1 Introduction and Fundamental Concepts

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1.1 Electrokinetic Mechanisms for Microﬂuidic and Nanoﬂuidic Transport

1.1.1 Introduction to Microfluidic and Nanofluidic Systems The enormous potential of microscale and nanoscale technology for chemical and biological analysis is reflected by the recent explosive growth in research on laboratory-on-a-chip and miniature diagnostic devices (van den Berg and Harrison, 1998; Manz and Becker, 1998). An example of such a device is depicted in Fig. 1.1. Advances in microfabrication, nanofabrication and microelectromechanical systems (MEMS) over the past few decades has allowed miniaturized devices of growing complexity and sophistication to be developed for various applications, in particular, for the biomedical and pharmaceutical industries. Microchip devices have already been developed for drug screening, electrochemical immunoassays, drug delivery, point-of-care medical diagnostics, bacteria detection, flow cytometry (e.g., cell culture and manipulation), proteomics, genomics (e.g., polymerase chain reaction, or DNA sizing and sequencing), environmental monitoring, and the detection of explosives and biological warfare agents, among others. Although commercially successful devices have just begun to appear, it is anticipated that they will spur new biotechnologies in the next decade. The ability to miniaturize, automate, and parallelize large-scale batch processes represents a distinct advantage that not only reduces the amount of expensive samples and reagents used but also allows the process to be carried out at a fraction of the cost and duration as a result of shorter residence, reaction, and response times. Although high-throughput production is probably unlikely with microdevices, the production rate is nevertheless higher than that of batch processes of similar dimensions, particularly if continuous and parallel-chip structures are used. Such mas- sively parallel chips that operate with small-volume samples are intended for gene sequencing, proteomics, drug development, antibiotic screening, and other laboratory processes. The same microfluidic technologies for transporting fluids in small devices may also find applications in portable direct-methanol fuel cells and future microscale versions of such battery-replacement technology.

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2 Introduction and Fundamental Concepts

Sample Flow loading metering Mixing Reaction Separation Detection Sample inlets

Electrokinetic pumps Photodetector or Air purge Optical sensor Manifold or Microvalves Separator or Micromixer Chromatographic Sample Thermal Reactor column outlet Figure 1.1. Schematic illustration of an integrated laboratory-on-a-chip device to perform scaled-down biological and chemical analyses involving ﬂow metering, mixing, reaction, separation, and detection.

We believe, however, that miniaturized diagnostics will constitute the most commercially relevant development among the entire range of microfluidic devices. Rapid and portable integrated miniature diagnostic kits that do not require laboratory equipment and trained technicians are extremely attractive to the medical, antiterrorism, environmental, and animal-care diagnostic industries. Specifi- cally, future multiplexed miniature diagnostic kits will use antibody functionalized immunobeads for pathogen detection and DNA probe functionalized genetic beads for DNA identification. Such bead-based diagnostic kits probably will not be able to compete with hybridization microarrays in terms of library volume (target number). However, microfluidic devices, in particular, those driven by electrokinetics, with the ability to manipulate these nanocolloids at the microscale level, have response times that are orders of magnitude faster. In these instances, the transport limita- tions for any docking event will be reduced by the shear number of nanocolloids and by the judicious application of electrokinetic forces, delivered by microelec- trodes and nanoelectrodes, on the colloids. The colloids offer an overall surface area that is orders of magnitude larger than that of a microarray pixel or even that of a microfluidic channel. They can hence capture all the targets and, if the detection is carried out with fluorophores on the targets or with other target-labeled reporters, can provide significantly lower detection thresholds than those obtained with current microarray technology. Diagnostic bead assays, like all other surface assays, involve several steps of mixing, rinsing, and hybridization. The beads can be transported from one reservoir to the next on the chip for such multiplex operations, or they can be confined to one reservoir and the rinsing–hybridization buffer is pumped through the reservoir. A combination of these two strategies can also be used. There is therefore a need to develop bead manipulation as well as fluid transport and metering platforms. Chip- based membranes that can confine or trap these beads may also be necessary, but such membranes tend to produce large hydrodynamic resistance (pressure drop) and require high pressures above those accessible by micropumps. Electrokinetic bead traps or filters, on the other hand, do not produce such significant pressure drops and are hence much more desirable.

