The Physics of Microdroplets E-Book

Jean Berthier

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 517

Veröffentlichungsjahr: 2012

Beschreibung

The Physics of Microdroplets gives the reader the theoretical and numerical tools to understand, explain, calculate, and predict the often nonintuitive observed behavior of droplets in microsystems. Microdrops and interfaces are now a common feature in most fluidic microsystems, from biology, to biotechnology, materials science, 3D-microelectronics, optofluidics, and mechatronics. On the other hand, the behavior of droplets and interfaces in today's microsystems is complicated and involves complex 3D geometrical considerations. From a numerical standpoint, the treatment of interfaces separating different immiscible phases is difficult. After a chapter dedicated to the general theory of wetting, this practical book successively details: * The theory of 3D liquid interfaces * The formulas for volume and surface of sessile and pancake droplets * The behavior of sessile droplets * The behavior of droplets between tapered plates and in wedges * The behavior of droplets in microchannels * The effect of capillarity with the analysis of capillary rise * The onset of spontaneous capillary flow in open microfluidic systems * The interaction between droplets, like engulfment * The theory and application of electrowetting * The state of the art for the approach of 3D-microelectronics using capillary alignment

Leseprobe

Contents

Cover

Half Title page

Title page

Copyright page

Acknowledgements

Preface

Introduction

From Conventional Single-phase Microfluidics to Droplets and Digital Microfluidics

Domains of Application

Organization of the Book

References

Chapter 1: Fundamentals of Capillarity

1.1 Abstract

1.2 Interfaces and Surface Tension

1.3 Laplace’s Law and Applications

1.4 Measuring the Surface Tension of Liquids

1.5 Minimization of the Surface Energy and Minimal Surfaces

1.6 References

Chapter 2: Minimal Energy and Stability Rubrics

2.1 Abstract

2.2 Spherical Shapes as Energy Minimizers

2.3 Symmetrization and the Rouloids

2.4 Increasing Pressure and Stability

2.5 The Double-Bubble Instability

2.6 Conclusion

2.7 References

Chapter 3: Droplets: Shape, Surface and Volume

3.1 Abstract

3.2 The Shape of Micro-drops

3.3 Electric Bond Number

3.4 Shape, Surface Area and Volume of Sessile Droplets

3.5 Conclusion

3.6 References

Chapter 4: Sessile Droplets

4.1 Abstract

4.2 Droplet Self-motion Under the Effect of a Contrast or Gradient of Wettability

4.3 Contact Angle Hysteresis

4.4 Pinning and Canthotaxis

4.5 Sessile Droplet on a Non-ideally Planar Surface

4.6 Droplet on Textured or Patterned Substrates

4.7 References

Chapter 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

5.3 Droplet in a Corner

5.4 Conclusion

5.5 References

Chapter 6: Microdrops in Microchannels and Microchambers

6.1 Abstract

6.2 Droplets in Micro-wells

6.3 Droplets in Microchannels

6.4 Conclusion

6.5 References

Chapter 7: Capillary Effects: Capillary Rise, Capillary Pumping, and Capillary Valve

7.1 Abstract

7.2 Capillary Rise

7.3 Capillary Pumping

7.4 Capillary Valves

7.5 Conclusions

7.6 References

Chapter 8: Open Microfluidics

8.1 Abstract

8.2 Droplet Pierced by a Wire

8.3 Liquid Spreading Between Solid Structures – Spontaneous Capillary Flow

8.4 Liquid Wetting Fibers

8.5 Conclusions

8.6 References

8.7 Appendix: Calculation of the Laplace Pressure for a Droplet on a Horizontal Cylindrical Wire

Chapter 9: Droplets, Particles and Interfaces

9.1 Abstract

9.2 Neumann’s Construction for Liquid Droplets

9.3 The Difference Between Liquid Droplets and Rigid Spheres at an Interface

9.4 Liquid Droplet Deposited at a Liquid Surface

9.5 Immiscible Droplets in Contact and Engulfment

9.6 Non-deformable (Rigid) Sphere at an Interface

9.7 Droplet Evaporation and Capillary Assembly

9.8 Conclusion

9.9 References

Chapter 10: Digital Microfluidics

10.1 Abstract

10.2 Electrowetting and EWOD

10.3 Droplet Manipulation with EWOD

10.4 Examples of EWOD in Biotechnology – Cell Manipulation

10.5 Examples of Electrowetting for Optics – Tunable Lenses and Electrofluidic Display

10.6 Conclusion

10.7 References

Chapter 11: Capillary Self-assembly for 3D Microelectronics

11.1 Abstract

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

11.3 Real Case: Spreading and Wetting

11.4 The Importance of Pinning and Confinement

11.5 Conclusion

11.6 Appendix A: Shift Energy and Restoring Force

11.7 Appendix B: Twist Energy and Restoring Torque

11.8 Appendix C: Lift Energy and Restoring Force

11.9 References

Chapter 12: Epilogue

Index

The Physics of Microdroplets

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Scrivener Publishing Collections Editors

James E. R. Couper

Ken Dragoon

Richard Erdlac

Rafiq Islam

Norman Lieberman

Peter Martin

W. Kent Muhlbauer

Andrew Y. C. Nee

S. A. Sherif

James G. Speight

Publishers at Scrivener

Martin Scrivener ([email protected])

Phillip Carmical ([email protected])

Copyright © 2012 by Scrivener Publishing LLC. All rights reserved.

Co-published by John Wiley & Sons, Inc. Hoboken, New Jersey, and Scrivener Publishing LLC, Salem, Massachusetts. Published simultaneously in Canada.

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The cover shows a Surface Evolver simulation of a liquid plug in an enclosed rectangular hydrophobic channel, with square pillars added to constrict the channel (top). The pinched regions between the pillars have high curvature, hence high pressure, so liquid flows into the low-curvature, low-pressure larger blobs (middle), until the neck pinches off to form two stable blobs (bottom).

Library of Congress Cataloging-in-Publication Data:

ISBN 978-0-470-93880-5

Acknowledgements

J. Berthier

I would like to thank all my colleagues who have contributed with photographs and sketches: R. Berthier, N. Sarrut, P. Dalle, J. Wagh, P. Clementz, P. Pouteau, F. Baleras, N. Chiarutinni, M. Bestehorn, and G. Morris.

I am grateful to E. Berthier and his colleagues B. Casavant and D.J. Beebe at University of Wisconsin for their discussion on “Suspended microfluidics” and sketches for chapter 8 “Open microfluidics”.

