Microfluidics for Biotechnology, Second Edition

Microfluidics for Biotechnology Second Edition

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Microfluidics for Biotechnology Second Edition Jean Berthier Pascal Silberzan

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10 9 8 7 6 5 4 3 2 1

Contents Preface

xi

Acknowledgements

xv

  Chapter 1   Dimensionless Numbers in Microfluidics 1.1  1.2  1.3  1.4 

Introduction Microfluidic Scales Buckingham’s Pi Theorem Scaling Numbers and Characteristic Scales 1.4.1  Micro- to Nanoscales 1.4.2  Hydrodynamic Characteristic Times 1.4.3  Newtonian Fluids 1.4.4  Non-Newtonian Fluids 1.4.5  Droplets and Digital Microfluidics 1.4.6  Multiphysics 1.4.7  Specific Dimensionless Numbers and Composite Groups References

  Chapter 2   Microflows 2.1  Introduction   2.1.1  On the Importance of Microfluidics in Biotechnology   2.1.2  From Single Continuous Flow to Droplets 2.2  Single-Phase Microflows   2.2.1  Navier-Stokes (NS) Equations   2.2.2  Non-Newtonian Rheology   2.2.3  Laminarity of Microflows   2.2.4  Stokes Equation   2.2.5  Hagen-Poiseuille Flow   2.2.6  Pressure Drop and Friction Factor   2.2.7  Bernoulli’s Approach   2.2.8  Modeling: Lumped Parameters Model   2.2.9 Microfluidic Networks: Worked Example 1—Microfluidic Flow Inside a Microneedle 2.2.10 Microfluidic Networks: Worked Example 2—Plasma Extraction from Blood 2.2.11 Hydrodynamic Entrance Length: Establishment of the Flow 2.2.12  Distributing a Uniform Flow into a Microchamber

1 1 1 1 3 3 3 4 6 8 11 13 15

17 17 17 17 18 19 24 32 35 38 40 45 48 50 56 58 60 

vi

Contents

2.2.13  The Example of a Protein Reactor 2.2.14  Recirculation Regions 2.2.15  Inertial Effects at Medium Reynolds Numbers: Dean Flow 2.2.16  Microflows in Flat Channels: Helle-Shaw Flows 2.3  Conclusion References

61 62 65 69 70 70

  Chapter 3   Interfaces, Capillarity, and Microdrops

73

3.1  Introduction 3.2  Interfaces and Surface Tension 3.2.1  The Notion of Interface 3.2.2  Surface Tension 3.3  Laplace Law and Applications 3.3.1  Curvature Radius and Laplace’s Law 3.3.2  Examples of the Application of Laplace’s Law 3.4  Partial or Total Wetting 3.5  Contact Angle: Young’s Law 3.5.1  Young’s Law 3.5.2  Young’s Law for Two Liquids and a Solid 3.5.3  Generalization of Young’s Law—Neumann’s Construction 3.6  Capillary Force and Force on a Triple Line 3.6.1  Introduction 3.6.2  Capillary Force Between Two Parallel Plates 3.6.3  Capillary Rise in a Tube—Jurin’s Law 3.6.4  Capillary Rise Between Two Parallel Vertical Plates 3.6.5  Capillary Pumping 3.6.6  Force on a Triple Line 3.6.7  Examples of Capillary Forces in Microsystems 3.7  Pinning and Canthotaxis 3.7.1  Theory 3.7.2  Pinning of an Interface Between Pillars 3.7.3  Droplet Pinning on a Surface Defect 3.7.4  Pinning of a Microdroplet—Quadruple Contact Line 3.7.5  Pinning in Microwells 3.8  Microdrops 3.8.1  Shape of Microdrops 3.8.2  Drops on Inhomogeneous Surfaces 3.9  Conclusions References

73 73 73 76 80 80 84 86 87 87 90 91 93 93 93 95 97 98 99 100 101 101 101 103 104 105 105 105 118 126 128

  Chapter 4   Digital, Two-Phase, and Droplet Microfluidics

131

4.1  Introduction 4.2  Digital Microfluidics

131 131

Contents

vii

4.2.1  Introduction 4.2.2  Theory of Electrowetting 4.2.3  EWOD Microsystems 4.2.4  Conclusion 4.3  Multiphase Microflows   4.3.1  Introduction   4.3.2  Droplet and Plug Flow in Microchannels   4.3.3  Dynamic Contact Angle   4.3.4  Hysteresis of the Static Contact Angle   4.3.5  Interface and Meniscus   4.3.6  Microflow Blocked by Plugs   4.3.7  Two-Phase Flow Pressure Drop   4.3.8  Microbubbles   4.3.9  Liquid-Liquid Extraction 4.3.10 Example of Three-Phase Flow in a Microchannel: Droplet Engulfment 4.4  Droplet Microfluidics 4.4.1  Introduction: Flow Focusing Devices (FFD) and T-Junctions 4.4.2  T-Junctions 4.4.3  Micro Flow Focusing Devices (MFFD) 4.4.4  Highly Viscous Fluids—Encapsulation 4.5  Conclusions References

170 173 173 174 182 187 194 195

  Chapter 5   Diffusion of Biochemical Species

201

5.1  Introduction 5.2  Brownian Motion 5.3  Macroscopic Approach: Concentration 5.3.1  Fick’s Law 5.3.2  Concentration Equation 5.3.3  Spreading from a Point Source—1D Case 5.3.4  Semi-Infinite Space: Ilkovic’s Solution 5.3.5  Example of Diffusion Between Two Plates 5.3.6  Radial Diffusion 5.3.7  Diffusion Inside a Microchamber 5.3.8 Diffusion Inside a Capillary: The Example of Simultaneous PCRs 5.3.9  Particle Size Limit: Diffusion or Sedimentation 5.4  Microscopic (Discrete) Approach 5.4.1  Monte Carlo Method 5.4.2 Diffusion in Confined Volumes: Drug Diffusion in the Human Body 5.5  Conclusion References

131 131 151 160 161 161 161 162 163 164 164 167 168 168

201 201 202 203 203 207 208 209 211 213 214 220 222 222 226 235 236

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Contents

  Chapter 6   Transport of Biochemical Species and Cellular Microfluidics

237

6.1  Introduction 6.2  Advection-Diffusion Equation   6.2.1  Governing Equation for Transport   6.2.2  Source Terms   6.2.3  Boundary Conditions   6.2.4  Coupling with Hydrodynamics   6.2.5 Physical Properties as a Function of the Concentration of the Species   6.2.6  Dimensional Analysis and Peclet Number   6.2.7  Concentration Boundary Layer   6.2.8  Numerical Considerations   6.2.9  Taylor-Aris Approach 6.2.10  Distance of Capture in a Capillary 6.2.11  Determination of the Diffusion Coefficient 6.2.12  Mixing of Fluids 6.3  Trajectory Calculation 6.3.1  Trajectories of Particles in a Microflow 6.3.2  Ballistic Random Walk (BRW) 6.4  Separation/Purification of Bioparticles 6.4.1  The Principle of Field Flow Fractionation (FFF) 6.4.2   Chromatography Columns 6.5  Cellular Microfluidics 6.5.1  Flow Focusing 6.5.2  Pinched Channel Microsystems 6.5.3 Deterministic Arrays—Deterministic Lateral Displacement (DLD) 6.5.4  Lift Forces on Particles 6.5.5  Dean Flows in Curved Microchannels 6.5.6  Bifurcation Channels 6.5.7  Recirculation Chambers 6.6  Conclusion References

289 291 294 294 297 298 299

  Chapter 7   Biochemical Reactions in Biochips

303

7.1  Introduction 7.2  From the Principle of Biorecognition to the Development of Biochips 7.2.1  Introduction to Biorecognition 7.2.2  Biorecognition 7.2.3  Biochip Technology 7.3  Biochemical Reactions 7.3.1  Rate of Reaction 7.3.2  Michaelis Menten Model

237 237 237 240 241 242 244 247 248 251 253 258 264 265 271 272 275 279 279 280 282 283 287

303 303 303 304 306 309 309 315

Contents

ix

7.3.3  Adsorption and the Langmuir Model 7.3.4  Biological Reactions 7.4  Biochemical Reactions in Microsystems 7.4.1  Homogeneous Reactions 7.4.2  Heterogeneous Reactions 7.5  Conclusion References

322 325 327 328 332 357 357

  Chapter 8   Experimental Approaches to Microparticles-Based Assays

361

8.1  A Few Biological Targets 8.1.1  Biopolymers 8.1.2  Some Aspects of Cells 8.2  Microparticles as Biotechnological Tools 8.2.1  Fluorescent Particles 8.2.2  Other Micro- and Nanoparticles 8.2.3  Chemical Modification of Surfaces 8.3  Experimental Methods of Characterization 8.3.1  Microscopies 8.3.2  Physical Characterization: Light Scattering 8.3.3  Biochemical Characterization 8.4  Molecular Micromanipulation 8.4.1  Force Measurements 8.4.2  Optical Tweezers 8.4.3  Flow-Based Techniques References Selected Bibliography

361 362 367 368 369 371 375 376 376 386 387 391 391 392 393 394 396

  Chapter 9   Magnetic Particles in Biotechnology

397

  9.1  Introduction   9.1.1  The Principle of Functional Magnetic Beads   9.1.2  Composition and Fabrication of Magnetic Beads   9.1.3 An Example of Displacement by Magnetic Beads for Biodetection   9.1.4  The Question of the Size of the Magnetic Beads   9.2  Characterization of Magnetic Beads   9.2.1  Paramagnetic Beads   9.2.2  Ferromagnetic Microparticles   9.3  Magnetic Force   9.3.1  Paramagnetic Microparticles   9.3.2  Ferromagnetic Microparticles   9.4  Deterministic Trajectory   9.5  Example of a Ferromagnetic Rod   9.5.1  Governing Equations   9.5.2  Analytical Solution for the Magnetic Field

397 397 398 400 401 402 402 403 403 404 404 405 406 407 408



Contents

  9.5.3  Trajectories (Carrier Fluid at Rest)   9.5.4  Trajectories (Carrier Fluid Convection)   9.6  Magnetic Repulsion   9.7  Magnetic Beads in EWOD Microsystems   9.8  Example of a Separation Column   9.9  Concentration Approach 9.10  Example of MFFF   9.10.1  Trajectories   9.10.2  Concentration of Magnetic Beads   9.10.3  Results and Comparison 9.11  Assembly of Magnetic Beads—Magnetic Beads Chains 9.12  Magnetic Fluids   9.12.1  Introduction   9.12.2  Magnetic Force on a Plug of Ferrofluid   9.12.3  Notes 9.13  Magnetic Micromembranes   9.13.1  Principle   9.13.2  Deflection of Paramagnetic Micromembranes   9.13.3  Oscillation of Magnetic Membranes 9.14  Conclusion   References

409 410 412 413 415 417 419 420 422 423 423 428 428 429 430 430 431 431 433 436 436

  Chapter 10   Micromanipulations and Separations Using Electric Fields

439

10.1  Action of a DC Electric Field on a Particle: Electrophoresis   10.1.1  The Debye Layer   10.1.2  Electro-Osmosis   10.1.3  Electrophoresis of a Charged Particle   10.1.4  Electrophoresis of DNA   10.1.5  Electrophoresis of Proteins   10.1.6  Cell Electrophoresis 10.2  Dielectrophoresis   10.2.1  Theoretical Basis   10.2.2  The Clausius-Mossoti Factor   10.2.3  Optimization of the Electric Field   10.2.4  Characterization of Particles   10.2.5  Electrorotation and Traveling Wave   10.2.6  Instabilities   10.2.7  DEP-Based Separations References

439 439 442 443 445 451 453 453 453 456 457 458 460 462 464 468

  Chapter 11   Conclusion

473

List of Symbols About the Authors Index

475 477 479

Preface Since the concept of the first DNA biochip, biotechnologies have soared and have deeply changed the world of biology; and they have already direct implications on any of us. Starting from the very beginning of this science in the 1980s, spectacular advances have been made, such as the completion of the determination of the human genome sequence, and dramatic changes have broken out in the field of proteomics and, more recently, new breakthroughs have been made in the domain of cellular analysis. The field of investigations of biotechnology has constantly increased, from the first biochips built to analyze sequences of DNA and investigate its mutations to protein analysis and the study of role of proteins in the human life and the comprehension of the complex mechanisms that take place inside the cells. Biotechnology is a science that is not only dedicated to assist biologists in their desire to understand the complexity of life. It also has very practical applications, especially in bioanalysis and biodetection. For example, progress in the rapidity of detection of viruses has been spectacular, and it is expected that direct analysis of viruses may soon be performed in a few minutes at the doctor’s office. Biotechnology is not restricted to in vitro analysis, but has direct implications for in vivo treatments. Concerning the in vivo domain, the impact of the new technologies is manifold. First, the monitoring of the correct functioning of some vital organs in patients at risk is going to be possible. Second, miniaturization techniques will greatly reduce the invasiveness of external interventions inside the human body. Third, new biotechnological devices may help internal drug guidance to find their targets inside the human body. Fourth, new cell encapsulation techniques are going to improve considerably the human organs grafting. Finally, a trend that many of us will experience in the years to come is towards automated medical help and monitoring right at home. Biotechnological microsystems are called different names having more or less the same physical meaning, such as biochips, or bioMEMS—for microelectromechanical systems, or lab-on-a-chip—meaning that many of the different operations performed in a lab are done on a single microdevice and sometimes mTAS (micrototal analysis systems). It is surprising how the concept and development of the first DNA biochip opened the way to a completely new domain of technology. It soon appeared that many other concepts could be imagined and that miniaturization had many advantages in biochemical science. A first advantage resides in the automation and streamlining of biological processes, as shown with the DNA biochip: using microchips with thousands of wells, each one testing a specific DNA sequence, considerably reduces the time needed for xi

xii

Preface

the recognition process. Another example is that of the proteomic reactors breaking proteins into peptides by enzymatic catalyst inside microchannels; the peptides are then transported by a buffer fluid to a spray injector and then to a mass spectrometer where the peptides are identified. Such a protein chip realizes many operations in sequence which otherwise would have needed many different manipulations and a lot of time. Another advantage of biotechnological microdevices is the reduction of costs of biological analysis due not only to the streamlining and parallelization of the operations, but also to the reduction of the quantities of reactants. Because the reactants needed to perform the sequence of biological reactions are usually quite expensive, it is important that they be used in very limited quantities. Of course, biochips may still be somewhat expensive, especially if etched silicon is used, but it is more than compensated by the gain in the mass of reactants. It has been also found that the danger of working with toxic, dangerous bacteria, or even explosive substances—in chemistry—is greatly reduced by the minia­ turization of the reaction scale. Explosive substances are not dangerous anymore at very low concentrations, and dangerous viruses and toxic bacteria can be more easily confined in microsystems. It is also expected that biochips or bioMEMS can provide higher sensitivity than usual macroscopic systems. For example, some diseases caused by a virus can be detected earlier, at a number of viruses much smaller than the usual diagnostics, leading to better treatment and a reduction of the contagion possibility. On the research point of view, there are also many advantages brought by biotechnological microsystems. For example, in vivo interventions are facilitated by the small size of the new biotechnological devices, reducing the invasiveness of the drug delivery system. Another example is the technology of encapsulated active microparticles targeted specifically in the human body. It is expected that biochips will also contribute to the discovery of new drugs, by testing many new molecules at the same time on living cells isolated in lab-on-a-chip for cells. In that sense, biotechnology is increasingly considered a very useful complement to biology itself, as it may contribute to discover new drugs by automatically testing many molecules at the same time. As mentioned with the complementarity between biotechnology and biology, we point out here that the central theme of biotechnology is the control, displacement, and guidance of the different micro-sized objects that are present in a biologic buffer liquid. In reality, there are three types of biological objects. The first type of biological objects is the “natural” ones like DNA, proteins, antibodies, antigens, peptides, cells, bacteria, and red and white blood cells, which constitute the biological targets or the objects to study. Their sizes range from about 20 nm (short strands of DNA) to 200 mm (for the larger cells). The second type of biological object is constituted by micro- and nanoparticles that we may consider “artificial” or “accessory” and that are used as tools to perform specific tasks. In this category, we can list magnetic beads, different fluorophores (CY3, CY5, FITC), quantum dots, gold microparticles, polypyrolles, carbon nanowires, and surfactants. These objects are generally smaller than the previous ones, ranging from 10 nm to 2 mm. Recently, a third type

Preface

xiii

of biological objects has appeared: the encapsules. These objects are new composites—like polymeric or gelled capsules containing cells, bacteria, or proteins—and they bear great hopes in medical treatments; their size can vary between a few microns and 500 mm. Remark that some “natural” biological objects can also be used as tools, especially for biorecognition processes. A DNA strand can be considered as a tool to immobilize a complementary DNA strand. We will refer to all these objects under the names of micro- and nanoparticles and macromolecules. A simplified statement is that biologists study the functional and chemical behavior of biological objects, whereas biotechnologists focus on the mechanical and chemical behavior of these objects. A book on a subject as rapidly evolving as microfluidics for biotechnology necessarily reflects the state of the art at a certain time. In the first edition of this book in 2005, the focus was on microfluidics for lab-on-chips dedicated to DNA analysis and immunoassays for protemics. Since that time, cellomics has seen considerable developments. Many efforts have been dedicated to the study of cells, which is the key to understanding the functioning of the complex human system and to the development of new drugs. Attention has been given to single-cell studies and communication between cells. Hence, transport and manipulation of cells have become an important topic. In the wake of the development of cellular mechanics and cellular microfluidics, triggered by cell transport and encapsulation applications, digital and especially droplet microfluidics have seen a considerable boost. Another recent evolution is the development of the use of biological liquids—whole blood and alginates, for example—in in vitro biochips. This evolution has promoted the study of the rheology of biopolymers and their non-Newtonian, viscoelastic behavior in microsystems. This second edition reflects this evolution and incorporates new concepts in cellular microfluidics and cell manipulation, along with rheological considerations on viscoelastic liquids. A new chapter devoted to digital and droplet microfluidics has been introduced. However, it has seemed important to the authors to keep the theoretical fluid mechanics basis in order to maintain the coherence of the text and to provide a stand-alone book. Hence, this new edition has globally conserved the frame of the first edition. Physical laws do not change between macroscopic and microscopic scales, but the relative importance of the different forces is considerably changed between these two scales. In Chapter 1, the scaling of the different forces as a function of the dimension of the system is presented and the dominating forces and phenomena are pointed out. Nondimensionless numbers and characteristic times of the different phenomena associated to microfluidics are presented and discussed. Chapters 2, 3, and 4 are dedicated to the microfluidics aspects of the buffer fluid in biochips. In order to predict correctly the behavior of particles, the physical behavior of the buffer (carrier) fluid must be first determined. Chapter 2 treats continuous single-phase microflows. Theoretical bases are given first. A section dedicated to the rheology of non-Newtonian fluids in biotechnology has been incorporated, taking the example of alginate solutions, which are now widely used. Because of the growing importance of cell separation devices, emphasis has been placed on microfluidic networks.

xiv

Preface

New microfluidic solutions make use of droplets. For this reason, it has been found that a chapter dedicated to the notions of interface, capillarity, and static droplet behavior was needed. This is the object of Chapter 3. To go further in the investigations of new multiphase microfluidic solutions, Chapter 4, devoted to digital and droplet microfluidics, has been added to this second edition. These new techniques appear to be promising ways of transporting biological objects in extremely small liquid volumes. In digital microfluidic applications, microdrops of a few tenths of micrometers are moved individually step by step on a flat surface. In droplet microfluidic applications, same liquid volumes are transported within an immiscible continuous microflow. Because the micro-and nanoparticles and macromolecules in which we are interested are much larger than the fluid molecules, their behavior differs from that of the fluid. Therefore, Chapters 5 and 6 focus on the mechanical behavior of the particles themselves, under the action of diffusion (Chapter 5) and transport by advection (Chapter 6). Different numerical approaches are presented, such as continuum-based numerical and discrete methods. A special addition concerning cellular microfluidics (i.e., transport of cells in a carrier flow) is included in Chapter 6. All the studies on the buffer (carrier) fluid flow and the behavior of the particles in this flow are aimed at controlling the motion of the particles of interest to have them placed at some specific location in order to be able to perform the desired reaction or analysis. Chapter 7 is dedicated to the study of biochemical reactions. First, the principle of biorecognition is presented and the different biochemical reactions to recognize DNA sequences and antibodies are studied. Next, to take into account the transport of the reactants by the buffer fluid, a coupled approach including diffusion/advection of reactants and the biochemical reaction itself is examined. It has been found useful to precisely determine the nature and the characteristic of the most used targets in biotechnology (biological targets and synthetic particles). Chapter 8 describes the characteristics of these particles and introduces an experimental aspect by presenting the methods used to manipulate or characterize them. Because the transport by the buffer fluid is often not specific enough, complementary methods have been developed. In Chapter 9, we present the principle of labeled magnetic microbeads and show how these beads are used to bind with the targeted biological objects and to transport them into specifically designated areas. Another usual way of controlling the motion of microparticles is based on the use of electric fields. In Chapter 10, we present the different ways electric fields act on the particles, like electrophoresis and dielectrophoresis. We finally conclude in Chapter 11 by recalling the main recent developments and the future trends.

Acknowledgements We are grateful to Ken Brakke and to the COMSOL support team in Grenoble for their precious help and counsel. We would like to thank J-M Grognet at the French Ministry of Industry, L. Malier, director of the LETI, and J. Chabbal, director of the Biotechnology Department at the LETI, for their encouragement and support for this project. We thank our colleagues, particularly N. Sarrut, for their contribution with photographs, and A. Buguin who has been kind enough to review and comment on some chapters. Our discussions with A. Ajdari, R. Austin, D. Chatenay, J.-F. Joanny, F. Perraut, J. Prost, and L. Talini have fueled many parts of this book, particularly Chapters 8, 9, and 10. We are grateful to the editing team at Artech House for their help, especially Penny Comans, Erin Donahue, Kevin Danahy, and Vicki Kane. Finally, we wish to express our gratitude to our spouses, Susanne and Isabelle, for their patience and support during the long hours of writing of this book.

xv

Chapter 1

Dimensionless Numbers in Microfluidics 1.1

Introduction Scaling analysis and dimensionless numbers play a key role in physics. They indicate the relative importance of forces, energies, or time scales in presence and lead the way to simplification of complex problems. Besides, the use of dimensionless parameters and variables in physical problems brings a universal character to the system of equations governing the physical phenomena, transforming an individual situation into a generic case. The same remarks apply to microscale physics. Only forces, energies, and time scales are different, and, even if some dimensionless numbers are the same as the one used at the macroscale, many are specific to microscales. In this chapter, we present the most widely used dimensionless numbers in microfluidics, after having recalled the fundamental Buckingham’s Pi theorem.

1.2

Microfluidic Scales Let us characterize the dimension of a system by the length scale L. Areas then scale as L2 and volume scales as L3. Surface forces are in general proportional to the surface area and volume forces—like weight or inertia—are proportional to the volume. The most typical change when switching from macroscopic to microscopic scales is that the ratio between surfaces forces and volume forces increases as 1/L. In microsystems, surface forces tend to be dominant over volume forces. The scaling laws of different physical quantities that frequently appear in the physics of microsystems as function of the length scale L are given in Table 1.1. By looking at Table 1.1, it is deduced that when miniaturizing fluidic systems (i.e., L ® 0), inertia and gravity become less important, whereas capillarity and interface phenomena become dominant (Laplace pressure, capillary rise, and Marangoni force all scale as 1/L). Note the huge increase in hydraulic resistances (1/L4) and the importance of viscoelasticity at small scales, with the Deborah and elastocapillary numbers varying respectively as L-3/2 and L-2.

1.3

Buckingham’s Pi Theorem Buckingham’s pi theorem is a key theorem in dimensional physics [1]. The theorem states that for a system of equations involving n physical variables, depending only on k independent fundamental physical quantities (unities, for example), the system depends only on p = n − k dimensionless variables constructed from the original variables. 



Dimensionless Numbers in Microfluidics Table 1.1  Scaling Law of Typical Physical Quantities Intervening in Microfluidics Physical Quantity Area Volume Velocity Time Gravity force Inertia Hydrostatic pressure Hydraulic resistance Stokes drag Diffusion constant Reynolds number Péclet number Diffusion time (mass or temperature) Laplace pressure Bond number Capillary rise Capillary number Weber number Ohnesorge number Deborah number Elastocapillary number Marangoni number Marangoni force Knudsen number Electric field

Scale L2 L3 L L0 L3 L3 L L−4 L L−1 L2 L2 L2 L–1 L2 L–1 L L3 L–1/2 L–3/2 L–2 L L–1 L–1 L–1

Reference

Chapter 2 Chapter 6 Chapter 5 (1.9) (1.25) Chapter 5 Chapter 3 (1.20) Chapter 3 (1.10) (1.11) (1.12) (1.17) (1.19) (1.23) Chapter 3 (1.4) Chapter 7

At the same time, the use of the theorem is very powerful because it does not involve the form of the equation or system of equations, just the variables intervening in the problem. Also, because the choice of dimensionless parameters is not unique, it only provides a way of generating sets of dimensionless parameters. The user still has to determine the meaningful dimensionless parameters corresponding to the specific problem. More formally, in mathematical terms, if we have an equation such as

f (q1, q2 ,..., qn ) = 0

(1.1)

where the qi are the n physical variables, expressed in terms of k independent physical units, (1.1) can be restated as

(

)

F π 1, π 2 ,..., π p = 0

(1.2)

where the πi are dimensionless parameters constructed from the qi by p = n − k equations of the form

m

m

m

π i = q1 1 q2 2 .... qn n

(1.3)

where the exponents mi are integer numbers. For example, if we consider the Navier-Stokes equations for a flow around an obstacle, the variables are the obstacle dimension L, the flow velocity far from the obstacle U, the fluid density

1.4  Scaling Numbers and Characteristic Scales



ρ, and the fluid viscosity m. Hence, n = 4. The units intervening in the problem are to the number of 3: kilogram, meter, and second. The Buckingham theorem then states that there is only 4 − 3 = 1 dimensionless parameter characterizing the problem. This parameter is the well-known Reynolds number (1.9).

1.4

Scaling Numbers and Characteristic Scales 1.4.1

Micro- to Nanoscales

The Knudsen number defines the transition between micro- and nanoscales [2]. This transition is extremely important; it defines the lower limit where the continuum hypothesis can be used. The Knudsen number is defined as Kn = λ L



(1.4)

where L is a representative physical length scale and l the mean free path. Kn is small at the microscale, and is larger than 1 at the nanoscale. For a gas at normal conditions, λ is of the order of 1 mm, whereas it is smaller for a liquid (5–10 nm). 1.4.2

Hydrodynamic Characteristic Times

In microhydrodynamics, four characteristic times are usually defined. These times will be used in the following sections to establish some dimensionless groups:



1. The convective (or viscous) time scales the time for a perturbation to propagate in the liquid τC = R V (1.5)



where R is a dimension and V is the velocity of the liquid. Depending on the flow configuration (shear or elongational), the convective time may also be written as τ C = 1 γ� or 1 ε� where γ� , ε� are, respectively, a shear rate and an elongation rate. 2. The diffusional time is the time taken by a perturbation to diffuse in the liquid



τ D = R2 ν

(1.6)

where ν = m/ρ is the kinematic viscosity (units m /s). 2

Figure 1.1  Schematic of microscale (Kn <<1) and nanoscale (Kn >1) chambers.



Dimensionless Numbers in Microfluidics

3. The Rayleigh time is a time scale of the perturbation of an interface under the action of inertia and surface tension [3–5] τ R = ρ R3 γ



(1.7)

4. The capillary time—sometimes called the Tomotika time [3–5]—is the time taken by a perturbed interface to regain its shape against the action of viscosity τT =

1.4.3

η0 R η = 0 γ γ R

(1.8)

Newtonian Fluids

The terms of Newtonian and non-Newtonian fluids will be defined in Chapter 2. Let us mention here that usual liquids like water belong to the Newtonian category and polymeric fluids belong to the non-Newtonian category. The Reynolds number characterizes the relative importance of inertial and viscous forces. It is usually written under the form

Re = V R ν

(1.9)

where V is the average fluid velocity and R is a length characteristic of the geometry. Written under the form



Re =

ρV 2 µ V /R

the Reynolds number is the ratio of a dynamic pressure—linked to inertia—to a shearing stress—linked to viscous forces. In microfluidics, the Reynolds number is generally small, corresponding to a laminar flow regime. Very few microfluidic systems use turbulent flows. It is recalled here that, in enclosures, a number of Reynolds of 2,000 is required to reach the transition to turbulence [6]. A more subtle subclassification can be made for the laminar regime. A very low Reynolds number (less than 0.5) indicates a creeping flow, for which the Stokes approximation is valid (see Chapter 2); this is the general case. However, recently, with the development of microsystems for cells handling, larger velocities are often used corresponding to Reynolds numbers in the range from 1 to 20. In these two cases, the physical phenomena for the convective transport are different, as will be shown in Chapter 2. Figure 1.2 schematizes the definition of the Reynolds number for some geometrical configurations. The capillary number is very important in two-phase microfluidics. It compares viscous/elongational forces to surface tension forces. The capillary number can take different forms, depending on the physics of the problem. It can be built using the average velocity, the shear rate, or the elongational rate: Ca = η V γ Ca = ηγ� R γ

Ca = ηε� R γ

(1.10)

1.4  Scaling Numbers and Characteristic Scales



Figure 1.2  Reynolds numbers: (a) Reynolds number in a channel and (b) Reynolds number around a spherical particle.

Remarking that the shear stress can be written η V R , ηγ� , η ε� depending on the specific configuration, and the capillary pressure γ /R, one immediately sees that the capillary number is the ratio of the viscous forces to the capillary forces. Let us illustrate this remark by two examples. In flow focusing devices (FFD), the capillary number is a relevant criterion to predict liquid thread breakup [Figure 1.3(a)]. In this case, the capillary number is built on the shear rate (or elongation rate) [7]. As the filament stretches and thins down, the capillary number decreases and when it becomes less than a critical value Cacrit ~ 0.1 to 0.01, the surface tension forces break the liquid filament into droplets, minimizing the interfacial area (hence the surface energy); this phenomenon is usually called the Rayleigh-Plateau instability [8, 9]. In a plug flow, the friction of the plugs with the walls is a function of the capillary number of the liquid of the plugs [Figure 1.3(b)] [10]. The Weber number is used to predict the disruption of an interface under the action of strong inertial forces. More specifically, the Weber number is the ratio of inertial forces to surface tension forces



We =

ρ V2R ρV 2 = γ γ /R

(1.11)

The numerator corresponds to a dynamic pressure ρV 2 and the denominator to a capillary pressure γ /R. A strong surface tension maintains the droplet as a unique

Figure 1.3  (a) Capillary number for an elongating liquid thread in a flow focusing device (FFD) and (b) capillary number for plug flow in a tube.



Dimensionless Numbers in Microfluidics

microfluidic entity with a convex interface. If the inertia forces are progressively increased, the interface is first deformed by waves, becomes locally concave, and is finally disrupted [11]. The Ohnesorge number relates the viscous and surface tension forces. It is defined as Oh =



η We = Re ργ L

(1.12)

The Ohnesorge number has been shown to be a good criterion to predict the breakup of liquid jets in a gas [12]. Another expression—seldom used—for the comparison of viscous and surface tension forces is the Laplace number defined by La = 1 Oh2



(1.13)

In terms of characteristic times, the different dimensionless numbers can be derived from the characteristic times defined in Section 1.4.2 (see also Figure 1.4). This schematic diagram brings a new light on the different roles of these dimensionless numbers in fluid dynamics. 1.4.4

Non-Newtonian Fluids

As soon as the viscosity of a fluid does not depend only on temperature and concentration but also on the internal stress, the fluid is categorized as non-Newtonian. In biotechnology, such fluids are principally polymers (alginate, xanthan) or body fluids. Their viscosity decreases with an increase of the shear rate. These fluids are said to be viscoelastic, or shear-thinning.

Figure 1.4  Hydrodynamic dimensionless numbers for Newtonian fluids based on the different characteristic times.

1.4  Scaling Numbers and Characteristic Scales 

Viscoelastic fluids are characterized by a relaxation time. This time can be seen as the inverse of the critical shear rate, that is, the shear threshold that starts to change the value of the viscosity

τ = 1 γ�cr

(1.14)

or as a time for the polymer chains to change its configuration—from stretched to coiled, for instance. Even if this time may seem short (0.05 second for alginate polymers), it has important consequences for the liquid behavior as will be shown in Chapter 2. The Weissenberg and Deborah numbers are characteristic of non-Newtonian flows. Depending on the authors, the definition of these two dimensionless numbers may differ somewhat. Both numbers are the ratio of the relaxation time of the polymeric liquid and a specific process time or a characteristic time frame. If we follow Bird, Armstrong, and Hassager [13], the Weissenberg number is the equivalent of the Reynolds number for viscoelastic fluids. It is defined as the ratio of the relaxation time τ—characterizing the elasticity of the fluid—to the convective time τC, defined in (1.5) as τC = R/V

Wi = τ V R

(1.15)

In a shear or elongational flow, the Weissenberg number can be defined as Wi = τ γ�

Wi = τ ε�

(1.16)

A large Weissenberg number indicates a strong viscoelastic behavior. Figure 1.5 shows the definition of the Weissenberg number in the different flow configurations of a flow focusing device (FFD). The Deborah number—which name was given by one of the founder of rheology, Markus Reiner—is usually defined by the ratio of the relaxation time and the Rayleigh time defined by (1.7) as τ R = ρ R 3 / γ

Figure 1.5  Weissenberg numbers in different flow configurations: (a) in a convergent, (b) in a free falling jet [13], and (c) in a flow focusing device [7].



Dimensionless Numbers in Microfluidics 1

De = τ (γ ρR3 ) 2



(1.17)

The smaller the Deborah number, the more fluid the material appears [5]. The elasticity number is the ratio of the Weissenberg number to the Reynolds number; it represents the ratio of elastic to inertial stresses El =



Wi η0 τ = Re ρ R2

(1.18)

The elasticity number does not depend on the process kinematics, only on the fluid properties and geometry of interest. It can also be seen as the ratio of the relaxation time τ to the diffusional time R2/v established in (1.6). For example, extrusion of polymer melts corresponds to El >>1 (Figure 1.6), whereas flow of dilute polymeric liquids correspond to El <<1 [5]. The elastocapillary number characterizes the relative importance of elastic and capillary effects as compared to viscous stresses Ec =



τγ η0 L

(1.19)

In terms of characteristic time, it is the ratio of the relaxation time τ and the Tomotika time defined in (1.8) by τT = η0R/γ. Figure 1.7 shows how the different dimensionless numbers for viscoelastic behavior can be deduced from the characteristic times. 1.4.5

Droplets and Digital Microfluidics

The Bond number is a measure of the importance of surface tension forces compared to body (gravity) forces. A high Bond number indicates that the system is relatively unaffected by surface tension effects; a low number indicates that surface tension dominates (Figure 1.8).

Figure 1.6  Elasticity of a liquid alginate filament during droplet formation (the encapsules are Jurkat cells) (photo CEA-LETI).

1.4  Scaling Numbers and Characteristic Scales 

Figure 1.7  Dimensionless numbers and characteristic times for non-Newtonian fluids.

The Bond number is defined by

Bo = Dρ g R2 γ i

(1.20)

where ∆ρ is the volumic mass difference between the two liquids, g is the gravity acceleration, γ is the surface tension, and R is a characteristic dimension (radius or contact radius). The capillary length is a characteristic length for fluid subject to gravity and surface tension

λ=

γ Dρ g

The Bond number compares the characteristic dimension of the fluid element to the capillary length Bo = R2/λ2 (Figure 1.9). For clean water at standard temperature and pressure, the capillary length is ~2 mm. Note that the definition of the Bond number can be extended to other forces acting on the droplet. When an electrically

Figure 1.8  Numerical simulations of a microdrop (Bo <<1) and a larger drop (Bo >>1) obtained with Surface Evolver software (not to scale) [9].

10

Dimensionless Numbers in Microfluidics

Figure 1.9  The capillary length λ for a large droplet corresponds to the curved part of the droplet. 

conductive droplet is submitted to an electric field, its deformation is linked to the electrical Bond number defined as Boe =



ε E2 R γ

where E is the electric field and e the electrical permittivity of the liquid. In biotechnology, when working with droplets, evaporation must be avoided as much as possible [14]. Closed atmosphere and sacrificial droplets are often used to maintain a constant vapor pressure and limit evaporation [Figure 1.10(a)]. The evaporation number indicates whether the quantity of water contained in the sacrificial droplets is sufficient to limit the evaporation from the droplets of interest [15]. More specifically, recalling that the evaporation rate of a microdrop is proportional to its radius, the relative evaporation of droplets of interest compared to the total relative evaporation is given by

å Ei Ev =

Vi = åE

Vloss

å Ri

ρ Vi åR

D ρsat Va

=

å Ri åR

D ρsat Va ρ Vi

(1.21)

where Ri is the radius of the droplets of interest, Vi and Ei are the initial total volume of liquid of interest and its evaporation rate, Vloss is the liquid volume that is

Figure 1.10  (a) The evaporation from droplets of interest is less than 10% when the evaporation number is smaller than 0.1. (b) Evaporation from sacrificial droplets maintains the water vapor pressure in a closed box and prevents noticeable evaporation of droplets of interest. Reprinted with permission from [15]. Copyright 2008 Royal Society of Chemistry.

1.4  Scaling Numbers and Characteristic Scales 11

Figure 1.11  Change of contact angle when increasing the electric potential (electrowetting number).

evaporated, Va is the evaporated volume and ∆ρsat is the density change from the liquid to vapor phase. Figure 1.10 shows the evaporation amount as a function of temperature and number of sacrificial droplets in a closed petri dish. The evaporation number is closely related to the evaporated mass of liquid. 1.4.6

Multiphysics

Biotechnological microsystems most of the time combine the use of different physical phenomena, like hydrodynamics, electrics, magnetics, thermics, and chemical reactions. In this section, we present some dimensionless numbers useful for microsystems incorporating multiphysics, and we recall some others that are well known from the macroscale. The electrowetting number is used for electrowetting applications [10]. It is the ratio of the electric force acting on a droplet to the capillary force

ηew = C U 2 γ

(1.22)

Figure 1.12  Schematic view of Marangoni convective motion in a droplet sitting on a heated plate.

12

Dimensionless Numbers in Microfluidics

The Lipmann-Young law [16] relates the change of contact angle to the electrowetting number: cosθ − cosθ0 = ηew /2 (Figure 1.11). The Marangoni number determines the magnitude of convective motions in a droplet. Even if they do not move, free-standing droplets are seldom at rest. Internal motions are frequent. These motions are essentially caused by interfacial forces, especially by a gradient of interfacial tension. The most common case is that of a gradient of interfacial temperature, which can be due to surrounding temperature conditions or to evaporation [17] (Figure 1.12). Another cause of Marangoni convection is a gradient of concentration on the interface, which may occur when surfactants are added to the liquid. For example, a gradient of temperature results in a gradient of surface tension according to γ = γ0(1 + β/T) where β = −1/TC, TC being the critical temperature in Kelvin. Marangoni convection occurs if the gradient of the surface tension force dominates the viscosity forces. A dimensionless number—the Marangoni number—determines the strength of the convective motion [18]



Ma =

Dγ R ρ ν α

(1.23)

In the domain of bioreactions, like DNA immobilization [19], it is essential to determine whether the process is limited by the reaction time or by the time of transport of the species involved in the reaction. The Damköhler number is defined as the ratio of these two characteristic times

Da = τ R τ Tr

(1.24)

where τR and τTr are, respectively, the times taken by the reaction and the transport of species. For a purely diffusive situation, the transport time is of the order of τTr ≈ L2/D, where D is the diffusion coefficient. In such a case, the Damköhler number can be written as Da = DτR /L2. The Peclet number is a fundamental dimensionless number in convective transport as well at the macroscale as at the microscale. It relates the rate of advection of a flow to its rate of diffusion or thermal diffusion. It is equal to the product of the Reynolds number with the Prandtl number ν/α in the case of thermal conduction, and the product of the Reynolds number with the Schmidt number ν/D in the case of mass dispersion Peth = V L α

Pemass = V L D

(1.25)

In the general problem of mass transfer at a solid wall, the Sherwood number represents the ratio of convective to diffusive mass transport. It is defined by the expression

Sh =

kL D

(1.26)

where L is a characteristic distance, D is the diffusion coefficient, and k is a mass transfer coefficient (unit m/s). Usually the Sherwood number is correlated to the

1.4  Scaling Numbers and Characteristic Scales 13

Peclet number via a power law. In a typical Levêque problem (see Chapter 6), this correlation is [20] 1

Hö3 æ Sh = 1.615 ç Pe ÷ è Lø



where H is the channel width and L is its length. 1.4.7

Specific Dimensionless Numbers and Composite Groups

Recently, new dimensionless numbers have been introduced to answer new specific problems. Others have been introduced with the development of complex multiphysics phenomena. In this section we present some of them, relative to microsystems for biotechnology. Microdrop impact on liquid or solid surfaces is of great interest for ink-jet printing and spray cooling. The K number has been defined to separate the splashing and spreading regimes. The K number is a combination of the Weber and Ohnesorge numbers [21]

K = We Oh- 2 5

(1.27)

A high K number value indicates splashing. In microreactors, the Graetz number estimates the relative importance of diffusion and convection length. For example, in a straight channel, it compares the mean axial length traveled by a particle/molecule to the mean transverse length traveled by the particle/molecule. The axial displacement L is principally due to convective transport and the radial length w is due to the effect of Brownian diffusion [22], and the Graetz number is

Gr = L w

(1.28)

In micro-exchanger, the Graetz number is a function of the Peclet number [23].

Figure 1.13  Vortices in channel curve enhance mixing at large Dean number. Reprinted with permission from [25]. Copyright 2006 Royal Society of Chemistry.

14

Dimensionless Numbers in Microfluidics

Figure 1.14  Microflow at a sudden widening of a channel cross-section: (a) Newtonian, or slightly non-Newtonian, behavior, and (b) non-Newtonian behavior at a viscoelastic Mach number larger than 1.

Microflow in curved channel may have recirculating regions. These recirculations can be used to concentrate cells or particles [24], or to enhance the mixing of solutes [25]. The Dean number characterizes the possibility of having recirculations

De = U R ν R Rc = Re R Rc

(1.29)

where R is the dimension of the channel and Rc is the curvature radius. For a sufficiently large Dean number, a pair of vortices forms in the cross-section, carrying flow from the inside to the outside of the bend across the center and back around the edges (Figure 1.13). Recently, the viscoelastic Mach number was introduced in order to predict the distortion of fluid streamlines in microsystems under the effect of viscoelasticity [26]. This number is defined by

Mv = Wi Re = V Cs = V

η0 ρτ = V

ν τ

(1.30)

where Cs is the viscoelastic wave speed. Distortions of streamlines appear for Mv > 1–2. An illustration is shown in Figure 1.14. Walls of microsystems very frequently have electric charges; these charges spontaneously create an electrical double layer (EDL) in the adjacent fluid. Electro-osmosis makes use of this EDL to motion the fluid. The Dukhin number characterizes the surface conductivity and deformation of the electric field by the conductivity of the double layer; the Dukhin number is the ratio of the conductivity of the double layer to that of the bulk fluid [27]

Du = σ EDL σ bulk

(1.31)

1.4  Scaling Numbers and Characteristic Scales 15

References   [1] Hart, G. W., Multidimensional Analysis: Algebras and Systems for Science and Engineering, New York: Springer-Verlag, 1995.   [2] Hardt, S., “On the Validity of the Continuum Assumption in Nanofluidics,” Summer School, CISM Udine, Italy, September 1–5, 2008.   [3] Steinhaus, B., A. Q. Shen, and R. Sureshkumar, “Dynamics of Viscoelastic Fluid Filaments in Microfluidic Devices,” Physics of Fluids, Vol. 19, 2007, p. 073103.   [4] Eggers, J., “Nonlinear Dynamics and Breakup of Free-Surface Flows,” Review of Modern Physics, Vol. 69, 1997, pp. 865–929.   [5] Spiegelberg, S. H., D. C. Ables, and G. H. McKinley, “The Role of End Effects on Measurements of Extensional Viscosity in Filament Stretching Rheometers,” J. Non-Newtonian Fluid Mech., Vol. 64, 1996, pp. 229–267.   [6] Warhaft, Z., The Engine And The Atmosphere: An Introduction to Engineering, New York: Cambridge University Press, 1997.   [7] Le Vot, S., et al., “Non-Newtonian Fluids in Flow Focusing Devices: Encapsulation with Alginates,” Proceedings of the 1st European Conference on Microfluidics, Bologna, December 10–12, 2008.   [8] Chabert, M., and J-. L. Viovy, “Microfluidic High-Throughput Encapsulation and Hydrodynamic Self-Sorting of Single Cells,” PNAS, Vol. 105, No. 9, 2008, pp. 3191–3196.   [9] Berthier, J., Microdrops and Digital Microfluidics, New York: William Andrew Publishers, 2008. [10] Berthier, J., and P. Silberzan. Microfluidics for Biotechnology, Norwood, MA: Artech House, 2005. [11] Xiong, R., and J. N. Chung, “An Experimental Study of the Size Effect on Adiabatic GasLiquid Two-Phase Flow Patterns and Void Fraction in Microchannels,” Physics of Fluids, Vol. 19, 2007, p. 033301. [12] Pan, Y., and K. Suga, “A Numerical Study on the Breakup Process of Laminar Jets into a Gas,” Physics of Fluids, Vol. 18, 2006, p. 052101. [13] Bird, R. B., R. C. Armstrong, and O. Hassager, Dynamics of Polymeric Liquids: Volume 1, Fluid Mechanics, New York: John Wiley & Sons, 1987. [14] Jackman R. J., D. C. Duffy, E. Ostuni, N. D. Willmore, and G. M. Whitesides, “Fabricating Large Arrays of Microwells with Arbitrary Dimensions and Filling Them Using Discontinuous Dewetting,” Anal. Chem., Vol. 70, No. 11, 1998, pp. 2280–2287. [15] Berthier, E., J. Warrick, H. Yu, and D. J. Beebe, “Managing Evaporation for More Robust Mi­croscale Assays. Part 1. Volume Loss in High Throughput Assays,” Lab Chip, Vol. 8, No. 6, 2008, pp. 852–859. [16] Berge, B., “Electrocapillarity and Wetting of Insulator Films by Water,” C. R. Acad. Sci. Paris, Vol. 317, 1993, pp. 157–163. [17] Hegseth, J. J., N. Rashidnia, and A. Chai, “Natural Convection in Droplet Evaporation,” Physical Review E, Vol. 54, No. 2, 1996, pp. 1640–1644. [18] Baroud, C. N., and H. Willaime, “Multiphase Flows in Microfluidics,” C. R. Physique, Vol. 5, 2004, pp. 547–555. [19] Okkels, F., and H. Bruus, “Scaling Behavior of Optimally Structured Catalytic Microfluidic Reactors,” Physical Review E, Vol. 75, 2007, p. 016301. [20] Yarin, A. L., “Drop Impact Dynamics: Splashing, Spreading, Receding, Bouncing,” Annu. Rev. Fluid Mech., Vol. 38, 2006, pp. 159–192. [21] Holzbecher, E., “Numerical Solutions for the Lévêque Problem of Boundary Layer Mass or Heat Flux,” Proceedings of the Hannover COMSOL Conference, Hannover, November 4–6, 2008.

16

Dimensionless Numbers in Microfluidics [22] Gervais, T., and K. F. Jensen, “Mass Transport and Surface Reactions in Microfluidic Systems,” Chemical Engineering Science, Vol. 61, 2006, pp. 1102–1121. [23] Tran, V. M., J. Berthier, R. Blanc, O. Constantin, N. David, and N. Sarrut, “Micro-Extractor for Liquid-Liquid Extraction, Concentration and In Situ Detection of Lead,” IMRET-10: 10th International Conference on Microreaction, AIchE 2008 Spring National Meeting, New Orleans, LA, April 6–10, 2008. [24] Di Carlo, D., D. Irimia, R. G. Tompkins, and M. Toner, “Continuous Inertial Focusing, Ordering, and Separation of Particles in Microchannels,” PNAS, Vol. 104, 2007, p. 1882. [25] Sudarsan, A. P., and V. M. Ugaz, “Fluid Mixing in Planar Spiral Microchannels,” Lab Chip, Vol. 6, 2006, pp. 74–82. [26] Rodd, L. E., et al., “Role of the Elasticity Number in the Entry Flow of Dilute Polymer Solutions in Micro-Fabricated Contraction Geometries,” Journal of Non-Newtonian Fluid Mechanics, Vol. 143, No. 2-3, 2007, pp. 170–191. [27] Lyklema, J., “On the Slip Process in Electrokinetics,” Colloids and Surfaces A: Physicochemical and Engineering Aspects, Vol. 92, No. 1-2, 1994, pp. 41– 49.

Chapter 2

Microflows 2.1  Introduction 2.1.1  On the Importance of Microfluidics in Biotechnology

Biotechnology is closely linked to microfluidics. Biological targets are nearly always transported by a buffer fluid or carrier fluid, in vitro and in vivo. In the human body, any bioMEMS has to deal with body fluids. With in vitro microsystems, the target molecules/particles are nearly always transported by a buffer fluid for several reasons. First, the target molecules/particles are, most of the time, extracted from a liquid (e.g., DNA and cells); second, the biochemical reactions on these targets are performed in an aqueous environment; and third, the confinement of the targets is easier in a liquid than in a gas. Very few examples of biotechnological microsystems exist that do not require the use of microfluidics. One counterexample might be the electronic nose, in which the detection of target molecules transported by ambient air is done directly on a dry contact surface by mass spectrometry. The standard procedure to detect bacteria carried by air, like legionella, is to capture and concentrate them in a water-based solution. In this chapter, we discuss liquid microflows, but we will not consider gas flows. For the reader whose concern is gas microflows, useful information can be found in [1]. If liquids are the most frequent carrier of microparticles, the flow pattern can be very different. In Section 2.2, we present the different types of microflows currently used in biotechnology. 2.1.2  From Single Continuous Flow to Droplets

In biotechnology, microfluidics is present under various forms depending on the different applications. The most usual form is single-phase microflow in channels and capillary tubes (Figure 2.1). This is the general case of buffer liquids carrying biological targets and circulating in microchannel networks where different biological processes successively take place. Different reactions, biochemical analysis, and detection can be done in microchambers placed alongside the flow. Usually, the fluid is moved under the effect of pressure (from a syringe or a micropump), sometimes by electric forces (electro-osmotic flow). In order to accelerate the speed of biorecognition (for high throughput screening, for example) samples of different buffer solutions have to be treated simultaneously and continuously. In such a case, the solution is to convey successive buffer fluid plugs in capillary tubes; the plugs are separated by a nonmiscible, biocompatible liquid (Figure 2.2). Such a type of flow is commonly called the multiphase flow, or, if there are only two liquids, a two-phase flow. One of the fluids may be a gas. 17

18

Microflows

Figure 2.1  Continuous flow microfluidics: microchannels designed for liquid-liquid extraction. Two immiscible liquids are moving in parallel, separated by pillars aligned in the middle. Micro- and nanoparticles or macromolecules carried by one of the liquid migrate through the interface into the other liquid. If the flow rate of this last liquid is small, the particles are automatically concentrated. (Photo courtesy of N. Sarrut, LETI.)

For example, alternate plugs of liquid separated by air bubbles are used to create microemulsions or sprays. A similar, but slightly different, category of flow results from the breakup of one phase into droplets (Figure 2.3). It is still a multiphase flow, but the discontinuous phase is more or less dispersed in the continuous phase. This way of proceeding may be interesting to concentrate and isolate some biological targets in very small volumes of buffer fluid. Finally, microdrops can be considered as separate entities and manipulated individually (Figure 2.4). This type of fluid motion is often called digital microfluidics. The aim here is to manipulate with precision extremely small quantities of fluid—of the order of tens of nanoliters—containing the biological target (DNA, cells, and so forth). In this chapter, we deal with continuous and two-phase microflows. Digital microfluidics and droplet microfluidics will be treated in Chapters 3 and 4.

2.2  Single-Phase Microflows As we have seen in the preceding section, the first category of microflows is the single-phase continuous microflow. It is the most widely used in biotechnology for transporting biological targets and detection probes. In biochips, the fluid is flowing under the effect of a driving pressure—imposed by a micropump—or under the

Figure 2.2  Liquid plugs in a capillary tube. Alternate plugs of buffer solution—separated by immiscible silicon oil plugs—circulate inside a capillary tube. (Photo courtesy of LETI.)

2.2  Single-Phase Microflows

19

Figure 2.3  Water droplets in oil. Transition from a continuous microflow (left part of the photograph) to microdrops (right part of the photograph) by shearing through an aperture in a solid wall. (Courtesy of Harvard.)

action of a piston (syringe or deflected membrane), or under the action of an electric field inducing an electro-osmotic effect [1]. Different apparatus are design to monitor and actuate the flow as valves, compliance chambers, and passive or active micromixers. One of the major difficulties is to integrate many different functions in a miniaturized system, due to limitations brought by the fabrication process, the various connections between the fluidic elements, and the handling and packaging of the components. In this section, we will not go into the details of the many microfluidics apparatus, but we present the equations that govern fluid flows and we focus on the physics and particularities of microflows. 2.2.1  Navier-Stokes (NS) Equations

The macroscopic approach for the calculation of the velocities and pressure in a fluid is based on the continuum hypothesis, that is, in every elementary volume of

Figure 2.4  Digital microfluidics: photographs of (a) water microdrops on an electrode, (b) a water microdrop in silicon oil submitted to a pulsed electric field, and (c) a water drop in silicon oil in an EWOD microdevice. (Courtesy of LETI/Biosoc.)

20

Microflows

the fluid there are enough molecules to define statistical properties like velocity and pressure. The continuum hypothesis works well at a microscopic scale for liquids; for gases, the hypothesis breaks down at the nanoscopic scale where the characteristic Knudsen number (Kn) becomes of the order of 1

λ (2.1) L where λ is the mean free path of the molecules and L is the characteristic dimension of the channel. For gases, the limit for L is about 1 mm. In liquids, the mean free path is much smaller and the continuum hypothesis is applicable to any microsystem. In the most general point of view, fluid flows are determined by the knowledge of velocities U = {ui, i = 1, 3}, pressure P, density ρ, viscosity m, specific heat Cp, and temperature T. For each fluid, density, viscosity, and specific heat are related to pressure and temperature (or enthalpy) via characteristic equations of state (EOS) Kn =

ρ = f ( P ,T ) µ = g (P,T )

(2.2)

Cp = h(P,T )



Pressure and temperature characterize the number and the state of the molecules that are present in a given volume. Equations of state are generally complicated, but they can be approximated by analytical functions if the domain of variation of the parameters (P and T) is not too large. Thus we are left with five unknowns: ux, uy, uz, P, and T. These unknowns are related by a system of three equations: (1) a scalar equation for the mass conservation, (2) a vector equation for the conservation of momentum, and (3) a scalar equation for the conservation of energy. In biotechnology, fluid flows are often isothermal or variation of temperature is negligible. Note that this is not the case for microchemistry where chemical reactions are seldom isothermal. If temperature is constant or nearly constant, we have to deal with four unknowns ux, uy, uz, and P, with the help of the mass conservation equation and the conservation of momentum equation, plus the EOS ρ = f(P), m = g(P). Some authors give the name Navier-Stokes equations to the whole system; others confine this name to the second equation (momentum). 2.2.1.1  General Case: Governing Equations

The first equation is the mass conservation (or continuity) equation. For simplicity we demonstrate here only the two dimensional form of this equation. Assume a velocity field (u, v) and an element of volume (Dx, Dy) as sketched in Figure 2.5. The mass conservation equation requires



é é ∂ (ρ ∆ x ∆ y ) ∂ ( ρu ) ù ∂ ( ρ v) ù = ρ u D y + ρv D x - êρu + ú D y - ê ρv + úDx ∂t ∂x û ∂y û ë ë dividing by Dx Dy



∂ ρ ∂ (ρ u) ∂ ( ρ v ) + + =0 ∂t ∂x ∂y

(2.3)

2.2  Single-Phase Microflows

21

Figure 2.5  Two-dimensional conservation of mass.

This equation may be written as



é ∂u ∂v ù ∂ρ ∂ρ ∂ρ +u +v +ρê + ú=0 ∂t ∂x ∂y ë∂x ∂y û

(2.4)

and, under a vector form � Dρ + ρ Ñ.V = 0 Dt

(2.5)

D ∂ ∂ ∂ ∂ = +u +v +w Dt ∂ t ∂x ∂y ∂z

(2.6)

where the operator D/Dt is





in a three-dimensional Cartesian coordinate system. Liquids may generally be considered as incompressible and the mass conservation equation is then reduced to � (2.7) Ñ.V = 0 In Cartesian coordinates we have



∂u ∂v ∂w + + =0 ∂x ∂y ∂z and in cylindrical coordinates, the axisymmetric form of (2.7) is



∂vr vr ∂vz + + =0 ∂r r ∂z The second equation is the momentum conservation equation (or the NavierStokes equation). The change of momentum in a fluid element is equal to the balance between inlet momentum, outlet momentum, and exerted forces [2]. Figure

22

Microflows

Figure 2.6  Momentum in the x-direction.

2.6 shows the inlet and outlet momentum in the x-direction, and Figure 2.7 represents the forces on the same fluid element. Projecting all these forces on the x-direction, and using the mass conservation equation, we obtain



ρ

¶ τ xy Du ¶σ =- x + + Fx Dt ¶x ¶y

(2.8)

Normal stress and tangential stress are for most fluids (called Newtonian fluids) given by the constitutive relations

Figure 2.7  Force balance in the x-direction. Normal stress σx, tangential stress τxy, and body forces per unit volume Fx on a Dx Dy element.

2.2  Single-Phase Microflows

23

σ x = P - 2µ



∂ u 2µ æ ∂ u ∂ v ö + + ∂x 3 çè ∂ x ∂ y ÷ø

(2.9)

æ ∂ u ∂vö + τ xy = µ ç è ∂ y ∂ x ÷ø

where m is the dynamic viscosity. Combining (2.8) and (2.9) and extending the formulation to the three-dimensional case yield the Navier-Stokes equation æ ∂ 2u ∂ 2u ∂ 2u ö æ ∂u ∂u ∂u ∂uö ∂P +u +v +w + µ çç + + ρç ÷ + Fx ÷=2 ∂x ∂y ∂z ø ∂x ∂ y 2 ∂ z 2 ÷ø è ∂t è ∂x æ ∂ 2v ∂ 2v ∂ 2v ö æ∂v ∂v ∂v ∂vö ∂P +u +v +w + µç + + +F ρç ÷=ç ∂ x 2 ∂ y 2 ∂ z 2 ÷÷ y ∂x ∂y ∂z ø ∂y è ∂t è ø



(2.10)

æ ∂ 2w ∂ 2w ∂ 2w ö æ∂w ∂w ∂w ∂wö ∂P +u +v +w = + + + ρç µ çç ÷ + Fz ÷ 2 ∂x ∂y ∂z ø ∂z ∂ y 2 ∂ z 2 ÷ø è ∂t è ∂x The vectorial notation is



ρ

� � � DV = -Ñ P + µD V + F Dt

(2.11)

� where V is the velocity vector (u, v, w) and F is the body force per unit volume. 2.2.1.2  Axisymmetric Formulation of the NS Equations for Incompressible Liquids

It is useful to have the Navier-Stokes equation written in a cylindrical axisymmetric coordinates system æ∂v ∂ vr ∂ vr ö ∂P + vz + ρ ç r + vr ÷=∂r ∂z ø ∂r è ∂t



æ ∂ 2vr 1 ∂ vr vr ∂ 2vr + + µ çç 2 r ∂ r r2 ∂ z2 è ∂r

æ ∂ 2vz 1 ∂ vz ∂ 2vz æ∂v ∂ vz ∂ vz ö ∂P + vz + µ çç + + ρ ç z + vr ÷=2 ∂r ∂z ø ∂z r ∂r ∂ z2 è ∂t è ∂r

ö ÷÷ + Fr ø

ö ÷÷ + Fz ø

(2.12)



2.2.1.3  Energy Equation

In case there is a change in the liquid temperature, an energy conservation equation is added to complete the preceding system. This energy equation is, in Cartesian coordinates, æ ¶T ¶T ¶T ¶T ö ¶ æ ¶T ö +u +v +w = k ρ Cv ç ÷ ¶x ¶y ¶ z ø ¶ x èç ¶ x ÷ø è ¶t ¶ æ ¶T ö ¶ æ ¶T ö k k + + +q ¶ y èç ¶ y ø÷ ¶ z èç ¶ z ø÷

(2.13)

24

Microflows

where Cp is the specific heat (in J/kg/K), k is the conduction coefficient (in W/m/K) and q is a source or sink term (in W/m3). 2.2.2  Non-Newtonian Rheology

Usual liquids have a viscosity m depending on the temperature T and, to a lesser extent, on the pressure P because pressure is most of the time close to atmospheric pressure in microsystems. If nano- or microparticles are transported by the fluid, their concentration c has an important influence on the viscosity. The general expression of the viscosity is µ = f (P,T , c )



(2.14)

This general category of fluids is called Newtonian. Some liquids are more complex, such as polymeric solutions. 2.2.2.1  Introduction

The recent developments of microfluidic systems for biotechnology have widened the field of applications to more complex fluids like polymeric solutions and body fluids. Blood is one example. When flowing inside the human body, blood behaves like a viscoelastic non-Newtonian fluid, which viscosity decreases in high shearrate regions, for instance, the vicinity of the walls of arteries, veins, and capillaries [3]. This property explains why the heart manages to push this very viscous fluid through the whole body. Ex vivo, blood is most of the time diluted to be handled by fluidic systems. However, it is a very large gain in time if whole blood can be manipulated without dilution. Hence, the non-Newtonian character has to be taken into account in the conception of such systems [4]. Concentrated polymer solutions are another example of non-Newtonian fluid in microsystems. Agarose and alginate solutions are now being used to encapsulate living objects like bacteria, cells, and cluster of cells [5, 6]. The principle of such devices is shown in Figure 2.8. Depending on the concentration, alginate solutions show a more or less pronounced non-Newtonian behavior [7]. In the following

Figure 2.8  Principle of encapsulation in alginate: droplets of alginate are first formed in the FFD, and then the droplets are polymerized by the calcium ions to form capsules.

2.2  Single-Phase Microflows

25

sections, we first introduce the notion of non-Newtonian fluid, and present the example of biological polymeric solutions. 2.2.2.2  Viscosity of Non-Newtonian Fluids

The usual definition of the shear viscosity is

η = τ γ�

(2.15)

where τ is the shear stress (unit Pa) and γ� is the shear rate (unit s–1). Usual liquids like water and oil are said to be Newtonian because the relation between the shear stress and the shear rate is linear. The viscosity η is a constant for a Newtonian fluid (at a given temperature and pressure). Polymeric solutions behave differently. The shear stress is not proportional to the shear rate and viscosity depends on the shear rate. The rheological behavior of polymeric liquids depends on the conformation of the polymeric chains (Figure 2.9). It is obvious that the global rheological properties of the polymeric liquid will be different if the chains are in majority stretched or coiled. In a shear flow the polymeric chains stretch when the shear increases, resulting in a decrease of the viscosity. Such a behavior is called shear-thinning. Typical shear–thinning behavior is shown in Figure 2.10. In the case of blood, the red blood cells deform and elongate under shear stress. Note that shear is not the only cause of change of polymer structure in the liquid. Elongational flows also modify the polymer’s morphology [8]. Basically, the viscosity of a visco-elastic polymeric liquid depends on the concentration in polymers, the temperature, and the flow characteristics. Dropping the pressure, we simplify the approach by writing

η = η (c ,T , γ� )

(2.16)

2.2.2.3  Specific and Intrinsic Viscosity: Viscosity as a Function of Concentration

In rheology of polymers, it is usual to define the specific viscosity as



ηsp =

η - ηs ηs

(2.17)

Figure 2.9  Different morphologies of a polymeric chain calculated with a molecular dynamic method (MD). (a) At a low shear rate, the polymer is coiled in a nearly spherical shape, and (b) at a high shear rate, the polymer stretches (MD calculation by Katz et al.).

26

Microflows

Figure 2.10  Comparison between Newtonian (NWT) and non-Newtonian, shear thinning behavior: (a) shear stress versus shear rate (one-dimensional case), and (b) viscosity against shear rate. Viscosity remains constant until the critical shear rate; above this shear rate, viscosity decreases quickly.

where η is the dynamic viscosity of the solution (units Pa.s or kg/m/s) and ηs is that of the solvent (carrier fluid). The specific viscosity has no unit; at a small concentration, the specific viscosity goes to zero. One also defines the intrinsic viscosity by æ



ö

[η] = limc ®0 ç η η- ηs 1c ÷ è

ø

s

(2.18)

It is shown that the very general Martin relation [9] usually applies for solutions like alginates solutions

k¢ η c ηsp = (c [ η]) e [ ]

(2.19)

where k¢ is the Huggins coefficient. A simplified expression for (2.19) stems from a Taylor expansion 2



ηsp = c [ η] + k ¢ (c [ η]) +

3 k¢ 2 (c [ η]) + ... 2!

(2.20)

Truncature at the rank 2 yields the Huggins law 2



ηsp = c [η] + k ¢ (c [ η])



(2.21)

Because the Taylor expansion has been limited to the second order, Huggins law applies for very dilute solutions only. For semidilute solutions, more terms in the expansion (2.20) should be kept. However, it has been shown that the specific viscosity can generally be approached by the power law

ηsp = a (c [η]) n

(2.22)

where a is a coefficient. Taking alginate solutions as an example, it can be shown that relation (2.22) fits well the experimental results with a = 0.1, [η] of the order of 3–7 L/g and n of the order of 3–4 depending on the type of alginate [9, 10]. For

2.2  Single-Phase Microflows

27

Figure 2.11  Specific viscosity of alginate solution versus concentration: the dots correspond to the experimental values, the curve to the power law (alginate Keltone HV 1%). Insert: values of [η] and n have been determined by taking the logarithm of relation (3.8) and fitting with a straight line.

Keltone HV alginates, it has been found that [η]sp = (c[η])4.1 and [η] = 450 mL/g (Figure 2.11). Using relations (2.17) and (2.19), the viscosity of the solution is given by



n η = ηs é1 + a (c [η ]) ù êë úû

(2.23)

2.2.2.4  Viscosity Variation with Temperature

As a general rule, the viscosity of a polymeric liquid decreases with temperature. The Vogel-Fulcher-Tamman-Hess (VFTH) hyperbolic relation is often used to describe the thermal dependency of the viscosity [11] (Figure 2.12) and writes



logη = A +

B T - T0

(2.24)

where A and B are experimentally determined coefficients. The VFTH law stems from the fact that the density of the polymeric solution depends linearly on temperature and intermolecular distance. The VFTH relation states that the logarithm of the viscosity depends on the intermolecular distance d as



logη = A¢ +

B¢ d - d0

(2.25)

The change of viscosity with temperature of a polymeric liquid is such that it is important to always check the temperature before performing experiments with polymeric liquids.

28

Microflows

Figure 2.12  Thermal dependency of the viscosity of a polymeric liquid.

2.2.2.5  Viscosity and Shear Rate—Viscoelastic Behavior

Besides its dependency on concentration and temperature, the viscosity of polymeric solutions decreases with the shear rate of the flow, because, in high shear regions, the long polymer chains align with the flow; such a behavior is called shear thinning. Many different laws have been proposed for the non-Newtonian viscosity, depending on the carrier liquid and polymers. Small Shear Rates

In the domain of small shear rates, the Ostwald [12] relation is a good approximation of the viscosity η = m γ� n -1



(2.26)

and, using (2.15), the shear stress can be expressed as τ = m γ� n



(2.27)

When n = 1, the viscosity does not depend on the shear stress and the liquid is Newtonian. A value 0 < n <1 corresponds to a shear thinning liquid. Medium Shear Rates

In the domain of medium shear rates, for shear thinning liquids, it is common to use the Carreau-Yasuda relation [13]



.

η = η0 [(1 + (τ γ )α ]

m -1 α



(2.28)

where η0 is the viscosity at zero shear rate given by (2.23) η0 = ηs(1 + ηsp), is the shear rate, τ is the relaxation time (we will discuss the relaxation time in the next section), α is a constant, and m depends on the concentration of the solution. Again,

2.2  Single-Phase Microflows



29

taking alginates as an example, the Carreau-Yasuda relation fit well the experimental measurements as shown in Figure 2.13. Another relation sometimes used for visco-elastic liquids is the RabinowitchEllis relation [14] αη0 η= (2.29) 1 + (τ γ� ) β where α and β are again some constants to be determined. Relaxation Time

Due to the chains of polymers, droplets of alginate solutions show elasticity. The physicist Weissenberg used to present the image of some line tension—like an elastic filament—that brings elasticity to the flow streamlines. The elasticity of an alginate solution is characterized by a relaxation time τ [15]. Different characteristic times can be defined for polymers depending on the complexity of the polymeric structure and the forcing flow (shear or elongational flow) [16, 17]. In the case of a shear flow, the relaxation time appears in the Carreau-Yasuda . and Rabinowitch-Ellis models. As a matter of fact, the term τ γ is the Weissenberg number. A fit of the experimental data by the Carreau-Yasuda law produces τ. From a physical point of view, the relaxation time is approximately the inverse of the critical shear rate τ » 1 γ�cr, and the critical shear rate corresponds to the situation

Figure 2.13  Viscosity of Keltone HV alginate solutions versus shear rate: the dots correspond to the experimental results, and the continuous line corresponds to the Carreau-Yasuda model. The four curves correspond to four alginate concentrations: 1, 1.25, 1.5, and 1.75 wt%. Alginate viscosity increases with the concentration and decreases with the shear rate. The relaxation times are deduced from a fit of (2.28) on the different experimental curves.

30

Microflows

when the polymer chain starts stretching. The relaxation time usually depends on the concentration according to a power law τ » (c [ η]) p



(2.30)

where the exponent p depends on the liquid. 2.2.2.6  Non-Newtonian Microflows Physical Behavior

The flow field of viscoelastic liquids is sometimes surprising and very different from that of Newtonian liquids. Some striking examples, at the macroscopic scale, are shown in [16]. At the microscopic scale, there are many examples of shear thinning behavior. Friction on the walls of a channel is very important due to the small dimensions of the channel cross section. Even if the average velocity is small, shear rate can be large due to the vicinity of the walls. A typical example is that of blood flowing in the human body. It has been shown that streamlines at the entrance of a constriction can be very different for non-Newtonian fluids if the viscoelastic Mach number is larger than 1. Fluidic diodes can be made using visco-elastic liquids [18]. Finally, encapsulation of biologic object in gelling polymers is conditioned by the viscoelastic behavior of the encapsulating liquid [19]. Hence, it is important to be able to predict the behavior of visco-elastic fluids in microsystems. In this section, we give some insights for the computation of visco-elasticity liquid shear flows. Modeling Non-Newtonian Microflows

Liquid flows are characterized by their deformation tensor D (or rate-of-deformation or rate-of-strain tensor)

D=

1 (ÑV + ÑV T ) 2

(2.31)

where V is the velocity vector. One defines the shear rate associated to a fluid deformation by

γ� = 2 D : D

(2.32)

Let us recall that the 1D shear rate corresponding to a flow along a planar solid surface is γ� =



∂ V// ∂y

(2.33)

where V// is the fluid velocity close to the wall (and parallel to it), and y is the normal distance. For a 2D Cartesian coordinate system, the expression of the shear rate is 2



2

2

æ ∂u ö æ ∂u ∂v ö æ ∂v ö + γ� = 2 ç ÷ +ç ÷ + 2ç ÷ è ∂x ø è ∂y ∂x ø è ∂y ø

(2.34)

2.2  Single-Phase Microflows

31

Figure 2.14  Comparison between Newtonian and non-Newtonian flows (with the same zero-shear viscosity): the same flow rate is imposed in the two channels. The pressure drop is much smaller in the non-Newtonian fluid. Note the differences in the velocity profile (COMSOL calculation).

where u and v are the x and y components of the velocity. This value of the shear rate can be introduced in the Carreau-Yasuda law to produce the shear viscosity, which replaces the Newtonian viscosity in the Navier-Stokes equations. Examples

Non-Newtonian Microflow in a Microchannel  Let us compare the flow in a straight microchannel of two liquids, the first one Newtonian and the other nonNewtonian. Let us consider that the Newtonian fluid has a constant viscosity η0 = 2.63 Pa.s and the non-Newtonian fluid has a shear thinning viscosity defined by a . Carreau-Yasuda law η = η0 [(1 + (0.047γ )1.2 ]-0.41. If two identical channels have the same imposed inlet velocity, the pressure contour plots are very different in the two cases, as shown in Figure 2.14. The pressure drop is much smaller for the viscoelastic fluid because the viscosity at the wall is much smaller at the wall due to the shear. A viscoelastic fluid can be gel-like at rest (zero velocity), and suddenly flow if a large driving pressure is applied; once the fluid is in motion, the driving pressure is substantially reduced. Non-Newtonian Microflow in a Microchannel with a Constriction  Consider a microchannel with a constriction (Figure 2.15), like a microscopic Venturi [20]. Due to mass conservation, the velocity is larger in the narrow cross section. Because of the small transverse dimension of the constriction, the shear rate is important and the viscosity is small at the wall. These two effects add, and the velocity of the non-Newtonian fluid is larger than that of the Newtonian fluid in the constriction.

Figure 2.15  (a) Viscosity contour plot in a constriction for 1.25% Keltone HV alginate flowing at an average velocity of 1 mm/s. (b) Ratio of the elongation rate to the shear rate. Polymers stretch in the converging region and in the constriction and relax in the divergence region. (COMSOL calculation).

32

Microflows

In this geometry, the polymeric chains stretch in the entrance of the tube and relax at the outlet of the constriction (Figure 2.15). The normalized velocity profile in a cross section differs considerably from the Poiseuille-Hagen parabolic profile, as shown in Figure 2.16. Non-Newtonian Microflow in a Microfluidic Network  Microfluidic networks are commonly used in biotechnology. Later we will present some examples of microfluidic networks, such as a microneedle with multiple outlets and a network for extracting plasma from blood. Such networks are difficult to balance (e.g., to design in order to have the expected flow rate in each branch). Usually such networks are designed considering Newtonian fluids. It is emphasized here that such networks will not work properly for non-Newtonian fluids. Figure 2.17 shows the change of flow rates in a network depending on the visco-elastic character of the fluid. Using driving pressure conditions at the inlet, the computation shows that the flow rate in the small “branches” is reduced in the non-Newtonian case. 2.2.3  Laminarity of Microflows

Regardless of the size—macroscopic or microscopic—a fluid flow is said to be laminar when viscous forces dominate inertia. When this is the case, turbulences cannot develop and the fluid flow lines are, at least locally, parallel. One can picture it by considering that the flow is locally laminated. On the other side, a turbulent flow

Figure 2.16  The non-Newtonian velocity profile differs from the Poiseuille-Hagen parabolic profile (COMSOL calculation).

2.2  Single-Phase Microflows

33

Figure 2.17  Comparison of flow rates in a microfluidic network. The same values of the contour levels of the velocity show the difference in the flow rate in the small branch of a bifurcation between a Newtonian (NWT) and non-Newtonian (NNWT) fluid. (a) After one bifurcation, the change in flow rate can reach 5–10% for an inlet velocity of 2–3 mm/s, and (b) after two bifurcations, the change in flow rate can reach 20–30%.

presents fluctuating random vertices, even at very small scales. It is intuitive to think that microflows will predominantly be laminar since there is a strong limitation for vertices development and randomness due to the proximity of the solid walls. One important point here is that flow recirculation (vortex) does not mean that the flow is turbulent. In Figure 2.18, turbulence only starts at the transition Reynolds number of 100. Stable recirculation vortices at a Reynolds number of 50 are still laminar. Turbulence is associated with time fluctuations even with steady state boundary conditions. A nondimensional number (the Reynolds number) determines the ratio of inertia (convective forces) and viscous forces

Re =

UD ν

Figure 2.18  Different patterns of laminar and transitional flows behind a cylinder.

(2.35)

34

Microflows

where U is the average fluid velocity, D is a characteristic dimension of the chanµ nel (or the obstacle), and ν = is the kinematic viscosity (expressed in m2/s). The ρ Reynolds number naturally appears by performing a dimensional analysis of the NS equations. For simplicity we consider the two-dimensional Cartesian NS equation with no external forces 1 ∂ p µ é ∂ 2u ∂ 2u ù ∂u ∂u ∂u +u +v =+ ê 2 + ú ρ ∂x ρ êë ∂ x ∂t ∂x ∂y ∂ y 2 úû 1∂p µ ∂v ∂v ∂v +u +v =+ ρ ∂y ρ ∂t ∂x ∂y



(2.36)

é ∂ 2v ∂ 2v ù ê 2 + ú ∂ y 2 ûú ëê ∂ x

If U is a velocity reference (the average velocity), and D is a length reference (a characteristic dimension of the flow, for example, the diameter of the tube for a microflow in a capillary), we use the following scaling u* =

u , U

v* =

v , U

x* =

x , D

y* =

y , D

t* =

t , D /U

p* =

p ρU 2

Note that ρU2 has the same unit as the pressure. Then, the system (2.36) becomes ν é ¶ 2 u* ¶2 u* ù ¶ u* ¶ u* ¶ u* ¶ p* + u* + v* * = - * + + ê ú * * U D ë ¶ x*2 ¶ y*2 û ¶t ¶x ¶y ¶x



* * ν ¶ v* ¶ p* * ¶v * ¶v u v + + = + * * * * UD ¶t ¶x ¶y ¶y

é ¶2v* ¶2v* ù ê *2 + *2 ú ¶y û ë¶x

(2.37)

The system (2.37) is nondimensional with only one nondimensional parameter, the Reynolds number. Note that this conclusion agrees with Buckingham’s theorem [21], which states that if a problem depends on N dimensional parameters containing M different units, the nondimensional form depends on N - M nondimensional numbers. In the present case, there are N = 4 dimensional parameters ρ, η, D, and U. The M units contained in these four parameters are kilos, meters, and seconds. From Buckingham’s theorem, it results that there is N - M = 1 nondimensional number for the dimensionless system. The characteristic scales D and U depend on the geometry of the problem. For a flow inside a tube, U is the average axial velocity and D is the tube diameter. For an obstacle in a fluid flow, U is the velocity far from the obstacle and D is the obstacle characteristic dimension (usually its hydraulic diameter). The criterion for laminar flow has the form



Re =

UD < Retrans ν

(2.38)

Retrans is the transition threshold between laminar and turbulent flow. For flow in tubes and pipes, Retrans is of the order of 1,000–2,000, and for a flow past an obstacle, Retrans is of the order of 64–100 [22].

2.2  Single-Phase Microflows

35

The main difference between macroscopic and microscopic flow is that macroscopic flows are most of the time turbulent whereas microscopic flows are laminar. In biotechnology—or microchemistry—velocities are most of the time small and it is very seldom that the flow is turbulent. In fact, the “laminarity” of the flow is usually high. Typical fluid velocities are of the order of 1 mm/s at the most in channels of cross dimensions of 1 mm maximum. The kinematic viscosity of water being ν = 10–6 m2/s, the Reynolds number is—at the most—of the order of 1. Typical Reynolds numbers vary from 10-4 to 1. Thus, the character of the flow is very laminar, meaning that the streamlines are locally parallel (Figure 2.19) and that even obstacles in the flow will not induce any turbulence. We will see in the next section that for very small Reynolds numbers the Navier-Stokes equations may be simplified and are reduced to the Stokes approximation. 2.2.4  Stokes Equation

For a stationary flow, at very low velocities, inertial forces become very small compared to the viscous forces. The Reynolds number is smaller than 1 and the inertia terms on the left of (2.36) may be neglected. In this regime, the Navier-Stokes equation reduces to the Stokes equation



∂p =ν ∂x

é ∂ 2 u ∂ 2 u ∂ 2 u ù Fx + ê 2+ ú+ ∂ y2 ∂ z2 û ρ ë∂ x

∂p =ν ∂y

é ∂ 2v ∂ 2v ∂ 2v ù Fy + ê 2+ ú+ ∂ y2 ∂ z2 û ρ ë∂ x

∂p =ν ∂z

é ∂ 2w ∂ 2w ∂ 2ww ù Fz + ê 2 + ú+ ∂ y2 ∂ z2 û ρ ë∂ x

(2.39)

In the case where the external force is just the gravity force, the simplification is considerable because the system (2.39) is now linear � 1 Ñp = ν DV + Ñ z 2 2



(2.40)

where we have used the notation D = Ñ 2 for the Laplacian operator. By taking the rotational of (2.40), and using the following mathematical relations curl (grad P) = Ñ ´ Ñp = 0 � � D (curl A ) = curl (D A )

we obtain

� � D (Ñ ´ V ) = D ω = 0

(2.41)

where ω is the vorticity of the flow. Thus, in the Stokes formulation, vorticity is a harmonic function [23] and the problem can be solved in the vorticity-streamline formulation as soon as the values of the vorticity on the boundaries are known.

36

Microflows

Figure 2.19  High laminarity of microflows. The different fluid streams flow in parallel, without mixing. Note the location of the stagnation points that depend on the flow rates.

Stokes formulation for creeping flows is very attractive because an apparently complex problem can be simplified to a linear formulation. Besides linearity, the Stokes equation is reversible [24], that is, a change of the velocity u to its opposite –u on the boundaries of the domain will result in a change of all the velocities to their opposite. An example of this reversibility property can be done by considering a microdeflector in a microflow, as sketched in Figure 2.20. The calculation of the flow has

Figure 2.20  Computational domain and calculation grid of a 2D microchamber with a deflector. The maximum width is 2 mm and the length of the chamber is 2.6 mm. Note that the meshing depends on the shape of the solid walls.

2.2  Single-Phase Microflows

37

Figure 2.21  Flow lines in the microchamber. The deflector does not induce any recirculation (COMSOL calculation).

been performed with the numerical software COMSOL using the complete NavierStokes equations. The flow lines are shown in Figure 2.20, for an inlet velocity of 1 mm/s from left to right. If the flow is reversed, the pattern of the flow lines is exactly the same (Figure 2.21). This is typically a case where the Stokes equations are sufficient to describe the flow. The property of reversibility is very important because it shows that, at very low velocities, it is not possible to design a microfluidic “diode” where the pressure drop would be small in one direction and large in the opposite direction. If one wants to design such a diode, the flow velocity must be sufficiently large to be outside the Stokes hypothesis or, as mentioned earlier, be non-Newtonian [17]. For example, micropipes can give directionality to micropumping, based on a dissymmetrical design, as shown in Figure 2.22, if the fluid flow is sufficiently important. A simulation of such a design has been performed using the COMSOL software (Figure 2.23). For a flow of 1 mm/s from left to right, the pressure drop is 296 Pa; it is 324 Pa if the flow is reversed. For a flow of 100 mm/s, the pressure drop is 29

Figure 2.22  Directionality of pumped flow obtained by dissymmetrical pipes. Oscillations of the piezo-electrically actuated membrane trigger a directional flow only in the case of a sufficiently high Reynolds number.

38

Microflows

Figure 2.23  Modeling a fluidic “diode”: stream lines in dissymmetrical convergents (COMSOL software). The flow is laminar but with recirculating regions.

or 31 Pa depending on the direction of the flow. Thus, such a design functions as a diode for flow velocities larger than approximately 1 mm/s. Besides linearity and reversibility, the Stokes equation has a unique solution, meaning that there cannot be bifurcations of the solution linked to the development of instabilities. It is worth noticing that care must be taken when deciding to apply Stokes’ simplification. The condition Re << 1 must be verified everywhere in the fluid domain. If there is only one location, however small, where this condition is not realized, then the simplification may not be valid and will bias the whole solution. 2.2.5  Hagen-Poiseuille Flow

In practice, it is common to deal with cylindrical tubes or rectangular microchannels of different aspect ratios. In these cases, when the flow is laminar, there exists an analytical exact solution—for the cylindrical duct and the parallel plates—and an approximated solution—for the rectangular duct [2]. This solution is of interest because it simplifies considerably the PDE system governing the convection of particles (Chapter 6). 2.2.5.1  Cylindrical Tube

Navier-Stokes equations can be solved analytically in the particular case of a cylindrical duct. This solution is classical [2, 25] and we indicate here only the result. The velocity in the axial z-direction is given by



é æ r ö2 ù u( r ) = 2U ê1 - ç ÷ ú R ëê è ø ûú

where U is the average velocity. Equation (2.42) shows that the flow profile is parabolic and the same in any cross section. The mean velocity U is the averaged velocity in a cross section R

U=

(2.42)

1 ò u ( r ) 2 π r dr π R2 0

2.2  Single-Phase Microflows

39

In the case of a tubular duct U=



umax 2

Instead of using the mean velocity U to integrate the NS equations, we could have used the pressure difference between inlet and outlet, and we would find a relation between U and the pressure drop D P = Pin - Pout DP =



8 µU L R2

(2.43)

where L is the length of the tube. Equation (2.43) is sometimes called Washburn’s law [25]. 2.2.5.2  Parallel Plates

The same reasoning may be done for a laminar flow limited by two parallel plates. If the distance between the plates is D and the mean velocity is U, the velocity field is u( y ) =

2 3 é æ y ö ù U ê1 - ç ÷ ú 2 ê è D/2 ø ú ë û

(2.44)

where y is the transverse direction. Again, the profile does not depend on its location and is parabolic, with a maximum velocity of umax =



3 U 2

and the pressure difference between inlet and outlet is given by DP =



12 µ U L D2

(2.45)

2.2.5.3  Rectangular Ducts

Generally, in microtechnologies, capillaries of a circular cross section are used to link a fluid reservoir to the microsystem. Due to the microtechniques of etching in silicon, glass, or plastic, capillaries in bioMEMS are often rectangular [1]. An approximated, closed form solution exists for laminar flows in rectangular channels. The real flow profile is given by a series expansion [26], which is not always practical for applications. An approximation to this expansion was given by Purday [27]. The flow velocity in the z-direction inside a rectangular channel of dimensions 2a and 2b is approximated by



é æ x ös ù é æ y ö r ù u( x , y ) = umax ê1 - ç ÷ ú ê1 - ç ÷ ú êë è a ø úû êë è b ø úû

(2.46)

40

Microflows

where the exponents s and r depend on the aspect ratio α = b/a of the channel. A good approximation of these exponents is s = 1.7 + 0.5 α



-

1 4

ìï2 r=í ïî 2 + 0.3( α - 1/ 3)

for α £ 1/ 3 for α > 1/ 3

(2.47)

Relation (2.47) shows that the values of r and s are generally close to 2. Integration of (2.46) over the duct cross section yields umax é s + 1 ù é r + 1 ù =ê U ë s úû êë r úû



Taking into account relation (2.47), the value of the maximum velocity collapses to the value 3/2U of the two parallel plates solution when one dimension of the rectangle (for example, b) tends to infinity. Using the exponents given by (2.47), the relation (2.46) approximates the velocity within an accuracy of a few percent. Equation (2.46) combined with (2.47) shows that the velocity profile is nearly parabolic in the planes defined by the unit vectors (x, z) and (y, z) (Figure 2.24). 2.2.6  Pressure Drop and Friction Factor

Pressure has the dimension of an energy per unit volume. All along a flow, there is a redistribution of energy between pressure, inertia, and gravity. However, there is a loss of energy due to the friction at the wall. Next we give estimates of pressure drops in tubes of different shapes. 2.2.6.1  Friction Factor

Consider a duct of length L and cross section S. The momentum theorem in this duct yields

S D P = τ w p L

(2.48)

Figure 2.24  Left: velocity profiles in a rectangular duct; right: flow in a cross section (COMSOL).

2.2  Single-Phase Microflows

41

where p is the perimeter of the cross section (wetted perimeter). We can replace the unknown wall shear stress by a dimensionless unknown, the friction factor f defined as f =

τw 1 ρU2 2

(2.49)

Combining (2.48) and (2.49) yields



DP = f

pLæ 1 2ö ç ρU ÷ S è2 ø

(2.50)

Equation (2.50) may be transformed into a standard form by introducing the notion of a hydraulic diameter. 2.2.6.2  Hydraulic Diameter

Tubes and ducts are compared through their hydraulic diameter defined as



DH = 4

S p

(2.51)

where S is the cross section and p is the wetted perimeter. Some values of the hydraulic diameter are indicated in Figure 2.25.

Figure 2.25  Hydraulic diameter of different tubes.

42

Microflows

2.2.6.3  Pressure Drop for Different Duct Shapes

Using the definition of the hydraulic diameter (2.51) and substituting it into (2.50) yield DP = f



4L æ 1 2ö ç ρU ÷ Dh è 2 ø

(2.52)

For a cylindrical duct, the comparison between (2.42) and (2.52) yields f =



16 ReDh

(2.53)

and for the two parallel plates configuration, from (2.44) and (2.52), we obtain f =



24 ReDh

(2.54)

Friction in rectangular ducts of different aspect ratio λ = b/a (λ < 1) has been the object of numerous investigations. The expression of Shah and London [26] is



f =

24 (1 - 1.3553 λ + 1.9467 λ 2 - 1.7012 λ3 + 0.9564 λ 4 - 0.2537 λ 5 ) (2.55) ReDh

Another expression for the pressure drop in a rectangular capillary of cross dimensions a and b is [27]



DP =

8µL V a2

g (λ )

(2.56)



where the function g(λ) is defined by using the Heaviside function H 2



3 æ1+ λ ö g (λ ) = ç ÷ H (4.45 - λ ) + H (λ - 4.45) 2 λ è ø and the friction factor is f =

(

(2.57)

)

4 Dh2 a2 g ( λ ) ReDh

The most reliable and workable expression is probably [28]



DP =

4µ LU 1 min(a, b)2 q(ε )

q(ε ) =

æ π ö 1 64 - 5 ε tanh ç ÷ 3 π è 2ε ø

ε = min(b / a, a / b) and the friction factor is f =

(



)

2 Dh2 min(a, b)2 ReDh

(2.58)

1 q(η )

2.2  Single-Phase Microflows

43

For a square duct f =



14.2 ReDh

(2.59)

In the case of an equilateral triangular duct f =



13.3 ReDh

(2.60)

In the literature, there exist catalogs of friction factors for very different shapes of ducts, with or without obstacles [29]. 2.2.6.4  Laminar Pressure Drop

We have just seen that in the laminar case, the pressure drop is proportional to the mean velocity or to the flow rate. The hydraulic resistance of a fluidic channel can then be defined as DP = RQQ or D P = RV V . The two expressions are equivalent if we remark that RV = RVS, where S is the cross-section area. Next, for simplicity, we give expressions of RV. For an arbitrary cross section, with x- and y-dimensions of the same order, a general approximation of the hydraulic resistance is



RV » 2 µ L

p2 S2

(2.61)

where S is the cross section area and p is the wetted perimeter. Exact or approximate values have been found for particular shapes. These values are listed in Table 2.1. Note that the resistances correspond to an interior flow without free surface (all the walls are taken into account). 2.2.6.5  The Effect of Roughness

In the preceding sections we have derived expressions for the pressure drop in microchannels assuming that the walls were perfectly smooth. By “perfectly smooth,” we assume that the hypothesis for the derivation of the Poiseuille-Hagen flow is satisfied, that is, the flow has a unique axial component and no transverse component. If the walls have roughness of a sufficient size to induce some 3D component of the flow near the wall, then the Poiseuille-Hagen hypothesis is not met and the expression for the pressure drop must be revised. It has been observed that friction factors were usually higher than the values predicted by the conventional theory for smooth walls. Besides, the theoretical results are approximately valid for relatively large channels, but deviate significantly from experimental observations in the case of small channels. The effect of roughness on pressure drop can be approached by a perturbation method where the roughness is modeled by a spatial sinusoidal function [30–32]. It is concluded that the additional friction is related first to the roughness ratio ε = h/w, to the value of the Reynolds number and to the wave number of the wall surface (Figure 2.26). A general conclusion is that the roughness can be neglected for ε < 1–3%, but increases quickly above this threshold.