1.1 Electrokinetic Mechanisms for Microﬂuidic and Nanoﬂuidic Transport 3

Strategies are subsequently required for detecting the molecules, pathogens, or other organisms that have docked onto the beads. Currently, bead-based and other surface assays utilize optical-detection methods, similar to those used in cytometry. However, optical detection may not be the most robust technique for incorporation into chip-based devices, particularly if they require elaborate optical-detection facil- ities such as confocal microscopy. Although on-chip light sources, waveguides, and optical sensors are currently being developed, we envisage that these will remain expensive within the near future and incompatible with the concept of disposable chips – the devices require rinsing for reuse, which would typically render on-chip optical sensing impractical or commercially unfeasible. Electrical impedance sensing, with bare electrodes or electrodes functionalized with chemical or molecular probes, on the other hand, may prove to be significantly more practical, as it does not require fluorophore tagging and optical-detection schemes. In addition, electrical impedance sensing is perhaps most effective for nanocolloidal systems because the impedance signature of the colloids is sensitive to surface molecular-docking events. Significant efforts are currently underway in electrode functionalization and material synthesis to enhance selective molecular or colloidal docking. Whatever the detection method, microfluidic devices have the ability to trap these colloidal particles or beads for sensitivity enhancement in immunoassay diagnostic testing. The precision offered by such devices is also driving an emerging field in microfluidics, namely the synthesis of nanotextured materials, biological membranes, and biofilms. The microdevices allow self-assembly or directed-assembly of inorganic or biological materials one filament and one layer at a time by means of a modular and sequential assembly process at the molecular level. Designed complex structures with rich morphologies beyond spontaneous and uncontrolled self- assembly processes are the targets of such microfluidic material synthesis. At the very heart of these devices is core microfluidic technology that is indispensable to its operation, consisting of a set of integrated components that are required to actuate, regulate, mix, manipulate, separate, and detect the samples, reagents, and biological species involved. These components comprise pumps, valves, mixers, separators, and sensors fabricated onto a microchip and connected by a series of small capillary channels. Typically, the characteristic length scale L of these components and the channels range between 1 and 102 µm. Because the surface-area-to-volume ratio increases inversely proportional to L, interfacial effects become dominant at these small length scales. Although the large surface- area-to-volume ratio provides significant advantages over macroscopic flows such as enhanced heat and mass transfer and small sample volumes, the design of microfluidic devices requires proper consideration of the associated interfacial phenomena. For example, microscopic effects such as disjoining pressure and wall slip or wall shear present significant design challenges in the scale down to dimensions commen- surate with microfluidic devices. These effects become more acute when nanocolloids are used – not only is the surface-area-to-volume ratio even larger in these systems, new physical phenomena that are foreign to substrate assays are also observed when the colloid sizes approach molecular and electrical Debye double layer (thin region close to the surface in which concentration polarization of the ions occur; see Chapter 2) dimensions.

4 Introduction and Fundamental Concepts

As the boundaries of microfluidic technology are extended further, the capabil- ity of fabricating even smaller devices with dimensions of the order of 10 to 100 nm becomes increasingly promising. In these nanofluidic devices in which the Knud- sen number Kn ≡ λ/L describing the ratio between the molecular mean free path λ and the characteristic channel dimension L is typically above 10−3, molecular, nonequilibrium, and rarefaction effects become significant. Between Kn ∼ 10−3 and 10−1, continuum models can be used but require some form of slip boundary con- dition. However, in the free-molecular-flow limit Kn 10−1, continuum mechanics and mean-field theories break down and the assumptions adopted to retain the use of continuum models in microfluidic theory are no longer applicable (Karniadakis et al., 2005). Statistical and molecular theories are not, however, considered in this book. Instead, we restrict our scope mainly to microfluidic theories in the continuum limit; where nanofluidic devices are discussed, these will be assumed to be at the larger-scale limit at the lower end of the microfluidic boundary where the validity of continuum theories still hold. Nevertheless, even when local continuum equations are valid, it would be advantageous to “average” out smaller length-scale effects that are due to transverse variations, surface roughness, the electrical Debye double layer, colloid distribution, the presence of surfactant, etc., to obtain “effective” macroscopic equations that are easier to analyze and visualize. A good example is the channel-averaged longitudinal dispersion phenomenon in microchannels. Although the dispersion magnitude is sensitive to the channel cross section, such details are of concern only during the derivation of the dispersion coefficient and do not appear in the final macroscopic equation that contains the dispersion coef- ficients. Another example is the Smoluchowski electrokinetic slip velocity, which is derived in Chapter 3. The slip velocity is applied to a microchannel surface without explicit consideration of the electrical Debye double layer responsible for the slip on the microchannel walls. When the device and channel dimensions reach nanometer scales, the scale separation necessary to carry out such averaging is often violated, and the derivation of the proper effective equations needs to be carried out with care. Many microfluidic technologies are currently being designed with the goal of developing a generic integrated platform for the wide range of applications previously described. With advances in microfabrication and nanofabrication technologies, it is possible to drive microscale fluid and particle motion by use of tiny mechanical components, such as microgears and microactuators. Such micromachinery, known as MEMS (Ho and Tai, 1998), although already commercially available, however, has not been widely implemented in microdevices because of the complexity, cost, reliability, and wear issues related to the tiny mechanically moving parts. The development of soft lithography techniques in poly(dimethlysiloxane) or PDMS, which is a soft elastomer that is optically transparent and biocompati- ble, has also led to the design of mechanically actuated peristaltic pumps and mixers (Whitesides, 2006); alternatively, a sublayer of electrorheological fluids is also used in place of the mechanical actuators. Controlled wetting, centrifugal, thermocapillary, and magnetohydrodynamic devices were also developed. It is likely that these different platforms may have specific uses for various applications, depending on the particular design requirements of that application, for example, precision, speed, cost, and portability. Electrokinetically driven microfluidic devices,