Many thanks to J.C. Mourrat (EPFL) for the fruitful discussion on the analytical model for the twist mode of chapter 11 “Capillary Self-assembly for 3D Microelectronics”. I also am indebt to L. Di Cioccio, S. Mermoz and C. Frétigny for their help for this same chapter.

I am also grateful to D. Peyrade for the interesting discussions on capillary alignment of gold nanospheres in chapter 9 “Droplets, Particles and Interfaces”.

I would like to thank my company for having given me encouragements and support for this project, especially L. Malier director of the Leti, and C. Peponnet my group leader.

Finally I thank my children Erwin, Linda and Rosanne for constantly encouraging me during this work.

K. Brakke

I would like to thank my Ph.D. advisor Fred Almgren for introducing me to the mathematics of soap films and liquid surfaces and his support and encouragement during the early development of my Surface Evolver. I would also like to thank the Geometry Center at the University of Minnesota for numerous summer visits and a sabbatical, which greatly contributed to Evolver development.

Preface

Capillary phenomena are intriguing. During the many years I (Jean) have spent with my colleagues working on microsystems for biotechnology, I have observed the difficulty to predict – and sometimes understand – the behavior of droplets and interfaces at the micro scale. First, optical observation is not straightforward – it is not easy to locate an interface seen from above in the microscope. Second, the analysis of the observed phenomena is complicated. In my personal experience, that was the case for pancreatic cells encapsulation in micro-flow-focusing devices, liquid-liquid extraction systems, digital microfluidics, capillary valves, spontaneous capillary flows in closed and open channels, in cracks, and between fibers.

And the difficulty is even more important for the conception of new microsystems. Questions such as “where is the interface going to anchor?” or “will the particles cross the interface?” or “will the interface de-pin when the capsule arrives?” or “will the capillary force be sufficient?” are repeatedly being asked. Although illustrious pioneers such as P-G. de Gennes, D. Quéré, G.M. Whitesides, and others have contributed to the knowledge of interface behaviors on a theoretical standpoint, much is left to understand for the engineer having to design a microchip or the student behind his computer or the biologist at his lab bench.

In this book, Ken and I have attempted to give the reader the tools for solving these capillary and surface tension problems, present theoretical tools derived from previous works of colleagues and our personal experience, as well as provide calculation tools through the Surface Evolver numerical program.

I first heard about Evolver at a Nanotech Conference in 2004 and its potential for two-phase microflows and droplets behavior. Although it cannot treat the dynamics of a flow, it can be used to predict the stable shape and location of droplets and interfaces. A typical example is that of a capillary valve where the bulging out of the interface directly depends on the applied pressure. Besides, useful information can be gained by considering that an interface or a droplet has not reached its equilibrium position: this is for example the case of spontaneous capillary flows or droplets moving up a step or a slope. Finally, at the microscale, interfaces are restored nearly immediately by capillary and surface tension forces, which frequently dominate the other forces like weight, viscosity, and inertia. This applies for example to self-alignment problems.

I started to work with Evolver for predicting the behavior of droplets in digital microfluidic systems. Because the electrowetting effect can often be translated into a capillary effect (capillary equivalence), Evolver is well suited to treat such problems. I had the fortune that the author of Surface Evolver, Kenneth Brakke, agreed to assist me with the handling of the numerical program and our cooperation was extremely fruitful. After a few years of working on this topic, as well on the theoretical, numerical and experimental aspects, I had the opportunity to write the book Microdrops and Digital Microfluidics in 2008.

But many capillary problems were still to be tackled outside the domain of digital microfluidics. I continued to use Evolver, again with Ken’s help. When our Evolver tool box was sufficient, we thought that it could be useful to make it available to the scientific community and decided to write this new book with my publisher Martin Scrivener. The Evolver files corresponding to the examples and problems of this book are available for the reader at the internet address http://www.susqu.edu/brakke/physicsofmicrodrops.

We hope that our work will be useful to boost the developments of microfluidic systems and that this book will find an echo in the micro and nanotechnology world.

Jean Berthier, Grenoble, February 22, 2012

Kenneth A. Brakke, Susquehana University, February 22, 2012

Introduction

From Conventional Single-phase Microfluidics to Droplets and Digital Microfluidics

Starting in the year 1980, microfluidics was at first a mere downscaling of macrofluidics. Its development was triggered by the emergence of biotechnology and materials science, imagined by visionary pioneers like Feynman [1], deGennes [2], Whitesides [3] and others. In particular, biotechnology was as a new science at the boundary of physics and biology. The goal was to give biological, medical and pharmaceutical research new automation tools to boost the development of new drugs, fabricate new body implants and increase the potentialities of fundamental research. In reality, this plan imagined by these first researchers has been extremely effective and produced even more discoveries than what was first expected. In a way, biotechnology developments bloomed according to Feynman’s words: “The best way to predict the future is to invent it.” The foreseen goals have required the downscaling of fluidic systems to the “convenient” size to work at the proper scale characteristic of a population of biologic targets. At the same time, it was found that the downscaling brought economy in costly materials, fluids, and devices; that sensitivity was increased and operating times were greatly reduced by the integration of many functions on the same microchip. Gradually, as microsystems based on microflows become conventionally used, new approaches were investigated that required even less volume of sample fluids. This trend to downscaling has promoted the development of new microfluidic approaches such as droplet and digital microfluidics. Reduction of the liquid vessel containing the biological targets was found to be possible by the use of microdroplets. New systems based on the confinement of biologic targets in extremely small vessels like microdrops are emerging. In such approaches the liquid volumes are reduced to a few picoliters.

Domains of Application

Historically, genomics and proteomics were the first beneficiaries of the development of biotechnology, and now it is the turn of cellomics. Also, these developments have spread beyond the domain of biotechnology and created a “cloud” of new applications in other domains such as bioinformatics, bioengineering, tissue engineering, etc. At the same time, microfluidic techniques reached other domains, such as materials science, microelectronics and mechatronics. It has been quickly demonstrated that biochemical reactions such as PCR for the recognition of DNA can be performed with the same efficiency in droplets, with a lesser amount of replicas [4–6]. Proteins can be crystallized in droplets, resulting in a greater ability to investigate their structures by X-ray crystallography [7]. In biology, single cell research has become feasible, after encapsulating the cell in a droplet or a gelled (polymerized) droplet [8–10] or manipulating cells on a digital microfluidic chip [11]. Chemical reactions can also be performed with very small amounts of chemical species inside droplets [12–14]. The use of droplet and digital microfluidics soon extended beyond the limits of biotechnology. Electrowetting droplets are now commercially used in optics as tunable lenses [15] and screen displays [16]. In mechatronics, electrowetting switches (or CFA, for “capillary force actuators”) have been shown to be much more effective than electrostatic switches of the same size [17]. Self-assembly techniques using capillary forces produced by a droplet surface are currently used in materials science for manipulating gold nano-spheres for coating applications [18]. Self-alignment using capillary forces is also a promising approach to 3D-microelectronics, which is required to circumvent the present limitations of 2D assembly [19–21]. The examples are many showing the interest in microdrops.

