Dielectrophoresis: Theory, Methodology and Biological Applications

By Ronald R. Pethig

Comprehensive coverage of the  basic theoretical concepts and applications of dielectrophoresis from a world-renowned expert.

• Features hot application topics including: Diagnostics, Cell-based Drug Discovery, Sensors for Biomedical Applications, Characterisation and Sorting of Stem Cells, Separation of Cancer Cells from Blood and Environmental Monitoring
• Focuses on those aspects of the theory and practice of dielectrophoresis concerned with characterizing and manipulating cells and other bioparticles such as bacteria, viruses, proteins and nucleic acids.
• Features the relevant chemical and biological concepts for those working in physics and engineering

• Language: English
• Publisher: Wiley
• Release date: Mar 2, 2017
• ISBN: 9781118671412

Book preview

Index of Worked Examples

1.1Dielectrophoretic Force Acting on a Cell at a Low Frequency

1.2Dielectrophoretic Force Acting on a Cell at a High Frequency

1.3EWOD – Voltage Control of Contact Angle

1.4Magnetophoretic Force Acting on Cells

1.5Magnetophoretic Velocity

1.6Magnetic Labelling of Cells

1.7Magnetic Labelling of Cells

2.1Net Charge on a Cell Membrane

2.2Electric Force Acting on a Cell

2.3Sedimentation Force Acting on a Cell

2.4Stokes' Law has Limited Applicability to Cell Electrophoresis

2.5Application of the Helmholtz–Smoluchowski Equation

2.6Magnitude of Dipole Moment Induced in a Cell

2.7Magnitude of the Field Factor (E·∇)E

2.8Magnitude of the Dielectrophoretic Force

3.1Lines of Force around Three Charges

3.2Electrostatic Force Calculation

3.3Electrostatic Vector Force Calculation

3.4Electric Field of a Point Charge

3.5Electric Field of a Line of Charge

3.6Electric Field of a Charged Ring and Circular Disk

3.7Electric Flux through a Small Surface Area

4.1Work Done and Electrical Potential Energy

4.2Work Done in Assembling a Distribution of Charges

4.3Potential at a Point in a Field

4.4Potential outside a Charged Metal Sphere

4.5Equipotential Surfaces around a Spherical Electrode

4.6Relationship between Electrode Surface Charge Density and the Potential Difference between Parallel Electrodes

4.7Potential close to a Long Linear Distribution of Charge

4.8Potential Close to a Charged Metal Rod

5.1Field outside a Charged Metal Rod

5.2A Plot of a Vector Field

5.3A Plot of the Gradient of a Potential Field

5.4Solving Laplace's Equation in Spherical Coordinates

5.5Hemispherical Electrodes

5.6Dipole Moment of an Assembly of Point Charges

5.7A Macroscopic (Classic) Dipole Moment

5.8Dipole Moment of a Cell with Induced Surface Charge

5.9Electric Field of a ‘Pure’ Dipole

7.1Polarizability of Solid Hydrogen

8.1Onsager's Equation and the Dielectric Dispersion for Water

8.2Electrical Conductivity of Pure Water

8.3Composition of an Electrolyte for a Dielectrophoresis Experiment

8.4Relative Permittivity of KCl Solution

8.5Preparing a 0.02 Molar Fraction of Glucose in Water

8.6Converting Dielectric Data from Molar Fractions to Molar Concentrations

8.7Calculating the Dipole Moment of a Protein from its β-Dispersion

8.8The Conductivity Increment accompanying the β-Dispersion for Cytochrome-c

8.9Magnitude of the Conductivity Increment for DNA Solutions

9.1Trans-Membrane Ion Transfer and the Membrane Potential

9.2Deriving the Volume Fraction of Cells in a Suspension using Hanai's Mixture Equation

10.1Estimation of the Peak Angular Velocity of Electrorotation

12.1Average Separation Distance and Mean Free Path Length of Gas Molecules

12.2Fluid Flow through a Channel of Reducing Cross Section

12.3Pressure Drop across a Fluid Constriction

12.4Lift Force Acting on an Object in a Flowing Fluid

12.5Fluid Flow Resistance of a DEP Chamber

12.6Applying Kirchhoff's Laws to the Fluidics of a DEP Device

Preface

Herb Pohl's seminal book, Dielectrophoresis: The Behavior of Neutral Matter in Nonuniform Electric Fields, was published in 1978. The aim of this present text is to describe the development since then of the theory and practice of this subject. The primary focus is on the biomedical applications of dielectrophoresis (DEP), so many of the chapters are written with a multidiscipliniary readership in mind. However, the theories and techniques described here are valid for all types of particles – animate and inanimate. The subject has changed dramatically since 1978. Up to that time only 16 scientific reports on biological applications of DEP had appeared in the scientific literature, with 12 of them describing work performed by Herb and his postgraduate students at Oklahoma State University. One of these papers deserves special mention, namely that written in 1966 with his MSc student, Ira Hawk. They describe, in the journal Science (vol. 152, 1966), the first demonstration of a purely physical technique (i.e., DEP) that can be used to distinguish and separate live and dead cells simultaneously. Furthermore, the live cells that had been exposed to the DEP field for several minutes were found to be viable and capable of cell culture. A macroscopic pin-plate electrode arrangement, composed of a rounded 0.66 mm stainless-steel wire facing a flat steel plate, was used in these experiments. Microfabrication and microfluidic techniques, taken for granted now in this subject, were not available to Herb in 1978. Apart from the impact of microtechnologies, this present book has also to take into account the fact that, at its time of completion (July 2016), more than 300 published papers are devoted solely to the DEP behaviour of yeast cells, with more than 3000 other papers of relevance to biomedical applications of DEP. Herb's initial interest in the motion of particles induced by nonuniform AC fields (an effect he was later to term dielectrophoresis) was directed towards industrial applications such as the removal of carbon-black filler from polyvinyl chloride samples. However, as I was privileged to witness at first hand, he gained most amusement from observing the DEP behaviour of bioparticles. In this way, Herb was able to describe in his book, in some detail, the DEP characterization of yeast cells and several types of bacteria, as well as preliminary results for blood cells, chloroplasts, green algae and mitochondria. These results act as the springboard for this book.