1.1 Electrokinetic Mechanisms for Microﬂuidic and Nanoﬂuidic Transport 5

which use electric fields to impart a force on the liquid or the particles, for example, the nanocolloidal beads previously discussed, are also extremely popular for driving chip-scale pumping, mixing, and metering of liquids as well as colloidal trapping, sorting, and sensing. In particular, there have been considerable developments using direct-current (DC) electric fields. More recently, however, significant attention has turned to its alternating-current (AC) counterpart because of the many advantages over DC electrokinetics that it has to offer.

1.1.2 Microscale and Nanoscale Electrokinetic Transport The term electrokinetics concerns the use of applied electric fields to impart a net electrostatic (Maxwell) force in polarized surface regions such that fluid motion or the motion of particles suspended within the bulk of a liquid is induced. Electro- kinetics was proposed as the preferred method of transporting fluids in microfluidic devices because of the ease and low cost of electrode fabrication (e.g., standard lithographic masking and wet etching) and because electrokinetic mechanisms involve no moving mechanical parts that are subject to cost, reliability, and noise concerns. Mechanical microfluidic pumps, for example, remain prohibitively expensive, and many microfluidic devices still rely on large and expensive syringe pumps to transfer external fluid into the device, thus defeating many of the apparent advantages of these microfluidic devices. In contrast, electrokinetic flows produced by strategically placed embedded electrodes can be integrated into the device or chip and allow for easier, more precise control and handling of the fluid over other interfacial transport mechanisms such as centrifugal, Marangoni (surface- tension gradient), and thermocapillary flows (Stone and Kim, 2001; Sharp et al., 2002; Stone et al., 2004). There is no fluid transfer from the periphery, which is a major practical advantage. Other than single-phase fluids, digitated bubble and drop platforms using electrokinetics (Chapter 10) have also become increasingly popular. However, the largest advantage that electrokinetic devices, in particular those that use AC fields, offer is the ability to manipulate and sort microcolloids and nanocolloids (Morgan and Green, 2003). When integrated with the bead-based assays discussed earlier, AC electrokinetics could potentially offer the most accu- rate and rapid method for chip-based immunoassay diagnostics without requiring fluorophore labeling and optical detection (Chang, 2007), as is described in Chapter 8. With advances in nanochannel fabrication technology, it is now possible to apply electric fields across a fabricated nanochannel of dimensions similar to those of ion channels. The corresponding electrokinetic pumping and ion transport through these channels then mimic features unique to natural membranes such as ion selectivity, voltage gating, and action potential oscillations, which could further enhance the sensitivity of molecular and chemical diagnostics (see Chapters 2, 3, and 6, for example). However, given that the samples are injected at the micrometer (micron) scale, micro–nano interfaces need to be designed and fabricated. More- over, as the nanochannels approach dimensions of the order of the electrical double layer and hence behave similarly to ion-selective membranes, these interfaces can result in significant charge buildup. Proper design considerations are therefore necessary, given that such multiscale devices require an even greater understanding

