The Physics of Microdroplets -- Contents -- Acknowledgements -- Preface -- Introduction -- 1 Fundamentals of Capillarity -- 1.1 Abstract -- 1.2 Interfaces and Surface Tension -- 1.2.1 The Notion of Interface -- 1.2.2 The Effect of Temperature on Surface Tension -- 1.2.3 The Effect of Surfactants -- 1.2.4 Surface Tension of a Fluid Containing Particles -- 1.3 Laplace's Law and Applications -- 1.3.1 Curvature and Radius of Curvature -- 1.3.2 Derivation of Laplace's Law -- 1.3.3 Examples of Application of Laplace's Law -- 1.3.3.1 Pressure in a Bubble -- 1.3.3.2 Liquid Transfer From a Smaller Drop to a Bigger Drop -- 1.3.3.3 Precursor Film and Coarsening -- 1.3.3.4 Pressure in Droplets Constrained Between Two Parallel Plates -- 1.3.3.5 Zero Pressure Surfaces: Example of a Meniscus on a Rod -- 1.3.3.6 Self Motion of a Liquid Plug Between Two Non-Parallel Wetting Plates -- 1.3.3.7 Laplace's Law in Medicine: Normal and Shear Stress in Blood Vessels -- 1.3.3.7.1 Shear Stress in Vascular Networks -- 1.3.3.7.2 Normal Stress in Vascular Networks -- 1.3.3.8 Laplace's Law in Medicine: the Example of Lung Alveoli -- 1.3.3.9 Laplace's Law in a Gravity Field -- 1.3.3.10 Generalization: Laplace's Law in Presence of a Flow Field -- 1.3.4 Wetting - Partial or Total Wetting -- 1.3.5 Contact Angle - Young's Law -- 1.3.5.1 Young's Law -- 1.3.5.2 Droplet on a Cantilever -- 1.3.5.3 Contact Between Three Liquids - Neumann's Construction -- 1.3.5.4 Nanobubbles on Hydrophobic Walls, Line Tension and the Modified Young Law -- 1.3.6 Work of Adhesion, Work of Cohesion and the Young-Dupre Equation -- 1.3.6.1 Work of Adhesion -- 1.3.6.2 Work of Cohesion -- 1.3.6.3 The Young-Dupre Equation -- 1.3.7 Capillary Force, Force on a Triple Line -- 1.3.7.1 Introduction -- 1.3.7.2 Capillary Force Between Two Parallel Plates -- 1.3.7.3 Capillary Rise in a Tube.

1.3.7.4 Capillary Rise Between Two Parallel Vertical Plates -- 1.3.7.5 Capillary Rise in a Pipette -- 1.3.7.6 Force on a Triple Line -- 1.3.7.7 Examples of Capillary Forces in Microsystems -- 1.4 Measuring the Surface Tension of Liquids -- 1.4.1 Using Pressure (Bubble Pressure Method) -- 1.4.1.1 Using the Capillary Rise on a Plate - Wilhelmy Plate -- 1.4.2 Using Gravity: the Pendant Drop Method -- 1.4.2.1 Bond Number -- 1.4.2.2 Method -- 1.4.2.3 Using Shear Stress in a Microflow -- 1.4.3 Surface Free Energy -- 1.4.3.1 Introduction -- 1.4.3.2 Method of Good-Girifalco -- 1.4.3.3 Fowkes Method -- 1.4.3.4 Critical Surface Tension and Surface Free Energy -- 1.4.3.4.1 Zisman Plot -- 1.4.3.4.2 Disjoining Pressure -- 1.5 Minimization of the Surface Energy and Minimal Surfaces -- 1.5.1 Minimization of the Surface Energy -- 1.5.2 Conclusion -- 1.6 References -- 2 Minimal Energy and Stability Rubrics -- 2.1 Abstract -- 2.2 Spherical Shapes as Energy Minimizers -- 2.2.1 The Sphere Theorems -- 2.2.2 Sphere -- 2.2.3 Spherical Cap on a Plane -- 2.2.4 Drop Between Parallel Plates -- 2.2.5 Droplet in a Wedge -- 2.2.6 Droplet on an Exterior Corner -- 2.3 Symmetrization and the Rouloids -- 2.3.1 Steiner Symmetrization -- 2.3.2 Rouloids -- 2.3.2.1 The Sphere -- 2.3.2.2 The Cylinder -- 2.3.2.3 The Catenoid -- 2.3.2.4 Unduloids -- 2.3.2.5 Nodoids -- 2.3.3 Rouloid Summary -- 2.4 Increasing Pressure and Stability -- 2.4.1 Wedge -- 2.4.2 In a Square -- 2.4.3 Round Well -- 2.4.4 Square Well -- 2.5 The Double-Bubble Instability -- 2.5.1 Conditions for the Double-bubble Instability -- 2.5.2 Plateau-Rayleigh Instability -- 2.5.3 Plateau-Rayleigh Instability in Corners and Grooves -- 2.5.4 Instability of a Cylinder on a Hydrophilic Strip -- 2.5.5 Double-bubble Instability in Bulging Liquid Bridge -- 2.5.6 Lower-pressure Comparison Theorem -- 2.6 Conclusion -- 2.7 References.