I suspect that I am not alone in finding more enjoyment in writing and reading about the biomedical applications of DEP than of its use to separate carbon black from PVC, or particulate matter from petroleum, for example. How can other such studies (potentially important as they may be) induce the same ‘buzz’ as viewing the geometrical distinction between life and death in the form of the Argand plots shown in Figure 11.9 of this book? Can studies of inanimate particles be as amusing as observing viable Giardia rotating in the opposite sense to nonviable ones in a rotating electric field? Such entertainment will not occur with particles extracted from oil, for example, unless they are bacteria such as oil-eating Alcanovorax. This explains, in part, why this present text is restricted to the DEP behaviour of biological particles. An exception is the inclusion of polymer beads because they are used widely in biomedical and biosensor devices, with DEP able to monitor the extent of attachment to them of target bioparticles. There is also a pragmatic reason for focussing on biomedical applications of DEP. A search in the autumn of 2015, using the Web of Science Core Collection and other library data bases, revealed the existence of at least 4000 publications on the theory, technology and application of DEP. Of relevance to the subject matter of this present text are also the many hundreds of scientific papers on the theories of dielectric phenomena, as well as those that describe the dielectric and electrokinetic properties of cells, bacteria, viruses together with bio-macromolecules such as proteins and nucleic acids. By largely excluding conference abstracts for possible citation, as well as papers not addressing a bio-related topic or not readily available through normal library resources, the number of candidates for citation was reduced to around 3000 publications. To avoid the text assuming the character of a list of disjointed citations, an attempt has been made to summarize the development of bio-DEP over the past half-century through only around 800 references to relevant work. This does not completely mirror important contributions to the subject made by my own co-workers and many researchers from other laboratories. I apologize to those who inspect the index of cited authors and are disappointed to find that their innovative work has either not been described adequately or is not cited at all. Among past colleagues not cited at all is John Morgan, who submitted his PhD thesis ‘Dielectrophoretic Studies of Biological Materials’ in 1978, whilst for Paul Carnochan only one image (Figure 11.2) from his PhD thesis ‘Dielectric Properties of Biological Cell Suspensions’, submitted in 1982, records his valuable contribution. An omission of work from the citation index does not reflect its perceived lack of novelty or importance – it has simply suffered from the culling exercise performed for reasons explained above (or from an unfortunate oversight on my part).

Another objective of this book is to make large parts of its content agreeably accessible to those trained in the biomedical sciences – not just engineering and physical science graduates. For those engaged in biomedical applications of DEP, the guidance and involvement of those trained in the molecular and life sciences is greatly desired and in most cases can be considered as essential. However, most published works on DEP appear in journals of engineering or the physical sciences and are largely unhelpful in addressing the ‘so what, who cares?’ questions of interest and relevance to those trained in the life and medical sciences. Chapter 1 addresses a common question about how the technique of DEP can compete against other microfluidic methods for cell manipulation and separation, such as flow cytometry, electrophoresis and magnetophoresis. Electrophoresis is a method well understood by biologists, but its similarity to the term dielectrophoresis is not helpful in discouraging the impression that DEP represents no more than an esoteric extension of what they already know. The purpose of Chapter 2 is to describe, in broad terms, how the special features of DEP lend to it the promise of providing important contributions to cell biology, particularly to such areas as drug discovery, medical diagnostics and regenerative medicine. As already stated, bearing in mind that an increasing number of scientists trained in the biomedical fields are entering the subject area, the nontheoretical sections of the text, throughout this book, are written in a style that is hopefully suitable for an interdisciplinary readership. To assist this and to help maintain the narrative, separate boxes and worked examples are used throughout the book to act as pedagogical material and to divert the more formal and quantitative details away from the main text.

In the preface of a special issue of the Journal of Electrostatics (Vol. 21, 119–364, 1988) to honour the memory of Herb Pohl, I mentioned that his devotion to science and generous nature had once been revealed to me by his statement that ‘senior scientists should act rather as oak trees, to give shelter and provide growing conditions to the younger ones’. I also suggested that he would have gained much satisfaction and pleasure to see how some of his acorns had matured. In this spirit, I wish to take this opportunity to thank and acknowledge the contributions that the following, as young researchers at Bangor, made to my own understanding of DEP and to the content of this book: Talal Al-Ameen, W. Michael Arnold, Julian P. H. Burt, Paul Carnochan, Ka-Lok Chan, Colin Dalton, Peter R. C. Gascoyne, Andrew D. Goater, Clair Hodgson, Michael P. Hughes, Ying Huang, Richard S. Lee, Gary M. Lock, Zu-Hong Lu, Gerard H. Markx, Anoop Menachery, Hywel Morgan, John R. Morgan, Jonathan A. R. Price, Mark S. Talary, Xiao-Bo Wang and Xiao-Feng Zhou. It is with some pride that I know the DEP community will recognize the names of some fine oak trees in this list. We benefited from having the following with us on sabbatical leave or year-long fellowships: Ralph Hölzel, Takashi Inoue, Thomas B. Jones, Juliette Rousselet, Miguel Sancho, Herman P. Schwan and Junya Suehiro. A special mention should be given to John Tame, who operated the photolithography and clean-room facilities at Bangor. In the summer of 1986 he was asked if he could fabricate for us an array of gold, interdigitated, microelectrodes on a microscope slide. After being informed what we intended to do with it, he impishly responded: We usually keep our electronic devices away from water, but I'll give it a try. For nearly 20 years thereafter (until his untimely death in 2004) he provided various microelectrode arrays for the DEP and electrokinetic studies of the researchers mentioned above. In 2005 a new clean-room facility at Bangor was named and dedicated to his memory.