44

Microflows

Table 2.1  The Different Expressions of the Pressure Drop as a Function of the Geometry and Dimensions Shape

Geometry

Area, Perimeter

Expression DP = RV

Circle

R=

Ellipse R= Parallel plates

(

R=

[26]

S = π a2 p = 2π R

)

4µL 1+ ε2

R=

Rectangle

8µ L a2

Reference

[28]

S = π ab

b2

p = 4 a E 1 - (b a)

12 µ L h2

S = hw

4µL 1 h2 q(ε)

2

[26]

p = 2w [28]

S = hw p = 2 (w + h)

æ π ö 1 64 q(ε) = - 5 ε tanh ç 3 π è 2 ε ÷ø Square

Triangle (equilateral)

28.3 µ L h2

R=

R= 2a 3

2a 3

[28, 30]

S = h2 p = 4h

60 µ L a2

S = a2

3

p = 6a

3

[28, 30]

2a 3

Circular sector

R=

Half-circle

S = φ a2 p = 2 a(1 + φ )

64 µ L 3 a2

S = π a2 2 p=πa

R=

Annular torus

R=

Parabola

(

8µ L 1 a2 β

β = ε2 -1+

ε = b/a or ε = h/w and g (φ ) =

µL 1 a2 g(φ)

S = π a 2 - b2 2 ln (1 ε ) + ε 2 - 1 ln (1 ε )

R=

128 φ 3 tan (2 φ ) 2 - φ π5 16 φ

¥

[28]

p = 2 π (a + b )

35 µ L 4 h2

S=

é

1 ù ú ( 2n - 1 )2 (2n - 1 + 4 φ π ) 2 ( 2n - 1 - 4 φ π)ú ë û

å êê 1

)

[28]

E is the complete elliptic integral of the second kind. When two dimensions are present, ε is their ratio (ε < 1).

hw 3

[30]

2.2  Single-Phase Microflows

45

Figure 2.26  Sketch of a channel with rough upper and lower walls.

2.2.6.6  Slip Velocity

It is usual to consider a velocity at the contact of a solid surface to be zero. In reality, solid surfaces are not perfect, and there is usually a velocity slip at the wall of the order of a few m/s. We consider here a nanoscopic roughness smaller than that of the preceding section. At the macroscopic scale, or at a sufficiently large microscopic scale, the slip can be ignored because the velocities are much larger than the slip velocity. At the nanoscopic limit, the slip at the wall is not negligible in front of the average velocity. The slip is linked to the geometry and chemical properties of the surface. Velocity slip at the wall is usually related to nanobubbles trapped in nanocavities along the wall. This is why hydrophobic surfaces have the largest slip length, usually of the order of 1 mm. The slip at the wall is characterized by a slip length b (Figure 2.27). The slip length is usually defined by



æ ∂v ö v wall = b ç ÷ wall è ∂y ø

(2.62)

For a Poiseuille flow, the correction of velocity is given by



v real - v no -slip 6 b = v no -slip h

(2.63)

where h is the channel length. For h = 10 mm, the error is 60%. Usual microfluidics applications have dimensions of the order of h = 100 mm, and the error is of the order of 6%, which is to compare to the added friction linked to roughness. 2.2.7  Bernoulli’s Approach

Bernoulli’s work in hydraulics dates back to 1738. However, Bernoulli’s equation is probably the most frequently used in engineering hydraulics today, and it has recently found interesting applications in microfluidics [15]. This equation relates

Figure 2.27  Slip length at the wall.

46

Microflows

the pressure, velocity, and height in a steady motion of an ideal fluid, but its energy form has extended its domain of application to viscous fluids. However, there is an important restriction to the applicability of Bernoulli’s equation: the flow has to be one-dimensional. This is not very restrictive in microfluidics, because microflows in capillary and microchannels are mostly one-dimensional. There are three forms of Bernoulli’s equations. The first form derives directly from the Navier-Stokes equation under rather severe restrictions. First, suppose that the fluid is an ideal fluid, so that the diffusion terms in the NS equation may be neglected (Euler’s form of the NS equations)





� ∂u � � � 1 + u . Ñu = F - ÑP ∂t ρ

(2.64)

Equation (2.64) is a vector equation having the dimension of the flow field. Suppose also that the velocity field derives from a potential � u = -Ñφ If the external force field is conservative, that is, derives from a potential � F = -ÑΩ



and if the fluid is supposed incompressible ρ = const .



then (2.64) may be cast under the form



∂ � � P (-Ñφ ) + u . Ñu = -ÑΩ - Ñ ∂t ρ Then the following gradient is identically zero



é ∂ φ u2 Pù Ñ ê+ +Ω+ ú =0 2 ρ úû ëê ∂ t Thus, the function must be constant



-

∂ φ u2 P + +Ω+ =C ∂t 2 ρ

If we assume that the flow is stationary, we obtain the Bernoulli’s equation



u2 P +Ω+ =C 2 ρ

(2.65)

This first approach requires quite severe conditions; other forms of Bernoulli’s equations have been derived with less restricting hypotheses. The second form of Bernoulli’s equation arises from the fact that in a steady flow, the particles of fluid move along streamlines, as on rails, and are accelerated or decelerated by the forces acting tangent to the streamlines (Figure 2.28).

2.2  Single-Phase Microflows

47

Figure 2.28  Bernoulli’s equation along a streamline.

This formulation does not require the irrotational hypothesis for the flow, and we are left with the equation u



du dΩ 1 dP =ds ds ρ ds

(2.66)

where s is the distance along the streamline and u is the velocity directed along the streamline. The integration of (2.66) yields u2 P +Ω+ =C 2 ρ



The third form of Bernoulli’s equation is derived from the conservation of energy. Energy balance is often a powerful and elegant method in physics and that was the initial Bernoulli’s approach. In the case described in Figure 2.29, an element of fluid is transferred from one point to another in a duct with impermeable, rigid boundaries. We can look at it as if there were imaginary pistons moving with the speed of the fluid. The energies per unit volume, made up of kinetic, potential, and pressure, are equated to obtain



U=

ρ u12 ρ u22 + ρ g z1 + P1 = + P2 2 2

(2.67)

so that, by taking arbitrary points



U=

ρ u2 2

+ ρ g z + P = const .



(2.68)

The advantage of the energy approach is that it is general. First, it contains the streamline equation, just by taking imaginary pistons corresponding to the streamline as shown in Figure 2.30, second, the fluid may be assumed compressible; and

Figure 2.29  Sketch of the displacement of a fluid element in a duct.

48

Microflows

Figure 2.30  Bernoulli’s law: energy conservation along a streamline. P is the pressure and V is the velocity.

third, friction can be accounted for, by assuming a loss of energy due to friction on the rigid wall between any two cross sections. Streamlines do not account for wall friction since they are not bounded by walls, but energy conservation in a cross section does. As we have seen before, the pressure drop caused by friction on the wall is usually expressed by



DP = f

4L æ 1 2ö ç ρU ÷ Dh è 2 ø

where f is the friction factor (nondimensional). The value of f depends on the particular geometry of that part of the duct where the conservation equation is applied. In some cases, such as for laminar flows in circular or rectangular ducts, the coefficient f can be calculated; most of the time it is given by algebraic expression using one or more parameters derived from experiments. Bernoulli’s equation incorporating friction pressure drop is then



ρ u12 ρ u22 + ρ g z1 + P1 = + ρ g z 2 + P2 + DP1,2 2 2

(2.69)

where DP12 is the pressure loss by friction on the walls between points 1 and 2. Written in such a form, Bernoulli’s equation is a powerful tool for many applications in microfluidics, as we will see in the following sections. 2.2.8  Modeling: Lumped Parameters Model

Because it takes into account the complex geometry of the boundaries, the finite element method is the preferred method of modeling microfluidic flows. We will not deal here with the finite element method and its application to microfluidics; this would be a book itself, and some aspects are already detailed in the literature [33]. However, we present the lumped parameters model, which is a simplified calculation method and gives very interesting and accessible results in some cases.

2.2  Single-Phase Microflows

49

For complex hydraulic circuits, including many parts having different functions, modeling with finite elements numerical software can quickly become impossible because too many nodes are required and the capacity of the computer is exceeded. Besides, the description of the geometry can be quite difficult. In such a case, the circuit may be decomposed in connecting parts, with each part—or branch— corresponding to a precise function, such as connecting channels, microchambers, micropumps, and valves. Such a model is called a lumped model (Figure 2.31) [34, 35]. The model requires defining nodes {i = 1, N} and branches {j = 1, M}, the nodes being the extremity of the different branches. As we deal with a flow field calculation, the unknowns are the average velocities Uj and the pressure at the nodes Pi. Thus, the vector of unknowns is of the dimension N + M



ì U1 ü ï ï ïU M ï A=í ý ï P1 ï ï PN ï î þ A first set of equation is constituted by the mass conservation equations at each node i of the net. At a node i, the equation for the conservation of the flow rate is



åU j ji

i

S ji = 0

(2.70)

where ji is the index corresponding to all the branches connected to node i. Sji and Uji are, respectively, the cross section of these branches and the average velocities. Note that the velocities Uji are signed. The second set of equations is constituted pressure drop relations. For the branch [i - 1, i], this relation can be cast into the form

Pi -1 - Pi = f (U i -1,i , Si -1,i )

(2.71)

If the branch [i - 1, i] is simply a channel of constant section, or any type of channel where Bernoulli’s equation applies, the preceding relation collapses to Pi -1 - Pi =

8η Li -1,i U i -1,i Rh2,i -1,i

Figure 2.31  Schematic view of a microfluidic circuit.

(2.72)

50

Microflows

Many different functions may be taken into account by the lumped model, such as reduction or widening of the cross section, Venturis, and a change of the direction of the circuit and pumps. The system of (2.70) and (2.72) constitutes a system of N + M equations, because there is an equation of type (2.70) at each node and an equation of type (2.72) for each branch. With proper boundary conditions, such a system can be solved either by matrix inversion, if it is a very complex system, or by direct calculation for the simplest cases. Next we give an example of the usefulness of lumped models. Attention should be given to the liquid flowing in the channels. We saw in Section 2.2.2 that non-Newtonian networks behave differently than Newtonian fluid networks. 2.2.9  M  icrofluidic Networks: Worked Example 1—Microfluidic Flow   Inside a Microneedle

An example of the emergence of new technologies in medical science is microneedles. For external uses on human skin, the classical needle with syringe has found a replacement with the patch of microneedles [36, 37]. For internal delivery, new concepts of needle are currently being developed [38, 39]. One of these new concepts is that of a needle of smaller cross section—to be less invasive—and with many microscopic side channels in order to diffuse more efficiently the injected molecules. Flow motion in this last type of microneedle is a good example of microflow and an example of the utility of lumped models. For this reason, in this section we detail the flow distribution and the approach to the dimensioning of such a system. 2.2.9.1  Drug Delivery and Injection System

Diffusion in cell clusters and tumors will be presented in Chapter 4 concerning the mechanism of diffusion. We recall the principle of drug injection (Figure 2.32).

Figure 2.32  View of drug injection in a cell cluster (obtained by calculation [40, 41]). The ECS is delimited by the white dots.

2.2  Single-Phase Microflows

51

Figure 2.33  Schematic view of the drug dispense needle.

The active molecules transit through the needle tip into the cells extracellular space (ECS) and finally inside the cells. Let us focus now on the injection device. The rapidity of dispense and its efficiency can be increased by a microneedle, which has many exits uniformly distributed on the sides (Figures 2.33 and 2.34). The advantage of this type of needle is that drug delivery starts from all side exits disposed along the needle. For the best possible efficiency of the microneedle, all side exits should have the same flow rate. Eventually, the needle may be used as an electrode to increase the uptake rate. It has been observed that an electric field increases the uptake rate [42], so the needle can be electrically actuated. However, we discuss here only the microfluidic aspects. 2.2.9.2  Lumped Model

The statement of the problem is: What should be the dimensions of the different side exits if we want them to deliver the same flow rate? Intuitively, it is clear

Figure 2.34  Schematic view of the central channel and the side exits. All side channels must deliver the same flow rate.

52

Microflows

that the side exits close to the needle tip should have larger cross sections than those in the vicinity of the flow inlet. If all the side exits were to have the same section, then the flow would exit in the first side branches and not reach the needle tip. One solution might be to assign initial cross sections to the side exits, run a calculation with a numerical software, memorize the values of the flow rates at each exit, calculate a “distance” to the expected uniform flow solution, find an optimization algorithm to change the dimensions of the side exits, and run the process iteratively until convergence. However, this way of proceeding is long and costly. Optimization over N parameters (the N widths of the 2N side channels [ai, i = 1, N]) is very complex. In such a case, it is preferable to search for an inverse algorithm based on a lumped model formulation. Using a lumped element model to take care of all the different microfluidic segments in the needle, and imposing the constraint that the flow rate at all outlets is the same, a recurrence relation for the pressure at the nodes can be derived and solved to obtain the desired channel widths. 2.2.9.3  Algorithm

The algorithm has three steps: (1) calculation of the velocities in the central channel by a recurrence relation starting from the needle tip, (2) establishment of a recurrence relation for pressure at the nodes starting from the needle tip, and (3) calculation of the pressure at each node using the first two steps. Step 1: Velocities in the Axial (Central) Channel

Let the letters P, Q, V stand for, respectively, the pressure, flow rate, and fluid velocity. The density of the liquid is ρ. Because of the process of fabrication of the needle, the vertical dimension of the microchannels b is the same for all the channels. For simplicity, it is assumed that the spacing L (axial distance) between the side channels is constant and that the width a0 of the main channel is a given constant. Consequently, the length of the side channels is also a constant, Ls. A schematic view of the flow channels is given in Figure 2.35. Because there are 2N exits, the total mass conservation equation can be written as

Qin = (2 N )Qexit

(2.73)

By a recurrence approach, starting from the far end of the needle and progressing to the front end, we obtain

Figure 2.35  Schematic view of the main and secondary channels.

2.2  Single-Phase Microflows

53

Vi =



2Qs ( N - i ) , i = 0,... N - 1 ρ a0 b

(2.74)

Step 2: Pressure at the Axial Nodes

We have already seen in relations (2.49) and (2.50) that Washburn’s law for a rectangular capillary of cross dimensions a and b (λ = b/a) is DP =



8η L V a2

g (λ )

where the function g(λ) is defined by using the Heaviside function H 2

3 æ1+ λ ö g (λ ) = ç ÷ H (4.45 - λ) + H (λ - 4.45) 2 λ è ø



At an intersection, there is a distortion of the laminar flow lines. This problem is complex in a rectangular geometry and we simplify by D Pint er =



8η (13a )V a2

g (λ )

(2.75)

and in a side branch of length L, the linear pressure drop is reduced to D Plinear =



8η (L - 4 a ) V

g (λ )

a2

(2.76)

so that the total pressure drop of a side branch is D P = D Plinear + D Pint er =



8η (L + 9 a ) V a2

g (λ )

(2.77)

Again, starting from the last node in the axial channel and applying (2.77), we obtain the pressure at the last node PN = Po +



8η (Ls + 9 aN ) VN* aN 2

g (λN )

where VN* is the velocity in the Nth side branch and Po is the pressure at the outlets (atmospheric pressure). By replacing the flow rate by the flow velocity in the Nth side branch, the pressure can be cast into the form PN = Po +

8η (Ls + 9 aN ) Qo b a3N

g (λN )

(2.78)

Now, we progress towards the front end of the axial channel and deduce the recurrence relation Pi = Po +

8η (Ls + 9 aN) Qo 3 b aN

g (λN) +

8η L Qo g (λ 0) (N - i )(N - i + 1) b a03

(2.79)

54

Microflows

Equation (2.79) is a recurrence relation for the pressure at the nodes versus the widths of the side channels; this relation has been obtained by considering the velocities in the axial main channel only (with the exception of the last side channel). The pressures at the nodes are now directly calculated using the side channels. Step 3: Pressure at the Nodes (From Side Channels)

The pressure at a node is directly related to the outside pressure by the relation Pi = Po +

8η (Ls + 9 ai ) Qo b ai3



g (λi )

(2.80)

and the solution is obtained by equating the values of the pressure at the nodes from (2.80) and (2.79)

(Ls + 9 ai) g

ai3

(Ls + 9 aN ) g

( λi) =

3 aN

(λ N) +

L g (λ 0) (N - i )(N - i + 1) a03

(2.81)

For any given value of the width of the last side channel aN, the right-hand side of the previous relation is known, and we find the implicit relation for the ai of the type

(Ls + 9 ai) g ai3



( λi) = Bi

(2.82)

where Bi is the value of the right-hand side (which depends only of the width of the last side channel). The inverse solution is then reduced to an implicit solution of an analytical function, and this is tractable. Depending on the value of the λi, one has to solve either one of these two third-order polynomials. For λi > 4.45, 2 Bi ai3 - 9 ai - Ls = 0 3



(2.83)

For λi < 4.45,

(B b - 9) a i

2

3 i

- (Ls + 18b) ai2 - b ( 2 Ls + 9 b) ai = Ls b 2

(2.84)

It can be shown that these two polynomials have one real root and two imaginary roots, so that the real root is the desired solution. It is interesting to note that the fluid viscosity does not appear in (2.82), and neither does the value of the total flow rate (Qi), so that the calculated dimensions of the microsystem will satisfy the constraint of delivering the same flow rate at the outlets for different fluids and different inlet flow rates. That shows a generality in the system. Another advantage of this method is that it is straightforward to change the angle between the main and side channels by changing the pressure drop expression at an intersection and replacing (2.77) by

2.2  Single-Phase Microflows

55

Figure 2.36  Calculated width of side channels as a function of the channel number. The width increases from the entrance to the tip of the needle.



DP =

(

)

8η L + 9 a sin2 α V a

2

g (λ )

(2.85)

For α = 90°, formula (2.85) is the same as (2.77). Figure 2.36 shows that the distribution of the ai in the case of 2 ´ 10 side channels, for a value of a10, equals 30 mm. We now can use a direct model to verify the accuracy of the algorithm. Here, the direct model is the SABER code from the COVENTOR package [43]. Calculated widths are used as inputs in the SABER calculation. If the algorithm is correct, the results should agree. It is checked in Figure 2.37 that the SABER results agree well with the lumped model results. Microneedles based on the results of the dimensioning algorithm have been realized in silicon using microtechnologies for silicon etching and assembling (Figure 2.38). Needles of different sizes (from 500 mm to 300 mm) and with different side channel angles have been fabricated (Figures 2.39 and 2.40). Using methyl blue colored water and disposing the needle on a flat blotter, it is checked that all the flows at all exits are identical (Figure 2.41).

Figure 2.37  Comparison of side exit velocities. Continuous line: present algorithm, and dots: SABER results.

56

Microflows

Figure 2.38  View of a microneedle.

2.2.10  M  icrofluidic Networks: Worked Example 2—Plasma Extraction   from Blood

Separating plasma from blood is the first step of blood analysis. A promising method is based on the Zweifach-Fung bifurcation effect [44, 45]. The bifurcation law describes that, in the microcirculation, when erythrocytes (red blood cells) flow through a bifurcating region of a capillary blood vessel, they have a tendency to travel into the daughter vessel, which has the higher flow rate, leaving very few cells flowing into the lower flow rate vessel. The critical flow rate ratio between the daughter vessels for this cell separation is approximately 2.5:1 when the cell-to-vessel diameter ratio is of the order of 1. This effect will be investigated in more detail in Chapter 6. The principle is to design a network having the form of a tree where a fixed fraction of the main flow is diverted into side branches [46] (Figure 2.42). Let us note α this fixed fraction. The problem is to find the hydraulic resistances of all the side branches. 2.2.10.1  Flow Rates

Consider first the flow rates. At the first intersect,



Q0 = 2QB1 + QM 1 and, dividing by QM1, we find Q0/QM1 = 2α + 1. Then Q0 QM 1 = 2α + 1 α Q0 QB1 = 2 α + 1

(2.86)

(2.87)

Figure 2.39  Detailed view of the microneedle: main channel and oblique side channels. (Courtesy of F. Rivera, CEA/LETI.)

2.2  Single-Phase Microflows

57

Figure 2.40  Microscope views of an axial cut of the needle. (Courtesy of F. Rivera, CEA/LETI.)

A recurrence relation shows immediately that QM n = QBn =

Q0 ( 2 α + 1)

n

α Q0

(2.88)

n

( 2 α + 1)

2.2.10.2  Hydraulic Resistances

Let us do a backward recurrence. At the end rank n, the pressure difference between the intersect n and the outlets can be expressed by Pn - Patm = RQM n = RBn QBn

Then

RB n = R α

At the n - 1 intersect

R (QM n + QM n -1) = RBn -1 QBn -1

Figure 2.41  Visualization of equal flow rates at all exits. (Courtesy of F. Rivera, CEA/LETI.)

(2.89)

58

Microflows

Figure 2.42  Sketch of the separation device. Note that the main channel hydraulic resistances are all equal to R.

and RB n -1 =



Ré 1 ù 1+ ê α ë 1 + 2 α úû

(2.90)

Finally RB 1 =

ù Ré 1 1 ê1 + ú + .... + n -1 α ê 1+ 2α ú + 1 2 α ( ) ë û

(2.91)

The system is now fully known. Note that we have not taken into account the pressure drop at the intersections. It should be treated as in the preceding section. We insist that the dimensioning we have done corresponds to a Newtonian fluid, such as diluted blood, not whole blood. 2.2.11  Hydrodynamic Entrance Length: Establishment of the Flow 2.2.11.1  Theory

When a fluid enters a tube, at the entrance of a tube, there is a length where the flow in not yet established. The shape of the velocity profile evolves until it reaches the established profile (Hagen-Poiseuille profile for a laminar flow), as shown in Figure 2.43. The evolution of the profile is due to the progressive increase of the boundary layers at the wall. When the boundary layers merge at the center axis, the flow is established. The length of the establishment region may be approximated by using the boundary layer theory. If we suppose that the entrance length corresponds to the

2.2  Single-Phase Microflows

59

Figure 2.43  Evolution of the velocity profile at the entrance of a tube. Outside the boundary layer, the flow field presents no shearing.

length where the boundary layer grows till it reaches the value R = D/2, we can use Blasius’ boundary layer correlation [47] 1

δ @ 5Rex 2 x



(2.92)

where δ is the boundary layer thickness and x is the axial distance. With the entrance length h being the distance at which the two boundary layers merge, h is obtained from (2.92) by setting δ = R= D/2 h @ 0.01 ReD D





(2.93)

Actually, this approximation does not take into account the acceleration of the fluid in the core region. A more accurate expression has been obtained by Sparrow and Schlichting [47] h (2.94) @ 0.04 ReD D There is an important difference between macroscopic and microscopic flows. At a macroscopic scale, the establishment length may be long, whereas it is very short in microfluidics. An approximate entrance length for a 100 mm radius tube is of the order of 8 mm for a Reynolds number of 1, which is quite small. It is often experimentally observed that during biochemical reactions in capillary tubes, there is an anomalous region at the entrance of the tube, and the reason is observed wrongly attributed to the flow pattern in the entrance region. It is much more likely that there is an anomalous concentration of immobilized chemical species—or tracers—in the entrance region which results in anomalous reaction or detection (see Chapter 7). 2.2.11.2  Modeling

When modeling a flow in a microchannel, one wants usually wants to avoid the problem of the establishment length, for example, if one wants to determine precisely the linear pressure drop in the channel. In such a case, it is sufficient to specify for inlet conditions the established (or nearly established) velocity field (Figure 2.44). For a cylindrical tube, the velocity field is given by (2.42).

60

Microflows

Figure 2.44  Vorticity contour plot of an axial flow in a cylindrical tube showing the establishment length. (a) Uniform inlet velocity and (b) inlet velocity field given by (2.42).The continuous lines are streamlines.

2.2.12  Distributing a Uniform Flow into a Microchamber

BioMEMS and immunoassays currently use microfluidic networks feeding the microchambers where the biological processes take place. Inside the capillary channels, the velocity profile is a Hagen-Poiseuille flow profile, parabolic for cylindrical and rectangular channels, showing an important transverse gradient. For the purposes of bioanalysis, labeled surfaces are installed inside the microchamber and it is important that flow velocity should be as uniform as possible above all the labeled surfaces. Labeled surfaces affected by a weak part of the main flow do not function correctly. A simple design of a divergent cone between the capillary channel and the microchamber cannot be satisfactory because it would result in a very nonuniform velocity flow profile in the microchamber. We have used the COMSOL software [48] to solve the incompressible Navier-Stokes equation. Figure 2.45 shows that the velocity profile in a cross section of the microchamber is nearly parabolic. The conditions for correct functioning of the device are then not met. If the divergent cone is separated in subchannels as shown in Figure 2.46, the velocity profile is very much improved, at least in the middle part of the cross sec-

Figure 2.45  (a) Widening from a microfluidic channel to a microchamber. (b) Nearly parabolic velocity profile in a cross section, showing that the microchamber will not be fed by a uniform fluid flow.

2.2  Single-Phase Microflows

61

Figure 2.46  (a) Widening from a microfluidic channel to a microchamber using subchannels. (b) Velocity profile in a cross section of the microchamber. The profile is improved compared to Figure 2.45.

tion of the microchamber. Taking into account the requirements of lithography techniques, we obtain the classical solution schematically represented in Figure 2.47. From left to right, each branch is divided into two subbranches, so that the velocities are the same in all the branches of same size. The velocity profile is very flat in the middle part of the microchamber cross section. This principle is applied to design proteomic reactors (Figure 2.48), which will be described in the following section. We have shown here that by diverting the flow in many secondary flows, a satisfactory widening can be achieved. However, the price to pay is an increased friction, and the pressure drop is much higher in the solution of Figure 2.47 than that of Figure 2.45. 2.2.13  The Example of a Protein Reactor

In a proteomic reactor, proteins are broken into peptides (or peptidic segments) by the action of enzymes. The peptides are then separated according to their size inside a chromatography column and finally expulsed into a mass spectrometer. The proteins’ composition is reconstructed after analysis of the identified peptides by the reconstruction algorithms.

Figure 2.47  (a) Widening from a microfluidic channel to a microchamber using lithography technology. (b) Velocity profile in a cross section of the microchamber.

62

Microflows

Figure 2.48  Inlet channels of a microfabricated proteomic reactor. The flow rate is divided in two at each step. (Courtesy of N. Sarrut, LETI.)

The part of the microreactor where the proteins are “digested” by the enzymes must be carefully designed in order to have a complete reaction. The proteins are transported by the flow and the enzymes have been previously immobilized on the walls of the reactor. In order to have a complete reaction, a maximum surface area for the contact between proteins and enzymes must be available as shown in the upper part of Figure 2.48 [49]. This is typically a microfluidic problem coupled with a technological challenge. The microfluidic problem resembles, at another scale, to the problem of heat exchanger where obstacles are introduced to increase the contact surface. However, the technological constraints much more demanding at the microscopic level, with the fabrication of micropillars or the use of textured surface. Typical feasible pillars are shown in Figures 2.49 and 2.50. The use of computer simulation is essential to compare the efficiency of the different designs proposed by the etching technology. In Figures 2.51 and 2.52, a comparison of the flow in two different arrangements of micropillars has been made (the size of the pillars is less than 10 mm). The flow field has been computed with the help of the finite element numerical software COMSOL. The differences between the two arrangements are obvious. There are poorly irrigated channels in Figure 2.51, but not in the case of Figure 2.52. After computation of different motives, the configuration of Figure 2.52 has been adopted for the reactor (Figure 2.48). 2.2.14  Recirculation Regions

Recirculation regions are a very useful feature in microfluidic systems. They may be used to mix and homogenize a fluid [50], to promote biochemical reactions

2.2  Single-Phase Microflows

63

Figure 2.49  Detail of the fluid inlet in a proteomic reactor. Protein digestion takes place on the walls of the hexagonal pillars. (Courtesy of N. Sarrut, LETI.)

Figure 2.50  Nanostructuring of a proteomic reactor. (Courtesy of N. Sarrut, LETI.)

64

Microflows

Figure 2.51  Simulation results obtained with the numerical software COMSOL showing the poorly disposed pillars. The transverse channels have a very low fluid velocity and do not contribute to the biochemical reaction.

at the wall, such as DNA hybridization [51], and to trap biological objects [52]. Recirculation in microfluidic systems is usually not easy to obtain passively. One of the reasons is the small inertia of the flow associated to a small Reynolds number; another reason is the large value of the friction on the walls (especially bottom and

Figure 2.52  Correctly disposed channels. The flow affects all channels and a maximum of surface of reaction is obtained.

2.2  Single-Phase Microflows

65

Figure 2.53  (a) Sketch of the recirculation chamber; (b) 3D calculation with COMSOL.

top cover). In the literature, one finds mostly active (electric, acoustic, magnetic) methods [53, 54]. Note that recirculation is facilitated if there is a free surface—and less friction at the wall. This is the case of an open microbeaker actuated by an ionic wind needle [55]. However, in some cases, recirculation can be passively achieved by a careful design of the system. Next we indicate a rule to achieve recirculation in a microsystem derived from a computational approach. Let us take the example of a microchamber placed along a straight channel, such as that of Shelby et al. [52] (Figure 2.53). It has been experimentally observed that the Reynolds number of the flow is strongly linked to the occurrence of recirculation [52]. The COMSOL numerical program has been used to investigate the occurrence of recirculation. Because of the importance of the friction on the bottom and top walls, a 3D calculation is necessary. Two main parameters stem from the results: a channel Reynolds number defined by



Re =

Vmax DH Vmax 2 w d = ν ν w +d

(2.95)

where w and d are, respectively, the channel width and depth; and the gating ratio is defined by

G = d / h

(2.96)

where h is the opening length of the microchamber. Figure 2.54 shows the recirculation conditions. On one hand, the Reynolds number must be sufficiently high; on the other hand, the gating ratio should be sufficiently large (e.g., the depth of the channel should be large in front of the width of the opening of the microchamber). A very interesting feature is the assurance to have a recirculation when d/h > 0.8, even with a small Reynolds number. However, in this case, the recirculation velocity is quite small. 2.2.15  Inertial Effects at Medium Reynolds Numbers: Dean Flow

Usually, Reynolds numbers are very small in microsystems for biotechnology, most of the time smaller than 1, and the flow is highly laminar. However, recently, medium ranges of Reynolds numbers (Re ~ 1 to 20) have been investigated. It has

66

Microflows

Figure 2.54  Recirculation conditions as functions of the channel Reynolds number and the gating ratio.

been shown that such flows in curved tubes have spiral streamlines in the curved regions caused by a centrifugal effect. We shall see in Chapter 6 that these spiraling streamlines are useful to guide and concentrate particles or cells [57], and to enhance mixing [58]. Let us consider the example of Figure 2.55. The main component of the velocity is directed along the direction of the tube, but the transverse component of the velocity (here vy), which is zero at low Reynolds number, is positive in two quadrangles and negative in the two others. Hence, there is a recirculating component of the velocity that induces a spiral flow. If we define the Dean number by

De = U R ν

R Rc = Re R Rc

(2.97)

where Rc is the curvature radius of the channel. The inertia-induced spiral motion is noticeable when the Dean number is larger than 1. Another way of pinpointing this recirculation motion is by plotting the zvorticity contour, as shown in Figure 2.56. For a cylindrical, curved channel such as that of Figure 2.55, the rotational velocity components can be analytically computed [58]. Dean showed that, in a toroidal coordinate system (Figure 2.57), secondary flow velocities are given by the following equations

2.2  Single-Phase Microflows

67

Figure 2.55  Vy contour plot for a flow in a cylindrical curved tube (Re = 5) (COMSOL numerical software).

Figure 2.56  Vorticity contour plot for a flow in a cylindrical curved tube, Re = 5 (COMSOL numerical software).

68

Microflows

Figure 2.57  Toroidal coordinate system.

u=





v=

A 2 De 2 1- x2 - y2 4468 2

2

(

) (4 - 5 x

(

)

(

2

)

- 23 y 2 + 8 x 2 y 2 + x 4 + 7 y 4

(2.98)

)

A De 1- x2 - y2 x y 3 - x2 - y2 1152

where A is a dimensionless parameter linked to the overall pressure gradient driving the flow, defined by R2 ∂P A= µ V Rc ∂ θ At the center of the tube (x = 0, y = 0) there is an outward component of 2 2 the velocity u (0,0 ) = A De . Note that the pressure isosurfaces are still planes 1117 perpendicular to the channel axis (Figure 2.58).

Figure 2.58  Pressure isosurfaces of a Dean flow are still planes perpendicular to the channel axis.

2.2  Single-Phase Microflows

69

2.2.16  Microflows in Flat Channels: Helle-Shaw Flows

Flat channels—channels whose aspect ratio d/w is very small—are common in biotechnology. There are used to force biologic targets to come to contact to the wall, and to bind to immobilized ligands on the solid surface. A 3D approach to such problems is not practical. In order to describe the vertical velocity profile, very small meshes are required, and a large horizontal domain cannot be covered with such small meshes. Clearly, the situation is close to 2D, but the friction on the two walls cannot be disregarded, it is even the dominant friction force. In such a case, the 2D Navier-Stokes equations need to be modified. For a steady state flow æ ∂ 2u ∂ 2u ö 12 µ � ∂P � � + µ çç 2 + 2 ÷÷ - 2 u ρ (u . Ñ) u = ∂x ∂y ø d è ∂x



(2.99)

The additional force in the right-hand side of (2.99) is the friction on the upper and bottom solid surfaces. Indeed, integrating this term on a w d Dx parallelepiped domain leads to



1 wd

w x +Dx d / 2

d /2

0

-d / 2

ò ò x

12 µ 12 µ Dx ò d 2 u dx dy dz = d 2 d -d / 2

ò

u dy =

12 µ D x U d2

(2.100)

which is the friction force on two parallel plates on a length Dx (Figure 2.59). Using (2.99) in a 2D formulation produces identical flow rates and pressures than would a 3D calculation. Note that the maximum velocities are not identical because the 2D calculation averages the velocity in the vertical z-direction. Hence, the maximum 2D velocity is only two-thirds of that obtained by the 3D calculation.

Figure 2.59  Helle-Shaw flow in flat microchannels.

70

Microflows

2.3  Conclusion Microflows are an immense subject of study and research. So far, we have only presented their most basic aspects. There is much more to be said, for example, on liquid films and on velocity slip at a solid wall, but the aim of this chapter is to give the reader the bases to tackle the usual microflow problems and to have the prerequisites to deal with more complex situations. Biotechnology heavily relies on microfluidics and especially on microflows. From continuous single flows for bioanalysis and biorecognition, to plug flows for high throughput screening, and to microsprays for mass spectrometer analysis, microflows are constantly present. In the next chapter, we complete the approach to microfluidics with a more recent and fast-developing branch of microfluidics called digital microfluidics, which details the behavior of microdrops.

References   [1]  Nguyen, N. -T., and S. T. Wereley, Fundamentals and Applications of Microfluidics, Norwood, MA: Artech House, 2002.   [2]  Bejan, A., Convection Heat Transfer, New York: Wiley-Interscience, 1984.   [3]  Bremmell, K. E., A. Evans, and C. A. Prestidge, “Deformation and Nano-Rheology of Red Blood Cells: An AFM Investigation,” Colloids and Surfaces B: Biointerfaces, Vol. 50, No. 1, 2006, pp. 43–48.   [4]  Franklin, R. K., et al., “Microsystem for Determining Clotting Time of Blood and LowCost, Single-Use Device for Use Therein,” U.S. Patent 7291310, November 6, 2007.   [5]  Hong, J. S., et al., “Spherical and Cylindrical Microencapsulation of Living Cells Using Microfluidic Devices,” Korea-Australia Rheology Journal, Vol. 19, No. 3, 2007, pp. 157– 164.   [6]  Luo, D., et al., “Cell Encapsules with Tunable Transport and Mechanical Properties,” Biomicrofluidics, Vol. 1, 2007, p. 034102.   [7]  Le Vot, S., et al., “Microfluidic Device for Alginate-Based Cell Encapsulation,” XVI International Conference on Bioencapsulation, Dublin, Ireland, September 4–6, 2008.   [8]  Smith, D. E., and S. Chu, “Response of Flexible Polymers to a Sudden Elongational Flow,” Science, Vol. 281, 1998, pp. 1335–1339.   [9]  Donati, I., et al., “Synergistic Effects in Semidilute Mixed Solutions of Alginate and Lactose-Modified Chitosan (Chitlac),” Biomacromolecules, Vol. 8, 2007, pp. 957–962. [10]  Morris, E. R., et al., “Concentration and Shear Rate Dependence of Viscosity in Random Coil Polysaccharide Solutions,” Carbohydrate Polymers, Vol. 1, 1981, pp. 5–21. [11]  Nakheli, A., et al., “The Viscosity of Maltitol,” J. Phys. Condens. Matter, Vol. 11, 1999, pp. 7977–7994. [12]  Ostwald, W. O., “The Velocity Function of Viscosity of Disperse Systems,” Kolloid Zeitschrift, Vol. 36, 1925, pp. 99–117. [13]  Shibeshi, S. S., and W. E. Collins, “The Rheology of Blood Flow in a Branched Arterial System, “Appl. Rheol., Vol. 15, No. 6, 2005, pp. 398–405. [14]  Nijenhuis, K., et al., Non-Newtonian Flows, New York: Springer, 2007. [15]  Nickerson, M. T., and A. T. Paulson, “Rheological Properties of Gellan, K-Carrageenan and Alginate Polysaccharides: Effect of Potassium and Calcium Ions on Macrostructure Assemblages,” Carbohydrate Polymers, Vol. 58, 2004, pp. 15–24.

2.3  Conclusion

71

[16] Bird, R. B., R. C. Armstrong, and O. Hassager, Dynamics of Polymeric Liquids, Volume 1: Fluid Mechanics, New York: John Wiley & Sons, 1987. [17] Lindner, A., J. Vermant, and D. Bonn, “How to Obtain the Elongational Viscosity of Dilute Polymer Solutions?” Physica A, Vol. 319, 2003, pp. 125–133. [18] Groisman, A., M. Enzelberger, and S. R. Quake, “Microfluidic Memory and Control Devices,” Science, Vol. 300, 2003, pp. 955–958. [19] Le Vot, S., et al., “Non-Newtonian Fluids in Flow Focusing Devices: Encapsulation with Alginates,” 1st European Conference on Microfluidics, Microflu’08, Bologna, Italy, December 10–12, 2008. [20] Berthier, J., S. Le Vot, and F. Rivera, “On the Behavior of Non-Newtonian fluids in Microsystems for Biotechnology,” Proceedings of the NSTI Nanotech Conference, Houston, TX, May 3–7, 2009. [21] Buckingham, E., “On Physically Similar Systems: Illustrations of the Use of Dimensional Equations,” Phys. Rev., Vol. 4, 1914, pp. 345–376. [22] Rosenow, W. M., and H. Y. Choi, Heat, Mass, and Momentum Transfer, Englewood Cliffs, NJ: Prentice Hall, 1961, p. 48. [23] Landau, L., and E. Lifchitz, Mécanique des Fluides, Editions Mir, 1971. [24] Tabeling, P., “Introduction à la Microfluidique,” Belin, 2003. [25] Washburn, E. W., “The Dynamics of Capillary Flows,” Phys. Rev., 1921, pp 273–283. [26] Shah, R. K., and A.L. London, “Laminar flow forced convection in ducts,” Academic Press, 1978, pp 197. [27] Bendib, S., and O. Français, “Analytical Study of Microchannel and Passive Microvalve; Application to Micropump Simulation,” Proceeding, Design, Characterisation, and Packaging for MEMS and Microelectronics 2001, Adélaide, Australia, 2001, pp. 283–291. [28] Bahrami, M., M. M. Yovanovich, and J. R. Culham, “Pressure Drop of Fully-Developed, Laminar Flow in Microchannels of Arbitrary Cross Section,” Proceedings of ICMM 2005, 3rd International Conference on Microchannels and Minichannels, Toronto, Ontario, Canada, June 13–15, 2005. [29] Idel’cik, I. E., “Memento des pertes de charge,” Eyrolles, Paris, ed. 1960. [30] Bruus, H., Theoretical Microfluidics, Oxford, U.K.: Oxford Master Series in Condensed Matter Physics, 2008. [31] Wang, H., and Y. Wang, “Influence of Three-Dimensional Wall Roughness on the Laminar Flow in Microtube,” International Journal of Heat and Fluid Flow, Vol. 28, 2007, pp. 220–228. [32] Bahrami, M., M. M. Yovanovich, and J. R. Culham, “Rough Pressure Drop of Fully Developed, Laminar Flow in Rough Microtubes,” Transactions of the ASME, Vol. 128, 2006, p. 632. [33] Zimmerman, W. B., Process Modeling and Simulation with Finite Element Methods, New York: World Scientific Publishing, 2004. [34] Adjari, A., “Steady flows in networks of microfluidic channels: building on the analogy with electrical circuits,” C.R. Physique 5, pp. 539–546, 2004. [35] Pietrabisa, R., et al., “A Lumped Parameter Model to Evaluate the Fluid Dynamics of Different Coronary Bypasses,” Med. Eng. Phys., Vol. 18, No. 6, 1996, pp. 477–484. [36] Gardeniers, H. J. G. E., et al., “Silicon Micromachined Hollow Microneedles for Transdermal Liquid Transport,” Journal of Microelectromechanical Systems, Vol. 12, No. 6, 2003. [37] Luttge, R., et al., “Microneedle Array Interface to CE on Chip,” 7th International Conference on Miniaturized Chemical and Biochemical Analysts Systems, Squaw Valley, CA, October 5–9, 2003. [38] Rivera, F., et al., “Microdispositif de Diagnostic et de Thérapie In Vivo,” Patent EN 0350919, November 27, 2003.

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Microflows [39] Rivera, F., et al., “In Vivo Transfection Microsystems,” Proceedings of the 26th IEEE Engineering in Medicine and Biology Conference, San Francisco, CA, September 1–5, 2004. [40] Berthier, J., F. Rivera, and P. Caillat, “Numerical Modeling of Diffusion in Extracellular Space of Biological Cell Clusters and Tumors,” Proceedings of the Nanotech 2004 Conference, Boston, MA, March 7–11, 2004. [41] Berthier, J., F. Rivera, and P. Caillat, “Dimensioning of a New Micro-Needle for the Dispense of Drugs in Tumors and Cell Clusters,” Proceedings of the Nanotech 2005 Conference, Anaheim, CA, May 8–12, 2005. [42] Widera, G., and D. Rabussay, “Electroporation Mediated and DNA Delivery in Oncology and Gene Therapy,” Drug Delivery Technology, Vol. 2, No. 3, May 2004. [43] Coventor, http://www.coventor.com/microfluidics. [44] Svanes, K., and B. W. Zweifach, “Variations in Small Blood Vessel Hematocrits Produced in Hypothermic Rats by Micro-Occlusion,” Microvascular Res., Vol. 1, 1968, pp. 210–220. [45] Fung, Y. C., “Stochastic Flow in Capillary Blood Vessels,” Microvascular Res., Vol. 5, 1973, pp. 34–48. [46] Kersaudy-Kerhoas, M., R. Dhariwal, and M. P.Y. Desmulliez, “Blood Flow Separation in Microfluidic Channels,” Proceedings of the 1st European Conference on Microfluidics, Microfluidics 2008, Bologna, December 10–12, 2008. [47] Schlichting, H., Boundary Layer Theory, New York: McGraw-Hill, 1960, p. 169. [48] COMSOL software package, http://www.comsol.com. [49] Sarrut, N., et al., “Enzymatic Digestion and Liquid Chromatography in Micro-Pillar Reactors—Hydrodynamic Versus Electroosmotic Flow,” SPIE San Jose Photonics West— MOEMS-MEMS, 2005. [50] Xia, Z., et al., “Fluid Mixing in Channels with Microridges,” ASME IMECE, the Proceedings of IMECE 2007 ASME International Mechanical Engineering Congress and Exposition, Seattle, WA, November 2007. [51] Lambert, R. A., et al., “Moving Flap Model for Fast DNA Hybridization in Microchannel Flow,” Workshop of the California-Catalonia Alliance for Miniaturization Science and Engineering, Barcelona, Spain, September 18, 2006. [52] Shelby, J. P., et al., “High Radial Acceleration in Microvortices,” Nature, Vol. 425, 2003, p. 38. [53] Xu, J., and D. Attinger, “Control and Ultrasonic Actuation of a Gas–Liquid Interface in a Microfluidic Chip,” J. Micromech. Microeng., Vol. 17, 2007, pp. 609–616. [54] Lammertink, R. G. H., et al., “Recirculation of Nanoliter Volumes Within Microfluidic Channels,” Anal. Chem., Vol. 76, 2004, pp. 3018–3022. [55] Atencia, J., and D. J. Beebe, “Magnetically-Driven Biomimetic Micro Pumping Using Vortices,” Lab Chip, Vol. 4, 2004, pp. 598–602. [56] Yeo, L. Y., J. R. Friend, and D. R. Arifin, “Electric Tempest in a Teacup: The Tea Leaf Analogy to Microfluidic Blood Plasma Separation,” Applied Physics Letters, Vol. 89, 2006, p. 103516. [57] Lavine, M., “Microfluidics: Streams Swirled by Dean,” Science, Vol. 312, 2006, p. 1281. [58] Sudarsan, A. P., “Multivortex Micromixing: Novel Techniques Using Dean Flows for Passive Microfluidic Mixing,” Ph.D. dissertation, Texas A&M University, December 2006.

Chapter 3

Interfaces, Capillarity, and Microdrops

3.1  Introduction Microfluidics in biotechnological systems are not limited to microflows. Each time that a liquid is in contact with another fluid or liquid, such as air or another immiscible liquid, an interface forms. This interface is associated to surface tension forces which can be very important at the microscale, compared to the other forces such as gravity and inertia, as shown in Chapter 1. At the contact of a solid surface, capillarity forces appear and play an important role. They may even dominate or govern the flow. In this section we present the notion of interface and the theory of capillarity, and we apply it to the physics of microdrops, which are frequently found in microfluidic systems.

3.2  Interfaces and Surface Tension Fluids can be miscible or immiscible. When they are immiscible, an interface separates the two fluids. 3.2.1  The Notion of Interface

An interface is the geometrical surface that delimits two immiscible fluid domains. This is a mathematical definition which implies that an interface has no thickness and is smooth (i.e. has no roughness). As practical as it is, this definition is a schematic concept. The reality is much more complex, and the separation of two immiscible fluids (water/air, water/oil, and so forth) depends on molecular interactions between the molecules of each fluid [1] and on Brownian diffusion (thermal agitation). A microscopic view of the interface between two fluids looks more like the scheme of Figure 3.1. In the presence of a wall, the interface contacts the wall along a triple line. However, in engineering applications, the mathematical concept regains its utility. At a macroscopic size, the picture of Figure 3.1(b) can be replaced by that of Figure 3.2, where the interface is a mathematical surface without thickness and the contact angle q is uniquely defined by the tangent to the surface at the contact line. In a condensed state, molecules attract each other. Molecules located in the bulk of a liquid have isotropic interactions with all the neighboring molecules; these interactions are mostly van der Waals attractive interactions for organic liquids and 73

74

Interfaces, Capillarity, and Microdrops

Figure 3.1  (a) Schematic view of an interface at the molecular size. (b) At the contact of a wall, a triple line forms.

hydrogen bonds for polar liquids such as water [1]. 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 the molecules of the other fluid or gas (Figure 3.3). 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 2. Of course, a molecule is also in contact with gas molecules, but, due to the low densities of gases, there are less interactions and less interaction energy than in the liquid side. The result is that there is locally a dissymmetry in the interactions, which results in a defect of surface energy. If a molecule at the interface is pushed outwards by Brownian motion, it is immediately pulled back towards its bulk phase by molecular interactions. 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 dimension of energy per unit surface 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 loss of a molecule at the interface is U/2. Surface tension is a direct measure of this energy loss, and if d is a characteristic molecular dimension and d 2 is the associated molecular surface, then the surface tension is approximately



γ »

U 2δ 2

Figure 3.2  Macroscopic view of the interface of a drop.

(3.1)

3.2  Interfaces and Surface Tension

75

Figure 3.3  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.

This relation shows that surface tension is important for liquids with a large cohesive energy and a 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 areas. 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 capillary constraints at the contact of solids, they take a convex rounded shape, as close as possible to that of a sphere. Another consequence is that it is energetically costly to create or increase an interfacial area. The same reasoning applies to the interface between two immiscible liquids, except that the interactions with the other liquid will usually be more energetic 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 3.1). 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 water and saltwater. Salt molecules will diffuse freely across a boundary between fresh water and saltwater. The 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, but usually these molecules do not

76

Interfaces, Capillarity, and Microdrops Table 3.1  Values of Surface Tension of Different Liquids at the Contact with Air at a Temperature of 20°C (Middle Column) and Thermal Coefficient a (Right Column) Liquid γ0 α Acetone 25.2 -0.112 Benzene 28.9 -0.129 Ethanol 22.1 -0.0832 Ethylene-glycol 47.7 -0.089 Glycerol 64.0 -0.060 Methanol 22.7 -0.077 Mercury 425.4 -0.205 Perfluoro-octane 14.0 -0.090 Polydimethylsiloxane 19.0 -0.036 Pyrrol 36.0 -0.110 Toluene 28.4 -0.119 Water 72.8 -01514

“stick” to the wall because of the Brownian motion. However, impurities contained in the fluid, such as particles of dust or biological polymers like proteins, may well adhere permanently to the solid surface because, at the contact with the solid interface, they experience more attractive interactions. The reason is that the size of polymers is much larger than water molecules and van der Waals forces are proportional to the number of contacts. Usually surface tension is denoted by the Greek letter g with subscripts referring to the two components on each side of the interface, for example, gLG 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 at the contact with a gas, and “interfacial tension” for a liquid at the 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 E is E = γ S



(3.2)

where S is the interfacial surface area. 3.2.2  Surface Tension

In the literature or in the Internet there exist tables for surface tension values [2, 3]. Typical values of surface tensions are given in Table 2.1. 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 g ~ 20–30 mN/m, while for water, g ~ 70 mN/m. The highest surface tensions are for liquid metals; for example, liquid mercury has a surface tension g ~ 500 mN/m. 3.2.2.1  The Effect of Temperature on Surface Tension

The value of the surface tension depends on the temperature. Observing that the surface tension goes to zero when the temperature tends to the critical temperature

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77

TC (e.g., the temperature where gas and liquid phase are indiscernible), Eötvös and later Katayama and Guggenheim [4] have worked out the semi-empirical relation



n

æ T ö γ = γ ç1 è TC ÷ø *

(3.3)

where g * is a constant depending on the liquid and n is an empirical factor, which value is 11/9 for organic liquids. Equation (3.3) produces very good results for organic liquids. If the temperature variation is not very important, and taking into account that the exponent n is close to 1, a good approximation of the GuggenheimKatayama formula is the linear approximation

γ = γ * (1 + α T )

(3.4)

where a is a constant. It is often easier and more practical to use a measured reference value (g0, T0) and consider a linear change of the surface tension with the temperature

(

)

γ = γ 0 1 + β (T - T0 )

(3.5)

The coefficient b can be obtained by remarking that g = 0 when T = Tc: b = –1(Tc/T0). Relations (3.4) and (3.5) are shown in Figure 3.4. The value of the reference surface tension g0 is linked to g * by the relation g0 = g *(Tc/T0)/Tc. Typical values of surface tensions and their temperature coefficients a are given in Table 3.1. The value of the surface tension decreases with temperature. This property is at the origin of a phenomenon called the Marangoni convection or thermocapillary instabilities (Figure 3.5). Suppose that an interface is locally heated (for example, by radiation) and locally cooled (for example, by conduction). The value of the surface tension is smaller in the heated area than in the cooled area. A gradient of surface tension is then induced at the interface between the cooler interface and the warmer interface. This imbalance creates tangential forces on the interface, pushing the fluid from the warm region (smaller value of the surface tension) towards the

Figure 3.4  Representation of the relations (3.4) and (3.5).

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Interfaces, Capillarity, and Microdrops

Figure 3.5  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 the viscous forces.

cooler region (larger value of the surface tension). This surface motion propagates to the bulk under the action of viscosity. If the temperature source is temporary, the motion of the fluid tends to homogenize the temperature and the motion progressively 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. 3.2.2.2  The Effect of Surfactants

Surfactant is the short form 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. In biotechnology, very often surfactants are added to biological samples in order to prevent the formation of aggregates, maintain particles in suspension, and prevent target molecules from adhering to the solid walls of the microsystem (remember that microsystems have extremely large ratios between the wall surfaces and the liquid volumes). Like any small-sized particles, surfactants diffuse in liquids; when they reach an interface, they are captured because their amphiphilic nature prevents them from escaping easily from the interface. Consequently, they gather on the interface as in Figure 3.6, lowering the surface tension of the liquid. As the concentration in surfactants increases, the surface concentration increases too. Above a critical value of the concentration, called CMC (critical micelle concentration), the interface is saturated with surfactants and surfactant molecules in the bulk of the fluid group together to form micelles. The surface tension decreases with the concentration in surfactants as shown in Figure 3.7. At a very low concentration, the slope is nearly linear. When concentration approaches the CMC, the value of the surface tension drops sharply. Above CMC, the value of the surface tension is nearly constant [1]. For example,

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79

Figure 3.6  Schematic view of surfactants in a liquid drop.

pure water has a surface tension of 72 mN/m and water with surfactant (Tween 10 for example) at a concentration above the CMC has a surface tension of only 30 mN/m. In the limit of small surfactant concentration (c << CMC), the surface tension can be expressed as a linear function of the concentration

(

)

γ = γ 0 1 + β (c - c0 )

(3.6)

where β is a constant. Equation (3.6) is similar to (3.5). We have seen how a temperature gradient results in a gradient of surface tension leading to a Marangoni

Figure 3.7  Evolution of the value of the surface tension as a function of the surfactant concentration.

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Interfaces, Capillarity, and Microdrops

Figure 3.8  Schematic of Marangoni convection induced by a gradient of concentration.

type of convection. Similarly, a concentration gradient results in a gradient of surface tension and, consequently, to a Marangoni convection (Figure 3.8). Note that the direction of the motion is always towards the largest value of the surface tension.

3.3  Laplace Law and Applications 3.3.1  Curvature Radius and Laplace’s Law

Laplace’s law—sometimes called the Young-Laplace-Gauss law—is fundamental when dealing with interfaces and microdrops. It relates the pressure inside a droplet to the curvature of the droplet. Structurally, Laplace’s law derives from the Gibbs approach and is valid for a sufficiently small curvature [5]. Let us first describe the notion of curvature. 3.3.1.1  Curvature and Radius of Curvature

For a planar curve the radius of curvature R is the radius of the osculating circle, the circle that is the closest to the curve at the contact point (Figure 3.9). The curvature of the curve is defined by

κ = 1 R

(3.7)

Note that the curvatures as well as the curvature radii are signed quantities. Curvature radius can be positive or negative depending on the orientation (convex or concave) of the curve. Different expressions exist depending on the expression of the curve. In the case of a parametric curve c(t) = (x(t), y(t)), the curvature is given by the relation [6]

3.3  Laplace Law and Applications

81

Figure 3.9  Radius of curvature and osculating circle.

κ=

� �� - yx � �� xy 2

( x� +



3 y� 2) 2

(3.8)

where the dot denotes a differentiation with respect to t. For a plane curve given implicitly as f(x, y) = 0, the curvature is æ Ñf ö κ = Ñ. ç ÷ è Ñ f ø



(3.9)

that is, the divergence of the direction of the gradient of f. And for an explicit function y = f(x), the curvature is defined by d 2y d x2

κ=

3 2ö 2

(3.10)

æ æ dyö ç1 + ç ÷ çè è d x ÷ø ÷ø



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. The mean curvature of the surface is defined using the principal (maximum and minimum) curvatures k1 and k2 (Figure 3.10) in the whole set of curvatures H=



1 (κ 1 + κ 2) 2

(3.11)

Introducing the curvature radii in (3.11) leads to



H=

1 1æ 1 1 ö (κ 1 + κ 2 ) = ç + 2 2 è R1 R2 ÷ø

(3.12)

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Interfaces, Capillarity, and Microdrops

Figure 3.10  Schematic view of the radius of curvature of a surface.

For a sphere of radius R, the two curvatures are equal to 1/R and the mean curvature is H = 1/R. For a cylinder of base radius R, the maximum curvature is R and the minimum curvature 0; hence, H = 1/2R. For a plane, the two curvatures are 0 and H = 0; a plane has no curvature. At a saddle point of a surface (Figure 3.11), one of the curvature radii is positive because it corresponds to a convex arc, whereas the other one is negative, because it corresponds to a concave arc. If |R1| = |R2|, the mean curvature H is zero, and we have what is called a minimal surface: 1æ 1 1 ö 1æ 1 1 ö H= ç + = ç = 0. ÷ 2 è R1 R2 ø 2 è R1 R2 ÷ø

Figure 3.11  Mean curvature at a saddle point is zero if |R1| = |R2|.

3.3  Laplace Law and Applications

83

3.3.1.2  Laplace’s Law

Suppose a spherical droplet of liquid surrounded by a fluid (gas or liquid). The Laplace law links the curvature of the interface to the pressure difference across the interface � D P = Pi - Pe = γ Ñ. n



(3.13)

®

where n is the unit normal to the surface, Pi and Pe are, respectively, the internal and external pressures. This expression derives immediately from (3.9). It can equivalently be written as



1 ö æ 1 D P = Pi - Pe = 2 γ H = γ ç + è R1 R2 ÷ø

(3.14)

For a sphere R1 = R2 = R, and Laplace’s law is simply

D P = Pi - Pe = 2 γ R and, for a cylindrical interface (Figure 3.12), one of the two radii of curvature is infinite, and Laplace’s law reduces to



DP = γ R We do not indicate here the derivation of the Laplace law. The reader can refer to [7, 8]. Let us mention here that the reasoning is based on an energy balance between the pressure and surface energies



DP = γ

dA dV

Figure 3.12  Laplace’s law for a cylindrical interface.

(3.15)

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Interfaces, Capillarity, and Microdrops

When another force, such as gravity, intervenes, the pressure difference at the interface is a balance between the potential and the surface energy



1 ö æ 1 DP = ρgh - γ ç + è R1 R2 ÷ø

(3.16)

Note that the Laplace law is not valid for extremely small curvature radii. Applying the Laplace law for a nanobubble (R ® 0), the resulting pressure will be extremely high inside the bubble. This is not physical, and at this scale, a generalized Laplace law must be used that takes into account an extremely small curvature radius [5]. Next we give some examples of the application of Laplace’s law. 3.3.2  Examples of the Application of Laplace’s Law 3.3.2.1  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 3.13). 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. 3.3.2.2  Self-Motion of a Liquid Plug Between Two Nonparallel Wetting Plates

It has been first observed by Hauksbee [9] that a liquid plug limited by two nonparallel wetting plates moves towards the narrow gap region. Laplace’s law furnishes a very clear explanation of this phenomenon. Suppose that Figure 3.14 is a wedge (2D morphology); let us write Laplace’s law for the left side interface

Figure 3.13  Fluid flow from the smaller bubble/droplet to the larger is a direct application of Laplace’s law.

3.3  Laplace Law and Applications

85

Figure 3.14  Sketch of a liquid plug moving under capillary forces between two plates. The contact angle is θ < 90°.



P0 - P1 =

γ R1

(3.17)

γ R2

(3.18)

with the notations of Figure 3.15. For the right side interface



P0 - P2 =

Subtraction of the two relations leads to



1ö æ 1 P1 - P2 = γ ç - ÷ è R2 R1 ø Next, we show that R2 < R1. Looking at Figure 3.15, we have



R2 sin β = d2 where d is the half-distance between the plates. The angle β is linked to θ and α by the relation



β=

π + α -θ 2

Figure 3.15  Curvature of the interface in a dihedral.

(3.19)

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Interfaces, Capillarity, and Microdrops

Finally, we obtain R2 =



d2 cos (α - θ )

(3.20)

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



d1 cos (α + θ )

(3.21)

Comparing relations (3.20) and (3.21) and noting that d2 < d1 and cos(α - θ) > cos(α + θ), we deduce that R2 is 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. This is due to the fact that the difference of the curvatures in (3.19) is increasing when the plug moves to a narrower region. Bouasse [9] noted 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.

3.4  Partial or Total Wetting So far, we have discussed interfaces between two fluids and shown the importance of the surface tension. In the reality, except for droplets floating in a liquid, interfaces must attach somewhere, in general, to a solid surface or sometimes to a third liquid (Figure 3.16). The intersection of the three domains is called the triple line. Let us consider first the case of an interface contacting a horizontal solid surface. 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 (Figure 3.17). For example, water spreads like a film on a very clean and smooth glass substrate, whereas it forms a

Figure 3.16  Sketch of a liquid/air interface contacting another material (solid or liquid), forming a triple line.

3.5  Contact Angle: Young’s Law

87

Figure 3.17  Wetting is said to be total when the liquid spreads like a film on the solid surface.

droplet on a plastic substrate. In the case of partial wetting, the line where all three phases come together is the triple line. A liquid spreads on a substrate like a film if the total energy of the system is lowered by the presence of the liquid film (Figure 3.18). 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 determines the type of spreading (total or partial) S = γ SG - (γ SL + γLG)



(3.22)

If S > 0, the liquid spreads on the solid surface; if S < 0, the liquid forms a droplet. When a liquid does not totally wet the solid, it forms a droplet on the surface. Two situations can occur. If the contact angle with the solid is less than 90°, the contact is said to be hydrophilic if the liquid is aqueous, or more generally wetting or lyophilic. In the opposite case of a contact angle larger than 90°, the contact is said to be hydrophobic with reference to water or more generally not wetting or lyophobic (Figures 3.19 and 3.20).

3.5  Contact Angle: Young’s Law 3.5.1  Young’s Law

Surface tension is not exactly a force; its unit is N/m. However, it represents a force exerted tangentially to the interface. Surface tension can be looked at as a force per unit length. This can be directly seen from its unit, but it may be interesting to give a more physical feeling by making a very simple experiment (Figure 3.21) [10]. Take a solid frame and a solid tube that can roll on this frame. If we form a liquid film of

Figure 3.18  Comparison of the energies between the dry solid and the wetted solid.

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Interfaces, Capillarity, and Microdrops

Figure 3.19  Water spreads differently on different substrate.

soap between the frame and the tube by plunging one side of the structure in a watersoap solution, the tube starts to move towards the region where there is a liquid film. The surface tension of the liquid film exerts a force on its free boundary. On the other hand, we can increase the film surface by exerting a force on the tube. The work of this force is given by the relation

δ W = F dx = 2 γ L dx

(3.23)

The coefficient 2 stems from the fact that there are two interfaces between the liquid and the air. This relation shows that the surface tension γ is a force per unit length, perpendicular to the tube, in the plane of the liquid and directed towards the liquid. By extension, we can draw the different forces that are exerted by the presence of a fluid on the triple line (Figure 3.22). At equilibrium, the resultant of the forces must be zero. We use a coordinate system where the x-axis is the tangent to the solid surface at the contact line (horizontal) and the y-axis is the direction perpendicular (vertical). At equilibrium, the projection of the resultant on the x-axis is zero and we obtain the relation

γ LG cosθ = γ SG - γ SL

Figure 3.20  Silicone oil has an opposite wetting behavior than water.

(3.24)

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89

Figure 3.21  A tube placed on a rigid frame whose the left part is occupied by a soap film requires a force to be displaced towards the right; this force opposed the surface tension that tends to bring the tube to the left.

where γLG, γSG, γSL are, respectively, the liquid-gas, solid-gas, and solid liquid surface tensions. This relation is called Young’s law and is very useful to understanding the behavior of a drop. It especially shows that the contact angle depends on the surface tensions of the three constituents. For a microdrop on a solid, the contact angle is given by the relation



æ γ - γ SL ö θ = arccos ç SG è γ LG ÷ø

(3.25)

In experimental situations when we deal with real biological liquids, one observes unexpected changes in the contact angle with time. This is just because biological liquids are inhomogeneous and can deposit a layer of chemical molecules on the solid wall, thus progressively changing the value of the tension γSL, and consequently the value of θ, as stated by Young’s law. Young’s law can be more rigorously derived from free energy minimization. The change of free energy due to a change in droplet size can be written as [11]

dE = γ SL d ASL + γ SG d ASG + γ LG d ALG = (γ SL - γ SG + γ LG cos θ )d ASL

(3.26)

where the As are the surface areas and θ is the contact angle. At mechanical equilibrium dE = 0 and

γ SL - γ SG + γ LG cos θ = 0 Equation (3.27) is the same as (3.24).

Figure 3.22  Schematic of the forces at the triple contact line.

(3.27)

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Interfaces, Capillarity, and Microdrops

3.5.2  Young’s Law for Two Liquids and a Solid

Suppose that we know the contact angles of a liquid 1 and a liquid 2 on a substrate S in air. What is then the contact angle if liquid 2 is immersed into liquid 1 (Figure 3.23)? Young’s law for the first liquid is

γ L1, G cos (θ L1,G,S ) = γ S,G - γ S,L1

(3.28)

and for the second liquid is

γ L2,G cos (θ L2,G,S ) = γ S,G - γ S,L2

(3.29)

The difference of (3.28) and (3.29) yields [12]

γ L1,G cos (θL1,G,S ) - γ L2,G cos (θL2,G,S ) = γ S,L2 - γ S,L1

(3.30)

If liquid 2 is immersed into liquid 1, Young’s law yields

γ L1,L2 cos (θ L1,L2,S ) = γ S,L1 - γ S,L2

(3.31)

From (3.31) and (3.30), we deduce γ L2,G cos (θ L2,G,S ) - γ L1,G cos (θL1,G,S )

γ L1,L2

= cos (θL1,L2,S )



(3.32)

Surface tensions can easily be measured (by the pendant drop method, for example), and if the two contact angles in air θL1,S,G and θL2,S,G are known, the contact angle θL1,L2,S is given by



é γ L2,G cos (θ L2,G,S ) - γ L1,G cos (θL1,G,S ) ù θL1,L2,S = arccos ê ú γ L1,L2 ëê ûú

(3.33)

Figure 3.23  (a) Contact of droplets of liquid 1 and liquid 2 surrounded by air or gas. (b) Contact of a droplet of liquid 2 immersed in liquid 1.

3.5  Contact Angle: Young’s Law

91

This remark is important to predict the contact angle when using two liquids in a microfluidic system. It can be seen that very often films can form (i.e., one of the liquid totally wets the wall), especially when surfactants are used. A graphical construction explains this phenomenon (Figure 3.24). Relation (3.32) is scalar, but can be considered as the projection a vector equation. If we note the radii R0 = 1, R1 = γL1G/γL1L2, and R2 = γL2G/γL1L2, and draw the circles R0, R1, and R2, relation (3.32) is the projection on the x-axis of the vector OM¢ = OM1 - OM2 = OM1 + M1M¢. It the projection of M¢ falls inside the interval [-1, 1] there is a partial wetting (i.e., the two liquids have a Young contact angle at the wall); in the opposite case, there is a film on the wall. Because often γL1G/γL1L2 >> 1 and γL2G/γL1L2 >> 1—this is the case when surfactants are added to one of the fluids—the projection easily falls outside the [-1, 1] interval, and a film forms indicating the total wetting of one of the liquids. 3.5.3  Generalization of Young’s Law—Neumann’s Construction

Let us come back to the derivation of Young’s law. Young’s law has been obtained by a projection on the x-axis of the surface tension forces, but the force balance applies also to a y-axis projection. On a solid, fixed surface, the resulting constraints on the solid substrate cannot be seen. However, there are two cases where the yprojection of Young’s law is of importance: the cantilever and the contact between three liquids.

Figure 3.24  Geometrical construction of the resultant contact angle.

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Interfaces, Capillarity, and Microdrops

Figure 3.25  Cantilever deformed by the presence of a water droplet.

3.5.3.1  Droplet on a Cantilever

In the case of a microcantilever, the presence of a droplet induces capillary forces along the triple line (Figure 3.25). The deformation results from the resultant of the capillary forces perpendicular to the cantilever. At rest, this resultant bends the cantilever, as shown in Figure 3.25. The calculation is lengthy and has been given by Yu and Zhao [13]. 3.5.3.2  Contact Between Three Liquids—Neumann’s Construction

Take two immiscible liquids, denoted 1 and 2, with the droplet of liquid 2 deposited on the interface between liquid 1 and a gas. Even if the density of liquid 2 is somewhat larger than that of liquid 1, the droplet may “float” on the surface, as shown in Figure 3.26. The situation is comparable to that of Young’s law with the difference that the situation is now two-dimensional. It is called Neumann’s construction, and the following equality holds

� � � γ L1L2 + γ L1G + γ L2G = 0

(3.34)

Note that the density of the two liquids condition the vertical position of the center of mass of the droplet, but at the triple line, it is the y-projection of (3.34) that governs the morphology of the contact [14]. In Figure 3.27 we show some pictures of floating droplets obtained by numerical simulation (Surface Evolver software [15]).

Figure 3.26  Droplet on a liquid surface.

3.6  Capillary Force and Force on a Triple Line

93

Figure 3.27  Numerical simulations of different positions of a droplet (1 mm) on a liquid surface depending on the three surface tensions. (a) The surface tension of the droplet is very large. (b) The surface tension of the droplet with the other liquid has been reduced; the drop is at equilibrium due to the balance of buoyancy and surface tensions.

3.6  Capillary Force and Force on a Triple Line 3.6.1  Introduction

Capillary forces are extremely important at a microscale. We have all seen insects “walking” on the surface of a water pond (Figure 3.28). Their hydrophobic legs do not penetrate the water surface and their weight is balanced by the surface tension force. More than that, it is observed that some insects can walk up a meniscus (i.e., can walk on a locally inclined water surface). The explanation of this phenomenon was recently given by Hu et al. [16] and refers to a complex interface deformation under capillary forces. In the domain of microfluidics, capillary forces are predominant; some examples of the action of capillary forces are given in the following sections. 3.6.2  Capillary Force Between Two Parallel Plates

A liquid film placed between two parallel plates makes the plates very adhesive. For instance, when using a microscope to observe objects in a small volume of liquid deposited on a plate and maintained by a secondary glass plate, it is very difficult to

Figure 3.28  (a) Capillary forces make the water surface resist the weight of an insect. (b) An insect walking up a meniscus. (From: [16]. Courtesy of David Hu.)

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Interfaces, Capillarity, and Microdrops

Figure 3.29  Film of water between two glass plates.

separate the plates. This situation is schematized in Figure 3.29. In the first place, it is observed that the meniscus has a round shape (in order to minimize the free energy). Let us write Laplace’s law at the free interface. The first (horizontal) radius of curvature is approximately R. The second (vertical) radius of curvature, shown in Figure 3.30, is given by R2 =



h 2 cos θ

(3.35)

Laplace’s law states that



æ 1 2 cos θ ö DP = γ ç ÷ èR h ø

(3.36)

In (3.36) the minus sign derives from the concavity of the interface. Because the vertical gap h is much less than the horizontal dimension R, we have the approximation



DP » -

2 γ cos θ h

Figure 3.30  Sketch for the calculation of the vertical curvature.

3.6  Capillary Force and Force on a Triple Line

95

And the capillary force that links the plates together is



F»

2 γ cos θ π R2 h

(3.37)

This capillary force can be quite important if the contact angle θ is small; for h = 10 mm and R = 1 cm, the force F is of the order of 2.5N. However, if θ = π/2, there is no cohesion between the two plates. Conversely, if θ > π/2 the liquid droplet pushes apart the two plates, and the droplet can be used as a load carrier [17]. 3.6.3  Capillary Rise in a Tube—Jurin’s Law

When a capillary tube is plunged into a volume of wetting liquid, the liquid rises inside the tube under the effect of capillary forces (Figure 3.31). It is observed that the height reached by the liquid is inversely proportional to the radius of the tube. This property is usually referred to as Jurin’s law. Using the principle of minimum energy, one can conclude that the liquid goes up in the tube if the surface energy of the dry wall is larger than that of the wetted wall. If we define the impregnation criterion I by

I = γ SG - γ SL

(3.38)

The liquid rises in the tube if I > 0. Upon substitution of the Young law in (3.38), the impregnation criterion can be written under the form

I = γ cos θ

Figure 3.31  Capillary rise is inversely proportional to the capillary diameter.

(3.39)

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Interfaces, Capillarity, and Microdrops

When the liquid rises in the tube, the system gains potential energy—because of the elevation of a volume of liquid—and loses capillary energy—due to the reduction of the surface energy. The balance is [10]

1 1 1 E = ρ g hVliquid - Scontact I = ρ g h (π R2 h) - 2 π RhI = ρ g π R2 h2 - 2 π Rhγ cos θ 2 2 2 (3.40) 1 1 ρ g h (π R2 h) - 2 π RhI = ρ g π R2 h2 - 2 π Rhγ cos θ ct I = 2 2 Note that the detailed shape of the meniscus has not been taken into account in (3.40). The interface stabilizes when ∂E =0 ∂h

which results in

h=

2 γ cos θ ρg R

(3.41)

Equation (3.41) is the Jurin law. The capillary rise is inversely proportional to the tube radius. It can be also applied to the case where the liquid level in the tube decreases below the outer liquid surface; this situation happens when θ > 90°. The maximum possible height that a liquid can reach corresponds to θ = 0: h = 2γ /ρgR. In microfluidics, capillary tubes of a 100-mm diameter are currently used; if the liquid is water (γ = 72 mN/m), and using the approximate value cosθ ~ ½, the capillary rise is of the order of 14 cm, which is quite important at the scale of a microcomponent. What is the capillary force associated to the capillary rise? The capillary force balances the weight of the liquid in the tube. This weight is given by

F = ρ g π R2 h

Figure 3.32  Sketch of the capillary force of a liquid inside a tube.

3.6  Capillary Force and Force on a Triple Line

97

Replacing h by its value from (3.41), we find the capillary force F = 2 π R γ cos θ



(3.42)

The capillary force is the product of the length of the contact line 2πR times the line force f = γ cosθ. This line force is sketched in Figure 3.32. 3.6.4  Capillary Rise Between Two Parallel Vertical Plates

The same reasoning can be done for a meniscus between two parallel plates (Figure 3.33) separated by a distance d = 2R. The same reasoning as that of the preceding section leads to h=



γ cos θ ρg R

(3.43)

By substituting in (3.43) the capillary length defined by



κ -1 =

γ ρg

(3.44)

h = κ -2

cos θ R

(3.45)

one obtains

Note that the expressions for the two geometries (cylinder and two parallel plates) are similar. If we use the coefficient c, with c = 2 for a cylinder and c = 1 for parallel plates [18], we have

h = cκ -2

cos θ R

(3.46)

where R is either the radius of the cylinder or the half-distance between the plates.

Figure 3.33  Capillary rise between two parallel vertical plates.

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3.6.5  Capillary Pumping

If the microchannel is placed horizontally instead of vertically, the weight of the liquid cannot balance the capillary force and a continuous flow is set up that lasts as long as there is liquid available in the entry port (Figure 3.34). Let us assume that the reservoir is large so that the curvature of the nearly flat horizontal interface can be neglected. The pressure at x = 0 is then P0, the atmospheric pressure. Let us assume also that the channel is rectangular (width w, depth d, with d < w). Following Bruus [19], we assume that the continuous flow in the horizontal channel is from the Hagen-Poiseuille type and the flow rate can be symbolically written (Chapter 2)

D P = RV V =

ηL V f (w, d)

(3.47)

where f is a function depending on the aspect ratio. The velocity of the flow V is given by V = dL dt

and the driving pressure DP is

D P = 2 γ cos θ d



After substitution, we find a differential equation for L



L dL =

2 γ cos θ f (w, d ) dt ηd

(3.48)

which can easily be integrated, yielding



L =2

γ cos θ f (w, d ) t ηd

(3.49)

γ cos θ f (w, d ) 1 ηd t

(3.50)

The flow velocity V is then



V =

Figure 3.34  Principle of capillary pumping.

3.6  Capillary Force and Force on a Triple Line

99

Figure 3.35  Schematic of the capillary force on a triple line.

The velocity of the flow decreases like t . This decrease results from a balance between the constant Laplace driving pressure and the increasing flow resistance in the channel. 3.6.6  Force on a Triple Line

The analysis of capillary rise in tubes and capillary pumping has shown the expression of the capillary force on the triple contact line [20]. This expression can be generalized to any triple contact line [21]. For a triple contact line W — as sketched in Figure 3.35—the capillary force is



� � � F = ò f dl = ò γ cos θ n dl W

W

(3.51)

Suppose that we want to find the value of the resultant of the capillary forces in a particular direction, say, the x-direction (Figure 3.36). The projection along the x-direction of (3.51) is

Figure 3.36  Capillary force on a triple contact line in the x-direction.

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�� �� Fx = ò f .i dl = ò γ cos θ n.i dl W



W

(3.52)

Equation (3.52) can be simplified and cast under the form Fx =

ò

W

�� �� γ cos θ n.i dl = γ cos θ ò n.i dl = γ cos θ W

ò

W

e

cos α dl = γ cos θ ò dl¢ 0

Finally we obtain the expression

Fx = eγ cos θ

(3.53)

Equation (3.53) shows that the resulting force on a triple contact line in any direction does not depend on the shape of the interface [21]; it depends only on the distance between the two ends of the triple line normal to the selected direction. 3.6.7  Examples of Capillary Forces in Microsystems

It is very common in biotechnology to use plates comprising thousands of microholes or cusps. The position of the free surface of the liquid in the cusps is of utmost importance. In particular, the liquid must not exit the holes under the action of capillary forces. As an example, Figure 3.37 shows a free liquid interface in a square hole, calculated with the Surface Evolver numerical software [15].

Figure 3.37  The surface of a liquid in a microwell is not flat due to capillary forces. The figure is a simulation with the Surface Evolver software. (a) Case of water in a hydrophilic well (contact angles of 140°). (b) Case of water in a hydrophilic well (contact angles 60°). The “free” surface is tilted downwards or upwards depending on the contact angle. The walls have been dematerialized for clarity.

3.7  Pining and Canthotaxis

101

3.7  Pinning and Canthotaxis 3.7.1  Theory

Solid surfaces are not always smooth or chemically homogeneous. They can have edges and chemical heterogeneities. These surface discontinuities (geometrical or chemical) modify the behavior of an interface. The shape of an interface is modified locally by a point or line inhomogeneity. Let us suppose that an interface is coming to contact a straight edge (Figure 3.38), and that the Young contact angle θ is the same on both sides of the edge. If the liquid is slowly pushed over the edge, the contact line on the angle stays fixed or pinned as long as the contact angle is not forced over the limit α + θ, where α is the angle between the two planes. The condition for pinning is then

θ £ φ £ α + θ

(3.54)

where θ is the contact angle. In the case where the two planes have a different chemical surface, the Young contact angles can be denoted θ1 and θ2, and the condition (3.54) becomes

θ1 £ φ £ α + θ 2

(3.55)

The pinning condition between the two angles θ and α + θ is called canthotaxis. 3.7.2  Pinning of an Interface Between Pillars

Microsystems for biotechnology often make use of pillars to perform microfluidic functions. Let us consider the example in which the role of the pillars is to block and maintain fixed an interface between two immiscible fluids [22, 23] (Figure 3.39). This is the case of capillary valves, liquid-liquid extraction devices, and so forth.

Figure 3.38  Droplet pinning on an edge: the droplet is pinned as long as the contact angle varies between the natural Young contact angle θ to the value θ + α. Above this value, the interface moves over the right plane and the interface is suddenly released.

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Figure 3.39  Sketch of an interface pinned by pillars. (a) Microfabricated pillars; (b) interface between two slowly flowing liquids stabilized by pillars; and (c) detail of an interface between two pillars. Depending on the pressure difference P1-P2 the interface bulges more or less. If the pressure difference is too large, the interface breaks down.

Let us examine the case of triangular (or diamond shaped) pillars with hydrophobic surface. According to the Laplace theorem, the curvature radius of the interface is related to the pressure difference across the interface. When the pressure on one side of the interface is increased, the curvature increases until the pinning limit is reached. Then the interface is disrupted and the high pressure liquid penetrates into the low-pressure channel (Figure 3.40). Take an interface pinned between the two facing edges of two similar micropillars, and suppose that the pressure P1 in one liquid is progressively increased. Two conditions govern the pinning: the first condition is related to capillarity and to the phenomenon of canthotaxis [8], for example the pinning is effective if the condition

θ £ θ C

(3.56)

is met, and the interface does not slide on the pillar walls. In (3.56) θC is the (static) contact angle. Above this value, the interface slides irreversibly along the two facing walls of the pillars (Figure 3.41). The second condition is geometrical and corresponds to the minimum possible curvature of the interface. This curvature is obtained when the interface has the shape of a half-circle with a radius δ/2. In such a case, θ = α + π/2. The second condition is then

θ £ α + π 2

(3.57)

Figure 3.40  When the water pressure is increased, the interface is disrupted and water invades the solvent channel.

3.7  Pining and Canthotaxis

103

Figure 3.41  Under the effect of the pressure P1, the interface bulges out. The interface stays pinned until θ > θ C. Case θ £ α + π/2: position 1 corresponds to P1 = P2; positions 3 is the canthotaxis limit, and 4 is the sliding of the interface.

Above this value, the interface cannot withstand the pressure difference and irreversibly slides along the walls. The general condition for pinning when the pressure P1 is larger than P2 is π ö æ θ £ θ lim = min ç α + , θC ÷ è ø 2



(3.58)

Relation (3.58) states that when the curvature increases under the action of the pressure P1, the contact angle increases, and when it reaches θlim, the interface starts sliding, like in Figure 3.40. When the interface stays anchored to the edges, a geometrical analysis shows immediately that the curvature radius is given by R (θ ) =

δ δ = ( 2 sin θ - α) æ ö π æ ö 2 cos ç α - ç θ - ÷ ÷ è 2øø è

(3.59)

The maximum pressure that the interface can withstand, corresponding to the minimum curvature radius of the interface, is D Pmax =



γ Rmin

(3.60)

where γ  is the surface tension between the two phases. Using the Heaviside function H, the maximum pressure difference across the interface is



D Pmax =

ù γ 2γ é πö æ = sin (θC - α) + H çθ c - α - ÷ (1 - sin(θC - α ))ú ê è Rmin δ ë 2ø û

(3.61)

3.7.3  Droplet Pinning on a Surface Defect

Local chemical and/or geometrical defects locally modify the contact angle. If the defect is sufficiently important, or if there are a sufficient numbers of defects, the

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Figure 3.42  Pinning of a drop moving from a hydrophobic area towards a hydrophilic surface due to a defect of the surface.

droplet cannot move even if capillary forces are applied on it. We show in Figure 3.42 a Surface Evolver numerical simulation of pinning of a droplet during its motion from a hydrophobic to a hydrophilic substrate. 3.7.4  Pinning of a Microdroplet—Quadruple Contact Line

Pinning may also occur at a transition line between two surfaces with different chemical coatings, inducing a sharp transition of wettability [8]. This effect is due to the fact that, when the contact line reaches the separation line, we have a four-phase contact line. The canthotaxis condition states that there is equilibrium as long as the contact angle is comprised between the Young angles on both sides θ1 and θ2 as shown in Figure 3.43 [8, 9]

θ1 £ θ £ θ 2

(3.62)

Equation (3.62) can be written in terms of surface energy



æγ æγ - γ S1L ö - γ S 2L ö arccos ç S1G £ θ £ arccos ç S 2G ÷ γ γ è ø è ø÷

(3.63)

Figure 3.43  Quadruple contact line pinning on a wettability boundary. (a) Vertical cross section of the droplet with the limiting contact angles θ1 and θ2. (b) An Evolver simulation of the droplet.

3.8  Microdrops

105

where the indices S1 and S2 denote the left and right solid surfaces. If the external constraint is such that θ continues to increase, the triple line is suddenly depinned and the liquid is released and invades the lyophobic surface. 3.7.5  Pinning in Microwells

We have already presented the morphology of liquid in a microgroove. It is important that the liquid does not spread out of the well. The maximum liquid volume that a well can contain is that corresponding to an interface pinned to the rim (Figure 3.44). The canthotaxis limit states that if θ is the Young contact angle on the upper surface, the interface can bulge up to this limit. On the other hand, if the liquid volume decreases—by evaporation, for example—the liquid withdraws progressively to the inner corners before completely receding.

3.8  Microdrops 3.8.1  Shape of Microdrops

In this section, the shape of microdrops in different situations typical of microsystems is investigated, assuming that these microdrops are in an equilibrium state (i.e., at rest) or moving at a sufficiently low velocity that the inertial forces can be neglected. Different situations will be examined: sessile droplets deposited on a plate, droplets constrained between two horizontal planes, pendant droplets, droplets on lyophilic strips, in corners and dihedrals, and in wells and cusps. 3.8.1.1  Sessile Droplets

It is easily observed that large droplets on horizontal surfaces have a flattened shape, whereas small droplets have a spherical shape (Figure 3.45).

Figure 3.44  Pinning of liquid in a groove: the interface stays attached to the rim as long the volume of liquid is such that it is comprised between the two limits 1 and 2 defined by the continuous black lines.

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Interfaces, Capillarity, and Microdrops

Figure 3.45  Comparison of the shape between microdrops and macrodrops (not to scale). Microdrops have the shape of spherical caps, whereas larger drops are flattened by the action of gravity and their height is related to the capillary length.

This observation is linked to the balance between gravity and surface tension. A microscopic drop is governed solely by surface tension, whereas the shape of a larger droplet results from a balance between the two forces. The scale length of this transition , is the capillary length (see Chapter 1). We recall that this length is defined by the ratio of the Laplace pressure to the hydrostatic pressure. If we compare the two pressures for a drop, we obtain



γ D PLaplace » � D Phydrostatic ρ g �

(3.64)

where γ is the surface tension, ρ is the density, and g is the gravitational constant. The two pressures are of the same order when



�»

γ ρg

(3.65)

, is called the capillary length. A drop of dimension smaller than the capillary length has a shape resembling that of a spherical cap. A drop larger than the capillary length is flattened by gravity. Note that a dimensionless number—the Bond number—can be derived from (3.64) yielding a similar meaning. The Bond number is expressed by



Bo =

ρ g R2 γ

(3.66)

where R is of the order of the drop radius. If Bo < 1, the drop is spherical, or else the gravitational force flattens the drop on the solid surface. A numerical simulation of the two shapes of droplets obtained with the numerical software Surface Evolver is shown in Figure 3.46. The capillary length is of the order of 2 mm for most liquids,

Figure 3.46  Numerical simulations of a microdrop (Bo <<1) and a larger drop (Bo >>1) obtained with Surface Evolver software (not to scale).

3.8  Microdrops

107

even for mercury. In the following sections we analyze successively the characteristics of drops having, respectively, large and small Bond numbers. Case 1: Large Droplet, Bo >> 1

According top the observation of the preceding section, a large droplet has a flat upper surface and its shape is shown in Figure 3.47. Let us calculate the height of such a droplet as a function of contact angle and surface tension. Take the control volume shown in Figure 3.47 and write the balance of the forces that act on this volume. The surface tension contribution is S = γ SG - (γ SL + γ LG)



(3.67)

and the hydrostatic pressure contribution is



e

P = ò ρ g (e - z )dz = *

0

1 ρ g e2 2

(3.68)

The equilibrium condition yields P* + S = 0, which results in the relation



1 ρ g e 2 + γ SG - ( γ SL + γ LG ) = 0 2

(3.69)

Recall that Young’s law imposes a relation between the surface tensions

γ SG - γ SL = γ LG cos θ

(3.70)

Upon substitution of (3.70) in (3.69), we obtain



γ LG (1 - cos θ ) =

1 ρ g e2 2

Using the trigonometric expression 1 – cosq = 2sin2(q/2), we finally find



e=2

γ LG θ θ sin = 2 � sin ρg 2 2

(3.71)

Relation (3.71) shows that the height of a large droplet is proportional to the capillary length. With the capillary length being of the order of 2 mm, the height of large droplets is less than 4 mm.

Figure 3.47  Equilibrium of the forces (per unit length) on a control volume of the drop.

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Case 2: Microscopic Drops, Bo << 1

As mentioned earlier, a microdrop has the form of a spherical cap. A spherical cap is a surface of minimum energy if only surface tension is taken into account. This can be checked by using the Surface Evolver software. The result is shown in the Figure 3.48 for a contact angle of 110°. Figure 3.49 shows a cross section of the droplet on nonwetting and wetting surfaces. The volume V of such a droplet is a function of two parameters in the set of the four parameters {θ, a, R, h}, where θ is the contact angle, a is the contact radius (i.e., the radius of the circular base), R is the curvature radius (i.e., the sphere radius), and h is the height of the droplet. We shall not demonstrate again the derivation of the liquid volume; it has already been done in [8]. The most used formula is V (R, θ) =



π 3 R ( 2 - 3cos θ + cos3 θ ) 3

(3.72)

The other ones are π h (3 a2 + h2 ) 6

V (a, R) =

πæ 2 2 æ 2 æ 2 2 èR ± R - a ç 3 a + èR ± R - a 6 è

æ è

V (a, h) =

2ö

÷ ø (3.73)

æ è

π (2 - 3cos θ + cos3 θ ) V (a, θ) = a3 3 sin3 θ V (h, θ) =

π 3 (2 - 3cos θ + cos3 θ ) h 3 3 (1 - cos θ )



Note that in (3.73) the plus sign corresponds to a nonwetting case (lyophobic) and the minus sign corresponds to the wetting case (lyophilic). The surface area of the spherical cap is also of importance since the surface energy is proportional to this surface.

Esurf = γ S

Figure 3.48  Shape of a microdrop calculated with the software Surface Evolver (θ = 110°).

(3.74)

3.8  Microdrops

109

Figure 3.49  Cross section of a microdrop sufficiently small to be a spherical cap. (a) Nonwetting droplet. (b) Wetting droplet. Notice that α = θ − π/2 in the first (nonwetting) case and α = π/2 − θ in the second (wetting) case.

