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

viii

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 (Figure 1.6), whereas flow of dilute polymeric liquids correspond to El 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.

72

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

3.2  Interfaces and Surface Tension

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).

78

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,

3.2  Interfaces and Surface Tension

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 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.

88

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)

3.5  Contact Angle: Young’s Law

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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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.

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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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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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Interfaces, Capillarity, and Microdrops

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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Interfaces, Capillarity, and Microdrops

�� �� 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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Interfaces, Capillarity, and Microdrops

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) 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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Interfaces, Capillarity, and Microdrops

Case 2: Microscopic Drops, Bo 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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Interfaces, Capillarity, and Microdrops

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

(3.80)

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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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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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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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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123

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.

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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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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 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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Digital, Two-Phase, and Droplet Microfluidics

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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Digital, Two-Phase, and Droplet Microfluidics

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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Digital, Two-Phase, and Droplet Microfluidics

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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147

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
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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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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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.

4.2  Digital Microfluidics

159

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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163

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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Digital, Two-Phase, and Droplet Microfluidics

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.

4.3  Multiphase Microflows

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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Digital, Two-Phase, and Droplet Microfluidics

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.

4.3  Multiphase Microflows

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).

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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 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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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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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.

184

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.

186

Digital, Two-Phase, and Droplet Microfluidics

if Pe = - 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
= 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 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

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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

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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 = 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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Diffusion of Biochemical Species

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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Diffusion of Biochemical Species

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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Diffusion of Biochemical Species

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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Diffusion of Biochemical Species

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.

5.4  Microscopic (Discrete) Approach

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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Diffusion of Biochemical Species

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.

234

Diffusion of Biochemical Species

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

238

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); 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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Transport of Biochemical Species and Cellular Microfluidics

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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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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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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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 ÷ > 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 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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Transport of Biochemical Species and Cellular Microfluidics

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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Transport of Biochemical Species and Cellular Microfluidics

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.

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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

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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.

6.5  Cellular Microfluidics

285

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 2b cannot stay in channel 1 when passing

Figure 6.66  Schematic of a DLD device with e = 1/3.

6.5  Cellular Microfluidics

291

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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293

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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295

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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297

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).

6.6  Conclusion

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.

References   [1] Nguyen, N. -T., and S. T. Wereley, Fundamentals and Applications of Microfluidics, Norwood, MA: Artech House, 2002.   [2] Tabeling, P., Introduction à la Microfluidique, Paris: Editions Belin, 2004.   [3] Starkey, T. V., “The Laminar Flow of Suspensions in Tubes,” British Journal of Applied Physics, Vol. 6, January 1955, pp. 34–37.   [4] Tanner, R. I., Engineering Rheology, Oxford, U.K.: Oxford Engineering Series, 2000.   [5] Levich, V. G., “Physiocochemical Hydrodynamics,” Am. J. Phys., Vol. 31, No. 11, 1963, p. 892.   [6] Wissler, E. H., “On the Applicability of the Taylor-Aris Axial Diffusion Model to Tubular Reactor Calculations,” Chemical Engineering Science, Vol. 24, 1969, pp. 527–539.   [7] Batycky, R. P., D. A. Edwards, and H. Brenner, “Thermal Taylor Dispersion in an Insulated Circular Cylinder—1. Theory,” Int. J. Heat Mass Transfer, Vol. 36, No. 18, 1993, pp. 4317– 4325.   [8] Berthier, J., Van Man Tran, F. Mittler, and N. Sarrut, “The Physics of a Coflow Micro-Extractor: Interface Stability and Optimal Extraction Length,” Sensors and Actuators A, Vol. 149, 2009, pp. 56–64.   [9] COMSOL, Multiphysics modeling. http://www.comsol.com/. [10] Dehghan, M., “Numerical Solution of the Three-Dimensional Advection-Diffusion Equation,” Applied Mathematics and Computation, Vol. 150, 2004, pp. 5–9. [11] Kamholz, A. E., and P. Yager, “Theoretical Analysis of Molecular Diffusion in PressureDriven Laminar Flow in Microfluidic Channels,” Biophysical Journal, Vol. 80, 2001, pp. 155–160. [12] Kamholz, A. E., and P. Yager, “Molecular Diffusive Scaling Laws in Pressure-Driven Microfluidic Channels: Deviation from One-Dimensional Einstein Approximations,” Sensors and Actuators B Chemical, Vol. 82, Issue 1, 2002, pp. 117–121. [13] Hessel, V., H. Löwe, and F. Schönfeld, “Micromixers—A Review on Passive and Active Mixing Principles,” Chem. Eng. Sci., Vol. 60, No. 8–9, 2005, pp. 2479–2501. [14] Berthier, J., and F. Ricoul, “Numerical Modeling of Ferrofluid Flow Instabilities in a Capillary Tube at the Vicinity of a Magnet,” Proceedings of the MSM 2002 Conference, Puerto Rico, 2002. [15] Fowler, J., H. Moon, and C. -J. Kim, “Enhancement of Mixing by Droplet-Based Microfluidics,” Proceedings of the IEEE Conference MEMS, Las Vegas, NV, January 2002, pp. 97–100. [16] Berthier, J., “Interfacing Continuum and Discrete Methods: Convective Diffusion of Microparticles and Chemical Species in Microsystems,” Proceedings of the 2008 COMSOL European Conference, Hannover, November 3–5, 2008. [17] MATLAB, The Language of Technical Computing, The MathWorks Inc., version 6.2, 2000. [18] Giddings, J. C., “Field-Flow Fractionation,” C & E News, Vol. 66, 1988, pp 34–45.