At the School of Engineering in Edinburgh I have appreciated moral support and helpful interactions with Professors Alan Murray, Ian Underwood, Anthony Walton, as well as Drs Andrew Downes, Stewart Smith, Adam Stokes and Jon Terry. I enjoyed working with Colin Chung, Massimo Muratore and Srinivas Velugotla during their PhD research projects. I also acknowledge very fruitful research interactions at Edinburgh with Dr Paul de Sousa of the Centre for Clinical Brain Sciences, Dr Steve Pells of the MRC Centre for Regenerative Medicine and Professor Christopher D. Gregory of the MRC Centre for Inflammation Research. Chris Gregory deserves special mention – with the objective of making the text accessible to biologists he undertook the herculean task of reading many of the draft chapters, cleaning out much bio-unfriendly material. His research is directed towards understanding how apoptotic cancer cells condition their microenvironment. His input was thus especially valuable for those sections of Chapter 11 dealing with cell death – in so doing he performed what was not even required of Hercules, namely to replace the removed material. During the writing of this book I have appreciated valuable e-mail exchanges with Massimo Camarda, Cesare Cametti, Rodrigo Martinez-Duarte, Nic Green, Mike Hughes, Ralph Hölzel, Hywel Morgan and Joel Voldman. Special thanks also go to Anne Parkinson for translating Mossotti's paper of 1850 into English and so helping to clarify certain aspects of Chapter 6; Professor Peter Sarre of Nottingham University for facilitating my access to that university's original copy of George Green's Essay. I have greatly appreciated the efforts of the international team working for John Wiley & Sons, especially the meticulous attention to detail given by David Michael in copyediting the manuscript.

Those with sharp eyes might notice a ‘smiley face’ in Chapter 6. This is by way of a personal tribute to Herb Pohl, who in 1975 introduced me to this symbol and how to use it. Finally, the last lines on the dust cover of Herb Pohl's book of 1978 read: A far wider range of potential applications exists than Professor Pohl has been able to include. The book should thus provide stimulating reading for imaginative research workers in the physical, medical and biological sciences. It is probably pushing an ambition too far – but hopefully this present effort goes part way to achieving the same sentiment.

The electronic supplemental content to support use of this text is available online at http://booksupport.wiley.com.

Ronald Pethig

Edinburgh

Nomenclature

1

Placing Dielectrophoresis into Context as a Particle Manipulator

1.1 Introduction

For those interested in etymology, deciphering the origin and hence probable meaning of compound words ending in phoresis is relatively straightforward. Based on Greek translation, such compound words imply something to do with ‘carrying things around’. For example, stating that electrophoresis describes an object being carried (i.e., moved) by an electrical effect is therefore an acceptable definition. For a particle to be set into motion requires the imposition on it of an external force. An example is the buoyancy force acting on a particle suspended in a fluid – the particle will either sink or rise under the action of gravity, depending on whether its specific density is greater or less than that of the surrounding fluid. If the particle finds itself in a flowing fluid, it will also experience a viscous drag force and be accelerated to the speed of the local fluid flow. The particle can be solid or take the form of a fluid droplet or gas bubble. This book's focus is the use of dielectrophoresis as a means to spatially manipulate bioparticles such as cells, bacteria, viruses, proteins and nucleic acids. In May 2013, in the United States, two sessions were devoted to this subject at an international conference on Advances in Microfluidics and Nanofluidics. In the flyer that promoted the conference, it was stated:

As dielectrophoresis (DEP) is arguably one of the fundamental pillars of microfluidic manipulation and given the continued advances in this mature field, we will be organising special sessions on dielectrophoresis with the aim of promoting interaction between researchers that work on fundamentals and applications of DEP across different communities and disciplines.

Various methods can be used to manipulate particles in a microfluidic device, so what justification is there to state that dielectrophoresis can be singled out as ‘one of the fundamental pillars’? Why is DEP considered an important topic for a conference on microfluidics and nanofluidics? Why devote a whole book to the subject? Some answers are provided in this chapter by reviewing those forces that can be used to manipulate bioparticles in microfluidic devices. It is not intended as a comprehensive review, but covers sufficient ground to set dielectrophoresis into context and highlight some of its special features and advantages. Bearing in mind that an increasing number of scientists trained in the biomedical fields are entering the subject area, the text is written in a style intended to be suitable for an interdisciplinary readership. To help maintain the word flow, boxes and worked examples are used in this chapter (and throughout the book) to divert the more formal and quantitative details away from the main text.

Dielectrophoresis is the induced motion of a particle when it is placed in an electric field gradient. In Chapter 2, we find that one advantage of this method is that it scales favourably with a reduction in dimensions of the electrodes used to generate the electric field. It is therefore ideally suited for applications in microfluidic devices designed to perform, for example, as an electronically controllable ‘laboratory on a chip’ or ‘micro-total analysis’ system. Although the terms are often used interchangeably, lab on chip is used to describe devices that integrate several laboratory processes, whereas micro-total analysis systems are considered to integrate all laboratory processes required for an analysis. For both cases, fluid flow in one or more channel networks, fabricated into or from a single solid substrate, is an essential element of the analytical or preparative function of the device [1–6]. It is also generally accepted that to qualify as a microfluidic device, at least one of its fluidic dimensions should be in the range 1 μm ∼1 mm.

The fundamental features and potential advantages of using microfluidic devices for biomedical assays and processes will now be outlined.

1.2 Characteristics of Micro-Scale Physics

A simple form of a microfluidic device would be, for example, a channel etched into a glass substrate of length 1 cm and internal cross section 10 μm × 10 μm, equipped with an inlet and outlet fluid port. One envisaged application of such a simple structure would be to study how thrombocytes (platelets) in a flowing fluid interact with immobilized proteins. The proteins can be immobilized by coating them onto the internal surfaces of the channel. With our specified dimensions the channel has an internal volume of 10−9 dm³ (1 nL). One small droplet of water that leaks from a tap has a volume about 20 000 times larger than this! Physical effects or forces, such as surface tension that controls the size of a water droplet, may play relatively minor roles in our macro-scale world of activity, but can dominate in microfluidic devices. The ability to accommodate such forces, either by minimizing their disruptive effects or using them to advantage, is an important aspect of the design and operation of a microfluidic device.