Again, we shall not give the derivation of the surface area [8]. The usual expressions are S(θ , h) =

2 π h2 1 - cos θ

S(a, h) = π (a2 + h2 )

S(a, θ) =

(3.75)

2 π a2 1 + cos θ

3.8.1.2  Droplets Constrained Between Two Plates

It happens very often in biotechnology that droplets are constrained between two horizontal solid surfaces. Such droplets have a relatively smaller free energy than sessile droplets and are easier to handle. This is particularly the case for electrowetting. We consider here only the case of microsystems where the vertical gap δ is small (usually 50 to 500 mm), their Bond number, given by Bo = ρg2/γ, is less than 0.1, and the free interfaces have circular cross sections.

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Interfaces, Capillarity, and Microdrops

Shape of a Droplet Constrained Between Two Horizontal Planes

A droplet squashed between two parallel plates has two types of contact: one with the bottom plate and another with the top plate. These contacts can be independently either wetting or nonwetting depending on the material of the substrates. Figure 3.50 shows different possibilities and the corresponding shapes calculated using the Surface Evolver numerical software. Let us derive the expression of the curvature radius in the vertical plane. This curvature radius is shown in Figure 3.51 for the morphology of Figure 3.50(d). Remarking that angles with perpendicular sides are equal, we can write Rcosθ1 = H − δ, where H is the distance between the curvature center and the upper plate and δ is the gap between the plates; similarly Rcos(π- θ2) = Rcosθ2 = H. We deduce the value of the curvature radius R

R=-

δ cos θ1 + cosθ 2

(3.76)

In the case of a concave interface corresponding to θ1 < π/2, θ2 > π/2, and θ1 + θ2 < π [Figure 3.51(b)], similar reasoning leads to the negative curvature radius

R=

δ cos θ1 + cosθ 2

In the particular case where θ1 + θ2 = π [Figure 3.51(c)], the vertical profile of the interface is flat (the interface has a conical shape) and the curvature radius is infinite.

Figure 3.50  Sketch of the shape of a drop between two horizontal plates. Four different cases are observed: (a) hydrophobic contact with both plates; (b) hydrophilic contact on both plates; (c) hydrophilic contact on bottom plate, hydrophobic contact on top plate and a concave interface (θ1 < π/2, θ2 > π/2, and θ1 + θ2 < π); and (d) the same situation as (c), but the interface is convex (θ1 < π/2, θ2 > π/2, and θ1 + θ2 > π).

3.8  Microdrops

111

Figure 3.51  Schematic of the geometry of a droplet constrained between two parallel planes: (a) case of a convex interface θ1 < π/2, θ2 > π/2, and θ1 + θ2 > π, (b) case of a concave interface θ1 < π/2, θ2 > π/2, and θ1 + θ2 < π, and (c) case of a flat interface θ1 + θ2 = π.

The volume of such a droplet is often useful to know. The calculation is complicated except in the case where the two contact angles are equal (θ1 = θ2 = θ). In this case, the exact formula has been derived in [8]



3 ìï é δ 1æ δö π sin (2 θ - π )ù üï V = 2 π í(R2 - 2 r R + 2 r 2 ) - ç ÷ + (R - r) r 2 êθ - + úý 2 3 è 2ø 2 2 ë û ïþ ïî

(3.77)

In the literature, the Nie et al. correlation is sometimes used [24]

V=

π é 3 (2 R) - (2 R - δ )2 ( 4 R + δ )ûù ë 12

(3.78)

However, this formula does not take into account the contact angle. Discrepancy up to 8% can result, depending on the Young contact angle. The difference between the exact and approximate expressions is shown in Figure 3.52.

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Figure 3.52  The different approaches to the calculation of the volume of a droplet between two parallel horizontal plates.

3.8.1.3  Droplet in a Corner: The Concus-Finn Relation

Microfluidic channels and chambers are etched in silicon, glass, or plastic. Let us investigate first the effect of a corner—or a wedge—on the droplet interface. Take the case of a 90° wedge. The shape of the droplet is shown in Figure 3.53 depending on the Bond number Bo = ρgR2/γ where R is a characteristic dimension of the droplet, which can be scaled as the 1/3 power of the value of the volume of liquid R3 = 3Vol/π. Let us consider now microdrops in which the Bond number is small. It has been observed that liquid interfaces in contact with highly wetting solid walls forming a

Figure 3.53  The shape of a liquid drop in a 90° wedge. (a) A small volume droplet of 0.125 ml tends to take the form of a sphere despite the different contact angles on the two planes, with a Bond number of the order of 0.04. (b) A larger droplet of 1.25 ml—Bond number 4—is flattened by gravity (Surface Evolver calculation).

3.8  Microdrops

113

wedge tend to spread in the corner (Figure 3.54). This motion results from the fact that the interface curvature is strongly reduced in the corner. In the case of Figure 3.54, the vertical curvature radius is small; the Laplace pressure is low in the corner and liquid tends to spread in the corner. Concus and Finn [25] have investigated this phenomenon and they have derived a criterion for capillary motion in the corner of the wedge. If θ is the Young contact angle on both planes and α is the wedge halfangle, the condition for capillary self-motion is



θ<

π - α 2

(3.79)

This case corresponds to wetting walls. Conversely, when the walls are nonwetting, the condition for de-wetting of the corner is



θ>

π + α 2

(3.80)

In Figure 3.55, the Concus-Finn relations have been plotted in a (θ, α) coordinates system. One verifies that, for a flat angle, the Concus-Finn relations reduce to the usual capillary analysis. The Concus-Finn relations can be numerically verified using the Surface Evolver software. Figure 3.56 shows the spreading of the liquid in the corner when condition (3.79) is met. In microtechnology, wedges and corners most of the time form a 90° angle, so that a droplet disappears in the form of filaments if the wetting angle on both planes is smaller than 45°. One must be wary that, when coating the interior of microsystems with a strongly wetting layer, in order to have very hydrophilic (wetting) surface, droplets may disappear; they are transformed into filaments in the corners. The converse can also be verified. For a rectangular channel, if the coating is strongly hydrophobic, and the contact angles on both planes are larger than

Figure 3.54  A liquid interface is deformed in the corner of a wedge made of two wetting plates. This phenomenon is due to a decrease of curvature at the edge.

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Figure 3.55  Plot of the domains of self-motion in a corner, according to the Concus-Finn relations.

135°, the drop detaches from the corner and does not wet the corner anymore (Figure 3.57). A generalization of the Concus-Finn relation has been derived by Brakke and Berthier in [8]. When the two planes do not have the same wettability (contact angles θ1 and θ2), the relation (3.79) becomes

θ1 + θ 2 π < -α 2 2

(3.81)

Remember that α is the wedge half-angle. An important consequence of relation (3.81) applies to trapezoidal microchannels, a form easily obtained by microfabrication (Figure 3.58).

Figure 3.56  A droplet spreads in a corner when the contact angles verify the Concus-Finn condition (Surface Evolver calculation).

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115

Figure 3.57  Droplet in a strongly hydrophobic corner: the droplet does not wet the tip of the corner, in accordance to the Concus-Finn relation.

When a glass cover is sealed on top, the upper corners form 45° angles and the extended Concus-Finn condition indicates the following limit



θ1 + θ 2 π < - α = 67.5° 2 2 The glass cover may be quite hydrophilic, say, θ1 ~ 60°, and if the channel is also hydrophilic, say, θ2 ~ 70°, then (θ1 + θ2)/2 ~ 65° and the liquid spreads in the upper corners, leading to unwanted leakage. The same extension applies to the nonwetting case. The extended Brakke-Concus-Finn de-wetting condition for a corner (3.80) is



θ1 + θ 2 π > +α 2 2

(3.82)

In a general way, the surface tension does not appear in the Brakke-ConcusFinn relations. Thus, these relations also apply for two-phase liquids. If we consider water and oil, the contact angles with hydrophilic and hydrophobic surfaces, respectively, are of the order of θ1 ~ 60°, respectively θ2 ~ 130°. Concus-Finn relations show that a droplet of oil surrounded by water is likely to form in hydrophilic channels, because the water spreads on the solid wall; conversely, oil is the continuous phase in a hydrophobic channel, while water is dispersed into droplets.

Figure 3.58  Cross section of a trapezoidal microchannel.

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3.8.1.4  Droplet in a Wetting/Nonwetting Corner

Let us examine the case of a corner dihedral with one surface lyophilic (wetting) and the other one lyophobic (nonwetting). It is expected that, if the contact angle on the wetting side is small and that on the nonwetting side is large, the drop will be positioned on the wetting side. By referring to [26], a criterion for the drop to be positioned on the wetting side only is θ2 - θ 1 > π - 2 α



(3.83)

with α being the wedge half-angle. Relation (3.83) can be verified by numerical simulation as shown in Figure 3.59. 3.8.1.5  Droplet in a Groove

The behavior of liquid or liquid droplets in grooves has become a subject of research with the development of open microfluidics. Grooves and cusps present the advantage of being easily accessible and easily washable and they confine the liquid in small volumes, due to pinning of the upper edges. Seemann et al. [27] and Lipowsky et al. [28] have observed that two parameters govern the morphology of the liquid in a groove: (1) the aspect ratio X of the groove geometry (i.e., the ratio of the groove depth to the groove width); and (2) the contact angle θ of the liquid with the solid substrate. Basically, there are three morphologies for a liquid in a groove: filaments, wedges, and droplets. Filaments correspond to the case where the liquid spreads in the groove, either in corners if the volume of liquid is small, or in the whole groove if the volume of liquid is sufficient (Figure 3.60); filaments are obtained for contact angles smaller than 45°, according to the Concus-Finn relation θ £ π/2 - α, where α is the corner half-angle. The liquid remains in the form of droplets or stretched droplets if the contact angle is larger than 45°. If the volume of liquid is small compared to the dimensions of the groove, the liquid goes to the corners of the groove, forming wedges. For more details on these morphologies, a complete diagram has been found by Seemann et al. [27]. 3.8.1.6  Droplet on a Lyophilic Strip

In this section we present the different morphologies of a droplet sitting on a wetting (lyophilic) band on the surface of an otherwise nonwetting (lyophobic) horizontal

Figure 3.59  Droplet in a corner with wetting and nonwetting sides. (a) The droplet stays attached to the corner. (b) The droplet is at equilibrium on the wetting side (Surface Evolver calculation).

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117

Figure 3.60  The different morphologies of liquid in grooves: (a) filament, (b) droplet, and (c) wedges.

plate. In Figure 3.61 we have schematized the forces on the triple line and we observe that their resultant on the wetting region (band) tends to elongate the droplet; on the other hand, the forces exerted on the nonwetting region tend to pinch the droplet. However, there is a resisting force to this phenomenon: it is the surface tension, whose contribution is to bring back the surface towards that of a spherical cap. Then the question is: Can a droplet be stretched by capillary forces to a point where it is completely resting on the wetting band? This question has been answered in a series of references [29–33]. They have shown that four morphologies are possible depending mostly on the lyophilic contact angle θ and the volume of the drop V. These four morphologies are shown in Figure 3.62. In Figure 3.62(a) when the liquid volume is small, the droplet has a spherical shape and is totally located on the lyophilic band. For larger volumes, the morphology depends on the lyophilic contact angle. If the contact angle is smaller than a threshold value θ < θlim(V), the droplet spreads on the lyophilic surface without overflowing on the lyophobic surface. If θ > θlim(V), the droplet stays localized in a bulge state (i.e., does not spread), and two morphologies are possible depending on the volume of the droplet: the volume is sufficiently small and the droplet is constrained by the lyophilic surface limits, and the volume is sufficiently large and the droplet spreads over the transition line onto the lyophobic surface. The cross-sectional profiles of the droplet in the different types of morphologies are shown in Figure 3.63. These profiles are circle arcs in all cases, with different curvatures. The curvature (and the internal pressure according to Laplace’s law) is maximal in the bulge morphology, where the ratio of the height of the drop to the base width is maximal.

Figure 3.61  Sketch of the capillary forces on the triple contact line.

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Figure 3.62  (a–d) Different shapes of a droplet on a lyophilic band: results of a numerical simulation with Surface Evolver [14].

3.8.2  Drops on Inhomogeneous Surfaces

Young’s law has been derived assuming a perfectly flat homogeneous surface. This is somewhat an abstraction, and surfaces—even when carefully microfabricated— have some roughness and may not be chemically homogeneous. We investigate in this section the modifications to Young’s law linked to inhomogeneous surfaces. 3.8.2.1  Wenzel’s Law

It has been observed that roughness of the solid substrate modifies the contact between the liquid and the solid. The effect of roughness on the contact angle is not intuitive. It is surprising that roughness amplifies the hydrophilic or hydrophobic property of the contact. We shall not present the derivation of the Wenzel law here. It is classical and the reader can refer to many books [8, 10, 34]. Let us denote θ * as the contact angle with the surface with roughness r and θ as the angle with the smooth surface (in both case, the solid, liquid, and gas are the same).

Figure 3.63  Transverse shape of the droplet in the different morphologies: (a) the drop is a spherical cap with Young contact angle, (b) the drop has a circular transverse shape with h ~ L/2, (c) the droplet bulge over the hydrophobic band, but its contact surface lies within the hydrophilic band, and (d) the curvature is reduced by liquid overflowing onto the lyophobic surfaces.

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Figure 3.64  Left: contact on a smooth surface; right: contact on a rough surface.

The Wenzel law states that

cos θ * = r cos θ

(3.84)

which changes the Young law to the following form

γ LG cos θ * = ( γ SG - γ SL ) r

(3.85)

Taking into account that r > 1, relation (3.84) implies that

cos θ * > cosθ

(3.86)

We can deduce that if θ is larger than 90° (hydrophobic contact), then θ * > θ and the contact is still more hydrophobic due to the roughness (Figure 3.64). If θ is smaller than 90° (hydrophilic contact), then θ * < θ and the contact is still more hydrophilic. In conclusion, surface roughness increases the wetting character (Figure 3.65).

Figure 3.65  Contact of a liquid drop on a rough surface.

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Figure 3.66  Large-scale roughness: schematic view of a drop located on an angle of the solid surface. The position of the drop might not be stable.

An important remark at this stage is that the scale of the roughness on the solid surface is very small compared to that of the drop [35]. Indeed, if not, it would not be possible to define a unique contact angle θ*; the drop would not be axisymmetrical anymore, and the contact could be sketched as in Figure 3.66, with many different contact angles depending on the location of the droplet. 3.8.2.2  Cassie-Baxter Law

The same analysis was done by Cassie and Baxter for chemically heterogeneous solid surfaces. For simplicity we analyze the case of a solid wall constituted of microscopic inclusions of two different materials. We shall not present the derivation of the Cassie law. It is classical and the reader can refer to many books [8, 10, 34]. Let us denote θ1 and θ2 as the contact angles for each material at a macroscopic size, and f1 and f2 are the surface fractions of the two materials (Figure 3.67). The Cassie-Baxter relation states that

cos θ * = f1 cos θ 1 + f2 cos θ 2

(3.87)

This relation may be generalized to a more inhomogeneous material



cos θ * = å fi cosθ i i

(3.88)

Note that f1 + f2 = 1 or å fi = 1 if there are more than two components.

Figure 3.67  (a) Contact on a homogeneous substrate. (b) Contact on a heterogeneous surface.

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121

The Cassie-Baxter relation shows that the cosine of the contact angle on a microscopically inhomogeneous solid surface is the barycenter of the cosine of the contact angles on the different chemical components of the surface. The CassieBaxter law explains some unexpected experimental results: Sometimes a microfabricated surface may present chemical inhomogeneity and the wetting properties are not those that were intended. Take the case where a uniform layer of Teflon is deposited on a substrate to make it hydrophobic. However, if the layer is too thin, the Teflon layer may be porous and the coating inhomogeneous; the wetting properties are then modified according to the Cassie-Baxter law and the gain in hydrophobicity may not be as large as expected. As for Wenzel’s law, an important remark at this stage is that the scale of the heterogeneities of the different chemical materials of the solid surface is very small compared to that of the drop [35]. Indeed, if not, it would not be possible to define a unique contact angle anymore. This latter type heterogeneity is related to drop pinning as we have seen in Section 3.7. 3.8.2.3 Contact on Microfabricated Surfaces—Superhydrophobic and Superhydrophilic Substrates

According to the Wenzel and Cassie laws, the contact angle of a liquid on a solid surface depends on the roughness and the chemical homogeneity of the surface. By combining the effect of these two laws, special superhydrophobic and superhydrophilic substrates have been developed. First, the Wenzel law shows that the hydrophobic or hydrophilic property of a surface can be increased by increasing the roughness of this surface. Many techniques have been developed to increase the roughness of a surface. A widely used method consists in growing nanocrystals on the surface and deposing a coating on top by chemical vapor deposition (CVD). Teflon, silanization, or fluorated coating (CFx) have hydrophobic properties and are widely used in biotechnology (often to reduce the unwanted adsorption of biologic objects on the surface); on nanocrystal-treated surfaces these coatings produce very hydrophobic substrates [36]. It is even possible to switch a hydrophilic surface—such as gold—to a very hydrophobic surface. An example is the deposit of a coating of CF4-H2-He by plasma discharge on a gold plate (Figure 3.68) [37]. The

Figure 3.68  A flat gold plate treated with CF4-H2-He plasma deposition becomes very hydrophobic. (a) A water droplet bounces back from the surface treated with CF4-H2-He plasma deposition. (b) A water droplet spreads on the original gold surface [37]. Reprinted with permission from [37]. Copyright 2005 American Chemical Society.

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Figure 3.69  AFM image of a gold surface treated with CF4-H2-He plasma discharges. The number of discharges is (a) 0, (b) 3, (c) 7, and (d) 11. After 10 discharges, the surface is superhydrophobic and the rugosities are still very small, of the order of 10 nm [37]. Reprinted with permission from [37]. Copyright 2005 American Chemical Society.

hydrophobic character increases with the numbers of discharges. Note that in this case, the increase in roughness of the surface is moderate. The rugosities created by the plasma coating are only 10 nm (Figure 3.69). This method can be applied to very different substrates, such as silicon, gold, or even cotton. A liquid at rest on a Wenzel surface contacts the entire surface. It has been found that the use of the Cassie law would be still more efficient to reinforce the hydrophobic or hydrophilic property of a surface. The idea is to pattern the surface with extremely pronounced rugosities, such as micropillars or grooves. Figure 3.70 shows an example of patterning a silicon surface with micropillars [38]. The roughness r of such surfaces is very large. If Wenzel’s law is applicable, it is expected that the hydrophilic/hydrophobic character will be very pronounced. Thus, the question is: Can Wenzel’s formula, taking into account a roughness based on the shape of the microstructures, be used to derive the contact angle? The answer is not that straightforward. It has been observed that the droplet does not always contact the bottom plate and sometimes stays on top of the pillars, which is called the fakir effect (Figure 3.71). In such a case, should not the Cassie law, based on a juxtaposition of solid surface and air, have been used? Also, what is the limit be-

Figure 3.70  (a) Surface patterned with micropillars, and (b) surface patterned with microgrooves.

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Figure 3.71  (a) Droplet penetrating between the pillars. (b) Droplet sitting on top of the pillars (fakir effect). θW is the Wenzel contact angle and θC is the Cassie contact angle.

tween a Wenzel droplet and a Cassie droplet? All these questions are discussed next. Let us first investigate the case of a hydrophobic substrate. Hydrophobic Substrate

The contact angle of a sessile drop sitting on microfabricated pillars has been the subject of many investigations recently. As we have seen previously, Young’s law defines the contact angle on the substrate material



cos θ =

γ SG - γ SL γ LG

(3.89)

If the drop penetrates between the pillars, one can write the Wenzel angle as

cos θW = r cos θ

(3.90)

where θW is the Wenzel contact angle and r is the roughness of the surface. If the drop stays on top of the pillars, one can write the Cassie law under the form

cos θC = f cos θ + (1 - f ) cosθ 0

(3.91)

where θC is the Cassie contact angle, θ0 is the contact angle with the layer of air, and f is the ratio of the contact surface (top of the pillars) to the total horizontal surface. If the pillars are not too far from each other, the value of θ0 is roughly θ0 = π (Figure 3.72). Equation (3.91) then simplifies to

cos θC = -1 + f (1 + cos θ )

(3.92)

The two relations (3.90) and (3.91) can be plotted in a [cos θYoung cos θreal] diagram (Figure 3.73) [39–44]. In such a representation, the two equations correspond

Figure 3.72  Sketch of a Cassie drop (fakir effect). The interface between the pillars is roughly horizontal.

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Figure 3.73  Plot of the Wenzel and Cassie laws for a sessile droplet sitting on a surface textured with micropillars.

to two straight lines, the first one with a slope r, and the second one with a slope f. The two lines intersect, because



r=

Stotal Shorizontal

>f =

Stop Shorizontal

The two lines intersect at a Young contact angle θi defined by θC =θW, so that

cos θi =

f -1 r-f

(3.93)

In the diagram of Figure 3.72, for a given Young angle, there are two contact angles. Which one is the real one? From energy considerations—for example, by using Laplace’s law—it can be deduced that the real contact angle is the smaller one, so that when the Young contact angle is not very hydrophobic (θ < θi), the contact corresponds to a Wenzel regime and the drop wets the whole surface. When the Young contact angle is more hydrophobic (θ > θi), the drop is in a Cassie regime and sits on top of the pillars. Note that the situation we have just described does not correspond always to the reality. It happens that a droplet is not always in its lowest energy level and that they are sometimes in metastable regimes. One example was given by Bico et al. [39–41]. A drop deposited by a pipette on a pillared surface, even if it should be in a Wenzel regime, does not necessarily penetrate between the pillars; it may stay on top of the pillars. An impulse—mechanic, electric or acoustic—is necessary for the drop to regain the expected Wenzel regime. A surface is said to be superhydrophobic when the contact angle of aqueous liquid is close to 180°. In nature, some tree leaves in wet regions of the globe have

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125

superhydrophobic surfaces in order to force water droplets to roll off the leaves, preventing rotting of the leaves [45]. It can be shown that the best situation for superhydrophobicity for a geometrically textured surface is having f as small as possible and r as large as possible (Figure 3.74). Hydrophilic Surface

In the preceding section, we discussed nonwetting textured surfaces. Here we examine the case of a hydrophilic (wetting) textured surface. This case refers to the theory of impregnation [41]. A droplet on a rough wetting surface has a smaller contact angle than the Young contact angle, according to Wenzel’s law. However, it has been observed that in some cases, imbibition occurs (i.e., a part of the liquid forms a film on the substrate). With the same notations, r as the roughness of the surface and f as the Cassie ratio, one can define a critical contact angle by



cos θcrit =

1- f r-f

(3.94)

If the Young contact angle θ is such that θ < θcrit, then the liquid wets the surface (i.e., a liquid film spreads on the surface). In the opposite case, the drop is in the Wenzel regime with a contact angle given by the Wenzel law. The two possible morphologies are shown in Figure 3.75. Relation (3.94) is very similar to (3.93) for hydrophobic substrates. We verify that, for a flat surface r ® 1, the surface is wetted only if the Young contact angle is θ = 0. For a microporous substrate (r ®¥), θcrit = π/2. More generally, (3.94) defines

Figure 3.74  Superhydrophobicity requires a Cassie/Wenzel diagram with a very small coefficient f and a large coefficient r.

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Figure 3.75  The two possible morphologies of a droplet on a wetting textured surface.

a critical angle comprised between 0 and π/2. In the Wenzel regime (θ > θcrit), the contact angle is cos θ * = r cosθ



(3.95)

If θ < θcrit, a film forms, but the remaining droplet has a contact angle defined by cos θ* = 1 - f (1 - cos θ)



(3.96)

This expression shows that the presence of a film improves the wetting (θ* < θ), but it is not possible to induce a wetting transition (total wetting) by texturing a solid: (3.96) shows that complete wetting θ* = 0 requires θ = 0. Conclusion/Discussion

A complete diagram of wetting transitions is shown in Figure 3.76 [46, 47]. In the case of a hydrophobic substrate, if the Young angle θ is such that θ > θi where θi is defined by cosθi(f - 1)(r - f ), the droplet stays on the pillar tops (fakir effect), producing a superhydrophobic situation. If π/2< θ < θi, the droplet is in the Wenzel regime, completely in contact with the surface of the pillars, with a contact angle larger than the Young contact angle. In the case of a hydrophilic substrate, if the Young contact angle θ is such that π/2> θ > θcrit, the droplet is in the Wenzel regime, with a real contact angle θ* smaller than θ. For θ < θcrit where θcrit is defined by cosθ crit(1- f )(r - f ), the liquid spreads between the pillars and leaves a droplet above the pillars with a contact angle smaller than θ.

3.9  Conclusions This chapter is devoted to the study of surface tensions, capillary forces, and microdrops in microsystems. Starting with the notion of surface tension, the fundamental

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127

Figure 3.76  Plot of the relation between cos(θreal) and cos(θYoung) for patterned surfaces. Reprinted with permission from [47]. Copyright 1996 American Chemical Society.

Laplace relation has been derived. Next, the Young law for the contact angle of an interface on a solid has been presented. From these two relations, an expression for the capillary force on a triple line has been deduced. Such an expression has a key role in determining the behavior of droplets on different substrates and geometry of microsystems. This chapter has shown the essential role of surface tension and capillarity at the microscale. These forces often screen out forces such as gravity or inertia, which are predominant at the macroscopic scale. Although we have taken the stance of presenting capillarity and surface tension from an engineering point of view by considering global effects, one has to keep in mind that interactions at the nanoscopic scale are the real underlying causes of these global effects. Finally, it is stressed here that liquid-liquid or liquid-gas interfaces adopt a shape that minimizes the interfacial area, taking into account the constraints at the contact with the solid parts. Such surfaces encompass the concept of minimal surfaces—surfaces with mean zero curvature (Figure 3.77) [48]—and extend it to minimal energy surfaces, given the constraints acting on them. The prediction of the shape of an interface results from the minimization of the energy of the system (surface, gravitational, and so forth) under some constraints imposed by external conditions, such as walls, wires, fixed volume, or fixed pressure. When gravity is negligible, these surfaces have a constant mean curvature [49]. Before a droplet is deposited on a solid surface, the surface energy of the system is

ESG,0 = γ SG SSG,0

(3.97)

After deposition of the droplet, the surface energy is the sum of the three surface energies

E = ELG + ESL + ESG,1

(3.98)

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Figure 3.77  Example of minimal surface: one surface formed by the interface of a liquid attached to a vertical rod (Surface Evolver calculation).

where ESG,1 is the surface energy of the solid surface in contact with the gas. Then, we have E = γ LG SLG +

òò (γ SL - γ SG )dA + ESG,0

SSL

(3.99)

The last term on the right-hand side of (3.99) does not depend on the drop shape. Thus, we have to minimize E = γ LGSLG + òò (γ SL - γ SG )dA SSL



(3.100)

Taking into account Young’s law, the energy to be minimized is [50] E = γ LGSLG - γ LG òò cos θ dA SSL



(3.101)

As mentioned earlier, the parameters intervening in (3.101) are θ and γLG. Thanks to Young’s equation, we do not need the surface tension of the solid with the liquid or the gas. This is a real simplification that θ and γLG are difficult to measure.

References   [1] Israelachvili, J., Intermolecular and Surface Forces, New York: Academic Press, 1992.   [2] Table of surface tension for chemical fluids, http://www.surface-tension.de/.   [3] Navascués, G., “Liquid Surfaces: Theory of Surface Tension,” Rep. Prog. Phys., Vol. 42, 1979, pp. 1133–1183.   [4] Guggenheim, E. A., “The Principle of Corresponding States,” J. Chem. Phys., Vol. 13, 1945, pp. 253–261.   [5] Pasandideh-Fard, M., et al., “The Generalized Laplace Equation of Capillarity. I. Thermodynamic and Hydrostatic Considerations of the Fundamental Equation for Interfaces,” Advances in Colloid and Interface Science, Vol. 63, 1996, pp. 151–178.

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  [6] Weisstein, E., http://mathworld.wolfram.com/Curvature.html.   [7] deGennes, P. G., “Wetting: Statistics and Dynamics,” Rev. Mod. Phys., Vol. 57, 1985, p. 827.   [8] Berthier, J., Microdrops and Digital Microfluidics, New York: William Andrew Publishing, 2008.   [9] Darhuber, A. A., and S. M. Troian, “Principles of Microfluidic Actuation by Modulation of Surface Stresses,” Annu. Rev. Fluid Mech., Vol. 37, 2005, pp. 425–455. [10] deGennes, P. G., F. Brochart-Wyart, and D. Quéré, Drops, Bubbles, Pearls, and Waves, New York: Springer, 2005. [11] Wang, J. Y., S. Betelu, and B. M. Law, “Line Tension Approaching a First-Order Wetting Transition: Experimental Results from Contact Angle Measurements,” Physical Review E, Vol. 63, 2001, pp. 031601-1, 031601-10. [12] Li, W., et al., “Screening of the Effect of Surface Energy of Microchannels on Microfluidic Emulsification,” Langmuir, Vol. 23, 2007, pp. 8010–8014. [13] Ying-Song Yu, Ya-Pu Zhao, “Deformation of PDMS Membrane and Microcantilever by a Water Droplet: Comparison Between Mooney–Rivlin and Linear Elastic Constitutive Models,” Journal of Colloid and Interface Science, Vol. 332, 2009, pp. 467–476. [14] Bormashenko, E., Y. Bormashenko, and A. Musin, “Water Rolling and Floating Upon Water: Marbles Supported by a Water/Marble Interface,” Journal of Colloid and Interface Science, Vol. 333, 2009, pp. 419–421. [15] Brakke, K., “The Surface Evolver,” Exp. Math., Vol. 1, 1992, p. 141. [16] Hu, D. L., and J. W. M. Bush, “Meniscus-Climbing Insects,” Nature, Vol. 437, 2005, pp. 733–736. [17] Suzuki, K., “Flow Resistance of a Liquid Droplet Confined Between Two Hydrophobic Surfaces,” Microsyst. Technol., Vol. 11, 2005, pp. 1107–1114. [18] Tsori, Y., “Discontinuous Liquid Rise in Capillaries with Varying Cross-Sections,” Langmuir, Vol. 22, 2006, pp. 8860–8863. [19] Bruus, H., Theoretical Microfluidics, Oxford, U.K.: Oxford University Press, 2008. [20] Zeng, J., and T. Korsmeyer, “Principles of Droplet Electrohydrodynamics for Lab-on-aChip,” Lab Chip, Vol. 4, 2004, pp. 265–277. [21] Berthier, J., et al., “Actuation Potentials and Capillary Forces in Electrowetting Based Microsystems,” Sensors and Actuators, A: Physical, Vol. 134, No. 2, 2007, pp. 471– 479. [22] Günther, A., et al., “Micromixing of Miscible Liquids in Segmented Gas-Liquid Flow,” Langmuir, Vol. 21, 2005, pp. 1547–1555. [23] Berthier, J., et al., “The Physics of a Coflow Micro-Extractor: Interface Stability and Optimal Extraction Length,” Sensors and Actuators A, Vol. 149, 2009, pp. 56–64. [24] Nie, Z. H., et al., “Emulsification in a Microfluidic Flow-Focusing Device: Effect of the Viscosities of the Liquids,” Microfluid Nanofluid., Vol. 5, 2008, pp. 585–594. [25] Concus, P., and R. Finn, “On the Behavior of a Capillary Surface in a Wedge,” PNAS, Vol. 63, No. 2, 1969, pp. 292–299. [26] Brakke, K., “Minimal Surfaces, Corners, and Wires,” J. Geom. Anal., Vol. 2, 1992, pp. 11–36. [27] Seemann, R., et al., “Wetting Morphologies at Microstructured Surfaces,” PNAS, Vol. 102, 2005, pp. 1848–1852. [28] Lipowsky, R., et al., “Wetting, Budding, and Fusion—Morphological Transitions of Soft Surfaces,” J. Phys.: Condens. Matter, Vol. 17, 2005, pp. S2885–S2902. [29] Gau, H., et al., “Liquid Morphologies on Structured Surfaces: From Microchannels to Microchips,” Science, Vol. 383, 1999, pp. 46–49. [30] Lenz, P., and R. Lipowsky, “Morphological Transitions of Wetting Layers on Structured Surfaces,” Phys. Rev. Letters, Vol. 80, No. 9, 1998, pp. 1920–1923. [31] Brinkmann, M., and R. Lipowsky, “Wetting Morphologies on Substrates with Striped Surface Domains,” Journal of Applied Physics, Vol. 92, No. 8, 2002, pp. 4296– 4306.

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Interfaces, Capillarity, and Microdrops [32] Klingner, A., and F. Mugele, “Electrowetting-Induced Morphological Transitions of Fluid Microstructures,” Journal of Applied Physics, Vol. 95, No. 5, 2004, pp. 2918–2920. [33] Gau, H., S. Herminghaus, P. Lenz, R. Lipowsky, “Liquid morphologies on structured surfaces: From microchannels to microchips,” Science, Vol. 383, 1999, pp. 46– 49. [34] Tabeling, P., and S. Chen, Introduction to Microfluidics, New York: Cambridge University Press, 2007. [35] Berthier, J., and P. Silberzan, Microfluidics for Biotechnology, Norwood, MA: Artech House, 2005. [36] Uelzen, T., and J. Müller, “Wettability Enhancement by Rough Surfaces Generated by Thin Film Technology,” Thin Solid Films, Vol. 434, 2003, pp. 311–315. [37] Kim, S. H., et al., “Superhydrophobic CFx Coating Via In-Line Atmospheric RF Plasma of He-CF4-H2,” Langmuir, Vol. 21, 2005, pp. 12213–12217. [38] Zhu, L., et al., “Tuning Wettability and Getting Superhydrophobic Surface by Controlling Surface Roughness with Well-Designed Microstructures,” Sensors and Actuators, A Physical, Vol. 130-131, 2006, pp. 595–600. [39] Bico, J., U. Thiele, and D. Quéré, “Wetting of Textured Surfaces,” Colloids and Surfaces A: Physicochemical and Engineering Aspects, Vol. 206, 2002, pp. 41– 46. [40] Bico, J., C. Marzolin, and D. Quéré, “Pearl Drops,” Europhys. Lett., Vol. 47, No. 2, pp. 220–226, 1999. [41] Bico, J., C. Tordeux, and D. Quéré, “Rough Wetting,” Europhys. Lett., Vol. 55, No. 2, 2001, pp. 214–220. [42] Patankar, N. A., “Transition Between Superhydrophobic States on Rough Surfaces,” Langmuir, Vol. 20, 2004, pp. 7097–7102. [43] Patankar, N. A., and Y. Chen, “Numerical Simulation of Droplet Shapes on Rough Surfaces,” Proceedings of the 2002 Nanotech Conference, Puerto Rico, April 21–25, 2002, pp. 116–119. [44] Patankar, N. A., “On the Modelling of Hydrophobic Contact Angles on Rough Surfaces,” Langmuir, Vol. 19, 2003, pp. 1249–1253. [45] Barthlott, W., and C. Neinhuis, “Purity of the Sacred Lotus, or Escape from Contamination in Biological Surfaces,” Planta, Vol. 202, 1997, pp. 1–8. [46] Onda, T., et al., “Super Water-Repellent Surfaces,” Langmuir, Vol. 12, No. 9, 1996, pp. 2125–2127. [47] Shibuichi, S., et al., “Super Water-Repellent Surfaces Resulting from Fractal Structures,” J. Phys. Chem., Vol. 100, 1996, pp. 19512–19517. [48] Brandeis University, http://rsp.math.brandeis.edu/3D-XplorMath/Surface/gallery_m.html. [49] Hewgill, D. E., “Computing Surfaces of Constant Mean Curvature with Singularities,” Computing, Vol. 32, 1984, pp. 81–92. [50] Patankar, N. A., and Y. Chen, “Numerical Simulation of Droplets Shapes on Rough Surfaces,” Nanotech 2002, Technical Proceedings of the 5th International Conference on Modeling and Simulation of Microsystems, 2002, pp. 116–119.

Chapter 4

Digital, Two-Phase, and Droplet Microfluidics

4.1  Introduction Due to the miniaturization trend and the need for handling smaller volumes of liquids, new types of microfluidics have emerged, like digital and droplet microfluidics. These fields have seen remarkable developments during the last few years. Digital microfluidics is used to move, merge, and mix droplets on a paved grid of a solid planar substrate. This is a powerful tool for extremely precise droplet handling with applications in the domain of DNA recognition and analysis. On the other hand, droplet microfluidics is particularly suited for cell encapsulation, and this is the engine driving today’s medical replacement of defective organs in the body. It has become so popular that Hübner et al. [1] have published a paper in the journal Lab-on-a-Chip entitled: “Microdroplets: A Sea of Applications.” One could categorize these two fields by two-dimensional and three-dimensional microfluidics, respectively, for digital and droplet microfluidics. In this chapter we present the basics of each approach and indicate their applications.

4.2  Digital Microfluidics 4.2.1  Introduction

Digital microfluidics is sometimes called planar microfluidics because it consist in moving, merging, and dividing droplets on a planar—or at least locally planar— surface. There are two means of actuation: acoustic and electric. In this chapter we only deal with the electric actuation, called electrowetting, and more specifically electrowetting on dielectrics (EWOD) because it is the principle of most digital microfluidics systems. Acoustic actuation has also seen some interesting developments, and the reader can refer to the publications of Augsburg University and to [2]. The advantages of such microsystems are very small liquid samples (less than 100 nl) and extremely precise control of the droplets due to the “digital” actuation. A sketch of such systems is shown in Figure 4.1. 4.2.2  Theory of Electrowetting 4.2.2.1  Berge-Lippmann-Young (BLY) Equation

In the presence of an electric field, electric charges gather at the interface between conductive and nonconductive (dielectric) materials. Theses electric charges exert 131

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Digital, Two-Phase, and Droplet Microfluidics

Figure 4.1  Schematic of a digital microfluidic chip (courtesy of CEA-LETI).

a force on the interface, and if the interface is deformable—like that of a conductive liquid and a nonconductive fluid or gas—this force can distort the interface. This especially occurs with electric forces exerted on a liquid-gas interface at the vicinity of the contact line with a solid, resulting in a change of the contact angle (Figure 4.2). This property was first observed by Gabriel Lippmann in 1857, but the real start of electrowetting techniques is recent with the developments of microsystems and Berge’s equation frequently referred as the Lippmann-Young law [3]. We shall denote in this book the Berge-Lippmann-Young equation the BLY equation. This equation describes the change of contact angle with the applied voltage



cos θ = cosθ 0 +

C V2 2 γ LG

(4.1)

where q is the real contact angle, q0 the Young contact angle (the one observed without any electric actuation) g LG the surface tension, C the specific capacitance, and V the voltage. In fact, the BLY equation recovers only a part of the physics of electrowetting. However, it is a clever and convenient engineering approach to convert the effect of the electric forces into an observable change of contact angle. 4.2.2.2  The Different Theories for Electrowetting

The underlying physics behind the electrowetting change of contact angle has been the object of many investigations, and different approaches have been pursued,

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which all lead to the derivation of the BLY law. Basically three approaches exist: thermodynamical [4], energy minimization [5], and electromechanical approaches [6, 7]. We present here the thermodynamical and electromechanical approaches because they shed an interesting light on the physics of the phenomenon. Energy approach is more mathematical and the reader can refer to the work of Shapiro et al. [5]. Thermodynamical Approach

The thermodynamical approach is based on the existence of an electric double layer in the conductive liquid along the substrate surface. This layering of charges stretches the droplet. First, we assume a perfectly smooth solid surface at the contact of the conductive liquid. The solid is a metal directly at the contact of the liquid and the potential difference is small enough so that no electric current is flowing through the liquid (no hydrolysis if the liquid is aqueous). Upon applying an elementary electric field, an elementary potential difference builds up at the interface and an electric double layer forms in the liquid at the contact of the surface. Gibbs’ interfacial thermodynamics yields eff dγ SL = -ρ SL dV



(4.2)

where g eff denotes the effective surface tension at the liquid-solid interface, rSL the surface charge density in counter-ions, and V the electric potential. If we make the Helmholtz simplifying assumption that the counter-ions are all located at a fixed distance dH from the surface (dH is of the order of a few nanometers), the double layer has a fixed specific capacitance (capacitance par unit area) CH =



ε0 εl dH

(4.3)

where el is the relative permittivity of the liquid and e0 is the permittivity of vacuum: e0 = 8.8541878176 × 10−12 F/m. Integration of (4.2) yields eff γ SL (V ) = γ SL -



V

ò

Vpzc

V

ρSL dV = γ SL -

ò

Vpzc

CH V dV = γ SL -

CH (V - Vpzc )2 2

(4.4)

Figure 4.2  (a) in absence of electric charges, a droplet of water shows a contact angle larger than 90° on a hydrophobic solid substrate. (b) the contact angle of the water with the substrate notably decreases when the electrode is actuated.

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where Vpzc is the potential at no charge: spontaneous charges appear at the surface of a solid when immersed into electrolyte solutions at zero voltage, and the potential at no charge is the voltage required to compensate this spontaneous charging. Equation (4.4) is the original Lippmann’s law. Using Young’s law, we can translate eff

the change of g  SL in a change of the contact angle. Young’s law applied successively at zero potential and at potential V can be written γ SG - γ SL = γ LG cosθ 0

eff γ SG - γ SL (V ) = γ LG cos θ

(4.5)

where q and q0 are the actuated and not-actuated contact angles. Upon subtraction of these two equations and a substitution in (4.4), we obtain the Berge-LippmannYoung law



cos θ = cosθ 0 +

CH (V - Vpzc )2 2 γ LG

(4.6)

Equation (4.6) shows that the contact angle decreases with an increase of the applied voltage. However, direct applications of Lippmann’s law to a liquid contacting a metallic surface are of little use because of the limitation of the voltage due to hydrolysis phenomena. For water, dH ~ 2 nm, el ~ 80, gSL ~ 0.040 N/m, and the maximum voltage difference is of the order of 0.1V, so that the relative change of the value of the surface tension is DgSL/gSL ~ 2%. In terms of contact angle, using gLG = 0.072 N/m, we find (cos q – cos q0) < 0.01. Since Berge [3], modern electrowetting applications circumvent this problem by introducing a thin dielectric film, which insulates the liquid from the electrode. The specific capacitance is decreased by the presence of the dielectric, but this effect is compensated by much larger working voltages. This technique is called electrowetting on dielectric (EWOD) (Figure 4.3). In this new configuration, the

Figure 4.3  Scheme of the electrowetting setup used to verify the Lippmann-Young equation. The specific capacitance C of the system is the sum of the specific capacitances of the different layers between the electrode and the liquid. The zero potential electrode may be placed anywhere in the conducting drop. Upon actuation of a voltage V, the droplet spreads on the substrate. The value of the contact angle depends on the value of the actuation potential V, according to the BYL law.

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electric double layer builds up at the surface of the insulator. The system now comprises two capacitors in series, namely the double layer at the solid surface specific capacitance CH and the dielectric layer specific capacitance CD given by CD =



ε0 ε D d

(4.7)

Comparing CH (4.3) and CD (4.7), we find CD ε D dH = CH εl d



This relation shows that CD << CH because dH << d and eD < el. Thus, the total capacitance C, given by the relation 1 1 1 = + C CD CH



can be approximated by C » CD. This relation shows that the voltage drop occurs within the dielectric layer and (4.4) is replaced by



eff γ SL (V ) = γ SL -

C 2 ε ε V = γ SL - 0 D V 2 2 2d

(4.8)

where the potential at no charge Vpzc has been neglected assuming that the insulating layer does not give rise to spontaneous adsorption of charge. Using Young’s law, and considering that the potential at no charge on a dielectric is zero, we find the Berge-Lippmann-Young law for electrowetting on dielectric



cos θ = cosθ 0 +

C V2 2 γ LG

(4.9)

The last term on the right hand side of (4.9) is dimensionless, and is called the electrowetting number h



η=

C ε ε V2 = 0 D V2 2 γ LG 2 d γ LG

(4.10)

For a usual dielectric like Teflon or parylene 1 mm thick and a droplet of water, the coefficient e0 eD 0/2dgLG is of the order of 10-4 V-2. In order to obtain a substantial change of contact angle, applied electric potentials should be of the order of 30 to 80V. Electromechanical Approach

The electromechanical approach was introduced by Jones at al. [7], Kang [8], and recently reviewed by Zeng and Korsmeyer [9]. Let us start from the Maxwell stress tensor



1 æ ö Tik = ε 0ε ç Ei Ek - δ ik E2 ÷ è ø 2

(4.11)

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�2 along the surface of dV. In (4.11), the notation E2 corresponds to E and dik is the Kronecker delta function: dik = 0 if i ¹ k and dii; = l; and i, k = x, y, or z. On a general standpoint, the net force acting on the liquid volume element is

Fi = � ò W Tik nk dA

(4.12)

where we have used the Einstein summation convention (summation on the repeated indices). At the surface of a perfectly conducting liquid on the gas side, the electric field is perpendicular to the surface (Figure 4.4), and related to the surface density of electric charges by Gauss’ law � � σ s = ε 0 E . n



(4.13)

where n� is the outward unit normal vector. Moreover, the electric field vanishes in the conducting liquid. If we consider the {x,y,z} axis system such as the x-axis is aligned with n, the electric field is E = (Ex,0,0) in the gas domain, and E = (0,0,0) in the liquid domain. In the gas domain (e = 1), the Maxwell tensor is



æ1 ε E2 ç2 0 x ç [T ] = çç 0 ç ç 0 è

0 1 - ε 0 Ex2 2 0

ö ÷ ÷ ÷ 0 ÷ ÷ 1 - ε 0 Ex2 ÷ ø 2 0

(4.14)

and vanishes in the liquid domain



[T ] = [0]

(4.15)

We can now integrate (4.12); the cross terms xy, yz, and zx are all zero, the forces in the y (respectively z) direction cancel out, and we find that the only nonvanishing contribution is a force directed along the outward normal n� � F � ε � σ � (4.16) = Pe n = 0 E2 n = s E 2 2 δA where d A is an elementary surface area of the interface. In (4.16), Pe is the electrostatic pressure defined by Pe = e0 /2 E2. The electrostatic pressure Pe acts on the

Figure 4.4  Electric force acting at the interface of a conducting liquid.

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Figure 4.5  Schematic of the electric charge distribution in the vicinity of the triple contact line. Electric charges are located at the solid-liquid interface and at the liquid-gas interface, approximately within a distance d from the solid surface. In reality, this is a very simplified view since the liquid interface is distorted by the electric forces very close to the triple contact line.

liquid surface and brings a negative contribution to the total pressure within the liquid. The liquid interface is distorted by the electric forces, according to (4.16). Electric charges at the liquid-gas interface are located close to the triple contact line, as sketched in Figure 4.5, within a distance roughly equal to the dielectric thickness d. This problem has been investigated by Kang [8] and Vallet et al. [10] using the Schwarz-Christoffel conformal mapping [11]. In our case, the electric potential f satisfies the Laplace equation Ñ2 f = 0 and consequently is a harmonic function. The theory of analytic and harmonic function shows that a conformal mapping exists (the Schwarz-Christoffel mapping, or SC mapping) that transforms the functions E and f for a half plane into the same functions for a wedge (Figure 4.6). This mapping shows that the charges concentrate at the tip of the wedge, and, after some complicated algebra, produce the resultant of the electric forces

Figure 4.6  Principle of the Schwarz-Christoffel conformal mapping. The two orthogonal fields E and f are transformed from a simple half plane geometry (with an evident solution) to the geometry of a wedge. Note the concentration of the electric field around the tip of the wedge.

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Fhorizontal =



Fvertical

ε 0 εD V 2 2d

ε ε V2 1 = 0 D 2 d tan θ

(4.17)

The SC mapping shows that the electric forces distribution on the liquid-gas interface caused by the Maxwell-stress is limited to a very small region close to the triple line. At a macroscopic scale, we can picture the situation as in Figure 4.7, and assume that the forces are point forces acting on the triple line. A balance of the forces—electric plus capillary—in the horizontal direction at the triple line produces the BLY equation



cos θ = cosθ 0 +

εD V 2 2 γ LG d

The electromechanical approach shows that the Lippmann term (electrowetting number h) comes directly from the Maxwell stress on the liquid-gas interface. This term can also be called electrostatic pressure. Another consequence of (4.17) is that there is a vertical force on the liquid interface near the triple line. This vertical force increases quickly when the contact angle decreases. This could be an explanation for the phenomenon of the contact angle saturation that we present next. Remark that, in reality, the interface angle with the wall is still the Young contact angle (Figure 4.8), the BLY contact angle being the global angle of the interface with the wall. For large values of the applied voltage the interface profile is then very distorted. 4.2.2.3  Lippmann-Young Law and the Electrocapillary Equivalence

Another slightly different way of writing the Lippmann-Young law is



γ LG cosθ - γ LG cosθ 0 =

1 CV 2 2

Figure 4.7  Resultant of the electric forces on the liquid-gas interface.

(4.18)

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Figure 4.8  Real shape of the interface at the vicinity of the wall [12].

The first interpretation of the Lippmann-Young law stems from this equation: when actuated, the contact angle of the droplet changes, resulting in an “apparent” change of capillarity. It is very important here to note the word “apparent”; in reality, the contact angle at the wall is not the Lippmann contact angle, but, at some—not well defined—distance to the surface, the macroscopic contact angle has the value defined by the BLY law. This explanation fails for dynamic motion, as mentioned by Jones and coworkers. Indeed, the situation is quite different during rapid dynamic motion of droplets under EWOD actuation; the contact angles are not the equilibrium contact angles. This interpretation is not formally rigorous, but it is very convenient in an engineering approach to explain many static or quasistatic phenomena occurring in EWOD microsystems. The second, more rigorous interpretation of the BLY law stems from (3.38). In Chapter 3 we have seen that the capillary line force on a triple line with contact angle q0 is given by

fcap = γ LG cosθ 0

(4.19)

This line force is directed in the solid plane perpendicularly to the triple contact line and oriented outward. Following Jones et al. [7], Kang [8], and Zeng and Korsmeyer [9], we have seen that the line tension electrostatic force is



fEWOD =

C 2 V 2

(4.20)

The Lippmann-Young law—under the form (4.18)—can then be written as

ftot = fcap + fEWOD

(4.21)

Under such a form, the electrowetting effect is to add an electric force to the capillary force on the triple line. 4.2.2.4  Saturation—Modified BLY Law

Equation (4.1) predicts that a total wetting is obtained for a sufficiently high voltage. This is not the case, and (4.1) breaks down above a certain voltage called the satura-

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tion voltage and denoted Vsat. This phenomenon is not yet completely understood; some explanations have been put forward. In this section we present briefly a few of them. Possible Explanations for the Saturation Effect

The different explanations for the saturation limit of the contact angle are the trapping of charges in the dielectric [13], an ionization at the triple line [10], an electric resistance [5], a zero surface-liquid energy limit [14, 15], and—this is our theory— the rapid increase of the vertical component Fy of the electric force according to (4.17). A review of all these approaches is presented in [16]. Let us focus on the zero surface energy limit—which is a very useful approximation—pioneered by Peykov, Quinn, Ralston, and Sedev, which will be referred as the “PQRS model.” The PQRS model considers that the saturation phenomenon is a thermodynamic limit of stability. In this approach, the effective solid-liquid surface tension decreases with the voltage according to eff γ SL (V ) = γ SL -



1 CV 2 2

(4.22)

and the voltage dependant Young’s law can be cast under the form



cos θ (V ) =

eff γ SG - γ SL (V ) γ LG

At zero voltage, the force balance is that defined by the classic Young’s law. As the voltage increases, the effective solid-liquid surface tension decreases, and the contact angle decreases (Figure 4.9). The lower limit for the effective solid-liquid surface tension is zero; when this value is reached, the minimum contact angle is obtained. This minimum value is the saturation contact angle qsat. At saturation



cos θ sat = cos θ (Vsat ) =

γ SG γ LG

(4.23)

which is equivalent to



æγ ö θ sat = arccos ç SG ÷ è γ LG ø

(4.24)

Figure 4.9  Sketch of the different contact angles depending on the applied voltage. (a) at zero potential, the contact angle is determined by the classical Young law; (b) the contact angle decreases when the applied voltage increases; (c) the lower limit of the contact angle is obtained when the solid-liquid surface tension vanishes.

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Using (4.22), we derive 1



Vsat

1

æ 2 γ SL,0 ö 2 æ 2(γ SG - γ LG cosθ 0 ) ö 2 =ç =ç ÷ø è è C ø÷ C

(4.25)

Equations (4.20) and (4.21) require the knowledge of the solid surface tension gSG. In the case of apolar surfaces, the Zisman’s criterion produces a good approximation [16, 17]. In the particular case of EWOD, the coating of the substrate is realized with apolar materials like Teflon, parylene, PET, PTFE, and so fourth, and we can reasonably use the value of the wetting surface tension (Zisman’s criterion) for gSG. The PQRS model predicts values of the saturation contact angle in reasonably good agreement with the experimental observations; however these are sometimes somewhat overestimated. The other plausible explanation for the saturation of the contact angle stems from (4.17). When the vertical Fy force on the triple line becomes sufficiently larger than the horizontal Fx force (i.e., Fy /Fx = 1/tanq >>1) the droplet cannot spread further on the substrate. Modified BLY Law

To take into account the saturation limit, the BLY law can be modified to [18]



æ cos θ - cosθ 0 =Lç cos θ S - cosθ 0 è 2γ

ö CV 2 ÷ (cosθ S - cosθ 0 )ø

(4.26)

where L is the Langevin function L(X) = coth (3X) – 1/3X [19], and qs is the saturation angle. Equation (4.22) reduces to the BLY law for small and moderate values of the potential V. At large potentials, it satisfies the saturation asymptote. Equation (4.22) is called the “modified” or “extended” Lippmann-Young law. It has been verified that this function fits the experimental results [18]. Figure 4.10 shows the fit between the experimental points and the modified Lippmann law.

Figure 4.10  Fit of the experimental results for (cos q – cos q0) versus V 2 obtained by Langevin’s functions.

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4.2.2.5  Hysteresis

As we have seen in the preceding section, at large values of the potential there is the saturation limit. At small values of the potential there is the hysteresis limit. Hysteresis is defined as the deviation of the contact angle from its mean value due to physical phenomena like microscopic surface defects and roughness. During dynamic motion of an interface, dynamic hysteresis refers to the difference between advancing and receding contact angles. Hysteresis is currently observed in electrowetting. When we have established the BLY law, the value of the macroscopic contact angle is in reality the average between an advancing and a receding value. If we start with a nonactuated droplet and we increase the value of the voltage, the droplet spreads. The contact angle is then an advancing contact angle. When the voltage decreases, the droplet regains its initial shape and the observed contact angles are the receding contact angles. The advancing and receding contact angles usually differ (Figure 4.11). Another manifestation of electrowetting hysteresis occurs in EWOD Microsystems during the motion of a droplet on a substrate paved with electrodes. Below a minimum actuation voltage Vmin, the droplet does not move. In the following section we produce the relation between hysteresis and minimum actuation voltage. Hysteresis and Minimum Actuation Potential

Let us consider the example of electrowetting on dielectric microsystems (EWOD) schematized in Figure 4.12. It is observed that a droplet of conductive liquid does not move from one electrode to the next as soon as an electric actuation is applied. A minimum voltage threshold is required in order to obtain the motion of the drop [20]. This minimum electric potential (Vmin) depends on the nature of the conductive liquid/surrounding fluid/solid substrate triplet. In this section, we relate the value of the minimum potential to the hysteresis contact angle. We show that the force balance on the droplet produces an implicit relation linking the minimum potential Vmin to the value of the hysteresis contact angle a.

Figure 4.11  Experimental evidence of electrowetting hysteresis. Case of a sessile droplet of deionized water immersed in silicon oil (Brookfield) and contacting a SIOC substrate. The arrows show the advancing and receding phases.

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Figure 4.12  Sketch of a droplet starting to move towards the actuated electrode.

Experiments have been conducted using different substrates (Teflon and SiOC) and different conductive liquids and surrounding gas/fluids (deionized water in oil or air, biological buffers with surfactants in oil or air, ionic liquids in air). It will be shown that the results of the models are in agreement with the experimental results. Our starting point is the BLY law



cos θ - cosθ 0 =

C 2 V 2γ

(4.27)

At the onset of the motion, there is no dynamic effect; hence, we will interpret the BLY law as a pseudocapillary effect. A droplet starts to move under an “apparent wettability gradient” between an actuated and a nonactuated electrode. We are remind here that the electrowetting line force density on a triple line is given by



fEWOD =

C 2 V = γ (cos θ - cos θ0 ) 2

(4.28)

This line force acts on that part of the triple line located above the actuated electrode (Figure 4.12). On the part of the triple line located on the initial nonactuated electrode, the forces are just the capillary line forces. Usually the substrate is hydrophobic so the forces are exerted in the same direction as the electrowetting forces. We recall that the capillary line force acts on the triple line in the plane of the substrate, perpendicularly to the triple line. In the case of a hydrophilic contact, cosq > 0 and the line force points away from the surface, whereas in the case of hydrophobic contact, the line force is negative and points inside the droplet. The EWOD line force acts on the triple line and has a component located in the plane of the substrate perpendicular to the triple line, and also a vertical component. If the surface is perfect, using the pseudocapillary equivalence for the electrowetting force, the drop will immediately move as soon as the neighboring electrode is actuated. However, experiments have shown that there is an electric potential threshold below which the droplet does not move (i.e., there is a pseudogradient of wettability below which the droplet does not move). Figure 4.13 shows experimental results for a microdrop of deionized water immersed in silicon oil and placed on a SiOC substrate. The contact angle is not the same when the droplet is spreading on the substrate (advancing) or receding from the substrate. After having performed different plots using different substrates and liquids, it is concluded that the plot of Figure 4.13 is typical. The vertical shift between the two curves defines the electrowetting hysteresis angle. Hysteresis angles are usually of the order of a 2 to 15°.

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Figure 4.13  Sketch of the hysteresis curves. Actuated contact angles are not identical if the voltages are increasing or decreasing. Advancing and receding curves are approximately shifted by the hysteresis angle a.

Let us assume now that the two contact angles are the actuated and not actuated Young contact angles, plus or minus the hysteresis angle, as sketched in Figure 4.14 [20]. The advancing and receding limit contact angles are then q + a and q0 − a where q is the actuated contact angle and q0 the nonactuated contact angle. This notation stems from the Hoffman-Tanner law [21] indicating that the advancing and receding contact angles are respectively larger and smaller than their Young values. The minimum actuation potential is then the potential required for obtaining a net positive electrocapillary force

θ (Vmin ) + α £ θ 0 - α

(4.29)

This relation is illustrated by the sketch of Figure 4.15. The electro-capillary force in the direction x (unit vector i) on the hydrophilic electrode is given by



�� Fx = ò γ cos θ dl n.i L

(4.30)

Figure 4.14  Sketch of the advancing and receding contact angles with and without the hysteresis angle.

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Figure 4.15  Sketch of the droplet at onset of actuation. If V > Vmin, q(V) + a < q0 – a and the drop moves to the right under the action of capillary and electrocapillary forces.

where dl is a unit element of the contour line, and n the normal unit. Equation (4.30) can be integrated



�� Fx = γ cosθ ò dl n.i = γ cos θ e L

(4.31)

where e is the width of the electrode, as shown in Figure 4.16. Equation (4.31) shows that the shape of the triple line above an electrode has no effect on the capillary force. If we remark that the x-direction force on the triple line outside the electrodes vanishes, we conclude that the x-direction capillary force on the droplet, whatever its shape, is

Fx = e γ (cosθ - cosθ 0 )

(4.32)

This force remains constant during the motion of the droplet between two electrodes. Now, if we take into account the contact angle hysteresis, we obtain the advancing and receding capillary forces Fa, x = e γ cos(θ + α )

Fr, x = -e γ cos(θ 0 - α)

(4.33)

Figure 4.16  (a) views of droplets on electrodes; (b) sketch of the contact of a drop with the substrate.

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Using sine and cosine functions expansions, the total capillary force is then

Fx = e γ [cosθ - cosθ 0 ] - e γ α [sinθ + sinθ 0 ]

(4.34)

The first term on the right hand side of (4.34) is the usual “Lippmann” force. The second term is a resistance force depending on the value of the hysteresis contact angle. It can be shown that this second term is always negative, because sinq and sinq0 are positive. A consequence is that hysteresis reduces the capillary force, as expected. Because the minimum potential corresponds to the linear part of the BLY relation, the “Lipmmann” force can be expressed by FEWOD =



eC 2 V 2

(4.35)

A criteria for drop displacement is then



eC 2 V - e γ α [sinθ + sinθ 0 ] > 0 2

(4.36)

Without hysteresis (a = 0) the drop would move even with an infinitely small electric actuation. Taking into account the hysteresis (a ¹ 0), (4.36) shows that the minimum electric potential is given by



2 Vmin =

2γ α [sin θ (Vmin ) + sinθ 0 ] C

(4.37)

Using the Lippmann-Young law, (4.37) can be cast under the form

C 2 V = α [sin θ (Vmin ) + sinθ 0 ] 2 γ min

(4.38)

Equation (4.38) is somewhat cumbersome because it is an implicit equation due to the fact that q depends on V. In the case of a sufficiently small Vmin (4.38) can be simplified Vmin = 2



γ α sinθ 0 C

(4.39)

A large capacitance, a low liquid surface tension, and a small value of the hysteresis angle minimize the value of the voltage required to move droplets. 4.2.2.6  Working Range

Equation (4.39) gives an expression of the minimum actuation that is required to move droplets. On the other hand, there exists also a maximum actuation voltage—noted Vmax—above which the electrocapillary force on a drop does not increase anymore due to the saturation phenomenon. For the moment we do not take into account the dielectric breakdown voltage, which is closely related to saturation and will be treated later on. Hence Vmax = Vsat. Thus, for a given type of EWOD microdevice characterized by its capacitance C and its surface properties, and for a given electrically conductive liquid immersed in a surrounding nonconductive gas

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Figure 4.17  Comparison of Vmax between experimental results and PQRS model.

or fluid, the electrowetting principle will only be effective between the two limits Vmin and Vmax. It is very convenient here to use the PQRS model to express Vmax as indicated in (4.25) 1



Vmax

1

æ 2 γ SL,0 ö 2 æ 2(γ SG - γ LG cosθ 0 ) ö 2 =ç =ç ÷ø è è C ø÷ C

The advantage of this model is to produce an analytical relation for Vmax, which, combined with the Vmin model, indicates the maneuverability interval for any droplet in open EWOD systems. The maximum actuation potential depends on the microfabrication of the chip through the capacitance C and the surface tension of the substrate gc, and on the interfacial properties between the liquid and the surrounding fluid, through the term g cosq0. A good agreement between experimental and calculated values for Vmax has been found in [20]. It is interesting to bring together the equations defining the values of Vmin and Vmax. The minimum potential corresponds to the potential required to overcome the contact angle hysteresis and displaced drops by EWOD; the maximum potential is linked to the saturation limit. The domain for EWOD workability is then given by



2 γ α sinθ 0 <

C V2 < γ C - γ cosθ 0 2

(4.40)

At this stage, we remark that the interval [Vmin, Vmax] depends on the capacitance C as shown in Figure 4.18. Microdevices with thinner layers of dielectric have a larger capacitance and require a lower level of actuation. There is a clear

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Figure 4.18  Interval Vmin - Vmax as a function of the capacitance C.

advantage in trying to increase the specific conductance of the dielectric layer: the actuation requires lower potentials as well for Vmin as for Vmax. Equation (4.40) can be equivalently written

fmin = 2 γ α sinθ 0 < fEWOD < γ C - γ cosθ 0 = fmax

(4.41)

showing the range of the electrowetting force. The optimal solution would be the lowest possible fmin and a largest possible fmax. First, let us analyze the conditions for obtaining the smallest possible fmin. For aqueous liquids, a very hydrophobic contact angle q0 is optimal for having a very low minimum force for droplet motion, because sin q0 ~ (p - q0) ~ 0. On the other hand, the maximum electrowetting force is fmax = g SL0. The hydrophobic substrate that maximizes g SL is the most suitable. 4.2.2.7  Substrate Capacitance

Usually the substrate comprises of an electrode, a dielectric layer, and a hydrophobic layer (Figure 4.19). The dielectric layer guaranties the electric insulation in such

Figure 4.19  Total capacitance includes the contribution of the dielectric layer (Parylene, SiO2, or Si3N4) and the hydrophobic layer (Teflon, SIOC, and so fourth); the contribution of the electric double layer can be neglected.

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149

a way that there is no electric current (and hydrolysis) in the liquid. Usual materials for electrowetting dielectrics are parylene, SiO2, Teflon, and Si3N4. In order to have a large apparent change of contact angle between actuated and nonactuated modes, it is best to use a hydrophobic coating. This hydrophobic layer contributes to increase the amplitude of the change of contact angle between the nonactuated and actuated state. Hydrophibic layers are often made of Teflon and spread by spin coating; sometimes they are made of SiOC spread by plasma deposition. We have seen in the thermodynamical approach that the conductance of a series of layers is ε0 C= d (4.42) å εii i =1,n Using the values of usual electrowetting substrates (Table 4.1), it is seen that the contribution of the electric double layer to the total capacitance is usually negligible (less than 3/1000). Hence the double layer can be taken out of (4.42) and we obtain



1 1 1 = + C Cdielectric Chydrophobic -layer

(4.43)

An order of magnitude of the capacitance of current EWOD microsystems is C ~ 2.2 10-5 F/m2. According to (4.42) the capacitance is increased by reducing the thicknesses of the layers of the substrate. However, there is a limit to this reduction given by the dielectric breakdown limit, which will be presented in the following section. 4.2.2.8  Dielectric Breakdown

Breakdown of the dielectric occurs when the electric field in the dielectric exceeds a limit value called the critical electric field, denoted here EBD. Above this value, the material is disrupted. This threshold is also called the theoretical dielectric strength of a material. It is an intrinsic property of the bulk material. At breakdown, the electric field frees bound electrons. If the applied electric field is sufficiently high, free electrons may become accelerated to velocities that can liberate additional electrons during collisions with neutral atoms or molecules in a process called avalanche breakdown. Breakdown occurs quite abruptly (typically in nanoseconds), resulting in the formation of an electrically conductive path and a disruptive discharge through the material. For solid materials, a breakdown event severely degrades, or

Table 4.1  Values of Thickness and Relative Permittivity for the Different Usual Materials Material Thickness Relative Permittivity SiOC 3.36 1.2 mm Teflon 400 nm 1.9 Si3N4 400 nm 7.8 Teflon 1000 nm 2.2 Water (electric double layer) < 30 nm 80

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Digital, Two-Phase, and Droplet Microfluidics Table 4.2  Values of the Breakdown Voltage for Some Typical Materials Material Paraffin Oil Teflon

Critical electric filed EBD [MV/m or V/mm]

10

15

Glass

Mica

100

197

59

Note that the values indicated here are given for perfect materials. Real values are usually less than that.

even destroys, its insulating capability. The detailed physical explanation of dielectric breakdown is not the subject of this book and is well documented in [22]. For a dielectric of thickness d, the critical electric field EBD is related to the dielectric breakdown voltage VBD by

VBD = d EBD

(4.44)

Indications of the value of the critical electric field are given in Table 4.2 for some typical materials. In Section 4.2.2.4, we have developed the notion of saturation potential. Typically saturation potentials are of the order of 80V for a chip of capacitance C ~ 2.2 10-5 F/m2, obtained with a total dielectric/insulating layer of approximately 1.5 mm thickness. Thus the electric field at saturation is of the order of 55 V/mm. This value is just below the breakdown value of Teflon. In other words, typical chips have been designed to function unto the saturation potential. However, dielectric breakdown is sometimes observed at lower values of the potential, as shown in Figure 4.20. It seems that breakdown frequently occurs when there are defaults in the substrate surface or when objects like cells or proteins adhere to the substrate. A possible explanation could be the anomalous value of the electric field at the vicinity of geometrical inhomogeneities. The contact of an object with the substrate is sketched in Figure 4.21. Assuming that the liquid is perfectly conductive and that the object is insulating—which is a coarse assumption in the case of a cell or a protein, and if e1 and ed denote the relative permittivity of the object and the solid dielectric, respectively, the electric potential is given by the Laplacian equation

Figure 4.20  (a) Dielectric breakdown at the vicinity of a spherical object; the breakdown is materialized by the formation of cracks having the shape of tree branches; (b) close up on the “tree effect”; (c) electrostatic breakdown model [22] showing the growth of “failure tree” due to an electron avalanche.

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Figure 4.21  Electric scheme of the contact of an object with the substrate.



ε1 Ñ2 φ 1 = ε d Ñ2 φ d

(4.45)

On a theoretical point of view, the electric potential can be obtained through the Schwarz-Christoffel conformal mapping, in a similar mathematical approach as that of Section 4.2.2.2. If the angle is sharp, there is a pole of the transform at the tip of the wedge (Figure 4.22). This anomalous, localized value of the electric field may increase the voltage above the dielectric breakdown voltage [22]. This could explain why dielectric breakdown is sometimes observed when cells or proteins adhere to the substrate, or when the surface gets crackled after a long period of use. Work is ongoing to reinforce the level of the dielectric breakdown voltage without raising the value of the capacitance. 4.2.3  EWOD Microsystems

The principle of EWOD (electrowetting on dielectric) is to use a paving of electrodes embedded in a substrate as displacement paths that the droplets follow step by step, digitally, upon actuation of the electrodes (Figure 4.23) [23–26].

Figure 4.22  Contour plot of the magnitude of the electric field below a protein sticking to the surface (Comsol software).

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Figure 4.23  View of an EWOD system. The light surfaces are the electrodes (Photo courtesy of LETI).

4.2.3.1  Open and Covered EWOD Systems

Two types of EWOD devices are currently used: open and covered systems. In the following we present these two types of systems and then we analyze their architecture; finally we investigate how fundamental operations on droplets—like motion, coalescence (addition), division, dispensing—are performed. In open EWOD systems, sessile droplets are moved directly on the paving of electrodes. The zero potential electrodes can be a catena parallel to the surface or special electrodes embedded in the substrate. In covered EWOD systems, the catena is replaced by a top plate covering the droplet [Figure 4.23 and 4.24(b)]. In such system the free surface [i.e., the surface area between liquid and surrounding fluid (usually air, sometimes silicone oil)] is reduced. The substrate at the bottom is similar to that of open EWOD systems. The top plate is constituted by a plate electrode usually made of ITO coated with a thin layer of Teflon. Thus, without electric actuation, the contact is hydrophobic with both plates, as shown in Figure 4.24. In reality, in a covered EWOD system, the droplet closely follows the shape of the electrodes (Figure 4.25). In such a configuration, the vertical gap is very small compared to the horizontal dimension of the droplet, and the surface area of the free interface (liquid-air) is very small compared to that of the solid-liquid interface.

Figure 4.24  Schematic view of open and covered EWOD microsystems; (a) sessile droplet on electrodes; (b) covered EWOD microsystem fabricated by the LETI.

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Figure 4.25  (a) photograph of a “square” droplet in a covered EWOD microchip; (b)the droplet simulated with Evolver software.

Consequently, the energy of the liquid-air interface is much smaller than that of the solid-liquid interface and the droplet adopts the shape of the underlying electrode. This is why a droplet can adopt a nearly square shape. 4.2.3.2  Basic Fluidic Operations

In order to build a biochip, basic fluidic functions must be achievable by EWOD. In the following, operations such as droplet motion, merging (addition), division, dispense, and mixing are presented. Droplet Motion

The principle of droplet motion due to a difference—or a gradient—of wettability is well known [19, 27, 28]. When a droplet is located on the boundary between a lyophobic region and a lyophilic region, it moves to the lyophilic region, at least if hysteresis of contact angle is not too large. The situation is similar in the case of electrowetting. If a conductive droplet is located on the boundary between an actuated and a nonactuated electrode, an electrowetting force is applied on the contact line located above the actuated electrode, and a capillary force is exerted on the contact line located on the lyophobic surface. The resultant of the forces parallel to the surface is directed towards the electrically actuated region. The droplet is out of equilibrium, and if the resulting force is sufficient to overcome hysteresis, the droplet moves (Figure 4.26).

Figure 4.26  (a) View of a droplet of ionic liquid moving to the right; (b) numerical simulation with Surface Evolver [29]: the droplet moves until it finds an equilibrium state.

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Using the capillary equivalence, the electrowetting force exerted on the droplet by an electrode line of width e is fEWOD = e



C 2 V = e γ (cos θ - cosθ 0 ) 2

Division of Droplets

A very important operation requested for the digital manipulation of droplets in EWOD microsystems is the splitting or cutting of droplets (i.e., the division of a droplet into two “daughter droplets”). This operation is needed because the merging of droplets increases at two times the volume of liquid that the system must treat. Division in two half-volumes restores the possibilities of the system. The principle of droplet cutting/splitting/division is shown in Figure 4.27. The liquid is at the same time stretched by two actuated electrodes at both ends of the droplet, creating two lyophilic (hydrophilic) areas with electro-capillary forces pulling in opposite directions. The nonactuated lyophobic (hydrophobic) electrode exerts a pinching force on the triple contact line. Depending on the force balance and the elasticity of the interface, the droplet can be cut in two. Splitting a drop into two same daughter drops require an energy input, which is a function of the increase in free surface. It has been checked by experiments that the division of a sessile water droplet is impossible by electrowetting actuation, whereas this operation is feasible in covered EWOD microsystems. Numerical results confirm the experimental impossibility of splitting sessile water droplets with EWOD. The electrowetting forces cannot elongate the droplet sufficiently. Figure 4.28 shows that a droplet in the position of Figure 4.27 cannot be sufficiently elongated and escapes randomly to the left or right on a hydrophilic patch. Furthermore, numerical simulations show that, in an open EWOD configuration, a drop may be split if it has been previously elongated by any other means (Figure 4.29). From all the liquid that we have tested, it appears that only ionic liquids may be split in an open system EWOD, because they have very small contact angles with the actuated electrodes and their elasticity is larger than aqueous liquids. Drop division potentially depends on two dimensionless parameters: the initial elongation ratio A between drop length and drop width, and the “cutting” ratio CR between pinching length and initial drop length. The domain of possible splitting

Figure 4.27  Drop division. Scheme of the forces on the contact line; the principle is to apply a stretching force in one direction combined with a pinching force in the other direction.

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Figure 4.28  A spherical drop cannot be split; it just escapes to any one of the two hydrophilic regions. The band in the middle is hydrophobic (Evolver numerical software [29]).

is plotted in Figure 4.30 for two very different liquids. For an usual buffer liquid (continuous line), with a surface tension gLG = 70 mN/m and hydrophilic and hydrophobic contact angles of 70° and 115°, the optimal cutting ratio is CR = 1/2 and the elongation required is A > 5. An “open” water droplet has to be stretched on at least 5 electrodes to be split by electrowetting. The domain of possible splitting depends on the values of the contact angles and on the elasticity of the interface. In Figure 4.30, the dotted line corresponds to an ionic liquid with 60° and 93° contact angles. Numerical simulation confirms the possibility of splitting a droplet in a covered system provided that the vertical gap is sufficiently small. Figure 4.31 shows how a droplet confined between two parallel plates is easily cut in two by electrowetting forces. We have already seen that in open EWOD systems, division is very difficult for most liquids. So it is understandable that for closed EWOD system, there exists a limit value for the vertical gap d lim above which division is not possible. For a square electrode of dimension e, an approximate criterion for the limit vertical gap is



δ lim = - cos θ0 e

(4.46)

where q0 is the nonactuated contact angle (note that q0 is larger than p/2, so that cos q0 is negative). This relation may be derived by using the Laplace law in the

Figure 4.29  Division of an initially stretched droplet predicted by the Evolver numerical software.

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Digital, Two-Phase, and Droplet Microfluidics

Figure 4.30  Domain of possible drop division for open EWOD systems obtained by a series of Evolver calculations. Continuous line: deionized water with contact angles 70° and 115° and surface tension 70 mN/m. Dotted line: ionic liquid with contact angles 60° and 93°, and surface tension 40 mM/m.

pinching region. The pressure inside the liquid is related to the two curvature radiuses by



1 ö æ 1 DP = γ LG ç è R1 R2 ÷ø

(4.47)

where the minus sign takes into account the concavity of the drop surface. The vertical curvature radius R1 is

R1 = -

δ 2 cos θ0

(4.48)

and, because the width of the pinching region goes to zero

Figure 4.31  Simulation of splitting a droplet confined between two horizontal plates (the upper plate has been dematerialized for visualization). The hydrophobic contact angle is 115°, the hydrophilic contact angle is 75°, and the liquid/gas surface tension is 70 mN/m.

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R2 »



e 2

(4.49)

After substitution of (4.49) and (4.48) in (4.47), the pressure difference between the drop interior and exterior is æ cos θ0 1 ö DP = 2 γ LG ç - ÷ è δ eø



(4.50)

Equation (4.57) indicates that the pressure inside the drop decreases when the vertical gap d increases. The lowest possible pressure difference is zero, so that we find the maximum vertical gap dlim defined by (4.46). Equation (4.46) produces a rule for scaling up or down covered EWOD devices: if the ratio d/e is kept constant, and the same materials are used, drop division will still be possible. Equation (4.46) also confirms that a very hydrophobic contact angle is best for the efficiency of drop splitting. In the typical case of e = 800 mm, and q0 = 115°, then the vertical gap d should not exceed a value of about 340 mm. Droplet Dispensing

At the beginning of any EWOD process, microdrops have to be extracted from a reservoir. This step is called droplet dispensing [30, 31]. In the following section we investigate the conditions for satisfactory droplet dispensing. First, we observe that experimental and numerical simulations show that drop dispensing in an open EWOD system is not possible for usual buffer fluids (aqueous solutions and biological buffers). In consequence, we analyze the dispensing in a covered EWOD microsystem. As shown in Figure 4.32, to be effective, dispensing is constituted by three steps: 1. Liquid is extruded from the reservoir onto the electrode row by applying an electric potential on the electrode row and by switching off the reservoir electrodes. Extrusion occurs because there is an electrowetting force driving the liquid onto the electrode row and a hydrophobic force pushing the liquid out of the reservoir. 2. A pinching effect shrinks the liquid filament at the level of the cutting electrode when this latter has been switched off. This pinching effect has already been analyzed in the preceding section. This pinching step is sometimes enough to separate a droplet from the reservoir, but it has been observed that a third step, called the “back pumping” step was useful to easily extract well calibrated droplets. 3. Final dispense is obtained by “back pumping” the liquid into the reservoir after reactuation of the reservoir electrodes. The role of back pumping is to decrease the droplet pressure so that pinching becomes more effective. It has been checked that the dispense process is facilitated if the reservoir is separated from the processing electrodes by a solid wall made of plastic. Simulation results obtained with the Evolver software are very close to the experimental results and confirm the key role of back pumping for drop dispense (Figure 4.33). In fact, back pumping consists in switching the contact angle from a hydrophobic

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Digital, Two-Phase, and Droplet Microfluidics

Figure 4.32  (a, b) liquid extrusion from reservoir; (c) pinching of the liquid extrusion; (d) separation by back pumping. [Photo courtesy of Y. Fouillet (CEA-LETI)].

Figure 4.33  Comparison between the numerical and experimental results for drop dispensing in a closed EWOD microsystem.

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value—which results in pressurizing the drop—to a hydrophilic (< 90°) or neutral (90–95°) value in order to reduce the internal pressure. This decrease in pressure facilitates the pinching effect on the cutting electrode. Figure 4.34 shows the time evolution of the drop internal pressure during the different phases of the extraction process. If the decrease in pressure due to back pumping is sufficient, the pinching effect on the cutting electrode becomes efficient and separation occurs. 4.2.3.3  Electrode Shape and System Design Crenellated Electrodes

In reality, the motion of a droplet from an electrode to the next is not straightforward. Microfabrication imposes a gap separating the electrodes. This gap is usually of the order of 10 to 30 mm, depending on the precision of the lithography process, compared to an electrode size of the order of 800 mm. This gap creates a permanent hydrophobic region between two neighboring electrodes. If the droplet has a volume such that it is limited by the boundaries of the electrode, it cannot move to the next electrode when the latter is actuated. This is frequently the case in covered EWOD microsystems where droplet volumes are carefully controlled, and the size of the electrodes determines the volume of liquid in each droplet within a margin of a few percents. In order to remedy to such a caveat, jagged or crenellated electrodes have been designed as shown in Figure 4.35. The idea behind such a design is that the droplet contact line with the electrode plane extents over onto the dents of the next electrode. As soon as the next electrode is actuated, electro-capillary forces act to produce the motion of the droplet. Such jagged electrodes require more complicated

Figure 4.34  Pressure evolution during drop dispense. Each time an electrode in the electrode row is actuated, internal pressure decreases and the drop spreads on the new electrode. When the reservoir electrode is actuated for back pumping with a contact angle of 80°, the pressure decreases to a level where the pinching effect becomes effective and drop is dispensed (gLG = 40 mN/m).

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Figure 4.35  (a) typical electrode shape; (b) the interface is not sufficiently elastic to adjust to the electrode boundaries.

microfabrication, but are very efficient for droplet motion provided the dimensioning of the dents is correctly done. Centering Electrodes

Solid substrates of EWOD microsystems are microfabricated using extra care to make them as smooth as possible. Surface defects can lead to unwanted pinning, resulting in the malfunctioning of the microchip. A consequence of the smoothness of the surface is that microdrops, if not anchored by a boundary line, may not always be positioned at the same location on the surface. They show an unstable positioning and tend to drift until they find an anchored position by pinning to a singular point or to a boundary line. To maintain a microdrop at a given location, star-shaped electrodes are used (Figure 4.36). Dispensing Electrodes

In order to maintain the liquid close to the inlet port of the system, the reservoir liquid is maintained at the system entrance by a special electrode. The principle is similar to that of the star-shaped electrode (Figure 4.37). 4.2.4  Conclusion

EWOD designs, especially covered, can complete the fluidic operations required to build a lab-on-a-chip. The have proved to be able to perform PCR (polymerase chain reactions) as well as conventional systems, but in a much smaller volume.

Figure 4.36  Principle of star-shaped electrodes (Surface Evolver calculation).

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161

Figure 4.37  Principle of dispensing an electrode.

4.3  Multiphase Microflows 4.3.1  Introduction

Two-phase or multiphase microflows have gained a lot of attention recently. They have proved to be unavoidable for applications like extraction of targets from a carrier fluid, or for making microemulsions, and above all for encapsulation. This section presents the generality of two-phase microflows. The next section focuses on droplet microfluidics. 4.3.2  Droplet and Plug Flow in Microchannels

It is common to have two immiscible fluids flowing in a microchannel. They can be a gas and a liquid or two immiscible liquids, like water and oil. In such a case, it is common to speak of the water or oil phase. Often, the biologic targets are transported by the aqueous phase; the second organic or gas phase is used to separate the droplets or plugs. There are different two-phase flow regimes, but the most common are droplet and plug flows (Figure 4.38).

Figure 4.38  Microdrops and plugs in a capillary tube.

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4.3.3  Dynamic Contact Angle

The contact angle formed between a flowing liquid front (advancing or receding) and a solid surface is not constant but reflects the balance between capillary forces and viscous forces. The relative importance of these forces is often expressed by the nondimensional capillary number Ca defined by µU γ

Ca =



(4.51)

where m is the dynamic viscosity of the moving fluid (unit kg/m/s), U its velocity (m/s), and g its surface tension (N/m). The capillary number is a scale of the ratio between the drag force of the flow on a plug and the capillary forces. In a cylindrical tube of radius R, the friction pressure drop for a plug of length L is given by the Washburn law DP =



8 µU L R2

(4.52)

We deduce an order of magnitude of the drag force (force necessary to push the plug in the tube)

Fdrag » DP π R2 » µ U L

(4.53)

On the other hand, the capillary/wetting force is given by Fcap » γ R



(4.54)

From (4.53) and (4.54) we deduce



Fdrag µ U L L » » Ca γ R Fcap R

(4.55)

Hoffman first proposed an expression for the dynamic contact angle as a function of the capillary number Ca based on experimental observations [32]. However, this correlation is rather complicated and Voinov and Tanner have established the more workable correlation θd 3 - θ s3 = A Ca



(4.56)

where qd and qs are the dynamic and static contact angles. The value of the coefficient A is A ~ 94 when q is expressed in radians. Tanner’s law is plotted in Figure 4.39. For microflows, using the approximate values m ~10-3 kg/m/s, U ~ 10 m m/s to 1 cm/s, and g ~50 10-3 N/m, the typical values of the capillary number are in the range 2.10-7 to 2.10-4. The capillary number is then small and corresponds to the linear part of the Tanner law. Linearization of (4.56) yields [33]



θd = (θ s3

+

1 ACa) 3

æ 1 ACa ö » θs ç 1 + 3 θ s3 ÷ø è

(4.57)

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Figure 4.39  Experimental results for the dynamic contact angle versus the capillary number (dots) and Tanner relation (continuous line).

or



θ d - θs »

1 ACa 3 θ s2

(4.58)

Note that the capillary number is signed. Equation (4.58) shows that qd - qs is of the sign of Ca; the values of the advancing and receding contact angles are then given by θ a » θs +



1 A Ca 3 θ s2

1 A Ca θd » θs 3 θ s2

(4.59)

confirming the experimental observation that an advancing contact angle is larger than the static contact angle and a receding contact angle is smaller than the static contact angle (Figure 4.40). 4.3.4  Hysteresis of the Static Contact Angle

Young’s law predicts the value of the static contact angle as a function of the surface energy of the different materials (liquid plug, surrounding liquid and solid

Figure 4.40  Sketch of advancing, static, and receding contact angles. The advancing contact angle is larger than the static contact angle, which is in turn larger than the receding contact angle.

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substrate). Apparently, it should result in a unique value of the static contact angle. However, it happens frequently that the static contact angle is not uniquely defined, because a static angle is obtained after stopping a moving interface [33–35]. It can be comprised between two values, the first obtained by slowing down to a stop at an advancing front qs,a, and another value (smaller) obtained by slowing down to a stop at a receding front qs,r as shown in Figure 4.41. 4.3.5  Interface and Meniscus

The shape of liquid plug in a capillary tube depends on the capillary forces. A liquid plug moving inside a capillary tube (or between two parallel plates) is limited by two meniscus, one corresponding to the advancing front (index a), the other one corresponding to the receding front (index r) as shown in Figure 4.42. In microcapillaries, because the gravity force is negligible, menisci have spherical shapes. Note that receding, advancing, and static contact angles are not identical. 4.3.6  Microflow Blocked by Plugs

In this section we analyze the motion of one or more liquid plugs inside a cylindrical capillary tube. We use a lumped model and we show that Bernoulli’s equation combined with Tanner’s law explains the main features of the behavior of liquid plugs moving inside capillary tubes [36]. Flow regions may be decomposed in two steps (Figure 4.43); first the regions where a fluid moves inside the capillary, inducing a friction pressure drop; second, the interfaces that induce a capillary pressure drop. The total pressure drop in the capillary is then

DPchannel = DPcap + DPdrag

(4.60)

Figure 4.41  Hoffman-Tanner law for advancing and receding contact angles versus capillary number. The advancing contact angle is larger than the receding contact angle and there is a static hysteresis at zero velocity.

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165

Figure 4.42  Schematic view of a liquid plug in a capillary tube with the advancing and receding contact angles.

The pressure drop due to friction on the solid walls is given by the Washburn law [37]



DPdrag =

8U (µ1 L1 + µ 2 L2 ) R2

(4.61)

where indices 1 and 2 address to liquid 1 (liquid plug) and liquid 2 (surrounding carrier fluid). R is the radius of the capillary, U the average liquid velocity and L1, L2 the total length of contact of liquid 1, 2 with the solid wall (L1 + L2 = L, total length of the tube). Each interface—advancing and receding—contributes (positively or negatively in function of the contact angles) to the capillary pressure drop. The capillary pressure drop derives directly from the Laplace law, which relates the pressure difference at a spherical interface of curvature radius a by



DP =

2γ a

(4.62)

The meniscus has a spherical shape (if the capillary is small enough); as shown in Figure 4.44. The contact angle is related to the tube radius R and the curvature radius a by



cos θ = -

R a

Substitution of this equation into (4.62) yields



DPa = -

2γ cos θ a R

(4.63)

Similarly, the receding front contribution is given by

Figure 4.43  Decomposition of a two-phase flow in a lumped element. Between points A and B, C and D, and E and F, the pressure drop is due to friction; between B and C, and D and E the pressure drop results from capillary forces.

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Figure 4.44  Schematic view of the meniscus in a cylindrical capillary tube.



DPr =

2γ cos θr R

(4.64)

Remark that in our convention the pressure drop is always taken following the fluid flow. Consider the two configurations of Figure 4.45; if qa is larger than p/2, there is a positive pressure drop associated with the advancing interface. If qr is smaller than p/2, the receding front contributes positively to the pressure drop [Figure 4.45(b)], and negatively in the opposite case [Figure 4.45(a)]. The capillary pressure drop is due to the difference of the capillary forces between advancing and receding fronts because of the two different contact angles (advancing and receding) qa and qr

DPcap =

2γ (- cos θ a + cosθ r ) R

(4.65)

Equation (4.65) shows that too many plugs in the capillary may rapidly block the flow. For N plugs in the flow the capillary pressure drop may be larger than the driving pressure

Figure 4.45  Two possible configurations for a plug moving inside a capillary tube: (a) at a low velocity, the receding angle is larger than p/2 and the contribution to the pressure drop is negative; (b) at a high velocity, qr is smaller than p/2 and the contribution to the pressure drop is positive. The slope of the pressure drop inside the different liquid is due to the friction pressure drop.