3 Droplets: Shape, Surface and Volume -- 3.1 Abstract -- 3.2 The Shape of Micro-drops -- 3.2.1 Sessile Droplets - the Bond Number -- 3.3 Electric Bond Number -- 3.4 Shape, Surface Area and Volume of Sessile Droplets -- 3.4.1 Height of a "Large" Droplet: Bo » 1 -- 3.4.2 Microscopic Drops: Bo « 1 -- 3.4.2.1 Shape of the Droplet -- 3.4.2.2 Volume of a Sessile Droplet -- 3.4.2.3 Surface Area of a Spherical Cap -- 3.4.2.4 Example: Measurement of the Volume of a Droplet from a Single Image (Hydrophobic Case) -- 3.4.3 Droplets Between Two Parallel Plates -- 3.4.4 Shape of a Droplet Flattened Between Two Horizontal Planes -- 3.4.5 Curvature Radius of the Free Interface -- 3.4.6 Convex Droplet Shape -- 3.4.6.1 Volume of a Droplet Flattened Between Two Parallel Planes and Having the Same Contact Angle with Both Planes (Convex Case) -- 3.4.7 Volume of a Droplet Flattened by Two Horizontal Planes with Different Contact Angles with the Two Planes (Convex Case) -- 3.4.8 Surface Area (Convex Case) -- 3.4.9 Concave Droplet Shape -- 3.4.9.1 Volume of a Droplet Flattened Between Two Parallel Planes and Having the Same Contact Angle with Both Planes (Concave Case) -- 3.4.9.2 Volume of a Droplet Flattened by Two Horizontal Planes with Different Contact Angles with the Two Planes (Concave Case) -- 3.4.9.3 Surface Area (Concave Case) -- 3.5 Conclusion -- 3.6 References -- 4 Sessile Droplets -- 4.1 Abstract -- 4.2 Droplet Self-motion Under the Effect of a Contrast or Gradient of Wettability -- 4.2.1 Drop Moving Over a Sharp Transition of Wettability -- 4.2.2 Drop Moving Uphill -- 4.2.3 Dynamic and Quasi-static Approach -- 4.2.4 Drop Moving Up a Step -- 4.2.5 Drop Moving Over a Gradient of Surface Concentration of Surfactant -- 4.2.6 Conclusion -- 4.3 Contact Angle Hysteresis -- 4.4 Pinning and Canthotaxis -- 4.4.1 Droplet Pinning on a Surface Defect.