Figure 1 Different applications using microdroplets: (a) droplets moved with magnetic beads for PCR application [6]; (b) protein crystallization in a microdrop (from [7], ©Wiley-VCH Verlag GmbH & Co. KGaA.

Reproduced with permission); (c) encapsulated cells in a polymerized alginate matrix (photo courtesy CEA-LETI); (d) tunable lenses by Varioptics (from [15], courtesy Varioptics); (e) screen displays by Liquavista, (from [16], courtesy Liquavista); (f) schematic of a capillary force actuator (not to scale) [17].

Organization of the Book

This book is dedicated to the study of droplets and interfaces principally in a steady or quasi-steady state, although some dynamic considerations have been added when it was judged useful. The first chapter presents the general considerations leading to the concepts of surface tension and capillary forces, associated to the notions of surface energy and contact angle. Young’s and Laplace’s laws, which are the two “pillars” of any capillary approach, are described, commented and exemplified. The second chapter presents the theory of liquid surfaces in space, including some ways to prove certain surfaces are minimums of energy. Chapter 3 is devoted to the determination of the shape, surface area, and volume of droplets. In chapter 4, the shape and behavior of sessile droplets (droplets place on a solid surface) is investigated for many different configurations of chemical and geometrical surface inhomogeneities: drops at the boundaries between hydrophilic and hydrophobic substrates, or on geometrical inhomogeneities such as steps or grooves or corners. The fifth chapter concerns the behavior of droplets in asymmetric geometries; in a first part, the Hauksbee problem is treated and an extension to hydrophobic surfaces is given. In a second part, the Concus-Finn relations are presented. In chapter 6, the behavior of droplets in microwells and closed microchannels is investigated. The cases of wetting and non-wetting plugs are treated as well as that of trains of droplets. Chapter 7 is dedicated to the phenomena of capillary rise, capillary pumping and capillary valving. In the first two parts, we analyze how capillary forces can contribute to moving a liquid in horizontal or vertical tubes. In the third part, we analyze the opposite: how to find a geometry that can stop a capillary flow. The focus of chapter 8 is open microfluidics, i.e. microflows partially guided by a solid wall, but also in contact with air or another liquid, which is becoming a very important issue in biotechnology; this type of microflow rely mainly on capillary forces and if necessary on electrowetting forces to move the fluid. Chapter 9 deals with the contact and potential engulfment of droplets and particles by interfaces. Examples pertaining to encapsulation of polymerized droplets and capillary assembly are presented. Chapter 10 is on digital microfluidics, a convenient way to manipulate droplets on a planar, or locally planar surface, which has seen many developments lately. We present the state of the art and new developments in this technique. In chapter 11, we treat an example of the use of capillary forces: the ongoing approach to 3D-microelectronics by assembling stacks of chips on a wafer. A promising approach to achieve chip positioning and alignment is that of capillary self-assembly.

References

[1] R. Feynman, Chap 6 in Building biotechnology by Y.E. Friedman, third edition, Logos press, 2008.

[2] P-G de Gennes, F. Brochart-Wyart, D. Quéré. Capillary and wetting phenomena: drops, bubbles, pearls, waves. Springer, 2002.

[3] G.M. Whitesides, Chap 9 in Biotechnology and Materials Science – Chemistry for the future, by L.M. Good, ACS publications, 1988.

[4] P.-A. Auroux, Y. Koc, A. deMello, A. Manz and P. J. R. Day, Miniaturised nucleic acid analysis, Lab Chip 4, pp. 534–546, 2004.

[5] E. Wulff-Burchfield, W.A. Schell, A.E. Eckhardt, M. G. Pollack, Zhishan Hua, J. L. Rouse, V. K. Pamula, Vijay Srinivasan, J. L. Benton, B. D. Alexander, D. A. Wilfret, M. Kraft, C. Cairns, J. R. Perfect, and T. G. Mitchell, “Microfluidic Platform versus Conventional Real-time PCR for the Detection of Mycoplasma pneumoniae in Respiratory Specimens,” Diagnostic microbiology and infectious disease67(1), pp. 22–29, 2010.

[6] http://www.quantalife.com/technology/ddpcr

[7] Bo Zheng, L. Spencer Roach, and R. F. Ismagilov, “Screening of Protein Crystallization Conditions on a Microfluidic Chip Using Nanoliter-Size Droplets,” JACS125, pp. 11170–11171, 2003.

[8] T. Thorsen, R. W. Roberts, F. H. Arnold, S. R. Quake, “Dynamic pattern formation in a vesicle-generating microfluidic device,” Phys. Rev. Lett.86, pp. 4163–4166, 2001.

[9] S.L. Anna, N. Bontoux, and H.A. Stone, “Formation dispersions using flow focusing in microchannels,” Appl. Phys. Lett.82(3), pp. 364–366, 2003.

[10] J.F. Edd, D. Di Carlo, K.J. Humphry, S. Köster, D. Irimia, D.A. Weitz, M. Toner, “Controlled encapsulation of single cells into monodispersed picoliter drops,” Lab Chip8(8), pp. 1262–1264, 2008.

[11] D. Witters, N. Vergauwe, S. Vermeir, F. Ceyssens, S. Liekens, R. Puers and J. Lammertyn, “Biofunctionalization of electrowetting-on-dielectric digital microfluidic chips for miniaturized cell-based applications,” Lab Chip11, pp. 2790–2794, 2011.

[12] H. Song, J. D. Tice, R. F. Ismagilov, “A microfluidic system for controlling reaction networks in time,” Angew. Chem.42, pp. 767–771, 2003.

[13] A. Gnther, K.F. Jensen, “Multiphase microfluidics: from flow characteristics to chemical and material synthesis,” Lab. Chip6, pp. 1487–1503, 2006.

[14] J. Atencia, D.J. Beebe, “Controlled microfluidic interfaces,” Nature437, pp. 648–655, 2005.

[15] VarioticsTM: http://www.varioptic.com/en/tech/technology01.php

[16] LiquavistaTM: http://www.liquavista.com/files/LQV060828XYR-15.pdf

[17] C. R. Knospe and S.A. Nezamoddini, “Capillary force actuation,” J. Micro-Nano Mech.5 p. 5768, 2009.