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167

DPcap =



2γ N(- cos θ a + cos θr ) > Pi - Po R

(4.66)

and the flow will come to a stop (Pi and Po are the inlet and outlet pressures). Let us introduce now the linearized Hoffman-Tanner law to find a more workable expression of the capillary pressure drop [38]



æ 1 ACa ö θ a = θ s, a ç 1 + 3 θ s3,a ÷ø è

(4.67)

æ 1 ACa ö θ r = θ s, r ç 1 3 θ s3,r ÷ø è

(4.68)

and

with

Ca =



U µ1 γ

(4.69)

where the index s stands for the static contact angle, and qs,r and qs,a are the two static contact angles. They are equal if there is no static hysteresis (i.e., if the surface is perfectly smooth). The minus sign in (4.68) derives from the fact that we consider Ca as positive. After a substitution of (4.68) and (4.67) in (4.66), using some algebra, and keeping the higher order terms only, the capillary pressure drop can be cast under the form



DPcap @

2γ 2 ANU µ1 æ sin θ s,a sin θs,r ö N (- cos θ s,a + cos θs,r ) + ÷ ç 2 + R 3R θ r2,s ø è θ a, s

(4.70)

4.3.7  Two-Phase Flow Pressure Drop

In the preceding section, the pressure drop for plug flow has been derived. An extremely important factor is the pressure drop due to droplet flow, which is complex and still a subject of investigation. The theories developed for macrofluidic applications, like that of flow homogeneization, do not apply to microscopic two-phase flows since the pressure drop depends of the precise number of droplets circulating in the microchannel. In this problem, many parameters intervene: relative size of the droplets (r/rcyl), surface tension of the droplets (gas bubbles or solid spheres do not behave similarly), the viscosity of the carrier fluid and that of the droplet, the frequency (number of droplets per unit time), the spacing between droplets, and so fourth. In a general manner, two-phase flow hydraulic resistance is larger than that of the single-phase flow. If we assume a droplet regime, we follow the approach of Engl et al. [39], and the pressure drop is equal to the single phase pressure drop plus a corrective term to take into account the effect of the droplets



DP =

8 ηsp LQ æ Ld ö ÷ ç1 + D ø π R4 è

(4.71)

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where Ld is a length related—but not equal—to the size of the droplet, and D the spacing between the droplets. Figure 4.46 illustrates (4.71) for an oil flow carrying water droplets. A general conclusion is that small droplets hardly alter the hydrodynamic resistance of the channel and droplets with a large spacing do not modify substantially the pressure drop [40]. 4.3.8  Microbubbles

Uncontrolled microbubbles are a drawback in microflow systems, and one usually wants to get rid of them; it is currently done by degassing before starting the biological protocol. However, there are some interesting cases where microbubbles are deliberately used: as in the cases of bubble actuated micromixers [41] and sonoporating lysis sytems [42]. In the first case, the interface of a pinned bubble is pulsed by an acoustic field, and the standing waves on the interface induce recirculating vortices in the liquid (Figure 4.47). In the second case, the collapse of the bubble by cavitation induces a pressure wave that lysis the cells present in the liquid. Air bubbles can also be used as valves, sealing a microfluidic channel [43]. Such valves prevent the diffusion of species. 4.3.9  Liquid-Liquid Extraction

Perhaps the most interesting application of two-phase coflows or counterflows is liquid-liquid extraction (LLE), which is used to extract chemical or biochemical species present at extremely small concentrations (ppb, parts per billions) in a liquid. The principle consists in having two immiscible liquids flowing side by side; one liquid is the carrier liquid, the other one is a concentrating liquid immiscible with the other, and circulating at a very low speed—or even at rest—in order to

Figure 4.46  The pressure drop coefficient increases linearly with 1/D, and depends on the volume of the droplet in the tube.

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169

Figure 4.47  (a) Mixing in a liquid by acoustic actuation of the air-water interface [41]; (b) cell lysis caused by the collapse of a gas bubble [42].

achieve the concentration of the targeted species [44–48]. Usually targets do not spontaneously cross the interface separating the two liquids; they are captured at the interface by a chemical reaction called complexation (e.g., they bind to ligands transported or diffused in the secondary liquid). Figure 4.48 shows the principle of LLE for the extraction of lead ions from water. In this particular case, the interface between the two fluids is stabilized by micropillars [48]. The principle is very attractive, but the difficulty lies in the interface stability. Interface stability is not granted: instabilities of the Rayleigh-Taylor type contribute to

Figure 4.48  Principle of LLE of lead in water: when lead ions transported by the water phase meet dithizone ligands transported by the secondary phase at the water/solvent interface, they form a complex that is trapped in the solvent.

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disrupt the interface. Different systems have been imagined to maintain the integrity of the interface, at least unto a certain level of velocities: the two main categories are the grooved-channel design [44, 45] and the micropillar-row design [46, 47]. They are shown in Figure 4.49. On a general basis, stability depends on the ratio of the capillary numbers of the two phases



Ca1 η1 U1 γ η U η Q w = = 1 1 = 1 1 2 Ca2 γ η2 U 2 η2 U 2 η2 Q2 w1

(4.72)

Thus stability requires adapted ratios of the capillary numbers; but because the secondary phase is flowing at a very low velocity, it is not possible to have a ratio between the two capillary numbers close to 1 unless the dominant velocity of the carrier phase is limited and the length of the device not too long. This leads to a second difficulty, which is the efficiency of the system. A limited length of channel means a limited interfacial area and a limited transfer to the secondary phase. An analysis of this trade-off is in [49]. In general, the efficiency of the system is related to the length of the channels by an exponential relation of the type



æ DL ö eff = 1 - exp ç -K 2 ÷ w1 U1 ø è

(4.73)

where D is the diffusion coefficient of the targets in the carrier phase, L the length of the system, w1 the width of the carrier channel, U1 the velocity of the carrier fluid, and K a nondimensional coefficient related to the geometry of the system. 4.3.10  Example of Three-Phase Flow in a Microchannel: Droplet Engulfment

Plugs transported by an immiscible carrier fluid have been used to perform biological and chemical reactions in microvolumes [50]. Figure 4.50 shows the principle of such reactions. Different reagents transported by independent plugs are successively mixed with a solution containing a chemical or biochemical species. A condition for a proper functioning of such plug flow reactions is that the plugs do not coalesce. Coalescence would bring contamination between the liquids. In order to keep the

Figure 4.49  (a) Microgutters for LLE [45]; (b) immiscible microflows separated by a row of pillars [46] (photo by N. Sarrut, CEA-LETI).

4.3  Multiphase Microflows

171

Figure 4.50  Principle of three-phase flow reactions: spacer plugs of immiscible liquid prevent coalescence of droplets.

plugs separated, Chen et al. [51] use spacer plugs constituted by a third immiscible liquid. Obviously a first condition for the efficiency of spacer plugs is that the liquid plugs do not engulf each other. Figure 4.51 shows the satisfactory arrangement of the plugs (a), and engulfment (b). The condition for the stability of plugs in contact is given by the balance of the surface tension forces at the triple line (Neumann’s construction)

� � � γ ct + γ tr + γ rc = 0

(4.74)

Equation (4.74) can be satisfied only if the magnitude of every force is smaller than the sum of the magnitudes of the other two forces. This statement can be easily

Figure 4.51  Sketch of two plugs in contact. (a) Plugs stay distinct; (b) spacer liquid plug engulfing reagent liquid; (c) Neumann’s construction for the equilibrium of the interfaces.

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� � Figure 4.52  Assuming that g1 is larger� than g2 + g3, the resultant of the forces γ 2 + γ 3 projected on the direction of g1 cannot equilibrate γ 1 .

verified by remarking that if the magnitude of a force is larger than the sum of the magnitudes of the two others, equilibrium cannot be reached (Figure 4.52). Hence, it can be shown that the condition for nonengulfment is γ rc < γct + γ tr and

(4.75)

γ ct < γ rc + γ tr



A more strict approach of engulfment of a liquid droplet can be done by using energy considerations. Let us examine the case where a spherical droplet or solid particle is at the interface between oil and water (Figure 4.53). Let the symbols A, W, and O, respectively, stand for the sphere, water, and oil. If we assume that gravity can be neglected because the droplet Bond number is small Bo =



g Dρ R2 << 1 γ

We are left with a purely capillary problem. The surface energy of the system is

E = EAW + EAO - EWO

(4.76)

The third term on the right hand side of (4.76) corresponds to the exclusion of the interface SWO. Equation (4.76) yields

E = γ AW SAW + γ AO SAO - γ WO SWO

Figure 4.53  Sketch of the sphere at the interface.

(4.77)

4.4  Droplet Microfluidics

173

where the g ’s are the surface tensions and S’s the surface areas. Using spherical cap surface expressions, we find

E = γ AW [2 π R (R - h)] + γ AO [2 π R (R + h)] - γ WO π (R2 - h2 ) where h denotes the distance between the sphere center and the interface. The sphere places itself in a minimum energy configuration, corresponding to ¶E = 0 = -R γ AW + R γ AO + γ WO h ¶h



(4.78)

The equilibrium position is given by h=



R (γ AW - γ AO ) γ WO

(4.79)

If gAW - gAO < 0, h is negative and the capsule moves into the water phase. A condition for total engulfment is h=



R (γ AW - γ AO ) < -R γ WO

which leads to γ AW + γ WO < γ AO



(4.80)

If this is the case, the sphere is engulfed in the water phase. Note that the sphere is engulfed in the oil phase if h > R, that is, γ AO + γ OW < γ AW



(4.81)

More generally, the capsule will stay at the interface if

γ AO + γ OW > γ AW > γ AO - γ OW

(4.82)

4.4  Droplet Microfluidics 4.4.1  Introduction: Flow Focusing Devices (FFD) and T-Junctions

In the first section of this chapter we have presented digital microfluidics, [i.e., the manipulation of droplets on (locally) planer surfaces]. One can speak of these as “2D droplets.” In this section, we focus on the formation and behavior of droplets in a microflow, which can be viewed as “3D droplets.” It is an extremely important topic in biotechnology to be able to produce monodispersed droplets in a continuous flow. It is the key to the production of controlled emulsions, and to encapsulation techniques. We shall see that such droplets can be produced either in T-junctions of in flow focusing devices (FFD) [52–57]. We successively investigate the mechanisms of droplet formation in T-junctions and FFDs. Finally, we present applications of such devices in biology and biotechnology.

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Digital, Two-Phase, and Droplet Microfluidics

Two different systems corresponding to two types of instabilities are used to create microdrops in a microflow. The first type of instability occurs in T-junctions at a low velocity—small Capillary and Weber numbers—where the detachment of a droplet is governed by the pressure drop created by the merging droplet [Figure 4.54(a)]. This phenomenon is called squeezing. The second type is obtained in FFDs and T-junctions with a higher flow velocity, where a flowing liquid is reduced to a filament under the action of the other fluid [Figure. 4.54(b)]; because of the surface tension forces, the filament cannot be indefinitely stable; it breaks down in droplets at some distance of the channel entrance. In such a case, we shall see that there are two regimes of flow: dripping and jetting. In the following sections, we present these two types of microdevices and show their applications for the encapsulation of liquids and particles. An important remark at this point is that multistep devices using T-junctions and FFDs can be realized, as in Figure 4.55. This principle is the key to multilayering encapsulation. 4.4.2  T-Junctions 4.4.2.1  Droplet Detachment in T-Junctions

T-junctions that we consider are typical of microfluidics; due to microfabrication constraints they are formed by two rectangular channels of the same depth b, usually merging at an angle of 90°. It has been observed that droplet detachment in Tjunctions depends on the flow velocity: the instability leading to droplet detachment is different whether Ca < 10-2 or Ca > 10-2. Microsystems for biotechnology usually function with small flow rates, so we assume here that the carrier fluid (continuous phase) flows at a low speed (Ca < 10-2). This regime is called the “squeezing” regime. Large velocities have been treated by Nisisako et al. [58] and Thorsen et al. [59]. For proper droplet formation, it is required that the carrier fluid wets the walls.

Figure 4.54  Two different types of instabilities leading to droplet break-up: (a) in a T-junction; (b) in a FFD. Photos reprinted with permission from [55, 56]. Copyright 2006, Royal Society of Chemistry and copyright 2003, American Institute of Physics.

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175

Figure 4.55  The two types of instabilities can be used successively. Reprinted with permission from [55]. Copyright 2006, Royal Society of Chemistry.

Principle of Fluid Segments Formation

T-junctions are one of the most frequently used microfluidic geometries to produce immiscible fluid segments and droplets. The droplet formation proceeds in several steps: the liquid penetrates the main channel, forms a blob, and develops a neck. The neck elongates and becomes thinner as the blob advances downstream. It eventually breaks-up and the droplet detaches. At low Capillary and Weber numbers, interfacial forces dominate shear stress, and break-up is triggered by the pressure drop across the droplet (or the bubble). In such a flow regime, the size of the droplets is determined solely by the ratio of the volumetric rates of flow of the two immiscible fluids. For rectangular cross-sections, if L is the length of the fluid segment, a the width of the channel, Qdisp and Qcont the flow rates of the discontinuous and continuous phase respectively, it has been observed that the relation



L Q = 1 + α dis a Qcont

(4.83)

links the length L to the flow rates [60]. In (4.83), the constant a is positive and of the order of 1. Hence the length of the droplet L is always larger than a, and the droplet is in reality a fluid segment. Note that (4.83) is not valid for the entire domain of variation of the ratio Qdisp/Qcont. For small values of this ratio, L is constant, as indicated in Figure 4.56. A more accurate formulation is



L Q = 1 + α dis H(Qdis - Qcont ) a Qcont

(4.84)

where H is the Heaviside function. The physics behind (4.83) or (4.84) is complex. The process can be broken down into four steps (Figure 4.57). In the first phase, the stream of discontinuous fluid enters the main channel. In the second phase, it forms a blob, which has approximately the size of a main channel width (L ~ a). If the flow rate of discontinuous liquid Qdis is sufficiently large compared to the flow rate of the continuous

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Digital, Two-Phase, and Droplet Microfluidics

Figure 4.56  Graph of L/a versus Qdis/Qcont: the relation is independent of the dynamic viscosity h showing that the shear stress has no influence if the capillary number is smaller than a critical value Cacrit ~ 10-2.

liquid Qcont (Qdis > Qcont), the droplet elongates in the main channel. This third phase does not take place in the opposite case. Finally, the droplet detaches. During the two first phases, the droplet reaches a length L ~ a. If Qdis < Qcont the droplet does not have the time to elongate, and separation occurs immediately. Hence, L ~ a when Qdis < Qcont. Conversely, if Qdis > Qcont, the droplet elongates. Let us calculate the elongation length. During the elongation phase, the droplet growth rate is approximately given by

Figure 4.57  The four phases of droplet formation in a T-junction: (a) the stream of discontinuous fluid enters the main channel; (b) the stream blocks the main channel (almost totally, except for a very small gap e); (c) the droplet elongates downstream; (d) the droplet separates from the inlet.

4.4  Droplet Microfluidics

177

ugrowth »



Qdis ab

(4.85)

Remember that b is the height of the channel. On the other hand, the neck shrinks at a rate



usqueeze »

Qcont ab

(4.86)

If d denotes the width of the neck, the time needed to achieve the squeezing is approximately



τ squeeze »

d d ab » usqueeze Qcont

The total length of the droplet when it detaches is then



L » a + ugrowth τ squeeze » a + d

Qdis Qcont

If we note a = d/a, and scale by a, we nearly recover (4.84)

L d Qdis Q » 1+ = 1 + α dis a a Qcont Qcont

(4.87)

However, at this point, a is not a constant (a = d/a) whereas in (4.84) a is a constant, with a value close to either 0 or to 1. In the case where Qdis < Qcont, we have seen that L ~ a, which is equivalent to d = 0. In the case where Qdis > Qcont, ugrowth > usqueeze, which means that the growth velocity is larger than the squeeze velocity. It is observed that the width of the neck d does not vary quickly during the elongation phase; it suddenly goes to zero at the breakup. This is due to a little gap between the blob and the wall (e in Figure 4.57) that vanishes suddenly at breakup. Hence, the squeezing velocity is somewhat smaller than its value from (4.86). These observations explain why the ratio d/a can be approximated by a constant a, of the order of 1. Finally, we can approximate



d » α H(Qdis - Qcont ) a and (4.87) collapses to (4.84). Droplets Formation: Frequency Control of the Droplet Size

According to (4.84), the size of the droplet is of the order of the channel width, in any case always larger than the channel width. The question is: how can monodispersed droplets be produced smaller then the channel width? It has been found [61, 62] that smaller droplet sizes can be produced if the incoming rate of liquid was modulated in frequency. Without frequency modulation, there is a natural frequency f0 of droplet detachment. By superposing a tunable forcing frequency, resonances can be found leading to the formation of monodispersed droplets whose sizes differ notably from that of (4.84). Such regimes are called synchronized regimes. In such

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Digital, Two-Phase, and Droplet Microfluidics

Figure 4.58  Arnold tongues: the grey areas correspond to the synchronized regimes. ff is the forcing frequency.

regimes, the system delivers regular drops at regular time intervals. One must be careful to avoid quasi-periodic regimes where droplets are emitted irregularly and have irregular sizes. The physics behind the frequency actuated droplet emission is not yet completely understood. It involves complex nonlinear fluid dynamics. However, it has been observed that, depending on the forcing frequency, there were domains of synchronized regimes. Such domains are shown in Figure 4.58, and are called Arnold tongues. A very important experimental observation is that droplet volumes vary as the inverse of the emission frequency (Figure 4.59). Hence droplet volumes can be considerably reduced, approximately by an order of 10, and the range of droplet size is extended to the interval [a/3, a]; moreover, the size of the droplet can be adjusted in line by varying the frequency. 4.4.2.2  Mixing in T-Junctions

As we have mentioned above, T-junctions are particularly well suited for biochemical and chemical reactions. However, the crux for obtaining a high efficiency of such reactions is that the constituents that react are well mixed in a very short time. For example, in the case of rapid polymerization, the components should be rapidly mixed in order to obtain a homogeneous polymerization. Mixing in liquid plugs has been thoroughly studied by Handique et al. [63], Song et al. [64], Tice et al. [65], and Bringer et al. [66].

Figure 4.59  Droplet size is inversely proportional to the frequency.

4.4  Droplet Microfluidics

179

Figure 4.60  Striation thickness.

In the process of folding and stretching, the striation thickness is the distance between the filets of the diffusing species (Figure 4.60); the diffusion time is then given by Fourier’s law



t»

st 2 2 D

(4.88)

Ottino [67] has shown that the striation thickness is reduced after each folding of the filet according to

st (n) = st (0)σ - n

(4.89)

We analyze two types of channels: straight channels and channel with turns (Figure 4.61). We show that the mixing process is much more efficient in channels with turns. Suppose first that the microchannel is straight. Because the walls are fixed and the plug is moving, two recirculation patterns form inside the plug [Figure 4.62(a)]. Striation thickness in the plug is given by [67]



st = st (0)

L d

(4.90)

Figure 4.61  Two different types of microchannels; the channel with turns is much more efficient for mixing the components inside the droplets.

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Digital, Two-Phase, and Droplet Microfluidics

Figure 4.62  (a) Internal motion in a moving plug; (b) striation pattern in a moving plug. (c) Once diffusion has homogenized the two halves of the plug, the mixing is not complete.

where L is the length of the plug and d the distance traveled by the droplet. Each time the plug has traveled a distance d = 2L, the striation thickness is divided by two. Striation patterns are schematically shown in Figure 4.62(b). The two halves homogenize first due to the reduction of the striation. Taking into account that si(0) » a and using (4.90), the time for homogenization of each 1/2 plug is approximately



t»

a2 L2 2 D d2

(4.91)

However, at this time the concentration in the plug is not uniform and the situation is schematized in Figure 4.62(c). It has been observed that winding microchannels reduce homogenization time [68]. We analyze here the role of the turns of the capillary tube. Suppose a capillary tube constituted of n linear segments of length d ~ 2L, the segments being individualized by sufficiently pronounced turns. First, the recirculation flow inside the plug is modified by the turns, as shown in Figure 4.63. Second, the dissymmetry of the recirculation flow in the turns induces a reorientation of the fluid domains as shown in Figure 4.64. This reorientation is essential for the mixing of liquids in the plug. This phenomenon is called the Baker’s transform and is schematized in Figure 4.65. Reorientation is necessary to increase the number of striations. Using Ottino’s formula (4.89), the striation thickness is st(n) = a s –n; using (4.90) with d ~ 2L, we find s = 2. The diffusion time at step n is then derived from (4.91)

Figure 4.63  Dissymmetry of internal recirculation flow is induced by turns of the capillary tube.

4.4  Droplet Microfluidics

181

Figure 4.64  Sketch of the effect of stretching and folding in the straight parts and reorientation in the turns.



tdiffusion (n) =

a2 σ -2 n 2D

(4.92)

On the other hand, if we remark that d(1) is of the order of L, the convection time is given by



tconvection (n) »

d(n) L =n U U

(4.93)

Following Bringer et al. [66] and Stroock et al. [68], the mixing time is obtained by equating the diffusion and convection time, so that s



tconvection (n) » n

L a2 σ -2 n » = tdiffusion (n) U 2D

(4.94)

After rearrangement, we find



2 nσ 2n »

aU a a » Pe D L L

(4.95)

where Pe = aU/D is the Peclet number. Equation (4.95) produces the value of n (number of segments) necessary to obtain mixing. If we remark that the solution of

Figure 4.65  Baker’s transformation and reduction of the striation length.

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Digital, Two-Phase, and Droplet Microfluidics

the equation xex = q is given by the Lambert-W function defined by xex = q Þ x = W(q), then n is given by aö æ W ç ln σ Pe ÷ è Lø n» 2 ln σ



(4.96)

Suppose now that the flow rate Qdis < Qcont, (4.84) gives L ~ a and introduce the value s = 2 of the Lyapunov coefficient. Equation (4.96) becomes n»



W (0.7 Pe) 1.4

Using typical values of D ~ 10-9 m2/s, U ~ 1 mm/s and a = 100 mm, the value of the Peclet number is 100 and n ~ 3. In the case where D ~ 10-10 m2/s, the number of segments required to obtain mixing is 5. 4.4.3  Micro Flow Focusing Devices (MFFD) 4.4.3.1  Introduction

Controlling the size of droplets and their monodispersity is fundamental in biotechnological applications. Repeatability is needed for the automation of any system; however, calibration is required by biologic protocols. A free jet breaks up somewhat randomly under the effect of the Plateau-Rayleigh instability and gives birth to polydispersed droplets (Figure 4.66). A T-junction already helps to produce

Figure 4.66  Comparison between jet instability (a), T-junction (b) and FFD (c).

4.4  Droplet Microfluidics

183

droplets with an increased monodispersity; microflow focusing device (MFFD) is actually the best tool to achieve such a goal [69, 70]. In the following, we present the principles of a MFFD and the different geometries that are currently used. 4.4.3.2  Mechanism

The principle of a flow focusing device is shown in Figure 4.67. In the most common flow regime (i.e., the droplet regime—we shall see later that different flow regimes may appear) the formation of a droplet can be described by the following four stages. First a “tongue” of dispersed phase liquid enters the orifice. It forms an obstacle to the flows coming from each of the sides and pinching of the tongue occurs in a second step. A liquid blob is formed, which is linked to the incoming dispersed phase by an elongating thread (stage 3). Finally, when it thins, the thread becomes instable and breaks, liberating a droplet. On a theoretical point of view, the physics of a MFFD is complex. Currently there are no general quantitative theories for two-phase flow in a geometry such as a flow-focusing configuration. There are many parameters to the problem: The actuation parameters: Qi and Qe or Pi and Pe, respectively, the dispersed phase and continuous phase flow rates or pressures. The fluids characteristics: hi, he, g, ri, and re, denoting respectively the viscosity of the dispersed and continuous phases, the surface tension between the two liquids, and the density of the two liquids. The geometry parameters, like the dimensions of the channels wi, we, wn, ws, d and the flow resistances of the channels Ri, Re and Rs, which are respectively the widths of the dispersed phase channel, continuous phase channel, nozzle, outlet channel, depth of the channels, and hydraulic resistances of the dispersed phase, continuous, and outlet channels. The droplet parameters, fr, L, Vd (or fd), and Ud, corresponding to the droplet release frequency, the distance between two droplets, the volume of a droplet (or its diameter), and the velocity of the droplet in the outlet channel.

Figure 4.67  (a–d) Different geometries of MFFD. Type 1: systems for encapsulation; type II, systems for producing emulsions.

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Digital, Two-Phase, and Droplet Microfluidics

However there have been some flow models in special cases that we will present later in this section. When facing considerable theoretical difficulties, an approach based on nondimensional scaling numbers helps reveal the physical behavior. Throughout this section we will use the scaling numbers characteristic of FFDs already presented in Chapter 1. Actuation of MFFDs

There are two ways of making the fluids circulate in a MFFD. The flows are either driven by syringe pumps or by micropumps. In the first case, it is referred as flow rate actuation; in the second case, it is referred as driving pressure actuation. In the literature flow rate, actuation is more common because syringe pumps are largely used in the laboratories. However, recently [71, 72] with the development of reliable pressure pumps, pressure actuation has started to be used, with the advantage of very constant flow rates. It is common to make experiments keeping the ratios q* = Qi /Qe or p* = Pi /Pe constant. It is emphasized here that there is no equivalence between theses two types of actuation: we shall see that, in the first case, the capillary numbers ratio Cai /Cae  = hi Qi /he Qe —characterizing the flow behavior—depends on the viscosity ratio Cai /Cae  = hi /he, while in the second case, the ratio Cai /Cae is constant. The Different Flow Regimes

Two different categories of flow regimes exist in a FFD: dripping and jetting. Some authors, according to their particular geometry of FFD, subdivide these two categories. In the dripping regime, the flow rates are small enough so that the droplet forms immediately after the nozzle. In the jetting regime, a thread or filament stretches far into the outlet channel (Figure 4.68). In the first case, drops are larger, with a small coefficient of variation (CV) of the order of a few percents. This is why the dripping regime is preferred in biotechnology. The dripping regime occurs at low values of the flow rates. Upper limits of this regime have been investigated in [73, 74]. It appears that two nondimensional numbers pinpoint the transition to jetting regime: the critical Weber number Wec of the dispersed phase, and the critical capillary number Cac of the continuous phase. The first condition for a dripping regime is



We =

ρiUi2Rthread < Wec γ

(4.97)

Figure 4.68  Dripping and jetting regimes. (a) Dripping regime: drops form at the nozzle; (b) jetting regime: drops form at the tip of a long thread.

4.4  Droplet Microfluidics

185

where ñi, Rthread are the density of the alginate phase and the radius of the alginate thread. This dimensionless number characterizes the balance between inertial and interfacial tension forces for the dispersed phase. The second condition is



Ca =

ηe Ve < Cac γ

(4.98)

This capillary number scales viscous and interfacial tension forces for the continuous phase. Equations (4.97) and (4.98) can be written as:





Qi < Qic = d wi

Qe < Qec =

γ Wec ρi Rthread

d we γ Cac ηe

(4.99)

(4.100)

where wi and we are the width of the channels, and d the depth. Let us focus now on the dripping regime. Different flow configurations appear, according to the external actuation. These configurations are shown in Figure 4.69. Suppose a pressure actuation of the system. As shown in Figure 4.69, depending on the relatives values of the driving pressures Pi and Pe, the regimes can be (a) a flow reversal in the central channel if Pe >> Pi; (b) a droplet regime; (c) a plug regime—large droplets touching the walls; (d) annular flow regime—dispersed phase flowing inside the continuous phase; (e) reversal of the flow in the external channels

Figure 4.69  The different types of flow regimes in a MFFD.

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Digital, Two-Phase, and Droplet Microfluidics

if Pe << Pi. Similar regimes occur with flow rate actuation, except that flow reversal is not observed due to the conditions imposed by the syringe pumps. It is practical to plot the flow domains in a [Qi, Qe] or [Pi, Pe] coordinates system. The flow chart usually has the aspect shown in Figure 4.70. Volume, Frequency, and Spacing of the Droplets

Volume Vd, frequency f, and spacing D of droplets produced in the MFFD are the three main parameters that one would like to control. There are two relations linking these numbers. The first relation links the flow rate of the dispersed phase to the frequency and the droplet volume

Qi = f Vd

(4.101)

The second relation links the velocity in the outlet channel Us to the frequency and spacing

U s = f D

(4.102)

The volume of the droplets is an important topic as encapsulation or emulsification applications require a precise, given volume of droplets. This raises the question of the dependency of the droplet size on the different parameters of the FFD. It has been experimentally observed that the size of the droplets depends on the flow conditions, the geometry of the FFD, the viscosities of the fluids, and to some extend to the surface tension between the fluids. Quantitative theory does not exist for two-phase flows in FFDs, so we have to rely on the different experimental results and observed trends.

Figure 4.70  Typical flow chart for FFDs. The droplet regime is the regime of choice for MFFDs applications.

4.4  Droplet Microfluidics

187

Droplet Size Dependency on Flow Rates

For a given geometry and a viscosity ratio l, the size of the droplets depends on the ratio p* of the applied pressures Pi and Pe or the ratio q* of the applied flow rates Qi and Qe. In all cases—even for T-junctions—the droplet volume decreases with decreasing p* or q* [75–79]. This is justified by observing that when the continuous phase velocity increases, the pinching of the dispersed phase tongue is more efficient and a smaller droplet is expelled. Conversely, when the velocity of the dispersed phase increases, the tongue progresses faster, and a larger blob of liquid is expelled downstream from the nozzle. Dependency on Viscosity

The influence of the viscosity on the droplet volume is still debated. We have screened the literature in order to appreciate the dependency of the droplet size on the viscosity. Different behaviors appear: relatively low-viscosity fluids and highly viscous fluids behave oppositely. If we denote l the viscosity ratio l = hi /he, the droplet volume Vd is a growing function of l for relatively low-viscosity fluids [78–86]. It seems to be the opposite with highly viscous fluids, like semidilute solutions of alginates [68, 76, 85]. In the case of low-viscosity liquids, the continuous-phase capillary-number Cae indicates the ratio between the viscous forces—or viscous shear—and the surface . tension: Cae = heRg e /g » heUe /g . The relation between the droplet diameter and the capillary number is then [79]



æ 1 ö 1 φ»fç µ ÷ è Cae ø Cae

(4.103)

More precisely, for T-junctions, Serra et al. [78] have shown that the droplet diameter is a function of the ratio of the internal (dispersed phase) and external (continuous phase) capillary numbers



æ Ca ö æ Ca ö φ » f ç i ÷ µç i ÷ è Cae ø è Cae ø

0.22



(4.104)

Such a relation leads to an increase of the droplet diameter with the ratio l = hi /he. Highly viscous fluids (or viscoelastic fluids) do not satisfy (4.104). The reason for such a behavior is not yet known; however, it is thought that highly viscous liquids and non-Newtonian fluids present a smaller incoming tongue and a larger retracting thread, leaving a smaller volume for the expelled blob. 4.4.4  Highly Viscous Fluids—Encapsulation

Encapsulation of biologic components—cells, proteins, biochemical species—has become fundamental in biotechnology and medicine, especially for in vivo applications. Usually cells are encapsulated in a biocompatible polymer, like alginates, polysaccharides, or hydrogels. Once polymerized, the capsule must be microporous in order to let the nutrients diffuse through, but prevent large, dangerous molecules to diffuse inside. For this reason, relatively concentrated polymeric solutions are

188

Digital, Two-Phase, and Droplet Microfluidics

used; in other words, the polymeric solution is highly viscous. Viscosity can reach 2500 times that of water; however droplets are still being produced by the FFD. We analyze next the behavior of highly viscous flows in FFDs, especially the alginate solution, which viscosity has been described for in Chapter 2. 4.4.4.1  High-Viscosity Fluids in a MFFD

In this section we derive a relation between the flow rates and the driving pressures of alginate and oil flow. Model

Using the notations of Figure 4.71 and the hydraulic resistances of each branch (Re, Ri, Rs), the pressure drops in each branch can be written under the form Pe - P+ = Re Qe Pi - P+ = Ri Qi

(4.105)

P+ = Rs (2 Qe + Qi ) f (Qi Qe ) where f is a function of the ratio Qi /Qe, which takes into account the increase of pressure drop due to the droplet flow in the outlet channel [39, 40]. We have made the assumption that the pressure at the intersection P+ is not fluctuating much; this assumption is justified by the experimental results. Denoting q* = Qi /Qe and p* = Pi /Pe, the solution of (4.105) is



q* =

p*(C + 2) - 2 (1 + D) - p*

C=

Re Rs f

D=

Ri Rs f



(4.106)

Figure 4.71  Sketch of the hydraulic resistances of the different branches of the FFD. The indices i, e, and s, stand for internal (alginate) phase, external (oil) phase, and outlet.

4.4  Droplet Microfluidics

189

When the internal (dispersed) fluid is very viscous—like concentrated alginate solutions—its flow rate is much smaller than that of the external (continuous) fluid. Under this assumption, (4.106) can be simplified, yielding q* = A (p* + B - 1)



(4.107)

where A = (Re + 2 Rs f ) Ri

B = Re (Re + 2 Rs f )

(4.108)

The coefficients A and B are deduced from the expressions of the hydraulic resistances Re, Ri, and Rs, which in term are computed by the Washburn law adapted for rectangular channels [87]. At this stage, two important remarks should be made: all the parameters in (4.107) and (4.108) are known except the alginate viscosity hi and the two-phase additional pressure drop coefficient f. First, the flow velocity—hence the shear rate—of the alginate solution in the feeding channel is very small. In consequence, a zero shear alginate viscosity hi0 can be used for the determination of the hydraulic resistance of the alginate flow in the feeding channel. Second, the value of the function f is determined by remarking that the droplets have a size approximately equal to that of the cross section of the channel, and are separated by relatively large distances. In such a case, one can expect excess pressure drop due to the droplets to be small [39, 40]. We can then assume f ~1, which is experimentally verified. An important consequence of f being close to unity is that it can be taken as a constant; then the coefficient B in (4.107) is purely a geometrical factor (the viscosity he cancels out), while A is proportional to he /hi and D is proportional to hi /he. The model is confirmed by experimental observations. Figure 4.72 shows the different flow regimes in the [Qi, Qe] coordinates system together with a comparison between model and experimental results.

Figure 4.72  Diagram of flow regimes for alginate (1.25 wt%)/oil binary system with flow-rate actuation (same legends as Figure 4.71).

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Digital, Two-Phase, and Droplet Microfluidics

In particular the model explains why the upper limit of the droplet domain is the same for any couple of liquids. The boundary corresponds to the limit between droplet regime and flow reversal inside the central (alginate) channel (i.e., Qi = 0). By setting q* = 0 in (4.107), we find p* = 1 – B; B being purely a geometrical parameter—under the condition that f stays close to unity—the upper limit of the droplet regime in the pressure diagram is a fixed straight line, independent of the viscosity hi of the discontinuous phase. This property has been checked experimentally using deionized water (DIW), as well as 1 to 1.75% Keltone alginate solutions. Scaling Rules

Are there scaling rules between FFDs? In this section, we focus on the geometrical scaling-up between a small FFD (50 mm) and a larger one (100 mm to 1 mm). We determine the driving pressure conditions to obtain the droplet regime at any size. It is shown that a similar behavior of the device is obtained by a scaling-up of the dimensions by a ratio k and a scaling-down of the driving pressures by the ratio 1/k—or a scaling-up of the flow rates by k2 [72]. There are two conditions to fulfill to produce monodispersed droplets. First, the flow rates or driving pressures must be adapted to correspond to the droplet regime. Second, the flow rates or driving pressures must be chosen to obtain a dripping regime, avoiding the jetting regime. It is recalled that the jetting regime is obtained at larger velocities and is characterized by a continuous thread of alginate that extends far into the outlet channel, breaking randomly and producing rather polydisperse droplets (Figure 4.73). Hence, the dripping regime is chosen in order to be certain to obtain monodispersed droplets, which is a requirement for cell encapsulation. In the following we shall see that geometrical scaling rules are determined by the condition of a droplet regime, and the flow conditions, e.g. driving pressures (or flow rates) scaling rules by the condition of a dripping regime. Let us begin by the first condition. Geometrical Scaling

We have shown that there are two limits for the droplet regime. One is flow reversal in the central channel (e.g., when the flow rate in the central channel changes direction). This limit is given by Qi = 0 or q* = 0 , or, in a pressure actuated system Pi /Pe = p* = cste = p*1,2. The second is the droplet-plug regime limit. This limit only

Figure 4.73  (a) Dripping regime: monodisperse droplets are produced at the nozzle; (b) jetting regime: polydispersed droplets are produced at the end of an alginate jet (photo courtesy of S. Le Vot, CEA-LETI).

4.4  Droplet Microfluidics

191

depends on the flow rate ratio q* = cste = q*2,3, or, in a pressure actuated system, Pi /Pe = p* = cste = p*2,3. Using (4.107), these transitions can be written in terms of hydraulic resistance ratios: æ Re ö p *1,2 = 1 - B = ç 1 + 2 Rs f ÷ø è



p *2,3

æ Ri ö = ç 1 + q *2,3 2 Rs f ÷ø è

-1

-1

æ Re ö çè 1 + 2 R f ÷ø s

(4.109)

For rectangular channels, the hydraulic resistances can be expressed under the form [87]

R » 4 η L / (w d 3ζ )

(4.110)

where z is a coefficient depending on the aspect ratio. In order to conserve the limits of the droplet regime for new channel dimensions, the following conditions must be respected if the additional friction factor f is assumed to be always close to unity (e.g., if the droplet frequency is not too high), Re Le ws ξs = = cste Rs Ls we ξe

Ri ηi Li ws ξ s = = cste Rs ηe Ls wi ξ i

(4.111)

An immediate solution for this system is the homothetic scaling, where all the dimensions are scaled by the same constant k. Using the superscripts 1 for the original geometry and 2 for the scaled up geometry, we have w(2) L(2) d(2) = = =κ w(1) L(1) d(1)



ξ(2) α (2) = =1 ζ (1) α (1)

(4.112)

and the ratio of the hydraulic resistances is constant in the scaling-up. This homothetic scaling up based on the analytical model is verified by the experiment. Figure 4.74, corresponding to a pressure plot, shows that the limits of the droplet regimes respect the similarity rules, even in the nonlinear domain when f starts to slightly depart from unity. Flow Conditions—Pressure and Flow Rate Scaling

To limit polydispersity, the droplets should be produced in the dripping regime. The transition from dripping to jetting regime has been subject of many investigations [73, 74]. It appears that two nondimensional numbers pinpoint this transition: the critical Weber number Wec of the discontinuous phase and the critical capillary number Cac of the continuous phase. The first condition to observe a dripping regime can be expressed as

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Digital, Two-Phase, and Droplet Microfluidics

Figure 4.74  In a homothetic scaling-up, the boundaries of the dripping droplet regime are similar for the depths 50, 100 and 200 mm. The close square symbols, the open triangular symbols, and the stars, respectively, represent the depths of 50 mm, 100 mm, and 200 mm.

We =

ρiVi2Rthread < Wec γ

(4.113)

where ri, and Rthread are respectively the density of the alginate phase and the radius of the alginate thread. This dimensionless number characterizes the balance between inertial forces and interfacial tension on the alginate phase. The second condition is Ca =



ηe Ve < Cac γ

(4.114)

This capillary number scales the viscous forces with the interfacial tension for the oil phase. Equations (4.113) and (4.114) yield Qi < Qic = d wi



γ Wec ρi Rthread

(4.115)

and

Qe < Qec =

dwe γ Cac ηe

(4.116)

Assuming that Rthread is always approximately of the order of magnitude (~a few tenth of microns) at break-up, (4.115) and (4.116) imply that the critical flow

4.4  Droplet Microfluidics



193

rates scale is k2 because the product (dw) scales as k2. A natural choice for the flow rates is then Q(2) = κ 2 Q(1) (4.117) An interesting consequence is that the velocities of each phase remain constant in the scaling: V(2) = V(1). Using the model, we can transpose (4.115) and (4.116) in terms of pressure Pi < Pic = (Ri + Rs f )Qic + 2Rs f Qec



Pe < Pec = (Re + 2Rs f )Qec + Rs f Qic

(4.118)

As the hydraulic resistances R scale like 1/k3 and the flow rates Q scale as k2, the critical pressures Pc governing the dripping-jetting transition should be of the ratio 1/k . In consequence, we have for the pressure the relation

P(2) = (1 κ ) P(1)

(4.119)

When the dimensions are scaled homothetically by k, and the driving pressures are in the ratio 1/k. It has been experimentally observed that the droplet emission frequency fr is in the ratio 1/k



fr(2) =

1 (1) fr κ

(4.120)

and the droplet diameter f in the ratio k

φ(2) = κφ (1)

(4.121)

It is not coincidental that the droplet diameter and the geometrical dimensions scale identically; the same applies for the pressure and droplet frequency. These experimental scaling rules have been experimentally verified. Figure 4.75 shows the droplet diameter f as a function of the dimension of the device. In Figure 4.76 the inverse of the frequency 1/fr has been plotted against the device depth.

Figure 4.75  Droplet diameter f varies linearly with the device dimension. The reference system is a 50 mm device functioning with driving pressures Pi = Pe = 200 mbars.

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Digital, Two-Phase, and Droplet Microfluidics

Figure 4.76  The inverse of the frequency 1/fr varies linearly with the device dimension.

4.4.4.2  Encapsulating a Single Cell

Encapsulation of single cell has been identified as an important process for the study of cells. It is also important for in vivo applications where capsules are reimplanted in the human body. In order to perform a single-cell encapsulation, incoming cells must be spaced by a time difference synchronized with the droplet formation frequency (Figure 4.77). 4.4.4.3  Gelling, Polymerization, Biocompatibility, and Viability

For encapsulation purposes, gelling of the capsules is mandatory. Conventional methods polymerize capsules in microwells or beakers. Gelling inside a microsystem remains a difficult problem, because it usually occurs quickly and gelled capsules clog microchannels. Chemicals used to perform the polymerization process must be totally biocompatible, or else the viability of the cells may drop drastically. This research area is somewhat outside the scope of this book, and we refer the reader to the abundant literature in this domain.

4.5  Conclusions This chapter was devoted to the physics of droplets and two-phase flows in microchannels. The applications of these different types of flows are extremely important in biotechnology. Droplets are the smallest containers that can be imagined, and mastering their manipulation is the key to new developments.

Figure 4.77  Single-cell encapsulation: a spacing of the incoming cell according to the droplet formation frequency leads to the formation of capsules containing at most one cell each [88].

4.5  Conclusions

195

Figure 4.78  Controlled emulsion obtained with a FFD [89].

It is emphasized here that digital microfluidics (planar microfluidics) and droplet microfluidics are, in fact, complementary; they just do not address the same problems. For example, digital microfluidics is well adapted to biorecognition and biodiagnostics because only an extremely small volume of liquid is required and droplets can be carefully monitored until the detection step. Droplet microfluidics is well adapted to continuous processes, as for example the production of a large number of encapsulated biological objects. From a general standpoint, digital (planar) microfluidics can treat very small volumes of liquid (stored in reservoirs) in parallel, whereas droplet microfluidics treats small liquid volumes in series (continuous on line process). The prospects of droplet microfluidics are especially promising: they are the key to the production of controlled emulsions, encapsulation of living cells or biologic agents, and the synthesis of complex particles (Figure 4.78).

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[39] Engl, W., et al., “Droplet traffic at a simple junction at low capillary numbers,” Phys. Rev. Lett., Vol. 95, 2005, p. 208304. [40] Adzima, B.J., S.S. Velankar, “Pressure drops for droplet flows in microfluidic channels,” J. MicroMec. MicroEng., Vol. 16, 2006, pp. 1504–1510. [41] Jie Xu, D. Attinger, “Control and ultrasonic actuation of a gas–liquid interface in a microfluidic chip,” J. Micromech. Microeng., Vol. 17, 2007, pp. 609–616. [42] Le Gac S, E. Zwaan, A. van den Berg, C-D Ohl, “Sonoporation of suspension cells with a single cavitation bubble in a microfluidic confinement,” Lab Chip, Vol.7, 2007, pp. 1666– 1672. [43] van der Wijngaart, W., T. Frisk, and G. Stemme, “A micromachined interface for transfer of liquid or vapor sample to a liquid solution,” Proceedings of the 13th International Conference on Solid-State Sensors, Actuators and Microsystems, Seoul, June 5–9, 2005. [44] Aota, A., M. Nonaka, A. Hibara, and T. Kitamori, “Countercurrent laminar microflow for highly efficient solvent extraction,” Angew. Chem. Int. Ed., Vol. 46, 2007, pp. 878–880. [45] Miyaguchi, H., et al., “Microchip-based liquid–liquid extraction for gas-chromatography analysis of amphetamine-type stimulants in urine,” Journal of Chromatography A, Vol. 1129, 2006, pp. 105–110. [46] Ehrfeld, W., V. Hessel, H. Lowe, Microreators, Wiley-VCH, 2000, pp. 126–130. [47] Van-Man Tran, J. Berthier, R. Blanc, O. Constantin, C. Vauchier, N. Sarrut, “Micro-extractor for liquid-liquid extraction, concentration and in situ detection of lead,” Proceedings of the 2008 AIChE Spring Meeting, 10th International Conference on Microreaction Technology (IMRET-10), 2008. [48] Jason G. Kralj, Hermantkumar R. Sahoo, Klavs F. Jensen, “Integrated continous microfluidic liquid-liquid extraction,” Lab Chip, Vol 7, 2007, pp. 256–263. [49] Berthier, J., et al., “The physics of a coflow micro-extractor: Interface stability and optimal extraction length,” Sensors and Actuators A: Physical, Vol. 149, No. 1, 2009, pp. 56–64. [50] Song, H., D.L. Chen, and R.F. Ismagilov, “Reactions in droplets in microfluidics channels,” Angewandte Chemie, Vol. 45, 2006, pp. 7336–7356. [51] Chen, D.L., Liang Li, S. Reyes, D.N. Adamson, and R.F. Ismagilov, “Using three-phase flow of immiscible liquids to prevent coalescence of droplets in microfluidic channels: criteria to identify the third liquid and validation with protein crystallization,” Langmuir, Vol. 23, 2007, pp. 2255–2260. [52] Tice, J.D., H. Song, A. D. Lyon, and R. F. Ismagilov, “Formation of droplets and mixing in multiphase microfluidics at low values of the Reynolds and the Capillary numbers,” Langmuir, Vol. 19, 2003, pp. 9127–9133. [53] Garstecki, P., et al., “Formation of droplets and bubbles in a microfluidic T-junction: scaling and mechanism of break-up,” Lab Chip, Vol. 6, 2006, pp. 437–446. [54] Joanicot, M., and A. Ajdari, “Droplet Control for Microfluidics,” Science, Vol. 5, 2005, pp. 887–888. [55] Cabral, J.T., and S.D. Hudson, “Microfluidic approach for rapid multicomponent interfacial tensiometry,” Lab Chip, Vol. 6, 2006, pp. 427–436. [56] Anna, S.L., N. Bontoux, and H.A. Stone, “Formation dispersions using ‘flow focusing’ in microchannels,” Appl. Phys. Lett., Vol. 82, No. 3, 2003, pp. 364–366. [57] Takeuchi, S., et al., “An axisymmetric flow-focusing microfluidic device,” S Adv. Mater., Vol. 17, No. 8, 2005. [58] Nisisako, T., T. Torii, and T. Higuchi, “Novel microreactors for functional polymer beads,” Chemical Engineering Journal, Vol. 101, 2004, pp. 23–29. [59] Thorsen, T., et al., “Dynamic pattern formation in a vesicle-generating mirofluidic device,” Phys. Rev. Lett., Vol. 86, 2001, pp. 4163–4166. [60] Garstecki, P., et al., “Formation of droplets and bubbles in a microfluidic T-junction: scaling and mechanism of break-up,” Lab Chip, Vol. 6, 2006, pp. 437–446.

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Chapter 5

Diffusion of Biochemical Species

5.1  Introduction In Chapters 2, 3, and 4 we have analyzed the motion of a liquid under the form of microfluidics flow and the form of microfluidics drops (digital and droplet microfluidics). In reality, the fluid that has been studied in these chapters is a buffer fluid — or carrier fluid — containing the micro- and nanoparticles, macromolecules, or cells that are of interest. In the rest of this book, we focus on the behavior of the microparticles themselves in the buffer fluid. Different forces may act on the particles. At a microscopic scale, diffusion is always present, assuming that the particles are sufficiently small. This is the case for DNA, micromagnetic beads, and so fourth, but much less so for large biological objects like cells. Often other physical phenomena superpose with diffusion. In Chapter 6 we deal with convective transport problems, in Chapter 7 with biochemical reactions, in Chapter 9 with magnetic forces on magnetic beads, and in Chapter 10 with electric forces on charged and neutral microparticles. In this chapter we present the basis of the diffusion theory and the different approaches to solve diffusion problems. First, we analyze the basis of diffusion that is the random walk of particles due to the Brownian agitation of the fluid molecules. Next we introduce the diffusion equation of concentration and present some examples of applications in the biotechnology domain, and then we introduce the discrete Monte-Carlo approach and some applications to the diffusion of macromolecules in the human body.

5.2  Brownian Motion Because Brownian motion is a microscopic scale movement, it is no wonder that the Brownian motion was first discovered by biologists J. Ingenhousj and R. Brown [1], the latter after the observation of pollen grains floating at the surface of a drop of water. In a gas or a liquid, there is an agitation of the molecules linked to temperature. A molecule moves in a straight line until it collides with another molecule resulting in a change of direction. The average linear displacement between two collisions is called the mean free path. In biotechnology, we deal with macromolecules and microparticles larger than the fluid molecules. The basic scheme of displacement is the same; the molecules of the carrier liquid collide with the macromolecules to make them perform a random walk (Figure 5.1). 201

202

Diffusion of Biochemical Species

Figure 5.1  Brownian motion of a microparticle in a fluid at two different times. The continuous line shows the trajectory of the particle.

Brownian motion and the random walk theory is the basis of diffusion: imagine a very small volume where the diffusing particles are initially confined. Each particle originated in this volume effects a random walk with the time, and the particles are progressively dispersed in the buffer liquid (Figure 5.2). It can be shown that the concentration follows a Gaussian profile that smears out with time. There are two different approaches to calculate the diffusion of these macromolecules/microparticles. The first one is the concentration approach based on the continuum hypothesis, and the second one involves discrete methods where particles are followed individually.

5.3  Macroscopic Approach: Concentration Suppose an elementary volume of liquid dV located at a coordinate (x,y,z) as shown in Figure 5.3. The concentration c of biological species or microparticles contained in this volume at the time t is defined by



c(x, y, z, t) =

δm δV

(5.1)

and a mass flux of substance through a surface is defined by [2]



J=

δm δt δA

(5.2)

Figure 5.2  Example of diffusion from a point source. The dots are the final location of the diffusing particles after a time interval.

5.3  Macroscopic Approach: Concentration

203

Figure 5.3  Schematization of an elementary volume with a concentration c of diffusing substance.

 where the mass flux J traverses the elementary surface dA in a time interval dt. The international unit for concentration is the kilo per cubic meter (kg/m3). However, biologists and chemists often express a concentration in mole/mL. In this case, the concentration is defined by the number of moles inside an elementary volume. The SI unit for mass flux is kg/m2/s, and we will use a more adapted unit (i.e., mole/ mm2/s). There is a fundamental law that links the mass flux to the concentration gradient called Fick’s law. 5.3.1  Fick’s Law

Fick’s law can be expressed as � J = -D Ñ c



(5.3)

where D is the diffusion constant or coefficient. The SI unit of D is m2/s, the same as for the cinematic viscosity n or the thermal diffusivity a. All of these quantities are coefficients in a transport equation either of concentration, velocity, or enthalpy and characterized by a flux. 5.3.2  Concentration Equation 5.3.2.1  Differential Diffusion Equation

Using Fick’s law (5.3) and evaluating the mass balance of a substance in an elementary volume of carrier liquid, yields the diffusion equation



¶c = div (D Ñc) + S ¶t

(5.4)

In (5.4), the terms S stands for a source or sink term of concentration. For example, if there is a biochemical in some part of the domain, the concentration of substance may locally appear or disappear. We will come back to this point in Chapter 6.

204

Diffusion of Biochemical Species

Equation (5.4) is sometimes called Fick’s second law [3]. In every subdomain where D is constant (does not depend on the spatial coordinates), (5.4) may be rewritten as ¶c = D D c + S ¶t



(5.5)

showing that the diffusion equation of a parabolic nature. Equation (5.5) is a differential equation with a solution that describes the concentration of a system as a function of time and position. The solution depends on the boundary conditions of the problem as well as on the parameter D. If the concentration c in diffusing particles or molecules is small—which is usually the case—the diffusion coefficient D does not depend on c and (5.5) is linear. Note that the magnitude of D is 10-9 m2/s for self-diffusion (diffusion of the molecules of the buffer fluid) and typically 10-11 m2/s for colloidal substances. 5.3.2.2  Diffusion Coefficient

An expression of the diffusion coefficient of a particle in a carrier fluid was first obtained by Einstein. This expression may be derived by two different approaches. The first one is based on thermodynamics: the starting point is the Gibbs free energy [3]. The magnitude of the driving force of diffusion is



Fdiffusion = -

1 ¶µ NA ¶ x

(5.6)

where m is the chemical potential of the of the diffusing species and NA the Avogadro number. Thermodynamics show that

µ = µ 0 + RT ln(γ c)

(5.7)

where g is the activity coefficient. For dilute systems, g = 1 and we obtain, after substitution of (5.7) in (5.6)



Fdiffusion = -

kB T ¶ c c ¶x

(5.8)

Under stationary state conditions, the diffusion force is balanced by the viscous resistance



Fdiffusion = -

kB T ¶ c = Ffriction = CD v c ¶x

(5.9)

where CD is the friction factor and v the stationary velocity. Thus



v=-

kB T ¶ c CD c ¶ x

If we remark that the flux of material through a cross section is � � J = cv

(5.10)

5.3  Macroscopic Approach: Concentration

205

then by comparison with Fick’s law, we obtain the important result kB T CD

D=



(5.11)

The second approach [1] is based on Langevin’s formula (similar to Newton’s law, but with a complementary term for the Brownian motion) � � � dv = - CD v + F(t) m dt



(5.12)

where m is the mass of the particle, and F(t) a randomly fluctuating force. By multiplying (5.12) by x and taking the time average — symbolized by the brackets — we can rewrite (5.12) under the form 2

m<



æ d xö d æ d xö dx x > -m < ç > = - CD < x > + < x F(t) > ç ÷ ÷ dt è dt ø dt è dt ø

However, < x F(t) > = < x > < F(t) > = 0



The isotropic distribution of the energy yields 2

æ d xö 1 1 m< ç > = kBT ÷ 2 d t 2 è ø



then the Langevin equation is reduced to a differential equation m<



d æ d xö dx x > = kBT - CD < x > d t çè d t ÷ø dt

(5.13)

The solution of (5.13) for instances larger than CD/m is [1] < x2 > = 2



kBT t = 2 Dt CD

and D=



kBT CD

5.3.2.3  Anisotropic Media

In free space, diffusion is isotropic. However, in a confined space, diffusion may be anisotropic if the media containing the fluid is anisotropic [4]. Anisotropic media have different diffusion properties in different directions. Some common examples are textile fibers, polymer films and laminated microlayers in which the molecules have a preferential direction of orientation (Figure 5.4). It is also shown [5] that at the very vicinity of a surface, diffusion becomes anisotropic.

206

Diffusion of Biochemical Species

Figure 5.4  Anisotropic media. The preferred direction of diffusion is the x-direction.

The diffusion constant D must be replace by a coefficient matrix [D] where



é D11 [D] = êêD21 êë D31

D12 D22 D32

D13 ù D23 úú D33 úû

(5.14)

The mass flux is then anisotropic and Fick’s law can be written under the form [4] ì Jxü ï ï í J y ý = [D]Ñc ïJ ï î zþ



(5.15)

Finally the diffusion equation is ¶c ¶2 c ¶2 c ¶2 c ¶c = D11 2 + D22 2 + D33 2 + (D23 + D32 ) ¶t ¶y ¶z ¶x ¶y ¶z

+ (D31 + D13)

¶c ¶c + (D12 + D21) ¶z ¶x ¶x ¶y

if the Ds are taken constant. We may rotate the axis in order to transform the rectangular coordinates (x,y,z) into the rectangular coordinates (x,h,x) characterizing the principal axes of diffusion and the preceding equation becomes



¶c ¶2 c ¶2 c ¶2 c D = D1 2 + D2 + 3 ¶t ¶ξ ¶ η2 ¶ζ 2 It is possible to make the further transformation



ξ1 = ξ

D D D , η1 = η , ζ1 = ζ D1 D1 D1

where D is arbitrary, to obtain

(5.16)

5.3  Macroscopic Approach: Concentration



207

é ¶2 c ¶c ¶2 c ¶2 c ù =Dê + + ú 2 ¶t ¶ η12 ¶ ζ 12 û ë ¶ ξ1

(5.17)

Equation (5.17) has an isotropic value for the diffusion constant D to the price of a rotation plus a homothetic transformation of the axes. In Section 5.3.8, for reasons that we discuss, we proceed differently. We modify the computational domain by a homothetic transformation to the price of a change of the isotropic diffusion coefficient into a matrix of anisotropic diffusion coefficients. 5.3.3  Spreading from a Point Source — 1D Case

We analyze here the diffusion of a substance (tracers or nanoparticles) in a onedimensional geometry. Suppose that a very small spot of concentration of tracer particles has been initially placed in a rectangular capillary of a very small cross section (Figure 5.5). In such a case, the diffusion may be considered one-dimensional and depends one two variables: the time t and the axial coordinate x. The initial condition may be approximated by:

c (x, t0 ) = c0 δ (x)

(5.18)

where d (x)is the Dirac function. With such an initial condition, the solution to (5.5) is x2



c (x, t) =

c0 e 4 Dt 4 π Dt

(5.19)

The solution (5.19) shows that the distribution profile of concentration in tracers is Gaussian with x. In Figure 5.6, we have plotted the solution of (5.19) with the scaling c/c0, for D = 10-10 m2/s at three different times (0.2, 1, and 10 seconds). x2 appears in the soluRemark that the characteristic nondimensional group 4 Dt tion (5.19) to (5.5). This group represents in fact a characteristic diffusion length. A characteristic diffusion length may be defined by

xc » 4 Dt

Figure 5.5  Schematic view of the diffusion of tracers in a one-dimensional geometry.

(5.20)

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Diffusion of Biochemical Species

Figure 5.6  Gaussian profiles of diffusion from a point source according to (5.19).

In the preceding example, one finds by using (5.20): xc(t = 0.2) ~10 mm; xc(t = 1.0) ~20 mm, and xc(t = 10) ~60 mm. 5.3.4  Semi-Infinite Space: Ilkovic’s Solution

It is seldom that the diffusion equations (5.4) or (5.5) can be solved analytically. There are some one-dimensional cases where an analytical solution may be found (we have seen one in the preceding section); but usually, as soon as the geometry of the diffusion problem is two-dimensional, or if the one-dimensional problem presents complex boundary conditions, the use of a numerical approach is required to solve the diffusion equation (5.4). We expose here the analytical solution of the diffusion equation in the simple case of diffusion of species in a half space. Suppose a half space with an initial concentration c0. Suppose also that any microparticle or macromolecule that contacts the solid wall limiting the half-space domain is immediately immobilized. Then the concentration at the wall is zero at any time. The solution for the concentration equation is then



æ x ö c = c0 erf ç ÷ è 4 Dt ø

(5.21)

where x is the distance from the wall, and the error function erf is defined by x



2 2 erf (x) = e - u du ò π 0

This function has the characteristic values: erf (0) = 0  et  erf (¥) = 1 and its d erf 2 - x2 derivative is = e . Thus, the derivative of (5.21) is dx π

5.3  Macroscopic Approach: Concentration

209 x2



¶c 2 - 4 Dt = c0 e ¶x π

1 4 Dt

With this in mind, the mass flux per surface unit, given by Fick’s law, may be written under the form



� ¶c 2 J = -D Ñ c wall = -D = -D c0 ¶ x x=0 π

1 ˆ i = - c0 4 Dt

D ˆ i πt

(5.22)

where iˆ is the unit vector perpendicular to the wall. This latter relation is called the Ilkovic’s solution to the diffusion problem. It shows that the mass flux is proportional to the concentration far from the wall and to the square root of the diffusion coefficient. Strictly, from (5.22) the initial mass flux is infinite. In fact, from a practical point of view, such a situation is not possible: there is always a transition time during which the fluid with the concentration c0 is brought into contact with the wall and the initial time for diffusion is always approximate. 5.3.5  Example of Diffusion Between Two Plates

In this section we show the limitation of the Ilkovic’s solution for the problem of diffusion between two plates. Suppose that we insert a small volume of liquid (V = 2 ml) between two parallel horizontal glass plates (Figure 5.7) separated by a distance of 270 mm. The liquid contains nanoparticles (hydrodynamic diameter DH = 100 nm, diffusion coefficient D = 0.21 10-11 m2/s) in concentration c0. At the beginning, the particles are uniformly dispersed in the liquid at rest; progressively the particles closer to the walls are immobilized by contact with the walls under the action of the Brownian motion, and a concentration depletion progresses from the walls towards the drop center. It is possible to count the number of particles immobilized at any time on the photographs of the upper plate taken under the microscope (Figure 5.8). It may be shown that the particle size is so small that sedimentation can be completely neglected and we assume that there is statistically the same number of particles immobilized on the upper and lower plate. Assuming that the drop is cylindrical (which is the equilibrium shape—see Chapter 3), the diffusion process is governed by the two-dimensional axisymmetrical equation

Figure 5.7  Schema of the drop and the two glass plates.

210

Diffusion of Biochemical Species

Figure 5.8  Photographs of the upper plate taken under the microscope at 5 mn and 15 mn.

æ 1 ¶ c ¶ 2c ¶ 2c ö ¶c = div (D gradc) = D ç + + ÷ ¶t è r ¶ r ¶ r2 ¶ z2 ø



(5.23)

In this particular case, we can express the concentration in particles per unit volume, and the mass flux at the wall J is then expressed in particles per seconds per unit surface. The mass flux at each wall (defined by z = 0 and z = e) is given by Fick’s law J = J upper + Jlower = -D



¶c ¶z

z= e + D

¶c ¶z

z =0



(5.24)

If we suppose that the radial dimension R of the drop is large in front of the distance between the plates e, we can approach (5.23) by the one-dimensional equation ¶c ¶ 2c = div (D gradc) = D 2 ¶t ¶z



(5.25)

A first approach to the problem consists in solving (5.25) for times less than (e 2)2 . For these times, the depletion of the concentration has not reached the 4D center plane of the drop. The problem is then similar to that of Ilkovic for each plate, and the solution of (5.24) is τ=



J = 2 c0

D πt

(5.26)

This solution breaks down when the concentration at the center plane starts de2 creasing from its initial value c0. So when time is larger than τ = (e 2) , Ilkovic’s 4D solution is no longer valid. In Figure 5.9, we compare the experimental results (dots) to the Ilkovic’s solution and to the results of a simple 1D numerical scheme (finite differences method). Vertical concentration profiles at different times—obtained by the numerical method—are plotted in Figure 5.10. At the beginning, the profile is still flat at the center with a concentration c0. At times t larger than t, the concentration at the center plane decreases below the value c0.

5.3  Macroscopic Approach: Concentration

211

Figure 5.9  Wall concentration of immobilized nanoparticles (particles/mm2) as a function of time—a comparison between measurements (dots), Ilkovic’s solution, and numerical results. Before (e 2)2 t <τ = = 2166 s, all the results are close together. After the time t, the Ilkovic solution 4D departs from the experimental results whereas the numerical results still agree with the experimental results.

5.3.6  Radial Diffusion

In the case of a purely radial diffusion from a source point or a central sphere (Figure 5.11), the diffusion equation is



é ¶2 c 2 ¶ c ù ¶c =Dê 2 + ú ¶t r ¶r û ë¶r

Figure 5.10  Vertical profiles of concentration c/c0 versus time.

(5.27)

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Diffusion of Biochemical Species

Figure 5.11  Radial diffusion from an initial spherical volume.

On putting [4]

u = c r

(5.28)

¶u ¶2 u =D 2 ¶t ¶r

(5.29)

(5.27) becomes



This equation is formally the same as the one-dimensional diffusion equation, and can be solved by using the same methods. 5.3.6.1  Steady State Solution

In the case of a steady state problem, (5.29) becomes d æ 2 dc ö r =0 d r çè d r ÷ø



(5.30)

of which the general solution is c=



A +B r

(5.31)

where A and B are constants to be determined from the boundary conditions. 5.3.6.2  Transient Solution

In the case of a spherical surface with a constant concentration c1 (r = a) and an external concentration c0 (r > a), the solution is derived directly from the onedimensional Ilkovic’s solution

5.3  Macroscopic Approach: Concentration

213

Figure 5.12  Biodiagnostic detection device. Left: view of the main and detection microchambers. Right: enlargement of the detection chamber (courtesy of LETI/Biomérieux).



c - c0 aæ r-a ö = ç 1 - erf ÷ c1 - c0 r è 4 Dt ø

(5.32)

5.3.7  Diffusion Inside a Microchamber

The standard procedure for biodiagnostic DNA recognition is the polymerase chain reaction (PCR). However, recently there has been development of new microdevices to directly detect DNA by fluorescence in microchambers (Figure 5.12). The principle is to bring the DNA strands inside the microchamber (for example using magnetic particles) and then let them diffuse so that the DNA strands can hybridize on a labeled surface. Because the system must be very sensitive and work with very few DNA strands, it is important to block any back diffusion towards the inlet channel. A very simple analysis of the diffusion inside the microchamber may be done by considering the diffusion equation in the 2D geometry defined by Figure 5.13, and using standard numerical techniques. The results presented in Figure 5.14 have been

Figure 5.13  Schematic view of the computation domain.

214

Diffusion of Biochemical Species

Figure 5.14  Diffusion of macromolecules initially concentrated in the middle of the detection microchamber (the macromolecules are released from the aggregate of magnetic beads). The right side of the chamber is considered an exit towards the inlet channel.

obtained by discretization of the diffusion equation by using a Crank-Nicholson formulation [6]. The 2D diffusion equation may be written under the form n +1 n +1 n +1 n +1 n +1 n +1 cin, +j 1 - cin, j D é ci +1, j - 2 ci, j + ci -1, j ci , j +1 - 2 ci , j + ci , j -1 ù ú = ê + 2ê Dt (D x)2 (D y)2 úû ë

+

n D é ci +1, j

ê 2ê ë

- 2 cin, j + cin-1, j 2 (D x)

+

cin, j +1

(5.33)

- 2 cin, j + cin, j -1 ù ú 2 (D y)

úû



where i and j are the indices corresponding to the space location and n is the time index. Equation (5.33) can be written in a matrix form and inverted to obtain the solution for the concentration at each time step tn at every point of the computational grid. An analysis of the results shows that it is necessary to block back diffusion inside the inlet channel. This is done by injecting air in secondary reservoirs—by thermal expansion, for example (Figure 5.15). 5.3.8  Diffusion Inside a Capillary: The Example of Simultaneous PCRs 5.3.8.1  Introduction

In order to parallelize some biological operations [like polymerase chain reaction (PCR)] it has been thought to perform these operations at the same time at different

5.3  Macroscopic Approach: Concentration

215

Figure 5.15  Enlarged view of the detection chamber and diffusion blocking reservoirs (courtesy of LETI/BioMérieux).

locations in a capillary tube [7]. In this example of the simultaneous PCRs, the biological sample to be analyzed is brought into the tube under the action of capillary forces (or by pipetting). At the solid wall, at different locations, different primers (reverse and forward) have been grafted (Figure 5.16). When the liquid has filled the tube and is at rest, the primers are released by optical methods (insolation) and then diffuse locally inside the tube (Figure 5.17). The presence of both reverse and forward primers is necessary for the PCR amplification. In the regions of the tube where a sequence of the DNA contained in the liquid is corresponding to a type of primer, DNA amplification occurs and detection is made by fluorescence methods. Within the time frame of the PCR cycles for amplification, the primers should not diffuse to the next region in a substantial way (Figure 5.18). If they do, detection would be inaccurate. Thus, it may be necessary to introduce neutral gaps between the primers regions so that there is no diffusion of primers between two regions. However, the aim is a compact microsystem and these gaps should be reduced as much as possible. The problem consists of calculating the primers diffusion inside the tube and to determining their concentration as a function of time.

Figure 5.16  Schematic view of the capillary with the different labeled regions.

216

Diffusion of Biochemical Species

Figure 5.17  Schematic view of the primers concentration and the PCR reaction regions (drawing not to scale).

In the following we present two approaches to calculate the diffusion inside the capillary. The first one is an analytical approach, based on the simplification that the concentration in primers—after a few seconds—is uniform in a cross section of the tube. The second approach consists in numerically solving the diffusion equation after it has been rendered nondimensional. The advantage of this second solution is that it gives insight on the diffusion barrier. 5.3.8.2  Analytical Model for Diffusion Inside a Tube

First, taking into account that the ratio L/R between the tube length and the tube radius is large, we show that we can assume the concentration in primers uniform in any cross section in a very short time after their release from the wall. A characteristic diffusion time for a primer to diffuse on the length R is



τ»

R2 4 D

(5.34)

Primers have a diffusion coefficient of the order D = 10-10 m2/s, and suppose that the radius of the tube is R = 50 mm. The characteristic time is then t = 6 sec-

Figure 5.18  Schematic view of the concentration profile of the different PCR reaction products.

5.3  Macroscopic Approach: Concentration

217

onds. During the time t, axial diffusion is not significant as we can show by using a very simple Monte-Carlo approach [Figure 5.19(a)] and is confirmed by experimental observations [Figure 5.19(b)]. We may assume a uniform concentration in primers in any annular volume delimited by the initial functionalized regions (volume V = p R2 a, where a is the length of a region). After the time t, the primers are homogeneously scattered in the different annular volumes. The second step consists in calculating the concentration of the primers as a one-dimensional diffusion phenomenon. For each primer i, we have ¶ci ¶ 2c = Di 2i ¶t ¶z



(5.35)

An analytical solution to (5.35) is given by the following combination of error functions [4] ci =

é æ a -z ö æ a + z öù 1 + erf ç i c0,i êerf ç i ÷ ÷ú 2 è 2 Di t ø úû ëê è 2 Di t ø

(5.36)

In Figure 5.20, concentration profiles of two neighboring primers with different coefficients of diffusion have been plotted. It is immediately seen that a spacer gap must be introduced between the two regions to prevent cross mixing. The advantage of the analytical solution is to produce an expression of the required spacing between the functionalized regions. Suppose that the concentration

Figure 5.19  (a) Diffusion of primers from the wall after 6 seconds (obtained by a Monte-Carlo simulation). The starting location of the primers has been randomly chosen on the walls. The diffusion outside the annular volume is negligible in this time interval. (b) The experimental view of diffusing fluorophores.

218

Diffusion of Biochemical Species

Figure 5.20  Concentration profiles at different times. In the present case, the two types of primers have different diffusion coefficients.

in primers i in region i + 1 should not be larger than a threshold concentration cmax, at a time tf defined by the kinetics of amplification, then the minimum distance between the two regions i and i + 1 is given by the implicit relation



é æ ö æ a + zmin ö ù 2 cmax ê a - zmin ÷ + erf ç i = erf ç i ÷ú ê ç 2 Di t ÷ ú c0,i 2 D t i f è ø f ø ë è û

(5.37)

The solution of (5.37) requires finding the zero of a function, which is a standard procedure in most mathematical software. 5.3.8.3  Dimensional Analysis

The analytical method is a fast and simple method to find an approximate solution to the problem. However, a dimensional analysis reveals more of the physics of axial diffusion and will be the basis for a numerical approach. Start from the axisymmetrical diffusion equation for each primer



æ 1 ¶ c ¶ 2c ¶ 2c ö ¶c = div (D gradc) = D ç + + ÷ ¶t è r ¶ r ¶ r2 ¶ z2 ø

(5.38)

Remark that the capillary length L is very large before the capillary radius R. If we want to set up a numerical calculation, we have to deal with a computational domain with a very large aspect ratio L/R. We can introduce the new variables



z* =

z , L

r* =

r R

(5.39)

so that the transformed computational domain is defined by L* = 1, R* = 1. Let’s introduce the other nondimensional variables:

5.3  Macroscopic Approach: Concentration

c* =

219

c , c0



t* =

t RL D

(5.40)

It is straightforward to see that the nondimensional diffusion equation is



¶ c* L æ 1 ¶ c* ¶ 2c* ö R ¶2c* = ç + ÷+ ¶ t* R è r* ¶ r* ¶ r*2 ø L ¶ z*2

(5.41)

This equation is an axisymmetrical diffusion equation with the anisotropic diffusion coefficients D*z =



R , L

D*r =

L R

(5.42)

In order to compensate for the change in geometry, the diffusion coefficients are now strongly anisotropic; the equivalent diffusion coefficient in the direction r is large whereas that in the direction z is small. The ratio between the r and z diffusion coefficient is D*r L2 = >> 1 D*z R2



Remark that (5.41) verifies Buckingham’s Pi theorem [8]. There are four independent parameters in (5.38): c0, L, R, and D. These parameters are measured with three different units: kilos or moles (if we count the concentration in kilos or moles), meters, and seconds. According to Buckingham’s theorem, there should be a 4 - 3 = 1 dimensionless parameter in the nondimensional equation. This parameter is evidently L/R. 5.3.8.4  Numerical Solution

Equation (5.41) may be solved by using a standard finite element method. The computational domain is defined by r* Î {0,1}, z* Î {0,1}. At the wall, the condition ¶ c* ¶c of impermeability yields * r* =1 = = 0. An initial condition c*0 = 1 is im¶ r r =R ¶r posed in a very small volume at the periphery of the tube. Concentration contours obtained at two different times are shown in Figure 5.21. Note that the nondimensional calculations results have to be transformed back into dimensional values. It is a straightforward process if we use (5.39) and (5.40), and if we remark that the initial concentration c0 is obtained by converting the surface concentration in immobilized primers into a volume concentration. 5.3.8.5  Diffusion Barriers

It can be checked that the analytical and numerical results agree (Figure 5.22). Most of the time analytical results when sufficiently accurate should be preferred. However, in the present case, the numerical approach—although more complex to set up—is more powerful. For example, it is possible to investigate whether axial

220

Diffusion of Biochemical Species

Figure 5.21  Concentration contours at two different times calculated with the numerical software COMSOL [9], showing the same behavior as that of the analytical approach.

diffusion inside the tube can be reduced. Although this problem is relevant to Chapter 6 (biochemical reactions), it is a diffusion barrier problem and we will mention it here. We found that if the gap between the annular regions initially labeled with the primers was adequately functionalized, the recapture of the primers during diffusion would limit the axial diffusion. In Figure 5.23, we have compared the axial diffusion of primers for a simple, or labeled, gap. 5.3.9  Particle Size Limit: Diffusion or Sedimentation

The following question stems out of an inspection of (5.5): does the gravity force— which is not present in the equation—affect the diffusion of the microparticles? In

Figure 5.22  Comparison of concentration profiles between analytical and numerical model after the same time interval.

5.3  Macroscopic Approach: Concentration

221

Figure 5.23  Comparison of the concentration profiles between a labeled and a nonlabeled gap. Axial diffusion is remarkably reduced if the gaps are labeled.

other words, is the apparent weight of the particles negligible? One can conceive easily that if the particles are small enough they will not sediment and they will diffuse in the available volume; if they are sufficiently large—like cells or bacteriae—they will tend to sediment despite the molecular agitation [3]. We derive here a criterion to estimate the sedimentation of the microparticles and to decide if the diffusion equation is valid. The settling velocity is defined as the uniform vertical velocity of a particle in a liquid at rest. The settling velocity can be calculated by writing the balance between gravitational force and hydrodynamic drag. Let CD be the friction factor (hydrodynamic drag coefficient), CD is defined by

CD = 6 π η RH

(5.43)

Then, the hydrodynamic drag force on the particle is

Ffriction = CD v = 6 π η RH v

(5.44)

And the settling velocity VS is obtained by the force balance

CD VS = Dρ gVolP

(5.45)

where Dr is the buoyancy density (difference between the volumic mass of particle and liquid), h the dynamic viscosity of the fluid, and g the gravity acceleration (9.8 m/s2). For a spherical particle, the sedimentation velocity is given by



VS =

2 Dρ g R2 9 η

(5.46)

Suppose now that a typical dimension of the problem is d. For example, the vertical dimension of a biodiagnostic microchamber is of the order of d = 50 mm. Lets compute the times t1 and t2 for the particle to move on the distance d by sedimentation and Brownian motion

222

Diffusion of Biochemical Species



τ1 =

d VS

τ2 »

d2 4D

and

then, the ratio b =t1/ t2 is

β=

τ1 d 4D kBT k T 1 » =4 =4 B τ 2 VS d 2 D ρ g d Vol p Dm g d

(5.47)

If we introduce the characteristic Boltzmann length scale L = kBT/g D m [5], (5.47) becomes β=4



L d

The ratio (5.47) represents an energy ratio between the energy of the Brownian motion and the potential energy of the particle. If b << 1, sedimentation dominates and the particles will descend with the settling velocity. As a general rule, particles larger than 1 mm tend to be affected by sedimentation. For example, cells usually sediment because their size is larger than 10 mm. A more detailed analysis may be found in [3]. The preceding analysis shows that it is very important not to let the particles aggregate. If so, the Brownian motion ceases and the aggregate will sediment. This is why surfactants or PEG (polyethylene-glycol) [10] are usually added to buffer liquids.

5.4  Microscopic (Discrete) Approach In the preceding sections, we have followed an approach based on the continuum: this approach assumes that in every elementary volume of liquid there is a number of microparticles sufficiently important to define a concentration. This approach is very convenient because it introduces a partial differential equation (PDE) that can be solved by usual discretization techniques, like the finite element method or the finite volume method. However, very complex geometries of the diffusion domain may not be easily treated with such methods. We present here another approach very useful when dealing with microscopic scales. Contrary to the preceding continuum approach, this approach is discrete, meaning that the displacement of every particle is calculated. This approach is well adapted to complex geometrical domains and small number of particles. 5.4.1  Monte Carlo Method

The Monte Carlo method is based on the mimicking of the random walk of particles. Because the mean free path is very short, it is possible without changing the

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223

statistical randomness to allow for longer linear displacement steps, with the condition that they remain small compared to the free space defined by the surrounding geometry. 5.4.1.1  Two-Dimensional Case

In the two-dimensional case, a particle moves in a time step Dt from the location (x, y) to the location (x + Dx, y + Dy), where the space increments are defined by D x = 4 D D t cos(α ) D y = 4 D D t sin(α )

(5.48)

α = random (0, 2π ) In (5.48) the function “random” is a choice of uniformly distributed random numbers in the interval [0, 2p]. The validity of the method depends on the quality of randomness of the angle a. In the following we have used the Matlab command “rand” [11]. The length of the displacement has been scaled by the real diffusion length scale. Example of Random Walk from a Source Point

We show here two examples of random walk calculations. The first case is the 2D diffusion from a source point (Figure 5.24). Using the algorithm defined by (5.48), we find the images in Figure 5.25. Diffusion is isotropic around the initial spot and has a Gaussian shape along a radius. In Figure 5.26, it can be verified that the average square distance is related to the time by the relation

< d 2 > = 4 Dt Example of Random Walk in a Microchannel

The same algorithm can be applied to the case of a microchannel. The results are shown in Figure 5.27. In this case also, the particle distribution follows a Gaussian profile.

Figure 5.24  2D diffusion of tracers originated at a source point.

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Figure 5.25  Random walk of 10 nm particles originated at the location (0,0). Left, from top to bottom: trajectories for 1, 10, and 500 nanoparticles in a time interval of 50 seconds. Right, from top to bottom: end point of 1, 10, 500 nanoparticles at t = 50 seconds.

Figure 5.26  Square of average distance of particles versus time. From top to bottom: 1, 10, and 500 nanoparticles. In the third figure at bottom, the curve is nearly linear and its slope is approximately 2.10-4/50 = 400 mm2/s. If we relate this value to the relation = 4Dt, the value of the slope must be 4D and we find D = 10-10 m2/s, which is the input value in the model.

5.4  Microscopic (Discrete) Approach

225

Figure 5.27  Monte-Carlo diffusion in a quasi-1D geometry. Left, from top to bottom: trajectories of 1, 10, and 500 nanoparticles in a time interval of 50 seconds. Right, from top to bottom: end point of the trajectories at t = 50 seconds.

5.4.1.2  Three Dimensional Case

In the three-dimensional case, a particle moves in a time step Dt from the location (x, y, z) to the location (x + Dx, y + Dy, z + Dz). The random walk algorithm is D x = 4 Ddt cos(α ) sin(β ) D y = 4 Ddt sin(α ) sin(β ) D z = 4 Ddt cos(β)

(5.49)

α = random (0, 2π)

β = arccos (1 - 2 random (0,1)) The angles a and b are defined in Figure 5.28. Recall the definition of the angle b in (5.49). If we had taken simply b = random(0, 2p), the z-direction would be a preferred direction of displacement. If we want a uniformly distributed direction angle, we have to take a random a angle between 0 and 2p and a random z coordinate between –1 and +1. This random z-coordinate is obtained by the function 1-2 random(0,1) and the angle b is equal to arcos(1-2 random(0,1)).

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Figure 5.28  Schematic view of the linear motion M1M2 during the random walk.

5.4.2  Diffusion in Confined Volumes: Drug Diffusion in the Human Body 5.4.2.1  Introduction

One of the most useful applications of the Monte-Carlo method is the diffusion inside a confined domain. It is striking to see how diffusion occurs in very small volumes. In chemistry, there is the example of a solid alloy composed of aluminum with lead inclusions. At a temperature where the lead is molten and the aluminum still solid, one can follow the random walk of the lead molecules inside the aluminum matrix (Figure 5.29). In biology, there are many examples of diffusion in very confined media: proteins diffuse in cells [13], and macromolecules diffuse in cells’ interstitial spaces. Due to its importance, diffusion processes in confined media is the object of many studies, and there is abundant literature on this topic. We present here an important example in biology—that of diffusion in extracellular spaces of cell clusters.

Figure 5.29  Example of very confined diffusion. Lead molecules diffusing inside an aluminum matrix [12].

5.4  Microscopic (Discrete) Approach

227

Figure 5.30  Geometry of extracellular space from [15]—an electromicrograph of a small region of rat cortex. The ECS is in dark on the picture. “Lakes” or intercleft spaces can be seen at the bottom right where the extracellular space widens.

In the biological field, the delivery of drugs in the human body, especially in clusters of cells, is an utmost important problem requiring the knowledge of diffusion in a complex, confined geometry. In this case, there are two different types of media: the extracellular space (ECS) and the cells; these two types are separated by the cell membrane. Diffusion of biological molecules first takes place inside the extracellular space. After the molecules have penetrated the cells through the cell membranes (which is called uptake), the molecules diffuse inside the cells. In this section, we focus on the diffusion inside the ECS. Because cellular uptake is not immediate, the biological cluster of cells may be seen as porous media, where the cells are the “solid grains” and the extracellular space (ECS) the “pores” (Figures 5.30 and 5.31). In the particular case of tumoral cells, the extracellular path is called the tumor interstitial matrix (IM) [14].

Figure 5.31  Cell arrangement in the human skin from [16]. The shape of the cells is regular, but the anisotropy of the ECS changes from top to bottom. The typical width of the ECS is a few microns.

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We will show in the following that diffusion in the ECS is much slower than free diffusion. It is common to use an apparent (or effective) diffusion coefficient (ADC) to determine the speed of diffusion of drugs in the tumor ECS [17]. Speed of diffusion based on the apparent diffusion coefficient is equal to that of the real diffusion coefficient in the restricted geometry of the ECS. The apparent diffusion coefficient depends on the morphology of the ECS, especially on the tortuosity (Figure 5.32)— the ratio of the real distance to the straight line distance between two points—and also to special features of the ECS like intercleft spaces and constrictions. Real drug delivery time will be determined by adding to the diffusion characteristic time in the ECS the uptake characteristic time (time for the macromolecule to enter the cell), plus the diffusion time inside the cell [13]. Remark that it is of great importance in cancer treatment to be able to estimate the value of the apparent diffusion coefficient [18]. If the delivery time is too long—it may reach 48 hours—or if some cells are not delivered, the balance between destruction and multiplication of cancerous cells will be unfavorable. Note also that in some cases, a change of the ADC reflects a change in the cells shape and arrangement [19], so that an evolution of a disease may be followed. We suppose that the fluid flow in the ECS is negligible in front of the molecular diffusion, so we have to calculate the diffusion of a substance in a very complex geometry. Different types of numerical approaches have been proposed for regular repetitive patterns like squares and triangles: the homogenization theory [20], which is based on the calculation of the diffusion in a motif and extending the result to the whole domain, and the Monte Carlo method [21] in geometry where the boundaries are defined by analytical linear functions. It is thought that at a certain point regular patterns calculation could be sufficient to approximate an average ADC [22]. However this is not always the case if the ECS has intercleft spaces or constrictions, especially if one wants to estimate the local uptake rates [23] or if any change in cell shape and arrangement takes place [20]. So far, there have been

Figure 5.32  Schematic view of the diffusion paths in a porous media depending on the tortuosity.

5.4  Microscopic (Discrete) Approach

229

very few investigations for irregular and disordered clusters, mostly because of the difficulty in describing the geometry [24]. However, recently progress has been made to tackle the problem of diffusion in the ECS of clusters of cells. In the following, we present a 2D numerical approach based on a two-step calculation: first, the calculation of the cells boundaries, and second a Monte-Carlo numerical scheme for the diffusion in the ECS morphology defined in the preceding step. 5.4.2.2  Cell Boundaries

First, cell arrangement may be mimicked: cells rearrange inside the boundaries of the cell cluster in a function of constraints like the surface tension of the membranes and their volume (depending on the growth or the shrinking of cells). A numerical software like the Surface Evolver [25] (see Chapter 3), is well adapted to calculate the morphology of the cell cluster. In order to describe a cell cluster morphology—a given set of points (vertices)—segments (edges) delimiting the initial cells are introduced in the Evolver numerical program. Depending on line (surface) tensions and cells volumes, the shape of the cells evolves until the convergence to a minimum energy arrangement, mimicking real cell arrangement. It is assumed here that cell membranes behave similarly to an interface with surface tension. The initial edges are then refined and deformed depending on the specified constraints. A calculated arrangement of cells mimicking a real cluster of cells has been plotted in Figure 5.33. In the Evolver approach, the computational nodes are located on the cells edges and are referenced by their coordinates (x,y) and by the corresponding edge number. Besides, each cell is referenced by its oriented edges. In order to prepare step 2, this information is memorized and stored. 5.4.2.3  Monte Carlo Numerical Scheme

Particles—or macromolecules—are initially placed in a central microregion, simulating the injection point at the tip of the microneedle. Diffusion is then simulated by following the particles execute random walks inside the ECS. In a two dimensional system, the displacement (Dx, Dy) of any particle in the time step Dt is given by the relations

Figure 5.33  (a) Cell arrangement calculated with the Evolver numerical program; (b) cell cluster morphology is enlightened by the calculated location of pharmaceutical molecules after they have diffused in the ECS; (c) real cell cluster observed by fluorescence imaging.

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D x = 4 D D t cos(α ) D y = 4 D D t sin(α )

α = random (0, 2π) where D is the “free” diffusion coefficient, given by Einstein’s law:



D=

kB T 6 π η RH

where kB is the Boltzman constant (1.38 10-23 J/K), T the temperature (K), h the dynamic viscosity of the carrier fluid, and RH the hydraulic radius of the particle. 5.4.2.4  Uniformly Narrow ECS

Suppose first that the width of the ECS is approximately constant (as in Figure 5.33). The cell edges defined in the preceding step are widened to the desired width in order to define a real ECS. Particle location inside the cluster is permanently tracked and the particles are not allowed to cross the solid (cell) boundary. A random walk of particles may then be confined inside the ECS as shown in Figure 5.34. If the time allowed for the calculation is sufficiently large, the ECS is explored by the diffusing particles as shown in Figure 5.35. In a porous media, the distance between two points may be defined as the length of the shortest line in the fluid domain between two points (Figure 5.36), and tortuosity is then defined as the ratio between the distance in the liquid and the straight line distance. It may be theoretically shown [20] that for any 2D regular isotropic lattice of convex cells, tortuosity has a unique value

τ = 2

(5.50)

Figure 5.34  Random walk of two particles inside an ECS calculated by a Monte-Carlo method and constrained by ECS boundaries.

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231

Figure 5.35  Random walk of 200 particles inside the ECS of the cluster.

and for 3D lattices, the value of the tortuosity is

τ = 3

(5.51)

It is relatively easy to be convinced of the validity of (5.50) and (5.51). Because the media is isotropic, we can estimate the tortuosity on a diagonal direction (Figure 5.37). We can approximate the length LAB by a pixel discretization: By projection on the horizontal and vertical axis, we obtain

LAB = n D x + n D y = 2 n D x On the other hand, the Pythagore relation is



dAB2 = (n D x)2 + (n D y)2 = 2 n 2 D x 2 Combining the two preceding equations yields

Figure 5.36  Definition of tortuosity in regular and irregular lattices. The tortuosity t is equal to the ratio LAB/dAB ; in a free media t = 1.

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Figure 5.37  Estimation of the tortuosity in the case of convex, isotropic, porous media.

τ2 =



LAB2 4 n2 D x2 = =2 dAB2 2 n2 D x2

Finally



τ=

LAB = 2 dAB

The same reasoning may be applied to the 3D case. It has been also shown that there is a very simple relation between the effective and free diffusion coefficient involving the tortuosity



Deff 1 = 2 D τ

(5.52)

Thus, if the width of the ECS is narrow and remains uniformly constant in the 2D cluster, the value of the effective diffusion coefficient is half that of the free diffusion coefficient



Deff 1 = 2 D Assuming the same conditions for the ECS (isotropy and convexity), the MonteCarlo numerical model shows that (5.50) and (5.52) also apply for any isotropic cluster of irregular cells. Figure 5.38 shows the location of the diffusing particles initially starting from the injection point after a time interval of 15 seconds (D = 10-10 m2/s). Let us introduce the normalized diffusion length b by



β=

L 4 Dt

(5.53)

5.4  Microscopic (Discrete) Approach

233

Figure 5.38  Diffusion distance from the point source in an isotropic cell cluster.

where L is obtained by averaging the distance of each particle between their location at time t and at time t = 0 . In Figure 5.39, we have plotted the normalized diffusion length versus time for different cluster morphologies: ordered (square and hexagonal cells) and disordered, with narrow ECS. A narrow ECS is defined here by a constant width less than 1/10 of the average size of the cells but larger than about three times the mean free path of the particles. On very rare occassions, the diffusing particles execute random walk inside a small region of the ECS where the particles are initially placed, so that the value of b is that of free diffusion: b = 1 at t = 0. After a short time, the particles have explored all the available initial space and start diffusing inside the ECS. They are now constrained inside the ECS by the cells’ boundaries, and b reaches a nearly constant

Figure 5.39  Normalized diffusion length versus time for different (regular and irregular) clusters.

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value. The numerical results of Figure 5.39 are very similar to that of de Sousa et al. [26] obtained experimentally for regular triangle and square lattices. The asymptotic value of b is 0.7, thus



β=

L 1 » 0.7 » 4 Dt 2

(5.54)

By definition, the apparent diffusion coefficient satisfies L »1 4 Deff t



(5.55)

From (5.54) and (5.55), we find Deff 1 1 = β2 = 2 » D 2 τ



(5.56)

leading to the value τ = 2. 5.4.2.5  Intercleft Spaces and Channel Restrictions

Real ECSs in the human body are often more complex than those of a uniformly narrow gap between the cells, which we analyzed in the preceding section. Very often the spacing of the cells’ lattice is not uniform and there are intercleft spaces. It is shown that diffusion speed may be reduced by entrapment when the dimensions of the residual spaces are large and the connecting exits are sufficiently small. By “sufficiently small” we mean that the mean free path of the particle inside the carrier fluid is of the order of the cross dimensions of the ECS. An idealized example is that of a cluster of round cells. If the dimension of the gaps between the cells is decreased, the apparent diffusion coefficient becomes smaller than the value predicted by (5.6). In the case defined in Figure 5.40, we D 1 obtain eff » . 4 D

Figure 5.40  Normalized diffusion length versus time for a cluster of round cells with small gaps.