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Transport of Biochemical Species and Cellular Microfluidics [19] Scott, R. P. W., “Dispersion in Chromatography Columns,” Chrom-Ed Series, http://www. chromatography-online.org/Dispersion/Rate-Theory/rs1.html. [20] Martin, A. J. P., and R. L. M. Synge, Biochem. J., Vol. 35, 1941, p. 1358. [21] Schuler, M. L., and F. Kargi, Bioprocess Engineering: Basic Concepts, Upper Saddle River, NJ: Prentice-Hall, 2002. [22] Sarrut, N., S. Bouffet, F. Mittler, O. Constantin, P. Combette, J. Sudor, F. Ricoul, F. Vinet, J. Garin, and C. Vauchier, “Enzymatic Digestion and Liquid Chromatography in Micro-Pillar Reactors—Hydrodynamic Versus Electro-Osmotic Flow,” SPIE San Jose Photonics West, MOEMS-MEMS 2005, 2005. [23] Kennedy, M. J., S. J. Stelick, S. L. Perkins, Li Cao, and C. A. Blatt, “Hydrodynamic Focusing with a Microlithographic Manifold: Controlling the Vertical Position of a Focused Sample,” Microfluidics-Nanofluidics, Vol. 7, No. 4, 2009, pp. 569–578. [24] Hairer, G., and M. J. Vellekoop, “An Integrated Flow-Cell Full Sample Stream Control,” Microfluidics-Nanofluidics, Online advanced article. [25] Yamada, M., H. Nakashima, and M. Seki, “Pinched Flow Fractionation: Continuous Size Separation of Particles Utilizing a Laminar Flow Profile in a Pinched Microchannel,” Anal. Chem., Vol. 76, 2004, pp. 5465–5471. [26] Maenaka, H., M. Yamada, M. Yasuda, and M. Seki, “Continuous and Size-Dependent Sorting of Emulsion Droplets Using Hydrodynamics in Pinched Microchannels,” Langmuir, 2008. [27] Huang, L. R., E. C. Cox, R. H. Austin, and J. C. Sturm, “Continuous Particles Separation Through Deterministic Lateral Displacement,” Science, Vol. 304, 2004, pp. 987–990. [28] 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. [29] Inglis, D. W., J. A. Davis, R. H. Austin, and J. C. Sturm, “Critical Particle Size for Fractionation by Deterministic Lateral Displacement,” Lab on a Chip, Vol. 6, 2006, pp. 655–658. [30] Morton, K. J., K. Loutherback, D. W. Inglis, O. K. Tsui, J. C. Sturm, S. Y. Chou, and R. H. Austin, “Crossing Microfluidic Streamlines to Lyse, Label and Wash Cells,” Lab Chip, Vol. 8, 2008, pp. 1448–1453. [31] Rubinow, S. I., and J. B. Keller, “The Transverse Force on a Spinning Sphere Moving in a Viscous Fluid,” J. Fluid Mech., Vol. 11, 1961, pp. 447–459. [32] Leighton, D., and A. Acrivos, “The Lift on a Small Sphere Touching a Plane in the Presence of a Simple Shear Flow,” J. Applied Mathematics and Physics, Vol. 36, 1985, pp. 174–178. [33] Cherukat, P., and J. B. McLaughlin, “The Inertial Lift on a Rigid Sphere in a Linear Shear Flow Field Near a Flat Wall,” J. Fluid Mech., Vol. 263, 1994, pp. 1–8. [34] Park, J. -S., S. -H. Song, and H. -I. Jung, “Continuous Focusing of Microparticles Using Inertial Lift Force and Vorticity Via Multi-Orifice Channels,” Lab Chip, Vol. 9, 2009, pp. 939–948. [35] Davit, Y., and P. Peyla, “Intriguing Viscosity Effects in Confined Suspensions: A Numerical Study,” EPL, Vol. 83, No. 6, 2008, p. 64001. [36] Di Carlo, D., D. Irimia, R. G. Tompkins, and M. Toner, “Continuous Inertial Focusing, Ordering, and Separation of Particles In Microchannels,” PNAS, Vol. 104, No. 48, 2007, pp. 18892–18897. [37] Di Carlo, D., J. F. Edd, D. Irimia, R. G. Tompkins, and M. Toner, “Equilibrium Separation and Filtration of Particles Using Differential Inertial Focusing,” Anal. Chem., Vol. 80, No. 6, 2008, pp. 2204–2211. [38] Asgar, A., S. Bhagat, S. S. Kuntaegowdanahalli, and I. Papautsky, “Continuous Particle Separation in Spiral Microchannels Using Dean Flows and Differential Migration,” Lab. Chip, Vol. 8, 2008, pp. 1906–1914.

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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.

7.2 From the Principle of Biorecognition to the Development of Biochips

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.

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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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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.

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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.

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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.

7.3  Biochemical Reactions

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)

7.3  Biochemical Reactions

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).

318

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

320

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).

324

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).

326

Biochemical Reactions in Biochips

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.

328

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)

330

Biochemical Reactions in Biochips

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

x1 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 = 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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Biochemical Reactions in Biochips   [9] [10] [11] [12]

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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[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

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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

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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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Experimental Approaches to Microparticles-Based Assays

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* 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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Experimental Approaches to Microparticles-Based Assays

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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Experimental Approaches to Microparticles-Based Assays

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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Experimental Approaches to Microparticles-Based Assays

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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379

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)

384

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.

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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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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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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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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