[18] O. Lecarme, T. Pinedo-Rivera, K. Berton, J. Berthier, D Peyrade, “Plasmonic coupling in nondipolar gold collidal dimers,” Applied Physics Letters98, 083122, 2011.

[19] T. Fukushima, T. Tanaka, M. Koyanagi. “3D System Integration Technology and 3D Systems,” Advanced Metallization Conference Proceedings, pp. 479–485, 2009.

[20] K. Sato, T. Seki, S. Hata, A. Shimokohbe, “Self-alignment of microparts using liquid surface tension – behavior of micropart and alignment characteristics,” Precision Engineering27, pp. 42–50, 2003.

[21] J. Berthier, K. Brakke, F. Grossi, L. Sanchez and L. Di Cioccio, “Self-alignment of silicon chips on wafers: A capillary approach,” JAP108, 054905, 2010.

Chapter 1

Fundamentals of Capillarity

1.1 Abstract

In this first chapter, the fundamentals of capillarity are presented. We follow a conventional approach [1], first presenting surface tension of an interface, which is the fundamental notion in capillarity theory; this notion leads naturally to that of wetting, then to Laplace’s law, and to the introduction of Young contact angles and capillary forces. Next, different applications of capillary forces are shown, and the problem of the measurement of surface tensions is presented.

1.2 Interfaces and Surface Tension

1.2.1 The Notion of Interface

Mathematically speaking, an interface is the geometrical surface that delimits two fluid domains. This definition implies that an interface has no thickness and is smooth (i.e. has no roughness). As practical as it is, this definition is in reality a schematic concept. The reality is more complex, the boundary between two immiscible liquids is somewhat blurred and the separation of the two fluids (water/air, water/oil, etc.) depends on molecular interactions between the molecules of each fluid [2] and on Brownian diffusion (thermal agitation). A microscopic view of the interface between two fluids looks more like the scheme of figure 1.1. However, in engineering applications, it is the macroscopic behavior of the interface that is the focus of attention, and the mathematical concept regains its utility. At a macroscopic size, the picture of figure 1.1 can be replaced by that of figure 1.2, where the interface is a mathematical surface without thickness and the contact angle θ is uniquely defined by the tangent to the surface at the contact line.

Figure 1.1 Schematic view of an interface at the molecular size.

Figure 1.2 Macroscopic view of the interface of a drop.

In a condensed state, molecules attract each other. Molecules located in the bulk of a liquid have interactions with neighboring molecules on all sides; these interactions are mostly van der Waals attractive interactions for organic liquids and hydrogen bonds for polar liquids like water [2]. On the other hand, molecules at an interface have interactions in a half space with molecules of the same liquid, and in the other half space interactions with molecules of the other fluid or gas (figure 1.3).

Figure 1.3 Simplified scheme of molecules near an air/water interface. In the bulk, molecules have interaction forces with all the neighboring molecules. At the interface, half of the interactions have disappeared.

Consider an interface between a liquid and a gas. In the bulk of the liquid, a molecule is in contact with 4 to 12 other molecules depending on the liquid (4 for water and 12 for simple molecules); at the interface this number is divided by two. Of course, a molecule is also in contact with gas molecules, but, due to the low densities of gases, there are fewer interactions and less attraction than on the liquid side. The result is that there is locally a dissymmetry in the interactions, which results in an excess of surface energy. At the macroscopic scale, a physical quantity called “surface tension” has been introduced in order to take into account this molecular effect. The surface tension has the dimensions of energy per unit area, and in the International System it is expressed in J/m2 or N/m (sometimes, it is more practical to use mN/m as a unit for surface tension). An estimate of the surface tension can be found by considering the molecules’ cohesive energy. If U is the total cohesive energy per molecule, a rough estimate of the energy excess of a molecule at the interface is U/2. Surface tension is a direct measure of this energy excess, and if δ is a characteristic molecular dimension and δ2 the associated molecular surface area, then the surface tension is approximately

(1.1)

This relation shows that surface tension is important for liquids with large cohesive energy and small molecular dimension. This is why mercury has a large surface tension whereas oil and organic liquids have small surface tensions. Another consequence of this analysis is the fact that a fluid system will always act to minimize surface area: the larger the surface area, the larger the number of molecules at the interface and the larger the cohesive energy imbalance. Molecules at the interface always look for other molecules to equilibrate their interactions. As a result, in the absence of other forces, interfaces tend to adopt a flat profile, and when it is not possible due to boundary constraints or volume constraints, they take a rounded shape, often that of a sphere. Another consequence is that it is energetically costly to expand or create an interface: we will come back on this problem in Chapter 10 when dividing a droplet into two “daughter” droplets by electrowetting actuation. The same reasoning applies to the interface between two liquids, except that the interactions with the other liquid will usually be more attractive than a gas and the resulting dissymmetry will be less. For example, the contact energy (surface tension) between water and air is 72 mN/m, whereas it is only 50 mN/m between water and oil (table 1.1). Interfacial tension between two liquids may be zero: fluids with zero interfacial tension are said to be miscible. For example, there is no surface tension between fresh and salt water: salt molecules will diffuse freely across a boundary between fresh and saltwater.

Table 1.1 Values of surface tension of different liquids in contact with air at a temperature of 20 °C (middle column, mN/m) and thermal coefficient α (right column, mN/m/°C).

liquid

γ

0

α

Acetone

25.2

–0.112

Benzene

28.9

–0.129

Benzylbenzoate

45.95

–0.107

Bromoform

41.5

–0.131

Chloroform

27.5

–0.1295

Cyclohexane

24.95

–0.121

Cyclohexanol

34.4

–0.097

Decalin

31.5

–0.103

Dichloroethane

33.3

–0.143

Dichloromethane

26.5

–0.128

Ethanol

22.1

–0.0832

Ethylbenzene

29.2

–0.109

Ethylene-Glycol

47.7

–0.089

Isopropanol

23.0

–0.079

Iodobenzene

39.7

–0.112

Glycerol

64.0

–0.060

Mercury

425.4

–0.205

Methanol

22.7

–0.077

Nitrobenzene

43.9

–0.118

Perfluorooctane

14.0

–0.090

Polyethylen-glycol

43.5

–0.117

PDMS

19.0

–0.036

Pyrrol

36.0

–0.110

Toluene

28.4

–0.119

Water

72.8

–0.1514

The same principle applies for a liquid at the contact of a solid. The interface is just the solid surface at the contact of the liquid. Molecules in the liquid are attracted towards the interface by van der Waals forces. If the attractions to the solid are strong, the liquid-solid interface has negative surface energy, and the solid is said to be wetting or hydrophilic (or lyophilic for non-water liquids, but we will use the term hydrophilic for all liquids). If the attractions are weak, the interface energy is positive, and the solid is nonwetting or hydrophobic (or lyophobic).