The following are practical examples of dominant physical phenomena at the micro scale:

Microfluidic devices tend to have a large ratio of their surface area to volume. Consider a spherical chamber of radius R. This has a surface area of 4πR² and a volume of (4πR³)/3. The ratio of these two parameters is 3/R. Therefore, as the radius R decreases the ratio of surface area to volume increases. For example, a 10 dm × 10 dm × 10 dm cube has a surface-to-volume ratio of 40 m−1, whereas for the 1 cm × 10 μm × 10 μm channel considered above, this ratio increases to 4 × 10⁵ m−1. Scaling down the dimensions of a fluidic device thus provides the opportunity for suspended particles to interact with a large surface area. This can represent a desired outcome, as in the study of platelet-protein interactions, or lead to an undesirable result such as the adventitious adherence of cells to the internal walls of narrow-bore tubing.

In micro devices, capillary action and other surface energy effects can be greater than gravitational forces. This can result in an upward or transverse fluid movement, or even block downward fluid flow in a capillary.

A small drop of fluid placed in the inlet of a microfluidic device can evaporate very rapidly.

Fluids that are brought together in a microfluidic circuit do not mix easily. Any mixing that does occur arises mainly from the diffusion of solutes across the boundaries between separate laminar flows of fluid.

Solute particles that are heavier than the surrounding fluid settle to the channel bottom very quickly.

Small fluid volumes will almost immediately take on the temperature of the environment and cool down or heat up very quickly.

1.2.1 Exploiting Micro-Scale Physics

Some of the physical effects outlined above may be undesirable in the design and operation of microfluidic devices. However, they can also be exploited as powerful tools. Examples of such physical effects and their advantages include:

Fluid flow in microchannels is almost always laminar, characterized by the parallel flow of the individual lamellar elements of the fluid (see Chapter 12, section 12.4.5). The flow has a parabolic velocity profile across a channel, with zero velocity at each channel wall and a maximum velocity at the centre of the channel. These characteristics can be turned to advantage in the design of particle separation devices, where an external force drives target particles into different parts of the fluid velocity profile or across the boundary between adjacent fluid streams.

A large surface-to-volume ratio provides an intrinsic compatibility between the use of a microfluidic system and surface-based assays.

At micro dimensions, diffusion becomes a viable approach to move particles, mix fluids and control reaction rates. Small drug molecules, for example, can diffuse at rates of ∼10 mm/s at 25 °C in aqueous solutions. This allows the establishment of controlled concentration gradients in flowing systems, as well as rapid and complete equilibration of small molecular weight particles across a microchannel. Relatively fast reaction times are therefore possible when molecular diffusion lengths are of the order of the microchannel dimensions.

Unaided by centrifugation, sedimentation becomes a viable means to separate dispersed particles by density across small channel dimensions. For example, red blood cells will sediment in a 100 μm deep channel in about 1 minute and generate a 50 μm layer of plasma in the process.

Evaporation of small quantities of fluids can be extremely rapid because of a typically large surface-to-volume ratio. This effect can be used for the concentration of suspended particles.

The energy associated with surface tension can be used to drive liquids through microfluidic devices. By chemically treating the surfaces of a microchannel to be hydrophilic, water will be driven through the channel without any applied pressure. This flow is driven by the attractive energy between the water and the channel wall surface. Surface tension effects can be controlled electrically using the technique known as electrowetting on dielectric (EWOD) described later in this chapter.

It is possible to design passive fluidic devices that utilize inherent properties of the fluid and its microenvironment (e.g., capillary force, evaporation, heat transfer, diffusion) for fluid movement, mixing, heating, cooling and catalysing chemical reactions. Thus, disposable stand-alone devices can be designed that require no external power source or instrumentation, yet still perform many, if not all, of the functions typically associated with full-scale automated chemical analysis devices containing pumps, mixers and heating elements.

Other advantageous characteristics of microfluidic devices are associated with economic considerations. For example, polymer-based microfluidic structures can be mass produced at very low unit cost, allowing them to be disposable. Micro devices require only small volumes of sample and reagents (down to picolitres) and produce only small amounts of waste. They are also amenable to high throughput by processing multiple samples and assays in parallel.

1.3 Microfluidic Manipulation and Separation of Particles

A critical action for many biological and medical diagnostic procedures is the selective manipulation and separation of particles. By manipulation we mean the relocation of a particle with respect to its position within a fluidic device or to that of neighbouring particles. An extension of this is particle separation, which implies the physical isolation within or outside of a device of the target particles from a mixture of different particle types.

1.3.1 Defining the Performance of Cell Manipulators and Separators

The performance of a particle manipulator can be measured in terms of how quickly, precisely, reproducibly and how many target particles can be relocated to a specific site or sites. The concept of a particle separation device is quite straightforward – the input is a heterogeneous mixture of particles and the output consists of target particles that can be collected or totally isolated from the unwanted particles. The performance of such a device is often given in terms of its throughput. Throughput can be expressed in terms of either the volumetric flow rate that can be handled by the device (e.g., mL/s) or the number of particles that can be processed in a given time (e.g., 100 000 cells per second). However, we also want to know how well the device performs as a separator. If for every 100 000 cells in a sample there are 100 target cells, can it deliver 100 target cells per second with no contamination from unwanted cells? The language employed to evaluate this is not as straightforward as defining its throughput. For example, suppose we have cells that have been brought back to physiological temperature after a period of cryopreservation. It is common experience to find that many of these cells will have suffered as a result of being frozen for a long time and will be nonviable (dead). We will want our cell separation procedure to provide us with as many viable cells as possible and very few dead ones. An evaluation of this is variously called the recovery rate, the target capture efficiency or the yield, which should be close to 100%. If we define the viable, live, cells as the target cells and the dead ones as the unwanted cells, the yield (or recovery rate, capture efficiency) can be defined as the ratio of the number of viable cells collected at the output to the total number of viable cells contained in the original cryopreserved sample fed into the separator:

numbered Display Equation

We can determine the yield parameter by performing viability assays for the input and output samples and use this to compare the use of new buffers and cryopreservation procedures designed to maintain cell viability. However, if we are using a cell separator to isolate cancer cells from peripheral blood (to assess chemotherapy treatment, for example) we have (as yet) no accurate method to determine the number of cancer cells that exist in blood samples taken from patients. In this case, the use of terms such as yield, recovery rate or capture efficiency cannot be used to describe the performance of a separator used to extract cancer cells from blood. However, if a known number of cancer cells are ‘spiked’ into a blood sample of defined volume, it is possible to determine the yield of a procedure to isolate them.

