrication Techniques of μPADs 247

7.4.2.1 Hydrophobic Barrier 247

7.4.2.2 Hydrophobization of the Substrate 247

7.4.3 Functionalization and Loading of Reagents 249

7.4.4 Detection 249

7.4.4.1 Colorimetry 249

7.4.4.2 Electrochemistry(EC) 250

7.4.4.3 Chemiluminescence 251

8 Fiber-based Microfluidics 257

8.1 Introduction 257

8.2 Droplet on Fibers 259

8.2.1 Droplet on a Horizontal Fiber 259

8.2.2 Small Droplet 260

8.2.2.1 Effect of Gravity on Small Droplets 261

8.2.2.2 Large Droplet 261

8.2.3 Droplet Between Fibers 263

8.2.3.1 Droplet Between Two Parallel Fibers 263

8.2.3.2 Non-parallel Fibers in the Same Plane 264

8.2.3.3 Drop Between Two Fibers – General Case 265

8.2.3.4 Droplet Sliding Down a Fiber 266

8.3 SCF Guided by Fibers 268

8.3.1 Approximate General Condition for Spontaneous Capillary Flow in a Fiber Bundle 268

8.3.2 Geometrical Study: SCF Guided by Fibers 270

8.3.2.1 Homogeneous Bundle 271

8.3.2.2 Inhomogeneous Bundles 273

8.3.2.3 Numerical Example 279

8.3.2.4 Packed Bundle 281

8.3.2.5 Generalization to Large Bundles 282

8.3.2.6 Influence of the Parameter C=R 282

8.3.2.7 Conclusion 282

8.4 Examples of Microfluidics on Fibers 284

8.5 Electrochemical Detection on Fibers 284

8.6 Applications in Biology 285

8.6.1 Blood Typing Diagnostics 285

8.6.2 Woven Fibers 286

8.6.3 Smart Bandages 286

8.6.4 Smart Textiles 288

8.7 Capillary Rise in Fibers 288

8.7.1 Cylindrical Tubes: Jurin’s law 288

8.7.2 Capillary Rise Between Pillars 291

8.7.2.1 Capillary Rise in a Bundle of Four Vertical Square Pillars 291

8.7.2.2 Comparison of Capillary Rise Between a Wilhelmy Plate and Pillars 292

8.7.2.3 Comparison of Capillary Rise Between a Single Rod and a Bundle of Packed Rods 294

8.8 Conclusions 295

8.9 References 296

Appendix 8.1 Calculation of the Laplace Pressure for a Droplet on a

Horizontal Cylindrical Wire 298

Appendix 8.2 Perimeters 299

Appendix 8.3 Wonky Corners SCF 300

Appendix 8.4 Transition Between “All Wetted” and “All But Corners” Cases 301

9 Epilog 303

9.1 Open Microfluidics 303

9.2 References 305

Index 307