5.5  Conclusion

235

A limiting case is that of a gap width of the order of the mean free path of the particle, in such a case, the particles are trapped inside the intercleft space. The relevant theory is the “percolation theory” and there have been considerable efforts in this domain for biological applications. 5.4.3.6  Conclusion

We have modeled the diffusion of biochemical species in a cluster of cells by a three step algorithm: 1. Evolver generation of arrangement based cluster; 2. Monte Carlo random walk of the diffusing species; 3. particle tracking to constrain the diffusing species inside the ECS. The results of the model show that the ratio between the apparent diffusion coefficient and the free diffusion coefficient in dense cell clusters with small extracellular spacing is always the same, whatever the morphology of the cluster (ordered or disordered). In a 2D cluster Deff 1 1 = 2 » D 2 τ



where t is the tortuosity of the porous media. However the situation is much more complex in the extracellular space of irregular and anisotropic clusters of cells, especially if there exist intercleft spaces. Speed of diffusion can be considerably reduced by particle entrapment in the intercleft spaces or if the desired diffusion direction is not the same as the preferred direction of the anisotropic cluster.

5.5  Conclusion Diffusion is very probably the main phenomenon concerning microparticles and target macromolecules in biotechnological applications. Estimation of diffusion time may be performed by solving the partial differential equation for the diffusion of concentration, or by a discrete approach. The advantage of the first “continuum” approach is the availability of numerical software—finite element method is recommended because it adapts best to the shape of the boundaries—and the relative fast computational time (at least in a two-dimensional case). A discrete approach—like the Monte Carlo method—is perhaps more demonstrative because it mimics the behavior of the particles and is well adapted to very complicated geometries. The drawback of the method is the computational time. For the technological applications, diffusion is at the same time advantageous and not. For example, one takes advantage of the Brownian motion to make molecules recognize each other, which leads to the desired hybrization. However, diffusion may disperse the target molecules or mix these molecules with other undesirable molecules. The art of the design of biotechnological components resides in part in the clever uses of molecular diffusion.

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References   [1] http://scienceworld.wolfram.com/physics/BrownianMotion.html.   [2] Tabeling, P., Introduction à la microfluidique, Belin, 2003.   [3] Hiementz, P. C., and R. Rajagopalan, Principles of Colloid and Surface Chemistry, Marcel Dekker, 1997.   [4] Crank, J., The Mathematics of Diffusion, Second Edition. Oxford University Press, 1999.   [5] Faucheux, L. P., and A . J. Libchaber, “Confined Brownian Motion,” Physical Review E, Vol. 49, No. 6, 1994.   [6] Press, W.H., et al., Numerical Recipes, Cambridge University Press, Cambridge, 1987.   [7] Chatelain, F., and J. Berthier, “Microfluidic device for performing a plurality of reactions and uses thereof,” PCT/FR2004/01850, 2004.   [8] Buckingham, E., “On Physically Similar Systems: Illustrations of the Use of Dimensional Equations,” Phys. Rev., Vol. 4, 1914, pp. 345–376.   [9] COMSOL reference manual, Stockholm: COMSOL AB, http://www.comsol.com. [10] de Gennes, G., “Polymers at an interface: a simplified view,” Adv. Colloid Interface Sci., Vol. 27, No. 5, 1987, pp. 189–209. [11] http://www.mathworks.com/. [12] Johnson, E., et al., “Nanoscale lead-tin inclusions in aluminium,” Journal of Electron Microscopy, Vol. 51, 2002, pp. S201 –S209. [13] Pollack, G. H., Cells, gels and the engines of life, Ebner and Sons Publishers, 2001. [14] Rumanian, S., et al., “Diffusion and convection in collagen gels: implications for transport in the tumor interstitium,” Biophys. J., Vol. 83, 2002, pp. 1650–1660. [15] Nicholson, C., and E. Sykova; “Extracellular space structure revealed by diffusion analysis,” TINS, Vol. 21, No. 5, 1998, pp. 207–215. [16] M. Martin, “Conséquences d’une irradiation ionisante sur la peau humaine,” Clefs CEA, Vol. 48, 2003, pp. 53–55. [17] J. Lankelma, et al., “A mathematical model of drug transport in human breast cancer,” Microvascular Research, Vol. 59, 2000, pp. 149–161. [18] El-Kareh, A.W., Braunstein, S. L., and T.W. Secomb, “Effect of cell arrangement and interstitial volume fraction on the diffusivity of monoclonal antibodies in tissue,” Biophys. J., Vol. 64, 1993, pp. 1638–1646. [19] Herneth, A. M., Guccione, S., and M. Bednarski, “Apparent diffusion coefficient: a quantitative parameter for in vivo tumor characterization,” European Journal of Radiology, Vol. 45, 2003, pp. 208–213. [20] Chen, K. C., and C. Nicholson, “Changes in brain cell shape create residual extracellular space volume and explain tortuosity behavior during osmotic challenge,” Proc. Natl. Acad. Sci. USA, Vol. 97, No. 15, 1999, pp. 8306–8311. [21] Saxton, M. J., “Lateral diffusion in an archipelago, the effect of mobile obstacles,” Biophys. J., Vol. 52, 1987, pp. 989–997. [22] Blum, J. J., Lawler, G., Reed, M., and I. Shin, “Effect of cytoskeletal geometry on intracellular diffusion,” Biophys. J., Vol. 56, 1989, pp. 995-1005. [23] Szafer, A., Zhong, J., and J. C. Gore, “Theoretical model for water diffusion in tissues,” Magnetics Resonance in Medicine, Vol. 33, No. 5, 1995, pp. 697–712. [24] Berthier, J., Rivera, F., and P. Caillat, “Numerical modeling of diffusion in extracellular space of biological cell clusters and tumors,” Nanotech 2004, Boston, 7–11 March, 2004. [25] Brakke, K. A., “The Surface Evolver,” Experimental Mathematics, Vol. 1, No. 2, 1992, pp. 141–165. [26] de Sousa, P. L., Abergel, D., and J-Y Lallemand, “Experimental time saving in NMR measurement of time dependent diffusion coefficients,” Chemical Physics Letters, 2001, p. 342.

Chapter 6

Transport of Biochemical Species and Cellular Microfluidics 6.1  Introduction In general, biotechnology deals with the manipulation of biological targets, such as DNA strands, proteins, cells, or cluster of cells. One of the goals of biotechnology is the manipulation of very small amounts of targets, even a single target, for example a single cell. To do so, different methods are used successively to allow for more and more selectivity. Figure 6.1 schematizes the different methods from the less selective to the most selective. The first step is the transport by microfluidic means. For example, the targets are to be extracted and concentrated from a liquid sample, or they have to be guided towards a reactive surface, mixed with a reagent, dispersed in another liquid, or transported to a mass spectrometer. In any case, the knowledge of transport mechanism is mandatory. The next steps in selectivity depend on the particular application. Transport phenomena depend on the velocity of the carrier flow and on the size and nature of the biological objects. It is characterized by the Peclet number (Pe) that has been defined in Chapter 1. Figure 6.2 schematizes the different observed behavior. Very small particles diffuse while being transported; their location becomes stochastic. Larger particles have a lesser diffusion and are guided by the carrier flow streamlines. At larger velocities, they gain inertia and can abandon the streamlines when the curvature is important. Still larger (and heavier) particles sediment. We present first the governing equations of transport (advection-diffusion equation) under the continuum assumption and their nondimensional form, which introduces the characteristic Peclet number, and then we analyze some characteristic cases, such as the flow in a microchannel, and present the Taylor-Aris model. This model will lead us to the major problem of mixing in microfluidics. To complete the approach, Langevin’s equation is introduced for particles experiencing a strong Brownian motion and a particle trajectory approach for larger particles less affected by the Brownian motion. Applications of particle trajectory to field flow fractionation and chromatography columns are presented next. Finally, a section is devoted to cellular microfluidics.

6.2  Advection-Diffusion Equation 6.2.1  Governing Equation for Transport

As we have done for the mass conservation equation and for the momentum equation, we write the concentration balance in an elementary volume (Dx, Dy). For 237

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Transport of Biochemical Species and Cellular Microfluidics

Figure 6.1  Schematic approach to micromanipulation of biologic targets.

simplicity, we consider a 2D element, but the reasoning is the same for a 3D volume. The change in the mass of species in the volume is equal to the convective flux balance plus the diffusion flux balance (Figure 6.3). Thus, we can write



¶ Jy ¶c ¶c ¶c ¶J DxDy + u DxDy + v DxDy + x DxDy + DxD= 0 ¶t ¶x ¶y ¶x ¶y

(6.1)

Figure 6.2  Different behaviors for transport of targets: (a) diffusion plays an important role (Pe <<1); (b) targets follow streamlines; (c) inertial effects force targets to abandon the flow streamlines when curved (Pe >>1); and (d) large and heavy targets sediment.

6.2  Advection-Diffusion Equation

239

Figure 6.3  Concentration balance in an elementary volume.

where Jx and Jy are the diffusion fluxes given by the Fick’s law J x = -D

¶c ¶x

¶c Jy = -D ¶y



(6.2)

In (6.2) D is the diffusion constant. Dividing (6.1) by DxDy and substituting (6.2) yields



¶c ¶c ¶c +u +v = Ñ .(D Ñ c) ¶t ¶x ¶y

(6.3)

¶c � + U . Ñc = Ñ .(D Ñ c) ¶t

(6.4)

or



where U is the vector (u, v, w). Recall that the material derivative notation is



D ¶ ¶ ¶ ¶ ¶ = +u +v +w = + U .Ñ Dt ¶ t ¶x ¶y ¶z ¶t

(6.5)

and then (6.3) can be cast under the form



Dc = Ñ .(D Ñ c) Dt

(6.6)

Assuming that D is constant, the advection-diffusion equation becomes



¶c � + U . Ñc = D D c ¶t

(6.7)

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or, in a Cartesian coordinate system,



é ¶ 2c ¶ 2c ¶ 2c ù ¶c ¶c ¶c ¶c +u +v +w = Dê 2 + 2 + 2 ú ¶t ¶x ¶y ¶z ¶y ¶z û ë¶x

(6.8)

Suppose for an instant that the diffusion coefficient D = 0. In such a case Dc =0 Dt



(6.9)

so that the concentration c remains the same along a trajectory (Figure 6.4). This propriety is valid for short times where diffusion process has not had time to smear out the concentration. It is well known that very laminar flows, usually associated with biotechnological devices, are very unfavorable to diffusion, “short” times are often rather long, and mixing devices promoting mixing have been developed to enhance diffusion [1]. Now, let us come back to the general case where the diffusion coefficient is not zero. Let us define a concentration norm in the whole domain occupied by the fluid by the mathematical function Q = ò c 2 dx dy dz



Mathematically, this function represents a measure of the concentration. In the absence of source or sink of substance, it is possible to derive [2] ¶Q = -Dò (Ñc)2 dx dy dz ¶t



showing that Q always decreases with time. Thus the concentration smears out with time (Figure 6.5). If D were negative, which does not happen, the particles would concentrate and there would be antidiffusion, which, of course, does not exist. 6.2.2  Source Terms

If there are concentration source or sink terms, (6.7) becomes

Figure 6.4  Sketches of mass transport in the absence of diffusion. (a) Near a solid wall and (b) in the bulk of the flow.

6.2  Advection-Diffusion Equation

241

Figure 6.5  Sketch of transport of concentration in the real case where D ¹ 0.



¶c � + U . Ñc = D D c + S ¶t

(6.10)

where S is the source term. The terms S in the advection-diffusion equation is a source or sink term depending of its sign. The unit of S in the International Unit System is mole/m3/s or particles/m3/s. S may be a function depending on a volume, a surface, a contour, or a point. Usually creation or removal of concentration of a constituent is linked to chemical or biochemical reactions. In Chapter 7, we shall see some examples of source or sink terms. 6.2.3  Boundary Conditions

Many different forms of boundary conditions (BC) can exist for the advectiondiffusion equation. However, two boundary conditions are remarkable. The first one is the Dirichlet condition c = 0 at a solid wall. This condition means that the concentration of the studied macromolecules or nanoparticles vanishes at the solid wall. This is the case of total adhesion, when any particle of the concentration field that contacts the wall is immobilized and removed from the ensemble of the transported particles (Figure 6.6).

Figure 6.6  Dirichlet condition at a solid wall. Contour plot of concentration and mass flux. The particles are immobilized on the wall upon contact. A boundary layer of concentration develops along the solid wall.

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The second one is the condition is the homogeneous Neumann condition ¶c = 0, where n is the unit vector defining the normal direction to the wall. In ¶n this case there is no mass flux from the concentration field to (or through) the wall and it corresponds to a situation of no adherence of the transported species and the wall (Figure 6.7). Evidently, there exist more complicated boundary conditions, especially when the mass flux to the wall is governed by a biochemical reaction. In such a case, the mass flux at the wall is determined by the chemical reaction rate Jn = - D

where

¶c d G = ¶n dt

(6.11)

dG is the reaction kinetics. The boundary condition is the Neumann condition dt ¶c 1 dG =(6.12) ¶n D dt

and we obtain the scheme of Figure 6.8 where a boundary layer of concentration develops along the wall. This type of problem will be treated in Chapter 7. 6.2.4  Coupling with Hydrodynamics

The advection-diffusion equation is not sufficient to solve the complete problem by itself. The velocity field must be known. The complete formulation is � ¶ρ + Ñ.(ρ U) = 0 ¶t � � � � � ¶U + ρU.ÑU = -Ñ P + ηD U + F ρ ¶t

¶c � + U . Ñc = Ñ .(D Ñ c) + S ¶t

(6.13)



Figure 6.7  Homogeneous Neumann condition at the solid wall. Particles do not adhere on the contact surface.

6.2  Advection-Diffusion Equation

243

Figure 6.8  During some biochemical reactions, micro- and nanoparticles can be temporarily immobilized at the wall until equilibrium is found. Depletion layers similar to concentration boundary layers form in the vicinity of the solid wall.

The system (6.13) is a system of five scalar equations (in the three-dimensional case). There are five unknowns (u, v, w, P, c), two fluid properties r and m, the diffusion constant of the species D, and two external actions on the fluid: the body force par unit volume F and the concentration source or sink per unit volume S. Note that system (6.13) is only a weakly coupled system under the condition that the concentration is sufficiently small to not affect the buffer fluid viscosity and density. In the next section we treat the problem of the variation of the fluid properties with concentration. Usually, in a microfluidics microsystem, the flow of the buffer fluid is permanent (steady state) and only the concentration changes with time. In such a case, if we assume that r, m, and D are constant, and that there are no body forces (gravity is usually negligible in very small systems). In such a case, the system (6.13) collapses to � Ñ.U = 0 � � � 1 U .ÑU = - Ñ P + ν D U ρ

¶c � + U . Ñc = D D c + S ¶t

(6.14)



And if the hypothesis of a creeping flow is valid (i.e., the Stokes approximation is justified), the system (6.14) collapses to the linear system, under the condition that the function S is well behaved � Ñ.U = 0 � ÑP = η D U (6.15)



¶c � + U . Ñc = D D c + S ¶t

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Transport of Biochemical Species and Cellular Microfluidics

In the next section, we analyze the case where viscosity and density are not constant. 6.2.5  Physical Properties as a Function of the Concentration of the Species

When the concentration in transported species becomes substantial, the buffer fluid properties are modified. We analyze next the influence of concentration on viscosity, density, and diffusion. 6.2.5.1  Viscosity

The viscosity of the buffer fluid is a function of the concentration of the suspension. In the buffer liquid flow, micro- and nanoparticles are each animated with a rotation motion, so that molecular vortices form inside the buffer fluid by entrainment of the surrounding fluid. The result is an increase in viscosity of the fluid. A relative viscosity may be defined as ηr =



η η0

(6.16)

where h0 is the viscosity of the buffer fluid (with no particles) and h is apparent (real) viscosity, and a specific viscosity by ηsp =



η - η0 η0

(6.17)

The specific viscosity changes with the volume fraction of particles defined by



φ=

volume of particles volume of the fluid

(6.18)

For very dilute solutions, in which it can be assumed that the transported particles are independent (i.e., do not interact), the specific viscosity is given by Huggins’s law [3, 4]

ηsp = [η ] ϕ

(6.19)

and the apparent viscosity is then

η = η0 (1 + [η] ϕ)

(6.20)

In (6.19) and (6.20), [h] is the intrinsic viscosity which depends on the type of the particles. For spherical particles, the value of k is approximately [h] = 5/2 (Figure 6.9). Relation (6.20) is valid only for relatively small volume fraction. It is well known that there is a packing fraction—of the order of f = 0.65—at which the viscosity becomes infinite and the carrier liquid cannot flow anymore. In such case the relation (6.20) is just the linear part of the more complete relation

ηsp = [η ] ϕ + k [η]2 φ 2 + ...

(6.21)

6.2  Advection-Diffusion Equation

245

Figure 6.9  Relation between specific viscosity and volume fraction.

In (6.21) the constant k is called the Huggins’s constant. Relation (6.21) is represented in Figure 6.10. A rapid and approximate calculation shows that for many applications in biotechnology—such as the transport of DNA—the volume fraction of target macromolecules or nanoparticles is small. Suppose a concentration of substance c0 is expressed in M (mole/liters). Its value in mole per cubic meters is 103 c0. If we note RH, the hydraulic radius of a single element of the substance, then the volume of this element is V = 4/3pRH3 and the volume fraction of the substance is

ϕ = 103 c0 A VDNA

(6.22)

where A is the Avogadro number (A = 6.02 1023). Typically for DNA analysis, the maximum concentration is 1 mM, and by taking an approximate hydraulic radius of RH = 20 nm = 20 10-9 m, relation (6.22) gives the maximum volume fraction of f = 0.02. The value of the specific viscosity is then only 5%. However, there is an exception. With the development of cellular microfluidics, polymeric solutions are increasingly used, and the viscosity of the solution is

Figure 6.10  Apparent viscosity versus volume fraction of particles.

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Transport of Biochemical Species and Cellular Microfluidics

considerably changed by the presence of the polymers. At zero shear rate, the viscosity is given by an expression of the type η = ηs [1 + a (c [η ])n ]



Above the threshold c* = 1/[h], the effect of the transported polymers becomes important. This point is illustrated in Figure 6.11. 6.2.5.2  Diffusion Constant

In Chapter 5, we introduced the Einstein relation for the diffusion of the particles D=



kB T 6 π RH η

(6.23)

Substituting (6.19) in (6.23) yields D(φ) =

kB T 6 π RH



1 5 ù é η0 ê1 + φú 2 ë û

=

D0 5 1+ φ 2

(6.24)

The relative change in D is D(φ) - D0 5 »- φ D0 2



For the typical value f = 0.02, the relative change of D is -5%. 6.2.5.3  Density

The density of the buffer fluid is a function of the concentration. Using the definition of the volume fraction, the density is

ρ = φ ρP + (1 - φ) ρL

(6.25)

Figure 6.11  Comparison between concentration free and a concentration of polymers (alginate) c = 2 g/l. Pin = 100 Pa, w = 100 mm, d = 100 mm (COMSOL).

6.2  Advection-Diffusion Equation

247

The relative change in r is ρ - ρL ρ - ρL =φ P ρL ρL



For the typical value f = 0.02, rP= 2,000 and rL =1,000 kg/m3, the relative change is 2%. 6.2.5.4  Transport System of Equations

In the case where the concentration effect on the fluid properties is not negligible, the advection-diffusion equation is not decoupled anymore. It has been seen in Section 6.1.5.1 that concentration is proportional to volume fraction. Taking advantage of the linearity of the transport equation, we obtain the following system for a creeping flow (Stokes hypothesis) � Ñ.U = 0 � (6.26) Ñ P = η DU ¶φ � + U . Ñφ = D D φ + S ¶t



In the case where the concentration of species is sufficient to affect the properties of the liquid, we have to solve (6.26) using the constitutive relations 5 ù é η(φ) = η0 ê1 + φú 2 ë û D(φ) =

kB T 6 π RH

1 5 ù é η0 ê1 + φú 2 û ë

=

D0 5 1+ φ 2

(6.27)



The system is now strongly coupled and no more linear due to terms of the form 1 D φ. The numerical solution requires a coupled multiphysics approach where the φ unknowns are the vectors (u, v, w, P, f) at each node of the computational domain and the use of a nonlinear solver. 6.2.6  Dimensional Analysis and Peclet Number

Let us start from the usual form of the diffusion-advection equation without source or sink.



é ¶ 2c ¶ 2c ¶ 2c ù ¶c ¶c ¶c ¶c +u +v +w = Dê 2 + 2 + 2 ú ¶t ¶x ¶y ¶z ¶y ¶z û ë¶x

(6.28)

In this problem, there are four parameters: the velocity U¥, the length scale L, the incoming concentration c0, and the diffusion coefficient D. These four parameters contain three different units: m, s, kg (or mole). In such a case, Buckingham’s

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Transport of Biochemical Species and Cellular Microfluidics

Pi theorem implies that there is a 4 – 3 = 1 nondimensional number that governs the nondimensional equation and characterizes the phenomenon. Suppose that we take for references a velocity U¥, a length scale L, and a concentration c0. Relevant dimensionless variables may be defined as c* =

c * x * y * z * u v w * t ,x = ,y = ,z = ,u = , v* = , w* = ,t = L c0 L L L U¥ U¥ U¥ U¥

(6.29)

Substitution of (6.29) in (6.28) yields



¶ c* ¶ c* ¶ c* ¶ c* D é ¶ 2c* ¶2c* ¶2c* ù + u * * + v* + w* * = + + ê ú * * U ¥ L ë ¶ x*2 ¶y*2 ¶z*2 û ¶t ¶x ¶y ¶z

(6.30)

As was expected from Buckingham’s theorem, only one dimensionless parameter appears in (6.30). This parameter represents the ratio of inertia to diffusion and is referenced by Pe =



U¥ L D

(6.31)

The Peclet number is a key feature in the problems of dispersion under the action of diffusion and advection. We shall see in the following sections many examples where the Peclet number determines the solution of the advection-diffusion problem. Note that the Peclet number may be written as a function of the Reynolds number



Pe =

U¥ L ν = Re Sc ν D

(6.32)

where Sc is the nondimensional Schmitt number. 6.2.7  Concentration Boundary Layer

In Chapter 2, we have seen that the entrance length of microflows in capillary tubes is very short, because the hydrodynamic boundary layer develops and reaches very quickly the middle of the tube. It would be wrong to conclude that the same will happen to mass transfer boundary layer. In fact, the picture looks like that of Figure 6.12 where the hydrodynamic flow is established but not the concentration field. Figure 6.13 shows the calculated mass transfer boundary layer inside a usual detection chamber for a buffer fluid carrying DNA strands. The results are obtained by solving the advection-diffusion equation using a finite difference numerical scheme. It appears immediately that the vertical distance (1 mm) is too important because the compounds carried by the buffer flow (DNA strands) are mostly unaffected by the labeled wall and keep flowing through the chamber. An estimate of the boundary layer thickness may be found by a dimensional analysis. The starting point is the advection-diffusion equation, assuming a steady state concentration field, no source terms, and a two-dimensional problem

6.2  Advection-Diffusion Equation

249

Figure 6.12  Schematic view of the flow and concentration boundary layer in a tube.

u

é ¶ 2c ¶ 2c ù ¶c ¶c +v = D ê 2 + 2 ú ¶x ¶y ¶y û ë¶x

(6.33)

Boundary layer hypothesis assume that the vertical convection term is negligible as well as the axial diffusion term, so we are left with



u

¶c ¶ 2c =D 2 ¶x ¶y

(6.34)

After substitution of the Hagen-Poiseuille flow profile in (6.34), we obtain



3 é y2 ù ¶ c ¶ 2c U ê1 - 2 ú =D 2 2 ë ¶y d û ¶x

(6.35)

where U is the average velocity and d is the half vertical distance between the plates (Figure 6.13). Now, we introduce the following scaling

Figure 6.13  Typical mass transfer boundary layer on a partially labeled solid surface in an analysis chamber. Results obtained by solving the advection-diffusion equation with a finite difference algorithm (average velocity 1 mm/s, labeled distance 6 mm).

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Transport of Biochemical Species and Cellular Microfluidics

x* =



x y c , y* = , c* = L d c0

(6.36)

where L is a reference axial distance and c0 is a reference concentration. Introducing the Peclet number and taking into account (6.36), (6.35) can then be rewritten as 3 d ¶ c * ¶ 2c * Pe éë1 - y *2 ùû = 2 L ¶ x * ¶ y *2



(6.37)

Now, we follow Levêques’s approach, and change the vertical origin y� = 1 - y *



so that y� is zero at the wall. Equation (6.37) becomes 3 d ¶ c * ¶ 2c * Pe [2 y� - y�2 ] = 2 L ¶ x * ¶ y�2



In the boundary layer, the distance y� is small and we can assume ë2 y� - y�2û » 2 y. � If � we note δ , the reduced boundary layer thickness, and note that at a distance from the wall δ�, the advection and diffusion terms are of the same order, we obtain the scaling ¶ c * cw - c0 » ¶ x * c0 x * ¶ 2c * cw - c0 » c0 δ�2 ¶ y�2

and finally

3 Pe



d �3 δ »1 x

Thus, 1

δ æ x ö3 1 » 1 d çè 3 d ÷ø Pe 3



(6.38)

Take the case of Figure 6.11: d = 1 mm and Pe = 10,000. Equation (6.38) reduces to 1

δ » 0.32 x 3 d



δ » 0.08. This result agrees with the numerical result of Figure d 6.11 and confirms the sketch of Figure 6.10. To conclude, concentration boundary layers are often present in microfluidics and they are to be taken into account for the comprehension and calculation of the transport phenomena.

and for x = 1 cm,

6.2  Advection-Diffusion Equation

251

6.2.8  Numerical Considerations 6.2.8.1  Boundary Layer

The small thickness of the concentration boundary layer has consequences on the numerical computation. It is essential to model precisely the mass transfer in the boundary layer because it determines the mass transfer to the solid wall. To do so, the computational grid needs to have at least two meshes in the boundary layer. It is then necessary to reduce the side of the geometrical elements in the boundary layer, especially where the boundary layer starts (Figure 6.14). This reasoning often leads to very small values of the mesh size near the wall. A mesh size of the order of a few microns is usual. 6.2.8.2  The Question of the Mesh Size

A numerical Peclet number is defined, associated to the size of the mesh



æ u Dx v Dy ö Pem = max ç , ÷ è D D ø

(6.39)

For the numerical calculation to be stable, the numerical Peclet number must be smaller than 4



æ u Dx v Dy ö , Pem = max ç ÷ <4 è D D ø

(6.40)

This condition requires the following limitations on the mesh size Dx <



4D u

4D Dy < v

(6.41)

Figure 6.14  Nonuniform computational grid at the vicinity of the wall, especially at the beginning of the boundary layer.

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Transport of Biochemical Species and Cellular Microfluidics

Thus the mesh size must be quite small. For D = 10-10 m2/s and u =1 mm/s, one finds Dx = 4 mm. This is approximately the same size as that of the meshes in the boundary layer. The problem could be rapidly intractable, especially if it is a threedimensional problem. In order to avoid this difficulty and relax the size of the mesh outside the boundary layer, an artificial diffusion is added to the diffusion constant in the meshes outside the boundary layers. If this additional diffusion is correctly chosen, it reduces the value of the mesh Peclet number, allows for larger meshes, and does not affect the solution very much since it applies in a domain where the gradient of concentration is small. It is usual to choose the value of the added diffusion by Dadd , x = β



uDx 2

(6.42)

where b is a coefficient depending on the problem. Such a formulation respects the condition imposed by the boundary layer because u and Dx are small in the boundary layer. Note that the equivalent numerical diffusion coefficient is anisotropic

Dnum



uDx é êD + β 2 ê 0 =ê ê ê ê 0 êë

0 D+ β

vDy 2

0

ù ú ú ú 0 ú w D z úú D+β 2 úû 0

(6.43)

6.2.8.3  Time Step

Time step and mesh size are not independent. If an explicit formulation is chosen, the Courant condition yields u D t < D x



(6.44)

and introducing the limitations on the mesh size

æ 4 Dnum, x δ (x) ö u D t < D x < min ç , è u 2 ÷ø

(6.45)

æ 4 Dnum, x δ (x) ö , D t < min ç 2 u ÷ø è u2

(6.46)

Thus,

This condition is usually very restrictive. Typically it yields values of the time step of the order of 10-2 seconds. Having in mind that the duration of a biological reaction is of the order of 10 minutes to 10 hours, the number of time step is quite large. This is why semi-implicit or implicit algorithms are preferred for solving transient concentration problems in microsystems. To conclude, it is important for the numerical modeling of mass transfer in microsystems to have very small meshes in the boundary layers, to use an added diffusion coefficient to avoid numerical instabilities, and to choose a semi-implicit or implicit solution scheme.

6.2  Advection-Diffusion Equation

253

6.2.9  Taylor-Aris Approach 6.2.9.1  Taylor-Aris Model

At a time when flow velocimetry methods were developing, it was found that the measurement of flow velocity inside tubes may be achieved using small particles. The principle was to introduce a radioactive substance or an electrolyte in the flow at a certain cross section and to follow its translation inside the pipe. However, it appeared immediately that the substance introduced into the stream diffused at the same time as it moved with the fluid. A model was then developed by Taylor [5] and completed by Aris to take into account in a simple manner both diffusion and advection. This model, although it was born in the 1950s, has found a recent renewal in biotechnology where advection and diffusion of particles and macromolecules carried by a fluid are an everyday concern. The basic idea behind Taylor’s approach is that it is not possible to superpose advection and diffusion for a liquid flowing in a pipe under the action of pressure gradient. It is not correct to assume a mere translation where the particles would move as a whole with the fluid and diffuse with their usual diffusion coefficient. In a Lagrangian coordinate system moving at the average velocity of the fluid, the particles diffuse much more than it is predicted by the usual diffusion theory. As will be seen, this is due to the radial gradient of velocity in the flow. The strength of Taylor’s approach is to show that a superposition may be done by allowing the particles to move at the average velocity of the fluid and using an effective or apparent diffusion coefficient for the particle—larger than the usual coefficient derived from the Einstein’s formula. Suppose a capillary tube of radius R in which a liquid flows at a mean velocity U carrying particles with a concentration c (Figure 6.15). The advection-diffusion equation written under an axisymmetric (x, r) form is



é ¶ 2c 1 ¶ c ¶ 2c ù ¶c ¶c + u (r) = Dê 2 + + ú ¶t ¶x r ¶ r ¶ r2 û ë¶x

(6.47)

The local velocity u(r) is given by the Hagen-Poiseuille solution



æ r2 ö u (r) = 2U ç 1 - 2 ÷ R ø è

(6.48)

Note that the velocity is zero at the wall and is maximum and equal to 2U on the ¶ 2c in (6.47) may be omitted if it is assumed that the axial central axis. The term ¶ x2 change in concentration is much less than the radial change. After substituting (6.48) in (6.47), we obtain

Figure 6.15  Schematic view of the cylindrical capillary and the diffusion front.

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Transport of Biochemical Species and Cellular Microfluidics

æ é 1 ¶ c ¶ 2c ù ¶c r2 ö ¶ c + 2U ç 1 - 2 ÷ = Dê + 2ú ¶t R ø ¶x è ë r ¶ r ¶ r û



(6.49)

In a Lagrangian system of coordinate moving with the average velocity of the fluid (6.49) becomes æ é 1 ¶ c ¶ 2c ù é¶c ¶c ù 2 r2 ö ¶ c + + = + 2ú U U 1 D ê ê ¶t ç ÷ ¶ x úû R2 ø ¶ x è ë ë r ¶ r ¶ r û



(6.50)

The boundary condition at the capillary surface is ¶c ¶r



r =R

=0

indicating that there is no flow of substance through the wall of the tube. The reasoning may be decomposed in two steps: first, assume temporarily that the concentration gradient along the x-axis is linear ¶c ¶c = = cste ¶x ¶x



(6.51)

Here c is the average concentration over the cross section of the tube, defined by R

c=

R

1 2 c 2 π r dr = 2 ò c r dr ò S0 R 0

(6.52)

Along the axis of the capillary, the concentration must have a finite value. Then, the solution of (6.50) may be written as c = c0 +



U R2 ¶ c æ r 2 1 r 4 ö 4 D ¶ x çè R2 2 R4 ÷ø (6.53)

¶c =0 ¶t

where c0 is the concentration on the capillary axis (r = 0). In the following, remember that the time derivative is taken in the moving coordinate system. By using (6.52) and (6.53), we find the following relation between c and c c=c+

U R2 ¶ c æ 1 r 2 1 r 4 ö - + 4 D ¶ x çè 3 R2 2 R4 ÷ø

¶c =0 ¶t

The total mass flow of species through any cross section of the capillary is given

by



(6.54)

R æ R2 U 2 ö ¶ c Q = ò uc 2 π r dr = - π R2 ç ÷ è 48 D ø ¶ x 0

6.2  Advection-Diffusion Equation

255

The corresponding flux is j=



æ R2 U 2 ö ¶ c Q = ç 48 D ÷ ¶ x π R2 è ø

(6.55)

Relation (6.55) shows that the mass flux of the concentration c has the form of Fick’s law with the effective diffusion coefficient Deff =



R2 U 2 48 D

(6.56)

Now, if we assume that the axial concentration gradient is no more constant



¶c ¹ cste, we are entitled to write ¶x ¶c ¶j =¶t ¶ x

(6.57)

and we obtain ¶c ¶2 c = Deff ¶t ¶ x2



(6.58)

This last equation shows that the average concentration is governed—in the moving coordinate system—by the usual diffusion equation for a stationary medium with the effective diffusion coefficient Deff defined by (6.58). A numerical example of the Taylor model is shown in Figure 6.16. The diffusion front progresses with the flow and, at the same time, smears out due to molecular diffusion. One can make a simplified picture of the situation (Figure 6.17). In the case of the Taylor-Aris method, the concentration front has a parabolic shape; thus, the diffusional surface is much larger than if the front were flat. For this reason, the same situation for a flow induced by electro-osmosis presents less diffusion because the diffusional front is nearly flat (Figures 6.17 and 6.18). 6.2.9.2  Conditions of Applicability of the Method

One question remains: What are the conditions for the Taylor-Aris approach to be valid? ¶2 c First, we have neglected the axial diffusional in (6.47) in front of the radial ¶ x2 1 ¶ c ¶ 2c . This radial term is important if the radial velocity gradiffusional term + r ¶ r ¶ r2 dient is large (the velocity profile varies from 2U to 0 along the radius) (i.e., if the average velocity is sufficiently important). In such a case, we have D << Deff

which is equivalent to

UR >> 7 D

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Transport of Biochemical Species and Cellular Microfluidics

Figure 6.16  Concentration profiles obtained with (6.58) for a velocity of 1 mm/s in a channel of radius R = 100 mm.

or



Pe =

UF >> 14 D

(6.59)

Relation (6.59) requires that the Péclet number should be larger than 14. Second, the concentration c must be a slowly changing function of x (we have ¶c ¶c even supposed in a first approximation that = = cste); from (6.54), we derive ¶x ¶x

Figure 6.17  Comparison of advection/diffusion between a parabolic profile (pressure-induced flow) and a flat profile (electro-osmotic flow). In the moving coordinate system, diffusion is more important for the parabolic profile than for the flat profile.

6.2  Advection-Diffusion Equation

257

Figure 6.18  (a) Experimental view of diffusing particles in a Poiseuille flow and in an electro-osmotic flow. (b) Experimental velocity profile in both cases. Dispersion is reduced if the diffusion front is flat.



¶ c ¶ c U R2 ¶ 2 c = + 4 D ¶ x2 ¶x ¶x

æ 1 r2 1 r4 ö ¶ c ç- + ÷» è 3 R2 2 R 4 ø ¶ x

(6.60)

To be satisfied, relation (6.60) requires



¶c U R2 ¶ 2 c >> ¶x 4 D ¶ x2 If L is the length over which a noticeable change in c can occur, we may approximate the gradients by ¶c c » ¶x L ¶2 c c » 2 2 ¶x L



then the preceding inequality may be written as LD >> 1 U R2

(6.61)

L UR >> >> 7 R D

(6.62)

and, taking into account (6.59),

To these two conditions, we add the condition for a laminar flow



Re =

UF << 2000 ν

(6.63)

The three conditions (6.59), (6.61), and (6.63) give the limits of applicability of (6.57) for a cylindrical tube. R2 U 2 The value Deff = refers to cylindrical tubes only. Another value of Deff 48 D can be obtained by the same reasoning for a flow limited by two parallel plates [2]



Deff =

H2 U2 210 D

(6.64)

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Transport of Biochemical Species and Cellular Microfluidics

where H is the half-distance between the two plates. The general form for the equivalent diffusion coefficient using the Péclet number is Deff 1 2 = Pe D β



(6.65)

where b is a geometric coefficient depending on the shape of the cross section. Note that the lower limit for the Peclet number (6.59) is not universal and depends on the cross section of the capillary tube. The general formulation would be Pe > 2 β



(6.66)

6.2.9.3  Applicability to Microflows

The conditions defined in the preceding section are very often satisfied by microflows in microsystems. Generally, tube diameters are in the range of 100 mm to 1 mm, and velocities vary from a few microns per second to a few millimeters per second. Thus, for a water-based flow, the Reynolds number is smaller than 10, and the flow is strongly laminar. Because diffusion coefficients are very small (seldom larger than 10-10 m2/s), the Peclet number is at least of the order of 10 and most of the time larger than that. Finally, the condition L/R >> 7 requires that the microchannel be sufficiently long, which is usual. When applicable, the Taylor-Aris approach is very simple and useful. It has many applications in chemistry [6, 7] and for immunoassays, as we will see in Chapter 8. 6.2.10  Distance of Capture in a Capillary

In biotechnology, the capture of particles advected by a carrier fluid flowing inside a capillary tube is a fundamental question. For example, we may want to dimension an annular surface to capture a certain type of particles in the carrier fluid. In this section we do not deal with the capture itself (this will be done in Chapter 8), but with the contact of the particles with the solid wall. 6.2.10.1  Analytical Approach Scaling Analysis

A very simple approximation may be done by comparing an axial convection to a radial diffusion. Particles near the wall are not going to have a very long axial displacement before impacting the wall, whereas the particles initially located at the center of the capillary will follow the longest trajectory before impacting the wall (Figure 6.19). The average maximum time necessary for a particle to diffuse radially to the wall is



τ»

R2 4 D

(6.67)

6.2  Advection-Diffusion Equation

259

Figure 6.19  Sketch of particle trajectories depending on their starting point.

During this time t of radial diffusion, the particle has moved along the axial direction on a distance of L » 2 Vτ »



V R2 2D

(6.68)

The coefficient 2 in (6.68) corresponds to the maximum Hagen-Poiseuille velocity 2V at the center of the tube. With this reasoning, it can be deduced that after a distance L, statistically all the particles will have impacted the wall at least one time. If the wall property is such that there is a capture upon contact, then (6.68) is a good approximation of the dimension of the surface of capture. Equation (6.68) may be rewritten in an nondimensional form L VR 1 » = Pe R 2D 2



(6.69)

where Pe is the Péclet number. We see here another significance of the Péclet number. Suppose now that the flow rate is imposed by the experimental conditions (a syringe pump, for example), and using the relation between the average velocity and the flow rate Q = SV = π R2V

the length L becomes

L»



Q 2 π D

(6.70)

and the radius of the capillary does not appear in the equation anymore. For a given mass flow rate, if the radius is increased, the fluid velocity decreases; the convection distance is then shorter, but the radial diffusion distance is longer. The two effects balance each other (Figure 6.20). Spatial Fourier Series

Another approach to the problem of the capture distance in a capillary tube has been proposed in [8]. First, note that, in a coordinate system moving at the average velocity of the liquid, the governing equation becomes

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Figure 6.20  The length L is the same regardless of R if the flow rate is identical in both cases.

¶c ¶2 c =D 2 ¶t ¶y



(6.71)

A normalized uniform y-distribution of the particles at the channel entrance can then be expressed by the Fourier series expansion ¥ ææ 1 (-1)k 1ö π yö =å cos ç ç k + ÷ 1ö 2 k=0 æ 2 ø w ÷ø èè π çk + ÷ è 2ø



(6.72)

Upon substitution of (6.72) in (6.71) and integration, one finds the solution ¥

ææ (-1)k cos ç ç k + 1 èè k=0 π æ k + ö çè ÷ø 2

c (y, t) = 2 c0 å

æ æ exp ç - ç k + è è



1ö π yö ÷ 2 ø w ÷ø

ö 1ö π Dt ÷ ÷ø 2 w2 ø 2

(6.73)

2

Integration of (6.73) with respect to y produces the time dependence of the concentration 2 æ æ ö 1ö π2 exp k + Dt ç ÷ ç ÷ 2 2 ø w2 1ö è è ø k=0 2 æ π çk + ÷ è 2ø ¥

C(t) = 2 c0 å

(-1)k

(6.74)

Substitution of t by x/U brings back to the fix Eulerian coordinate system. We note that each mode is attenuated with the axial distance x by the factor ak =

Ck (t) =2 c0

2 æ æ 1ö π2 xö + exp k D ÷ ç çè ÷ø 2 2 2 w Uø 1ö è æ π2 çk + ÷ è 2ø

(-1)k

After some distance, all the modes are damped except the first mode corresponding to k = 0. The persistence of the first mode only is sketched in Figure 6.21. The attenuation in the x-direction is then

6.2  Advection-Diffusion Equation

261

Figure 6.21  At inlet, the uniform distribution is decomposed in spatial Fourier series; after some translation length L, only the first mode remains, and all the others are damped.



a=

æ π 2D x ö 8 exp ÷ çπ2 è 4 w 2U ø

(6.75)

The number a represents the fraction of targets still in suspension in the flow channel at the length x. The fraction of targets transferred to the solvent at the length x is then 1 - a. A reduction of 63% (corresponding to 8e-1/p2) of the number particles continuing to flow in the aqueous channel is reached at the length Le determined by



Le 4 Uw 4 = 2 = 2 Pe w π D π

(6.76)

Equation (6.76) has exactly the same form than (6.69). However, the coefficient before the Péclet number has been explicitly determined. The ratio Le /w is sometimes denoted as the Graetz number (see Chapter 1). Note that the preceding reasoning uses the assumption that the velocity profile is flat (velocity U). It has been shown numerically that the result is not changed by considering a quadratic velocity profile—with the same average velocity U. After some traveling distance, the profile of concentration becomes sinusoidal and indiscernible from the profile obtained with a uniform velocity. 6.2.10.2  Numerical Approach

Confirmation of the preceding approach can be done by using numerical modeling. We have two choices to set up a numerical approach. Either a Hagen-Poiseuille flow

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formulation can be introduced in the advection-diffusion equation and we have to solve in a cylindrical geometry (r, z) the equation



æ é ¶ 2c 1 ¶ c ¶ 2c ù ¶c r2 ö ¶ c + 2U ç 1 - 2 ÷ = Dê 2 + + ú r ¶ r ¶ r2 û ¶t R ø ¶r è ë¶z

(6.77)

or we start from the Navier-Stokes equations and solve the system of equations ¶u 1 ¶rv + =0 ¶z r ¶r u

1 ¶ P η é ¶2u 1 ¶ u ¶2u ù ¶u ¶u +v =+ ê + + ú ρ ¶ z ρ ë ¶ z2 r ¶ r ¶ r2 û ¶z ¶r

(6.78)

1 ¶ P µ é ¶2v 1 ¶ v ¶2v ù ¶v ¶v u +v =+ ê + + ú ρ ¶ r ρ ë ¶ z2 r ¶ r ¶ r2 û ¶z ¶r



Equations (6.77) and (6.78) treat the case of a permanent buffer fluid flow moving inside a cylindrical tube of constant diameter, with a transient concentration ¶u ¶v advected by the fluid. Thus, there are no transient terms æ = = 0 in the Navierè ¶t ¶t ¶c Stokes equations, but the term ¹ 0 is present in the advection-diffusion ¶t equation. This system can be decoupled if the velocity field does not depend on the concentration—this is usually the case since the concentration is assumed to be small and there are no considerable aggregation regions. In such a case, the system can be solved in two steps: step 1, hydrodynamics; and step 2, advection of particles. This process has the advantage of mobilizing less computational memory, but requires loading a file with the results of the velocity field and transmitting it to step 2. If enough computational memory is available and if the same meshing of the computational domain is possible, it may be advantageous to solve the system as a totally coupled system—even if it is not the case—and have only one (large) matrix to invert in the numerical algorithm. This last coupled approach was performed using the COMSOL numerical software [9]. Figure 6.22 shows the results of the calculation. The velocity field has the parabolic shape of the Hagen-Poiseuille solution except at the entrance where the flow is not yet established. Concentration flow lines have been plotted confirming that the particles entering in the middle of the channel have the longest axial trajectory (a flow line is the line defined by the gradient of the concentration function and collinear to the Fick’s concentration flux). The flow lines are perpendicular to the side walls because the boundary condition is c = 0 at the walls. Equation (6.76) is much easier to solve. Of course most numerical software aimed at the resolution of partial differential equations (PDE) will do the job, but a straightforward discretization can be performed and implemented with math-

æ è



é ¶ 2c 1 ¶ c ¶ 2c ù ¶c ¶c ¶c +u +v = Dê 2 + + ú r ¶ r ¶ r2 û ¶t ¶z ¶r ë¶z

6.2  Advection-Diffusion Equation

263

Figure 6.22  Results of the numerical modeling. Velocities are indicated by the arrows and show a Poiseuille parabolic profile, except at the entrance of the channel where the flow is totally established. A few flow lines for concentration have been plotted proving that the distance of capture depends on the initial position of the particle.

ematical software such as MATLAB if the geometry of the computational domain is simple (rectangle or cylinder). A finite volume method can be set up by using a semi-implicit Crank-Nicholson discretization scheme. Figure 6.23 shows the indices for r and z; the discretized equation at the node (i, j) is n +1 n +1 æ cin, j +1 - cin, j ù cin, +j 1 - cin, j rj2 ö é ci , j +1 - ci, j ê ú + U ç1 - 2 ÷ Dt Dr Dr R øê ú è ë û



=

n +1 n +1 n +1 n +1 n +1 n +1 n +1 n +1 D é ci +1, j - 2 ci , j + ci -1, j ci , j +1 - 2 ci , j + ci , j -1 1 ci , j +1 - ci , j ù (6.79) ê ú + + rj 2ê Dr (D z)2 (D r)2 ú ë û

+

n n n n n n n n D é ci +1, j - 2 ci , j + ci -1, j ci , j +1 - 2 ci , j + ci , j -1 1 ci , j +1 - ci , j ù ê ú + + 2ê rj Dr (D z)2 (D r)2 úû ë



with the precaution that on the centerline (r = 0), the terms in 1/r should be removed ¶c (because = 0). More on the numerical algorithm for the solution of the advection¶ r diffusion equation may be found in [10]. The results for a concentration “burst” of particles have been plotted in Figure 6.24.

Figure 6.23  Schematic view of the computational nodes and grid.

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Figure 6.24  Contour plot of concentration at four different times after injection. The calculation has been performed in a (r, z) coordinate system and only half of the channel is represented. The solid wall is located at the bottom of each picture. The velocity field is given by the Hagen-Poiseuille solution.

6.2.11  Determination of the Diffusion Coefficient

Measurement of liquid phase diffusion coefficients is based on the observation of the spreading of the diffusing substance/solute. Diffusion coefficients are very small, and measurements in macroscopic systems are not reliable because of uncontrolled fluctuations of velocity. Because they are very laminar, easily controllable, and not distorted, microflows are well adapted to the measurements of liquid phase diffusion coefficients. The experimental principle is based on the mixing of the buffer liquid alone and the buffer liquid with a concentration of the targeted substance, as shown in Figure 6.25. In the diffusing zone the streamlines are parallel and directed along the x-axis; the substance/solute progressively diffuse in the y direction and there is a growing distance d(x) of the concentration gradient. It can be shown that the concentration profile is given by the relation [2, 11, 12]. c (x, y) =

1 æ y U ö c0 ç 1 - erf ÷ 2 è 4D x ø

(6.80)

where U is the mean flow velocity. A fit of the relation (6.80) with the experimental concentration profile produces the value of the diffusion coefficient D (Figure 6.26).

6.2  Advection-Diffusion Equation

265

Figure 6.25  Experimental principle for the measurement of the diffusion coefficient.

6.2.12  Mixing of Fluids 6.2.12.1  Introduction

Mixing of liquid constituents is a major problem in biochips and bioMEMS. The high degree of laminarity of microflows delays the mixing of constituents. Two different liquids can flow side by side for a rather long distance before complete mixing occurs. This is a real difficulty for the miniaturization and compactness expected from a biochip. It is always possible to design a fluidic system with zigzags to have more capillary length in a compact surface as sketched in Figure 6.27 and according to the photograph of the microsystem of Figure 6.28. However, there is another difficulty linked to poor mixing in biochips. The time required to execute the different biological processes may be important. Besides miniaturization, another advantage expected from microsystems is the reduction of reaction time. It is then often necessary to accelerate the mixing process. Many different micromixers have been developed. They fall into two categories: active

Figure 6.26  Concentration profile of diffusing species marked with fluorescent markers at three different locations in the channel.

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Figure 6.27  Sketch of compaction of a long channel on a biochip to realize micromixing.

and passive. Active micromixers use actuated devices, such as piezoelectric actuated membranes [13], whereas passive micromixers use only the energy of the flow and special morphological design promoting mixing of the compounds [2, 13]. 6.2.12.2  Parallel Flows

In the preceding section, we established a relation for the concentration diffusion between parallel flows. Let us analyze this relation. Figures 6.29 and 6.30 show the concentration distribution in a half channel. The difficulty of mixing the flows is obvious: for typical dimensions, velocity, and liquids, the mixing length is very important and often not acceptable for compact microsystems. It is interesting to estimate the length L at which a relative concentration of c/(c0/2) = 90% at the wall is reached. Using the inverse erf function and (6.80), and y = R, we find



R U = erfinv (0.1) = 0.0889 4 DL

Figure 6.28  Example of a micromixer based on a long mixing length. The micromixer incorporated in the global design of a proteomic reactor. (Courtesy of N. Sarrut, CEA/LETI.)

6.2  Advection-Diffusion Equation

267

Figure 6.29  Contour plot of concentration c/c0 from relation (6.80) for R = 100 mm, L = 1 cm, D = 10-10 m2/s, and U = 1 mm/s. The four plots correspond to c/c0 =0.1, 0.2, 0.3, and 0.4.

where the function erfinv is the inverse of the erf function. Thus, the length L is



L 1 RU UR = » 32 2 R 4 (erfinv (0.1)) D D

(6.81)

and we see that the mixing length is a function of the Péclet number

Figure 6.30  Perspective view of the relation (6.80) showing the concentration profiles in the channel for R = 100 mm, L = 1 cm, D = 10-10 m2/s, and U = 1 mm/s. The four plots correspond to c/c0 = 0.1, 0.2, 0.3, and 0.4.

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L » 32 Pe R



(6.82)

It is worth comparing the mixing length from relation (6.82) with the entrance length in a channel (2.25) established in Chapter 2. There is an obvious similarity, if we write the two relations as Lmix » 32 R PeR » 8(2 R) PeD

h » 0.04 (2 R) ReD

(6.83)

In the case of the hydrodynamic entrance, it is the action of the viscosity that homogenizes the flow to reach a fully developed flow. In the case of the mixing of microflows, it is the action of the molecular diffusion that homogenizes concentration. The physics of the two problems is similar. In both cases the phenomenon is linked to the growth of a boundary layer. It is no wonder then that the form of (6.83) is similar. The major difference is that the cinematic viscosity is of the order of 10-6 m2/s, whereas the diffusion coefficient is only 10-10 m2/s. The hydrodynamic entrance is then very short, whereas the mixing length is very long. 6.2.12.3  Improving the Mixing of Parallel Flows

Usually in bioMEMS, the flow rate is imposed and, from (6.82), the only action way of reducing the mixing length for parallel flows is to reduce the radius R. This will have a considerable effect since the mixing length Lmix varies as the square of R. However, if the radius is reduced and if the flow rate is imposed, it is necessary to divide the flow in multiple branches. A typical design based on the reduction of the channel cross section is shown in Figure 6.31. 6.2.12.4  Chaoting Mixing

The principle of chaoting mixing is based on successive stretching and bending of fluid streamlines. In Figure 6.31, we show how a domain of liquid 1 immerged in a liquid 2 is deformed by chaoting mixing. If the succession of folding and folding is done rapidly, at a time scale much smaller than that of diffusion, the time interval for the stretching-folding process may be neglected and it is possible to compare the corresponding mixing zones in Figure 6.32. Suppose a time scale t; then diffusion length is approximately

λ » 4 Dτ

(6.84)

If we chose the value of t so that the distance l is approximately the distance between the folded regions, the mixing zones at the time t corresponding to Figure 6.32 are shown in Figure 6.33. The mixing zone after chaoting mixing has a more important surface than the original one. This proves the efficiency of chaoting mixing. The important thing here is that the stretching/folding deformations are performed in a short time compared to the diffusion time.

6.2  Advection-Diffusion Equation

269

Figure 6.31  Schematic view of a parallel micromixer by IMM. The two fluids are mixed together after in channels of reduced cross sections.

6.2.12.5  Mixing in Two-Phase Flows

Mixing is not a problem reserved to single phase microflows. Obtaining short time for mixing is also important in two-phase flow and plug flow. The mixing of plug flows has recently been a subject of interest. Qualitatively, the internal convective motion in a liquid plug is that sketched in Figure 6.34. Usually—but not always [14]—the plug keeps the same geometry during its motion because of the effect of surface tension. Due to the displacement of the plug and the friction at the solid walls, two convective cells forms in the plug.

Figure 6.32  Principle of chaoting mixing by successive stretching and folding of flow domain.

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Figure 6.33  The mixing zone is enlarged by a succession of stretching and folding.

Modeling the internal convection in the plug may be performed by considering the problem in the moving coordinates system. In this system, the plug has fixed boundaries (to the first order) and the solid walls are moving with a velocity –V. Specifying this value of the velocity on the solid walls, and symmetry conditions on the side surfaces with no contact of any wall, the numerical solution is straightforward. We show in Figure 6.35 the result obtained with the COMSOL numerical software [9]. 6.2.12.6  Mixing in Digital Microfluidics

The preceding examples concerned mixing in microflows. With the development of digital microfluidics (see Chapter 4), the mixing of fluids and substances in microdrops is now a growing subject of study. The understanding of the mixing phenomena in microdrops is only at a qualitative stage. In order to illustrate this problem and to familiarize the reader with internal microdrop motion, we show in Figures 6.36 and 6.37 the principle of mixing two fluids in a microdrop by moving the drop along a designated path [15]. It appears that mixing in digital microfluidics shows very special patterns. Right now it is a new topic of investigation.

Figure 6.34  Mixing in a plug flow due to the friction at the walls.

6.3  Trajectory Calculation

271

Figure 6.35  Modeling of the internal motion in a plug flow with COMSOL. (Courtesy of E. Favre, COMSOL.)

6.3  Trajectory Calculation Computation of transport of substance using the concentration equation requires that the molecules or particles composing this substance have a small size. Because gravity is not taken into account in the advection-diffusion equation, the particles are not allowed to sediment and their size/weight is limited by the sedimentation size/weight (Chapter 5). When transporting large biological objects such as cells, proteins, or heavy particles like some magnetic beads (Chapter 9), gravity has an important influence on the transport. In such a case, the influence of Brownian motion on the particles is

Figure 6.36  Mixing of two constituents in a drop by electrowetting (open EWOD) displacement from [15].

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Figure 6.37  Mixing of two microdrops confined by electrodes in an EWOD microsystem. (Courtesy of CEA/LETI.)

reduced. A first step in this approach is to calculate the trajectories of the particles in a deterministic (i.e., without taking account of the Brownian motion). 6.3.1  Trajectories of Particles in a Microflow







Larger particles experience less diffusion under the action of Brownian motion. In such a case, these particles follow have trajectories determined by the forces acting upon them. At a macroscopic scale, the kinematics theory relates the mass acceleration of a body to the resultant of the external forces that act upon it. This is the well-known Newton’s theorem. � � dVp (6.85) m = å Fe dt � � where m is the mass of the particle, Vp is the velocity, and Fe are the external forces. We will treat here the case of particles submitted to gravity force and hydrodynamic drag force. Newton’s equation can then be written under the form � � dVp � m = Fhyd + Fgrav dt

(6.86)

The hydrodynamic drag is derived from the velocity field according to the equation � � � � � Fhyd = -CD (Vp - Vf ) = -6 π η rh (Vp - Vf ) (6.87) where CD is the drag coefficient, h is the dynamic viscosity of the carrier fluid, rh is the hydrodynamic diameter of the particle, and Vf is the velocity of the carrier fluid. It is assumed here that the velocity field of the carrier fluid is not affected by the presence of the particles, which is the general case, except if the volume fraction of particles is important, leading to the formation of aggregates. Under this assumption, the velocity field of the carrier fluid must be calculated before attempting the calculation of the particles trajectories, using classical hydrodynamics equations (i.e., NavierStokes equations). A typical situation in microfluidics is the Hagen-Poiseuille flow between two plates or in a rounded capillary. The gravity term is given by

6.3  Trajectory Calculation

273

� Fgrav = g vol p D ρ yˆ



(6.88)

where g is the acceleration of gravity, volp is the volume of the particle, yˆ is the vertical unit vector (oriented downwards), and r is the difference between the volumic mass of the particle and that of the liquid. After substitution of (6.87) and (6.88) in (6.86), one obtains the equation for the particle’s velocity



� � � dVp m = -6 π η rh (Vp - Vf ) + g vol p D ρ yˆ dt

(6.89)

This relation can be decomposed along each coordinate (here we choose a 2D configuration) m

d up = -6 π η rh (up - uf ) dt

d vp m = -6 π η rh v p + g vol p D ρ dt



(6.90)

Using the notations c1 =

6 π η rh m

g vol p D ρ c2 = m



(6.91)

this system becomes d up = -c1 (up - uf ) dt



d vp = -c1 v p + c2 dt

(6.92)

and this system can be solved analytically up = up,0 e -c1t + uf [1 - e -c1t ]



v p = v p,0 e

- c1t

c + 2 [1 - e -c1t ] c1

(6.93)

By definition, x and r coordinates of the particle at a given time are linked to the velocity by the relations d xp = up = up,0 e -c1t + uf [1 - e -c1t ] dt



d rp c = v p = v p,0 e -c1t + 2 [1 - e -c1t ] dt c1

(6.94)

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If the starting velocity of the particle is zero, we obtain a simple relation between the coordinates of the particle d xP c = uf 1 d rP c2



(6.95)

where the ratio c1/c2 is c1 CD = c2 g D m



Assuming a Hagen-Poiseuille flow in the duct, (6.95) becomes rP2 ö c1 d xP V0 æ = 1 ÷ 2 çè d rP R 2 ø c2



(6.96)

Integration of this relation gives the relation xp =

rp3 rp,03 ù V0 c1 é êr - rp,0 + ú 2 c2 êë 3 R2 3 R2 úû

(6.97)

where (0, rp0) is the starting location of the particle. A particle starting from the middle of the duct (rp0 = 0) will contact the wall at an axial distance of

L=

CD V0 R g Dm 3

(6.98)

It is interesting to compare this result (6.98) with (6.68). The two results are quite different. In the present case, we calculate the distance of travel of a particle submitted to hydrodynamic drag considering that the particle is supposed sufficiently large (or heavy) to neglect the Brownian motion in front of the gravity force. In the previous case, we calculated the same distance for a particle submitted to hydrodynamic drag force but small enough to neglect gravity in front of Brownian motion. The ratio of the two calculated lengths is



Ldiff 3 R Dm 3 R 3 R = = gDm = 2 LB Lgrav 2 D CD 2 kB T

(6.99)

where LB is the Bolzman length. Interestingly, the average buffer fluid velocity has disappeared from (6.99), which is just a balance between gravity forces and Brownian motion energy (kBT). Depending on the relative particle mass Dm, the travel distance will be either the gravity model distance defined by (6.98) or the diffusion model distance given by relation (6.68). Note that it is very seldom that the trajectory equation can be solved analytically, such as in this case, but it is always interesting to spend some time investigating if an analytical solution may exist—even to the price of some simplification (initial velocities set to zero). Most of the time, a numerical approach is required. Different methods such as Runge Kutta or predictor-corrector can be used. We give an example of the predictor-corrector scheme in Chapter 9.

6.3  Trajectory Calculation

275

6.3.2  Ballistic Random Walk (BRW) 6.3.2.1  Model

In this section, we combine particles entrainment by the flow with Brownian motion. This approach is sometimes called the ballistic random walk (BRW) method. We have seen in Chapter 5 that discrete models such as the Monte Carlo model are interesting because they bring new insight to the understanding of the effect of Brownian motion. With this in mind, a similar approach may be done for microparticles transport. The behavior of the buffer (or carrier) fluid is still obtained by solving the Navier-Stokes equations. If the entrainment of the microparticles is strong enough, one can assume that the transported microparticles are following trajectories slightly modified by the effect of Brownian motion (Figure 6.38) The real force balance on a particle is given by Langevin’s equation [16]



� � � � dVc me = CD (Vf - Vc ) + F(t) dt

(6.100)

where the function F(t) represents the Brownian forces. Although this is not strictly correct, we approximate (6.100) by the superposition of a deterministic trajectory, modified by the effect of the Brownian motion modeled by a Monte Carlo method. This is approximately correct if the particle trajectory is not too much affected by Brownian motion (i.e., if the entrainment is strong in front of the Brownian motion). In this example, for simplicity we do not take into account gravity force. In such a case,



C æ - Dtö Vp = Vf ç 1 - e m ÷ çè ÷ø

(6.101)

According to (6.101), the velocity of an extremely small particle is that of the carrier fluid. Now we account for the Brownian motion by introducing the relations

Figure 6.38  Sketch of the superposition of advection by the buffer fluid and Brownian motion.

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Vp, x = Vf , x +

4D cos(α ) Dt

Vp,y = Vf ,y +

4D sin(α ) Dt

α = random (0, 2π)

(6.102)

and for the 3D modeling

Vp, x = Vf , x +

4D cos(α ) sin(β) Dt

Vp,y = Vf ,y +

4D sin(α ) sin(β) Dt

Vp,z = Vf ,z +

4D cos(β) Dt

(6.103)

α = random (0, 2 π)

β = a cos(1 - 2 random(0,1)) More explanation about these equations can be found in Chapter 5. Suppose now that a uniform concentration of target particles arrives at the entrance of the duct, meaning that they are uniformly dispersed in the entrance cross section. An easy way to obtain a uniform concentration in a circular cross section is to generate two sets of values uniformly distributed: x = R random (0,1)



y = R random (0,1) and to reject the values (x, y) located outside the circle of radius R. If the particles are referenced by their polar coordinates (r, f), then the distribution in f is uniform and linear in r, as shown by Figure 6.39. The buffer flow velocity profile is given by the Hagen-Poiseuille relation



2 æ æ rö ö V (r) = 2 V0 ç 1 - ç ÷ ÷ è Rø ø è

(6.104)

In the present model, we suppose that the wall is completely adherent (i.e., the particles contacting the wall are immobilized immediately), as shown in Figure 6.40. The model is then

6.3  Trajectory Calculation

277

Figure 6.39  Initial distribution of particles in the entrance cross section.

Vp, x = Vf , x +

4D cos(α ) sin(β) Dt

Vp,y = Vf ,y +

4D sin(α ) sin(β) Dt

2 æ æ rö ö Vp,z = 2 V0 ç 1 - ç ÷ ÷ + è Rø ø è





(6.105)

4D cos(β) Dt

α = random (0, 2 π) β = a cos(1 - 2 random(0,1)) 6.3.2.2  BRW in a Capillary Tube

Equation (6.105) has been implemented in MATLAB [17]. Figure 6.41 shows the trajectories of the particles in the capillary tube. The photograph of the particles at a given time is shown in Figure 6.42. The target particles are dispersed following the Hagen-Poiseuille parabolic profile of velocity. By superposition of images of location of the particles at different times, we understand the pattern of the transport of the microparticles (Figure 6.43). In a cross section, the location of the particles is given in Figure 6.44. The particles adhering to the wall are clearly seen on the periphery.

Figure 6.40  Sketch of a particle impacting the wall.

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Figure 6.41  Calculated trajectories of particles (100-nm diameter) transported by a buffer fluid flow (500 mm/s) in a 50-mm radius capillary tube.

It is interesting to compare these results to that of Figure 6.24 obtained by solving the advection-diffusion equation for the concentration. In Figure 6.45, we have placed the particles on the concentration contour lines. The comparison is rather good; Monte Carlo computation for more particles would have been still more accurate. 6.3.2.3  BRW in a Recirculating Chamber

The scope of BRW modeling can be extended by interfacing the BRW algorithm to the flow field calculated by a finite element model [16]. It can be shown that the method is equivalent to the full coupled advection and diffusion system, but brings a new light to the diffusion of species in a carrier flow (Figures 6.46 and 6.47). 6.3.2.4  3D Modeling of BRW

A similar interfacing BRW model—(6.103)—with a 3D FEM brings gives interesting insights to the advection of particles by a carrier liquid. In Figure 6.48 we show the typical results obtained with this method.

Figure 6.42  Calculated location of the particles at a given time. Note the similarity with Figure 6.18.

6.4  Separation/Purification of Bioparticles

279

Figure 6.43  Superposition of the location of particles at different times.

6.4  Separation/Purification of Bioparticles Separation and purification of bioparticles are required for many different applications and targets (purification of proteins, separation of DNA strands by length, and so forth). Several techniques have been developed to perform these processes. We present here, mostly qualitatively, the principle of field flow fractionation and chromatography columns. 6.4.1  The Principle of Field Flow Fractionation (FFF)

Field flow fractionation (FFF) is a group of techniques to separate different types of particles [18]. The principle is shown in Figures 6.49, 6.50, and 6.51. The principle here is that particles in a liquid flow separate according to their physical properties such as volume, mass, electric charge, or magnetic moment. Suppose a horizontal flow drag force depends on the size of the particle. If another force field is applied vertically (such as gravitation), the particles will gather at different places on the lower solid wall.

Figure 6.44  (a) Image of the particles in a cross section. (b) Radial distribution of particles corresponding to (a).

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Figure 6.45  Comparison of the concentration contour lines (from Figure 6.21) with the results of the Monte Carlo simulation. Radius R = 50 mm, diffusion coefficient D = 2 × 10-10 m2/s and average velocity V = 500 mm/s.

When the liquid flow is initiated, the solute zone is carried downstream at a rate depending on the particle size and mass. Figure 6.52 shows a correct separation when the immobilized particles form two peaks completely disjointed. In Chapter 9, an example of magnetic FFF is given, with the calculation of trajectories inside the channel. 6.4.2   Chromatography Columns

Chromatography is a separation technique mostly employed in chemical and biochemical analysis [19, 20]. In a single-step process, it can separate a mixture (buffer fluid carrying different types of bioparticles) into its individual components and

Figure 6.46  (a) COMSOL multiphysics model of a bolus of concentration in a recirculating flow. (b) Same calculation with the BRW method (1,000 particles).

6.4  Separation/Purification of Bioparticles

281

Figure 6.47  Random walk of particles trapped in a recirculation microchamber: (a) if the diffusion constant is small enough (D = 10-10 m2/s), the particles are trapped, and (b) they escape progressively when the diffusion coefficient is sufficiently large (D = 10-9 m2/s).

simultaneously provide an quantitative estimate of each constituent. In biotechnology, the analysis is usually carried out on a mass spectrometer (MS) placed behind the chromatography column. The name chromatography may look strange at first sight. Color has nothing to do with modern chromatography, but the name was given to this method of separation by the Russian botanist Tswett who used a simple form of liquid-solid chromatography to separate a number of plant pigments. The colored bands he produced on the adsorbent bed evoked the term chromatography for this type of separation. In a chromatography device, there are two phases: a mobile phase and a stationary phase. The mobile phase transports the sample with the targets and other compounds, and the stationary phase is designed to retain longer in the column the compounds which associate best with it. Targets and compounds are then separated in zones or bands [21]. A typical design of a chromatography column is that of the size exclusion chromatography sketched in Figure 6.53. In such a device, the smallsized proteins or peptides travel at a smaller speed through the column, because they enter inside the gel beads, whereas large-sized proteins are excluded and travel faster in the connected porosities. The result is the separation of the particles into two disjointed bands (Figure 6.54).

Figure 6.48  3D dispersion of a tracer: (a) straight channel and (b) turning channel

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Figure 6.49  Injection of the sample mixture in the FFF channel. Particles injected onto the column without the field or flow turned on are evenly distributed across the column.

An example of a different type of chromatography columns is that of the proteomic reactor [22]. Proteins are first digested into peptides; then all the peptides are immobilized in a microfabricated (Figure 6.55) chromatography column. A flow of acetonitril (CH3CN) is then used at different concentrations to elute progressively the peptides. The eluted peptides are transported to a spray nozzle and sprayed into the mass spectrometer. Figure 6.56 shows the detection by the mass spectrometer of the well-known peptide b-Galactosidase with its three characteristic peaks.

6.5  Cellular Microfluidics Transport and manipulation of cells are becoming of utmost importance in today’s biotechnology. Many new devices have been recently developed to separate, concentrate, and immobilize cells in microsystems. In this chapter, we present some remarkable features, such as single-phase flow focusing, pinched channels, bifurcation channels, and ratchets for cell separation, and Dean flows for cell alignment and recirculating chambers for cell trapping.

Figure 6.50  When a field is applied, the solute zone is compressed into a narrow layer against one wall.

6.5  Cellular Microfluidics

283

Figure 6.51  An applied velocity field in the channel exerts a different hydrodynamic drag on the two types of particles. The larger particles stay behind and are separated from the smaller particles.

6.5.1  Flow Focusing

In Chapter 4, we have seen how flow-focusing devices are used to produce droplets. In such a case, we can speak of two-phase flow focusing. Single-phase flow focusing has also a great interest, although for a totally different application. The principle is to focus an incoming carrier fluid containing targets, for example, cells, or colored in the case of optofluidics, inside a secondary fluid. Different types of single phase FFDs have been developed. Let us first present 2D focusing. 6.5.1.1  Single-Phase 2D Focusing

It is of interest to confine cells—or other relatively large biological particles—along a wall or in the middle of a stream. An example of the use of 2D focusing along a vertical wall is presented in Section 6.5.1.2. Figure 6.58 shows how the focusing

Figure 6.52  Sketch of the separation. Separation is correct when the two aggregates are completely disjointed.

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Figure 6.53  Proteins purification by size-exclusion chromatography.

is done. A secondary flow, miscible with the first one, flowing at a larger flow rate, confines the first flow. Let us analyze the case of flow focusing along a wall and denote Q1 and Q2 as the two flow rates, with Q as the total flow rate Q1 + Q2 and w1 and w2 as the two channel widths occupied by fluid 1 and fluid 2, as schematized in Figure 6.59. After a short establishment length, the new profile is a Poiseuille-Hagen quadratic profile. Clearly, the relative width e = w1/w is related to the flow rate ratio Q1/Q. Using (2.47) for the flow in a rectangular channel

Figure 6.54  Separation principle of substance traveling through a chromatography column.

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Figure 6.55  (a) Detailed view of a microfabricated chromatography column in a proteomic. (b) Microscope view of the spray nozzle. (Courtesy of N. Sarrut, CEA/LETI.) r s é s + 1ù é r + 1ù é æ y ö ù é æ x ö ù 1 1 Q=Uê ê ç ÷ úê ç ÷ ú ë s úû êë r úû êë è d / 2 ø úû êë è w / 2 ø úû

we derive

Q=U w d



d/2é r é s + 1 ù é r + 1 ù ïì æ y ö ù ïü Q1 = U ê 2 1 ê ú dy ý í ç ÷ ò ë s ûú ëê r ûú îï 0 ëê è d / 2 ø úû þï

d/2 r ïì é æ x ö ù ïü í ò ê1 - èç ÷ø ú dxý w / 2 úû ï îï 0 êë þ

(6.106)

Integrating (6.106) with the values s = r = 2 yields



Q1 » ε (2 + 3ε - 2ε 2 ) Q

(6.107)

Figure 6.56  Experimental result of chromatography separation of peptides in a proteomic reactor. Mass spectrometry trace of separation in the microfabricated column of Figure 6.47. Experiment using 50 fento-mol of a protein tryptic digest (b-Galactosidase) and liquid chromatography flow rate 300 nanoliters/min. Reconstructed chromatograms and corresponding mass spectra for a tryptic peptide of b-Galactosidase.

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Figure 6.57  Different types of single-phase flow focusing.

Equation (6.107) is an implicit equation relating the relative width e = w1/w to the flow rate ratio Q1/(Q1 + Q2). An efficient focusing is obtained for Q1 << Q2 and a first-order approximation of w1 is

w1 1 Q1 » w 2 Q

(6.108)

6.5.1.2  Single-Phase 3D Focusing

More sophisticated flow focusing may be obtained by a 3D geometry. In particular, focusing in a small defined region inside a channel can be achieved by using the principle of FFDs described in Chapter 4 (see also Figure 6.61). The designs and shapes must be compatible with the principles of microfabrication. Devices such as that of Figure 6.61 can realize interesting flow focusing [23, 24]. Two successive FFDs are used, the first one placed below the incoming flow and the next one placed above it. The evolution of the focusing of the flow is shown in Figure 6.61. After the first focusing, the flow is confined to a triangle at the upper part of the channel. After

Figure 6.58  2D flow focusing. (Courtesy of COMSOL.)

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Figure 6.59  Schematic of the flow focusing.

the second focusing, it is confined to a diamond-shaped region surrounded by the secondary liquid. By adjusting the secondary flow velocities, the liquid filament can be centered in the exit channel. Again, upon integration of (2.47), an approximated expression for the radius r of the “spot” is given by Q1 9 r 2 » Q 4 wd



(6.109)

6.5.1.3  Discussion

In the two preceding sections, the focusing of a flow has been presented. The precision of the focusing depends in the first place of the flow rate ratios, as indicated by (6.108) and (6.109). Note, however, that relatively large objects such as cells will continue along their streamlines, while smaller objects—DNA strands, for example—will have a convective-diffusive behavior and progressively smear out from their streamlines. 6.5.2  Pinched Channel Microsystems

Separation of macromolecules, particles, and, above all, cells is fundamental in biotechnology. For instance, extracting a well-defined population of cells in a wide variety of cells is at the basis of blood analysis. An interesting device is the pinched channel device (Figure 6.62). After focusing along a wall, the cells follow different streamlines according to their size [25, 26]. Large particles follow the streamline passing by their gravity center. After focusing, smaller particles are located closer to the wall and, after the sudden enlargement of the channel, they follow a streamline far apart from that of larger particles.

Figure 6.60  Different focusing obtained with the numerical software COMSOL for different flow conditions.

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Figure 6.61  (a) View of the flow focusing. (b) Calculated streamlines and focusing using COMSOL.

A schematic of the flow streamlines can be easily obtained numerically (Figure 6.63). If w1 and w2 correspond to the widths of the two channels, the deviation of a cell of diameter d initially focused along the upper wall is given by the geometrical relation y w -d 2 » 1 w2 w1

which leads to



y » (w1 - d 2)

w2 w1

(6.110)

If we write relation (6.110) for two different types of cells, characterized by a difference of diameter Dd, the increase of vertical distance between the two trajectories is



Dy »

D d w2 2 w1

(6.111)

Figure 6.62  Sketch of a pinched segment device. The focusing flow forces the cells along the upper wall, and the cells follow the streamlines passing by their gravity center.

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Figure 6.63  Schematic trajectories of two different types of cells or particles.

The larger the broadening rates w2/w1, the larger the separation distance after the enlargement. Figure 6.64 shows the result of a pinched channel system on the separation and extraction of the largest cell population. 6.5.3  Deterministic Arrays—Deterministic Lateral Displacement (DLD)

Deterministic arrays are another type of microdevices for sorting out particles and cells [27–30]. The principle of deterministic lateral displacement (DLD) is related to the particle shifting of streamline caused by an obstacle. The principle is illustrated in Figure 6.65. A small particle—or cell—slides between the rows of pillars and, on average, follows a straight line. A large particle cannot slide between the pillars, because it is trapped by its streamline and laterally shifted when approaching the pillars. On average, it follows an oblique path. Hence, small and large particles do not behave the same. A size threshold separates particles going straight and diagonally.

Figure 6.64  (a) Carrier flow before the pinched segment. (b) After the pinched flow system. Reprinted with permission from [26]. Copyright 2008, American Chemical Society.

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Figure 6.65  Sketch of a deterministic array. Small particles stay in their own stream channel, whereas large particles are forced to shift to the next stream channel at the approach of a post.

Let us analyze the situation into more details. In the schematic Figure 6.66, rows of posts have been placed with a shift e at each level (here the shift e = 1/3). The flow can be decomposed in n = 1/b streamlines (here n = 3, and the streamlines are noted as 1, 2, 3) having the same flow rate, equal to e times the flow rate between two pillars. Consider a small spherical particle in the stream channel 1. Always using the assumption that a particle or a cell follows the streamline passing by its centroid, because the diameter of the particle D is less than 2b, the particle is transported in its initial stream channel and globally has a straight trajectory. On the other hand, a particle of diameter D > 2b cannot stay in channel 1 when passing

Figure 6.66  Schematic of a DLD device with e = 1/3.

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through the gap between two pillars; it is forced into the next channel (e.g., channel 2). This motion is repeated at each row of pillars. Globally the particle follows a diagonal trajectory with an angle e. The critical particle diameter is then Dc = 2 β



(6.112)

Let us now calculate the value of b. Using the notations of Figure 6.62, we have β



g

ò u(x) dx = ε ò u(x) dx 0 0

(6.113)

Assuming a parabolic velocity profile between two pillars, the velocity u(x) can be expressed as



é g2 æ - çx u(x) = umax ê êë 4 è

gö ÷ 2ø

2ù

ú úû

(6.114)

Upon substitution of (6.114) in (6.113) and integration, the width b is the solution of the cubic equation 3



2

æ βö 3 æ βö ε çè g ø÷ - 2 èç g ø÷ + 2 = 0

(6.115)

and, using (6.112), the critical diameter is solution of 3



2

æ Dc ö æ Dc ö çè g ÷ø - 3 çè g ÷ø + 4 ε = 0

(6.116)

A plot of Dc/g versus e is shown in Figure 6.67. A very interesting application of DLD is given in [30] (Figure 6.68), where a cell is progressively deviated into a lysis solution and is eventually lysed with chromosome and cell contents being separated. 6.5.4  Lift Forces on Particles

In this section we assume medium to large velocities of the carrier flow (i.e., a flow Reynolds number approximately larger than 10). 6.5.4.1  Lift Force on a Particle or Cell

A relatively large rigid particle in a moderate or large Reynolds number flow is submitted to a drag force expressed by (6.87) and also to a lift force. There are two types of lift forces on the particle depending on its distance to the solid wall (Figure 6.69). The first lift force is called the shear-gradient induced lift and is linked to the flow velocity profile at the location of the particle. Due to its weight and size, the

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Figure 6.67  Plot of Dc/g as a function of e.



particle moves a little more slowly than the fluid and the relative velocity is larger on the wall side; a pressure difference acts on the particle to push it towards the wall, as well as a spin on the particle associated to the vorticity of the flow. The expression of this lift force was derived by Rubinow and Keller [31] for a spherical, rigid particle � � � (6.117) Flift = π R3ρ f ω ´ u � where rf is the density of the medium (carrier fluid), ω is the vorticity of the carrier � flow at the location of the particle, and u is its velocity. For a uniform laminar flow

Figure 6.68  View of the DLD of a cell towards a lysis solution resulting in the lyse of the cell and the separation of the chromosome from other cell contents.

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Figure 6.69  The two lift forces on a rigid particle: (a) shear-gradient lift and (b) wall-effect induced lift.

� field, the vorticity is equal to ω = ¶ u ¶y » U d where U is the average velocity and d is the channel depth. The lift force is then proportional to the square of the particle Reynolds number defined as Rep = U RH/vf. For the lift force to have an effect, the particle Reynolds must be sufficiently large. The second lift force is linked to the vicinity of the solid surface [32–34]. It is sometimes called a wake or wall effect induced lift force. The relative velocity near the wall side of the particle is reduced by the presence of the wall and the pressure on the wall side is larger than that on the centerline side. A lift force is exerted on the particle towards the channel center. This force can be expressed by

Flift = 9.22 γ� 2 ρf RH 4

(6.118)

which, in a Poiseuille-Hagen flow is equal to



æ U2 ö Flift = 9.22 ç 36 2 ÷ ρf RH 4 d ø è

(6.119)

where γ� is the shear rate at the location of the particle. The lift force may be expressed as a function of the particle Reynolds number as



Flift = 9.22 (36 Re p2 ) µ f ν f

RH 2 d2

(6.120)

showing that the boundary layer lift force is proportional to the square of the particle Reynolds number. Hence, for this lift force to be noticeable, the particle must have a relatively large radius and a relatively high velocity. 6.5.4.2  Focusing of Particles in a Straight Channel

Combining the effect of the two lift forces, a rigid particle of a given size tends to be focused at a constant distance from the wall as shown in Figure 6.70.

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Figure 6.70  (a) Particles dispersed in the flow submitted to lift forces and (b) particles at equilibrium are focused at a fixed distance from the wall.

6.5.4.3  Farhaeus Effect

At this point, we mention that the Farhaeus-Lindqvist effect should not be attributed to lift forces. The Farhaeus effect denotes the property of blood cells to move away from the walls [35]. This effect is linked to the non-Newtonian behavior of blood (as we have discussed in Chapter 2). With the viscosity being smaller at the walls due to the shear rate, the more liquid plasma circulates preferentially along the wall, pushing the cells towards the channel center. 6.5.5  Dean Flows in Curved Microchannels

The hydrodynamics of a Dean flow was presented in Chapter 2. In this section, we show what use can be made of Dean flows for the focusing of particles and cells [36, 37]. We recall here that the Dean effect is a vortex effect in curved microchannels. This rotational effect appears when fluid inertia is sufficient and curvature is large. It is characterized by the nondimensional Dean number defined as

De = U R ν R Rc = Re R Rc

(6.121)

where Rc is the curvature of the microchannel and R is its hydraulic diameter. A Dean number in the range of 0.1–1 realizes the rotational effect. Let us investigate how a Dean flow acts on particles and cells. Consider a spiral microchannel as shown in Figure 6.71 [38]. As we saw in Chapter 2, the effect of the curvature on the flow is the formation of two vortex tubes in the channel. Let us assume that the particles or cells are neutrally buoyant. Three forces are exerted by the flow on the cells (Figure 6.72): (1) hydrodynamic drag that contributes to transport the cells from the inlet to the outlet, (2) lift forces that tend to bring together the cells in four equilibrium positions (in a cylindrical tube, we have seen that lift forces maintain the particles on a tube a some distance of the walls), and (3) the Dean vortex that reduces the equilibrium positions to only one near the inner wall. 6.5.6  Bifurcation Channels

In this section we investigate the behavior of cells transported in microchannel networks, especially in a “branched” geometry such as the one schematized in Figure 6.73.

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Figure 6.71  Sketch of a spiral microchannel.

We shall distinguish two cases: low velocities, with negligible inertia forces, and larger velocities where inertial effects and lift forces intervene. 6.5.6.1  Bifurcation Channels at Low Flow Rates

At low flow rates, the assumption holds that a cell/spherical particle follows the streamline passing by its centroid [39, 40]. Let us consider spherical particles focused near a wall and approaching a bifurcation (Figure 6.74). For simplicity, let us assume a 2D situation and neglect the effect of the channel depth. The 3D calculation is similar, using the velocity expression given in Chapter 2. The velocity field can be approximated by the Poiseuille-Hagen quadratic profile



u(y) = 6U

y (w - y) w2

(6.122)

where U is the mean axial velocity, related to the flow rate by U + Q/(dw). At the bifurcation, the flow rate conservation equation requires that

dò

w1

0

u (y) dy = Q*1

(6.123)

Figure 6.72  The cells, initially dispersed in the channel, progressively concentrate under the effect of the Dean flow. (a) Initially the particles are dispersed in the channel, (b) lift forces impose four equilibrium positions, and (c) the Dean vortex leaves only one equilibrium position.

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Figure 6.73  Flow in a branched network.

where Q*1 is the flow rate in the secondary channel. Using (6.122) and (6.123) 3



2

Q* æw ö æw ö 2 ç 1 ÷ - 3ç 1 ÷ + 1 = 0 è wø è wø Q

(6.124)

Assuming that the flow rate distribution in the network is known—or has been calculated by the network theory of Chapter 2—(6.124) can be solved to produce the threshold width w1. Wall focused particles with a diameter D < 2 w1 will turn into the side channel. 6.5.6.2  Zweifach-Fung Bifurcation Law for Blood Flow

It has been experimentally observed that, at a bifurcation, blood cells have a tendency to travel into the daughter vessel that has the higher flow rate, leaving very few cells flowing into the lower flow rate vessel [41, 42]. The critical flow rate ratio between the daughter vessels for this cell separation is approximately 2.5:1 when the cell-to-vessel diameter ratio is of the order of 1. The first reason for this apportioning is that cells are drawn into the higher flow rate vessel because they are subjected to a higher-pressure gradient—due to the relative velocity difference, as in the case of the lift. The second reason is the torque on the cell/particle associated with the difference of relative flow velocity (Figure 6.75). This property, linked to the Farhaeus effect, has been used to design plasma extraction microsystems [43–45]. The principle is illustrated in Figure 6.76.

Figure 6.74  Trajectories of spherical particles at a bifurcation.

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6.5.7  Recirculation Chambers

Immobilization of cells within microfluidic channels is fundamental for the study of cellular behavior. A number of approaches such as encapsulation within photocrosslinkable polymers, adhesion to patterned proteins, and protein coatings have been widely investigated. However, immobilization of cells inside microfluidic devices is a promising approach for enabling studies related to drug screening and cell biology. In continuous microflow systems, microstructures that enable the capture of cells have been used to immobilize cells within fluidic channels [46, 47]. We give here the example of grooves etched perpendicularly to the continuous flow streamlines (Figure 6.77) [46]. The transported cells (cardiac muscle cell HL-1) are trapped in the cavities and remain trapped according to the cavity size. We recall from Chapter 2 that recirculation is obtained for a sufficient large Reynolds number and a small neck of the enclosure. Figure 6.77 shows the streamlines and the recirculation regions depending on the groove dimension. It is observed that the cells are immobilized along the downstream bottom edge of the larger grooves and along the upstream bottom edge of the smaller grooves. This is clearly linked to the recirculation pattern. Due to their aspect ratio, there is a complete recirculation in the smaller grooves, and the entering cells are carried backwards to the upstream edge. On the other hand, large grooves have very partial recirculation zones in the edges, and the entering cells are trapped downstream. The velocity of the main flow of course regulates the recirculation patterns. It is observed that cells trapped in recirculating grooves stay there much longer than those entering the larger, nonrecirculating grooves. The trapping of cells in enclosures in presently a very active topic of research and is very promising: it acts passively on cells and does not require sophisticated methods.

Figure 6.75  Analysis of the forces on a cell at a bifurcation.

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Figure 6.76  Sketch of a plasma extraction device using the Zweifach-Fung bifurcation law.

6.6  Conclusion In this chapter, we have presented the governing equations for the transport of biological substance. Two different approaches are possible depending on the size and mass of the molecules/particles of the substance. If they are sufficiently small, the advectiondiffusion equation is the one to choose. If the particles are larger and submitted to nonnegligible gravity forces, it may be interesting to calculate individually their trajectory, with or without the introduction of a Brownian perturbation. This is the case for the transport of cells in a continuous flow—for separation and/or immobilization purposes. To that extent, cellular microfluidics has become essential in biotechnology.

Figure 6.77  Contour plot of the y-velocity (transverse velocity) showing recirculating flows in the smallest enclosures only (25 and 50 mm). Particles are trapped differently in recirculating and nonrecirculating regions. (COMSOL calculation).

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299

Transport mechanisms of micro- and nanoparticles or macromolecules or cells are at the heart of any biotechnological microdevice. The ultimate goal being the handling or detection of the smallest possible number of these objects, it is necessary to have very precise control over the particles. This cannot be done in just one step. Transport by microflows and microdrops constitutes the first step to manipulate the particles. Other more specific steps are magnetic and electric methods and will be presented in the following chapters.

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6.6  Conclusion

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[39] Yamada, M., and M. Seki, “Microfluidic Particle Sorter Employing Flow Splitting and Recombining,” Anal. Chem., Vol. 78, 2006, pp. 1357–1362. [40] Yamada, M., and M. Seki, “Hydrodynamic Filtration for On-Chip Particle Concentration and Classification Utilizing Microfluidics,” Lab on a Chip, 2005, pp. 1233–1239. [41] Svanes, K., and B. W. Zweifach, “Variations in Small Blood Vessel Hematocrits Produced in Hypothermic Rats by Micro-Occlusion,” Microvascular Res., Vol. 1, 1968, pp. 210–220. [42] Fung, Y. C., “Stochastic Flow in Capillary Blood Vessels,” Microvascular Res., Vol. 5, 1973, pp. 34– 48. [43] Yang, S., A. Ündar, and J. Zahn, “A Microfluidic Device for Continuous, Real Time Blood Plasma Separation,” Lab Chip, Vol. 6, 2006, pp. 871–880. [44] Kersaudy-Kerhoas, M., L. Jouvet, and M. Desmulliez, “Blood Flow Separation in Microfluidic Channels,” Proceedings of µFLU08 European Conference, Bologna, December 2008. [45] Davis, J. A., D. W. Inglis, K. J. Morton, D. A. Lawrence, L. R. Huang, S. Y. Chou, J. C. Sturm, and R. H. Austin, “Deterministic Hydrodynamics: Taking Blood Apart,” PNAS, Vol. 103, No. 40, 2006, pp. 14779–14784. [46] Manbachi, A., S. Shrivastava, M. Cioffi, G. G. Chung, M. Moretti, U. Demirci, M. Yliperttula, and A. Khademhosseini, “Microcirculation Within Grooved Substrates Regulates Cell Positioning and Cell Docking Inside Microfluidics Channels,” Lab Chip, Vol. 8, 2008, pp. 747–754. [47] Shelby, J. P., D. S. W. Lim, J. S. Kuo, and D. T. Chiu, “High Radial Acceleration in Microvortices,” Nature, Vol. 425, 2003, p. 38.

Chapter 7

Biochemical Reactions in Biochips

7.1  Introduction In this chapter, we come to the very purpose of biochips. So far, we have dealt with the principles of microfluidic transport of macromolecules or microparticles and we have shown how these principles are used to displace and manipulate these objects inside microsystems. It is recalled here that the approach has been done in two steps, first, the study of the microfluidic flow as a carrier fluid; second, the study of the behavior of macromolecules and/or microparticles in such microfluidic flows. Up to now, we have only dealt with tools to perform a task. The essential question is: What task are we going to perform with such tools and what have all these techniques been developed for? This brings us to the purpose of biochips or bioMEMS. Basically, the main purpose of biochips is the analysis and recognition of macromolecules: DNA, proteins, and so forth. Ultimately, recognition process should be fast, sensitive and reliable, with the less possible false results, and largely parallelizable, allowing for simultaneous samples recognition. We will see in this chapter that biorecognition is based on a mechanism of key lock [1], which is in reality a biochemical reaction. This leads us to present the kinetics of chemical and biochemical reactions, with a special attention to some key reactions, like enzyme-catalyst reactions for proteins and adsorption reactions for DNA hybridization. Because in biochip technology, the targets to analyze are immerged and carried by a buffer fluid, recognition times do not depend only on the biochemical reaction kinetics but also on the presence of targets in the vicinity of the reagents. It is then necessary to treat the coupling between biochemical reactions and the advection-diffusion of targeted molecules. In conclusion, we point out that biorecognition is very dependent on detection sensitivity. The same care that is taken for developing an efficient bioreaction should be taken also to the detection process.