When using a separator to isolate specific cells for therapy, an important parameter to define is the purity of the output. This can be defined as the fractional content of target cells in the output sample:

numbered Display Equation

Flow cytometry can be used to determine the concentration of the target cells and the total cell count. In some cases a cell separator is used to increase the percentage of target cells in a heterogeneous cell population prior to PCR analysis, for example. This is referred to as enrichment and is quantified using the following relationship:

numbered Display Equation

The subtle relationship between enrichment, yield (or recovery rate, capture efficiency) and purity can be obtained by combining the relationships given above for yield and enrichment, to give:

numbered Display Equation

1.4 Candidate Forces for Microfluidic Applications

Some forces scale down well for applications in microfluidic devices and others do not. For example, centrifugation is used to separate particles based on differences in their size and density. A centrifugal force occurs as a reaction to centripetal acceleration (Rω²) acting on a mass moving with angular velocity ω (radians / s) along a path of radius of curvature R. This does not scale favourably with miniaturization, because with all other factors remaining constant a reduction of R results in a reduction of the centrifugal force. Another factor that does not favour the incorporation of centrifugation into lab-on-chip devices is that it requires a rotating component. To subject a cell suspension to the relatively gentle centrifugal acceleration of 100 times the gravitational acceleration (100 g) requires that the sample is positioned at the end of an 18 cm long arm that is rotating at ∼1000 revs/min. This rotation rate has to be increased a hundredfold if we wish to maintain the centrifugal force but reduce the rotor arm length to 18 μm for example.

Examples of forces that do scale well with miniaturization include those that induce electrophoresis and dielectrophoresis. These effects become more dominant as the applied electric field is increased and for a fixed voltage applied across an electrode pair the field increases with reduction of the spacing between them. Indeed, using microelectrodes and modest applied voltages of 10 V or less, electric fields as large as 10⁶ V/m can be generated, together with useful associated electrokinetic forces that are difficult to achieve at a macroscale. As the dimensions of a fluidic system, such as channel width and height, decrease the influences of laminar flow increase and can be exploited in various ways. The forces that can usefully be scaled down to operate in microfluidic devices are given in Table 1.1. Examples of how each one of these forces can be used to selectively manipulate and separate cells will now be outlined:

Table 1.1 Forces used to manipulate and separate cells in microfluidic devices.

Notes: aFACS: fluorescence activated cell sorting; bMACS: magnetic activated cell sorting.

1.4.1 Mechanical

Particles can be selectively separated according to their size by mechanical filtration or sieving. The particles are suspended in a fluid (in which they do not dissolve) and are then flowed through microstructured perforations or constrictions. Membranes composed of pure nylon fibres or polycarbonate containing pores of precise diameter have been widely used to separate blood cells by size and deformability. Such membranes can easily clog, so that the recovery of target cells from them is not straightforward. For such reasons efforts have been directed towards replacing conventional membranes with physical structures such as weirs and arrays of microposts or pillars that are built into fluidic channels. In a weir structure the microchannel takes a sudden reduction of cross section. Particles that are too big to pass through such structures are retained. In some cases, especially at high fluid flow rates, the deformability of a particle determines whether it is captured or not. Particle deformity is an important consideration for the selective retention of blood cells because, as shown in Table 1.2, there are overlaps of the size distributions of the various blood cells.

Table 1.2 The volume (fL = 10−15 L), diameter, density, concentration ranges and properties of cells in normal adult human blood.

Sources: Bain, B. J. (1995) Blood Cells: A Practical Guide. 2 edn. Blackwell Science, Osney Mead. De Waele, M., Foulton, W. and Renmans, W. (1988) Hematologic values and lymphocyte subsets in fetal blood. Amer. J. Clin. Prac. 89, 742–746. Lynch, D. C., Yates, A. P. and Watts, M. J. (1996) Haematology, Churchill Livingstone, New York, NY.

Notes: Cell counts for adults vary due to demographic factors (e.g., sex, age, ethnic origin and geographical location) and biological factors (e.g., diurnal variation, pregnancy, menstruation, menopause, exercise, cigarette smoking, alcohol intake). About 70% of the lymphocytes are T cells (approximately two-thirds CD4 and one-third CD8), 5–10% are B cells and the remainder are non-T, non-B-cells.

An informative investigation of the mechanical filtration of erythrocytes (red blood cells) from leukocytes (white blood cells) in whole blood was performed by Wilding et al. [7]. Blood was passed between microposts or over weirs etched into a silicon substrate in a chamber capped with a glass top. This is shown schematically in Figure 1.1. The objective was to demonstrate that isolation of leukocytes from the erythrocytes, followed by the polymerase chain reaction (PCR) for the DNA released directly from the trapped leukocytes, could be performed as sequential processes in a single microfluidic chamber. Removal of the erythrocytes from the whole blood sample was required because haemoglobin protein molecules that can leak from them inhibit the PCR process.

Image described by the caption

Figure 1.1 Schematic of a weir-type microfilter used to separate blood cells. A small gap between the top of the weir and a glass cover plate provides active filtration of cells based on size and deformability. In this example white blood cells (WBCs) are trapped on the weir whereas red blood cells (RBCs) flow freely over it. (Based on Wilding et al. [7].)