Usually surface tension is denoted by the Greek letter γ, with subscripts referring to the two components on each side of the interface, for example γLG at a Liquid/Gas interface. Sometimes, if the contact is with air, or if no confusion can be made, the subscripts can be omitted. It is frequent to speak of “surface tension” for a liquid in contact with a gas, and “interfacial tension” for a liquid in contact with another liquid. According to the definition of surface tension, for a homogeneous interface (same molecules at the interface all along the interface), the total energy of a surface is

(1.2)

where S is the interfacial surface area.

In the literature or on the Internet there exist tables for surface tension values [3,4]. Typical values of surface tensions are given in table 1.1. Note that surface tension increases as the intermolecular attraction increases and the molecular size decreases. For most oils, the value of the surface tension is in the range γ ≈ 20 – 30 mN/m, while for water, γ ≈ 70 mN/m. The highest surface tensions are for liquid metals; for example, liquid mercury has a surface tension γ ≈ 500 mN/m.

1.2.2 The Effect of Temperature on Surface Tension

The value of the surface tension depends on the temperature. The first empirical equation for the surface tension dependence on temperature was given by Eötvös in 1886 [5]. Observing that the surface tension goes to zero when the temperature tends to the critical temperature TC, Eötvös proposed the semi-empirical relation

(1.3)

where vL is the molar volume. Katayama (1915) and later Guggenheim (1945) [6] have improved relation to obtain

(1.4)

where γ* is a constant for each liquid and n is an empirical factor, whose value is 11/9 for organic liquids. Equation (1.4) produces very good results for organic liquids. If temperature variation is not very important, and taking into account that the exponent n is close to 1, a good approximation of the Guggenheim-Katayama formula is the linear approximation

(1.5)

It is often easier and more practical to use a measured reference value (γ0, T0) and consider a linear change of the surface tension with the temperature,

(1.6)

(1.7)

Relations (1.5) and (1.6) are shown in figure 1.4. The value of the reference surface tension γ0 is linked to γ* by the relation

Figure 1.4 Representation of the relations (1.5) and (1.6).

(1.8)

Typical values of surface tensions and their temperature coefficients α are given in table 1.1.

The coefficient α being always negative, the value of the surface tension decreases with temperature. This property is at the origin of a phenomenon which is called either Marangoni convection or thermocapillary instability (figure 1.5). If an interface is locally heated by any heat source (such as radiation, convection or conduction), the surface tension is reduced on the heated area according to equations (1.5) or (1.6). A gradient of surface tension is then induced at the interface between the cooler interface and the warmer interface. We will show in section 1.3.7 that surface tensions can be viewed as forces; as a consequence, there is an imbalance of tangential forces on the interface, creating a fluid motion starting from the warm region (smaller value of the surface tension) towards the cooler region (larger value of the surface tension). This surface motion propagates to the bulk under the influence of viscosity. If the temperature source is temporary, the motion of the fluid tends to homogenize the temperature and the motion gradually stops. If a difference of temperature is maintained on the interface, the motion of the fluid is permanent; this is the case of a film of liquid spread on a warm solid. Depending on the contrast of temperature between the solid surface and the liquid surface, the motion of the liquid in the film has the morphology of convective rolls, hexagons or squares. Figure 1.6 shows hexagonal patterns of Marangoni convection in a film of liquid heated from below [7]. The white streamlines in the left image show the trajectories of the liquid molecules.

Figure 1.5 Sketch of interface motion induced by a thermal gradient between two regions of the surface. The motion of the interface propagates into the bulk under the action of viscous forces.

Figure 1.6 Marangoni convection, due to thermocapillary instabilities, makes hexagonal patterns in a thin film of liquid.

Reprinted with permission from [7], ©AIP 2005.

1.2.3 The Effect of Surfactants

“Surfactant” is the short term for “surface active agent”. Surfactants are long molecules characterized by a hydrophilic head and a hydrophobic tail, and are for this reason called amphiphilic molecules. Very often surfactants are added to biological samples in order to prevent the formation of aggregates and to prevent target; molecules from adhering to the solid walls of the microsystem (remember that microsystems have extremely large ratios between the wall areas and the liquid volumes). Surfactants diffuse in the liquid, and when reaching the interface they are captured because their amphiphilic nature prevents them from escaping easily from the interface. As a consequence, they gather on the interface, as is sketched in figure 1.7, lowering the surface tension of the liquid.

Figure 1.7 Schematic view of surfactants in a liquid drop.

As the concentration of surfactants increases, the surface concentration increases also. Above a critical value of the concentration, called CMC for Critical Micelle Concentration, the interface is saturated with surfactants, and surfactant molecules in the bulk of the fluid group together to form micelles. The evolution of the value of the surface tension as a function of the concentration in surfactants is shown in figure 1.8. At very low concentration, the slope is nearly linear; when the concentration approaches the CMC, the value of the surface tension drops sharply; above CMC, the value of the surface tension is nearly constant [8]. For example, pure water has a surface tension 72 mN/m, and water with Tween 80 at a concentration above the CMC has a surface tension of only 30 mN/m.

Figure 1.8 Evolution of the value of the surface tension as a function of the surfactant concentration.

In the limit of small surfactant concentration (c CMC), the surface tension can be expressed as a linear function of the concentration

(1.9)

Equation (1.9) is similar to equation (1.6) (different β of course). We have seen how a temperature gradient results in a gradient of surface tension leading to Marangoni type of convection. Similarly, a concentration gradient results in a gradient of surface tension, and consequently to a Marangoni convection, as in figure 1.9. Note that the direction of the motion is always towards the largest value of surface tension. Spreading of surfactant molecules on an interface can be easily seen experimentally: an instructive example is that of a thin paper boat with a cavity at the rear (figure 1.10). When the boat is placed gently on the surface of water, it rests on the surface of water suspended by surface tension forces. Upon putting a drop of soap solution/detergent in the notch, boat accelerates rapidly. Soap molecules try to spread over the surface of water. Since they are confined in the cavity of boat with only way out, they reduce the surface tension only at the rear, creating a net force which drives the boat forward.

Figure 1.9 Schematic Marangoni convection induced by a gradient of concentration.

Figure 1.10 Soap boat: a floating body contains a small volume of soap. At first, the soap exits the rear of the boat under Marangoni stress. Hence a low surface tension region is created behind the boat, whereas the unsoaped region in front of the boat has a larger surface tension. This difference of surface tension pulls the boat forward

(courtesy MIT: http://web.mit.edu/1.63/www/Lec-notes./Surfacetension/Lecture4.pdf[9]).