7.2  From the Principle of Biorecognition to the Development of Biochips 7.2.1  Introduction to Biorecognition

The discovery of the recognition potential by the immune system—sometimes called immune specificity—has been a major milestone in the development of biology and has been awarded many Nobel prizes. The first step was the discovery of the model key-lock by Fisher in 1892, sketched in Figure 7.1. In such an approach a 303

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Biochemical Reactions in Biochips

Figure 7.1  Principle of molecular recognition imagined by Fisher in 1892.

macromolecule binds to a specific complementary macromolecule and with no other one—at least in theory—due to a sufficient interaction force. The discovery of the key-lock model was followed by the development of synthetics analogs by Landsteiner (1930), and by the principle of solid phase immunoassay by Langmuir and Shaefer (1942), associated to the detection by immuno-fluorescence introduced by Coonsz (1942). With the determination of the structure of antibodies and their three-dimensional structure (Yalow and Berson (1959), Poljak et al. (1973)), and the production of specific monoclonal antibodies (Köhler and Milstein), all the pieces of the puzzle were present to give birth to antigen, antibody or protein biorecognition by a process that is called now immunoassay. 7.2.2  Biorecognition

As we just have seen, biorecognition is a process based on the lock and key principle. We show here two examples of biorecognition which are the basis of DNA and proteins biochips. Start with the case of DNA. DNA has a well-known double helix structure as indicated in Figure 7.2. The two helices are linked by hydrogen bonding between two of the four groups of base: A with G and C with T. It has been observed that, given some favorable conditions of temperature or pH, the double helix can dissociate (denaturing). Then a single DNA strands having a known sequence can recognize a complementary sequence on another single strand of DNA. This process is called hybridization (Figure 7.3). Thus, if we can fabricate a given DNA sequence, and have many copies of this sequence, these strands will recognize specific DNA strands with complementary sequence. Remark that the bonding is reversible and it corresponds to an equilibrium reaction as we will see later in this chapter.

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305

Figure 7.2  (a) Double-helix DNA structure. (b) Hydrogen bonds between A and T (Adenine and Thymine), and C and G (Cytosine and Guanine).

A similar lock and key approach can be done for antigen (or proteins) [2]. Roughly speaking, an antibody is a very complex molecule having a Y shape as symbolized in Figure 7.4. An antibody can recognize a specific antigen approaching one of the two binding sites located on both ends of the Y. The bond is made of

Figure 7.3  DNA denaturing and hybridization with a complementary sequence. (a) DNA double strand, (b) denaturing, (c) RNA copy, (d) hybridization.

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Figure 7.4  Schematic view of antibody-antigen binding.

multiple, noncovalent, interactions, like hydrogen bonds, van der Waals forces or Coulombic interactions. As for DNA, it is a reversible equilibrium reaction, called immunoreaction. Biologists use the term affinity to name the interaction force between antibody and antigen. 7.2.3  Biochip Technology

Biochips derived directly from the principle of biorecognition. First, let us consider two Eppendorf tubes, each one functionalized with a specific antibody (Figure 7.5). In such a case, we want to detect a precise antigen that can be recognized by the grafted antibodies. First, the samples are filled into the Eppendorf tubes, and after

Figure 7.5  Principle of biorecognition (in an Eppendorf tube).

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307

Figure 7.6  Macroscale immunoreactions in Eppendorf tubes to microscale immunoreactions in a microfluidic channel.

a sufficient time to allow for recognition, the tubes are washed. Only the couples of specific antibody-antigen resist to the washing process. Fluorescent markers are then introduced in the tubes and, again after washing, there remain only the marked specific couples. Detection is made by comparison of the emitted light between the positive and negative Eppendorf tubes. Biochips or bioMEMS are just a miniaturization the preceding principle (Figure 7.6), plus a systematic recognition due to the many different spots grafted (functionalized) with different types of antibodies.

Figure 7.7  Principle of biorecognition of macromolecules submitted to molecular diffusion on a microplate.

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Figure 7.8  Principle of biorecognition of macromolecules submitted to diffusion-advection in a circulating microchannel.

At this stage we can make a distinction between two types of biochips: a first category where the buffer liquid (the liquid containing the samples) is at rest (Figure 7.7) or moving (Figure 7.8) inside a microchannel or microchamber. We will see later in this chapter the difference in the process of capture. Figure 7.9 schematizes one of the first biochip. At the beginning there were only a few recognition sites, but nowadays some biochips have more than 10,000 recognition sites and thus can recognize many DNA sequences or antigens in one run. The presentation of biochip technology would not be complete if the detection problematic was left aside. Both the requirements of miniaturization and of ultra precise sensitivity have led to improvements in detection methods. Ultra sensitive detection is a subject of many studies and is not the subject of this book. However, one has to keep in mind that any development of a biochip must take into account

Figure 7.9  Schematic view of one of the first “biochip”(Kharpo 1989). Miniaturization is not yet achieved, the dimensions of the plate are 2.2 ´ 2.2 mm.

7.3  Biochemical Reactions

309

Figure 7.10  Examples of DNA detection by fluorescence in a DNA biochip. Fluorescent spots correspond to a positive reaction, and can be treated by image processing.

the definition of a detection device. Optic methods by fluorescence are very widely used. The principle is to attach a fluorescent bead to the immobilized target molecule and to implement a sensitive reception of the emitted light (Figure 7.10).

7.3  Biochemical Reactions Biochemical reactions can be extremely complex. In the following, we will not go into the details of these reactions, but only treat the reactions kinetics. The precise chemical interactions leading to reaction is the domain of the biologists and chemists. In biotechnology, we are only interested in the kinetics of the reaction to know and improve its efficiency and reduce its duration. 7.3.1  Rate of Reaction 7.3.1.1  Definition

Consider a chemical reaction of the form

A + n B ® mC + D

(7.1)

And note the molar concentration of a participant J at some instant by the symbol [ J ]. The rate of consumption of a reactant at a given time is defined by –d[R]/dt, where R is either A or B; this rate is a positive quantity (Figure 7.11). The rate of formation of one of the products C or D—which we denote P—is defined by d[P]/dt and is also a positive quantity (Figure 7.12). By considering the stoichiometry of reaction (7.1), we deduce the relations



d [D] 1 d [C ] d [ A] 1 d [B] = ==dt m dt dt n dt

(7.2)

The rate of reaction is uniquely defined by



v = vD =

1 1 vC = vA = vB m n

(7.3)

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Biochemical Reactions in Biochips

Figure 7.11  Definition of rate as the slope of the tangent drawn to the curve showing the variation of concentration with time.

Figure 7.12  Definition of rate as the slope of the tangent drawn to the curve showing the variation of concentration with time.

7.3  Biochemical Reactions

311

where



vD =

d [D] d [C ] d [ A] d [B] ; vC = ; vA = ; vB = dt dt dt dt

(7.4)

7.3.1.2  Rate Laws

Rate of reaction is essential to determine the kinetics of the reaction; it is also a guide to the mechanism of the reaction, for any proposed mechanism should be consistent with the observed rate law. Formally, the rate of reaction is a function of the concentration of the species present in the reaction [3] v = f ([ A],[B],[C ],[D])



(7.5)

For gases, the rate of reaction can be deduced from gas kinetics theory [4] and is expressed by the simple expression v = k [ A][B]



(7.6)

In a liquid phase, the reaction rate is more empirical and is often—but not always—obtained by an expression of the type v = k [ A]a [B]b



(7.7)

where k, a, and b are coefficients independent of time. The parameter k is called the rate constant. 7.3.1.3  Reaction Order

If a reaction rate is described by a formula of the type (7.7)—and this is a frequent case—it is possible to define the order of the reaction with respect to a species by its exponent, and an overall reaction order by the sum of the exponents corresponding to each species

o = a + b

(7.8)

Note that the order is not necessarily an integer; for example, the reaction rate may be 1



v = k [ A]2 [B] and the order of the reaction is 3/2. Some reactions may obey a zero order rate law, i.e. the reaction rate does not depend on the concentrations of the species, just on the parameter k. Some comments on the “constant” k are needed at this stage. First, the unit of k depends on the order of the reaction. For a zero order reaction, k is expressed in mole/m3/s; for a first-order reaction, k is dimensionally a frequency and expressed in s–1; for a second order reaction, the unit of k is m3/mol/s.

312

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Second, the magnitude of k for a given reaction order may be very different; the reason is that for a chemical reaction to proceed, the concerned molecules must have a closely defined state (for example, orientation). This condition is related to a probabilistic behavior which in term depends on the activation energy – sometimes called “Arrhenius” factor. Because the range of activation energy is large, the rate constants take very different values. 7.3.1.4  Temperature Dependence of the Reaction Constant

A closer look at the “Arrhenius factor” is obtained by considering the dependency of k on the temperature T. This property is often used to modify the equilibrium state of biochemical reactions like DNA hybridization, as we will see later on. The dependency of the rate constant k on the temperature T is given by Arrhenius law



ln k = ln A -

Ea RT

(7.9)

The two parameters A and Ea /R are called the Arrhenius parameters; more specifically, A is the “frequency” factor and Ea /R is the activation energy. These parameters are usually determined graphically from experimental results as shown in Figure 7.13. The intercept is ln A and the value of the slope –Ea /RT. High activation energy corresponds to a very steep slope and a very important dependency of k on the temperature. For the rate constant k be really a constant, the activation energy must be zero. Another form of (7.9) is -



k= Ae

Ea RT



(7.10)

Figure 7.13  Schematic drawing of the relation (7.9) showing the Arrhenius parameters A and Ea/RT.

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313

Under this form, an interpretation of the rate constant is the rate of successful collisions between reacting molecules [3]. The activation energy represents the minimum kinetic energy that reactants must have in order to form products and the “frequency” term A corresponds to the rate at which collisions occur. 7.3.1.5  Rate Laws and Reaction Kinetics

Rate laws are differential equations and their integration is the concentration as a function of time (i.e., the reaction kinetics). However, their integration is seldom possible analytically. Take the example of the first-order unimolecular reaction A®P



We have the following reaction rate d [ A] = -k [ A] dt

and the reaction kinetics is

[ A] = [ A]0 e - k t

(7.11)

7.3.1.6  Near Equilibrium Reactions

Very often a reaction is partly reversible, that is, the product is formed and at the same time dissociate according to A®P P®A



The rate laws for the two reactions are



v=

d [ A] = k [ A] dt

v¢ =

d [P ] = k ¢ [P ] dt

The concentration [A] is reduced by the first reaction and increases by the reverse reaction and the net rate of change is



d [ A] = -k [ A] + k¢ [P] dt

(7.12)

If the initial concentration is [A]0, and if there is no initial concentration of [P], then

[ A] + [P] = [ A]0

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Biochemical Reactions in Biochips

and the reaction kinetics is given by d [ A] = -k [ A] + k¢ ([ A]0 - [ A]) = (-k + k¢)[ A] + k¢ [ A]0 dt



(7.13)

This type of kinetics is classic and its mathematical structure will be investigated more closely later in the section dedicated to the Langmuir model. 7.3.1.7  Consecutive Reactions

Some reactions proceed through the formation of an intermediate. Consider for example the consecutive reactions A®I ®P

with



v=

d [ A] = kA [ A] dt

v¢ =

d [I ] = kI [I ] dt

Let us consider the concentration in the intermediate product [I ] d [I ] = kA [ A] - kI [I ] dt



(7.14)

The concentration [P] is given by the differential equation dP = kI [I ] dt



(7.15)

The first of the rate laws is an ordinary first order decay and, from (7.11), we can write [ A] = [ A]0 e - kA t



(7.16)

Substitution of (7.16) in (7.14) yields



d [I ] = kA [ A]0 e - kA t - kI [I ] dt and upon integration, assuming that [I]0 = 0



[I ] =

kA (e -kA t - e -kI t )[ A]0 kI - kA

If we notice that, at all times,

[ A] + [I ] = [P]

(7.17)

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315

We obtain the kinetics of production of [P]



ïì k e -kI t - kI e -kA t ïü P[I ] = í1 + A ý [ A]0 kI - kA îï þï

(7.18)

Concentrations in [A], [I], and [P] are sketched in Figure 7.14. This example corresponds to the decay of radioactive elements, such as for example the reaction

239

U®

239

Np ®

239

Pu

7.3.2  Michaelis Menten Model 7.3.2.1  Presentation of the Model

Now, we can tackle a very important group of reactions in biotechnology which are the enzymatic reactions. Enzymatic reactions are of utmost importance in biotechnology for two reasons: first, they are used to break proteins into smaller pieces called peptides that can be analyzed by a mass spectrometer; second they are a powerful method for amplifying detection in biorecognition processes. In fact, enzymes are only catalyst for the reaction. In an enzyme-catalyst reaction, a substrate is converted into products and the reaction rate depends on enzyme concentration. The catalyst driven reaction can be written by the symbolic expression

E + S « ES ® P + E

Figure 7.14  Concentrations of A, I, and P as a function of time.

(7.19)

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Biochemical Reactions in Biochips

where E, S, and P refer respectively to enzyme, substrate, and product concentration. Note that the name “substrate” corresponds to the species that is undergoing the chemical reaction. The notation ES refers to an intermediate state where the substrate E is bonded to the enzyme E (Figure 7.15). If we note k1, k–1 an k2 the rate constant of the 2 reactions of equation (7.19), the kinetics of ES binding is given by



d [ES] = k1[E][S] - k2 [ES] - k-1[ES] dt

(7.20)

And the product concentration kinetics is



V=

d [P ] = k2 [ES] dt

(7.21)

The Michaelis-Menten approach is based upon the simplification that assumes that the rate of production of ES concentration is constant, i.e.





d [ES] = k1[E] [S] - k2 [ES] - k-1[ES] = 0 dt Then, we have the relation [ES] =

k1 [E] [S] k2 + k-1

(7.22)

(7.23)

If we note that the total (initial) concentration of enzyme is

[E]0 = [E] + [ES]

Figure 7.15  Schematic view of the enzymatic reaction.

(7.24)

7.3  Biochemical Reactions

317

We can eliminate [E] and [ES] from (7.22), (7.23), and (7.24) and we deduce the rate of the reaction V=

k2 [E]0 æ k2 + k-1 ö 1 1+ ç è k1 ÷ø [S]

(7.25)

Introducing the notations

Vmax = k2 [E]0

(7.26)

k2 + k-1 k1

(7.27)

and



Km =

We obtain the Michaelis-Menten law V=

Vmax K 1+ m [S]

(7.28)

By remarking that the concentration of substrate [S] decreases with the concentration of product [P] according to

[S] = [S]0 - [P]

(7.29)

and if we recall from (7.21) that V is the rate of production of P, integration of (7.28) gives the relation between the concentration of product [P] and substrate [S]



Vmax t = [P] + Km Ln

[S]0 [S]0 - [P]

(7.30)

The constant has been adjusted so to have [P] = 0 at t = 0. Relation (7.30) is implicit. The kinetics of P derived from (7.30) is schematically represented in Figure 7.16. It is easy to see that the Michaelis-Menten law can be cast under the form



1 1 = V Vmax

æ Km ö çè 1 + [S] ÷ø

(7.31)

This form is called the Lineweaver-Burk expression of the Michaelis-Menten relation. It is convenient to determine the kinetic constants Km and Vmax. If we rewrite (7.31) under the form

1 1 K 1 = + m V Vmax Vmax [S] we see that the plot of the reciprocal velocity 1/V against the reciprocal substrate concentration 1/[S] is linear; the intercept is 1/Vmax and the slope is Km/Vmax (Figure 7.17).

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Biochemical Reactions in Biochips

Figure 7.16  Michaelis-Menten kinetics for product concentration.

7.3.2.2  More Insight on the Michaelis-Menten Law

The Michaelis-Menten model is only approximate because of the hypothesis of a constant rate of production of the complex [ES]. In reality, the kinetics differ somewhat of that of Figure 7.16. By looking closely at a plot of the kinetics of production of concentration P, we find three different parts corresponding to three different regimes (Figure 7.18). In the following we investigate the physics behind the three different parts of the reaction kinetics [5, 6]. The first part corresponds to the early stage of the reaction t < t0. During this stage, the concentration [ES] may be considered small, and (7.20) collapses to

d [ES] = k1[E][S] = k1[E]0[S]0 = const dt

Figure 7.17  Lineweaver-Burk linear relation for an enzymatic reaction. A simple linear regression produces the two Michaelis-Menten parameters Km and Vmax.

7.3  Biochemical Reactions

319

Figure 7.18  The three regimes of the enzymatic reaction.

Integrating this relation and substituting the result in (7.21) yields the parabolic kinetics



[P] = k1k2 [E]0[S]0

t2 2

(7.32)

After this early stage, the reaction acquires a steady state rate, according to the Michaelis-Menten approach



d [ES] =0 dt We integrate (7.21) to obtain the linear form



[P ] = a t + b Using continuity considerations, the preceding equation can be rewritten as



[P] = k1Vmax [S]0

t ö æ t0 ç t - 0 ÷ è 2 ø

(7.33)

This linear form corresponds to the second regime in Figure 7.18. At the end of the reaction, the substrate S becomes depleted, the concentration [S] may be neglected in (7.20), and the equation collapses to



d [ES] = -(k2 + k-1) [ES] dt

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Biochemical Reactions in Biochips

This equation can be integrated as an exponential law, and using (7.21), we find

d [P ] k2 c1 = exp(-(k2 + k-1) t) dt (k2 + k-1) where c1 is a constant. Integrating once more, we obtain the form



[P ] = -

k2 c1 exp(-(k2 + k-1) t) + c2 (k2 + k-1)

Using the value of the asymptote [P]¥ and continuity at time t1, the “asymptotic” regime is defined by



[P] = [P]¥ -

Vmax [S]0 t0 exp [ -k1Km (t - t1)] Km

(7.34)

So, the three different regimes and their assumptions have been identified: parabolic at first when [ES] is small, then linear when d[ES]/dt = 0, finally asymptotic when [S] becomes small. The system of (7.32), (7.33), and (7.34) is more accurate that the Michaelis-Menten law, but requires the knowledge of four parameters instead of 2: Vmax, Km, k2 and [P]¥ (it can be shown that the times t0 and t1 may be deduced from considerations on the derivability of the kinetic curve). In the following section, we show on an example the difference between the two models. 7.3.2.3  Example of Enzymatic Reaction

In this example, we set up a catalyst reaction between a synthetic protein and an enzyme. Consider a substrate composed of molecules of BAEE (Benzoyl-ArginylEthyl-Esther) - which is a synthetic protein—reacting in presence of trypsine—which is an enzyme. The experiment consists in mixing the substrate S (synthetic protein) with the enzyme E (trypsine) in a small beaker (Figure 7.19). Reaction kinetics is measured by an optical method based on the absorbance of light by the reaction product B. Figure 7.20 shows the absorbance curves at different times for three different initial concentrations of BAEE. Kinetics plots are then deduced from light absorbance.

Figure 7.19  Mixing BAEE and trypsine.

7.3  Biochemical Reactions

321

Figure 7.20  BAEE (Benzoyl-Arginyl-Ethyl-Esther) reacting with trypsine in a beaker. (a) Measurements of absorbance of light at 500 nm. (b) Experimental kinetics curves (concentration of Benzoyl versus time). From top to bottom: initial concentrations of substrate 1, 0.7 and 0.4 mM.

Michaelis-Menten model and the piecewise analytical model from preceding section have been used to interpret the experimental data (Figure 7.21). The piecewise analytical model fits better with the experimental data. However, as we have mentioned earlier, it requires more physical parameters than the Michaelis-Menten model, which remains a good trade off between simplicity and precision.

Figure 7.21  Comparison of reaction kinetics between experiments (dots), Michaelis-Menten model (dotted line) and piecewise analytical model (continuous line).

322

Biochemical Reactions in Biochips

7.3.3  Adsorption and the Langmuir Model 7.3.3.1  Langmuir Model

Another very important class of reactions in biotechnology is the adsorption of molecules on a solid functionalized surface. In particular, it is the case of DNA hybridization. In such a reaction, there are three components: first, a “free” substrate in a buffer fluid sometimes called “target” or “analyte,” in concentration [S]; second, a surface concentration [G]0 of ligands—or capture sites—immobilized on a functionalized surface; third a product which is the surface concentration of adsorbed targets, that we denote [G] (Figure 7.22). Note that [S] is a volume concentration (unit mole/m3) whereas [G] and [G]0 are surface concentration (unit mole/m2). Such a kinetic is called a Langmuir-Hinshelwood mechanism. The reaction is weekly reversible because targets are constantly captured by ligands and they constantly dissociate (at a smaller rate). The reaction may be symbolized by S®G G®S



In the case of adsorption, the reaction rates are somewhat different to the definition of the usual chemical rates, mainly because the rate the immobilization of the substrate S depends not only on the volume concentration at the wall, but also on the available sites for adsorption. Thus, we can write v=-



d [S ] = kon ([G ]0 - [G ])[S]w dt

d [G ] v¢ = = koff [G ] dt

(7.35)

where kon and koff are called respectively the adsorption and dissociation rates and [S]w is the concentration at the wall. For simplicity we will note G=[G], c = [S] and c0 = [S]w. The net rate of adsorption is then

dG = kon c0 (G 0 - G) - koff G dt

Figure 7.22  Adsorption of targets on a surface functionalized with immobilized ligands.

(7.36)

7.3  Biochemical Reactions

323

This last equation can be rewritten under the form



dG = konc0 G 0 - (konc0 + koff ) G dt

(7.37)

Equation (7.37) can be integrated and we obtain



G konc0 é1 - e -(konc0 + koff ) t ù = û G 0 konc0 + koff ë

(7.38)

Using (7.38), we obtain the surface concentration kinetics shown in Figure 7.23. At small times, the exponential term in (7.38) can be developed in a Taylor expansion and the surface concentration kinetics is the linear function of the time defined by

G = konc0 G 0 t

(7.39)

Equation (7.39) indicates that the kinetics described by the Langmuir equation (7.36) is rapid if the term konc0 is large (i.e., when the adsorption constant on the surface and the concentration in molecules are large). For longer times, the surface concentration approaches an asymptotic value defined by



G¥ konc0 = G 0 konc0 + koff

(7.40)

It can be verified in (7.38) that in the case where koff is zero, the asymptotic value is then G0 and the surface is becomes totally saturated. The larger the coefficient koff, the smaller the value of G¥ /G0. 7.3.3.2  Adsorption and Desorption

Suppose that after the hybridization has reached its asymptotic value, the remaining targets or analytes in solution are suddenly washed out. Desorption is then the driving mechanism and the corresponding kinetics is schematized by Figure 7.24.

Figure 7.23  Kinetics of surface concentration from equation (7.38).

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Biochemical Reactions in Biochips

Figure 7.24  Kinetics of adsorption and desorption.

The starting time for desorption is the time ta, and the surface concentration at this instant is Ga

Ga =

kon c0 G0 kon c0 + koff

(7.41)

The Langmuir equation for desorption is dG = -koff G dt



(7.42)

and the kinetics of desorption is G - k (t - t ) = e off a Ga



(7.43)

Desorption kinetics follows an inverse exponential law (Figure 7.24). The tangent to the desorption kinetic curve at t= ta is given by



G = 1 - koff (t - t a ) Ga

(7.44)

and the derivative at t = ta is



koff kon c0 dG = -koff G a = G0 d t t = ta kon c0 + koff

(7.45)

This last formula may be written under the form dG =d t t = ta

1

G0

1 + kon c 0 koff

(7.46)

7.3  Biochemical Reactions

325

Figure 7.25  Different adsorption and desorption kinetics depending on the kinetic constants.

When desorption follows adsorption, the kinetics of desorption depends not only on the desorption coefficient koff, but also on the values of G0 and kon. This property in shown in Figure 7.25 where different desorption kinetics are sketched, depending on the value of the saturation level. 7.3.4  Biological Reactions 7.3.4.1  Introduction

In the preceding sections, we have dealt with chemical and biochemical reactions, in the sense where the reactants were chemical or biochemical. In biology, there are slightly different types of reactions mainly because one has to take into account the rate of birth or death of living organisms by introducing a source or sink term in the reaction equations. However, these reactions have basically a mathematical formulation similar to chemical and biochemical reactions. 7.3.4.2  Predator-Prey Systems and the Lotka-Volterra Equations

Volterra developed this model in 1925 to predict the evolution of populations of animals in biology (fish population in the Adriatic Sea); nearly at the same time Lotka derived the same model for some chemical reactions [7, 8]. In the frame of this book, we are mostly interested in the biochemical aspect of the model and we present it briefly to introduce the special form of the competition terms in the system of Lotka-Volterra equations. We will show later that competition-displacement reactions for immunoassays present similarity with the predator-prey model and we will use the competition terms extracted from the Lotka-Volterra model. Biologists have developed models to predict the evolution of two interconnected populations, especially if one population is the prey and the other is the predator. It has been observed that the fluctuations of the two populations are closely linked (Figure 7.26).

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Figure 7.26  Time evolution of populations of preys and predators.

The simplest—but very interesting—model is that of Lotka-Volterra. If A and B represent the populations of preys and predators, their time evolution is given by [9, 10]



¶A = a A - b AB ¶t ¶B = -c B + d AB ¶t

(7.47)

In (7.47) the term aA represents the growth of population A if predators were absent—a being the rate of birth, and the term bAB the decrease in the number of prey due to the action of the predators (for these reasons, it is proportional to A and to B). On the other hand, the term cB represents the mortality rate of predators (b being the rate of deaths) and the term dAB the prey contribution (as a source term) to the predator growth rate (proportional to A and B). Mathematically speaking, the system (7.47) is strongly coupled and nonlinear. It is also structurally not stable. However, it bears much of the physics of the evolution of the prey-predator system. A first step in analyzing the Lotka-Volterra model is to render the system nondimensional by introducing the new parameters τ = a t; α =

c a

A B u=d ; v=b c a

(7.48)

Substituting (7.48) in (7.47) yields



¶u = u (1 - v) ¶τ ¶v = α v (u - 1) ¶τ

(7.49)

7.4  Biochemical Reactions in Microsystems

327

In the (u, v) plane, we obtain



dv v (u - 1) =α du u (1 - v) The variables u and v can be separated



(1 - v) (u - 1) dv = α du v u

(7.50)

Integration of (7.50) produces the phase trajectories

α u + v - ln(uα v) = H

(7.51)

For a given H, the trajectories in the phase plane are closed as illustrated in Figure 7.27. The diagram of Figure 7.27 shows that the two populations are linked and form the shifted cycles of Figure 7.26. For our concerns here, we will keep in mind that the nonlinear terms bAB and dAB represents the interactions between species A and B, especially the first term bAB, which represents the competition between the species.

7.4  Biochemical Reactions in Microsystems In the preceding section, we have investigated the kinetics of biochemical reactions. However, in the reality they can seldom be considered alone without taking into account other physical phenomena like diffusion or transport. Indeed, the reactants are usually injected into the microchamber in which they later diffuse and react. Thus, it is important to consider the global problem of advection-diffusion coupled with the biochemical reaction itself. We will consider next the kinetics of these coupled problems.

Figure 7.27  Closed phase plane trajectories from (7.45) with various H, corresponding to the LotkaVolterra system. The arrows denote the direction of change with increasing time t.

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Biochemical Reactions in Biochips

The reaction itself may be performed in two different ways: first, in the whole volume of the reaction chamber; second, on a functionalized surface located on the wall of the reaction chamber. The first type is called homogeneous reaction, the second heterogeneous reaction. Thus, we will consider successively the reactions kinetics coupled with advection-diffusion phenomena for homogeneous or heterogeneous situations. 7.4.1  Homogeneous Reactions 7.4.1.1  Governing Equations

Let us consider a second-order reaction of the type

A + n B ® mC occurring in a fluid volume where the reactants A and B are transported by a flow of velocity u. If we recall the advection-diffusion equation (Chapter 5), and notice that there is now sink-source term for concentration, the governing equations are [11] ¶ cA + u ÑcA = DA DcA - k cA cB ¶t ¶ cB + u ÑcB = DB DcB - n k cA cB ¶t



(7.52)

¶ cC + u ÑcC = DC DcC + mk cA cB ¶t where DA, DB and DC are the diffusion coefficients of species A, B, and C, and k the reaction rate. In (7.52), we have adopted the concentration notations cA = [A], cB = [B], and cC = [C]. Remark that the sink-source term has the characteristic form of a second-order reaction k[A][B]. The advection-diffusion equations are in this case nonlinear due to the nature of the sink-source term. Moreover, the two first equations in [A]and [B] are strongly coupled via their sink term. The third equation for [C] is only weakly coupled to the two other. The solution of the system is not easy and requires a numerical approach. Typically, there are two main cases of problems. Note tC the characteristic time of the reaction and tM the mixing time—which depends on dynamic fluid motion or only diffusion. Define a nondimensional number by



Da =

τC τM

(7.53)

Da is called the Dammköhler number. For a purely diffusive situation, the diffusion mixing time tM is of the order of



τM »

L2 D

7.4  Biochemical Reactions in Microsystems

329

where L is the characteristic dimension of the microsystem and D the order of magnitude of the diffusion coefficients of the reactants. After substitution, one obtains Da =

·

Dτ C L2

If Da is large, the reaction time is much larger than the mixing time. The concentrations in [A] and [B] can then be considered uniform in the reacting volume, and system (7.52) collapses to ¶ cA » -k cA cB ¶t ¶ cB » - n k c A cB ¶t

(7.54)

¶ cC » mk cA cB ¶t



This system is considerably easier to solve since it does not requires the knowledge of the velocity field and of the diffusion process. ·

If Da is small, the picture is much more complicated. There are reaction fronts that form and diffuse progressively before obtaining a homogeneous final state [12]. Numerical treatment is usually required for such systems.

7.4.1.2  Reaction-Diffusion at a Front Separating Two Reactants

Start from the same second order reaction

A+B®C and suppose that it takes place in a volume at rest (no convective transport), as sketched in Figures 7.28 and 7.29. In the case of a one-dimensional space (Figure 7.29), we have indicated the solution for the concentration alone in Chapter 4. This solution was an error (erf) function and the concentration spreads proportionally to the square root of time. Now we add a second-order reaction. The equations governing this type of reaction diffusion in one dimension geometry are ¶ cA ¶ 2 cA = DA - kcAcB ¶t ¶ x2 2



¶ cB ¶ cB = DB - kcAcB ¶t ¶ x2

(7.55)

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Figure 7.28  Reaction A + B ® C on a diffusion front.

together with the initial conditions



t=0

cA (x,0) = cA,0

cB (x,0) = 0

for

x<0

t=0

cA (x,0) = 0

cB (x,0) = cB,0

for

x < 0

(7.56)

The system of equations (7.55) can be transformed under a more symmetrical form by using the following transformations 1

x* =

x , L

t* =

2

L t, D

D = DADB ,

æ ö2 D L= ç ÷ è k CA,0CB,0 ø

(7.57)

We obtain the equivalent system of equations ¶ cA ¶2 c A = - β -1 cAcB χ ¶ t* ¶ x*2



2 ¶ cB -1 ¶ cB = - β cAcB χ ¶ t* ¶ x*2

(7.58)

We see that the system depends only on the two nondimensional parameters c and b defined by



χ=

DA , DB

β=

CA,0 CB,0

Figure 7.29  Reaction A + B ® C in one-dimensional geometry.

(7.59)

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331

Contrary to the diffusion problem, there is no known analytical solution for the system (7.58). We just mention here that, for very long times, there exists an approximate solution [13] using the variable R = cAcB. 7.4.1.3  Reaction and Advection-Diffusion in a Microchannel

In biotechnology, chemical reactions are very often performed in microchannels with parallel flowing buffer fluids (Figure 7.30). If y is the cross direction and x the direction of the flow, the two reactants A and B produce the component C by reaction-diffusion in the transverse direction y (Figure 7.31). Suppose a reaction symbolized by A+B®C



For a steady state flow of reactants, the system of equations for the reaction is [14, 15] U

¶ cA ¶ 2 cA = DA - kcA cB ¶x ¶ y2

U

¶ cB ¶ 2 cB = DB - kcA cB ¶x ¶ y2

U

¶ cC ¶ 2 cC = DC + kcA cB ¶x ¶ y2



(7.60)

where, for the diffusion term, the second derivative in x has been neglected in front of that in y. We use a change of variables resembling to that of (7.57) 1

x* =

x , U L2 D

y* =

y , L

D = DADB ,

æ ö2 D L= ç ÷ è k cA,0cB,0 ø

(7.61)

Figure 7.30  Reaction of two components flowing in parallel in a microchannel.

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Biochemical Reactions in Biochips

Figure 7.31  Sketch of the homogeneous reaction in flowing solution.

and, if we use the notations



c*A =

cA , cA,0

c*B =

cB , cB,0

χ=

DA , DB

β=

cA,0 cB,0

the system (7.60) becomes ¶ c*A ¶2 c*A 1 * * χ = - cAcB β ¶ x* ¶ y*2



¶ c*B 1 ¶2 c*B = - β c*A c*B χ ¶ y*2 ¶ x*

(7.62)

This system is nondimensional and depends only on the two parameters c and b. Again, this system is nonlinear, and strongly coupled. The use of numerical methods is required to solve such systems. In Figure 7.32, we show the computed solution obtained with the numerical software COMSOL. 7.4.2  Heterogeneous Reactions

Biochemical reactions are said heterogeneous when a ligand is immobilized on the solid walls and the targets (also called analytes or reactants) are in solution. Heterogeneous reactions are widely used in microsystems [16]. They appear to

Figure 7.32  View of the flow velocities (left) and the reaction zones (right) in the T shape channel. Calculation performed with the FEMLAB numerical software.

7.4  Biochemical Reactions in Microsystems

333

be often more convenient than homogeneous reactions, because, for one thing, the ligand at the wall can be reused after washing of the reaction chamber. Also, it is easier to proceed in two steps: immobilization of the ligands on a “reaction” surface at the wall, and introduction of the analytes by a carrier fluid, rather than designing a complex micofluidic system where the targets and ligands merge and mix at the same time in the reaction chamber. Finally, it is also often more convenient to detect the binded couples (targets-ligands) when they are immobilized on a wall surface. In this section, we give some examples of how the kinetics of heterogeneous reactions is calculated. The first example is that of a diffusion-reaction problem of the Langmuir type with concentration depletion. 7.4.2.1  Example of Concentration Depletion

There are two general trends in the treatment of biochemical reactions in biotechnology. First, the volumes are getting smaller and the ratio between the reaction surface (functionalized or labeled surface) and the volume is increasing. Second, the number of target molecules or particles is getting smaller in order to increase the specificity and efficiency of the biochip. It follows from these two considerations that the concentration in targets may be affected by depletion during the reaction [17]. It is no more a constant as was supposed when we produced the solution of the Langmuir equation. We investigate here the solution to the Langmuir equation in the case of a uniform concentration in the liquid volume but decreasing with time and we show that there exists a closed form solution to the Langmuir equation in the case of depletion assuming the concentration is spatially homogeneous. Consider the schematic case of Figure 7.33. The mathematical formulation of the problem is obtained by replacing the constant concentration c0 by a variable concentration c in the Langmuir equation.

dG = kon c (G 0 - G) - koff G dt

(7.63)

Figure 7.33  Schematic view of the microchamber. Because of depletion, the concentration c in the chamber decreases.

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Biochemical Reactions in Biochips

The concentration c is obtained by the mass balance taking into account the depletion due to immobilization of the targets [18] S



dG dc = -V dt dt

(7.64)

where S is the functionalized surface and V the volume of the microchamber. Taking into account the initial conditions, integration of (7.64) leads to c = c0 -



S G V

(7.65)

where c0 = c(t = 0) is the initial (uniform) concentration. Upon substituting (7.65) in (7.63), one obtains the differential equation for the surface concentration G

dG S ö S æ = konc0 G 0 - ç konc0 + koff + kon G 0 ÷ G + kon G 2 è dt V ø V

(7.66)

and, for the concentration dc S ö æ = koff c0 + ç konc0 - koff - kon G 0 ÷ c - kon c 2 è dt V ø



(7.67)

By considering an infinite volume V = ¥, (7.66) collapses to the usual Langmuir equation. The two differential equations (7.66) and (7.67) are of the mathematically wellknown Ricatti type [19]. Ricatti equations can be solved if a particular solution is known. In such a case, a change of variable using the particular solution transforms the Ricatti equation in a Bernoulli equation, which has a closed form solution. Usually, in order to promote hybridization, the temperature of reaction is set to a value well beneath the “fusion” temperature (i.e., the temperature where dissociation dominates), so that koff is usually kept small. However, koff is not necessarily vanishing in front of kon c, especially in our case where the initial concentration c0 is small. Thus, all the terms in (7.67) have their importance. Let us mention that values of kon and koff for immunoassays have been investigated thoroughly in the literature [20, 21]. The algebraic manipulations to obtain to the solution are somewhat long. We will only briefly indicate the approach: First, we search the solution of the secondorder characteristic polynomial in c: S ö æ kon c 2 - ç konc0 - koff - kon G 0 ÷ c - koff c0 = 0 è V ø



(7.68)

The two roots of the polynomial are given by c- =



c+ =

2

koff S ö 1 æ koff S ö koff 1æ c0 c0 c0 - G0 ÷ - G0 ÷ + 4 ç ç kon V ø 2 è kon V ø kon 2è 2

koff S ö 1 æ koff S ö koff 1æ - G0 ÷ + - G0 ÷ + 4 c0 c0 c0 ç ç kon V ø 2 è kon V ø kon 2è

(7.69)

7.4  Biochemical Reactions in Microsystems

335

k Note that these two roots have the dimension of a concentration and that off is the kon equilibrium constant. In order to simplify the notations, we note 2



koff S ö koff æ - G0 + 4 cˆ = ç c0 c0 è kon V ÷ø kon cˆ has also the dimension of a concentration. Using the notations cˆ , c+, and c–, it is possible to show that the solution of (7.67) is c=



1 1 1 e - kon cˆ t + (1 - e -kon cˆ t ) cˆ c0 - c -

+ c-

(7.70)

The kinetics of surface concentration is obtained by substituting the concentration kinetics from (7.70) in (7.65) G=



V (c0 - c) S

(7.71)

At first glance (7.67) seems to be somewhat complicated, but this is not really the case. Let us examine three different cases: 1. First, it is easy to see that if V ® ¥, then S/V ® 0 and we obtain c = c0, which is the expected result for a semi-infinite case. 2. Second, suppose that koff = 0 and that the number of targets is larger than the number of ligands (initial hybridization sites).





N=

ntargets c0 V = >1 nligands G 0 S

The functionalized surface will end being saturated by the immobilized targets. One S finds first that c – = 0 and cˆ = c0 - G 0. Then, (7.62) collapses to V 1 c= S ö S ö æ æ æ -kon ç c 0 - G0÷ t ö 1 -kon çè c0 - V G0÷ø t 1 è V ø ÷ ç1 - e + e S öç c0 æ ÷ø çè c0 - G 0 ÷ø è V S By letting t ® ¥ in the preceding equation, c ® c0 - G 0 and G ® G0 V 3. Third, that koff = 0 and that the number of targets is smaller than the number of ligands (initial hybridization sites)



N=

ntargets c0 V = <1 nligands G 0 S

S The functionalized surface cannot be saturated. It is easy to see that c - = c0 - G 0 V S and cˆ = - æç c0 - G 0 ÷ö è V ø

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Biochemical Reactions in Biochips

Then, the solution is 1

c= 1

S G0 V

e

æS ö - kon ç G 0 - c0 ÷ t èV ø

-

æ 1 ç1 - e S öç æ è G c 0÷ çè 0 V ø

æS ö - kon ç G 0 - c0 ÷ t ö èV ø ÷

+ c0 -

S G0 V

ø÷

V c0 < G 0 S As for the classic Langmuir kinetics, the kinetic curve approaches an asymptote when time is sufficiently important. The value of the asymptote is obtained by taking t = ¥ in (7.70)

By letting t ® ¥ in the preceding equation, one finds c ® 0 and G ®

c¥ = cˆ + c -



After substitution of the value of cˆ , the asymptotic value of the concentration is c¥ = c +





(7.72)

This result shows that the asymptote is simply given by the second root of the polynomial. This result can be derived directly from (7.67): when a permanent regime is attained, the time derivative of c vanishes, and the asymptotic value for the concentration then verifies the zero right-hand side of (7.67). Thus, the positive root (c+) of the characteristic polynomial is the value of the asymptote. The asymptotic value of the surface concentration—if there is no saturation—is then given by



G¥ =

V (c0 - c + ) S

(7.73)

The kinetics at t = 0 is the same as that of the Langmuir model



dG dt

t =0

= konc0 G 0

As we have seen, the ratio between the surface of capture and the volume of the chamber is an important parameter. Targeted concentration levels in biochips are usually in the range c0 = 10–9 to 10–12 mole/l. Depending if the targets are DNA strands or antibodies, the surface density of ligands lies in the range G0 = 500 to 1,000 molecules/μm2. Typical range of values of the ratio S/V is obtained by considering two types of microsystems: the first one is that of Figure 7.34 with a round functionalized spot, the surface of the functionalized spot is approximately S = p R2, R = 2 mm and V is of the order of 10 x 10 x 1 mm3, so that S/V = 0.13 m–1. The second case is that of a capillary tube of radius R = 100 μm, functionalized along one-tenth of its length. The ratio S/V is then equal to S/V = 2/R = 2,000 m–1. The depletion model shows that precautions should be taken when the detection device is operated under a closed volume condition (no circulation of fluid) or under a closed-loop condition. In such cases, depletion in targets/analytes has to be taken into account in the kinetics. Langmuir kinetics and (7.70) do not agree if the S/V

7.4  Biochemical Reactions in Microsystems

337

Figure 7.34  Capture of marked DNA strands on a round functionalized spot.

ratio is sufficiently large. Comparison of the kinetics of binding are shown in Figure 7.35 using S/V=10 m–1 and 4 different values of koff. The other parameters correspond to DNA hybridization and typical values of hybridization of 32-mers DNA segments have been used: kon= 110 m3/mole/s, G0 = 1.668 10–8 mole/m2 and c0 = 0.3 10–6 mole/m3. Kinetics obtained by the Langmuir model and the depletion model are similar only in the case of a large koff. For such a koff depletion is not very important because of the large desorption (dissociation) process. On the other hand, if we plan to find the kinetic constants by a fit of the kinetic curves, care should be taken if depletion occurs. The fitted values will differ if a Langmuir model is used instead of a depletion-modified Langmuir model.

Figure 7.35  Comparison of adsorption kinetics between the semi-infinite Langmuiran model (dotted line) and the present model (continuous line) for four different values of koff: 10–6, 10–5, 10–4, and 10–3 s–1 and a ratio S/V=10 m–1.

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Biochemical Reactions in Biochips

7.4.2.2  Diffusion Limited Reaction Introduction

In a biochip, heterogeneous reaction kinetics depends not only on the reaction rate—as we have seen in the first section of this chapter—but also on the diffusion of species towards the surface where reaction occurs. If diffusion is fast, for example in the case of very small diffusing molecules, the flux of these reactant molecules at the wall is such that there is no delay in the reaction. On the other hand, if diffusion process is slow, there will be depletion near the reacting surface and the biochemical reaction will be slowed down. In the first case, the process is limited by the reaction rate, in the second case it is limited by the diffusional flux. This problem is the subject of an abundant literature [22–26]. In the preceding section, the concentration was supposed uniform (but timedependant) in the microchamber volume. In this section, we treat the case where a depletion layer is observed near the functionalized surface (Figure 7.36). Governing Equations

Concentration in the fluid volume is obtained through the usual diffusion equation



∂c = DDc ∂t

(7.74)

where c is the concentration of substrate, D its diffusion coefficient. In a twodimensional Cartesian system, this equation can be rewritten as



æ ∂2 c ∂2 c ö ∂c = Dç 2 + ÷ ∂t ∂ y2 ø è∂x

(7.75)

Figure 7.36  Schematic view of the diffusing targets and hybridization on the functionalized surface.

7.4  Biochemical Reactions in Microsystems

339

On the other hand, at the functionalized wall, the Langmuir model for binding yields dG = kon cw (G 0 - G) - koff G dt



(7.76)

where G is the concentration in immobilized analytes, G0 the initial concentration in available hybridization sites, kon the adsorption coefficient at the wall, koff the desorption coefficient at the wall—also called elution—and cw the concentration at the wall. Equations (7.76) and (7.75) are not independent. They are coupled by the Fick’s law dG = -D Ñ c w dt



(7.77)

Equations (7.76) can be substituted in (7.77) and we obtain the value of the wall concentration as a function of the wall concentration (and its derivative) G=



D Ñ c w + kon cw G 0 (kon cw + koff )

(7.78)

Equation (7.78) shows that there is some kind of equilibrium between the value of the concentration near the wall and the surface concentration in hybridized targets. If the concentration near the functionalized surface decreases—for any reason, such as an interruption in the arrival of targets—there will be a temporary depletion of targets near the wall, and the immobilized DNA strands will start to dissociate. On the other hand, if there is a large supply of targets near the wall, the rate of hybridization will increase. Numerical Approach

Numerical methods must be set up to solve such problems. If one has access to a finite element software, the numerical approach is straightforward. If not, and if the geometry of the microchamber is simple, a numerical formulation based on a finite difference approach can be set up using the following discretization based on the grid of Figure 7.37. First, using a Crank-Nicholson semi-implicit scheme [19], the advection-diffusion equation (7.75) can be discretized under the form n +1 n +1 n +1 n +1 n +1 n +1 cin, +j 1 - cin, j D é ci +1, j - 2 ci, j + ci -1, j ci , j +1 - 2 ci , j + ci , j -1 ù ú = ê + 2ê Dt (D x)2 (D y)2 ú ë û



n n n n n n D é ci +1, j - 2 ci, j + ci -1, j ci, j +1 - 2 ci, j + ci, j -1 ù ú + ê + 2ê (D x)2 (D y)2 úû ë



(7.79)

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Biochemical Reactions in Biochips

Figure 7.37  Schematic view of the discretization grid.

In (7.79), the subscripts n or n + 1 refer to the time step. Next, using an implicit scheme, (7.76) becomes G in +1 =

G in + kon cin,0+1 G 0 D t 1 + (kon cin,0+1 + koff ) D t

(7.80)

where the notation ci,0 refers to the concentration at the wall. Fick’s law can be discretized by



n +1 n +1 cin,0 - cin,1 ù G in +1 - G in D é ci,0 - ci,1 ê ú =+ Dt Dy ê 2 2 ú ë û

(7.81) n+1

After substitution of (7.81) in (7.80), we obtain the linear relation between ci,0 and n−1 ci,1 , and the whole system can be cast under the matrix form

[ A]{c n +1} = {sn }

(7.82)

where the vector {sn} depends on the concentrations at the preceding time step. By using the relevant boundary conditions, and by inversing the system [27], one obtains the concentration distribution at the new time step n + 1. Example of Diffusion Limited Reaction

Suppose a microchamber with a round functionalized spot, as shown in Figure 7.38. Hybridization kinetics are monitored by fluorescence (Figure 7.39). If the dimensions of the chamber are sufficiently large, and the diffusion coefficient sufficiently small, the reaction is slowed down by a depletion of targets in the vicinity of the reactive surface. This case is called diffusion limited reaction. It can be shown [24] that the nondimensional Dammkohler number characterizes the type of reaction

Da =

D/δ kon G 0

(7.83)

7.4  Biochemical Reactions in Microsystems

341

Figure 7.38  Sketch of a functionalized surface in a microchamber where the buffer fluid is at rest.

where d is the vertical dimension of the microchamber. If Da is large the reaction time is larger than the diffusion time and diffusion does not delay the reaction kinetics. On the other hand, if Da is small, the reaction kinetics is limited by diffusion. A very interesting observation can be made by analyzing the relation (7.83). If the vertical dimension of the microchamber is sufficientlty small, the Dammkohler number becomes large and diffusion does not limit the reaction kinetics. Thus, it is best to design a microchamber with a vertical size smaller than



δ <<

D kon G 0

(7.84)

Usually, for oligonucleotides, the dimension d should be smaller than 50 to 70 μm.

Figure 7.39  View under the microscope of a functionalized surface during the process of hybridization: the intensity of the laser light diffracted relates to the surface concentration of hybridized DNA. (Courtesy of CEA-LETI/BioMérieux.)

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Biochemical Reactions in Biochips

Figure 7.40  Concentration depletion in the case of a diffusion limited reaction. Caution: the representation is not to scale, the vertical axis has been extended for visualization.

When the reaction is diffusion limited, a region of concentration depletion forms at the vicinity of the reactive surface. Figure 7.40 shows a contour plot of the concentration in such a case. If the reaction were not diffusion-limited, the Langmuir kinetics would be the right one. If the reaction is diffusion-limited, the kinetics curves depart from that of the Langmuir model (Figure 7.41). The numeric model may be used to predict the reaction parameters. They can be adjusted to fit the experimental results (Figure 7.42).

Figure 7.41  (a) Average concentration of analyte at the wall (cp/cbulk) versus time. The dotted line is the Langmuir solution (kinetics limited only by the biochemical reaction). The three lines correspond respectively to D = 10-11, 3. 10-11 and 10-10 m2/s, and to the three Dammkoehler numbers 1/2, 1/0.66, and 1/0.2. (b) Same as (a) but for the surface concentration of hybridized sites (G/ G0) versus time. Note that the Langmuir solution does not depend on D.

7.4  Biochemical Reactions in Microsystems

343

Figure 7.42  Kinetics of hybridization of DNA segments on a solid flat surface functionalized with complementary DNAs. The dots correspond to the experimental results and the continuous line to the calculation.

7.4.2.3  Advection-Diffusion and Biochemical Reactions Introduction

Consider the case of a buffer fluid flowing through a microchamber comprising one or more reactive surfaces (Figures 7.43 and 7.44). Usually such experimental devices are used to find the kinetic constants kon and koff for a given sequence of DNA. Governing Equations

The advection-diffusion equation for concentration in the fluid volume is



∂c = DD c - v Ñc ∂t

Figure 7.43  Schematic view of DNA hybridization in a microchannel (side view).

(7.85)

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Biochemical Reactions in Biochips

Figure 7.44  Schematic view of a typical hybridization microchamber, with different types of “spots” for hybridization.

where c is the concentration of targets or analytes, D is its diffusion coefficient, and v is the flow velocity. Very often the channel is such that we can use a HagenPoiseuille velocity field (see Chapter 1). In the case of a flat channel limited by two parallel plates separated by a distance d, the flow velocity is v(y) =

3 v 2

é æ y ö2ù ê1 - ç ÷ ú êë è d / 2 ø úû

(7.86)

_ where v is the average velocity in the channel. Because the fluid velocity is directed along the x-axis, the advection-diffusion equation can be cast into the form



æ ∂2 c ∂2 c ö ∂c ∂c = Dç 2 + -v ÷ 2 ∂t ∂ x ∂y ø è∂x

(7.87)

At the functionalized wall, the Langmuir model for binding yields



dG = kon cw (G 0 - G) - koff G dt

(7.88)

where G is the concentration in immobilized analytes, G0 the initial concentration in available hybridization sites, kon the adsorption coefficient at the wall, koff the desorption coefficient at the wall—also called elution—and cw the concentration at the wall.

7.4  Biochemical Reactions in Microsystems

345

The coupling between the two equations is realized by Fick’s law dG = -D Ñ c w dt



(7.89)

Equations (7.89) can be substituted in (7.88) and we obtain the value of the wall concentration as a function of the wall concentration (and its derivative) G=



D Ñ c w + kon cw G 0 (kon cw + koff )

The same remarks as in the preceding section can be made concerning the fluctuating equilibrium near the wall. Numerical Approach

Numerical methods must be set up to solve such problems. If one has access to finite element software, the numerical approach is straightforward. If not, and if the geometry of the microchamber is simple, a numerical formulation based on a finite difference approach can be set up using the following discretization based on the grid defined in Figure 7.45. The method is very similar to that of the preceding section. First, using a Crank-Nicholson semi-implicit scheme [15], the advection-diffusion equation (7.87) can be discretized under the form n +1 n +1 n +1 n +1 n +1 n +1 cin, j+1 - cin, j D é ci +1, j - 2 ci , j + ci -1, j ci , j +1 - 2 ci , j + ci , j -1 ù ê ú = + 2 2 Dt 2ê D x D y ( ) ( ) ú ë û

+



n n n n n n D é ci +1, j - 2 ci , j + ci -1, j ci , j +1 - 2 ci , j + ci , j -1 ù ê ú + 2 2 2ê x y ( ) ( ) D D ú ë û

- vi, j

(c

n +1 i, j

- cin-+1,1j

(7.90)

)

Dx

Figure 7.45  Schematic view of the discretization grid.



346

Biochemical Reactions in Biochips

In (7.90), the subscripts n or n + 1 refer to the time step. Note that the velocity term must be discretized following the flow direction. Next, using an implicit scheme, (7.89) becomes G in +1

=

G in + kon cin,0+1 G 0 D t 1 + (kon cin,0+1 + koff ) D t

(7.91)

where the notation ci,0 refers to the concentration at the wall. Fick’s law can be discretized by



n +1 n +1 cin,0 - cin,1 ù G in +1 - G in D é ci,0 - ci,1 ê ú =+ Dt Dy ê 2 2 ú ë û

(7.92)

n+1 After substitution of (7.92) in (7.91), we obtain the linear relation between ci,0 n+1 and ci ,1 , and the whole system can be cast under the matrix form

[ A]{c n +1} = {sn }



(7.93)

where the vector {sn} depends on the concentrations at the preceding time step. By using the relevant boundary conditions, and by inversing the system [22], one obtains the concentration distribution at the new time step n + 1. Example of Advection-Diffusion-Reaction Kinetics

In this example, we show how experimental records of hybridization kinetics combined with the numerical model of the preceding section can be used to find the kinetics constants of different DNA strands. The experiment setup corresponds to that of Figure 7.44. In this experiment, a constant buffer fluid flow carries different types of DNA strands—with different sequences and length. The average flow velocity is 1 mm/s (10 μl/mn) and the dimensions of the microchamber are 10 × 10 × 1 mm. The flow is turned on during 50 minutes, then it is stopped during 310 minutes and it is again turned on for the rest of the experimental time. Hybridization kinetics is monitored by fluorescence. Four different kinetics are obtained for the four different types of oligonucleotides (Figure 7.46). The approach is to use the numerical model of the preceding section and to fit of the kinetics curves by varying the parameters c0 (bulk concentration), D (diffusion coefficient), kon (constant of hybridization), and koff (constant of desorption). At the experiment temperature, koff can be considered negligible, so that the fit is performed by varying three parameters only (Figure 7.47). A few trials are enough to find the values of c0, D, and kon. Table 7.1 shows the typical values for DNA strands of different length obtained by this approach. One of the conclusions of this analysis is that the adsorption constant depends not only on the length of the DNA strand but also on the nature of the basis pairs (A,C,G,T). 7.4.2.4  Displacement (Competition) Reactions Introduction and Principles

In the preceding sections, we dealt with a class of heterogeneous reactions that may be called sandwich reactions. Up to now, sandwich reactions have been the most

7.4  Biochemical Reactions in Microsystems

347

Figure 7.46  Experimental hybridization kinetics for different DNA strands. Fluorescence versus time (in mn).

common. This class of reactions is schematized in Figures 7.48 to 7.50. It is done in three steps: (1) functionalization of a zone on the solid wall, (2) hybridization or capture, and (3) detection by a fluorescent tag (or another method). The first step is schematized in Figure 7.48: a surface of the solid wall is functionalized with ligands having an affinity with the target macromolecule. Then, the buffer fluid sample is injected in the microchamber and the targets are adsorbed and immobilized (temporarily) by an equilibrium reaction on the available sites (Figure 7.49). Finally, detection of the concentration in immobilized targets is performed (Figure 7.50). Sandwich reactions were historically first. However, in the 1980s, a new, more sophisticated concept was invented to avoid tagging the biological targets with a marker, which is often a complicated process and which is not always possible (for example, biochips aimed at bioterrorism targets detection have no a priori knowledge of the target, which then cannot be tagged). This new type of reaction process is called displacement reaction, or sometimes competition reaction [28–31].

Figure 7.47  Fit of the experimental kinetics for 29 and 15 bp (basis pairs) DNA strands.

348

Biochemical Reactions in Biochips Table 7.1  Typical Values of Advection-Diffusion and Reaction Parameters for Short DNA Strands Number of Basis Pairs 32 14 15 29

D (m2/s) 0.85 10–10 0.80 10–10 0.75 10–10 0.70 10–10

Kon (m3/mole/s) 110 60 125 75

C0 (mole/m3) 0.34 10–5 0.24 10–5 0.46 10–5 0.29 10–5

The principle is shown in Figures 7.51 to 7.54. The first step is the same as before: functionalization of a surface of capture with ligands. The second step is most of the time the most difficult because it requires finding molecules “analog” to the targets, with an affinity to the ligand smaller than the target. This is the tricky part that biologists and chemists have to resolve. These analogs can be marked, usually by fluorescent marker. Then they are then immobilized to the ligands by a usual sandwich-type reaction. The higher the saturation rate, the better for the sensitivity of the detection. When the targets arrive in the vicinity of the surface of capture, they displace some of the analogs, due to their higher affinity with the ligands. It can be said they compete with the analogs to bind the ligands, hence, the name competition. The result of the displacement is a change in the fluorescence level. At the location of the surface of capture, the level of fluorescence decreases because of a decrease in the marked analog surface concentration. Further down the microchannel, the displaced analogs carry their fluorescent markers and there is a fluorescence increase at the microchannel outlet. In conclusion, the targets are not marked, just the analogs are marked, which is more convenient. However, the process requires finding the best analogs possible and is longer to set up. In a sense, displacement reactions are the negative of sandwich reactions. Displacement Reaction Using FRET

A very interesting type of displacement reaction uses the fluorescence resonance energy transfer (FRET) principle [32, 33]. When a fluorophore is excited by a light at the right wavelength, it emits light at a slightly shifted wavelength (Figure 7.55).

Figure 7.48  Step 1: Functionalization of the solid surface with ligand antibodies.

7.4  Biochemical Reactions in Microsystems

349

Figure 7.49  Step 2: Capture of the targets.

However, if another fluorophore is placed next to this fluorophore, it quenches the light emission by energy transfer (Figure 7.56). Using the FRET principle, it is possible to set up the protocol for a displacement reaction [33]. First, the surface of capture is functionalized with tripods formed by an antibody and a fluorophore (Figure 7.57). Then analogs marked with “quencher” fluorophores are immobilized on the tripods by chemical affinity (Figure 7.58). It results in a quenching of the fluorescence of the tripods. The target antigens in the buffer fluid then displace the analogs with their fluorophores, and the fluorescence increases again (Figure 7.59). The advantage of this type of reaction is that of a typical displacement reaction, plus the fact that the functionalized surface of capture can be regenerated just by reintroducing the analogs in the microchamber. The microsystem can thus be reused.

Figure 7.50  Step 3: Marking the targets with fluorophores.

350

Biochemical Reactions in Biochips

Figure 7.51  Functionalization of the surface of capture with ligands.

Modeling Displacement Reactions

If the technology of displacement reactions is now well known and biochips using this type of reactions are currently used [ ], the modeling of such reactions is still under development. The model presented here is based on the analogy with the competition term in the Lotka-Volterra equations [34]. First suppose that the analogs and the targets are together in solution and consider the Langmuir equations for each antigen A* and A (A* refers to the target and A the analog). d G1 = k1 c1 (G 0 - G1 - G 2 ) - k-1 G1 dt

d G2 = k2 c2 (G 0 - G1 - G 2 ) - k-2G 2 dt

(7.94)

The first equation in (7.94) considers that the analog A has the kinetics coefficient k1 and k-1 and can bind to ligands in a concentration of (G0 – G1 – G2). The second equation has the same meaning for the targets this time. In such an approach there is no displacement because each type of molecule binds to the ligands independently. In reality, because the analogs are immobilized first, and the targets arrive after washing of the channel, the Langmuir system of equations should be d G1 = -k-1 G1 dt

d G2 = k2 c2 (G 0 - G1 - G 2 ) - k-2G 2 dt

Figure 7.52  Saturation of the functionalized sites with analogs tagged with a marker.

(7.95)

7.4  Biochemical Reactions in Microsystems

351

Figure 7.53  Displacement of the analogs by the targets.

Still there is no competition between the two species in system (7.95). Now let us introduce a competition term by similarity with the Lotka-Volterra equations. By analogy, it can be assumed that the target A* is the predator and the analog A the prey. A competition term is then introduced in the equations to indicate that there is a competition between A* and A. The system (7.94) becomes d G1 = k1 c1 (G 0 - G1 - G 2 ) - k-1 G1 - β c2 G1 dt

d G2 = k2 c2 (G 0 - G1 - G 2 ) - k-2G 2 dt

(7.96)



Figure 7.54  Detection either by decrease in the fluorescence of the capture surface, either by the increase of fluorescence at the channel outlet.

352

Biochemical Reactions in Biochips

Figure 7.55  Schematic view of a tripod for functionalization of the solid surface. Fluorophore excitation and emission wavelengths are slightly shifted in frequency.

where the coefficient b has the same dimension than the adsorption constant k1 (m3/mole/s). Displacement of antigens A by the targets A* is then equal to the displacement rate b multiply by the concentration of free targets A* and immobilized analogs A. In the particular operation sequence that we have defined here, the displacement equations are



d G1 = -k-1 G1 - β c2 G1 dt d G2 = k2 c2 (G 0 - G1 - G 2 ) - k-2G 2 dt

(7.97)

To the difference to the Lotka-Volterra model, we have not introduced cross-terms in the second equation for A* because the targets are assumed to not be affected by the analogs A. The first equation can be integrated (c2 is assumed constant)

G1 = e -(k-1 + β c2 ) t Ga

(7.98)

Figure 7.56  Principle of fluorescence quenching by FRET. Most of the fluorescent activity is quenched by the proximity of TMR.

7.4  Biochemical Reactions in Microsystems

353

Figure 7.57  Step 1: Functionalization of the surface with tripodes.

where Ga is the initial surface concentration of immobilized analogs. Under the form (7.98), the model shows that desorption of analogs A under the action of competition of the targets A* is faster than if the analogs were alone. We are then left with one differential equation for the kinetics of adsorption of the targets



d G2 = k2 c2 (G 0 - G a e -(k-1 + β c2 ) t - G 2 ) - k-2G 2 dt

(7.99)

Equation (7.99) can be integrated under a closed form, using some algebraic developments. The method consists in writing

G 2 = G*2 + ε where G*2 is the solution of the equation



d G*2 = k2 c2 (G 0 - G*2 ) - k-2G*2 dt

Figure 7.58  Step 2: Quenching by FRET of the fluorescence.

354

Biochemical Reactions in Biochips

Figure 7.59  Step 3: Targets displace analogs; detection is associated with the restoration of fluorescence.

which means that G*2 is the solution of the Langmuir equation if A* was alone. It can be shown that the solution is given by G2 k2 c2 k2 c2 G a -k-1 t = (1 - e -(k2 c2 + k-2 ) t ) + (e - e -(k2 c2 + k-2 ) t ) G 0 k2 c2 + k-2 k-1 - (k2 c2 + k-2 ) G 0 (7.100) G*2 and the second term (always negative) is a correction G0 taking into account the occupation of surface sites by the analogs A. Note that the asymptotic value of G2 is

The first term is the term



G2 k2 c2 t®¥= G0 k2 c2 + k-2

Figure 7.60  Step 4: Regeneration of the functionalized surface. Fluorescence activity is quenched once more time.

7.4  Biochemical Reactions in Microsystems

355

which is the same asymptote as for G*2: at a very long time, all the analogs have dissociated and the targets hybridization kinetics follow a simple Langmuir law. Example of Displacement Reaction

Efficiency of displacement reactions for different targets, mostly biochemical, have been tested in microdevices using a continuous flow—hence the name CFI for continuous flow immunoassays. In order to estimate the displacement rate, the analytes (targets) are introduced in the channel by concentration “bursts” of different level (Figure 7.61). Each burst of analyte concentration displaces a certain number of analogs. So, at the microchannel outlet, bursts of concentration in marked analogs are detected. The displacement rate is determined by comparison between the concentration level in targets and the corresponding concentration level in analogs. The concentration in analogs is given by a series of differential equations of the type



d G1 = -k-1 G1 dt

for

t < ti-

d G1 = -k-1 G1 - β c2i G1 dt

for

t1- £ t £ ti+

d G1 = -k-1 G1 dt

for

t > ti+

(7.101)



where c2i is the burst concentration in analytes between times ti– and ti+. The solution of the system (7.101) for the first burst is G1 = G a e -k-1 t -

G1 = G a e β c21 t1 e -(k-1 + β c21) t

G1 = G a e

- β c21 (t1+ - t1- )

e -k-1 t

for

t < t1-

for

t1- £ t £ t1+

for

t > t1+



Figure 7.61  Schematic view of an experimental displacement reaction microdevice.

(7.102)

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Biochemical Reactions in Biochips

Figure 7.62  Calculated bursts of displaced analogs. Comparison with the case of desorption alone.

The general solution for an experiment with i successive bursts is é i -1 - β c (t + -t - ) ù G1 = G a e -k-1 t êÕ e 2 j j j ú e - β c2i (t -ti ) êë j =1 úû G1 = G a e -k-1 t

é i - β c2 j (t + -t j- ) ù j êÕ e ú êë j =1 úû

for

ti- £ t £ ti+ (7.103)

for

t > ti+

In Figure 7.62, we show a calculated kinetics of displaced analogs. In such a case, the bursts of targets were all of the same concentration. The base line is the kinetics of desorption of analogs alone (without the competition of the targets). The bursts of displaced analogs are decreasing due to the fact that the immobilized concentration in analogs is decreasing. In Figure 7.63, results of the model have been compared to experimental results [35, 36]. In this case, different concentrations of targets (RDX) are changed. These concentrations are clearly linked to the resulting displaced analogs concentrations.

Figure 7.63  Kinetic curves of displacement of RDX [36]. Comparison between experimental results and calculated kinetics. The base line is the desorption of RDX analogs alone.

7.5  Conclusion

357

Up to now, displacement reactions have been less sensitive than sandwich reactions, but their performances are steadily increasing.

7.5  Conclusion The most important application of biochips is biorecognition, and biorecognition is based on key-lock type of reaction. To this regard, we have investigated the physics of biochemical reactions and determined the kinetics of the most important reactions such as DNA hybridization and enzymatic reactions for proteins. These kinetics, however, can be modified by the concentration of reacting species (analytes or targets) and the coupling between biochemical reaction and advection-diffusion of reagents in the biochip is essential. Finally, we have distinguished between two types of reaction, the “sandwich” reaction that derives directly from the key-lock approach, and the displacement reactions that are more complex and require the use of an analog to the target. It is essential to point out that detection is an important part for the conception of any biochip. It is not sufficient to have a very efficient capture—by hybridization or immunorecognition—if there is no sensitive detection associated with it. The reading of the biochip reactive surface should be at least as sensitive as the reaction itself. Detection is not the subject of this book; let us just mention that the research on detection for biochip recognition is the topic of an abundant literature and is a field that is constantly improving. Developments are aimed in two directions: first, improvements in the detection method itself, such as improving the fluorescence by using new fluorophores (quantum dots, for example), or developing enzymatic amplification for detection, and so forth; and second, improvements of the design and materials, such as improved waveguide in the case of detection by fluorescence, or the use of CMOS detectors for photons emitted by the fluorophores or by the enzymatic revelation.

References   [1] http://www.cheng.cam.ac.uk/research/groups/laser/Teaching/metrology/immuno_label. pdf.   [2] Harlow, E., and D. Lane, Using Antibodies: A Laboratory Manual, New York: Cold Spring Harbor Laboratory Press, 1988.   [3] Atkins, P. W., Physical Chemistry, Oxford, U.K.: Oxford University Press, 1998.   [4] Laidler, K. J., Chemical Kinetics, New York: Harper and Row, 1987.   [5] Berthier, J., P. Combette, and L. Blum, “Numerical Calculation of a Microfluidic Protein Reactor: Is the Classical Michaelis-Menten Integral Relation Sufficiently Accurate?” Labon-a-Chip and Microarrays Conference, Zurich, Switzerland, January 14–16, 2002.   [6] Berthier, J., P. Combette, and L. Blum, “A Model for the Kinetics of Heterogeneous Enzymatic Reaction in a Protein Microreactor,” 4th LETI Annual Review, CEA Grenoble, June 2002.   [7] Sharov, A. A., Virginia Tech., http://www.gypsymoth.ento.vt.edu/~sharov/PopEcol/lec10/ lotka.html.   [8] Lotka, A. J., Elements of Physical Biology, Baltimore, MD: Williams & Wilkins Co, 1925.

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Edelstein-Keshet, L., Mathematical Models in Biology, New York: McGraw-Hill, 1987. Murray, J. D., Mathematical Biology, New York: Springer-Verlag, 1989. Tabeling, P., Introduction à la Microfluidique, Paris: Belin, 2003. Park, S. H., S. Parus, R. Kopelman, and H. Taitelbaum, “Gel-Free Experiments of Reaction-Diffusion Front Kinetics,” Physical Review E, Vol. 64, 2001, pp. 55102–6.

[13] Galfi, L., and Z. Racz, “Properties of the Reaction Front in an A+B®C Type ReactionDiffusion Process,” Physical Review A, Vol. 38, No. 6, 1988. [14] Baroud, C. N., F. Okkels, L. Ménétrier, and P. Tabeling, “Reaction-Diffusion Dynamics: Confrontation Between Theory and Experiment in a Microfluidic Reactor,” Physical Review E, Vol. 67, 2003, p. 060104. [15] Kamholz, A. E., B. H. Weigl, B. A. Finlayson, and P. Yager, “Quantitative Analysis of Molecular Interaction in a Microfluidic Channel: The T-Sensor,” Anal. Chem.,Vol. 71, 1999, pp. 5340–5347. [16] Butler, J. E.,”Solid Supports in Enzyme-Linked Immunosorbent Assay and Other SolidPhase Immunoassays,” Methods, Vol. 22, 2000, pp. 4 –23. [17] Ruckstuhl, T., M. Rankl, and S. Seeger, “Highly Sensitive Biosensing Using a Supercritical Angle Fluorescence (SAF) Instrument,” Biosensors and Bioelectronic, Vol. 18, 2003, pp. 1193–1199. [18] Berthier, J., L. M. Neuburger, H. Volland, and F. Perraut, “A Modified Langmuir Equation for Microfluidics Systems,” Proceedings of the 8th World Congress on Biosensors, Granada, Spain, May 24–26, 2004. [19] Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes, Cambridge, U.K.: Cambridge University Press, 1987. [20] Sapsford, K. E., Z. Liron, Y. S. Shubin, and F. S. Ligler, “Kinetics of Antigen Binding to Arrays of Antibodies in Different Sized Spots,” Anal. Chem., Vol. 73, 2001, pp. 5518–5524. [21] Winzor, D. J., “Determination of Binding Constants by Affinity Chromatography,” Journal of Chromatography A, Vol. 1037, 2004, pp. 351–367. [22] Mason, T., A. R. Pineda, C. Wofsy, and B. Goldstein, “Effective Rate Models for the Analysis of Transport Dependent Biosensor Data,” Mathematical Biosciences, Vol. 159, 1999, pp. 123–144. [23] Glaser, R., “Antigen-Antibody Binding and Mass Transport by Convection and Diffusion to a Surface: A Two-Dimensional Computer Model of Binding and Dissociation Kinetics,” Analytical Biochemistry, Vol. 213, 1993, pp. 152–161. [24] Hibbert, D. B., J. J. Gooding, and P. Erokhin, “Kinetics of Irreversible Adsorption with Diffusion: Application to Biomolecule Immobilization,” Langmuir, Vol. 18, 2002, pp. 1770–1776. [25] Stenberg, M., L. Stiblert, and H. Nyguen, “External Diffusion in Solid-Phase Immunoassays,” J. Theor. Biol., Vol. 120, 1986, pp. 129–140. [26] Lionello, A., J. Josserand, H. Jensen, and H. H. Girault, “Adsorption of Proteins in a Microchannel,” Lab-on-a-Chip, Vol. 5, 2005, p. 254. [27] MATLAB, The MathWorks, Inc., version 6, September 2000. [28] Narang, U., P. R. Gauger, A. W. Kusterbeck and F. S. Ligler, “Multianalyte Detection Using a Capillary-Based Flow Immunosensor,” Analytical Biochemistry, Vol. 255, 1998, pp. 13–19. [29] Selinger, J. V., and S. Y. Rabbany, “Theory of Heterogeneity in Displacement Reactions,” Anal. Chem., Vol. 69, 1997, pp. 170–174. [30] Kusterbeck, A. W., G. A. Wemhoff, P. T. Charles, D. A. Yeager, R. Bredehorst, C.-W. Vogel and F. S. Ligler, “A Continuous Flow Immunoassay for Rapid and Sensitive Detection of Small Molecules,” J. of Immunological Methods, Vol. 135, 1990, pp. 191–197.

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359

[31] Ligler, F. S., M. Breimer, J. P. Golden, D. A. Nivens, J. P. Dodson, T. M. Green, D. P. Haders, O. A. Sadik, “Integrating Waveguide Biosensor,” Anal. Chem., Vol. 74, 2002, pp. 713–719. [32] Volland, H., L. M. Neuburger, E. Schultz, J. Grassi, F. Perraut, and C. Creminon, “Solid-Phase Immobilized Tripod for Fluorescent Renewable Immunoassay. A Concept for Continuous Monitoring of an Immunoassay Including a Regeneration of the Solid Phase,” Anal. Chem., Vol. 77, 2005, pp. 1896–1904. [33] Neuburger, L. M., F. Perraut, E. Schultz, J. Berthier, C. Créminon, and H. Volland,“A New Concept for Continuous Flow Immunosensors,” Proceedings of the 8th World Congress on Biosensors, Granada, Spain, May 24–26, 2004. [34] Berthier, J., L-M. Neuburger, H. Volland, and F. Perraut, “An Analytical Model for CFIs,” Proceedings of the 8th World Congress on Biosensors, Granada, Spain, May 24–26, 2004. [35] Narang, U., “Fiber Optic-Based Biosensor for Ricin,” Biosensor & Bioelectronics, Vol. 12, 1997, pp. 937–945. [36] Wemhoff, G. A., S. Y. Rabbany, A. W. Kusterbeck, R. A. Ogert, R. Bredehorst and F. S. Ligler, “Kinetics of Antibody Binding at Solid-Liquid Interfaces in Flow,” J. of Immunological Methods, Vol. 156, 1992, pp. 223–223.

Chapter 8

Experimental Approaches to Microparticles-Based Assays The microscopic objects dealt with in this book can be distinguished in two categories. The first category is the particles of biological interest that are naturally present in the biological systems and on which it is necessary to obtain some information. The second category deals with artificial particles that are manufactured by chemical synthesis or by genetic modification, as tools to perform a function in the process (observation, characterization, or manipulation). Dealing with micronanoparticles means that the objects are not only smaller, they have intrinsic properties because of these length scales. In this chapter, which is more oriented toward practical experimental situations, we first present the biological objects, limiting ourselves to major biopolymers and to some aspects of cells. Then, we introduce a few basic physical notions and definitions, and review some of the synthetic particles and their use. Section 8.3 is devoted to techniques used to characterize these objects and we end with a few words on micromanipulation techniques. This chapter should be read as an introduction to these experimental techniques with practical aspects in mind. It gives an idea of what is possible along the main lines described throughout the book but is in no way an exhaustive picture. The interested reader is encouraged to go further into the matter with the classical books or review papers listed in the general bibliography at the end of the chapter or throughout the text. Furthermore, Chapter 9 details magnetic particles and related techniques, and Chapter 10 describes electric-field–based techniques.

8.1  A Few Biological Targets In this section, we focus on a few examples that are a major concern in many studies of this area. We only review here some aspects of three families of biological macromolecules: DNA, RNA, and proteins. In the second part, we deal with some aspects of live cells. With sequencing of the genomes of many organisms in the last 30 years, there has been a need to better understand not only the function of the different genes but also, more ambitiously, the way the different constituents of cells or organisms interact and organize themselves into complex networks. On this “functional genomics” point of view, physics and engineering are everywhere, from the concepts of DNA or protein arrays and their interpretation, to the modeling of the various functions and interactions in the biochemical networks.

361

362

Experimental Approaches to Microparticles-Based Assays

8.1.1  Biopolymers

Classically, protein expression is described by the following sequence: the genetic information carried on the DNA sequence is read by a protein assembly called RNA polymerase; this transcription gives rise to the RNA molecules. The messenger RNA finds its way out of the nucleus in the cytoplasm and is translated into functional proteins by the ribosome. We now describe a few elements on the structure of these macromolecules. 8.1.1.1  DNA Molecules

DNA (desoxyribonucleic acid) molecules are made of two strands twisted around each other. Each of these strands consists of four bases: adenine ([A]), thymine ([T]), guanine ([G]), and cytosine ([C]) on a phosphate backbone and they are arranged in a double helix where the bases are located inside and paired exclusively [A]-[T] and [G]-[C]; they are called Watson-Crick base pairs. The two strands are oriented and arranged in antiparallel directions (Figure 8.1). The arrangement of the base pairs along the strand bears the genome of an individual and contains all his or her genetic information. Above a certain denaturation temperature, the two strands separate. This property is used in the polymerase chain reaction (PCR) technique to amplify the number of copies of DNA molecules in a given sample. In this technique, after the denaturation step, the sample is cooled down and “primers” that are short complementary sequences bind to the beginning and end of the region of the DNA to be amplified. An enzyme (the polymerase) then reads the single strand and matches it with its complementary sequence using the free nucleotides in solution. So, starting from one double strand, we end up with two. The same process is cycled 30 to 40 times, leading to an exponential amplification of the number of copies of the initial sample. At a much larger scale, DNA is a polymer. When sufficiently diluted, DNA chains in solution adopt a coil configuration whose radius, called the radius of gyration Rg, is directly related to the size of the monomers b and their number N through the relation [1] (Figure 8.2):

Rg = b × N ν

(8.1)

For polymers in “good solvent” (meaning that the interactions between a monomer and a solvent molecule are favored compared with interactions between two monomers), this exponent ν is 3/5. In some cases however, these interactions are effectively comparable and the chain is said to be ideal, the exponent ν is then

Figure 8.1  Double helix structure of a DNA molecule

8.1  A Few Biological Targets

363

Figure 8.2  Coil configuration of a single polymer chain in solution.

1/2. Importantly, although for different physical reasons, the case of DNA doublestrand molecules falls into this last category. This arises because of the semiflexible nature of this chain and it can then be shown that only unrealistically long chains would be in good solvent [2]. The persistence length is the distance along the chain above which the position of two monomers behaves independently. For DNA, this length is 50 to 100 nm depending on the characteristics of the solution (pH, salinity, etc.) (Figure 8.3). This picture of DNA molecules is meaningful only in a buffer solution. In a cell’s nucleus, DNA in the form of chromosomes is packed extremely tightly by histones, a particular class of compaction proteins. DNA microarrays that have been used for several decades exist in different versions. Generically, it consists of depositing thousands of spots of single-strand DNA sequences on a solid substrate such as a glass slide and measuring the hybridization efficiency with DNA or RNA single strands, by fluorescence or radioactive labeling. When the target is itself DNA, these arrays can be used to identify genes such as those implicated in some diseases. The quantification of the level of these genes can be used for diagnostic [4, 5]. Compared to conventional techniques such as Southern blots that combine gel electrophoresis and hybridization for limited number of DNA fragments, the analysis of the whole genome of an organism can be completed in a single experiment. 8.1.1.2  RNA

Ribonucleic acid (RNA) molecules are similar to single-strand DNA from the point of view of their sequence. One of the four bases is different, thymine being replaced by uracyle. The phosphate backbone also presents some slight differences. The messenger RNAs (mRNAs) are the molecules resulting from the transcription process. They contain the same genetic information as DNA and come out of the nucleus to be translated into proteins in the cytoplasm. There are other smaller RNA molecules (transfer RNA : tRNA) that play a role in this translation process. The bases of RNA are similar to the ones of the DNA associate themselves, but because of the single-strand structure of RNA molecules, these interactions lead to partially folded

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Figure 8.3  Atomic force microscopy of DNA molecules adsorbed on a solid surface (see a description of the technique in Section 8.3.1.3). The image is color-coded in height so that the height difference between the white color and the dark color is ~ 1 nm. The size of the image is 1.5 mm. From such images the contour length of these molecules can be accurately measured as well as the localization of potentially interacting proteins. (From [3].)

structures. The shape imposed on these molecules by this folding plays an important role in the translation into proteins. The folding of these molecules is fixed by their sequence, hence by the DNA sequence. Their structures correspond to a minimum of energy and can now be accurately computed for reasonably long molecules (up to a few 1,000 nucleotides) [6] (Figure 8.4). Because it is an indicator of gene expression, mRNA has a central position in functional genomics and mRNA is one of the major targets of the DNA arrays described above. By hybridization with the small DNA sequences spotted on the array, the levels of particular RNAs are measured and, from there, one gets some information on the amount of the translated functional proteins. Practically, it is much easier to work with RNA than with proteins, and since their levels are strongly correlated, it is quite useful information. 8.1.1.3  Proteins

Proteins are the product of the translation of RNA by ribosomes in the cytoplasm. Along the RNA strand, three consecutive nucleotides, a codon, are translated into an amino acid in a very robust way following the so-called genetic code.

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Figure 8.4  Pairing of RNA bases leads to a complex molecular structure (Escherichia coli 16S ribosomal RNA secondary structure). (Courtesy of Prof. P. H. Noller.)

The primary sequence of proteins thus mirrors the one of the initial DNA molecule. This sequence also determines their 3-D structure since interactions between amino acids fold the protein. Compared to RNA, these interactions are more diverse than and not as specific as base pairing: Van der Waals interactions, electrostatic interactions, and hydrogen bonds combine together to shape the proteins into

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their functional form. There are several levels of organization; first the interactions between neighboring parts of the chain can lead to regular structures such as the α-helices or the β-sheets. These structures (the secondary structures) arrange themselves into more complex domains that are common to some extent to many different proteins. For a given protein, these domains are arranged in a particular way to define the full 3-D tertiary structure. Very often, these structures are not functional by themselves; they self-assemble, and the functional protein is a multimer (sometimes called the quaternary structure) of several of these units that are then called monomers (not to be mistaken with the monomer as a single unit of a polymer chain) (Figure 8.5). Some proteic assemblies result from only a few monomers (sometimes only one); some others, in the self-assembly of many of them (up to several hundred): actin filaments for instance are made of the helical arrangement of actin monomers; tubulin monomers organize themselves in a cylinder to form microtubules. Actin microfilament and microtubules form the cell cytoskeleton and have a very dynamic assembly-disassembly behavior inside the cells. As the 3-D structure imposes the protein function, it is of great interest to experimentally access it. Practically, the techniques used are X-rays diffraction, electron microscopy, and for smaller proteins, NMR. We have seen in the preceding part that it was already difficult to compute the shape of RNA molecules where there are only a limited number of possible interactions. This is of course even more the case here and it is actually quite difficult to predict the 3-D structure of a protein from its sequence [7]. Some proteins can also be engineered to become tools in the hands of biologists. Enzymes for instance are catalysts that are of particular importance since most of the biological processes are highly dynamic. Members of this family include restric-

Figure 8.5  Structure of α-hemolysin. This complex is a heptamer that forms pores in membranes; the “stem” crosses the lipid membrane and the “cap” is in contact with the extracellular medium. (From [8].)

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tion enzymes that are heavily used in molecular biology as molecular scissors able to cut DNA molecules at very specific spots. Another example where proteins are used as “workhorses” is antibodies: When a foreign body or a protein is injected in an animal, some of its cells produce particular proteins called antibodies that are highly specific to this “antigen.” This property is the basis of the immunological response and is often used to produce antibodies that can be isolated, purified, and tagged. They are then used to specifically localize and quantify the protein of interest for instance in fixed cells or in immunoassays. We have mentioned earlier the use of DNA microarrays to quantify the level of gene expression. Measuring directly the level and the activity of proteins with protein microarrays is the next step in this process. Here, the sequences of DNA spotted on the surface are replaced by proteins or by molecules with which these proteins can interact. One then gains access to protein-protein interactions, proteins interactions with small molecules, and so forth. This information is richer than that obtained by measuring the levels of RNA since proteins evolve after their production, they mature, are modified, interact with other components—all processes that can affect their function and/or activity [9]. The detection of the coupling between the proteins and the molecules immobilized on the surface is performed in different ways: by fluorescence, radioactivity, surface plasmon resonance, or mass spectrometry using the strategies illustrated in Chapter 7 and developed here in Section 8.3. 8.1.2  Some Aspects of Cells

Cells are extremely complex arrangements and, of course, living entities (the smallest there are). Prokaryotic cells do not have nucleus, they include bacteria and archae and have a simpler organization than eukaryotic cells in which DNA is packed in a nucleus. Both cell types have a barrier protecting them from the exterior: a soft phospholipid membrane for eukaryotes and a more rigid wall for bacteria. 8.1.2.1  Eukaryotic Cells

It is often required to sort and characterize particular cells. An interesting example of this process is the one of circulating cancer cells in the general framework of early cancer diagnosis. In some cancers such as breast cancer, isolated tumor cells disseminate in the body of the patients by being conveyed by the circulating blood. They can also be found in their bone marrow. The challenge becomes the detection of these cells estimated to 1–10 cells / 1 million nucleated cells. Their detection relies on specific markers in their cytoplasm or on their membrane and may use fluorescence-based techniques such as the ones reviewed later in this chapter. However, the number of pathologic cells is so low that not only does their detection require a particularly sensitive and specific technique, but it is also of prime importance to first enrich the medium with these cells. This results in a two-phase process where the enrichment does not have to be highly specific but should take all the suspect cells, and a second step where the true detection needs to be highly specific. An efficient strategy in this line is to use immunomagnetic enrichment: suspect cells are

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attached to magnetic beads by antigen-antibody coupling and sorted with a magnet. This example illustrates the current efforts to move and control populations of live cells, isolate them, and quantify their amount and their characteristics. Live cells can also be used as sensors: in cell-based biosensors, the cell itself is the detector. For instance, some cells are extraordinarily sensitive to particular chemicals and can be used as sensors for traces of these molecules. In others cases, if a drug has to be tested on a particular organ, the response of cultured cells of this organ are monitored by electrical or optical means and, although the details of the response are not always fully understood, they directly carry the biological information [10]. Cells arrays take advantage of the high parallelism of the microarrays technology. They consist of arrays of live cells or cells clusters that are each transfected with a different gene. The response of these different cells to external stimuli can then be monitored in parallel [11] for instance, by monitoring the expression of a GFP-labeled protein (see Section 8.2.1.2) or by measuring their electrical response using microelectrodes or patch-clamp techniques. 8.1.2.2  Bacteria

Bacteria are smaller than eukaryotic cells (typically a few micrometers vs. a few tens of micrometers). The bacteria family is very diverse; they display a wide range of morphologies and their mode of locomotion varies from one species to the next. Escherichia coli for instance is a flagellated bacterium that swims by a succession of “runs” and “tumbles.” Bacteria cell walls consist of peptidoglycans; two families are often distinguished according to the properties of this cell envelope: Gram (+) bacteria have a thicker cell wall compared to Gram(–) bacteria. Although very different from eukaryotic cells, some of the practical situations dealt with these cells are very similar to concerns mentioned earlier. For instance, the fight against bioterrorism has emphasized the importance of checking for the presence of a few virulent bacteria within relatively large samples. Bacteria are also used by biologists to produce useful molecular tools. Since the genome of some of them have been entirely sequenced, it has now become a routine work to modify it using molecular biology tools for instance by inserting exogenous genes that can express particular proteins. These bacteria (and E. coli in particular) are thus literally transformed in protein factories.

8.2  Microparticles as Biotechnological Tools Synthetic microparticles find a natural use either as a macroscopic “handle” for the manipulation of molecules or cells, or to add or to enhance a signal in the various detection schemes. This last category includes immunofluorescence or immunoelectron microscopy in which a fluorescent dye or a metal colloid is coupled to an antibody that specifically targets the molecule of interest.

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8.2.1  Fluorescent Particles

Fluorescence is one of the major tools used in biochemistry/biotechnology. The high sensitivity of the technique and the numerous available coupling strategies make it a very versatile routine tool. Before describing the probes used in this context, let us first recall a few notions of the physical principle 8.2.1.1  Fluorescence

Some molecules have the ability to absorb photons at a given wavelength and emit them back at a different (longer) wavelength corresponding to a lower energy (Figure 8.6). Fluorescein for instance is a very popular fluorescent dye that shines in yellow-green (λ ~ 520 nm) when excited by blue light (λ ~ 480 nm). For a given fluorophore, there is a range of wavelengths that are absorbed (the excitation spectrum) and a range of emitted wavelengths (emission spectrum) (Figure 8.7). This property is used extensively in fluorescence microscopy but also with other techniques such as flow cytometry (see Section 8.3.3.2). The reasons for such an extensive use are: · ·

·

The extreme sensitivity of fluorescence up to a single fluorophore detection. the huge panel of possible reactants that can be chemically synthetized. With the right coupling chemistry it is possible to couple a dye to virtually any biological object of interest. The advent of naturally fluorescent proteins such as GFP that can be synthesized by transfected or modified cells lines.

Immunofluorescence is a particular coupling strategy that uses biochemistry rather than chemistry. Antibodies are first covalently coupled to a fluorophore and then allowed to interact with the cell or the tissue so that only the protein of interest “becomes” fluorescent.

Figure 8.6  Jablonski energy diagram illustrating the principle of fluorescence.

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Figure 8.7  Excitation (black) and emission (gray) fluorescence spectra of fluorescein.

8.2.1.2  Fluorescent Molecules and Particles

Because of its versatility, fluorescence has been used in very different situations. Fluorescent particles can be chemically synthetized (exogenous) or directly synthetized by the cell or the bacterium (endogenous) by inserting the right gene in the sequence of the protein. In this last case, it is attached at a precise position on a given protein. Exogenous Fluorophores: Organic Molecules

It first has to be realized that the situation is different if one wants to deal with in vitro situations (isolated and/or purified molecules or fixed cells) or in vivo situations (live cells). Fluorophores are characterized by their excitation and emission spectra; their quantum yield, defined as the number of emitted photon per absorbed photon, quantifies their efficiencies. The fluorescence properties of these molecules depend on their immediate environments. For instance, they can become sensitive and local sensors for pH or viscosity. They can also detect ions in solutions: Calcium or other divalent ions for instance are readily and quantitatively detected by the FuraRed molecule even within live cells. Most of these organic fluorophores can be coupled to proteins and a routinely used technique is to attach them to a given antibody in order to specifically detect, localize, or measure the concentration of the corresponding protein in vitro or in fixed cells. However, they are often toxic and their use is consequently restricted mostly to fixed cells. Unfortunately, these molecules cannot switch indefinitely between their excited state and their ground state. After a certain number of these transitions (this number is extremely variable from one molecule to the other), they permanently lose their fluorescence properties, a phenomenon called photobleaching that is enhanced by the presence of dissolved oxygen in the solution. By carefully degassing solutions and keeping the excitation to a minimum, this effect can be minimized at the cost of a much decreased fluorescence intensity.

