The Physics of Microdroplets

The Physics of Microdroplets gives the reader the theoreticaland numerical tools to understand, explain, calculate, and predictthe often nonintuitive observed behavior of droplets inmicrosystems.

Microdrops and interfaces are now a common feature in mostfluidic microsystems, from biology, to biotechnology, materialsscience, 3D–microelectronics, optofluidics, and mechatronics. Onthe other hand, the behavior of droplets and interfaces in today′smicrosystems is complicated and involves complex 3D geometricalconsiderations. From a numerical standpoint, the treatment ofinterfaces separating different immiscible phases is difficult.

After a chapter dedicated to the general theory of wetting, thispractical book successively details:

The theory of 3D liquid interfaces

The formulas for volume and surface of sessile and pancakedroplets

The behavior of sessile droplets

The behavior of droplets between tapered plates and inwedges

The behavior of droplets in microchannels

The effect of capillarity with the analysis of capillaryrise

The onset of spontaneous capillary flow in open microfluidicsystems

The interaction between droplets, like engulfment

The theory and application of electrowetting

The state of the art for the approach of 3D–microelectronicsusing capillary alignment

Audience

The core market is broad including biotechnologists, biologists,bioengineers, biochemists, and materials scientists. In addition,engineers and scientists involved in 3D microelectronics,optofluidics, and mechatronics, will find much value in thisbook.

Acknowledgements xxi

Introduction 1

1. Fundamentals of Capillarity 5

1.1 Abstract 5

1.2 Interfaces and Surface Tension 5

1.3 Laplace s Law and Applications 12

1.4 Measuring the Surface Tension of Liquids 49

1.5 Minimization of the Surface Energy 61

1.6 References 62

2. Minimal Energy and Stability Rubrics 67

2.1 Abstract 67

2.2 Spherical Shapes as Energy Minimizers 68

2.3 Symmetrization and the Rouloids 73

2.4 Increasing Pressure and Stability 77

2.5 The Double–Bubble Instability 81

2.6 Conclusion 84

2.7 References 84

3. Droplets: Shape, Surface and Volume 85

3.1 Abstract 85

3.2 The Shape of Micro–drops 86

3.3 Electric Bonds Number 87

3.4 Shape, Surface Area and Volume of Sessile Droplets 87

3.5 Conclusion 105

  3.6 References 105

4. Sessile Droplets 107

4.1 Abstract 107

4.2 Droplet Self–motion Under the Effect of a Contrast orGradient of Wettability 107

4.3 Contact Angle Hysteresis 114

4.4 Pinning and Canthotaris 117

4.5 Sessile Droplet on a Non–ideally Planar Surface 124

4.6 Droplet on Textured or Patterned Substrates 125

4.7 References 142

5. Droplets Between Two Non–parallel Planes: from TaperedPlanes to Wedges 145

5.1 Abstract 145

5.2 Droplet Self–motion Between Two Non–parallel Planes 145

5.3 Droplet in a Corner 154

5.4 Conclusion 160

5.5 References 161

6. Microdrops in Microchannels and Microchambers 163

6.1 Abstract 163

6.2 Droplets in Micro–wells 163

6.3 Droplets in Microchannels 168

6.4 Conclusion 180

6.5 References 181

7. Capillary Effects: Capillary Rise, Capillary Pumping, andCapillary Valve 185

7.1 Abstract 185

7.2 Capillary Rise 185

7.3 Capillary Pumping 198

7.4 Capillary Valves 205

7.5 Conclusions 209

7.6 References 210

8. Open Microfluidics 213

8.1 Abstract 213

8.2 Droplet Pierced by a Wire 214

8.3 Liquid Spreading Between Solid Structures Spontaneous Capillary Flow 218

8.4 Liquid Wetting Fibers 241

8.5 Conclusions 250

8.6 References 250

8.7 Appendix: Calculation of the Laplace Pressure for a Dropleton a Horizontal Cylindrical Wire 251

9. Droplets, particles and Interfaces 253

9.1 Abstract 253

9.2 Neumann s Construction for liquid Droplets 253

9.3 The Difference Between Liquid Droplets and Rigid Spheres atan Interface 254

9.4 Liquid Droplet Deposited at a Liquid Surface 256

9.5 Immiscible Droplets in Contact and Engulfment 261

9.6 Non–deformable (Rigid) Sphere at an Interface 265

9.7 Droplet Evaporation and Capillary Assembly 278

9.8 Conclusion 291

9.9 References 292

10. Digital Microfluidics 295

10.1 Introduction 295

10.2 Electrowetting and EWOD 295

10.3 Droplet Manipulation with EWOD 306

10.4 Examples of EWOD in Biotechnology Cell Manipulation335

10.5 Examples of Electrowetting for Optics–Tunable Lenses andElectro fluidic Display 337

10.6 Conclusion 338

10.7 References 339

11. Capillary Self–assembly for 3D Microelectronics343

11.1 Abstract 343

11.2 Ideal Case: Total Pinning on the Chip and Pad Edges 344

11.3 Real Case: Spreading and Wetting 355

11.4 The Importance of Pinning and Confinement 358

11.5 Conclusion 359

11.6 Appendix A: Shift Energy and Restoring Force 360

11.7 Appendix B: Twist Energy and Restoring Torque 362

11.8 Appendix C: Lift Energy and Restoring Force 364

11.9 References 365

12. Epilogue 369

Index 369

Jean Berthier is a Scientist at the CEA/LETI and teaches at theUniversity of Grenoble, France. He is presently involved in thedevelopment of microdevices for liquid–liquid extraction (LLE),flow focusing devices (FFD) for bio–encapsulation of live cells,microfluidic resonators for high sensitivity biodetection andnumerical methods for the prediction of droplets and interfacesbehavior in microsystems. He is the first author of the bookMicrofluidics for Biotechnology published in 2005 with a secondedition in 2010. He is also the author of the book Microdrops andDigital Microfluidics, published in 2008.

Kenneth A. Brakke is Professor of Mathematics and ComputerScience at Susquehanna University in Pennsylvania. He received hisPhD in mathematics from Princeton University in the field ofgeometric measure theory. Since 1988, he has written and maintainedhis freely available Surface Evolver software, which shows computermodels of liquid surfaces.