1.2.4 Surface Tension of a Fluid Containing Particles

Pure fluids are seldom used, especially in biotechnology. Very often micro- and nano-particles are present and transported by the fluid or they are voluntarily added to the fluid. Depending on their concentration and nature, the presence of micro- or nano-particles in the fluid might modify considerably the value of the surface tension (figure 1.11). At the same time, the presence of micro-particles reduces the contact angle. The notion of contact angle will be discussed later on in this chapter.

Figure 1.11 Left, surface tension of two different kinds of 10 μm polystyrene particles; right, corresponding equilibrium contact angles.

This decrease in surface tension depends on the concentration, size and nature of the micro-particles; at the molecular scale, it is linked to the interactions between particles and liquid on one hand, and between particles themselves on the other hand [10].

1.3 Laplace’s Law and Applications

Laplace’s law is fundamental when dealing with interfaces and micro-drops. It relates the pressure inside a droplet to the curvature of the droplet. This section first describes the mathematical notion of the curvature of a surface, then how it relates to surface tension and pressure, followed by a number of applications.

1.3.1 Curvature and Radius of Curvature

For a planar curve the radius of curvature at a point is the radius R of the osculating circle at that point – the circle which is the closest to the curve at the contact point (figure 1.12). The curvature of the curve at the point is defined by

Figure 1.12 Radius of curvature and osculating circle.

(1.10)

(1.11)

(1.12)

(1.13)

The situation is more complex for a surface. Any plane containing the vector normal to the surface intersects the surface along a curve. Each of these curves has its own curvature, called a sectional curvature, signed with respect to the orientation of the surface. The mean curvature of the surface is defined using the principal (maximum and minimum) curvatures κ1 and κ2 (figure 1.13) in the whole set of curvatures:

(1.14)

It can be shown that the principal curvatures κ1 and κ1 are located in two perpendicular planes. In fact, it turns out that the sum of the sectional curvatures in any two perpendicular directions is the same. Introducing the curvature radii in (1.14) leads to

(1.15)

Figure 1.13 Schematic view of the curvature radii of a surface. The dashed and solid curves are the intersections of the surface with two planes perpendicular to the surface and each other.

(1.16)

1.3.2 Derivation of Laplace’s Law

Suppose a spherical droplet of liquid surrounded by a fluid. Let us calculate the work necessary to increase its volume from the radius R to the radius R + dR (figure 1.15). The part of the work due to the internal volume increase is

Figure 1.15 Schematic of a liquid drop immersed in a fluid; initially, the droplet radius is R and its surface area A0. An increase of its radius by a quantity dR corresponds to the new surface area A1 and the pressure P1.

(1.17)

where dV0 is the increase of the volume of the droplet,

(1.18)

The work to pull out the external fluid is

(1.19)

where dV1 is the decrease of the external volume, equal to -dV0. The work corresponding to the increase of interfacial area is

(1.20)

where dA is the increase of the surface area. The mechanical equilibrium condition is then

(1.21)

Substituting the values of the work found previously, it follows that

(1.22)

Equation (1.22) is the Laplace equation for a sphere. The reasoning we have done to obtain equation (1.22) can be generalized,

(1.23)

For simplicity, we have derived Laplace’s equation for the case of a sphere, but we can use (1.23) for an interface locally defined by two (principal) radii of curvature R1 and R2; the result would have been then

(1.24)

For a cylindrical interface, as sketched in figure 1.16, one of the two radii of curvature is infinite, and Laplace’s equation reduces to

Figure 1.16 Laplace’s law for a cylindrical interface.

(1.25)

Equation (1.24) is called Laplace’s law. Keep in mind that it is closely linked to the minimization of the energy. Laplace’s law is fundamental when dealing with interfaces, micro-drops and in digital microfluidics. In the following section, we give some examples of application of Laplace’s law.

1.3.3 Examples of Application of Laplace’s Law

Amongst other things, Laplace’s law explains many phenomena occurring during electrowetting actuation. We will talk about the use of Laplace’s law for electrowetting in chapter 10. In this section, we present some applications of Laplace’s law outside the electrowetting domain.

1.3.3.1 Pressure in a Bubble

The internal pressure in a bubble can be easily derived from the Laplace law (Fig. 1.17). If we assume that the thickness of the liquid layer is negligible in front of the bubble radius R, Laplace’s law yields

Figure 1.17 Comparison of the internal pressure in a bubble (a) and in a droplet of same radius (b).

(1.26)

The pressure in a bubble can be relatively high. For this reason, in children’s kits for blowing bubbles, surfactants are added to the soap solution to facilitate bubble inflation.

1.3.3.2 Liquid Transfer From a Smaller Drop to a Bigger Drop

It has been observed that when two bubbles or droplets are connected together, there is a fluid flow from the small bubble/droplet to the larger one (figure 1.18). This is a direct application of Laplace’s law: the pressure inside the small bubble/droplet is larger than that of the larger bubble/droplet, inducing a flow from towards the latter. This flow continues until the smaller bubble/droplet disappears to the profit of the larger one.

Figure 1.18 Fluid flow from the smaller bubble/droplet to the larger since the smaller bubble has higher curvature and thus higher pressure.

In biotechnology, this observation has been used to design microsystems where a microflow in a channel is set up by a difference of size of two droplets placed at both ends [12].

1.3.3.3 Precursor Film and Coarsening

At the beginning of this Chapter, we saw that the concept of an infinitely thin interface and a unique contact angle is a mathematical simplification of reality. When a partially wetting droplet is deposited on a flat solid surface, a very thin film of a few nanometers spreads before the contact line, and the contact between the liquid and the solid resembles the sketch of figure 1.19. The precursor film can be explained by thermodynamic considerations: because a jump between the chemical potential of the gas and of the solid is not physical, liquid molecules intercalate between the gas and the solid [13]. Molecules of the liquid progressively spread under the action of the “disjoining pressure” caused by the van der Waals interactions between the liquid and solid molecules [14]. Precursor films exist for hydrophilic contact for static droplets as well as for dynamic wetting. Mechanisms of the advancing precursor film are still a subject of investigation [15].

Figure 1.19 Interface with precursor film.

Such precursor films have been observed for different wetting situations, as shown in figure 1.20. Note the extreme thinness of the film in the photographs. When droplets of different size are deposited on a flat solid surface, if the droplets are sufficiently close to one another, it is observed that the smaller droplets disappear to the profit of the large droplets. This phenomenon is called “coarsening”. Experimental evidence of coarsening is shown in figure 1.21.