Wilding et al. [7] found that sieving of blood cells was influenced by several factors, namely: the deformability of the cells; their concentrations; the pressure applied to produce the fluid flow; the viscosity of the fluid; and the physical gap between microposts and above the silicon weirs. Erythrocytes readily passed through gaps as small as 3 μm, whereas the larger leukocytes (diameters in excess of 15 μm) could only squeeze through gaps larger than 7 μm. The filtration mechanism shown in Figure 1.1 was presumed to rely on trapping the leukocytes in the narrow gap between the top of the silicon weir and the Pyrex glass cover, but cell adhesion may also have played a role.

In other studies, Mohamed et al. [8] demonstrated the use of a micromachined silicon device for separating foetal cells from maternal blood, based on differences in cell size and deformability. The device consisted of four sections of successively narrower channels along the flow axis. These channels did not take the form of continuous structures with side walls, but as a series of pillars. In total the device contained more than three million of such ‘channels’. This design allowed the cells to deform and recover as they passed between channels and to migrate around regions where the cell flow was locally hindered or clogged. The nucleated foetal erythrocytes, ranging in diameter from 9 to 12 μm, could deform and pass through a channel as small as 2.5 μm wide and 5 μm deep. The larger leukocytes, ranging in diameter from 10 to 20 μm, could not deform to the same extent and were retained by the 2.5 μm wide and 5 μm deep channels. Later studies, using the same filtering device, demonstrated that cultured cancer cells spiked into whole blood could be recovered, based solely on their size and deformability [9].

1.4.2 Hydrodynamic

1.4.2.1 Basic Principles

The separation of particles using hydrodynamics often relies on the principle that macroscopic particles subjected to viscous drag forces in laminar fluid flow will stay within their fluid streamlines. An extreme example of this is the way that the coloured strands of a certain brand of toothpaste remain in place and do not mix together as the paste is squeezed slowly from its tube (see Figure 12.9). We can classify this as a deterministic effect – the individual streamlines of paste will flow in a predictable way. If particles within a streamline are small enough to be buffeted about by the thermally induced motions of the fluid's molecules, we have a stochastic process because they will diffuse in a random manner across adjacent fluid streamlines. In a nondeterministic, stochastic regime we are unable to separate particles according to the principles described in this section.

All fluid flow, whether in a channel or around an object, can be broadly classified as either laminar or turbulent. As described in Chapter 12 (12.4.5), which of these fluid flow conditions is dominant depends on the relative importance of the inertial forces versus viscous shear forces in the flow. An inertial force is the concept we use to understand the principle of inertia as embodied in Newton's First Law of Motion (an object not subject to any net external force moves at a constant velocity). In fluidics we can relate this to the translational momentum (mass × velocity) of a unit volume of a fluid element. This is given by the product (ρv) of the fluid density ρ and the bulk velocity v of the fluid flow. A fluid element can also have rotational inertia, which refers to the fact that the angular momentum of the fluid element will remain unchanged unless an external torque is applied. The shear forces that act to damp out translational and angular momentum of the fluid are viscous in nature. They occur at the channel walls and between fluid streamlines. The ratio of the inertial forces and viscous shear forces is a dimensionless parameter, known as the channel Reynolds number Re, given by the relationship:

(1.1) numbered Display Equation

in which η is the fluid's dynamic viscosity and the parameter L is the effective wetted (hydraulic) diameter of the channel. For low values of Re the viscous damping caused by shear at the channel walls and between fluid streamlines quickly removes translational and rotational kinetic energy from a fluid element and the flow is laminar. Laminar fluid flow is characterized by a parabolic velocity profile (see Figure 1.3). As a rough guide, described in more detail in Chapter 12, for a Reynolds number above ∼1000, the shear between streamlines is unable to dampen out the inertia of transverse and rotational fluid motions. As a result, the laminar streamline structure is destroyed and the fluid flow becomes turbulent throughout the channel. For aqueous fluids we have ρ ∼10³ kg m−3 and η ∼10−3 Pa s, so that the factor (ρ/η) in Equation (1.1) has a value of ∼10⁶ m−2 s. In microfluidic devices we typically have flow velocities much less than 1 cm/s and L rarely exceeds 10 cm. In Equation (1.1) we therefore have vL < 10−3 m²/s, giving us a Reynolds number of less than 1000. Unless a very high fluid flow velocity is achieved (driven by high pressure in a channel fabricated to withstand such pressure) it is not possible to induce high Reynolds number conditions. Therefore, as a working rule, we can assume that flow in a microfluidic device is laminar. In such flow the fluid stream velocity is zero at the boundary layer next to a channel wall or a wetted object's surface and increases with distance away from such boundaries.

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Figure 1.2 Schematic of the Zweifach–Fung effect where a particle with its centroid on the critical streamline is directed to the high flow rate channel in a bifurcation. (Yang et al. [13] reproduced with permission of the Chemical Society.)

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Figure 1.3 (a) Particles of various sizes are shown close to the wall of a channel that sustains a pressure-driven laminar flow of fluid. The parabolic profile of the fluid velocity results in the larger particles moving more rapidly and eluting from the channel more quickly than the smaller particles. This is the basis of steric field flow fractionation [14–16]. (b) If the fluid flow rate into a side channel is sufficiently low, only the smaller particles contained within the fluid flowing near the sidewall will be withdrawn into it from the main channel [17]. (c) Particles can be separated according to their size by a spreading flow profile at the exit of a pinched section of channel joining two flow streams. The extent of particle separation is controlled by adjusting the flow rates in the two inlets to the pinched section. (Based on Yamada et al. [19].)