4.4.2 Droplet Pinning on an Edge - Canthotaxis -- 4.4.3 Droplet Pinning at a Wettability Separation Line -- 4.4.4 Pinning of an Interface by Pillars -- 4.5 Sessile Droplet on a Non-ideally Planar Surface -- 4.6 Droplet on Textured or Patterned Substrates -- 4.6.1 Wenzel' s Law -- 4.6.2 Cassie-Baxter Law -- 4.6.3 Contact on Microfabricated Surfaces: the Transition Between the Wen-zel and Cassie Laws -- 4.6.3.1 Introduction -- 4.6.3.2 Contact Angle on a Microfabricated Substrate. Case of Hydrophobie Contact -- 4.6.3.3 Example of Square, Hydrophobic Micropillars -- 4.6.3.4 Wenzel-Cassie Hysteresis -- 4.6.3.5 Superhydrophobicity -- 4.6.3.6 Irregularly Textured and Fractal Surfaces -- 4.6.3.7 Hydrophilic Case -- 4.6.3.8 Simultaneous Superhydrophobic and Oleophobic Surfaces - the Concept of Omniphobicity -- 4.6.3.9 Conclusion -- 4.7 References -- 5 Droplets Between Two Non-parallel Planes: From Tapered Planes to Wedges -- 5.1 Abstract -- 5.2 Droplet Self-motion Between Two Non-parallel Planes -- 5.2.1 Identical Young Contact Angle with Both Plates -- 5.2.2 Different Young Contact Angles -- 5.2.3 Numerical Simulation - 2D and 3D Cases -- 5.2.4 A Reciprocal to the Hauksbee Problem -- 5.2.5 Example of Tapered System for Passive Pumping in Fuel Cells -- 5.2.6 Discussion -- 5.3 Droplet in a Corner -- 5.3.1 Dimensions of the Droplet and Effect of Gravity -- 5.3.2 Concus-Finn Relations -- 5.3.3 Numerical Approach -- 5.3.4 Example of a Liquid in a Micro-beaker -- 5.3.5 Extended Concus-Finn Relation -- 5.3.6 Droplet in a Wetting/Non-wetting Comer -- 5.3.7 Discussion -- 5.4 Conclusion -- 5.5 References -- 6 Microdrops in Microchannels and Microchambers -- 6.1 Abstract -- 6.2 Droplets in Micro-wells -- 6.2.1 Shape of the Liquid Surface in a Micro-well -- 6.2.2 Evaporation of Liquid in a Micro-well -- 6.2.2.1 Hydrophilic Case -- 6.2.2.2 Hydrophobic Case.

6.2.2.3 Controlling Evaporation in Microsystems -- 6.2.3 Filling a Micro-well -- 6.3 Droplets in Microchannels -- 6.3.1 Capillary, Weber and Bond Numbers -- 6.3.2 Non-wetting Droplets and Plugs -- 6.3.2.1 Silanization Surface Treatment -- 6.3.2.2 Rectangular Microchannels -- 6.3.2.3 T-shaped Microchannels - Abacus Groove -- 6.3.2.4 Plugs Slowed Down by Pillars -- 6.3.3 Wetting Droplets and Plugs -- 6.3.3.1 General Case: Contact Angle Between 45° and 135° -- 6.3.3.2 Contact Angle Smaller Than 45° -- 6.3.3.3 Contact Angle Larger Than 135° -- 6.3.4 Trains of Droplets - Compound Droplets -- 6.3.4.1 Trains Moving Inside a Microchannel -- 6.3.4.2 Packing of Droplets -- 6.4 Conclusion -- 6.5 References -- 7 Capillary Effects: Capillary Rise, Capillary Pumping, and Capillary Valve -- 7.1 Abstract -- 7.2 Capillary Rise -- 7.2.1 Cylindrical Tubes: Jurin's Law -- 7.2.2 Capillary Rise in Square Tubes -- 7.2.3 Capillary Rise on a Vertical Plate - Surface Tension Measurement by the Wilhelmy Method -- 7.2.4 Capillary Rise Between Two Parallel Vertical Plates -- 7.2.5 Capillary Rise in a Dihedral -- 7.2.6 Capillary Rise in an Array of Four Vertical Square Pillars -- 7.2.7 Comparison of Capillary Rise Between Wilhelmy Plate and Pillars -- 7.2.8 Oblique Tubes - Capillary Rise in a Pipette -- 7.3 Capillary Pumping -- 7.3.1 Principles of Capillary Pumping -- 7.3.2 Capillary Pumping and Channel Dimensions -- 7.3.3 The Dynamics of capillary Pumping: Horizontal Microchannel -- 7.3.4 The Dynamics of Capillary Pumping: General -- 7.3.5 Examples of Capillary Pumping -- 7.3.5.1 Spontaneous capillary flow in a composite channel -- 7.4 Capillary Valves -- 7.4.1 Principles of Capillary Valves -- 7.4.2 Valving Efficiency and Shape of the Orifice -- 7.4.3 Examples of Capillary Valves in Microsystems -- 7.4.3.1 Stop Valve -- 7.4.4 Delay Valves -- 7.5 Conclusions -- 7.6 References.

8 Open Microfluidics.