Figure 1.20 (a) Precursor film of spreading molten alloy (from [16], ©Elsevier, 2002); (b) precursor film of a liquid plug inside a carbon nanotube (from [17], ©AIP, 2005); (c) AFM scan on liquid crystal precursor film (from [18], courtesy Nanolane).

Figure 1.21 Experimental observation of coarsening: the number of droplets diminishes; only the largest droplets remain: (a) beginning of the observation, (b) increase in size of the large droplets and vanishing of the smaller droplets [19]

(Courtesy Michael Bestehorn).

The explanation of the phenomenon requires two steps: first, the existence of a precursor film (an extremely thin film on the solid surface spreading around each droplet) that links the droplets together; second, as in the previous example, the pressure is larger in a small droplet according to Laplace’s law, and there is a liquid flow towards the largest droplet (figure 1.22). The precursor film is very thin, thus the flow rate between droplets is very small and mass transfer is extremely slow. Hence experimental conditions require that the droplets do not evaporate.

Figure 1.22 When two droplets are linked by a precursor film, a fluid flow is established from the smaller droplet to the larger droplet. The smaller droplet progressively disappears.

A numerical simulation of the transfer of liquid from the smaller droplet to the larger droplet can be easily performed using the Evolver as shown in figure 1.23. In the model, the precursor film is not modeled; the two initial droplets are simply united in the same logical volume.

Figure 1.23 Evolver simulation of two droplets of slightly different volumes and supposedly in communication through the precursor film: liquid is transferred from the smaller droplet to the larger one.

1.3.3.4 Pressure in Droplets Constrained Between Two Parallel Plates

When using Laplace’s law, one should be careful of the orientation of the curvature. A convex surface has two positive radii of curvature. A “saddle” surface has one positive and one negative curvature radius. Take the example of a water droplet flattened between two horizontal plates (we will see in Chapter 3 that this situation is frequent in EWOD-based microsystems [20]). Suppose that the droplet is placed at the intersection of a hydrophobic band and a hydrophilic band. As a result, the droplet is squished by the hydrophobic band and elongated on the hydrophilic band (figure 1.24). The pressure in the droplet is given by the Laplace law

Figure 1.24 Sketch of a water droplet pinched by a hydrophobic surface. Case of a droplet constrained between two parallel planes (the upper plane has been dematerialized for visualization). Simulation performed with the Surface Evolver [21].

(1.27)

where R1, R2, R3 and R4 are respectively the horizontal curvature radius in the hydrophilic region, the vertical curvature radius in the hydrophilic region, the horizontal curvature radius in the hydrophobic region, and the vertical curvature radius in the hydrophobic region. Taking into account the sign of the curvatures, we obtain

(1.28)

The pressure in the drop being larger than the exterior pressure is equivalent to satisfying either of the relations

(1.29)

or

(1.30)

The vertical curvature radius R4 in the hydrophobic region is smaller than the concave horizontal radius R3 and the vertical curvature radius in the hydrophilic region R2 is larger than the convex horizontal radius R1. We shall see in Chapter 4 the use of a hydrophobic band to “cut” the droplet into two daughter droplets. For that to happen, the curvature radius R3 must be sufficiently small, so that the two concave contact lines contact each other. The inequality (1.29) then produces a condition on the level of hydrophobicity required to obtain droplet division.

1.3.3.5 Zero Pressure Surfaces: Example of a Meniscus on a Rod

Laplace’s law is often seen as a law determining a pressure difference on the two sides of the interface from the observation of curvature. But it is interesting to look at it the other way: knowing the pressure difference, what conclusion may be reached on the curvature of the interface?

Figure 1.25 Sketch of rod dipped into a liquid (wetting case).

(1.31)

where R1 and R2 are the two (signed) principal curvature radii. R1 (z) is the (negative) curvature radius of the vertical profile at the elevation z, and R2(z) is the (positive) radius of the osculating circle perpendicular to the vertical (which is a tilted circle, not the circular horizontal cross-section). Assuming that the system is small enough that the gravity term can be neglected compared to the surface tension term, we are left with

(1.32)

This is the equation of a zero curvature surface, also called a minimal surface. The equation of the surface can be obtained by writing that the vertical projection of the surface tension force is constant [22]. Using the notations of figure 1.26, we find

Figure 1.26 Vertical profile of liquid surface at the vicinity of a vertical rod.

(1.33)

Substituting the relation tan we are left with

(1.34)

Integration of (1.34) yields the equation of the vertical profile

(1.35)

(1.36)

(1.37)

(1.38)

Figure 1.27 shows the deformed surface obtained by a numerical simulation. Relation (1.38) shows that the elevation h0 along the wire increases with the surface tension. At first sight, this may seem a paradox because the surface is pulled tighter when the surface tension increases. However, we show later in this Chapter that the capillary force exerted by the wire is proportional to the surface tension. The force pulling the surface is thus larger for high surface tension liquids.

Figure 1.27 Vertical wire dipped into a fluid. The surface is deformed depending on the surface tension of the liquid and on the contact angle (Surface Evolver calculation). Left: view of the surface in the vicinity of the wire. Right: view from below showing the gain in elevation of the surface at the contact of the wire.

1.3.3.6 Self Motion of a Liquid Plug Between Two Non-Parallel Wetting Plates

It was first observed by Hauksbee [23] that a liquid plug between two non-parallel wetting plates moves towards the narrow gap. A. sketch of the plug is shown in figure 1.28. Laplace’s law furnishes a very clear explanation of this phenomenon. Suppose that figure 1.28 is a wedge (2D situation) and let us write Laplace’s law for the left side interface

Figure 1.28 Sketch of a liquid plug moving under capillary forces between two plates. The contact angle is θ < 90°.

(1.39)

and for the right side interface

(1.40)

Subtraction of the two relations leads to

(1.41)

Next, we show that R2 < R1. Looking at figure 1.29, we have

Figure 1.29 Curvature of the interface in a dihedral.

(1.42)

where d2 is the half-distance between the plates at the narrower contact point. The angle β is linked to θ and α by the relation

(1.43)

Finally we obtain

(1.44)

Using the same reasoning with a meniscus oriented in the opposite direction, we obtain the expression of R1

(1.45)

Comparing relations (1.44) and (1.45), noting that d2 < d1 and cos(α – θ) > cos(α + θ), we deduce that R2 is then smaller than R1, and P1 > P2. The situation is not stable. Liquid moves from the high pressure region to the low pressure region and the plug moves towards the narrow gap region. It has also been observed that the plug accelerates; it is due to the fact that the difference of the curvatures in equation (1.41) is increasing when the plug moves to a narrower region. Bouasse [24] has remarked that the same type of motion applies for a cone, where the plug moves towards the tip of the cone. In reality, Bouasse used a conical frustum (slice of cone) in order to let the gas escape during plug motion.