Equation (1.1) defines the Reynolds number for the fluid flow inside a channel in the absence of suspended particles or fixed objects such as posts. We can define a Reynolds number Rp that includes details of a particle by multiplying Re by the dimensionless parameter (R²/L²) where R is the radius of the particle:

(1.2) numbered Display Equation

For low values of the particle's Reynolds number (Rp ≪ 1) viscous drag of the fluid will act over its surface and accelerate it to the local velocity of a laminar fluid streamline. For Rp ∼ 1 inertial forces can lift the particle away from a channel wall and also cause it to cross fluid streamlines. If the walls of the fluid channel are not straight but curved, rotational flow called Dean flow [10], caused by fluid vortices induced by the channel curvature, can also cause particles to cross streamlines and alter their position in the channel. This effect is characterized by the Dean number De:

(1.3) numbered Display Equation

In Equation (1.3), δ is the ratio L/2Rc, with Rc being the radius of curvature of the channel of hydraulic diameter L. The position of a particle in a curved channel can be influenced by drag associated with Dean flow when De > 1. It has also been demonstrated that particles can be separated according to their size using slanted obstacles in a channel [11, 12]. Transverse flow streams perpendicular to the direction of the main fluid flow result from an anisotropic fluidic resistance of the slanted obstacles. Their top or bottom areas represent a higher flow resistance than their side areas. Particles subjected to the lateral pressure gradients induced by this flow resistance anisotropy are moved from the one sidewall of a channel to the other sidewall. This effect has been coined ‘hydrophoresis’ and is defined as the movement of suspended particles under the influence of a microstructure-induced pressure field [11].

Whether or not a particle (defined by its centroid) follows a fluid flow stream depends also on the rate at which it can diffuse in that fluid. The constant thermally induced motions of molecules in fluids ensures that when one fluid stream is placed adjacent to another one, as in laminar flow, its molecules percolate between flow streams in a process called diffusion. It is convenient to separate the actual diffusion process into two conceptual transport mechanisms: a molecular process modelled as a statistical random walk that is proportional to the degree of kinetic energy in the system and an advective process in which molecules are carried along by the average velocity of the flow. The common practice is to restrict the word diffusion to describe the first process and label the second process advection (convection if heat is being transferred). As described in Chapter 12 the relative importance of these two conceptual transport mechanisms is given by the Peclet Number Pe, the ratio of advection and diffusion:

(1.4) numbered Display Equation

where D is the particle's diffusion coefficient. If Pe < 1 the diffusion rate is smaller than the advection rate over the characteristic distance L. The flow is deterministic because a particle is confined to a fluid streamline. Cells suspended in aqueous fluids exhibit D values of the order 10−15∼10−16 m²/s (see Table 12.3). From Equation (1.4) such values of the diffusion coefficient ensure that when working with cells Pe will greatly exceed 1.0 in any practical microfluidic device. Thus, in low Reynolds number fluid flow and in the absence of an externally applied force cells will remain within their fluid streamlines.

1.4.2.2 Practical Examples of Applying Hydrodynamic Forces

When a microchannel splits into two channels that carry different fluid flow rates, the suspended particles will follow different streamlines depending on their locations in the channel leading up to the bifurcation. This is governed by the ratio of the fluid flow rates in the bifurcations and the difference in the shear force that acts on the surface of a cell when near to the channel wall compared to that when it is in the middle of the channel. This shear difference contributes to a lift force that pushes a cell away from direct contact with the channel wall. This is known as the Bifurcation Law or Zweifach–Fung effect, depicted in Figure 1.2 and has been explored by Yang et al. [13] as a method to separate blood plasma from blood cells.

As shown in Figure 1.3(a), in the situation where particles in a laminar fluid flow are forced close to a channel wall, the larger particles will have a larger velocity than the smaller ones. The larger particles will on average be in the faster flowing streamlines because their centroids cannot be closer to the wall than their radii. This is the basis of the various forms of field flow fractionation (FFF) pioneered by Giddings [14, 15] and extensively reviewed by Roda et al. [16]. In FFF a force (field) is applied perpendicularly to the parabolic flow to drive the particles (or analytes) to a channel wall or into different laminar flows due to differences in their size, density and other physical features such as shape, rigidity and surface properties. The applied field can be a centrifugal force or electrical field, for example. Because the larger particles will be swept downstream more rapidly they can be selectively collected from the fluid exit of the channel before the smaller particles are eluted. This mode of elution is referred to as steric or hyperlayer FFF, respectively, depending on whether the particles are all brought to a wall of the channel or distributed into different laminar streams, respectively. For submicron particles, where diffusion down their concentration gradient becomes a dominant process, the elution order of analyte size and mass is reversed. This is known as the normal elution mode.

Yamada et al. [17, 18] have exploited the properties of laminar flow for the size-selective filtration of leukocytes from blood and size-dependent separation of liver cells. The operating principle is outlined in Figure 1.3(b), which shows a narrow side channel branching off from a broad main channel. The fluid flow rates in the main and side channel will depend on their effective fluidic resistance values and the pressures applied to drive them. When the relative flow rate into the side channel is sufficiently low, only the fluid stream near the main channel wall will be withdrawn into the side channel. As shown in Figure 1.3(b), in this flow state particles whose diameters are larger than a certain value will not enter the side channel, even if they are flowing near the wall of the main channel and have a diameter smaller than the cross sectional size of the side channel. Depending on their size, shape and other factors such as surface roughness, the particles will also experience lift forces that push them away from the channel wall. This will influence the selective filtration of the particles.

Yamada et al. [19] also introduced the concept of pinched flow fractionation (PFF) outlined in Figure 1.3(c). A liquid containing the particles to be separated by size is continuously introduced into another channel containing particle-free fluid. By adjusting the flow rates of these two fluids, the particles can be restricted to flow at one sidewall of a ‘pinched’ section of the fluid channel. At the end of the pinched section, where the flow velocity profile spreads out, the larger particles are directed toward the centre of the channel and the smaller particles are directed within their slower moving stream lines towards the sidewall. Consequently, slight differences of particle elevations in the pinched channel section are significantly amplified in the broadened outlet. Particles are separated according to their size by deterministic lateral displacement in the laminar flow.

Davis et al. [20] have described an interesting version of size-dependent particle separation by deterministic lateral displacement in a process they refer to as bumping. As depicted in Figure 1.4, the particles flow through an array of microposts. Each row of posts is offset laterally with respect to the preceding row. Particles below a critical diameter follow streamlines cyclically through the gaps, moving in an average forward flow direction. Particles above this critical diameter cannot follow such a streamline and are ‘bumped’ by hydrodynamic lateral drag into the sequential streamline at each post. Thus, such particles do not move parallel to the fluid flow but at an angle determined by the ratio of post offset to row-row spacing. Davis et al. have demonstrated that this procedure can fractionate whole blood by separating the erythrocytes from the leukocytes and allow them to be collected in separate fluid exit ports [20]. By modifying this method of deterministic lateral displacement, Holm et al. [21] were able to separate from human blood the living parasites (trypanosomes) that cause sleeping sickness.