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This effect can however be used to one’s advantage to study the dynamics (see Section 8.3.1.1 for a description of the FRAP technique). Exogenous Fluorophores: Micro- and Nanoparticles

Fluorescent latex beads are plastic beads (typically a few hundred nm in diameter) loaded with organic fluorophores. These relatively large particles are quite useful to reveal the structures of larger structures or to probe flows in microchannels geometries (see Chapter 2). Their surface can be tailored to match one’s particular application by grafting particular molecules to them. For instance, immunoassays that use antibodies coupled to latex beads allow detecting specific proteins (see Section 8.2.2.1). Quantum dots (QDs) are fluorescent nanoparticles (typically 10 nm) whose use in biology oriented applications is relatively recent. These inorganic particles are made of semiconductors (very often ZnSe crystals surrounded by a thin ZnS shell), and besides their small size, have a few remarkable properties that explain their popularity. They all share a broad excitation spectrum in the blue and their emission wavelength depends only on their size. They are extremely bright and show practically no photobleaching. Moreover, they are small enough to be incorporated in many systems, even at the surface of live cells. For instance, by using two sizes, two different proteins can be labeled and excited with the same wavelength making colocalization experiments particularly easy. The applications of QDs in biology-related applications have long been delayed mainly because of the difficulties in dispersing these hydrophobic particles in water, not to mention their difficult coupling to biomolecules due to a particularly inert surface. However these difficulties have been solved in particular by their encapsulation with amphiphilic molecules such as block copolymers [12]. QDs are now commercially available with different surface groups and couplings. Because they are so bright and quite small, they can be used to track particular proteins at the surface of cells [13] or even within cells [14]. Endogenous Fluorophores

Green fluorescent protein (GFP) is a naturally fluorescent protein present in the jellyfish Aequorea Victoria. The GFP reporter gene can be fused by genetic engineering to the one of the protein of interest so that the resulting protein is a fusion of both, coupling the desired function with fluorescence. This way, proteins in living cells can be directly observed by fluorescence. Better efficiency and other colors have been developed by mutating the original GFP. Using these proteins, dynamic fluorescent imaging of proteins can be performed on live cells (for which the coupling with organic fluorophore would have been prohibited for toxicity reasons). Strategies exist to get transfected cell lines that have a transient response but it is also possible to get stable clones. 8.2.2  Other Micro- and Nanoparticles

We review here some of the particles used in tests or in biotechnology-related applications. Because of their potential, magnetic beads deserves a chapter by themselves and are described in Chapter 9.

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8.2.2.1  Latex Beads

Latex particles, which were mentioned earlier as potential fluorescent tracers, are commonly used because of their variety: they can be found commercially with different surface chemistries, in different sizes, and with other distinctive properties. For instance, one can easily get fluorescent or magnetic beads. These particles are synthesized by emulsion polymerization in which the hydrophobic organic monomer is encapsulated by surfactant molecules in a micelle (see Section 8.2.2.3) and then polymerized in the water phase. This produces nice monodisperse suspensions that unfortunately need the surfactant to stay stable. However, it is possible to add at the polymerization step a monomer that can play the role of the surfactant and that stays at the bead-water interface after copolymerization, stabilizing the particles by electrostatic interactions. Moreover, these chemical functions that are now at the surface of the particles can be used to initiate the coupling of biomolecules on the beads. For instance, the widely used latex agglutination tests consist in adsorbing antibodies to micron-sized latex particles. When the corresponding antigen is present, these beads interact with it and clump together. Because of their size, the particles and the clumps scatter light differently and a simple visual observation gives the answer on the presence of the antigen. 8.2.2.2  Gold Nanoparticles

The main use of gold nanoparticles is electron microcopy. They can be coupled to antibodies by electrostatic nonspecific adsorption. When the target protein is present, the nanoparticles couple specifically to it and appear as distinctive tiny black spots in transmission electron microscopy. The particular optical properties of these nanoparticles can also be used to improve agglutination tests classically performed with latex particles (see above). For instance, they are used in some commercial pregnancy tests. Because of their small size, these particles have a characteristic red color caused by a phenomenon called plasmon resonance that we will review in more detail in Section 8.3.3.1. The urine of pregnant women contains a particular hormone whose corresponding antibody is adsorbed both on the nanoparticles and on micron-sized latex particles. When the hormone is present the two types of particles coagglutinate and yield the formation of red clumps. In optical microscopy, gold (and other metals) nanoparticles are also used although not as frequently. In particular, relatively large metallic nanoparticles (a few tens of nanometers) can be easily detected by the apparition of a surface plasmon described above that enhances their diffusion by several orders of magnitude [15]. For smaller nanoparticles, photothermal heating by the laser illumination modifies the index of refraction very locally and allows their detection in live cells down to diameters of a few nanometers [16]. 8.2.2.3  Surfactants and Micelles

Surfactants (also called amphiphiles) are molecules composed of two antagonistic parts: a hydrophilic polar head and a hydrophobic nonpolar tail. Phospholipids

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that are the constituents of the cellular membrane belong to this category. In a water solution, the hydrocarbon tails minimize their interactions with the water, and the molecules self-assemble into structures exposing only the hydrophilic headgroups toward the water. In fact, they tend to aggregate into micelles such as the one depicted in Figure 8.8 as soon as their concentration is high enough. We can write the equilibrium between a solubilized surfactant molecule S and a micelle consisting of n of these molecules Sn (n>>1): nS « Sn

whose equilibrium constant is:

K = [Sn ]/[S]n

(8.2)

we call c, the total concentration in surfactants:

c = [S] + n[Sn ]

(8.3)

and we define c* as c*=(nK)1/n Combining (8.2) and (8.3), we immediately find:

If c << c*, [S] ∼ c



If c >> c*, [S] ∼ c* In other words, below c* called the “critical micellar concentration” (CMC), surfactant molecules are individually solubilized; above this concentration, they tend to aggregate in micelles. Micelles are very dynamic objects constantly exchanging molecules with the free surfactant molecules in the solution [17] (Figure 8.9).

Figure 8.8  Schematic view of a micelle (2-D cross section).

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Figure 8.9  Phase diagram of amphiphile molecules. Above the cmc, surfactants predominantly assemble in micelles.

Micelles are not restricted to small surfactant molecules. In particular, some block copolymers can be tailored to make micelles particularly well suited to drug delivery applications. In this last case, chemists have been quite imaginative in designing molecules with the right lengths and the right chemistries that self-assemble in micelles and can contain an organic drug in their core. These micelles can specifically target particular cells in an organ without releasing the drug too early or being destructed too soon by the defense mechanisms. The surface area per molecule in these aggregates results from a balance between attractive tail-tail interactions and repulsive head-head interactions. From geometric arguments, when the tail group of these molecules is too bulky for a good packing into micelles, bilayers can self-assemble and lead to the formation of vesicles. This is for instance the case for most of 2-chains surfactants such as phospholipids that are the major constituents of cell membranes (Figure 8.10). Cell membranes based on these structures incorporate membrane proteins that have a hydrophobic domain interacting strongly with the aliphatic chains of the lipids. We have seen earlier that the structure determination of proteins is often performed by diffraction techniques and thus needs them to be crystallized. However,

Figure 8.10  Schematic view of a bilayer vesicle.

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the membrane domains of these proteins are so hydrophobic that it makes them irreversibly precipitate as soon as they are removed from this lipidic environment. Their crystallization thus necessitates particular protocols in which the hydrophobic domain remains protected from water by surfactants molecules. These protocols have to be finely tuned for each particular protein and, for this reason, their structures are still unknown for most of them whereas soluble proteins whose crystallization conditions can be better rationalized, are better known in that respect [18]. 8.2.3  Chemical Modification of Surfaces

For these micronanoparticles that have a large surface/volume ratio, surface related problems and surface chemistry are of paramount importance. The first method that comes to mind to immobilize biomolecules on surfaces is to rely on nonspecific adsorption. Although such a coupling can be efficient enough in some situations, a better control is usually required. Very few surfaces are truly inactive. They very often bear chemical groups that can be used for further surface chemistry. Metal surfaces—in particular gold—and oxide surfaces—in particular SiO2—are good templates for chemical modifications. This last case is of particular interest because these surface treatments are also applied to glass by extension, although the chemistries of these two surfaces are not strictly identical. Polymer surfaces such as the surface of latex beads can also include a sophisticated surface chemistry by the right choice of the monomers used for their synthesis. However, even in this case, direct coupling may not be possible because some very reactive groups would readily hydrolyze in water where the coupling reaction is to take place. Intermediate coupling molecules are thus needed. Generically, these molecules have a reactive group at each extremity. One of them reacts on the solid surface: in the case of gold, it is a thiol group, and in the case of silica, it is a silane group. The end-group at the other extremity of these molecules is exposed toward the exterior world and is used for coupling to the proteins for instance. Chemical grafting on a plane surface and on a microparticle share some common features but also differ in a number of ways. On plane surfaces, the grafting of these molecules results from a collective mechanism where they interact together by Van der Waals interactions as they react on the surface [19]. The monolayer can be reinforced by a lateral polymerization illustrated by the case of silanes on silica where some of the silane groups react on the surface while the others react together forming a “net.” While it is relatively easy to qualitatively modify a surface, achieving good monolayers, which is the first step to a good surface coverage, is a delicate operation particularly in the silane/silica surface (even more so for the silane /glass system because of the defects and the different chemistry of the glass surface). With this strategy, it becomes possible to change the physical properties of the surfaces such as transform hydrophilic surfaces to hydrophobic ones. With particles, there is usually no need for a “perfect” monolayer and the grafting conditions are less drastic. A major use of this surface chemistry is to protect surfaces from nonspecific adsorption. In that case, long polyethylene glycol (PEG) molecules can be used. To adsorb on the surface, a protein would have to compress this layer, which is entropically very unfavorable [20].

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DNA chips are another example requiring surface modification. DNA oligomers are spotted onto a surface (usually glass) and need to be permanently anchored on it. Glass is negatively charged, so only a few of these molecules would naturally remain stuck on it once it is in contact with a water-based buffer. On the other hand, if the surface is coated with amine groups by silanization, the surface becomes positively charged and these oligomers then stick irreversibly to it. In the case of proteins, a covalent reaction is even better. Very often, a group able to react on amines is chosen because the exterior surface of proteins is rich in these chemical groups. The aldehyde group present in glutaraldehyde or the N-hydroxy-succidimide group (NHS) are common choices for this purpose. Other groups such as vinyl sulfone can react on thiols also often available on the proteins [21]. This strategy however has several drawbacks: it needs a high enough density of amine groups on the protein surface; it can interfere with the function of the protein if it reacts precisely on the functional site and, of course, even if it reacts on some other random place of the protein, the orientation information is lost. To overcome these difficulties, strategies that involve “molecular glues” by specific and sturdy interactions such as the one of streptavidin with biotin or the hexahistidine sequence (His)6 with Ni-NTA (nitrilo-tri acetic acid) are preferred [22]. Antigen-antibody interactions can also be used to the same end. These strategies are particularly seducing as groups such as biotin or (His)6 can be genetically included in the protein during its synthesis by cells and are actually often used to purify them after cells’ lysis. The position of these groups on the protein is thus well known and chosen to interfere as little as possible with their function. If the linker of the streptavidin or the NTA to the surface is sufficiently rigid, the orientation of the protein is preserved. On the other hand, with long spacers, the proteins can have all the possible orientations and their interaction with the surface is reduced.

8.3  Experimental Methods of Characterization 8.3.1  Microscopies

Within the last years, optical imaging techniques have seen extraordinary developments. The advances in computing techniques and the widespread use of lasers have made possible to image processes that were thought to be only indirectly accessible. Although they are all called microscopies and are all imaging techniques, there is little in common between optical microscopy, electron microcopy, and atomic force microscopy. 8.3.1.1  “Classical” Optical Microscopy

The first microscopy technique that comes to mind to characterize particles is optical microscopy. Optical microscopes, although all based on the same basic design, constantly improve, adding new potentialities that the use of lasers as light sources and the computer analysis of images have contributed to enhance. They are an invaluable compromise between ease of use, versatility, and performance.

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Although microscopes used to be designed in the way presented Figure 8.11, recent models are now corrected in a way to include a region within the microscope where all the rays are parallel. The reason for using such geometry is fairly easy to understand: what limits the performances of a microscope are the aberrations of the optics. For good optics, these imperfections are limited but it is often necessary to include optical components (filters, polarizers, etc.) in the optical path. In this case, it becomes necessary to position these elements in a region of the microscope where the rays are parallel. Modern research microscopes incorporate this feature and are called infinity corrected systems. Because of the diffraction, a single object appears as the convolution of the object shape and a function called the Airy function. This means that, because of the laws of far field optics, a point source will appear to have a finite size in the microscope. To be able to distinguish between two objects they have thus to be further apart than the width of this function whose order of magnitude is the wavelength of light. Thus, there is a separation criterion stating that the ultimate resolution of an optical microscope is given by the classical Rayleigh formula:

d = 1.22 (λ / 2 × N a )

(8.4)

Where d is the smallest possible spacing between the two objects, λ is the wavelength, and Na the numerical aperture of the microscope. If the objects are closer than d, they appear as a single larger “blob.” To achieve a better resolution, high numerical aperture objectives are needed (in particular oil immersion objectives) as well as the use of blue wavelengths. Still, any object regardless of its size can be observed by optical microscopy provided that it emits enough photons. If it is too small, its observed lateral size has nothing to do with the true one but if one is interested in its dynamic behavior or in a more macroscopic measurement such as concentration, this is not a concern. Fluorescence imaging of single molecules that is now routinely performed in many laboratories is a good illustration of these possibilities. In the same line, tiny displacements down to a few nanometers can be detected by optical interferometric techniques [23].

Figure 8.11  Optical path of a microscope. The object to be observed is before the focal point of the objective, close to it. Its image is formed at the focal point of the ocular, sending the final image to infinity.

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It is certainly one of the strengths of optical microscopy to be able to image samples in water-based solution. This way, live cells or tissues can be imaged. Fluorescence microscopy enables to observe only the objects carrying a fluorescent dye (Figure 8.12). The proteins of interest are then localized by comparing images in fluorescence mode to images in white light mode. By using two dyes and working with two colors, two different objects can be colocalized. Many solutions exist that can increase the contrasts of a particular image depending on the characteristics of the sample : The polarization of the light, its angle of incidence, the interference between two light rays are strategies used alone or in conjunction and that are commercially available [24]. The confocal microscope [25] works on a somewhat different principle. The image is formed point after point. Here, the light source is a point source. The point image in the sample emits light that is collected by a detector through a pinhole, confocal with the source. This way, only the light that one wants to collect is indeed collected; the light emitted by other parts of the sample whose fluorescence is also excited by the illumination (the parts of the sample in the cone of light produced by the microscope objective) is largely excluded from the detector (Figure 8.13). The sample is then scanned in order to make the full scale image. Why is this interesting? In classical microcopy, all the fluorescence light is collected, and although there is a maximum of intensity at the focal point, the image is blurred by this background. The confocal technique is very efficient in rejecting unwanted out-of-focus light. “Slices” at a given height are performed by scanning the sample in the x-y plane and three-dimensional images can then be reconstructed by combining these slices. As it is a scanning technique, it is somewhat slow and thus not very efficient to study fast dynamics. A further refinement of confocal microscopy is the two-photon microscopy [25] that uses a nonlinear effect to excite fluorescence only at the focal point leaving un-

Figure 8.12  Principle of fluorescence microcopy. In this particular configuration, the excitation light and the fluorescence emission go trough the objective (epifluorescence).

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Figure 8.13  Principle of fluorescence confocal microscopy. No image is formed on a screen but the fluorescence intensity is collected point by point by a sensor such as photomultiplier.

exposed the rest of the cone of illumination. To perform this task, an infrared (IR) light is used for excitation. No fluorescence is directly excited by these IR photons. However, if two of them combine, they can excite the dye to its excited state, so that a photon is emitted when the molecule returns to its ground state. The probability of such two-photon events is very small and needs a very high intensity to trigger a detectable fluorescence. This condition on intensity is met only at the focal point of the objective. Compared to classical confocal microscopy, there is no need for a pinhole and the corresponding optics. All the light coming from the excitation volume is collected. Furthermore, as the excitation volume is confined at the focus, there is no bleaching in the rest of the light cone and 3-D images can be reconstructed with better accuracy even for very diffusing sample. FRAP

As mentioned above, if a fluorophore is excited with a high intensity, it is irreversibly modified and loses its fluorescence. Therefore, if a high-energy light beam is focused on a localized area, the so-called “bleached” area will appear as a dark spot (note that this subsequent observation is performed with a lower intensity to avoid affecting the fluorescent yield). Upon diffusion of the other fluorophores within this dark area, it progressively recovers a higher level of fluorescence (Figure 8.14). Fluorescence recovery after photobleaching (FRAP) quantifies this recovery of fluorescence to extract dynamical characteristics of the system. Assuming a simple case of a single diffusive species, the classical diffusion equation applies:



¶c = DÑ2c ¶t

(8.5)

Where c is the concentration and D the diffusion coefficient. This equation is then solved to get the exact fluorescence profile in time and space. In practice, there

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Experimental Approaches to Microparticles-Based Assays

Figure 8.14  Principle of a FRAP experiment. After being bleached by a high-intensity laser beam, the fluorescence progressively increases by diffusion and/or active transport and this recovery is monitored. This intensity often does not come back to its initial value because of the immobile fraction of molecules of interest.

might be a fraction of the fluorescence that is never recovered (corresponding to immobile molecules). For the other molecules, the diffusion coefficient D is given by: D ∼ w2 / τ



(8.6)

where w is the width of the bleached spot, and τ the characteristic time of the fluorescence recovery. The proportionality coefficient depends on the geometry of the experiment and on the boundary and initial conditions; it is calibrated with known molecules. FRET

Fluorescence resonance energy transfer (FRET) consists of using two dyes in such a way that the emission spectrum of the first dye significantly overlaps the excitation spectrum of the second one. This way, when the two molecules are close enough, exciting the first dye results in a decrease of its emission and, conversely, to an increase of emission for the second one. Typically, the molecules cannot be further apart by more than a distance called the Forster radius that is of the order of 5 nm [27]. FRET is thus well adapted to intramolecular distance measurements or to other situations where the interacting molecules are very close. It is often described as a “molecular ruler.” TIRF

Total internal reflection fluorescence microscopy (TIRF) is very useful to image phenomena close to an interface. When light propagating in glass is totally re-

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381

flected at the glass-medium interface, a small fraction of the incident energy still penetrates into the medium. This energy decays exponentially over a distance given λ by l = [28] (λ is the wavelength of light, nglass and nwater the 2 2 2π nglass sin2 i - nwater indices of refraction of the glass and the water and i the angle of incidence) (Figure 8.15). This length is on the order of 100 nm. Since only the fluorophores within this penetration length are excited and emit light, only the molecules very close to the surface are visualized. The fluorescence background that comes from the bulk is suppressed and the contrast is enhanced. 8.3.1.2  Single-Object Microscopy, FCS, Superresolution Microscopies

As mentioned above, the advent of sensitive enough detectors and bright light sources have made relatively straightforward to image single molecules (GFPs or organic fluorophores). The local environments can thus be characterized down to molecular sizes. Even the dynamics of these molecules can be followed (see the example of single proteins diffusing along DNA strands (Section 10.1.4.4). The fluorescence correlation spectroscopy (FCS) aims at a local dynamical characterization. If a limited number of fluorophores is present in the volume excited by the laser, they are animated by Brownian motion and therefore, this number fluctuates. From the temporal fluctuations u(t) of the emission signal I(t), the autocorrelation function is given by: G(τ ) =

u(t) × u(t + τ ) I(t)

2

(8.7)

which can be solved in the simple diffusion case:



G(τ ) = 1 +

τ0 τ0 × N(τ 0 + τ ) τ 0 + S0τ

(8.8)

Figure 8.15  Schematics of the total reflection setup. The light is totally reflected at the glass/water interface and the energy decreases exponentially within the water.

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Experimental Approaches to Microparticles-Based Assays

Where N is the average number of fluorophores in the observation volume, S0 is a geometrical parameter, and τ0 is related to the diffusion coefficient D through the relation: τ 0 ∼1/ D



Other relaxation modes can also be accessed using this technique, still at a very local subcellular scale. We have in the preceding part insisted that imaging of single particles was relatively straightforward with sensitive enough detection equipment. By using this idea, several techniques have emerged in the last decade that reconstitute a complete image by the addition of many single object images. As the Rayleigh diffraction limit is not relevant for single object imaging, the resolution that can be obtained can be extremely good. Stimulated emission depletion (STED) microscopy was historically the first of these so-called superresolution microscopies. Here, by using nonlinear effects, the size of an activated fluorescent spot is decreased down to a few tens of nanometers. Many of these spots are collected to form an image [29] improving the resolution by an order of magnitude compared to the Rayleigh law. Photoactivated localization microscopy (PALM) and stochastic optical reconstruction microscopy (STORM) are two techniques that image dispersed nanoobjects. By choosing the right excitation only a few of them are randomly excited. Their precise localization is then extracted from the image and the procedure is repeated. The full image is then reconstructed from the superimposition of many of these single molecule images (up to millions of them). The drawback here is the dynamics. These operations take a long time, they are well suited to fixed cells on which a resolution down to 20 nm can be obtained but not to dynamic processes [30, 31]. Many developments are underway in this area. One of the next foreseen improvement will be to adapt these techniques to 3-D imaging while keeping the same resolution; another one, as mentioned before, is to make them faster mainly by using other fluorophores, to observe dynamical phenomena. 8.3.1.3  Nonoptical Microscopies Electron Microscopy

Electron microscopes use an electron beam to probe objects. The transmission electron microscope works on a similar principle as the optical microscope using electrons and not photons. Because of the much smaller wavelength of the electrons, the resolution obtained with these instruments is several orders of magnitude better than with optical microscopes (down to a fraction of a nanometer). The optical path is exactly the one used to describe optical microscopes: The source of electrons is a heated filament, and the deflection of the electron beam corresponding to the deflection of light by glass lenses is obtained by magnetic fields. Contrarily to the optical microscope, the resolution is never the theoretical resolution imposed by the Rayleigh formula (8.4) but is caused by aberrations inherent to the magnetic lenses. Because of the interactions of the electrons with air, the whole setup is placed under vacuum. This limits the applications of electrons microscopes: no live cell or even hydrated sample can be imaged with such instruments.

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383

Transmission electron microscopes (TEMs) image the density of electrons. The intrinsic absorption of every sample imposes to work on thin samples. Either it is naturally the case for instance with molecules or membrane patches that are then stuck on a thin carbon film, or thick samples have to be microtomed into thin slices before their observation. In a transmission microscope, areas richer in electrons appear darker in the image. As the biological sample samples are composed of light elements (carbon, oxygen, hydrogen) heavy elements have to be added to increase the contrast. It is common to use chemical staining or gold nanoparticles functionalized with antibodies that target particular proteins. They then appear as dark spots in the TEM images specifically located on the molecules of interest. If TEM works in the transmission mode, the scanning electron microscope (SEM) works in the reflection mode. Again the parallel with the optical path of an optical reflection microscope is tempting, the difference being that, in the present case, a SEM does not form the image of the reflected electron beam but analyzes the electrons scattered by the surface where the incident beam has been focused. Again the optics are magnetic lenses. These electron beam techniques have a very high resolution up to the point were TEM-based techniques can resolve some protein structures. They however need a sometimes tedious preparation of samples. Atomic Force Microscopy

The atomic force microscopy (AFM) works with a completely different principle [32]. Here, the surface to be analyzed is scanned under a fine stylus mounted on a flexible leaf-spring (a cantilever) in the same way as a stylus probes the surface of LP records in old-fashioned phonographs. When one is interested in microparticles or macromolecules, the first step is to strongly adsorb them on this surface. The deflection of the cantilever is then a direct measurement of the topography of the surface. To get some orders of magnitude, the radius of curvature at the apex of the tip is of the order of a few tens of nanometers, the cantilever spring constant is of the order of a few tens of mN/m. In most of the commercial instruments, not to say all of them, the detection of the position of the cantilever is performed optically by shining a laser beam on the back of the cantilever and measuring the reflected beam with a quadrant photodiode. The relative displacements of the sample versus the tip are performed by piezoelectric actuators in the three directions of space (Figure 8.16). In practice, the mode just described where the vertical position of the sample is fixed and the force of the tip acting on it varies, is seldom used for two main reasons. First, by using the microscope this way, the force is higher on the ridges or the bumps of the surface and lower in the valleys. As with any observation technique, applying a force is already a potentially perturbative process (the extreme case being scratching the surface), but having different forces on the surface may make the images very difficult to interpret. The second difficulty is more instrumental. Getting true vertical distances from the measurement of the deflection of the cantilever would necessitate an accurate calibration of the detector for each experiment, which is practically unreliable. There is however a way to circumvent these difficulties, which follows a very general instrumentation strategy: the force (given by the deflection of the cantilever)

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Experimental Approaches to Microparticles-Based Assays

Figure 8.16  Principle of the atomic force microscope.

is kept at a fixed value and the sample is dynamically moved up and down to keep this force constant during the scanning. This way, the two hurdles just mentioned above vanish: on the one hand, the force applied on the sample is constant so the perturbation to the sample is the same everywhere and, on the other hand, a constant force means a constant position of the cantilever extremity with respect to the laser. There is thus no more need for a calibration of the detector since the read value remains constant. As the sample vertical position is controlled by a piezoelectric actuator easily well calibrated in distance, the topography of the sample is directly given by these vertical displacements. The lateral resolution is limited by the exact shape of the tip and is hard to define in the classical way as the influence of this parameter depends on the size and shape of the imaged objects. It is certainly one of the strengths of the AFM to be able to image samples at high resolution in a liquid and in particular in a buffer solution. On this respect, it is a major advantage compared to other techniques such as electron microscopy. Because it is a simpler technique to use, AFM is often used also on dried samples. It is clear from Figure 8.17 that AFM imaging is resolutive enough to access some protein structures. Although the resolution is somewhat poor compared to traditional diffraction techniques and very partial, as only the surface can be imaged, there is no need for crystallization since single molecules or complex can be imaged, an invaluable advantage in the case of membrane proteins. Furthermore, these images are taken in buffer solutions, sometimes directly on cells, thus in conditions very close to the functional environments of the proteins. In biology, the samples one is interested in can have two characteristics that make them difficult to image with an AFM: they can be very soft and they may need to stay hydrated. Most of the time, it is both. The interaction of the stylus

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385

Figure 8.17  Image of bacterial aquaporin, a membrane protein still inserted in it native membrane [33]. Individual tetramers are clearly resolved. Lateral size of the image is 70 nm. (Image courtesy of Dr. Simon Scheuring.)

with the surface can then be too strong and the surface is scratched, or the molecules of interest are damaged or wiped out. To minimize these problems, other modes have been developed in which the spring sustaining the tip oscillates and periodically “taps” the surface. The signal used for the feedback is then not the average position of the tip but the amplitude of oscillation measured by a lock-in detection. Although the tip still interacts rather strongly with the sample when it touches it, there is no transverse force (friction) applied to it and a lot of damage is avoided. The AFM cantilever then behaves as a damped oscillator: it is characterized by a characteristic resonance frequency and a quality factor Q that witnesses the viscosity of the medium. In air (even more so in vacuum) the quality factor is high (easily 100–1,000) meaning that the resonance is well defined. In water, Q is much lower (1–10) and consequently, because the resonance frequency is not well defined, the feedback is less sensitive and more damage is brought to the sample. Recent electronic methods aimed at electronically increasing the quality factor may become a good alternative to these problems. With the microscopic sizes dealt with in this book, it is illustrative to describe the motion of a free cantilever subjected to thermal agitation. In that case, one can use the equipartition theorem on the potential elastic energy of this harmonic oscillator.

386



Experimental Approaches to Microparticles-Based Assays

1 1 mω 02u 2 = kBT 2 2 m is the mass associated with the oscillator, u its displacement, ω 0 = nance frequency, k is the spring constant of the cantilever. Therefore: k=



kBT u2

(8.9) k , its resom

(8.10)

Equation (8.10) is used practically to calibrate the spring constant of the cantilever: when k~50 mN/m, the thermal fluctuations are of the order of a few angstroms, which is readily measurable by the detector. Practically, a power spectral density is plotted and fitted with the theoretical lorentzian shape for a harmonic oscillator. The area under this curve is then used to access the spring constant. 8.3.2  Physical Characterization: Light Scattering

Light scattering is routinely used to get molecular weight information out of polymers solutions in its static version; it is also a powerful tool to directly measure hydrodynamic radii when in the form of dynamic light scattering. The increase in computing capabilities and the reduction in size and cost of lasers have greatly popularized the use of these techniques. 8.3.2.1  Static Light Scattering (SLS) [34]

It is a common observation that, when light hits a suspension, some of it is scattered along all directions. Rayleigh scattering describes quantitatively this scattering for particles smaller than the wavelength of the light. Here, we do not take into account the temporal fluctuations of the scattered light but average the signal over long times. The dynamic aspect will be treated in the next part. The theory proceeds by computing the interactions of the electric field associated with the incident light with the polarizability of the particle it interacts with. When applied to a solution of polymers of molecular weight M, the intensity I(θ) at an angle of incidence θ is then given by:



I(θ ) »

I0cα 2 (1 + cos2 θ) 2 4 r λ M

(8.11)

where α is the particle polarizability, I0 the incident beam intensity, c the concentration in particles, r is the distance to the detector, and λ the wavelength. As the polarizability varies linearly with the molecular weight, at a given angle, the intensity is thus also proportional to M:

I(θ) = K(θ ) × c × M

(8.12)

If we consider a mixture of polymers of different masses or a polydisperse sample, we have to sum over all the contributions:

8.3  Experimental Methods of Characterization

387

I(θ ) = K å ci Mi = Kc

å ci Mi

i

i

å ci

= KcMw

i

(8.13)

where Mw is the weight averaged molecular weight. This is an ideal formula and, particularly for polymers, the classical analysis of these experiments proceeds by plotting I for various concentrations and various angles (Zimm plot). For larger objects whose size becomes comparable with λ, the light can scatter from different places of the same object. This effect decreases the scattered light even more so for large angles and one has to take into account a structure factor that can be analytically expressed only for a few simple geometries. 8.3.2.2  Dynamic Light Scattering (DLS) [34, 35]

Also known as photon correlation spectroscopy, DLS consists in measuring the scattered light dynamically at a fixed angle. As the molecules diffuse within the observation volume, the emitted light resulting from the scattering interferes. The analyses of these time fluctuations are then used to deduce a diffusion coefficient. This technique is well suited to particles in the range 10 nm–1 mm. This analysis is performed by computing the correlation function G(τ ). For simple diffusive processes, G(τ ) can be accurately modeled and is found to be exponential:

G(τ ) = I(t) × I(t + τ ) = A[1 + B exp(-2G τ)]

(8.14)

where G = Dq², q = (4π n/λ) sin(θ /2), n being the refractive index of the sample and D the diffusion coefficient. When several different particles characterized by different diffusion coefficients are present, a multiexponential is used to fit the function G(τ ) and access these different diffusion constants. From these measurements, one gets the hydrodynamic radius Rh by inverting the Stockes-Einstein relationship:

Rh =

kT 6πηD

(8.15)

thus the dimension measured with this technique is actually the radius of the equivalent sphere that would diffuse similarly. This is a simplification that can have severe consequences in the case of nonspherical particles: an increase in the length or in the diameter of a rod for instance contributes very differently to its hydrodynamic radius. 8.3.3  Biochemical Characterization 8.3.3.1  Surface Plasmon Resonance [36]

The surface plasmon resonance (SPR) technique is used to quantify the amount of material on a surface. In biotechnology, this technique is used to detect and measure

388

Experimental Approaches to Microparticles-Based Assays

in real time the kinetic parameters of the interaction between two or more molecules. One of the strengths of the technique is that it does not require labeling the molecules. The SPR effect is based on the interaction of light with a metallic surface in the conditions of total internal reflection which means that, in the absence of a metallic layer, the light propagating in the solid (a glass prism) is totally reflected. In these conditions, an evanescent wave exists at the surface of the glass. As already described, the intensity of this nonpropagating field decreases normally to the surface over a typical distance of a fraction of a wavelength (see Section 8.3.1.1). A thin metallic layer present on the glass surface will not qualitatively modify this picture; most of the light is still totally reflected. However the evanescent wave can couple with the free electron clouds of the metal to create a plasmon (a cloud of excited electrons). For this phenomenon to occur, the energy carried by the incident photons has to exactly match the energy of the plasmon. As these plasmons are confined within the metal layer, there are drastic conditions of angle and wavelength that yield an efficient coupling. When these requirements are fulfilled, energy is effectively transferred to the plasmons and this results in a minimum in the reflected intensity (Figure 8.18). Since we are dealing with evanescent fields, the conditions for which this transfer is efficient are highly dependent on the immediate environment of the metallic layer (typically within a wavelength in depth; i.e., within a few hundred nanometers). A modification of the refractive index of the solution next to the metal surface then results in a change in these conditions. The instruments used for biological applications use a fixed wavelength and monitor the incidence angle corresponding to the minimum of the reflected intensity. For protein-protein interactions, this change in the index of refraction is proportional to the amount of material present on the surface. SPR analysis systems can thus compute in real time the surface excess and uses these measurements to get the kinetic parameters of the studied biomolecular interaction.

Figure 8.18  Surface plasmon resonance setup. The detector measures the angle of minimum reflection to access the surface excess.

8.3  Experimental Methods of Characterization

389

Practically, one of these two components is immobilized on the gold surface using a coupling strategy based on the ones described in Section 8.2.3. Therefore, the results obtained by this technique describe a particular situation where the ligand is anchored to, and thus influenced by, a solid surface. Even though this aspect can be minimized for instance by the use of long polymeric linkers or of biological gels, it is sometimes a severe limitation. In some other studies, the orientation is favored and an NTA-based approach is preferred. The analyte is then injected and the kinetics of association is followed by quantifying the amount of material on the surface. The shape of the evolution of this surface excess can be modeled by classical kinetic equations (for details, see Chapter 7):



d[LA] = kon × [L]× [ A] - koff × [LA] dt

(8.16)

“L” represents the ligand and “A” the analyte. kon is the association constant of the complex, koff is the dissociation constant. After a certain time, the system reaches a steady state described by equilibrium constants Ka and Kd that are given by :

Ka = kon / koff

and

Kd = koff / kon

(8.17)

After this steady state, analyte-free buffer is flown over the surface. As there is no more analyte in solution the mass action law imposes a desorption of the ligands. This step is described by kinetic equations very similar to the ones describing the association. Finally, a dissociating agent is injected to remove the remaining ligands and to regenerate the surface (Figure 8.19).

Figure 8.19  Typical sensorgram obtained by SPR illustrating the three steps of association, dissociation and regeneration.

390

Experimental Approaches to Microparticles-Based Assays

8.3.3.2  Flow Cytometry-Based Techniques [37]

Flow cytometry consists of flowing cells (although this principle can be used with any microparticle) one by one in front of a detector in order to measure their physical or chemical properties. According to this measurement, the population can be analyzed or even sorted in real time by addressing them toward the right container. The most popular of these systems is fluorescence activated cell sorting (FACS), where the signal measured by the detector is fluorescence. The cells are sorted according to their measured laser-excited fluorescence. The cells also scatter some of this light (see Section 8.3.2.1, “Light scattering”), giving additional information on their dimensions. After these measurements, the buffer stream transporting the particles breaks into droplets in a very controlled way upon the application of a vibration of the nozzle. Each of these droplets contains one cell, and according to the fluorescence measurement performed earlier, the drop is electrically charged with a positive or negative charge. A static electric field is then applied transversally to this stream of drops and deflects them in one direction or the other depending on their electric charge. Sometimes, several colors can be excited by a single excitation wavelength so different aspects of the cell content can be probed simultaneously (Figure 8.20).

Figure 8.20  FACS can sort cells according to their fluorescence properties.

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391

For instance, it is possible with these instruments to sort cells by their protein content provided the protein of choice is fluorescently labeled using immunofluorescence. In the same line, assays based on GFP fluorescence (Section 8.2.1.2) can be used to analyze protein expression of certain cells. In these situations, the cell can be kept alive and cultured after this sorting. Different aspects of cell phenotype or genotype can be tested with assays based on DNA fluorescence. Live/dead assays for instance are routinely performed. They are based on the membrane permeability: when cells die, their membrane is disrupted and dyes can freely penetrate them. Dyes that fluoresce only in the presence of nucleic acids are used for this purpose. They can penetrate dead cells membranes to bind to the nucleus DNA so dead cells become fluorescent while live cells, whose membranes are intact, are not. Drugs can drastically modify the biochemistry of cells and their influence can thus be monitored by FACS. Some of these drugs modify the cell cycle, a measurement of the DNA content of each cell of a population then gives a “signature” of this particular treatment. These measurements of DNA content are useful in other situations such as the expression of a particular gene. To perform these measurements, cells are first permeabilized to allow the entry of the dye to the nucleus and thus the measurement of the DNA content. Similar architectures of cell sorting devices have been implemented in microsystems with good results, although these lab-on-chip systems are still much slower than the traditional version.

8.4  Molecular Micromanipulation 8.4.1  Force Measurements

One of the main micromanipulation techniques at the molecular scale uses the AFM described earlier. With this instrument, forces between individual objects such as proteins can be measured. The principle is quite simple: one of the interacting proteins is immobilized on the tip of the instrument and the other one on a facing solid surface. The tip and the surface are first bought into contact and then separated. The force necessary to separate them is measured by the deflection of the cantilever (Figure 8.21). The small radius of the tip ensures that only events involving single molecules interactions are measured. Using this technique, antigen-antibodies interactions have been measured and the stretching of several biomolecules including DNA and proteins have also been characterized. Probing such interaction energies (close to kBT) necessitates a particular theoretical treatment. In particular, the lifetime of a bond under a force is affected by this force [38]. This model has been adapted to the problem of the measurement of rupture forces between single proteins and it has been shown that this force F varies logarithmically with the velocity of separation [39]:

F = (k B T / x β ) × Ln[rf x β /(koff k B T)]

(8.18)

Where rf is the so-called loading rate (product of the cantilever spring constant by the velocity of separation) and xβ a characteristic length of the bond.

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Experimental Approaches to Microparticles-Based Assays

Figure 8.21  Separation of a surface from the AFM tip in the presence of an adhesive interaction. In this particular case, adhesion proteins are grafted on the tip and on the surface. When the force acting on the cantilever is too high, it snaps back to its equilibrium position. The nonlinear part of the curve represents the stretching of a single PEG linker.

An extension of this direct force measurement technique is the biomembrane force probe (BFP) in which the cantilever is replaced with a very soft red blood cell (RBC). The RCB is aspired in a micropipette. The aspiration pressure in the pipette controls the stiffness of the RBC over a wide range. Similarly to the AFM, the interaction is measured between a bead glued at the apex of the RBC and a solid surface. The extension of the RBC spring is measured optically and the force deduced from it [48]. The very low adjustable stiffness of the force sensor makes it possible to use (8.18) over a wide range and to access to xβ and koff very accurately. 8.4.2  Optical Tweezers

Optical tweezers (OTs) use a highly focused optical beam to trap particles at the focus. When a laser is focused through a high numerical aperture microscope objective, it defines a well-defined light “cage” in which not only the intensity is maximal but where the gradients in light intensity are also extremely strong [40]. We detail in Chapter 10 how a spatial gradient of electric field can be used to trap particles of a different polarizability than the one of the surrounding medium (dielectrophoresis). The physical principle is the same in the present situation: the light intensity within a laser beam is not uniform but is maximum at its center. In fact the light distribution is Gaussian. The light intensity gradients then naturally drive particles of index of refraction higher than the solution toward the center of the beam. If the beam is tightly focused, the same effect drives the particles transversally toward the focus. The net effect is a trap localized close to the focal point. Close but not exactly at this point, because there is another force: the scattering force that tends to “push” the particle away. These traps can immobilize and transport particles with forces in the range of a few tens of piconewtons, which is well suited to many practical situations. The lasers used for applications in biology are usually in the infrared spectrum in a range for which there is no absorption of energy by the water molecules (and thus no heating). OTs have been used with many different objects: viruses, bacteria, and organelles within cells but the most well-known application is the trapping of micron-sized

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393

beads that are used as handles on biopolymers or on molecular force-generating systems such as biological motors [41]. Indeed, not only can OTs trap particles, they can also measure forces applied to them: as the object is pulled away by an external force, its position in the trap varies. Recent developments in optical microscopy have made it possible to track displacements of a few nanometers (see Section 8.3.1.1). The exact position of the bead then witnesses the force applied to it. In another configuration, the light intensity can be cranked up to balance for this external force so that the position of the bead remains unchanged. Both approaches need a calibration of the trap usually by moving the particle in the fluid (or vice versa) at a known velocity and using the Stockes Einstein friction to measure the applied force. We have discussed so far the use of these optical tweezers as a mean to handle a single particle. An interesting development is the possibility of creating arrays of traps in which many particles can be manipulated at will. The simplest way to perform this task is by defining two or more positions for the focus and having the beam rapidly switch between these positions. Another approach is to define a holographic array where the energy landscape the particles are submitted to (for instance an array of traps) is defined in the Fourier plane [42]. 8.4.3  Flow-Based Techniques

Flows can be used not only to transport particles or molecules but also to manipulate them. For instance, elongational flows where the velocity increases linearly in

Figure 8.22  Fluorescently labeled DNA stuck on a solid surface after “molecular combing.” (From [45]).

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Experimental Approaches to Microparticles-Based Assays

the direction of flow can be induced in microfluidic chambers. Single DNA molecules stretched in this type of flow can be observed by fluorescence microscopy and their shape compared with existing theoretical models [43]. Flows have also been used to align molecules: Elongated rigid objects such as microtubules orient naturally in the direction of the flow and a receding meniscus can be used to perfectly align DNA molecules in the direction of drying. This “molecular combing” is performed on a modified surface and in the right pH conditions in order to have one end of the DNA stick to the surface [44] (Figure 8.22). Once the DNA molecules are all aligned and stretched, their analysis is much easier. In principle, one should be able to analyze genomic DNA by analyzing the images and to identify genes or particular sequences after hybridization. Flow chambers are used to quantify the interactions of beads or cells with a surface. In the laminar flow conditions imposed by the geometry, the established Poiseuille flow imposes a constant shear rate near the solid surface. The force acting on the flowing object next to the wall can then be computed and is found to be [46]: F ∼ ηa2Q



(8.19)

Where Q is the flow rate, a the radius of the particle, and η the viscosity. The proportionality constant is imposed by the channel geometry and can be analytically calculated. In the framework of the theory of Bell mentioned above (8.18), the duration of arrests of particles interacting with a protein bound on the solid surface is related to the dynamic characteristics of the bond and to a characteristic length [46, 47].

References   [1] Flory, P., Principles of Polymer Chemistry, Ithaca, NY: Cornell University Press, 1971.   [2] Nakanishi, H., “Flory Approach for Polymers in the Stiff Limit,” J. Phys., Vol. 78, 1987, pp. 979–984.   [3] Antognozzi, M., et al., “Comparison Between Shear Force and Tapping Mode AFM-High Resolution Imaging of DNA,” Single Mol., Vol. 3, 2002, pp. 105–110.   [4] Lockart, D. J., and E. A. Winzeler, “DNA Array: Genomics, Gene Expression and DNA Arrays,” Nature, Vol. 405, 2000, pp. 827–836.   [5] Pollack, J. R., et al., “Genome-Wide Analysis of DNA Copy-Number Changes Using cDNA Microarrays,” Nature Genet., Vol. 23, 1999, pp. 41–46.   [6] Mathews, D. H., J. Sabina, and M. Zuker et al., “Expanded Sequence Dependence of Thermodynamic Parameters Improves Prediction of RNA Secondary Structure,” J. Mol. Biol., Vol. 288, 1999, pp. 911–940.   [7] Shea, J. E., and C. L. Brooks, “From Folding Theories to Folding Proteins: A Review and Assessment of Simulation Studies of Protein Folding and Unfolding,” Annu. Rev. Phys. Chem., Vol. 52, 2001, pp. 499–535.   [8] Song, L., et al., “Structure of Staphylococcal Alpha-Hemolysin, a Heptameric Transmembrane Pore,” Science, Vol. 274, 1996, pp. 1859–1866.   [9] Zhu, H., and M. Snyder, “Protein Chip Technology,” Curr. Opin. Chem. Biol., Vol. 7, 2003, pp. 55–63. [10] Stenger, D. A., et al., “Detection of Physiologically Active Compounds Using Cell-Based Biosensors,” Trends Biotechnol., Vol. 19, 2001, pp. 304–309.

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[11] Wu, R. Z., et al. “Cell-Biological Application of Transfected-Cell Microarrays,” Trends in Cell Biology, Vol. 12, 2002, pp. 485–488. [12] Dubertret, B., et al., “In Vivo Imaging of Quantum Dots Encapsulated in Phospholipid Micelles,” Science, Vol. 298, 2002, pp. 1759–1762. [13] Dahan, M., et al., “Diffusion Dynamics of Glycine Receptors Revealed by Single Quantum Dot Tracking,” Science, Vol. 302, 2003, pp. 442. [14] Courty, S., et al., “Single Quantum Dot Tracking of Individual Kinesins in Live Cells,” Nanolett., Vol. 6, 2006, 1491–1495. [15] Sheetz, M. P., et al., Nature, Vol 340, 1989, pp. 284–288. [16] Cognet, L., et al.,“Single Metallic Nanoparticle Imaging for Protein Detection in Cells,” Proc. Nat. Acad. Sci. USA, Vol. 100, 2003, pp. 11350–11355. [17] Israelachvili, J., Intermolecular and Surface Forces, San Diego, CA: Academic Press, 1992. [18] Caffrey, M., “Membrane Protein Crystallization,” J. Structural Biol., Vol. 142, 2003, pp. 108–132. [19] Ulman, A., An Introduction to Ultrathin Organic Films from Langmuir Blodgett to Self Assembly, San Diego, CA: Academic Press, 1992. [20] Jeon, S. I., et al., “Protein Surface Interactions in the Presence of Polyethylene Oxide. 1. Simplified Theory,” J. Colloid Interf. Sci.,Vol. 142, 1991. pp. 149–166. [21] Hermanson, G. T., A. Mallia, and P. K. Smith, Immobilized Affinity Ligand Techniques, San Diego, CA: Academic Press, 1992. [22] du Roure O., et al., “Functionalizing Surfaces with Nickel Ions for the Grafting of Proteins,” Langmuir, Vol. 19, 2003, pp. 4138–4145. [23] Denk, W., and W.W. Webb, “Optical Measurements Of Picometer Displacements,” Appl. Opt., Vol. 29, 1990, pp. 2387–2391. [24] http://microscopy.fsu.edu, April 2009. [25] Diaspro, A. (ed.), Confocal and Two-Photon Microscopy: Foundations, Applications, and Advances, New York: Wiley-Liss, 2002. [27] Stryer, L., and R. P. Haugland, “Energy Transfer: A Spectroscopic Ruler,” Proc. Natl. Acad. Sci. USA, Vol. 58, 1967, pp. 719–726. [28] Born, M., and E. Wolf, Principles of Optics (sixth edition), Oxford, UK: Pergamon Press, 1991. [29] Donnert, G., et al. “Macromolecular-Scale Resolution in Biological Fluorescence Microscopy,” Proc. Natl. Acad. Sci. USA, Vol. 103, 2006, pp. 11440–11445. [30] Betzig, E., et al. “Imaging Intracellular Fluorescent Proteins at Nanometer Resolution,” Science, Vol. 313, 2006, pp. 1642–1645. [31] Rust, M. J., M. Bates, and X. Zhuang, “Sub-Diffraction-Limit Imaging by Stochastic Optical Reconstruction Microscopy (STORM),” Nature Methods, Vol. 3, 2006, pp. 793–796. [32] Jena, B., and J. K. Horber, Atomic Force Microscopy in Cell Biology, San Diego, CA: Academic Press, 2002. [33] Scheuring, S. et al., “High Resolution AFM Topographs of the Eschrichia Coli Water Channel Aquaporin Z,” EMBO J., Vol. 18, 1999, pp. 4981–1987. [34] Johnson C. S., and D. A. Gabriel, Laser Light Scattering, Boca Raton, FL: CRC Press, 1981. [35] Berne B. J., Pecora R. Dynamic Light Scattering, New York, NY: Wiley, 1976. [36] Cooper M. A., “Optical Biosensors in Drug Discovery”, Natl. Review Drug Discov., Vol. 1, 2002, pp. 515–258. [37] Bonner, W. A., H. R. Hulett, and R. G. Sweet, “Fluorescence Activated Cell Sorting,” Rev. Sci. Instr., Vol. 43, 1972, pp. 404– 409.

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Experimental Approaches to Microparticles-Based Assays [38] Bell, G. I., “Models for the Specific Adhesion of Cells to Cells,” Science, Vol. 200, 1978, pp. 618–627. [39] Evans, E., and K. Ritchie, “Dynamic Strength of Molecular Adhesion Bonds,” Biophys. J., Vol. 72, 1997, pp. 1541–1555. [40] Ashkin, A., “Optical Trapping and Manipulation of Neutral Particles Using Lasers,” Proc. Natl. Acad. Sci. USA, Vol. 94, 1997, pp. 4853–4860. [41] Bustamante, C., et al., “Single Molecule Studies of DNA Mechanics,” Curr. Opin. Struc. Biol., Vol. 10, 2000, pp. 279–285. [42] Dufresnes, E. R., et al. “Computer Generated Holographic Optical Tweezers Arrays,” Rev. Sci. Instr., Vol. 72, 2001, pp. 810–816. [43] Perkins, T. T., D. E. Smith, and S. Chu, “Single Polymer Dynamics in an Elongational Flow,” Science, Vol. 276, 1997, pp. 2016–2021. [44] D. Bensimon, A. J. Simon, and V. Croquette, et al., “Stretching DNA with a Receding Meniscus: Experiments and Models,” Phys. Rev. Lett., Vol. 76, 1995, pp. 4754–4757. [45] http://www.pasteur.fr/recherche/unites/biophyadn/f-Fcombing.html. [46] Pierres, A., A. M. Benoliel, and P. Bongrand, “Measuring Formation and Dissociation of Single Bonds Interactions Between Biological Surfaces,” Curr. Opin. Coll. Interf. Sci., Vol. 3, 1998, pp. 525–533. [47] Alon, R., D. A. Hammer, and T. A. Springer, “Lifetime of the p-Selectin-Carbohydrate Bond and Its Response to Tensile Force in Hydrodynamic Flow,” Nature, Vol. 374, 1995, pp. 539–542. [48] Evans, E., K. Ritchie, and R. Merkel, “Ultrasensitive Force Technique to Probe Molecular Adhesion and Structure at Biological Interfaces,” Biophys. J., Vol. 68, 1995, pp. 2580– 2587.

Selected Bibliography Alberts, B., et al., Molecular Biology of the Cell, New York, NY: Garland Publishing, 1989. Collection of review papers on functional genomics, Nature, Vol. 405, 2000.

Chapter 9

Magnetic Particles in Biotechnology

9.1  Introduction In many biotechnological applications, the use of a carrier fluid to transport biological objects lacks specificity: for example, it is not always possible to bring by microfluidics transport a biological target to a specific location inside the biochip. A second complementary carrier is often needed. To this extent, magnetic beads are one of the most important categories of microparticles. Between 1990 and 2004 they were developed mostly for in vitro applications, principally for biodiagnostic and biorecognition and also for purification and separation operations. More recently their use has reached the domain of in vivo applications such as cancer treatment. In this chapter, we present first the nature of magnetic beads, their magnetic characteristics, and the force that can be applied on these beads. We then give examples of trajectory calculation for applications such as separation columns and magnetic field flow fractionation. (MFFF). Finally, we show how assembly of magnetic beads has been used to build new biological tools and we focus on chains of magnetic beads, ferrofluids, and magnetic membranes. 9.1.1  The Principle of Functional Magnetic Beads

At first sight it might seem strange to consider magnetic actuation of biological microsystems because neither DNA, proteins, antibodies, cells, nor bacteria (except just one kind, but that is just anecdotal)1 are magnetic. However, the principle of functionalization has totally changed the approach: as soon as it became possible to bind DNA strands—or other biological or biochemical macromolecules—on magnetic microparticles, these microparticles could be used to displace and manipulate complex biological molecules [1]. The principle of functionalization is schematized in Figure 9.1. The principle is to find a chemical linker between the bead surface and the target in order to attach the target to the bead. There are many types of functionalizations depending on both the surface of the particle and the target. For example it has been found that the chemical group streptavidin-biotin is a good linker for the capture of DNA. It is a very complex task to find the adequate functional coating of the bead. To facilitate the task, prefunctionalized beads are currently sold by specialized suppliers. 1

The bacteria Magnetospirillum magnetoacticum has magnetic microreceptors to use the Earth’s magnetic field for orientation.

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Figure 9.1  Schematic view of the principle of a functional magnetic microparticle. The bead is constituted by Fe2O3 nanoclusters embedded in a polymer sphere. The magnetic nanoclusters (or nanograins) have a size of 5 nm. A bead containing 20 nanoclusters—as shown here—has a diameter of about 150 nm. The surface is coated with streptavidin.

9.1.2  Composition and Fabrication of Magnetic Beads

Magnetic beads of 50 nm to 2 μm are available; the choice depending on the targets and the coating. The smaller beads are used to displace small-sized targets; for example 50-nm Miltenyi magnetic beads are well suited to manipulate 32-bp (basis pair) DNA strands. For larger targets, larger beads have to be used or a larger concentration of small beads (in this case there is more than one bead attached to a single target). The beads are fabricated to be superparamagnetic (i.e., they have a magnetization only if an external magnetic field is applied and they totally lose their magnetization if the external magnetic field is removed). These beads are obtained by embedding paramagnetic nanograins (magnetic domains) of iron oxide Fe2O3 or Fe3O4 (of about 5 nm in size) in a biologically compatible matrix of latex or polystyrene. Generally, one wants to avoid having a remanent magnetic field, because this remanent magnetization does not allow the dispersion of the beads by Brownian motion when the external magnetic field is switched off, resulting in unwanted aggregates in the carrier fluid. Because large-sized beads (1–2 μm) contain more magnetic material, they experience a larger magnetic force so that they can displace larger targets. For example, a 100-nm magnetic bead only contains about 13 to 15 paramagnetic nanograins of 5 nm. Figure 9.2 shows a perfectly spherical magnetic bead, but sphericity cannot always be achieved in the fabrication process. Usually large-sized beads have a more spherical shape than the smaller ones. In a general way, spherical beads are easier to manipulate because they interact with the others in a simpler way, as will be shown later in this chapter. With the development of diagnostic devices, it has been found that beads that are both magnetic and fluorescent would be very advantageous. Magnetic beads of 50 to 100 nm are not easily seen under a microscope, but this is much easier if they are fluorescent. Many efforts have recently gone into the development of magnetic fluorescent beads. Such beads have the advantage of combining two functions: displacement when an external magnetic field is applied, and detection when the beads are excited at a wavelength corresponding to the peak in the spectrum of

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Figure 9.2  Microscope view of fluorescent magnetic beads of 200 nm (a) white light, and (b) fluorescent image.

fluorescence. Two different approaches to obtain combined magnetic/fluorescent effect have been followed. The first approach consists of incorporating fluorescent markers inside the beads during their fabrication (Figure 9.2), but it was found rather difficult to obtain all the convenient properties of magnetic beads (sphericity, biocompatiblility, monodispersion, compactness). The second approach consists of assembling a complex magnetic bead-target-fluorescent bead (Figure 9.3) or in binding the fluorescent particle directly to the magnetic bead. The scheme of Figure 9.3 is widely used because it has the advantage of marking the target directly. Magnetic beads are used to bring the targets into a detection chamber, and they are usually removed after they have completed their task, so it is thus advantageous to have the targets linked to a fluorophore for the following processes, such as detection. Recently, the principle of a compound particle at the same time magnetic and fluorescent has been established by [2]. A spherical magnetic nanoparticle built around an iron-platinium core (FePt) is coated by a layer of cadmium and sulfur (CdS). This bead is heated so it can melt the outer layer of CdS. This

Figure 9.3  Schematic view of the association of magnetic bead carrying 72-bp.DNA and marked by a fluorescent quantum dot.

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Figure 9.4  Principle of fabrication of a bifunctional magnetic/fluorescent nanoparticle.

liquid layer, because of its contact angle with the underlying solid, is not stable and migrates to take the shape of Figure 9.4 where the liquid minimizes its surface energy. The drawback of this construction is that the quantic efficiency of the particle is much less than that of a “free” quantum dot. When excited by a light source, an important part of the emitted light is absorbed by the magnetic sphere. 9.1.3  An Example of Displacement by Magnetic Beads for Biodetection

Microsystems for biorecognition or biodiagnostic require different operating steps schematized in Figure 9.5. Suppose a fluid volume containing some target molecules (such as DNA, proteins, or cells). Because direct detection is not effective for few targets in a large volume, it is necessary to concentrate the targets in a small chamber (detection chamber). At this point functional magnetic beads are often used:

Figure 9.5  Principle of magnetic concentration of targets for biodiagnostics.

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Figure 9.6  The main chamber is the white sector at the top of the figure; the fluid entrance is the round circle. The concentration chamber is located at the bottom. (Courtesy Biomerieux/LETI.)

they diffuse in the large volume and bind to the targets on contact. After a binding time, magnetic force is used to concentrate the beads in a small chamber. The chemical linking of magnetic beads and targets can be broken if some conditions are changed in the chamber, for example an increase in the temperature. Finally, the magnetic beads are removed and the targets are concentrated in the detection chamber. A realization of such a microsystem is shown in Figure 9.6. Note that in the realization of such a device, in order to achieve a satisfactory design, a careful modeling of the trajectories and concentration of the beads has been performed. Modeling of magnetic beads motion will be presented later in this chapter. 9.1.4  The Question of the Size of the Magnetic Beads

A recurring question is to decide what type of magnetic bead is the most adapted to the problem one has to solve. Most of the time it is the smaller beads that are used, because even if the magnetic traction exerted is weak, they have the property to be dispersed by Brownian motion as soon as the magnetic field is shut down (Figure 9.7).

Figure 9.7  Dispersion under Brownian effect of small-sized magnetic beads in a liquid drop. (Courtesy D. Massé, LETI.)

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This is particularly useful because after the magnetic beads have done their job carrying target molecules, they must be separated from them (by thermal heating for example) before being removed out of the reaction chamber. A compact aggregate of magnetic beads cannot “free” the targets in the detection chamber. The weak magnetic traction that can exert a small magnetic bead is not an important drawback: if the target is small (like a 32-bp. DNA strand), the magnetic force need not be very important, and if the target is larger (a cell for instance), more than one bead can be attached to the target and the magnetic traction is the result of all the forces exerted by each bead.

9.2  Characterization of Magnetic Beads







The mechanical behavior of magnetic beads depends on their magnetic properties. Characterization of magnetic beads consists of determining the average magnetic properties of the beads. On a general point of view, Maxwell’s equations determine the electromagnetic behavior of any material. The magnetic induction in Maxwell’s equation is related to the magnetic field by [3] � � � (9.1) B = µ0 (H + M) where the magnetization is given by � � � M = χ H + Mr � where Mr is the remanent magnetization,so that � � � B = µ0µ r H + µ 0 Mr

(9.2)

with mr = 1 + c This shows that� the information we need is contained in the relation between � the magnetization M of the bead and the applied external magnetic field H. This relation is usually determined experimentally by making use of a superconducting quantum interference device (SQUID) or a Hall probe.

9.2.1  Paramagnetic Beads

Magnetic micro- and nanoparticles used in biotechnology are nearly always paramagnetic—even superparamagnetic—because the magnetic force should vanish when the external field is switched off. If not, the beads would agglomerate and it would be impossible to have them dispersed in the carrier fluid. Paramagnetic media follows Langevin’s law [4]



M æ 3χ H ö 1 = coth ç è Ms ÷ø 3 χ H Ms Ms

(9.3)

9.3  Magnetic Force

403

æ 3χ H ö χH 1 Note that, at low magnetic field, coth ç and (9.3) reduces = ÷ χ 3 H è Ms ø Ms Ms M 1 = 1to the usual expression M = cH. For large magnetic field, , and 2, 3χ H Ms Ms which states that saturation is reached. The diagram M(H) is plotted in Figure 9.8. If we assume that the paramagnetic beads are monodispersed (all the beads are identical), Langevin’s law may be applied to each bead. 9.2.2  Ferromagnetic Microparticles

The situation is more complex for ferromagnetic beads because ferromagnetic objects keep a remanent magnetization when the external field vanishes. There is generally no analytical function for the magnetization and one generally tries to fit the experimental curve by piecewise continuous polynomials.

9.3  Magnetic Force A general expression � of the magnetic energy of interaction of a particle immersed in a magnetic field H is [5, 6]





Em = -

� � 1 µ0 ò M.H dV 2

(9.4)

� � where M is the magnetization of the particle in the applied magnetic field H. The integration is taken over the particle volume. The magnetic force exerted by the magnetic field on the particle is the gradient of the interaction energy � Fm = -ÑEm (9.5)

Figure 9.8  Relation M(H) for the different types of materials.

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Because of the very small size of the magnetic particles, the integration in (9.4) may be replaced by the value of the fields at the center of the particle multiplied by the volume of the particle vp Em = -



� � 1 µ0 v p M.H 2

(9.6)

9.3.1  Paramagnetic Microparticles

Most of the time, particles used in biotechnologies are paramagnetic (one exception will be discussed in Section 9.13 on magnetic membranes), the magnetization is then aligned with the external field [7] � M=



� χ H 1 + Dm χ

where Dm is the demagnetization coefficient (Dm = 1/3 for a sphere). Because c << 1 for the considered paramagnetic particles, magnetization can be approximated by � � M = χH

and the energy of interaction is Em = -



1 µ0 v p ( χ p - χ f ) H 2 2

(9.7)

where cp is the magnetic susceptibility of the particle and cf is that of the carrier fluid. Thus, the magnetic force on a paramagnetic microparticle is given by the gradient of (9.7) � 1 Fm = µ0 v p (χ p - χ f ) ÑH 2 2



(9.8)

In the literature another equivalent expression is often found [7, 8]. Indeed, if no electric field is associated to the magnetic field (or if the electric field is constant), Maxwell’s equation implies that � Ñ´H = 0

and (9.8) can be written

� � � Fm = µ0 v p (χ p - χ f )(H.Ñ)H



(9.9)

9.3.2  Ferromagnetic Microparticles

In the case of ferromagnetic particles, we start from (9.5) and (9.6) [9] to obtain



Fm =

� � � � µ0 µ v p Ñ(M.H) = 0 v p Ñ((Mp - Mf ).H) 2 2

(9.10)

9.4  Deterministic Trajectory

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9.4  Deterministic Trajectory





At a macroscopic scale, kinematics theory relates the mass acceleration of a body to the resultant of the external forces that act upon it. This is the well-known Newton’s theorem. � � dV (9.11) m = å Fe dt � � where m is the mass of the particle, V the velocity, and Fe the external forces. At a macroscopic scale, Brownian motion (random hit by other molecules) is completely negligible. At a microscopic scale, the effects of the Brownian agitation are more visible and Newton’s formula should be replaced by Langevin’s law [10] � � dV m = å Fe + R(t) dt

(9.12)

where R(t) is a white noise due to the Brownian effect. However, it is often the case that an “average” trajectory can be calculated simply by using Newton’s equation, especially if the size of the beads is larger than 1 μm or if the forces that act on the particle dominate the Brownian motion. In such a case, because Brownian motion can be considered as a white noise, the real positions of the beads are close to the “average” trajectory, with a nearly symmetrical dispersion. This “average” trajectory is often sufficient to predict the behavior of the microsystem and to design the relevant component. Usually, for microfluidics systems using magnetic beads, three types of forces are present: gravity, hydrodynamic drag, and magnetic forces. In this case, Newton’s equation can be written under the form





� � � dVp � m = Fmag + Fhyd + Fgrav dt

(9.13)

� � � � � Fhyd = -CD (Vp - Vf ) = -6 π η rh (Vp - Vf )

(9.14)

� where Vp is the velocity of the particle. The hydrodynamic drag is derived from the velocity field according to the equation

where CD is the drag coefficient, h is the dynamic viscosity of the carrier fluid, rh the hydrodynamic diameter of the particle, and Vf the velocity of the carrier fluid. It is assumed here that the velocity field of the carrier fluid is not affected by the presence of the beads, which is the general case, except if the volume concentration of the beads is important leading to aggregation of the beads. Under this assumption, the velocity field of the carrier fluid must be calculated before attempting the calculation of the particles trajectories, using classical hydrodynamics equations (i.e., Navier-Stokes equations). A classical situation in microfluidics is the Poiseuille flow between two plates or in a rounded capillary. In such a case, the velocity field is obtained under a closed form.

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Again, if the concentration of magnetic beads is not too important, the external magnetic field is not affected significantly by the presence of the magnetic beads. In such a case, the magnetic field must be calculated before attempting the calculation of the trajectories. Under this assumption, the magnetic force on a particle of volume vP is given by



� ö æ1 Fmag = µ0v p D χ Ñ ç H 2 ÷ è2 ø

(9.15)

where Dc is the difference of magnetic susceptibility between a particle and the fluid, H the magnetic field. Equation (9.15) indicates that the magnetic force is aligned with the direction of the gradient of the square of the magnetic field. Finally, the gravity term is simply given by � Fgrav = g v p D ρ yˆ (9.16) where g is the acceleration of gravity, vp the volume of the particle, yˆ the vertical unit vector (oriented downwards) and Dr the difference between the volumic mass of the particle and that of the liquid. After substitution of (9.14), (9.15), and (9.16) in (9.13), one obtains the equation for the particles velocity



� � � dVp æ1 ö m = µ0 v p D χ Ñ ç H 2 ÷ - 6 π η rh (Vp - Vf ) + g v p D ρ yˆ è2 ø dt

(9.17)

x and y coordinates of the particle at a given time are linked to the velocity by the relations dx = Vp, x dt

dy = Vp,y dt

(9.18)

Equations (9.17) and (9.18) define the particle trajectory. It is very seldom that they can be solved analytically, but as we will see, it is always interesting to spend some time investigating if an analytical solution may exist—even for the price of some simplification. An example will be given in Section 9.10. Most of the time, a numerical approach is required. Different methods can be used such as Runge Kutta or predictor-corrector. We indicate in Section 9.9 a very simple first-order predictor-corrector method that is very efficient when the velocity of the carrier fluid is sufficiently low.

9.5  Example of a Ferromagnetic Rod A very didactic example of trajectories of magnetic microparticles is that of the ferromagnetic rod. In this particular case, a closed form solution exists for the magnetic field and the trajectories may be calculated easily [7, 10]. Thus, it is a good test for the verification of numerical models. Moreover, it bears the

9.5  Example of a Ferromagnetic Rod

407

principle of magnetic separators: we shall see that there are magnetic attraction zones and repulsion zones around the rod and particles are expelled from the repulsion zones towards the attraction zones and concentrate on some regions on the rod surface. Assembly of ferromagnetic rods located in a large external magnetic field are called high gradient magnetic separators (HGMSs) and are widely used in chemical and biological processes to remove magnetic particles from a carrier fluid [12]. 9.5.1  Governing Equations

Suppose a ferromagnetic rod of radius a, surrounded by a carrier fluid containing � paramagnetic microparticles and placed in a uniform external magnetic field H0 (Figure 9.9). In this problem, there is no electric current and the use of the magnetic scalar potential f is sufficient to solve the problem. Using the relation

� H = - grad(φ) the potential f is then the solution of the following equation



® � div(- µ0µ r grad(φ)) + div(µ 0 Mr ) = 0

(9.19)

where mr is the relative magnetic permeability of the different materials of the computational domain. The remanent magnetization may be supposed uniform in the rod, then (9.19) reduces to ®



div(- µ0µ r grad(φ)) = 0

(9.20)

Equation (9.20) might not be linear if the external magnetic field is large enough to saturate parts of the ferromagnetic regions. Note that f is continuous on the whole computational domain even at the boundaries between different materials. The

Figure 9.9  Schematic view of a ferromagnetic rod in an external magnetic field.

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� external field H0 is� a Neumann boundary condition for (9.20). After solving for f, ® � H is obtained by H = - grad(φ). � � � Note that �H is different from H0 in� the vicinity of the rod: H is the sum of the external field H0 and the induced field Hi. The � magnetic force field on the paramagnetic microparticles is then obtained from H by calculating � � � µ Fm = µ0 v p (χ p - χ f ) (H.Ñ)H = 0 v p (χ p - χ f ) Ñ H 2 2



(9.21)

and the value of the magnetic force field is then plugged into the trajectory equation



� � � dVp æ1 ö m = v p D χ Ñ ç H 2 ÷ - 6 π η rh (Vp - Vf ) + g v p D ρ yˆ è2 ø dt This is the general case. However, in the special case of the ferromagnetic wire, the calculation of the particles trajectories may be further detailed because the magnetic field may be calculated analytically [13]. 9.5.2  Analytical Solution for the Magnetic Field

The total magnetic field in the vicinity of the rod is the sum of the external magnetic field plus the induced field due to the presence of the rod. It can be shown [9] that, at a distance r from the rod center and at angle q from the external magnetic field H0, the magnetic potential is 1 æ ö φext = ç - H0 r + M a2 r -1 ÷ cos θ è ø 2



(9.22)

At this stage, two different cases may be distinguished: 1. At very � high external field, the rod is magnetically saturated and the total field H is given by Aö æ Hrext = cos θ ç H0 + 2 ÷ è r ø Hθext

where A =

Aö æ = sin θ ç - H0 + 2 ÷ è r ø

1 Ms a2. The magnetic force is then 2 Fm = -2 µ0 (χ p - χ f ) vP



(9.23)

æA ö A ç 2 + H0 cos 2θ ÷ r ÷ r3 ç è H0 sin 2θ ø

(9.24)

2. At sufficiently low magnetic field, the magnetization is linear homogeneous and (9.22) leads to

9.5  Example of a Ferromagnetic Rod

409

Hrext = H0 cos θ (1 + a2 L r -2 ) Hθext = H0 sin θ (- 1 + a2 L r -2 )



(9.25)

(µw - µ f ) and mw , mf are respectively the magnetic permeability of the rod (µw + µ f ) and the fluid. The magnetic force is then

with L =



æ é a2 ù ö 2 � a 2 ç L ê 2 ú + cos 2θ ÷ Fm = -2 µ0 (χ p - χ f ) v p 3 H0 L ç ë r û ÷ r çè ÷ø sin 2θ

(9.26)

Note the similarity between the two expressions of the magnetic forces (9.24) and (9.26). In both cases, in the vicinity of the rod, there are two attraction zones aligned with the external field (for q close to zero or to p) and two repulsion zones in the direction perpendicular to the external field (for q close to p /2 and –p /2). By taking the ratio between radial and azimuthal components of the force in either expression (9.24) or (9.26), it can be deduced that attraction forces are larger than repulsion forces. We show the magnetic field force in the vicinity of the rod in Figure 9.10. 9.5.3  Trajectories (Carrier Fluid at Rest)

We investigate first the case where the fluid is at rest. Using the algorithm for the calculation of trajectories described in Section 9.9, and the magnetic forces calculated in the preceding section, we find that the microparticles migrate from the two repulsion zones towards the two attraction zones, as shown in Figure 9.11. Theoretically, the repulsion and attraction regions extend to infinity, but the magnetic

Figure 9.10  Magnetic force field around a cylindrical rod placed in a uniform magnetic field (directed from left to right) showing to attraction zones and two repulsion zones.

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Figure 9.11  Trajectories of paramagnetic microparticles in the vicinity of the rod if the carrier fluid is at rest.

force decreases very rapidly from the rod surface, and particles located far from the rod will not move significantly. 9.5.4  Trajectories (Carrier Fluid Convection)

Suppose now that the carrier fluid is slowly moving around the rod. In this case, the flow is laminar and can be calculated either by solving the Navier-Stokes equations (Figure 9.12), either by using Lamb’s equation [14] é 1 öù æ ur = -C êln(Re) - 0.5 ç 1 - 2 ÷ ú sin θ è Re ø û ë



é 1 öù æ uθ = -C êln(Re) + 0.5 ç 1 - 2 ÷ ú cos θ è Re ø û ë

(9.27)

Figure 9.12  Velocity field of a laminar flow around a 2-mm diameter fiber. Velocity at infinity is 1 mm/s.

9.5  Example of a Ferromagnetic Rod

411

Figure 9.13  Trajectories of paramagnetic beads in the vicinity of a ferromagnetic rod under the action of a convective transport and a magnetic force field. Left, fluid velocity at infinity 0.1 mm/S; right, fluid velocity at infinity 1 mm/s.

with C=



Re =

uf0 æ 7.4 ö ln ç è Re ÷ø 2 a ρf uf0 µf

The velocity field is represented in Figure 9.12. Calculated trajectories for fluid velocities of 0.1 mm/s and 1 mm/s are shown in Figure 9.13, for a 1-mm radius fiber and 1-μm Dynal beads. The larger the carrier fluid velocity, the less the probability of capture by the rod.

Figure 9.14  Principle of magnetic repulsion.

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Figure 9.15  Force field created by two oblique symmetric ferromagnetic plates in a uniform external magnetic field. (From [15].)

9.6  Magnetic Repulsion In the previous section, it has been observed that the magnetic force field in the vicinity of a cylindrical rod placed in a uniform external magnetic field is highly nonuniform. The force is directed towards the rod in the regions aligned with the external field (from the rod center) and directed away from the rod in the regions perpendicular to the external field (from rod center). This observation can be extended to any shape of ferromagnetic body. We can even imagine more than one body in the external field as shown in Figure 9.14 and obtain a confinement zone for the magnetic particles in a region located between the ferromagnetic bodies. Note that the particles are not completely at rest in the confinement zone: they are repelled from the walls, and they move along the central axis to exit on both sides, as shown in Figure 9.15 from the calculation of the magnetic force field. Figure 9.15 Force field created by two oblique symmetric ferromagnetic plates in a uniform external magnetic field. (From [15].) A very simple experimental setup may be realized to illustrate the principle of magnetic repulsion [15] (Figure 9.16). Place two ferromagnetic wires on a glass

Figure 9.16  Experimental verification of the magnetic repulsion principle. Paramagnetic microparticles (1 μm) repulsed from two ferromagnetic parallel wires.

9.7  Magnetic Beads in EWOD Microsystems

413

support perpendicular to two magnets. Between the magnets, the magnetic field is approximately uniform. If a drop containing paramagnetic particles is deposited on the plate between the wires, then the microparticles move rapidly towards the central axis; later they migrate slowly along this central axis.

9.7  Magnetic Beads in EWOD Microsystems As mentioned in the introduction of this chapter, in biotechnology, magnetic beads are often used for the transport of macromolecules: superparamagnetic beads are functionalized or labeled to recognize and bind to a specific molecule. Functionalization is achieved by coating the magnetic bead with molecules having a chemical affinity for the target molecules. Hence, when dispersed in a liquid, the labeled beads bind to the target molecules, forming a composite macromolecule. In the absence of a magnetic field, the particles disperse in the liquid phase if they are sufficiently small. In presence of a magnetic field, they are attracted towards the magnetic pole, and aggregate together because of the formation of chains of magnetic dipoles. Techniques based on magnetic capture and concentration, similar to those using microflows, have been developed in digital microfluidic devices, with the all the advantages of smaller liquid volumes, allowing more precise and sensitive recognition and/or bioanalysis. Figure 9.17 shows an open EWOD device (with catena) and a droplet containing magnetic beads aggregated by a minimagnet placed below the substrate. On one hand, if the electrowetting forces are sufficient and on the other hand if the magnetic forces are sufficient, the magnetic aggregate separates from the droplet. Figure 9.18 demonstrates the principle of concentration of target molecules by paramagnetic labeled microbeads. First the beads are dispersed in the droplet and some bind on the target molecules (a); then the beads are aggregated by using a minimagnet (b); the droplet is then motioned by electrowetting actuation (c and d); if the magnetic and electrowetting forces are sufficient, the droplet continues its motion, leaving behind the aggregate with a small amount of liquid (e). Target molecules are now concentrated in a “nano” droplet.

Figure 9.17  Combination of magnetic forces exerted on an aggregate of magnetic beads and electrowetting forces exerted on a conductive liquid droplet. In this case, the magnetic forces are sufficiently important to pin the aggregate and the electrowetting forces are sufficiently large to move the droplet, leaving behind the aggregate with a small amount of liquid. (Photo courtesy N. Chiaruttini, CEA-LETI.)

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Figure 9.18  Principle of extraction/concentration using a combination of electrowetting and magnetic beads: at the end of the process (f ), the concentrated aggregate of magnetic beads attached to target molecules can be removed with a pipette.

Figure 9.19 shows the balance of forces applied on the receding contact line. In absence of magnetic and electrowetting forces, the balance of forces is given by the Young law (a). The aggregate is blocked near the tip of the magnet; the shift in the horizontal direction of the magnetic aggregate produces a horizontal magnetic force fx,mag; the equilibrium position is found to give the maximum force fx,mag (b). The

Figure 9.19  Balance of forces on the receding contact line. (a) Young’s law in absence of magnetic and electrowetting forces; (b) deformation of the droplet under the two actions of the magnetic and electrowetting forces; (c) balance of the forces at the rupture limit; (d) if the electrowetting forces are sufficient for pulling the interface and decreasing the contact angle below its limit value, rupture inevitably occurs.

9.8  Example of a Separation Column

415

Figure 9.20  Assembly of parallel ferromagnetic wires in a uniform external magnetic field; the carrier fluid circulates between the wires.

limit angle a of the droplet is given by the diagram (c). If the electrowetting force is larger than the equilibrium force of diagram (c), the contact angle decreases under the value a, and the droplet separates from the magnetic aggregate (d).

9.8  Example of a Separation Column It is important to be able to remove magnetic micro- and nanoparticles from a carrier liquid. This process may be performed in high-gradient magnetic separators (HGMSs). In this section, we investigate the behavior of magnetic beads submitted to a nonuniform magnetic field in the vicinity of an assembly of ferromagnetic rods or wires (Figure 9.20) and convected by a carrier fluid. Theoretically, this situation is just a generalization of the example of the ferromagnetic rod. The basic equations are the same [16]. The first step consists in calculating the magnetic field by using (9.20), then the magnetic force field is deduced from (9.21). Figures 9.21 and 9.22 show the magnetic vector field and magnetic forces in the assembly of wires. Magnitudes of these

Figure 9.21  Magnetic field in an assembly of ferromagnetic wires.

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Figure 9.22  Magnetic forces.

vectors are plotted in Figures 9.23 and 9.24. The situation is similar to the case of a single wire, but due to the packing of the rods, the repulsive forces are somewhat smaller. The second step consists of solving the Navier-Stokes equations for the flow in the bundle (Figure 9.25, left). As expected, fluid flow is accelerated in the horizontal gaps between the rods. The third step is the calculation of the trajectories following (9.17). Particles are principally captured on the surface of the rods facing the upcoming velocities (Figure 9.22, right). Few particles are captured in the attraction zones behind the rods; none are captured on the sides, where there are repulsive forces plus an acceleration of the flow. Note that we have supposed that the captured microparticles are not numerous enough to modify the magnetic field and the fluid flow circulation.

Figure 9.23  Magnitude of the magnetic field.

9.9  Concentration Approach

417

Figure 9.24  Magnitude of the magnetic forces.

9.9  Concentration Approach In this section, we present another approach to the behavior of magnetic particles. Particles are no longer considered as discrete entities but rather as a continuous component in the carrier fluid. The presence of magnetic beads is defined by their volume concentration c equal to the number of particles per unit volume (or moles per unit volume). In the International System of units c is expressed in particles/m3. Because they use very small volumes of liquids, biologists and chemists often use particles/μl or mole/ μl.

Figure 9.25  Left: norm of the carrier fluid velocities; right: trajectories of particles through the bundle. Magnetic particles are primarily captured on the side of the wires facing the external magnetic field, some are captured behind, and none are captured on the sides.

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Concentration repartition is calculated by solving the mass conservation equation for the particles in the liquid, taking into account convection due to the flow and to the magnetic forces [17]



� � � � ∂c = Ñ.(D Ñ c - c(Vf + u Fmag )) = D D c - Vf Ñ c - Ñ.(u Fmag c) ∂t

(9.28)

In (9.28) D is the diffusion coefficient (unit SI: m2/s) defined by Einstein’s equation



D=

kB T 6 π η rh

(9.29)

where kB is the Boltzmann constant, T the Kelvin temperature. The mobility u is defined by



u=

1 6 π η rh

(9.30)

Note that u is the inverse of the drag coefficient CD. In (9.28) the first term at the right-hand side is the diffusion term that takes into account the Brownian motion, the second term is the advection due to the motion of the carrier fluid, and the third term is the convection due to the magnetic forces. As for the calculation of particle trajectories, (9.28) requires the previous calculation of the velocity field of the carrier liquid and the magnetic force field. Generally, two types of numerical approach can be done to solve (9.28): finite elements method or finite differences/volumes method. In microfluidics, the boundaries of the domain have a very important impact and the finite elements method is well adapted for such problems. However, in a simple geometry (rectangular or axisymmetric), the finite differences method is very easy to use and one can write its own numerical program to solve such problems. Note that the problem is a weak coupled problem. It must be solved in three steps. First, compute the carrier fluid velocities, second, compute the magnetic force field, and third, solve for the concentration distribution in the domain. Using the same numerical frame to solve successively the three equations is the most straightforward solution. Note that boundary conditions must be specify to solve (9.28). At channel inlet, boundary conditions for the concentration are of the type c = c0 or c = 0 depend¶c = 0 at the channel ing on the location of particulate injection; and the condition ¶n outlet. The solid walls are impermeable to fluid and particle transport, so that the corresponding boundary conditions are

�� � � �� � � J .n = -D grad c.n + c v.n + c u Fmag .n = -D grad c.n + c u Fmag = 0 � where J is the flux of particles.

(9.31)

9.10  Example of MFFF

419

9.10  Example of MFFF In this section we illustrate how to calculate particle trajectory and concentration by taking the example of magnetic field flow fractionation. In biology and biotechnolgy, there is a constant need to separate particles, depending on their mass, electric charge, or magnetic properties. For example, this is the case for purification of proteins or for obtaining monodisperse magnetic beads. A very common method for separation of particles is called field-flow fractionation (FFF). In a typical FFF device, the carrier fluid flows horizontally in a channel and the particles experience a horizontal drag force; depending of the type of separation that is searched (mass, electric charge, magnetic properties) a relevant force field (gravity, electric field, magnetic field) is set up to act perpendicularly to the flow. Trajectories of the different particles differ in the FFF force field. Similar particles have similar trajectories and gather at the same location on the channel lower wall. A magnetic FFF is sketched in Figure 9.26. Much work has been done in the domain of sedimentation field flow fractionation (SdFFF) [18]; however, recently, new applications for biological processes— such as cell separation—have required the use of submicronic paramagnetic particles that are not much influenced by gravity but magnetic field-flow fractionation (MFFF or simply MF) is a well-suited method to separate these particles according to their size and magnetic permeability [19, 20]. The velocity field is well known in the case of a laminar flow between two horizontal parallel plates (the Reynolds number is much less than 1). From hydrodynamics considerations, it is well known that the velocity profile across any section is parabolic and given by



� � 3V0 æ y2 ö Vf = 1- 2÷ ç 2 è d ø

(9.32)

where d is half the distance between the channel walls and V0 the average velocity.

Figure 9.26  Schematic view of MFFF. The upper solid wall is called the depletion wall and the lower wall is the accumulation wall.

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9.10.1  Trajectories

Due to the design of the MFFF device, with a vertical external force field and a horizontal hydrodynamic drag, (9.17) can be decomposed on the x and y directions and we obtain the following system for the particle velocity dVp, x = -c1 (Vp, x - Vf , x ) dt dVp,y = -c1 Vp,y + c2 dt



(9.33)

where c1 and c2 are given by



c1 =

6 π η rh m

æ1 ö v p Dχ Ñ ç H 2 ÷ .yˆ g vp D ρ è2 ø c2 = + m m Equation (9.33) forms a noncoupled, first-order, differential system. To solve numerically for such a system is classical; almost any mathematical software possesses an algorithm to solve such a system. However, (9.33) admits a closed form solution for the velocity if we assume that c1 and c2 are constant [21]; this is not very restrictive since c1 is constant in an homogeneous fluid, and c2 also if the magnetic gradient is uniform—which is the case when the dimension of the magnet is sufficiently large compare to the dimension of the channel. The closed form solution is then Vp, x = Vp, x0 e -c1t + Vfx [1 - e -c1t ]



c Vp, y = Vp, y0 e -c1t + 2 [1 - e -c1t ] c1

(9.34)

where the subscript zero corresponds to the initial values at t = 0. We have thus derived an analytical expression for the particle velocity that depends on the values of the magnetic force, drag coefficient and gravity. In (9.34) the applied forces appear through the ratio c2/c1



æ1 ö v p Dχ Ñ ç H 2 ÷ .yˆ + g v p D ρ è2 ø c2 = c1 6π η r

(9.35)

It is seen in (9.35) that c2 / c1 represents the ratio between the applied external (vertical) forces and the hydrodynamic (horizontal) drag force (c2 / c1 has the dimension of a velocity). This ratio determines the trajectories. Two particles experiencing the same ratio c2 / c1 and starting for the same point at inlet will follow the same trajectory to some bias due to the Brownian motion. We now advance a step further and search for a solution for the trajectory. We have to solve the first-order differential system

9.10  Example of MFFF

421

y2 ö 3V æ d xP = Vp, x0 e -c1t + 0 ç 1 - P2 ÷ [1 - e -c1t ] 2 è dt d ø



d yP c = Vp, y0 e -c1t + 2 [1 - e -c1t ] dt c1

(9.36)

where xP and yP are the coordinates of the particle at time t. This time the system (9.36) is coupled because the y-coordinate appears in the first x-equation—which means that the trajectory of the particle will not be linear. 9.10.1.1  A Case of Analytical Solution to the Trajectory

If we examine in detail the system, it can be seen that, if the velocity of the particle is zero at the inlet Vp,x=Vp,y=0—this is the case if the particles start at the top, along the upper (depletion) wall—time can be eliminated from (9.36) and the differential equation has the following particular form



y2 ö c d xP 3V0 æ = 1 - P2 ÷ 1 ç 2 è d yP d ø c2

(9.37)

Equation (9.37) is easily solved (taking into account with x0 = 0, y0 = –d) and we find



xP = -

V0 c1 [yP (yP2 - 3d 2 ) - 2d 3 ] 2 d 2 c2

(9.38)

This is the equation of a cubic; the corresponding trajectory has been plotted in Figure 9.25.; one verifies that xP = 0 for yP = –d. By setting yP = d in (9.38) one finds the distance from inlet at which a particle meets the accumulation wall.



xd = 2 dV0

c1 c2

(9.39)

It can be readily checked that the larger the magnetic force, the smaller the distance xd and the larger the hydrodynamic drag, the larger the distance xd. Equation (9.39) can be used to define the separation efficiency of the MFFF. If the particles are injected at the top of the channel through a nozzle, the separation distance between the accumulation sites on the accumulation plate for two different types of particles (1 and 2) is given by (9.39). Provided that the particles are sufficiently small to neglect the gravity force before the magnetic force, one obtains a very simple relation [21]



2 L2 Dχ1 r1 = L1 Dχ 2 r22

(9.40)

where L is the distance from the channel entrance. Usually, smaller paramagnetic particles have lower magnetic susceptibility, so that we can expect an efficient separation between two populations of magnetic beads (i.e., the two types of magnetic beads are gathering in two distinct packets on the accumulation wall).

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Equation (9.40) for separation efficiency of a MFFF device shows the advantage of analytical methods when possible. They can produce a relation between parameters that is not always obvious when using numerical solutions. It is strongly recommended to always search first for analytical solutions—even at the price of simplifications—before turning to numerical methods. 9.10.1.2  General Case: Numerical Approach

The system (9.34) determines the velocity field of the particles in the carrier fluid. The question is now to solve the system (9.36) for particle trajectory. More or less sophisticated methods can be used. But if we take advantage of the very slow velocity of the carrier fluid, a very simple predictor-corrector method can be set up. Suppose that the particle has the coordinates xi and yi at time ti. The first step of the numerical scheme is to find a predictor �point at time ti+1. This predictor point is obtained by making use of the velocity Vp,i = (Vp, x,i ,Vp,y,i ) at time ti �i +1 = xi + Dt Vp, x,i x y�i +1 = yi + Dt Vp,y,i



(9.41)

� In reality, the particle velocity is not constantly Vp,i during the time interval [ti, ti+1]. �i +1, y�i +1), we now know the velocity at Because we have found the predictor point (x this point V�p,i +1 = (V�p, x,i +1,V�p,y,i +1) and a more accurate velocity in the time interval æ V� + Vp, x,i V�p,y,i +1 + Vp,y,i ö [ti, ti+1] is ç p, x,i +1 , ÷. 2 2 è ø The second step is then the following correction xi +1 = xi + Dt



yi +1

V�p, x,i +1 + Vp, x,i 2

V�p,y,i +1 + Vp,y,i = yi + Dt 2

(9.42)

This two-step predictor-corrector method can be schematized graphically (Figure 9.27). The particle is located at the point Mi at time ti, the predictor point is Pi+1 and the corrected point Mi+1. The distance between these two points is the first order error. The larger the carrier fluid velocity, the larger the distance [Pi+1, Mi+1]. A second-order method using the same principle can be easily set up to verify that the precision is satisfactory. 9.10.2  Concentration of Magnetic Beads

The starting point is (9.28) where the velocity field is given by (9.32) and the magnetic force by (9.21)



� 3V0 æ y2 ö ∂c æ1 ö = DD c Ñ c - u vp D χ Ñ ç H 2 ÷ Ñ c 1 ç ÷ 2 è2 ø 2 è ∂t d ø

(9.43)

9.11  Assembly of Magnetic Beads–Magnetic Beads Chains

423

Figure 9.27  Graphical scheme of the first-order predictor-corrector method.