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 A human body – or any mammalian organism – respects the rules of physics. Take the example of blood vessels. The arrangement of blood vessel networks very often satisfies Murray’s law (figure 1.30). In 1926, Murray observed the morphology of the blood system and found a very general relation between the dimensions of a “parent” branch and of a “daughter” branch, and he found that the same relation applies at any level of bifurcation. Soon after, he published this discovery [25]. Since that time, this relation is known as Murray’s law and can be written as

Figure 1.30 Left: Schematic of blood vessel system. Right: photograph of blood vessel division in chickens from [26].

(1.46)

(1.47)

(1.48)

(1.49)

These relations can be developed further to show that the wall friction is the same at each level [26,27]. This property simply stems from the expression of the shear stress of a cylindrical duct,

(1.50)

Murray showed that, on a physiological point of view, such a relation minimizes the work of the blood circulation. The important thing here is that the shear stress is constant in most blood vessel networks. Now what about the normal stress?

1.3.3.7.2 Normal Stress in Vascular Networks In the particular case of human or mammalian blood systems, the normal stress is simply the internal pressure, because, to a first approximation, the flow is purely axial and there is no radial component of the velocity. It has been observed that the thickness of the walls of blood vessels satisfies Laplace’s law (figure 1.31). In this particular case, the surface tension is replaced by the wall tension T, and Laplace’s law becomes

(1.51)

At a given distance from the heart, the pressure is approximately the same and Laplace’s equation (1.51) has the consequence that the wall tension increases together with the radius. As a consequence, arteries have larger wall thickness than veins, and similarly veins compared to capillaries. In medicine, an aneurysm is a localized, blood-filled dilation (balloon-like bulge) of a blood vessel caused by disease or weakening of the vessel wall. Aneurysms most commonly occur in arteries at the base of the brain and in the aorta (the main artery coming out of the heart, an aortic aneurysm). It has been observed that, as the size of an aneurysm increases, there is an increased risk of rupture, which can result in severe hemorrhage, other complications or even death. This expansion is a direct consequence of the preceding reasoning: if the thickness of the vascular wall is such as it withstands a given tension T, an increase of the radius will require a higher tension of the wall, and therefore the aneurysm will continue to expand until it ruptures (figure 1.32). A similar logic applies to the formation of diverticuli in the gut [28].

Figure 1.32 Sketch of an aneurysm. The wall tension required at the location of the aneurysm exceeds the wall tension that the thickness of the wall can withstand.

1.3.3.8 Laplace’s Law in Medicine: the Example of Lung Alveoli

It is very tempting to refer automatically to Laplace’s law because of its simplicity. But one should refrain from doing that uncritically. A striking example is that of lung ventilation. Ventilation of lungs has been widely studied for medical purposes. It was usual to consider the alveoli such as spherical balloons inflated during lung ventilation (fig. 1.33). The problem is that the alveoli are connected, and when applying Laplace’s law, the air in the smaller alveoli should be driven to the larger alveoli and a general collapse of the lungs would occur. Because the collapse of the alveoli does not – luckily – correspond to reality, it has been suggested that the concentration of surfactant in the alveoli is not uniform and compensates for the different pressures. Recently, a different, more realistic analysis has been made [29]: the alveoli are not “free” spheres but are packed together (figure 1.34) and there are pores in the alveoli walls. Alveoli cannot expand freely, and they are limited in their inflation. As a result, smallest alveoli do not collapse during ventilation, because large alveoli cannot grow indefinitely, and Laplace’s law is not the answer in this kind of problem.

Figure 1.33 Wrong sketch for the alveoli leading to an improper application of Laplace’s law.

Figure 1.34 Image of lung alveoli; Detail from an original slide generously provided by A. Mescher.

1.3.3.9 Laplace’s Law in a Gravity Field

(1.52)

The curvature is then smaller at the top of the drop than it is at the bottom of the drop. This effect can well be seen in an Evolver simulation (figure 1.35).

Figure 1.35 Large droplet flattened by the action of gravity: The vertical curvature radius R1’ is small at the bottom of the drop because the internal hydrostatic pressure is larger and the horizontal curvature radius R2’ relatively large. At the top, the curvatures R1 and R2 are equal and relatively large.

1.3.3.10 Generalization: Laplace’s Law in Presence of a Flow Field

The expression of Laplace’s law derived in section 1.3 assumed totally static conditions, or at the least that the shear rate of the flow field close to the interface is negligible in comparison with the surface tension. This is indeed often the case since usually capillary numbers are small in microfluidic systems. Recalled that the capillary number is a non-dimensional number characterizing the ratio between inertial forces and capillary forces. However, the flow field effect on the interface cannot be always neglected. For example, systems like flow-focusing devices are currently used in biotechnology to produce emulsions and encapsulates [30–32]. In such systems, the dynamic flow field exerts a considerable force on the interface, as has been shown by Tan et al. [33] (figure 1.36). In such a case the balance of the forces on the surface is more complicated: the complete stress tensor has to be taken into account, and we obtain the generalized Laplace law for liquid 1 and 2:

Figure 1.36 Left: sketch of the interface; right: photo of a flow focusing device; the generalized Laplace’s law applies for the incoming discontinuous phase (inside the dotted circle).

(1.53)

where ni is the unit normal vector, and σ′ik the viscous part of the stress tensor [34]. Note the implicit summation on the repeated index k on the right side of (1.53).

1.3.4 Wetting - Partial or Total Wetting

So far, we have dealt with interfaces between two fluids. Triple contact lines are the intersections of three interfaces involving three different materials: for example a droplet of water on a solid substrate in an atmosphere has a triple contact line. Liquids spread differently on a horizontal plate according to the nature of the solid surface and that of the liquid. In reality, it depends also on the third constituent, which is the gas or the fluid surrounding the drop. Two different situations are possible: either the liquid forms a droplet, and the wetting is said to be partial, or the liquid forms a thin film wetting the solid surface, the horizontal dimension of the film depending on the initial volume of liquid (figure 1.37). For example, water spreads like a film on a very clean and smooth glass substrate, whereas it forms a droplet on a plastic substrate. In the case of partial wetting, there is a line where all three phases come together. This line is called the contact line or the triple line.

Figure 1.37 Wetting is said to be total when the liquid spreads like a film on the solid surface.

A liquid spreads on a substrate in a film if the energy of the system is lowered by the presence of the liquid film (figure 1.38). The surface energy per unit surface of the dry solid surface is γSG; the surface energy of the wetted solid is γSL + γLG. The spreading parameter S