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Figure 1.4 The size-dependent separation by deterministic displacement of particles flowing through an array of microposts. Small particles stay within a flow stream that skirts the surface of posts in adjacent rows, whereas large particles are displaced laterally at each post. The extent of lateral separation of the particles is determined by the centre-to-centre post separation, the diameter of the posts, and the relative shift of the post centres in adjacent rows (based on Davis et al. [20]).

As the rate of fluid flow is increased in curved channels and the particle size is increased relative to the channel diameter, the particle Reynolds number given by Equation (1.2) increases so that inertial effects can become significant. Interesting examples of this have been demonstrated by Di Carlo et al. [22] for the flow of particles in curved channels. The superposition of lift forces from the channel walls with centrifugal forces arising from the fluid and particles was observed to induce precise ordering of initially scattered particles both longitudinally along the direction of fluid flow and laterally across the channel. This inertial self-ordering effect is shown schematically in Figure 1.5. A noteworthy application of the combined effects of inertial forces and Dean flow is a spiral microfluidic device, shown in Figure 1.5, for separating asynchronous mammalian cell lines according to their cell cycle [23]. This was achieved by exploiting the relationship between cell diameter and cell cycle and provided enriched subpopulations of viable cells in the G0/G1, S and G2/M phases. A comprehensive theoretical study and modelling of inertial focusing dynamics in spiral microchannels has been reported by Martel and Toner [24]. They conclude that the rich variety of inertial focusing dynamics observed in curved channels offers the potential of wide applications and advantages for future generations of microfluidic devices. Further study is also required to elucidate the underlying physical mechanisms and their associated limitations.

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Figure 1.5 (a) The continuous inertial focusing, ordering and separation of particles can be achieved by increasing the particle Reynolds number in a microchannel designed to have asymmetrically curved walls. The combination of lift forces at the channel walls and centrifugal forces acting on the particles and fluid can result in both longitudinal ordering and lateral focussing of particles (based on Di Carlo et al. [22]). (b) The spiral microfluidic design developed for cell cycle synchronization by Lee et al. [23]. (Reproduced with permission of the Chemical Society.)

1.4.3 Acoustic

An acoustic radiation force [25] can be exerted on a particle in a fluid stream using an ultrasonic transducer located at the wall of the channel. The form of transducer commonly consists of piezoelectric ceramic rings sandwiched and bolted between two metal blocks. When a DC voltage is applied to the piezoceramics, they expand and the pressure applied to the blocks is transmitted into the bulk of the fluid. An applied AC voltage causes the transducer to vibrate at the frequency of the applied voltage and this frequency is higher than that detectable by human ears. The longitudinal pressure wave created in the bulk fluid will travel at the speed of sound, which for water is around 1500 m/s. The wavelength λ is related to the frequency f and speed v by the relationship λ = v/f. So, for the case where f = 15 MHz and v = 1500 m/s, we have a wavelength of 100 μm. If the transducer faces a sound reflector, the interactions between the emitted and the reflected acoustic wave create regions of minimum and maximum pressure amplitude changes (nodes and antinodes, respectively). This effect is maximized by placing the sound reflector a multiple of half wavelengths from the transducer, to create a resonant standing wave [26].

The acoustic radiation force consists of two components. One component involves the gradient of the potential energy of the wave interacting with the compressibility difference between the particle and fluid, whilst the other involves the gradient of the kinetic energy interacting with the specific density differences between the particle and fluid [25]. The particles experience a force that is directly proportional to their volume – in other words large particles experience a greater acoustic force than small ones. Whether or not a particle is directed towards or away from a standing wave pressure node depends on its density and compressibility compared to that of the surrounding medium. Cells are of greater density and are less compressible than an aqueous medium and both of these factors result in their being directed to where minimal pressure amplitude changes occur – i.e., towards a pressure node. Particles of the same density as the surrounding medium will move towards a pressure antinode if they are more compressible than the medium. An illustration of the degrees of particle separation created in a half-wavelength acoustic standing wave is shown schematically in Figure 1.6. In this illustration the acoustic force is generated at right angles to the direction of the fluid flow. The suspended particles interact with the acoustic force as they flow along the channel and become spatially separated according to their size, density and compressibility. Different fractions of the particle mixture can then be collected downstream through different fluid exit ports [27, 28].

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Figure 1.6 A schematic of particles moving at different rates towards the pressure node at the centre of a half-wavelength acoustic standing wave, according to their size, density, compressibility, and acoustic pressure amplitude (based on Petersson et al. [27]). The acoustic force is generated across the channel, orthogonal to the fluid flow direction (into the page). The different particle fractions can be collected downstream from several exit ports [27, 28].

In an aqueous suspension of mixed viable and dead (nonviable) cells the dead ones tend to be more buoyant (less dense) and smaller than the viable ones. In an acoustic standing wave the viable cells should therefore have a greater tendency to move towards a pressure node than the dead ones. This has been demonstrated, as depicted in Figure 1.7, for the case of viable and dead breast cancer cells in a cell separation device constructed by Yang and Soh [29]. Acoustophoresis appears to be particularly well suited for processing fluids of high cell content in microfluidic devices. For example, whole blood is particularly difficult to flow through microchannels because of its high viscosity and clogging tendency. However, Lenshof et al. were able to produce plasma from whole blood in a sequential blood cell removal procedure in an acoustic force microdevice having a multiple outlet configuration [30]. The quality of the resulting plasma fulfilled the standard defined by the Council of Europe for plasma transfusion, namely an erythrocyte concentration less than 6 × 10⁶/mL. Furthermore, the plasma was directly linked in the device to a microarray for the detection of a prostate specific antigen via fluorescence readout