In (9.43), two terms on the right-hand side contribute to the convection of the particles: the drag force of the velocity field and the magnetic force term. Equation (9.43) must be solved by finite elements or finite differences/volumes numerical schemes. 9.10.3  Results and Comparison

We examine the case of an MFFF channel of 100-μm width and 1-mm length. The fluid carrier is water and its average flow velocity is 0.1 mm/s. Particles have a hydrulic radius of 1.4 μm and a magnetic susceptibility of 0.2, their diffusivity is 1.53 μm2/s and the magnetic force is 3.45 pN. In this particular case, the magnetic force is assumed constant. The velocity field is obtained under a closed form by (9.34) and the trajectory by (9.38)—if the injection point is located at the top plate—or else by a predictor-corrector scheme (9.42). Figure 9.28 shows contour plots of the particles concentration compared to the calculated trajectories. The location of the injection is either at the top of the channel or at 1/3 of the vertical height. As expected, the concentration contours are centered on the trajectories: Brownian motion slightly disperses the particles from their determinist trajectory. In a second step, the same method has then been applied to the case of the separation of a colloid mixture containing two different types of submicronic paramagnetic particles (Figure 9.29) differing by their magnetic susceptibility (0.2 and 0.8). We verify that the characteristic relation (9.40) applies even for trajectories not starting from the depletion (upper) wall.

9.11  Assembly of Magnetic Beads—Magnetic Beads Chains Superparamagnetic beads in an external magnetic field are similar to small induced magnetic dipoles [22, 23]. Their magnetic moment is given by

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Figure 9.28  Comparison between calculation of trajectory and concentration contours, showing that the “determinist trajectory” is the center line of the concentration contours. In the figure on top, the trajectory is the cubic curve given by (6.18), below the trajectory is obtained by a first-order predictor-corrector scheme. The channel width is 100 μm and the average fluid velocity 0.1 mm/s. Particles have a radius of 1.4 μm and a magnetic susceptibility of 0.2. They are submitted by the magnetic gradient to a force of 3.45 pN.

Figure 9.29  Two different families of trajectories for two different types of beads. In both cases, the starting point is the same, but the particles on the figure at the top have a magnetic susceptibility of 0.2 whereas those in the lower figure have a susceptibility of 0.8. One can verify that the relation (9.40) is also verified in the case of the numerical calculation of the trajectories.

9.11  Assembly of Magnetic Beads–Magnetic Beads Chains

� 1 µr - 1 3 � m= πa H 2 µr + 2



425

(9.44)

where a is the dipolar distance. This induced magnetization creates the induced magnetic field



� Hi =

� � � 1 � (3 i (i . m) - m) 3 4π r � � at a point defined by the vector ri (from the dipole center). In the total field H, the induced magnetic field of a microparticle of radius a is � 1 µ r - 1 a3 � Hi = H 4 µr + 2 r 3



(9.45)

Take two same microbeads—of radius a—referred to by indices 1 and 2. The magnetic field at the center of each bead is � � � H1 = H0 + Hi,12 � � � H2 = H0 + Hi,21



Using (9.45) with the corresponding indices, we find the coupled system � � 1 µ r - 1 a3 � H1 = H0 + H2 4 µr + 2 r 3 � � 1 µ r - 1 a3 � H 2 = H0 + H1 4 µr + 2 r 3

Let α =



(9.46)

1 µr - 1 , we can solve (9.46) and we find 4 µr + 2 � � æ 1 ö H1 = H0 ç ÷ è 1 - α u3 ø

with u = a/r. For r = a (when the two beads contact)



� � æ 1 ö H1 = H0 ç è 1 - α ÷ø Using the formulation of the magnetic force, the force exerted by one sphere on the other is



� a6 α2 F1 = -6 µ0 π 4 H02 3 3 r æ æ aö ö ç 1 - α çè ÷ø ÷ r ø è

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When the two spheres contact each other, r = a and � F1(contact) = -6 µ0 π a2 H02

α2 3

3 æ æ aö ö 1 α ç çè ÷ø ÷ r ø è

(9.47)

Equation (9.47) shows that the binding force between two similar beads is proportional to the square of the bead radius and to the square of the magnetic field. In the same way that we have shown the existence of magnetic attraction and repulsion regions at a wire surface, it can be shown that the same exists at the surface of a spherical bead [22]. The two regions around the bead aligned with the external magnetic field are magnetically attractive, and the region around the “magnetic equator” of the bead is a repulsion zone (Figure 9.30). Under the action of the Brownian motion, beads randomly contact each other; contacting beads tend to stick together by their attraction regions and progressively will form a linear chain of beads aligned in the external field (Figure 9.31). These binding forces are more efficient for larger beads (1 μm) than for smaller beads (50 nm), and nano-sized beads—those under 50 nm—are often dispersed by the Brownian motion. Magnetic chains are new tools in biotechnology (they can even be stabilized by polymer coating) and they have found an application for DNA separation: these chains are used to separate DNA segments in the same way as a gel. Let us first recall how DNA separation works in a gel. Under an electric field, the DNA segments migrate under electrophoretic forces (see Chapter 10) at a different speed depending on their size (Figure 9.32). The longer the strand of DNA, the more it encounters obstacles and the more it is delayed in its motion. This technique has been widely used to decrypt the human genome. The major drawback is that a characteristic time for the separation is of

Figure 9.30  Magnetic attraction and repulsion regions around a spherical bead. The external magnetic field is oriented from left to right. Only one-fourth of the space has been represented.

9.11  Assembly of Magnetic Beads–Magnetic Beads Chains

427

Figure 9.31  Chains of magnetic beads, aligned with the applied magnetic field, form in less than 1 minute.

the order of 24 hours. Some other solutions have been searched to obtain shorter separation times. The idea was to mimic the action of a gel by using lithography techniques to fabricate a lattice of micropillars. Due to the difficulty in fabricating this type of microcomponent, an interesting solution with magnetic beads has been set up [24, 25]. Magnetic beads (2.8 μm Dynal) initially dispersed in a flat microchannel limited between two plates form vertical columns (vertical chains) when an external magnetic field is applied vertically. The columns are naturally regularly spaced and they perform the same function as gels during DNA electrophoresis. However, the separation duration is much shorter than that of gels (100 mn instead of 24 hours). In this case, magnetic beads assembly is similar to a “calibrated” gel (Figure 9.33).

Figure 9.32  Left: schematic view of DNA fragments of different sizes migrate at different speed in an agarose gel under a constant electric field. Right: a longer strand encounters more difficulties to move in the porosities of the gel than a shorter one.

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Figure 9.33  Left: schematic view of a DNA separator constituted by magnetic beads aligned in vertical columns. (From [25].) Right: a long DNA strand moves from right to left; when meeting an obstacle, it stretches on both sides of the obstacle and the longer side slowly drags out all the DNA strand. DNA motion restarts until the next obstacle. (Courtesy J.L. Viovy, Institut Curie.)

9.12  Magnetic Fluids 9.12.1  Introduction

We have seen that when the concentration of magnetic micro- or nanoparticles in a carrier fluid is sufficient and the magnetic field uniform, magnetic chains will form. If the external field is not uniform, aggregates will form (Figure 9.34). But what if the particles are maintained dispersed by the use of surfactants or electric repulsion? In such a case, one obtains a magnetic fluid—or a ferrofluid— which is a coherent fluid formed by a stable suspension of magnetic nanoparticles (grains of magnetite Fe3O4 or maghemite) in a carrier fluid (Figure 9.35, right). There are two types of ferrofluids depending on their base (carrier fluid): the base may be organic and the particles are dispersed by surfactants, or the base may be polar and the particles are dispersed by electric charges (Figure 9.35, left). Note that no natural magnetic fluid exists. Liquid metals are not magnetic because their liquefaction temperature is above Curie temperature.

Figure 9.34  Image of an aggregate of magnetic microparticles (Dynal 1 μm).

9.12  Magnetic Fluids

429

Figure 9.35  Left: schematic view of an organic-based ferrofluid. Right: view of ferrofluid drops attracted by a magnet.

The advantage of magnetic fluids is that they can be actuated externally by a micromagnet or a microelectromagnet and that they can be inserted as “active plugs” in microsystems (Figure 9.36) to pump or regulate the flow [26]. 9.12.2  Magnetic Force on a Plug of Ferrofluid

Suppose that a ferrofluid plug in a capillary tube is controlled by an external magnetic field (Figure 9.37). The magnetic forces on the plug result in a magnetic pressure difference between the two edges of the plug according to Rosensweig’s formula [27]. If index a stands for the advancing front and r for the receding front, we have DPmag = µ0

Ha

ò

Hr

M dH +

µ0 (Mn2, a - Mn2,r ) 2

(9.48)

where H is the magnetic field along the capillary axis, M the magnetic moment, and Mn is the magnetic moment normal to the plug interface with water. Now if we note that a ferrofluid behaves like a paramagnetic media, in the limit of low magnetization M = cH and after substitution in (9.48), one obtains



D Pmagnetic =

1 µ (µ - µ 0 )(H a2 - Hr2 ) 2 µ0

Figure 9.36  Ferrofluid plug in a capillary tube (diameter 200 μm).

(9.49)

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Figure 9.37  Schematic view of a plug approaching a magnet. The fluid flow is directed from left to right. The magnetic force on the plug is always directed towards the magnet.

Note that an exact derivation may be done for any magnetic field by integrating the full Langevin’s formulation [28] M æ 3χ H ö 1 = coth ç è Ms ÷ø 3χ H Ms Ms



(9.50)

resulting in the expression Ha

D Pmag

é æ æ 3χ H ö ö ù sh ú 2 ê µ0 Ms ê ç çè Ms ÷ø ÷ ú µ M2 ÷ Ln ç = + 0 s 3χ ê ç 3χ H ÷ ú 2 ê ç ú ÷ M s ø û Hr ë è

Ha

2 éæ æ 3 χH öö ù ê coth ç ú è Ms ÷ø ÷ø ú êçè ë û Hr

(9.51)

This result shows that, at low magnetic field, it is the difference of the squares of the magnetic field at the interfaces that determines the magnetic force on the plug. The interface nearest to the magnet is dominant in (9.49) or (9.51) and the force on the ferrofluid plug is always directed towards the magnet. If the force is sufficient, the magnet will block the plug and the flow in the capillary will stop. 9.12.3  Notes

One has to be cautious with the use of magnetic liquids: they are not biocompatible— for example they block PCR—and they often must be separated from the biofluid by a secondary plug, usually made of oil, for biocompatibility.

9.13  Magnetic Micromembranes In this chapter, we are dealing with magnetic micro- and nanoparticles, and it seems interesting to present a very useful application of magnetic microparticles in biotechnology. In this section, we show how magnetic nanoparticles can be assembled inside a PDMS matrix to form a magnetic micromembrane. The advantage of such membranes is that they can be actuated externally (from outside the microdevice) by a time-varying magnetic field.

9.13  Magnetic Micromembranes

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9.13.1  Principle

It has been found that flexible, elastomer micromembranes could be used in microsystems to separate a biofluid from another fluid or gas. Increasing the pressure (by acoustic waves for example) or the temperature of the auxiliary fluid results in the pressurization or agitation of the biofluid [29]. Another actuation of elastomer micromembranes can be achieved if the membrane has magnetic elements attached to or embedded in it. Magnetic actuation has been obtained first by fixing permalloy microplates to an elastomer micromembrane [30]; more recently, magnetic microparticles have been embedded in a PDMS (poly-di-methyl-siloxane) matrix to obtain biocompatible, smooth-surfaced, and totally deformable magnetic micromembranes [31]. The principle is to mix paramagnetic (carbonyl iron) or ferromagnetic (ferrite) nanoparticles with liquid PDMS in a mass ratio of 25% to 50%. This matrix is spread by spin-coating and left to polymerize. Very uniform membranes can be obtained (Figure 9.38). An interesting property is that such membranes are nearly as deformable and flexible as pure PDMS membranes. Deflection of membranes is governed by two numbers: Young modulus (usually noted E and expressed in Pascals) and Poisson coefficient (usually noted n and without unity). Young modulus and Poisson coefficient for 100-μm thick PDMS only membrane are of the order of 8.0 105 Pa and 0.5, whereas that of PDMS membrane containing carbonyl iron in a mass ratio of 25% are of the order of 9.5 105 Pa and 0.55. 9.13.2  Deflection of Paramagnetic Micromembranes

Suppose now a circular paramagnetic micromembrane with a radius of 500- and 50-μm thick; this membrane is clamped to a solid wall by its circular edge (Figure 9.39). An estimate of the deflection of such membranes can be done by using the linear theory: if the maximum deflection is less than about 0.2 times the membrane thickness [32]



wmaw £

h 5

(9.52)

Figure 9.38  Two types of paramagnetic membranes. Left: 4 μm nanograins of carbonyl iron embedded in PDMS. Right:10-nm nanograins extracted from ferrofluid embedded in PDMS.

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Figure 9.39  Magnetic micromembrane clamped in a PDMS matrix.(right) and deformed under the action of a magnet.

then the maximum deflection is given by the relation P a4 64 D

(9.53)

E.h3 12(1 - ν 2 )

(9.54)

wmax =

where

D=



a is the membrane radius and P is the uniform applied pressure on the membrane. Using E = 9.105 Pa and n = 0.5, D is obtained through (9.54): D=1/8 10–7 and we find wmax=10 μm for an applied magnetic pressure of 128 Pa. This deflection is just at the upper limit of the linear regime given by (9.52). It is immediately seen that the limit of the linear regime is given by P£



13Dh a4

(9.55)

To this stage, the remaining question is how to calculate the applied magnetic pressure. An approximate solution is to use the formula previously derived for a magnetic microparticle alone (9.8) and to make the summation over all magnetic particles located inside the membrane. By doing this, we neglect the field interaction between the magnetic particles. However, this approximation is expected to be reasonably good because first, the particles are very small—the interaction forces are weak and very local—and second, the applied external field is perpendicular to the membrane whereas the interaction forces between the particles are contained in the plane of the membrane. Because the size of the membrane is generally small compared to the scale of the applied external magnetic field, we can assume a uniform magnetic gradient and we obtain � Fmag » µ0

å

grains

� � � � v p (M.Ñ)H = µ 0 N v p (M.Ñ) H

(9.56)

where vp is the volume of each magnetic microparticle and N is the number of particles embedded in the membrane. Let f be the ratio between the volume of the

9.13  Magnetic Micromembranes

433

magnetic material and the total volume of the membrane, then f V = N vp and the magnetic force on the membrane is:

� � � Fmag » µ0 f V (M.Ñ) H

(9.57)

where V is the total volume of the membrane. The magnetic pressure is then given by the expression



� � � µ Fmag Pm = » µ0 f h (M.Ñ) H = 0 f h χ pÑH 2 S 2

(9.58)

Using (9.53), we find that the membrane deflection is then linked to the gradient of the square of the magnetic field by



wmax

µ0 f χ p (1 - ν 2 ) a 4 2 » ÑH 2 2 64 Eh

(9.59)

The membrane maximum deflection is proportional to the content in magnetic particles f, to the magnetic susceptibility of the particles cp, to the foruth power of the radius a4 and to the gradient of the square of the magnetic field ÑH2. It is inversely proportional to the Young modulus E and to the square of the membrane thickness h2. Because micromagnets or microelectromagnets do not deliver an important magnetic field, and consequently an important gradient, the efficiency of a magnetic micromembrane depends on the use of very magnetizable particles that can be packed in the PDMS matrix without increasing too much the Young modulus and Poisson coefficient. To this extent it is likely that micromembranes containing nanoparticles will be more efficient than that containing metal microplates. 9.13.3  Oscillation of Magnetic Membranes

An application of magnetic micromembranes in biotechnology is the mixing of fluids. We analyze here briefly the principle of actuation of micromembranes by an oscillating (or pulsating) magnetic field. 9.13.3.1  Paramagnetic Membranes

In the previous section, we saw that the deflection of a paramagnetic membrane is proportional to the gradient of the square of the external magnetic field ÑH2. Suppose now that the source of the external magnetic field is�an electromagnet and that � this magnetic field is periodically reversed. Changing H into - H leaves the term ÑH2 unchanged. The magnetic force on a paramagnetic membrane is always attractive, and the deflection of the membrane is always directed towards the source of the external magnetic field. Another way of looking at this phenomenon is to consider Langevin’s law �for paramagnetic media (Figure 9.40). When the field is reversed to its opposite (H is

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Magnetic Particles in Biotechnology

Figure 9.40  Langevin’s law for paramagnetic material. Magnetization is aligned with the magnetic field.

� � changed in - H), the�magnetization is changed to its opposite (M is changed into � � - M), and the term (M.Ñ) H in the expression of the force remains identical. 9.13.3.2  Ferromagnetic Membranes

Ferromagnetic membranes show a different behavior when placed in an oscillating � � magnetic field. When the external field vanishes, during a change from H to - H, there is still a remanent magnetization as shown in Figure 9.41. An oscillating magnetic field is obtained by using a miniaturized electromagnet and the delivered magnetic field is often weak, so that the magnetization of the fer� romagnetic particles remains close to the remanent magnetization Mr. In this case, the expression of the force is approximately

Figure 9.41  Magnetic moment of a ferromagnetic membrane, measured in the plane of the membrane and along the axis perpendicular to the membrane.

9.13  Magnetic Micromembranes

435

� � � µ Fmag » 0 f V Ñ (Mr .H) 2



(9.60)

When the magnetic field is reversed, the force on the membrane is also reversed and the membrane oscillates with the external field. 9.13.3.3  Example

In order to obtain a larger deflection, ferromagnetic micromembranes have been fixed perpendicularly to a solid wall and oscillate like a “fish tail” under the actuation of a pulsating magnetic field (Figure 9.42). It has been shown that such motion greatly enhances the mixing in the microchamber [31].

Figure 9.42  (left) A view of the microchamber with the four micromembranes, the inlet capillary can be seen at the top of the picture; (right) view of the membranes during the oscillations (the electromagnet is the round disk behind the membranes).

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9.14  Conclusion In the miniaturization trend of biotechnological microsystems, microflows are often not selective or specific enough to correctly perform their expected role of carrier. Functional magnetic microbeads are then used as additional carriers, bringing the required selectivity. This is why magnetic beads have been found to be a very powerful complementary tool for manipulating biological objects and they appear now to be one of the most useful tools in biotechnology. Magnetic beads can bind to many different target molecules by the principle of functionalization and they can unbind by elution, so they can be used as temporary carriers. The applied force on magnetic beads can be precisely controlled by the action of an external magnetic field, leading to a simplified design of the microsystem. Moreover, with the combination of magnetism and fluorescence, beads are easily tracked and detected under the microscope. Magnetic beads can be assembled to form new biotechnological tools such as magnetic chains, magnetic fluids, and magnetic membranes. Magnetic chains can be arranged to form sieving matrices to separate long DNA strands, magnetic fluids are used primarily to perform micropumping, and magnetic membranes are well suited to enhance mixing in microsystems.

References   [1] Xie, X., et al., “Preparation and Application of Surface-Coated Superparamagnetic Nanobeads in the Isolation of Genomic DNA,” Journal of Magnetism and Magnetic Materials, Vol. 277, No. 1-2, June 2004, pp. 16–23.   [2] Gu, H., et al., “Facile One-Pot Synthesis of Bifunctional Heterodimers of Nanoparticles: A Conjugate of Quantum Dot and Magnetic Nanoparticles,” JACS, Vol. 126, 2004, p. 5664.   [3] Portis, A. M., Electromagnetic Fields: Sources and Media, New York: John Wiley & Sons, 1968.   [4] Chantrell, R.W., J. Popplewell, and S.W. Charles, “Measurements of particle size distribution parameters in ferrofluids,” IEEE Trans. on Magnetics, Vol. MAG-14, No. 5, September 1978, pp. 975–977.   [5] Oberteuffer, J.A., “Magnetic Separation: A Review of Principles, Devices, and Applications,” IEEE Trans. on Magnetics, Vol. MAG-10, No. 2, June 1974, pp. 223–238.   [6] Aharoni, A., “Traction Force on Paramagnetic Particles in Magnetic Separators,” IEEE Trans. on Magnetics, Vol. MAG-12, No.3, May 1976, pp. 234–235.   [7] Lawson, W. F., W. H. Simmons, and R. P. Treat, “The Dynamics of a Particle Attracted by a Magnetized Wire,” Journal of Applied Physics, Vol. 48, No. 8, August 1977, pp. 3213–3224.   [8] Kelland, D.R., “Magnetic Separation of Nanoparticles,” IEEE Trans. on Magnetics, Vol. 34, No. 4, July 1998, pp. 2123–2125.   [9] Zimmels, Y., “Effect of Concentration and Characteristic Distributions on Electromagnetic Separation of Polydisperse Mixtures,” IEEE Trans. on Magnetics, Vol. MAG-20, No. 4, July 1984, pp. 597–607. [10] Schlick, T., Molecular Modeling and Simulation, New York: Springer, 2000. [11] Murariu, V., et al., “Concentration Influences on Recovery in a High Gradient Magnetic Separation Axial Filter,” IEEE Trans. on Magnetics, Vol. 34, No. 3, May 1998, pp. 695–699.

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[12] Gerber, R., “Theory of Particle Capture in Axial Filters for High Gradient Magnetic Separation,” J. Phys. D: Appl. Phys., Vol. 11, 1978, pp. 2119–2129. [13] Pham,P., P. Massé, and J. Berthier, “Numerical Modeling of Superparamagnetic SubMicronic Particles Trajectories Under the Coupled Action of 3D Force Fields,” European Physical Journal, Vol. 12, 2000, pp. 211–216. [14] Zebel,G., “Deposition of Aerosol Flowing Past a Cylindrical Fiber in a Uniform Electric Field,” Journal of Colloid Science, Vol. 20, 1965, pp. 522–543. [15] Berthier J., et al., “Magnetic Confinement of Paramagnetic Micro and Nano-Particles Away from Solid Walls,” IEEE Trans. on Magnetics, Vol. 38, Issue 2, March 2002, pp. 913–916. [16] Berthier, J., et al., “Numerical Modeling of Paramagnetic Microparticles Trajectories in a Densely Packed Ferromagnetic Wire Bundle,” CFEC Conference, Milwaukee, WI, 2000. [17] Davies, L. P., and R. Gerber, “2D Simulation of Ultra-Fine Particle Capture by a SingleWire Magnetic Collector,” IEEE Trans. on Magnetics, Vol. 26, No. 5, September 1990, pp. 1867–1869. [18] R. Beckett, et al., “Measurement of Mass and Thickness of Adsorbed Films on Colloidal Particles by Sedimentation Field-Flow Fractionation,” Langmuir, Vol. 7, No. 10, 1991, pp. 2040–2047. [19] Rheinlaender, T., et al., “Different Methods for the Fractionation of Magnetic Fluids,” Colloid Polym. Sci., Vol. 278, 2000, pp. 259–263. [20] Rheinlaender, T., et al., “Comparison of Size-Selective Techniques for the Fractionation of Magnetic Fluids,” Journal of Magnetism and Magnetic Materials, Vol. 214, 2000, pp. 269–279. [21] J. Berthier, P. Pham, and P. Massé., “Numerical Modeling of Magnetic Field Flow Fractionation in Microchannels: A Two-Fold Approach Using Particle Trajectories and Concentration,” MSM 2001 Conference, Hilton Head Island, SC, March 19–21, 2001. [22] Alward, J., and W. Imaino, “Magnetic Forces on Monocomponent Toner,” IEEE Trans. on Magnetics, Vol. MAG-12, No. 2, 1986, pp. 128–133. [23] Eisenstein, I., “Magnetic separators: Traction Force Between Ferromagnetic and Paramagnetic Spheres,” IEEE Trans. on Magnetics, Vol. MAG-13, No. 5, September 1976, pp. 1646–1648. [24] Furst, E. M., et al., “Permanently Linked Monodispersed Paramagnetic Chains,” Langmuir, Vol. 14, No. 26, 1998, pp. 7334–7336. [25] Doyle, P. S., et al., “Self-Assembled Magnetic Matrices for DNA Separation Chip,” Science, Vol. 295, March 2002, pp. 2237. [26] Perez-Castillejos, R., et al., ‘The Use of Ferrofluids in Micromechanics,” Sensors and Actuators A (Physical), Vol. A84, No. 1-2; 1 August 2000, pp. 176–180. [27] Rosensweig, R. E., Ferrohydrodynamics, New York: Cambridge University Press, 1985. [28] Berthier, J., and F. Ricoul, “Numerical Modeling of Ferrofluid Flow Instabilities in a Capillary Tube at the Vicinity of a Magnet,” Technical Proceedings of the 2002 International Conference on Modeling and Simulation of Microsystems, Vol. 1, 2002, p. 764. [29] Cooney, C.G., and B.C. Towe, “A Thermopneumatic Dispensing Micropump,” Sensors and Actuators A, Vol. 116, 2004, pp. 519-524. [30] Khoo, M., and C. Liu, “Micro Magnetic Silicone Elastomer Membrane Actuator,” Sensors and Actuators A: Physical, Vol. 89, Issue 3, 15 April 2001, pp. 259–266. [31] Berthier, J., and F. Ricoul, “Development of Magnetic Micromembranes for the Actuation of Fluids in Biological and Chemical Microsystems,” Applied Nanoscience, Vol. 1, 2004. [32] Timoshenko, S., and S. Woinowski-Krieger, Theory of Plates and Shells, 2nd ed., New York: McGraw-Hill, 1959.

Chap te r 10

Micromanipulations and Separations Using Electric Fields

Because of their versatility, electric field based techniques are the most widely used for bioanalyses not only for the separation of biological objects through the wellknown electrophoresis-derived techniques, but also for the handling and micromanipulation of particles, cells or even biomolecules through dielectrophoresis. We will review in this chapter the principles of these techniques, their applications and their limitations. With the transfer of microfabrication techniques to this field and the advent of the lab-on-chip approach, particularly well suited to many of these experiments, new original tracks are now opened for potential developments. Although we certainly do not pretend to make an exhaustive list of these new routes, we will try to give a few examples illustrative of this approach.

10.1  Action of a DC Electric Field on a Particle: Electrophoresis Quite generically, objects dispersed in water bear a surface charge. The origin of this charge is diverse: it may come from the natural ionization of some of these colloids (e.g., proteins) that gives them a net charge depending on the pH of the solution, but other phenomena such as the adsorption of charged ions or the intrinsic ionic nature of some of these particles may come into play. However, it is important to remember that a charged particle should never be considered “naked” in the solvent. It is always surrounded by counterions so that its global charge is strictly zero (electroneutrality). Because of the thermal agitation, this layer of counterions is diffuse. Electrophoresis consists in applying to these charged objects a direct external electric field that sets them into motion. Their mobility is then indicative of some of their characteristics (charge, molecular weight, and so forth). This approach was pioneered by Tiselius [1] who got the Nobel prize in 1948 for his work on protein electrophoresis. In this part, we will first describe the structure of the counterions layer and how an electric field can affect them through the description of electro-osmosis. Then, by increasing complexity we will describe the electrophoresis of hard spheres, DNA and finally proteins or other biological objects. 10.1.1  The Debye Layer

Let us first calculate the distribution of the counterions around a charged colloidal particle [2] (Figure 10.1). 439

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Micromanipulations and Separations Using Electric Fields

Figure 10.1  Distribution of charges around a spherical particle of radius a. The gray halo represents the cloud of counterions.

This classical calculation starts by writing the Poisson equation that relates the potential y to charge density r:



Ñ2ψ = -

ρ ε 0ε r

(10.1)

where e0 is the vacuum permittivity (» 8.85·10–12 C2J–1m–1) and er the relative permitivity (» 80 for water). At equilibrium, r is given by summing the number of charges in the solution. The number ni of ions of each species is given by a Boltzmann distribution:

ni = ni 0 exp(- zi eψ / kBT )

(10.2)

zi is the valency of the ions, e is the elementary charge (1 eV = 1.6·10–19 C), kB the Boltzmann constant (~ 1.38·10–23 J.K–1) and T the temperature. ni0 is the number of these ions far from the surface. The charge density is then given by:



ρ = å ni zi e = n0 (- ze × exp(- ze ψ / kBT ) + ze × exp(ze ψ / kBT )) i

(10.3)

in the case of symmetric electrolytes. The Gouy-Chapman model combines (10.1) and (10.3), the boundary conditions being given by the global electro-neutrality condition and by the surface charge. In a plane geometry, this equation can be solved analytically but the case ey<
ψ = ψ 0 × exp(-κ × x)

(10.4)

10.1  Action of a DC Electric Field on a Particle: Electrophoresis

441

where:



κ2 = 2

z 2e 2n0 ε 0ε r kBT

(10.5)

k –1 is an important length that describes the range of these electrostatic interactions: it is known as the Debye length. y0 is the surface potential. x is the distance from the surface. Still in the framework of the Debye-Hückel approximation, the potential in the case of spherical particles of radius a is [2]:



ψ = ψ 0a

exp(-κ (x - a)) x

(10.6)

Here, x is the distance from the center of the sphere. It is good at this point to get an order of magnitude of the Debye length, which obviously depends strongly on the salinity: for a monovalent ion (z = 1) at a 0.1M concentration, k –1 ~ 1 nm; if c ~ 10–7M (which is the minimum that can be reached in ultrapure water where the salinity is determined by the H+ and OH– ions), k –1 ~ 1 mm. At a distance from the surface larger than k –1, the particles are neutral. There are more elaborate models that describe this double layer more realistically. In particular, the Stern model considers a more complex profile of the potential: a proximal zone where the counterions are immobilized and a diffuse layer where the above discussion is valid again (Figure 10.2). It is then common to introduce the z potential that gives the value of the potential at the point where the shear around the particle becomes significant. Practically, z is the only parameter that can be experimentally determined either by electrophoresis for colloids or by other means: electro-osmosis for walls,

Figure 10.2  Profile of the double layer in the presence of a proximal immobile layer (Stern layer) for a spherical particle. The potential takes the analytical form of (10.4) in the diffuse layer.

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Micromanipulations and Separations Using Electric Fields

sedimentation potential, or streaming potential (the electric potential developed by the fall of the objects or by flowing water on them) for heavy or tormented particles [3]. Note at this point that the simple Debye-Hückel description assumes that the z potential is located strictly on the surface of the particle (z = y 0). For a spherical particle (10.6), the total charge q is then given by: ¥



q = ò 4π r 2ρ dr = 4πε0 εr a(1 + κ a)ζ a

(10.7)

Equation (10.7) gives a relationship between the surface charge density s and z: σ = ε 0ε rζ (1 + κ a) / a



(10.8)

What are then the consequences of an electric field E on a charged particle? Although it is tempting to express the resulting force acting on the particle in the form F=qE, this is generally too simplistic an approach. Given the complex distribution of charges described above, this is not so surprising. Indeed, the electric field acts primarily on the charges of the double layer. For simplicity, we will first describe the consequences of an electric field on an immobile surface (electro-osmosis) before treating electrophoresis per se. 10.1.2  Electro-Osmosis

Let us consider a charged surface in an aqueous solution. The charge of the surface is balanced with the Debye layer of counterions extending over a distance k –1 (10.4) and (10.5). An electric field E parallel to this surface, acts on this excess of charges that move the fluid with them at a velocity v(z). We can then write the Navier–Stokes equation (see Chapter 1):



ρE + η

¶2v =0 ¶z 2

(10.9)

r and h are respectively the charge density and the viscosity of the fluid. The other equation is the Poisson equation (10.1) that gives r as a function of the electric potential y.

ρ = -ε l Ñ2ψ el = e0 . er is the fluid permittivity. substituting into (10.9) gives



Eε l

¶2ψ ¶2v = η ¶z 2 ¶z 2

(10.10)

On the surface, the usual boundary conditions apply: the velocity on the surface is 0 (no slippage) and the electric potential velocity is y = z. Far from the surface, ¶ψ ¶v = = 0. ¶z ¶z

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443

Integrating twice yields:



v(z) =

ε1E (ψ (z) - ζ ) η

(10.11)

On a plane surface, y decreases exponentially within the double layer and vanishes far from it. We can thus derive the electro-osmotic velocity Veo by writing (10.11) far from the surface and we get:

Veo = -ε1E ζ / η

(10.12)

Depending on the application, these electro-osmotic flows are either a nuisance by interfering with the electrophoretic motilities or a powerful means to set liquids into motion by a controlled plug-flow (Figure 10.3). The above calculation is also the basis of electrophoresis. We have considered the case of a fixed charged surface and have calculated the flows induced by an electric field. Electrophoresis is the exact opposite situation: the solid moves under the application of a dc electric field in an immobile fluid. 10.1.3  Electrophoresis of a Charged Particle

In electrophoresis as well, the electric field acts first on the counterions that are then driven in the direction opposite to the particle increasing therefore its hydro­ dynamic friction. Dealing with small particles, the dissipation in the fluid is principally viscous.

Figure 10.3  Velocity profile of an electro-osmotic plug-flow in a negatively charged channel such as fused silica. The thickness of the Debye layer (dotted lines) has been exaggerated.

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Micromanipulations and Separations Using Electric Fields

For weak electric fields, the velocity V of the particle can then be considered as linearly dependent on the electric field.

V = µ × E

(10.13)

where m is defined as the electrophoretic mobility. Depending on the salinity and the radius of the particle, two extreme cases can be analytically considered. If the charge density is high or for large particles, we have k a>>1 and we can use the “Smoluchowski” calculation on plane surfaces similar to the one derived for electro-osmosis (see Section 10.1.2) [4, 5]. In this case, m can be derived from the equilibrium of forces in the double layer, assuming that the charge profile is only marginally modified by the electric field and we get:



µ=

εζ η

(10.14)

In the framework of the Debye-Hückel approximation, plugging (10.7) in (10.14) gives:

µ=

q 4πηκ a2

(10.15)

The net result of (10.14) is that the mobility depends only on the z potential of the particle but not on its size nor on its shape. This result is a direct consequence of the confinement of the flow within the Debye layer. The physical mechanism underlying this motion is thus very different from the usual Stokes law where the bead directly submitted to an external force such as gravity experiences a viscous drag. We have so far focused on large particles or small Debye length, we can go to the other extreme case of a very small particle or a very small salinity (large Debye length). We then have k a<<1 and the particle can be seen as a pointlike object surrounded by a very diffuse cloud of counterions that extends many radii away from it. In that case, the flow lines extend far from the object over a characteristic length now close to the bead radius. Not surprisingly given the above discussion, this case is then described by the more familiar Hückel calculation and the mobility is accurately given by:

µ = q / 6πηa = 2εζ / 3η

(10.16)

by using (10.7). Between these limiting results, the behavior of the particles is more complex and best described by numerical simulations [5]. Furthermore, assumptions of the previous calculations are often caught out. For instance, the Smoluchowski calculation assumes that the charge distribution around the particle is only marginally disturbed by its motion. However, this is generally not true, the cloud of counterions is deformed by the electric field, this shape in turn modifies the friction. The same becomes true if the velocity of the particle becomes very high [5].

10.1  Action of a DC Electric Field on a Particle: Electrophoresis

445

In other words, out of the framework of the simplistic approximations presented above, electrophoresis becomes rapidly a very complicated phenomenon and it should be kept in mind that the practical interpretation of these experiments is often largely empirical. One last word in this rapid theoretical introduction, we have considered a particle in an infinite medium at rest. However, the suspension is contained in a physical cell. In the general case, the walls of this cell are not passive but the charge they bear responds to the external electric field by developing electro-osmotic flows (see Section 10.1.2). These induced flows superimpose a plug flow to the electrophoretic motion of the particles. They can dramatically affect the performances of the devices and should be well controlled. Double-stranded DNA is the molecule most commonly separated by electrophoresis. We will first focus on this case. However, there are many objects on which electrophoretic separation is used: proteins or single-stranded DNA are molecular examples. Cells or organelles are other more macroscopic examples. 10.1.4  Electrophoresis of DNA

DNA is a polyelectrolyte (a polymer chain whose monomers bear a charge). On top of the complexity of charged objects we have just discussed, we must add the one inherent to polymers. We will shortly present it and infer from these results the main characteristics of DNA electrophoresis. 10.1.4.1  Polymer Chains in Solution

We have seen in Chapter 8 that, in dilute solutions, DNA chains adopt a coil configuration whose radius, called the radius of gyration Rg, is directly related to the size of the monomers b and their number N through the relation:

Rg = b × N 1 / 2

(10.17)

To understand the behavior of a polyelectrolyte chain in an electric field, we can model it by a succession of charged beads of radius a connected together by springs (Rouse model) [6]. The Stokes friction coefficient of each bead with the solvent is 6pha. Without getting into the full calculation, it turns out that the coil is transparent to the hydrodynamic flow and the friction of the chain in the liquid comes exclusively from the individual frictions of the beads [7]. The effective friction over the whole length of the chain is then proportional to the number of beads and thus to its length. As the electric force applied to the chain is also proportional to its length, the net result is that the mobility of a long polyelectrolyte is independent of its length [8]. Now, we know that a polymer chain is not a succession of independent beads. However, more refined models and in particular the ones derived from the Zimm description where the hydrodynamic interactions between beads are taken into account, show that the added terms induce only small deviations to this law [8]. Practically, we can conclude that no size separation of long DNA will occur by simply applying an electric field to a solution of these molecules.

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Micromanipulations and Separations Using Electric Fields

10.1.4.2  Gel Electrophoresis

To achieve a separation, it is then necessary to use a sieving matrix that slows down the largest molecules. Most commonly this is a gel made from polyacrylamide or agarose whose mesh size (from a few tens of nanometers to a few hundred nanometers) conditions the separation. Practically, the solution containing the mixture to be analyzed is deposited at one extremity of a gel slab. The whole slab is immersed in the buffer and an electric field (up to ~ 100 V/cm) is applied via electrodes immersed in it. In these experiments, the electrodes are physically separated from the sample (although in electrical contact with it via the buffer solution) thus the electrochemistry at their surface (electrolysis in particular) has no consequence. After applying the electric field for a certain time, the molecules are stained and appear as bands (Figure 10.4). The migration of small molecules within the gel can be modeled by computing the free volume available to the particle [9–11]. In this approximation, the size Rg of the chain is smaller than the distance between the crosslinks of the gel and the DNA chain remains basically undeformed during the electrophoresis (Ogston regime). This volume depends on the gel characteristics as well as on the particle effective volume and one gets ultimately the following expression for the mobility:

µ µ 1/ log N

(10.18)

This equation is indeed well verified, and routinely used, for relatively short molecules. However, longer molecules deform in the gel and take the average orientation of the electric field. They can be modeled by reptating chains within an array of obstacles and their mobility becomes independent of the molecular weight again (Figure 10.5) [11–13].

Figure 10.4  Electrophoregram of a “DNA ladder” showing the logarithmic dependence of the mobility on the molecular weight (eq. 10.18).

10.1  Action of a DC Electric Field on a Particle: Electrophoresis

447

Figure 10.5  Electrophoretic migration of a DNA chain in a gel for three different DNA sizes: (a) Ogston regime; (b) and (c) reptating chains in the gel. (from [13])

For the molecules too large to be separated by direct gel electrophoresis, pulsed field electrophoresis in gel is the preferred technique [14]. In these experiments, the orientation of the electric field is periodically switched between at least two directions. The angle between these two directions can be finite (crossed field electrophoresis) or take the value of 180° so that the electric field changes its direction along the same orientation (field inversion electrophoresis). This approach is based on transitory states where the long molecules do not have time to fully extend in the direction of the electric field before its change of direction whereas small molecules do. Therefore these small molecules can reach a steady state in which they are elongated but switch their direction with the field. Long molecules that spend

Figure 10.6  Single molecule DNA image at a 120° change of orientation of the electric field (arrow) between (A) and (B). To allow for this single molecule image, DNA is fluorescently labeled. (from [16]).

448

Micromanipulations and Separations Using Electric Fields

most of their time to orient themselves along the field are considerably slowed down (Figure 10.6). This technique has literally been a breakthrough in the separation of long molecules (up to 10 Mbp) [13–15]. Extremely efficient, it remains also very time-consuming. 10.1.4.3  Capillary Electrophoresis and Electrophoresis on Chip

Capillary electrophoresis is a gel-free technique. It is becoming predominant over gel electrophoresis for challenging separations. In this case, the gel slab is replaced by a capillary whose typical dimensions are 100 μm in diameter and 50 cm in length. Since the capillary geometry is favorable to a good thermal dissipation, high electric fields can be used (up to 1 kV/cm). This strategy is much easier to automate and parallelize than the gels and has been massively used in “brute force” sequencing of various genomes including the human genome. The sieving matrix in this case is a polymer solution that acts similarly to a gel. The main difficulty in using this approach is the electro-osmotic flow (EOF) (Section 10.1.2) that superimposes to the electrophoretic motion and that is quite detrimental to the resolution. This effect is very dependent on the details of the chemistry of the surfaces and it is necessary to “hide” them to the electric field, for example by chemically grafting or by adsorbing neutral polymers on it. A strong tendency nowadays is to integrate these capillaries within “chips” where all the dimensions can be reduced and that have on the same chip, the injection, separation, and detection steps [17, 18]. With smaller devices, the analysis volumes are reduced and the times are much shorter. Furthermore, heat dissipation is even better than for standard capillaries enabling higher fields to be used. As it has been mentioned in the preceding chapters, the fabrication of these channels uses a technology stemming from the microelectronics industry. Many different materials can be used for the fabrication of the chips (plastics or plastic-silicon hybrids, for example) although it should be stressed that silicon, the preferred material in microfabrication, is generally too conductive to be compatible with the high electric fields required for this particular application. When it arrives at full maturity (which can be forecast within the next few years) this approach can be expected to lead to disposable, inexpensive devices. However, there are still hurdles to overcome: the use of new materials changes the electro-osmosis characteristics as well as the DNA-surface interactions. Furthermore, there seems to be a limit to the electric fields that can be used as too high values can have negative consequences such as the aggregation of long DNA chains by electrokinetic effects [19]. 10.1.4.4  New Exploratory Fields

Besides reproducing electrophoresis on a chip, micro- and nanofabrication has enabled new ideas and new concepts aiming at the separation of DNA molecules and more and more of proteins. Artificial Gel

As early as 1992, there have been seminal experiments where the gel commonly used as a separation sieving matrix has been replaced by a solid state microfabricated

10.1  Action of a DC Electric Field on a Particle: Electrophoresis

449

array of pillars [20]. The photolithography-derived techniques enable the fabrication of micron-size posts separated by a few microns, dimensions close to the gel’s pore size used for long DNA molecules. Combined with the ability to visualize single DNA molecules, these experiments have pioneered a whole school of experiments where the behavior of the chains could be unambiguously correlated to their extension, hooking and releasing from the posts, an invaluable help in the understanding of the phenomena. Indeed the first experiments of migration of DNA subjected to a low electric field in such arrays of posts have shown a lot of common features with their behavior in gels. Although this strategy uses well-known techniques derived from the microelectronics industry for the fabrication of the substrates, it has not yet come to the point where it is used for routine analysis. Similar to gels where the range of accessible molecular weights is dictated by the gel’s pore size and thus its degree of cross-linking, the efficiency of these devices is conditioned by the size and spacing of the posts, parameters that can be tuned during the fabrication step but not as readily as for a gel. However, for research purposes, this approach has been very successful and will be with no doubt a tool that can and will be used in particular situations. Indeed, after these proofs-of-concept experiments, pulsed-field electrophoresis has been performed in 2-D hexagonal arrays of pillars with extremely good results. Because of the high regularity of the array, DNA chains move more regularly than in a gel and sequences of electric field turned by 120° enable extremely fast separation of long molecules (up to a few 100 kbp in a few 10s which is 100 to 1,000 faster than pulsed field gel electrophoresis) [21]. Again, it seems relatively straightforward to integrate such structures in on-chip separation devices. More recently, arrays of columns made of magnetic beads that align in a regular hexagonal array upon the application of a magnetic field, have been used with good results. This strategy has some obvious advantages (no microfabrication and easy replacement of the medium), it does not allow however a range of variation in the spacing between posts as large as would be desirable [22]. Nanopores and Nanochannels

Not only DNA or other biopolymer chains can be forced through a micro-channel or a micro-aperture, it can be threaded through a nanopore. In the present version

Figure 10.7  Pulsed field electrophoresis in an hexagonal array of posts (diameter and spacing around 1 μm). Picture taken after a 120° change in orientation of the electric field. from [21].

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Micromanipulations and Separations Using Electric Fields

Figure 10.8  Schematic of a DNA strand translocating the a-hemolysin protein embedded in a lipid bilayer. The drawing is not to scale. The structure of this protein is represented Figure 8.5.

of these experiments, single strand DNA is driven by an electric field through a single biological pore, the a-hemolysin channel, that has been previously isolated and inserted in a lipid membrane (Figure 10.8) [23]. The structure of this proteic complex enables the translocation of a single single-strand DNA at a time. The electric field that drives the chain in the pore is also used for its detection: When the DNA is inside the pore, it effectively blocks the ionic electrical current increasing the electrical resistance. By analyzing the details of this time-resolved resistance, one can get the length of the molecule (by the duration of the pulse) and some information on its sequence (by the value of the resistance). It is hoped that, at some point, this technique will be used for ultra-fast DNA sequencing [24]. To achieve this goal, it is necessary however to develop a solid-state alternative to the a-hemolysin pore. Intense work is presently devoted to drill nanometer sized holes in thin inorganic membranes with sophisticated nanotechnology [25]. The next generation of these devices are expected to be more than simple pinholes. They should carry also in the same plane some kind of very local detection setup. This can be performed with the chemical grafting of molecules responding to the passing of some of the nucleotides (for instance by fluorescence energy transfer) or, maybe more realistically, in the form of transverse electrodes embedded in the membrane that will allow to achieve a single base pair resolution through the measurement of a tunneling current. Still in the nanoworld, nanochannels are the other field of application of this “nanoelectrophoresis”. In this case, double strand DNA is forced into nanochannels or nanoconstrictions with the electric field and observed by fluorescence microscopy. There are nowadays several techniques that can be used for this nanofabrication. The chains can considerably stretch in these structures (up to 60% of their fully extended length) and it is observed that their extended length is proportional to the number of base pairs. By a simple averaging of this projected length, an extremely accurate determination of the molecular weight can be obtained [26]. Another application of this confinement is the precise localization of proteins bound to the chain. If the protein and the DNA are labeled with fluorescent dyes of different

10.1  Action of a DC Electric Field on a Particle: Electrophoresis

451

Figure 10.9  LacI repressors bound to a DNA molecule stretched in a 200 nm wide nanochannel. Both the LacI and the DNA are fluorescently labeled with different shades of gray. There are ~ 20 LacI molecules in this particular case. From [27].

colors, a fluorescence image can reveal very rapidly and accurately the position of this protein along the chain (Figure 10.9) [27]. In the same spirit, structures alternating micro- and nanoconstrictions respectively 1.5-μm and 90-nm wide have been developed that induce different confinement regimes for the DNA molecules [28]. Upon the application of the electric field, molecules experience many transitions from thick regions to regions smaller than the radius of gyration. Although it was predicted that long molecules would be retarded by the constrictions by an entropic barrier, practically, their mobility turns out to be higher. A qualitative interpretation involves a large deformation of the coil. Strong interactions with the surface in these ultraconfined geometries are probably also to be taken into account [29]. 10.1.5  Electrophoresis of Proteins

After the genomics era and now that more and more genomes have been sequenced, proteomics that aims at identify the proteins expressed in living organisms is the next big challenge [30]. From a chemical physics point of view, proteins are not as well defined as DNA as their characteristics of charge and hydrophilicity varies a

452

Micromanipulations and Separations Using Electric Fields

lot. Besides, the separation of these biomolecules takes a different meaning than for DNA where the problem is to discriminate between identical objects differing only by their molecular weights. For proteins, completely different molecules differing by their primary sequence and therefore their length, charge, hydrophilicity, and so forth, have to be analyzed. The determination of the size is performed in Polyacrylamide gels in the presence of a surfactant, pH gradients are used to measure their charge. Two-dimensional gels combine these two techniques. 10.1.5.1  SDS PAGE

This technique is used for separating proteins according to their size. As they are globular charged objects taking various shapes, it is first necessary to unfold them. This step is performed in a denaturating solution of sodium dodecyl sulfate (SDS). This charged surfactant efficiently unfolds proteins so they can be considered as classical flexible polyelectrolytes whose charge is imposed by the bounded SDS. Since, on average, all the proteins have the ability to fix a similar amount of SDS molecules (per unit length), all the proteins of the mixture can be approximated to the same polyelectrolyte differing only in length even though their primary sequence is different. Polyacrylamide gel electrophoresis (PAGE) of these SDS-protein complexes is then conceptually similar to gel electrophoresis of DNA and is heavily used particularly in the Ogston regime m ~ 1/log(M) (10.18). 10.1.5.2  Isoelectric Focusing (IEF)

Because of the acidic and amine groups present in their structure proteins bear a net positive charge at low pH and negative at high pH. The pH value at which their charge is zero is called the isoelectric point. It is measured by creating a pH gradient in a gel and by measuring the point where the net charge of the proteins is strictly zero. When an electric field is applied to the gel, the mobility changes its sign at this isoelectric point. A given protein is thus electrofocused and accumulates at this point. 10.1.5.3  Two-Dimensional Electrophoresis

Electrophoresis can thus be used independently for molecular weight determination or for measurement of the IEF. In these complex systems, it is common to combine these two determinations by making 2-D electrophoresis (2-D PAGE) [31]. The principle is to make an IEF measurement over a pH gradient say along the horizontal direction. This step sorts the proteins with respect to their charge. The slab is then submitted to an SDS PAGE along the vertical axis that separates all the proteins of the same spot (i.e., having the same charge) with respect to their molecular weight (Figure 10.10). These analyses are performed routinely and despite the complex patterns due to the high diversity of the proteins of a particular sample, they give good and reliable results for instance in following the evolution of the expression of a given protein in different cell environments via the intensity of its related spot in the electrophoregram.

10.2  Dielectrophoresis

453

Figure 10.10  Principle of 2D protein electrophoresis. The first step is a IEF determination (charge sorting), the second step a SDS PAGE (molecular weight sorting). In the final electrophoregram, each spot is a single protein.

10.1.6  Cell Electrophoresis

Cells are difficult to separate with electrophoresis because, like for proteins, size is only one part of the information. The task is usually to separate cells that have a distinctive property from other cells. The equivalent of SDS has not been found for cells so even a size separation does not give very good results. Besides, gels are not well-suited for these large objects. Therefore, electrophoresis does not seem to be the technique adapted to this problem. Dielectrophoretic measurements, described in the next section, are more suitable [32].

10.2  Dielectrophoresis In contrast with the previously described electrophoresis, dielectrophoresis is not a transport technique, except in some cases that we will review in the end of this section; it is rather a micromanipulation tool. Based on a contrast between the polarizabilities of the particle and the medium, it can be used on any kind of particle charged or not. 10.2.1  Theoretical Basis 10.2.1.1  The Dielectrophoretic Force

Although the physics of dielectrophoresis (DEP) has been known and characterized for a long time, this effect has recently regained some interest in the biology-

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biotechnology community mainly because of its potential when coupled to microstructures. By definition, DEP is the motion induced by nonuniform electric fields and is due to a contrast of polarizabilities between the particle and its solvent. For the interested reader, there are some good textbooks that provide a far more detailed description than the present short chapter [33–35]. Let us consider a particle in a solvent in the presence of an electric field. Because of this field, charges accumulate nonuniformly at the interface with the surrounding medium. This charge distribution creates a dipole that itself interacts with the electric field. If the field is not homogeneous and thus different on both sides of the particle, a net force acts upon it that drives it toward the high electric field areas if its polarizability is higher than that of the medium, and in the other direction if it is smaller. More quantitatively, the basic Maxwell equations tell us that a particle of polarizability a and of radius a experiences a force F in the presence of an external electric field E given by: 2 3

F = π a3α Ñ E 2





(10.19)

The particles we are interested in are “lossy dielectrics.” This means that, on top of the intrinsic permitivity of the particles, one has also to consider their conductivity and the energy dissipated via this ionic conduction. In this framework, expressing the polarizability in (10.19) leads to: F = 2π a3ε 0 εr,l Re(fCM )ÑE2



(10.20)

where e0 is the vacuum permitivity, er,l is the relative permitivity of the solvent, fCM is the Clausius-Mossoti factor, and Re(fCM) is its real part: fCM =



ε*p - ε*l ε*p + ε*l

(10.21)

where ep* and e l* are respectively the complex permitivities of the particle and the solvent: ε * = ε 0ε r - j



σ ω

(10.22)

where er is the relative permitivity, s the conductivity and w the frequency of the electric field. We also note el = e0 · er,l and ep = e0 · er,p Several consequences can be immediately derived from this expression: · 

 he direction of the force exerted on the particle depends on the sign of the T real part of the Clausius Mosssoti factor: if Re(fCM) > 0, the particle is attracted to the regions where the field is maximum (as illustrated on Figure 10.11): this is what is usually called positive dielectrophoresis. In the other case, it is repelled from these areas (actually, it is the solvent that is attracted

10.2  Dielectrophoresis

455

Figure 10.11  Polarizable particles attracted toward high electric field region in positive DEP regime.

to these areas forcing the particles to be repelled from them): it is the negative DEP case where the particles appear to be driven towards the low electric field areas. ·  Another interesting point is that F ~ ÑE2. This dependence with the square of the amplitude of the electric field implies that DEP can be used with dc or ac fields. In the first case however, electrophoresis will compete with DEP in the motion of the particles. The use of high-frequency ac fields is particularly interesting because it suppresses electrolysis or, more generally, electrochemistry at the surface of the electrodes. ·  The force depends on the gradient of the electric field intensity. Therefore, to transport particles over large distances, it would be necessary to maintain a large gradient over such distances, which requires large electric fields (and thus high voltages) in the macroscopic world. On the other hand, large local gradients are more easily created in microstructures (small scales). This explains why, with the development of microfabrication and its use in life science, this effect has been effectively rediscovered recently to manipulate, characterize, or sort particles. However, to transport them over large distances remains a challenge. As we will see later in the text, a traveling wave or a succession of elementary displacements triggered by gradients over small scales can overcome this difficulty. One last word before detailing the consequences of these expressions. The DEP traps are formed and conditioned by the geometry of the electrodes that are in the solution (except for the electrodeless DEP detailed further in this chapter). This is markedly different from electrophoresis where the electrodes are physically placed out of the channel (although in electrical contact with it). Working with high frequencies minimizes the electrochemistry at the surface, however, we will see in particular in Section 10.2.6.2 that it cannot avoid electrohydrodynamic flows or a direct contact of metal electrodes with the objects to be manipulated. This last point

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Micromanipulations and Separations Using Electric Fields

can be very detrimental to the technique because of the interactions of metals with charged object in particular through the formation of an electric image that leads to irreversible sticking [34, 36]. Care has thus to be taken to coat these electrodes to minimize these effects [34, 36, 37]. 10.2.2  The Clausius-Mossoti Factor

If we want to go one step further in the calculation of the force induced by the electric field, we have to expand the Clausius-Mossoti factor. This discussion is very different whether we deal with low or high frequencies: At low frequencies (lower than 10 kHz), there is some dispersion of the dielectric constant of the particle. That is, ep is actually a function of the frequency of the electric field [38]. This dispersion is mainly due to the relaxation time of the polarization of the double layer surrounding the particle (this time depends on the radius of the particle and on the Debye length). The situation is quite complex and there is no satisfactory model that fully describes all the observed phenomena [39]. At higher frequencies, the counterions of the double layer do not have enough time to move and the particle is basically nondispersive meaning that ep and sp are independent on the frequency. The polarization is then only due to the contrast in dielectric constants between the particle and its surrounding medium. This interfacial polarization (Maxwell-Wagner effect [33]) is described by a unique relaxation time t that depends only on the permitivities and conductivities of the particle and the medium. We can then express the real part of the Clausius-Mossoti factor: Re(fCM ) =



(ε p - ε l )ω 2τ 2 2 2

(ε p + 2ε l )(1 + ω τ )

+

(σ p - σ l ) (σ p + 2σ l )(1 + ω 2τ 2 )

(10.23)

where



τ=

ε p + 2ε l σ p + 2σ l

(10.24)

in this case, Re(fCM) varies monotonously with the electric field frequency between σ - σl ε - εl the extreme values p (for w ®0) and p (w ® ∞) (Figure 10.12). σ p + 2σ l ε p + 2ε l There remain, however, some discrepancies between this expression and the forces quantitatively measured in diluted colloidal suspensions. It becomes important to take into account the ionic double layer that effectively modifies the conductivity of the particles. Some authors have treated this problem by considering an infinitely thin conductive layer on the surface of the particle. Empirically, one then adds a surface conductivity ls to the particle conductivity sp. The total effective conductivity then becomes [33]:

σ p¢ = σ p + 2λ s / a

(10.25)

the behavior of the particles is then well described by using sp¢ rather than sp.

10.2  Dielectrophoresis

457

Figure 10.12  Evolution of the real part of the Clausius-Mossoti factor with the frequency. The point s is the crossover frequency (see more details in section 10.2.4.3).

We have plotted on Figure 10.12 the expression of Re(fCM) in this highfrequency regime. The low-frequency regime down to dc behavior cannot be described by this curve. Below typically 1-10 kHz, Re(fCM) decreases and the particles may even take a negative DEP behavior. 10.2.3  Optimization of the Electric Field 10.2.3.1  Electrode Geometries

The exact three-dimensional landscape of the electric field intensity shapes the force acting on the particles. Depending whether the final application is to trap them, to set them into motion or in rotation, or to combine trapping and another force field (such as a hydrodynamic flow), the constraints on the geometry of the electrodes are different. For instance, one of the early geometries used “castellated electrodes” that provided a good array of traps [40]. The exact calculation of the electric field spatial variations can be performed via a finite elements analysis and the optimization of their shapes and disposition in space becomes possible. Planar four electrodes geometry can give precisely located trapping site in negative DEP but, when possible, three-dimensional arrangements of two sets of four electrodes facing each other give the best trapping cages (Figure 10.13) [41]. 10.2.3.2  Electrodeless DEP

Practically, DEP comes inevitably with electrohydrodynamic flows that we will detail in Section 10.2.6.2. In microstructures, even for modest applied voltages, electric field can become extremely high and can easily trigger electrochemical instabilities at the surface of the electrodes. This drawback is even worse when using 2-D electrodes in a channel or chamber having some depth (although there are some

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Figure 10.13  Numerical calculation of trapping of particles in an octopole geometry (from [42]).

geometries where the electrodes themselves are shaped in three dimensions: indeed the instabilities are much reduced in this geometry (see Section 10.2.7.2)). There is, however, a way to avoid some of these problems by using the electrodeless DEP and although it is not as flexible as the usual electrode-based methods, it offers enough advantages to be favorably considered in some circumstances [43–45]. This technique is based on insulating channels having constrictions. If one applies an electric field between the entrance and the exit of the channel, the field lines have to squeeze in the constrictions creating therefore high gradients. It is thus an easy way to use the DEP effect only by structuring the channel. No electrochemistry is involved and most of the instabilities disappear. This has a cost however: using this geometry, there is only one electric field that controls the displacements of the particles in the channel. Therefore, all the tasks have to be performed sequentially in contrast with in situ electrodes-based DEP in which many traps can be controlled independently in a parallel way (Figure 10.14). 10.2.4  Characterization of Particles 10.2.4.1  Real-World Particles

Of course, particles that need to be manipulated or separated particularly in bio­ micronano technology are generally not solid homogeneous spheres. They can actually be extremely diverse and go from DNA molecules [46] to cells [47] or viruses [48]. They are usually modeled by an effective sphere although multishell models may be more accurate to account for some of the particle characteristics, to the price of a higher number of parameters. More than absolute quantitative measurements, studies on complex particles are often relative between slightly different systems. 10.2.4.2  Collection Rate

Now that we know how to apply a force to a particle, and, actually design the energy well in which this particle is immersed, we can use this knowledge to char-

10.2  Dielectrophoresis

459

Figure 10.14  Electrodeless DEP of Live/dead bacteria. The posts are made of glass and the electric field is directed from left to right; Because of their different dielectric characteristics, dead bacteria are trapped at a different spot close to the constriction itself compared to live bacteria (in a wider region) (from [45]).

acterize it. Historically, the experiments were performed in the positive regime and the measurement was a collection rate that implicitly assumes that the rate at which the particles accumulate in this well is proportional to the force they experience [33]. Making this measurement at different frequencies gives access to a spectrum of Re(fCM) and thus to the various parameters it covers via multiparameters fits. 10.2.4.3  Levitation Height

However, it has been recognized more recently that these experiments in the positive DEP regime were difficult because of the many experimental difficulties caused by electrohydrodynamic flows (see Section 10.2.6.2). An alternative is to measure the height at which the DEP force can counterbalance gravity and levitate a particle in the negative DEP regime [49]. This height is given by equilibrating the DEP force and gravity:



Re(fCM )ÑE2 =

2(ρp - ρl )g 3ε l

(10.26)

We can express ÑE² as a function of the height z by using a phenomenological function q(z) that describes explicitly the dependency of the gradient of E² with the height: ÑE² = U²/q(z) where U is the rms value of the applied voltage. The levitation height h is then given by:



æ 2(ρp - ρl )g ö h = q -1 ç ÷ è 3ε l Re(fCM )U 2 ø

(10.27)

The function q(z) can be calibrated by levitating well-characterized particles and medium. Rather than static measurements, levitation can be combined with an external hydrodynamic flow (DEP-FFF) as we will see in Section 10.2.7.1.

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Micromanipulations and Separations Using Electric Fields

10.2.4.4  The Crossover Frequency

From (10.23) and as it is illustrated on Figure 10.12, the behavior of the particles switches from positive to negative DEP when the frequency of the applied field varies. This frequency is called the crossover frequency and is a particularly convenient (although far from complete) way to characterize particles. For instance a crossover frequency spectrum for different conditions such as variable solvent conductivities characterizes the dielectric constant and conductivity of the considered particle or complex bioparticles [35]. This frequency is easily calculated by solving the equation Re(fCM) = 0 and its solution is then given by:



f0 =

1 2π

(σ p¢ + 2σ l )(σ p¢ - σ l ) (ε p + 2ε l )(ε p - εl )

(10.28)

where sp¢ is given by (10.25). A fit of such spectra as a function of the conductivity gives an accurate description of some characteristics of the particle (Figure 10.15). sp¢ is also a function of the particle’s radius. Equation (10.28) has been tested for identical particles of various radii. The excellent agreement between theory and experiments is a good way to determine the surface conductivity of these particles (Figure 10.16). 10.2.5  Electrorotation and Traveling Wave

We have seen how DEP could be used to apply a force on polarizable particles in a solvent in order to trap them. A similar physical principle can be used to set them in rotation. Consider a polarizable particle in a rotating electric field such as the one

Figure 10.15  Fit of (10.27) (line) over experimental crossover frequencies measured at different medium conductivities (squares). Particles are 216 nm polystyrene latex beads. (from [50]).

10.2  Dielectrophoresis

461

Figure 10.16  Crossover frequency between positive and negative DEP for polystyrene latex beads of different radii in pure water. The line is a best fit of (10.28).

that can be formed using four electrodes successively addressed (Figure 10.17). If the angular velocity of the field is sufficiently high, the induced-dipole orientation makes some angle with the electric field and, as a consequence, a torque acts on the particle and makes it rotate. This torque experienced by the particle is then given by [33–35, 47]:

G = -4πε0ε r,l a3 Im(fCM ) × E2 with the same conventions as in the preceding part.

Figure 10.17  Physical principle of electrorotation.

(10.29)

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Micromanipulations and Separations Using Electric Fields

Although electrorotation presents some similarities with DEP, it also bears some fundamental differences in the sense that the direction of the ac electric field is not fixed but is continuously moving. The full calculation based on (10.29) shows that the direction of the torque depends on the electrical characteristics of the particle and the medium but not on the frequency: there is no equivalent to the crossover frequency (no change of sign in the rotation direction). However, the amplitude of this rotation is not monotonic and, depending on its sign, exhibits a maximum or a minimum precisely at the crossover frequency. Electrororation is successfully used as an analytical tool to probe the dielectric properties of particles or cells. By varying the frequency, dielectrophoretic spectroscopy can be performed [47, 46]. It can also be useful in complement of DEP cages to stabilize them or to get more information out of these trapping measurements. An extension of this calculation is the application of a similar sequence to electrodes aligned on a surface. Imagine this setup as the unfolding of the electrorotation setup. Instead of having a rotating electric field we now deal with a traveling wave of electric field [51]. This configuration therefore implies the successive addressing of many electrodes linearly arranged with phase shifted signals. Traveling waves have been effectively used for the pumping of liquids [53] and to induce the motion of particles or cells [47, 52]. As with electrorotation, high velocity switching between the electrodes leads to the asynchronous motion of the particles. This strategy can then been coupled with microfluidics and should lead to a directed motion in a channel. However, practically, a DEP force directly acting on the particles often comes into play and, to avoid sticking of the particles on the electrodes, care should be taken to work in negative DEP conditions. 10.2.6  Instabilities 10.2.6.1  Pearl Chaining

We have seen how the external electric field can induce a dipole in a particle. The interaction between this dipole and the electric field then drives the particle toward the high field region (for positive DEP). The story however does not stop here. If there is another particle in the vicinity of the first one, the electric field experienced by this second object is not the externally applied field but a field modified by the presence of the first particle (of course, the field experienced by the first particle is also modified by the presence of the second one). This leads to an arrangement of the particles in “pearl chains” where they touch each other and where this chain follows the fields lines (the whole assembly acts like a large dipole) (Figure 10.18) [34, 54]. This effect effectively increases the trapping efficiency [55]. The critical value of the electric field necessary to observe pearl chaining has been computed by Pohl [33]. The calculation leads to:



Ecrit »

ε p + 2ε l ε p - εl

kT 2πεl a3

A detailed review on pearl-chaining effects can be found in [34].

(10.30)

10.2  Dielectrophoresis

463

Figure 10.18  Interaction between particles. Each particle (in this case they are more polarizable than the medium) deflects the field lines and thus the effective electric field for the other particle.

Decreasing the concentration of particles in the solution is not an efficient way to avoid pearl chaining: As the high electric field region (close to the electrodes) collect the particles, their local concentration increases considerably in these area to reach values for which pearl chaining is observed even with extremely dilute initial solutions. 10.2.6.2  Electrohydrodynamic Instabilities

Most of the time, positive DEP brings hydrodynamic instabilities that can be strong enough to impair an efficient trapping or to modify the position of trapping. As a matter of fact, it has been often observed that the position of stable trapping is not at the point of maximal field but slightly away from it: on the electrodes themselves or on their edges. These effects can be qualitatively understood by taking into account field induced flows in the solution. Most of the time thermal gradients in the solution are the strongest contribution to these effects [53]. The dissipated heat is given by Q~sE2. Thus, the same field nonuniformities that give rise to the DEP effect, causes local thermal gradients that in turn, are the source of convective flows. In negative DEP regime, these flows actually stabilize the trapping. However, they tend to destabilize positive DEP trapping. Temperature rise has other consequences such as modifying the conductivities or the permittivities, which are both function of temperature (see [56] for a detailed review). In addition to these thermal effects, the electric field directly interacts with the double layer present at the surface of the electrodes. Effects similar to electro-osmosis will manifest themselves at time scales larger than the time necessary to establish this double layer. Thus, for sufficiently small frequencies (that depend on the characteristics of the medium but that can be roughly estimated 10–500 kHz), this effect, sometimes complicated by charge injection [57], manifests itself with a much higher intensity than the DEP itself. Low-frequency observations where the trapping site is at the center of the electrodes or precisely on their edges can be explained by a combination of flows and DEP.

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Micromanipulations and Separations Using Electric Fields

10.2.7  DEP-Based Separations 10.2.7.1  Combining DEP and a Hydrodynamic Flow

In contrast with electrophoresis, and except for traveling-wave DEP, DEP is thus a trapping and an analytical tool. Therefore, to achieve any kind of separation with this concept, another flow should be added. In that sense, DEP based separations are very similar to affinity chromatography where a mixture is brought in contact with a matrix by a flow in a column. Some of the components of this mixture having more affinity to the matrix are then trapped on it while the other components can flow with the solvent. The trapped species is then eluted in a second time by changing the solvent (for instance, its pH) or by adding a molecule having an even higher affinity for the matrix. DEP is the equivalent of a “smart” matrix in the sense that the affinity of particles for the traps can be tuned externally by changing the electric field characteristics. However, it still needs an eluant to flow the particles on the electrodes. The simplest idea that comes to mind is to flow the particles on the energized electrodes. By adjusting the characteristics of the electric field, one component of the mixture is trapped on the electrodes while the others flow with the liquid. There are quite a few examples of this approach that use different geometries for the electrodes or/and sequences of flow in one or two directions [58–60]. Another way of achieving a good separation of particles is to combine DEP and gravity in the field flow fractionation. This is the coupling between DEP levitation and a Poiseuille flow well-controlled by a difference of pressure in a microfluidic channel [61, 62]. The height at which the particles levitate is given by (10.27). As the Poiseuille flow is characterized by a parabolic velocity profile, the fluid velocity is different for each height and particles of different dielectrophoretic characteristics (different Re(fCM)) will travel at different speeds which results ultimately in different elution times. 10.2.7.2  Dielectrophoretic Ratchets

A good illustration of the use of DEP is the dielectrophoretic ratchet. This experiment can be declined in two versions (the Brownian ratchet or the shifted ratchets) that we briefly describe here. Brownian Ratchet

The theoretical models on which these particular experiments are based are described in [63]. Reference [64] provides a very complete review on these “force– free” motion phenomena. Let us imagine Brownian particles in a potential similar to the one described in Figure 10.19. They are trapped in the minima of this potential and, if the potential barriers are large enough, which is supposed here, their concentration profiles are very narrow and centered close to these minima (Figure 10.19(b)). Now, let us switch off this potential (i.e., we now impose to the particles a flat potential), the particles are going to freely diffuse and, as a consequence, the concentration peaks will broaden (Figure 10.19(c)). After a time toff, we cycle back to the saw-tooth potential (Figure 10.19(d)). The particles are again trapped in the minima of the potential. If toff is sufficiently large a nonnegligible fraction of

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Figure 10.19  Energetic potential (a) and concentration profile of the particles (b). The height of the energy barriers is much higher than thermal agitation. After a diffusion step of duration toff, the particles diffuse (c) and are trapped again when the potential is switched on (d).

them have diffused over a distance larger than the small side of the pattern of the potential, on the other hand, because of the asymmetry of this pattern, the fraction of the particles that have diffused over a distance larger than the large side of this pattern is much smaller. As a result, a fraction of these particles will be trapped in the minima next to the ones they previously occupied and, with the conventions of Figure 10.19, more of them will shift to the left than to right (shadowed area under the concentration peak in Figure 10.19(c)). By reiterating this process a large number of times, one can then set these particles into motion over potentially large distances whereas the only gradients present in the system are local. Hence, we have solved here one of the main limitations of DEP by making it a transport technique and not only an analysis technique. Furthermore, we have an adjustable “knob”: toff, the time during which we let the particles diffuse is a control parameter for these experiments. The macroscopic motion of the particles is expected to vary exponentially with their diffusion coefficient and thus with their size or their molecular weight which is very promising for separating particles of different sizes [63]. This sawtooth potential can be of DEP nature [37, 65]. In particular, it can be created by setting an ac. voltage between an electrode whose corrugations presents the “good” properties of periodicity and asymmetry and a planar one [37]. Qualitatively, by a simple “tip effect,” the electric field is of much higher intensity on the ridges than it is in the valleys. As a consequence, the asymmetry of the electrode reflects itself on the intensity of the electric field and thus on the energetic potential. This can be confirmed by a finite elements simulation of the electric field. When experiencing the electric field at high enough frequencies, the beads are confined in the valleys (negative DEP).

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Figure 10.20  Macroscopic velocities as a function of toff (diffusion step) for two different sizes of latex beads. In this particular case, setting toff = 3s leads to a factor of 10 between the velocities.

The macroscopic mean velocity V of a particle at a given toff is quantitatively described by the simple model exposed above (no adjustable parameter). Figure 10.20 plots the velocities of two different latex beads as a function of toff. These curves exhibit a maximum whose position at a given toff is very dependent on the size of the particles This is thus an extremely promising although quite slow technique. Shifted Ratchets

The previous idea (Brownian ratchet) relies on diffusion. It implies small velocities, a disadvantage that can be corrected if we use two potentials similar to the one described in the preceding part (Figure 10.19(a)). Here, these potentials are shifted by a fraction of their common period and addressed successively [66]. When the commutation time is too small, the particle cannot escape the corresponding trap and the macroscopic velocity is zero but, when both times are long enough, the particle has enough time to move by one total period per time cycle. This velocity Vopt is the optimal velocity. We can rephrase this statement in terms of mobilities instead of residence times: For identical residence times, particles will have either a zero velocity or an optimal velocity according to their mobilities. In other words, this device is a filter according to the mobility of the particles. Moreover, the mobility threshold of this filter can be chosen by tuning the two residence times. On a separation point of view, this filterlike situation is obviously a great improvement compared to conventional techniques as the velocity of some of the considered particles is exactly zero and even subtle differences in mobilities should be usable for a separation. Such characteristics can be obtained with the use of 2-D electrodes sputtered on glass (see the “Christmas tree” geometry in Figure 10.21). To get successively two of these potentials, two of these plates are stacked with their gold sides facing

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Figure 10.21  Stacked “Christmas-tree” asymmetric electrodes. These two pairs of electrodes are addressed successively.

each other. A high frequency ac voltage is successively applied between the two electrodes of each slide. This situation emphasizes some of the difficulties listed above: negative dielectrophoresis is difficult to deal with in this particular geometry as the regions of weakest electric field are located outside of the central channel, therefore expelling

Figure 10.22  Behavior of 0.5-μm latex beads when the voltage is switched from the lower set of electrodes (light gray) (a) to the upper set (dark gray) (b, c). (Photo courtesy of L. Talini.)

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the particles from it. It is a strikingly different situation from the one dealt with in the preceding part were both in the positive and in the negative dielectrophoresis regimes, particles were confined close to the grating surface. But the use of positive DEP regime is not trivial either: particles not initially present in the channel tend to be collected in it, increasing the concentration. Furthermore, electrohydrodynamic flows develop on the tips of the electrodes generating recirculations of particles around the trapping zones. In Figure 10.22, one can see a typical sequence of the migration of the particles (in that particular case 0.5-μm latex), as the potential is switched from one pair of electrodes to the other. The particles indeed move from one dielectrophoretic trap on one pair of electrodes to the next one on the other pair when switching the field from one plate to the other. The analysis of the experimental results is of the same nature as in the Brownian ratchet experiment. The expected velocity regimes V = 0 or V = Vopt are indeed observed for two different sizes [67]. Although the dielectrophoretic behaviors are more complex in this geometry than they are in the Brownian ratchets case, the observed filter effect along with the high velocities reached by the beads make this realization highly promising in separations problems. For this particular device as for many others presented here, an improvement in the performances will come from a better control of the microfluidics.

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C h a p t e r 11

Conclusion

At the end of this book, let us recall the different steps that we have made towards the comprehension and prediction of the mechanical behavior of micro- and nanoparticles, macromolecules, and cells. Because these biologic objects are immerged in a buffer liquid, we have focused first on the carrier fluid behavior, by studying microfluidic flows in microsystems. Microfluidics may be considered as a new science by itself. On many points it departs from the classic view of macroscopic fluid dynamics. We have presented and analyzed the different forms of microflows that can be found in microsystems for biology: continuous microflows (when a single phase liquid flows continuously in microchannels), digital microfluidics (when separated microdrops are displaced step by step on a solid surface), and two-phase flows (when droplets are transported by an immiscible carrier fluid flowing continuously in microchannels). Observing that the biologic objects of interest—macromolecules, DNA strands, proteins, cells, and so fourth—behave differently in the carrier fluid due to their size and weight, we have investigated the different transport mechanisms from molecular diffusion to isolated trajectories. Because our approach aims for in vitro problems and in vivo situations, we have given attention to the behavior of particles in confined volumes. Next, since the main purpose of biochips is the study of biological targets such DNA, proteins, and cells, the principle of key-lock recognition has been presented, showing that biochemical reactions are used to perform bioanalysis and biorecognition. Kinetics of the main reactions like DNA hybridization and protein enzymatic digestion have been carefully detailed from a theoretical standpoint and also in the different forms they take in a biochip environment. Because it has been observed that transport of particles by a buffer fluid is not always specific enough (i.e., the particles of interest cannot be all transported to the reactive surface), we have presented additional tools to transport and manipulate biological objects, for example, the use of magnetic particles as transport vectors and the specific effects of electric fields on these biological objects. Through this entire book, we have shown that precise handling and manipulation of micro- and nanoparticles, macromolecules, and cells are at the heart of biotechnology. The theoretical background and the modeling approach presented here are and will likely remain the basis for the comprehension of the phenomena involved in biochips and bioMEMS, even if biotechnology is rapidly evolving and new developments emerge constantly. Since the first edition of this book, digital and droplet microfluidics have seen a considerable development. Some other developments are

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already clearly foreseen, and others are still unpredictable. Among the foreseen developments, we can list: ·

·

·

·

·

·

Downscaling, with a continuous trend towards miniaturization of biochips and bioMEMS—for example the use of carbon nanotubes for improving the detection accuracy. This trend seems unavoidable since the ultimate goal is to work on single targets, Integration: Because miniaturization means also designing biochips containing a maximum of functions. Such designs simplify the problems of microfluidic connection between the different functions. Handling of living cells: Probably the most promising ongoing development. Supported by biology, biotechnology is presently pursuing the development of tools to facilitate the understanding of the complexity of cells, at the level of the single cell as well as at the level of group of interacting cells. The pharmaceutical industry is strongly implicated for the search and testing of drugs directly on living cells, which are expected to rejuvenate the production of new drugs. Cancerous protein markers: In cancerology, dramatic improvements are expected from the determination of marker proteins characterizing each specific type of cancer. In this field, new biotechnological tools for early detection of these markers are expected to totally change the prognosis of cancer. Drug guidance: The development of the techniques of drug guidance in the human body to directly address defective cells has become another major research topic. This is a domain where biotechnological developments could contribute to the efficiency of drug guidance. Self-assembly: Side applications of biotechnology have already appeared and are likely to increase with the techniques of self-assembly of nanoparticles and macromolecules.

An analysis of these listed trends shows that the theoretical background of this book will remain relevant in the future. On a modeling point of view, the progressive evolution towards smaller scales and smaller concentrations in particles is going to put forward the discrete numerical approaches. Even at a very small scale, the buffer (carrier) liquid still satisfies the continuum assumption and its behavior can be simulated by the classic Navier-Stokes formulation. On the other hand, discrete methods—such as those we have presented in Chapters 4 and 5—are more appropriate to model the behavior of the transported particles when the number of these particles becomes small. Numerical methods using a relevant coupling of the two approaches will progressively be preferred to model such types of problems. To conclude, the scientific domain of biotechnologies is complex and requires a wide and strong scientific background. However, it is rewarding because it addresses the human conditions, and it is also motivating because it is wide open to inventiveness and imagination.

List of Symbols Bo Boe C Ca D Da De DH Du E EBD Ec El Ev g Gr K Kn La Ma Mv Oh Q Re Sc Sh U V w We Wi

Bond number (gravitational) Bond number (electric) capacitance [F m–2] capillary number diffusion coefficient [m2 s–1] Damkohler number Dean number and Deborah number hydraulic diameter [m] Duckhin number evaporation rate [m3 s–1] break-down electric field [V m–1] elasto-capillary number elasticity number evaporation number gravitational acceleration [m s–2] Gratez number K number Knudsen number Laplace number Marangoni number Viscoelastic Mach number Ohnesorge number flow rate [m3 s–1] Reynolds number Schmidt number Sherwood number electric potential [V] average velocity [m s–1] width (microchannel) [m] Weber number Weissenberg number

α e e˙ γ γ˙ q

thermal diffusivity [m2 s–1] electric permittivity [F m–1] elongation rate [s–1] surface tension [N m–1] shear rate [s–1] Young contact angle [radians] 475

476

List of Symbols

qu qr λ m, η η ν ρ s τ τC τD τR τT

advancing contact angle [radians] receding contact angle [radians] mean free path and capillary length [m] viscosity (dynamic) [kg m–1 s–1 or Pa.s] electrowetting number viscosity (kinematic) [m2 s–1] volumic mass [kg m–3] electrical conductivity [S m–1] relaxation time [s] convective time [s] diffusional time [s] Rayleigh time [s] Tomotika time [s]

About the Authors Jean Berthier is scientist at the CEA/LETI, France. He received an engineering di­ ploma from the Institut National Polytechnique, an M.S. in mathematics from the University of Grenoble, and an M.S. in fluid mechanics at the same university. He has been working at the Sandia Laboratories and Los Alamos Laboratory for multiphase fluid flow computation. He is presently involved in the development of microsystems for liquid–liquid extraction and microflow focusing devices for bio­ encapsulation of cells. He teaches a masters course on mathematical methods in fluid mechanics at the University of Grenoble. He is the lead author of the book Microfluidics for Biotechnology (first edition) published by Artech House in November 2005 and the author of the book Microdrops and Digital Microfluidics published by William Andrew in February 2008. He is a member of the EON (European Observatory), OMNT (Observatory of Micro and Nano Technologies), and the editorial board of the Biomicrofluidics journal. Pascal Silberzan is a research professor at the “Physico-Chimie Curie” laboratory, a joint laboratory between the Centre National de la Recherche Scientifique and the Institut Curie in Paris, France, where he heads the “biology inspired physics at mesoscales” group. A graduate from the ESPCI, Paris, he received his Ph.D. from the Collège de France and the University Paris 6, on wetting phenomena. After his Ph.D., he gradually switched his interests from soft condensed matter to biology inspired physics. For his experimental work, he heavily relies on the use and devel­ opment of microfabrication techniques. Dr. Silberzan has been working at Cornell University and Princeton University, and joined the Institut Curie in 1993.

477

Index A Adherence, 241 Adhesion, 241 Adsorption, 233 Adsorption coefficient, 333 Advancing contact angle, 144 Advection-diffusion equation, 237 Affinity, 246 Alginate, 28 Analogs, 348 Analytes, 322 Anisotropic media, 205 Antibody, 306 Antigen, 306 Apparent contact angle, 138 Apparent diffusion coefficient, 234 Apparent viscosity, 244 Arnold tongue, 178 Arrhenius coefficients, 312 Avogadro number, 245

B Back pumping, 157 Baker’s transformation, 181 Ballistic Random Walk (BRW), 275 Bead (magnetic), 398 Berge-Lippman-Young law (BLY), 131 Bernoulli’s equation, 46 Bifurcation, 294 Biorecognition, 304 Biochemical reactor, 309 Blasius boundary layer, 59 Boltzmann constant, 418 Boltzmann length scale, 222 Bond number, 9, 106 Bond number (electrical), 10

Boundary layer, 249 Brownian motion, 201 Buckingham’s (Pi) theorem, 1

C Canthotaxis, 101 Capacitance, 135 Capillary electrophoresis, 448 Capillary forces, 99 Capillary length, 9, 106 Capillary number, 4 Capillary pumping, 98 Capillary rise, 95 Capsule, 194 Carreau-Yasuda relation, 29 Cassie-Baxter equation, 120 Cellular microfluidics, 282 Centering electrodes, 160 Chaotic mixing, 268 Chromatography column, 280 Clausius-Mosotti factor, 454 Cohesive energy, 74 Competition reaction, 346 Concentration equation, 237 Concentration boundary layer, 249 Concus-Finn relation, 112 Confocal microscopy, 378 Constriction, 31 Contact angle, 73 Contact line, 104 Continuous flow immunoassays (CFI), 355 Continuous phase, 183 Critical Micellar Concentration (CMC), 79 Curvature, 81 Curved microchannel, 294 479

480

D Dammköhler number, 12, 328 Dean flow, 65, 294 Dean number, 14, 65 Deborah number, 8 Debye layer, 439 Debye length, 441 Debye-Hückel approximation, 440 Deflection (membrane), 431 Deformation tensor, 30 Denaturing, 304 Depletion, 333 Desorption, 323 Desorption coefficient, 333 Deterministic trajectory, 405 Deterministic Lateral Displacement (DLD), 289 Dielectrophoresis, 453 Dielectric breakdown voltage (DBV), 289 Diffusion coefficient, 204, 246, 418 Diffusion (in a microchamber), 213 Diffusion (from a point source), 208 Diffusion barrier, 219 Digital microfluidics, 131 Diode (hydraulic), 37 Discrete approach, 222 Dispensing (of droplets), 157 Dispersed phase, 183 Displacement reaction, 346 Division (of droplets), 154 DNA, 304,362 Dripping mode, 190 Droplet microfluidics, 173 Droplet dispensing, 157 Droplet division, 154 Droplet motion, 153 Drug diffusion in human body, 227 Dukhin number, 14 Dynamic contact angle, 162

Index

Electrocapillary number, 8 Electron microscopy, 382 Electroosmosis, 442 Electrophoresis, 439 Electrowetting, 132 Encapsulation, 187 Engulfment (droplet), 170 Entrance length, 58 Enzymatic reaction, 316 Error (erf) function, 208 Establishment length, 58 Evaporation number, 10 ElectroWetting On Dielectric (EWOD), 132 Eötvös relation, 77 Extra-cellular space (ECS), 227

F FAC, 390 Fakir effect, 122 Farhaeus effect, 294 Ferrofluid, 431 Ferromagnetic bead, 403 Ferromagnetic rod, 406 Fick’s law, 203 Field Flow Fractionation (FFF), 279 Flow regime (FFD), 203 Flow focusing (single phase), 283 Flow focusing device (FFD), 182 Fluorescence, 319 Fluorescence resonance energy transfer (FRET), 348, 380 Force (on a triple line), 99 Force (EWOD), 139 FRAP, 379 Friction factor, 40 Functional magnetic beads, 397 Fusion temperature, 334

E

G

Einstein’s law, 418 Elasticity number, 8 Electroactuation, 460

Gating ratio, 65 Gravity (Bond’s number), 9, 106 Gel elecrophoresis, 446

Index

Gelling, 194 Graetz number, 13, 261 Green Fluorescent Protein (GFP), 371 Groove (droplet in a groove), 116 Guggenheim-Katayama relation, 77

H Hagen-Poiseuille flow, 38 Hauksbee problem, 84 Helle-Shaw chamber, 69 Heterogeneous raction, 332 Hoffman-Tanner law, 167 Homogeneous reaction, 328 Huggins law, 26, 244 Hysteresis, 142 Hydraulic diameter, 41 Hydraulic resistance, 43 Hydrodynamic characteristic time, 3

I Ilkovic solution, 208 Immunoassays, 346 Immunoreaction, 346 Interface, 73 Interstitial matrix (IM), 227 Intrinsic viscosity, 26

J Jetting mode, 186 Jurin’s law, 95

K Key-lock recognition, 306 Knudsen number, 3, 20 K-number, 13

L Lamb’s equation, 410 Lambert W function, 182

481

Laminarity, 33 Langmuir equation, 322 Langmuir kinetics, 322 Langevin’s equation, 205 Langevin’s function, 402 Laplace’s law, 80 Laplace number, 6 Latex microparticles, 372 Light scaterring, 386 Lineweaver-Burke relation, 317 Lippmann-Young law, 131 Liquid-liquid extraction (LLE), 168 Liquid plug, 161 Lotka-Volterra model, 325 Lumped models, 48

M Mach number (viscoelastic), 14 Magnetic beads, 397 Magnetic beads chain, 423 Magnetic Field Flow Fractionation (MFFF), 419 Magnetic fluid, 428 Magnetic force, 403, 429 Magnetic induction, 402 Magnetic membranes, 431 Magnetic repulsion, 412 Magnetic susceptibility, 404 Marangoni convection, 78 Marangoni number, 12 Martin’s relation, 26 Mass conservation equation, 20 Mass flux, 202 Material derivative, 239 Maxwell tensor, 136 Membrane, 431 Membrane deflection, 431 Michaelis-Menten model, 315 Microbubbles, 168 Microchamber, 60 Microdrops, 73 Microgroove, 116 Micromanipulation, 391

482

Micromixer, 105 Microneedle, 50 Microscopy, 370 Microwell, 105 Mixing, 268 Mobility, 418 Monte-Carlo model, 222 Multiphase microflow, 161

Index

Primers, 215 Protein, 305, 364 Protein reactor, 61

Q Quadruple contact line, 101 Quantum dot, 371 Quencher, 348

N Nanopores, 449 Navier-Stokes equations, 20 Network, 50 Neumann’s construction, 92 Newton’s law, 405 Newtonian fluid, 24 Non-Newtonian fluid, 24 Nozzle (FFD), 184 Ohnesorge number, 6 Optical tweezers, 392 Oscillations of magnetic membranes, 433 Ostwald relation, 28

P PAGE, 452 Parallel flow, 268 Paramagnetic bead, 402 Peclet number, 12, 248 Peykov-Quinn-Ralston-Sedev model (PQRS), 141 Photobleaching, 370 Pinched channel, 287 Plasma (blood) extraction, 56 Plug flow, 161 Poiseuille-Hagen flow, 38 Poisson coefficient, 432 Polymerase chain reaction (PCR), 362 Polymerization, 194 Polymeric liquid, 25 Predator-prey model, 325 Predictor-corrector model, 422 Pressure drop, 40

R Rabinowitch-Ellis relation, 29 Radius (curvature), 81 Random walk, 224, 275 Ratchet, 464 Rayleigh time, 4 Reaction constant, 311 Reaction front, 329 Reaction kinetics, 309 Reaction order, 311 Reaction rate, 309 Receding contact angle, 144 Recirculation chamber, 65, 297 Rectangular ducts, 39 Relaxation time, 29 Remanent magnetization, 402 Reversibility (flow), 35 Reynolds number, 4 Reynolds number (transition), 34 Ricatti equation, 334 RNA, 363 Rosensweig formula, 403 Roughness, 43

S Saturation (electrowetting effect), 140 Scaling (FFD), 190 Schwarz-Christoffel mapping, 137 Sedimentation, 220 Separation column, 415 Sessile drop, 105 Shear rate, 30

Index

Shear stress, 25 Shear thinning, 25 Shear viscosity, 25 Sherwood number, 12 Single phase pressure drop, 40 Single-phase flow focusing, 283 Slip length, 45 Slip velocity, 45 Sparrow-Schlichting relation, 59 Specific viscosity, 25 SQUID, 402 Star-shaped electrode, 160 Static contact angle, 88, 163 Stokes equation, 35 Striation thickness, 179 Surface chemistry, 375 Surface tension, 74 Surfactant, 78, 372

T Taylor-Aris diffusion, 253 Time step, 252 TIRF, 380 Tomotika time, 4 Trajectory, 272, 405 Transition (critical) Reynolds number, 34 Triple (contact) line, 93 Two-phase pressure drop, 167 T-junction, 173

483

U Uniform flow, 61 Uptake, 51

V Van der Waals force, 76 Viscoelastic fluid, 6, 25 Vicoelastic Mach number, 14 Viscosity, 24 Vogel-Fulcher-Tamman-Hess (VFTH) law, 27 Vortcity, 35 Washburn’s law, 39 Weber number, 5 Weissenberg number, 7, 29 Wenzel law, 118 Wetted perimeter, 41

Y Young law, 87 Young law (generalized), 91 Young modulus, 432

Z Zeta potential, 441 Zweifach-Fung bifurcation law, 296

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