6 Introduction and Fundamental Concepts

of the fundamental physicochemical hydrodynamics associated with microfluidic science. Electrokinetic devices do, however, suffer from several shortcomings. Conven- tional electrokinetic micropumps, for example, which are based on classical DC electro-osmosis (bulk liquid motion that is due to applied electric fields), have efficiencies that are an order of magnitude lower than those of mechanical micropumps. This is due to the confinement of the hydrodynamic shear within the electrical Debye double layer, which also leads to large wall shear and viscous dissi- pation (Zeng et al., 2001;Yaoet al., 2003; Wang et al., 2006a). The typical picoliter- or nanoliter-per-second flow rates produced by these electrokinetic micropumps are generally inadequate for microfluidic applications such as high-throughput drug screening and miniaturized medical diagnostic kits for which sample sizes are in the range between microliters and milliliters. At the flow rates achieved, diagnostic times would simply be too long, requiring several hours or even days. DC elec- trophoresis, the motion of charged particles under an applied electric field (see Chapter 4), also suffers from low velocities, typically around 100 µm/s, for field strengths of approximately 100 V/cm in common physiological and biological electrolyte samples. Particle aggregation and protein precipitation, particularly at channel corners where the field is more intense, also pose significant design issues in DC electrokinetic flows (Thamida and Chang, 2002; Takhistov et al., 2003). In addition to the safety concerns of using high DC voltages, these electric-field intensities can sustain large and steady DC currents that cause damage to molecular and biomolecular structures, protein denaturing, blood cell lysing, and particle aggregation when highly conducting liquids are used. In addition, the DC current also produces significant Joule heating effects and causes Faradaic reactions at the electrode that lead to the generation of bubbles and ionic contaminants. Trapped bubbles within a microchannel could also result in reduced throughput because of their enormous capillary pressure (Chang, 2005). As a consequence, microfluidic devices using DC electrokinetic machinery require an external electrode housing in open reservoirs for separating the electrodes from the working channels, thereby immediately ruling out integration of the components into miniature and portable diagnostic kits and defeating the particle manipulation and flow control advantages of electrokinetic devices with internal electrodes. Furthermore, a pH gradient often develops as a result of ion generation at the electrode reservoirs and will often produce a nonuniform electrokinetic potential or zeta potential (potential drop across the electrical double layer) because of its pH sensitivity. Undesirable pressure- driven back flow and long circulations can then occur because of such pH gradients (Minerick et al., 2002); this is discussed further in Chapter 3. Buffer solutions are therefore usually used to neutralize the pH gradient. However, the addition of such buffer solutions may often be difficult or undesirable, depending on the load and the sample. The absence of hydrodynamic shear outside the thin Debye double layer also presents significant challenges in micromixers. In addition to the low Reynolds num- bers in microflows, the DC electro-osmotic flow field is an irrotational potential flow in which the streamlines coincide with the electric-field lines in the absence of internal pressure gradients (Chapter 3). As a result, the mixing intensity is low; this is, in fact, exploited in capillary electrochromatography (CEC; see Chapter 4)

1.1 Electrokinetic Mechanisms for Microﬂuidic and Nanoﬂuidic Transport 7

to minimize sample dispersion. Nevertheless, low mixing efficiencies are undesirable in transport-limited reaction or molecular docking because the low diffusivities of some biomolecules or bioparticles result in transport times of over several hours. Many of these shortcomings can be overcome by use of high-frequency AC electric fields above 10 kHz. Instead of relying on the natural surface charges on a channel surface, AC electrokinetics utilizes the electric field supplied by the electrodes to induce polarization on the electrode. All electric forces are hence confined to the electrode surface, and embedded electrodes must be used to sustain a robust flow. The induced polarization on the electrode surface is nonuniform and back- pressure-driven vortices exist in almost all cases, whereas vortices for DC electro- osmotic flows are quite uncommon. Electrochemical reactions at the electrodes are absent if the root-mean-square (RMS) voltage does not exceed approximately 10 V at the high frequencies used, thus eliminating any Faradaic reactions that generate bubbles or ionic contaminants and pH gradients. Even in the absence of a Faradaic current, an AC charging current still exists at the electrode surface because of field- induced electromigration of ions from the bulk electrolyte. The electrodes can be covered with a thin dielectric film to completely eliminate the reactions altogether. Because the AC impedance decreases linearly with the applied frequency, Joule heating effects can be minimized. An exciting research direction, which is discussed in Chapter 7, is to use continuous coil or wire designs in place of disjointed electrodes such that the electrons preferably remain in the electrode instead of being transferred into the solution species by means of Faradaic reactions at the electrodes (Gagnon and Chang, 2005). In these cases, there still remains a significant field outside the coil or wire to drive a capacitive ionic current to the electrode surface and polarize it such that a Maxwell force is imparted on the liquid. Furthermore, careful design of the electrode geometry and RC (R being the electrical resistance and C the capacitance) characteristic time scales limits the field- penetration depth to below 10 µm, thus further reducing the total Joule heating and global AC current. With high frequencies, the localization of the AC current to the Debye double layer thickness minimizes penetration into the cell membrane and also stretching of the protein, thus preventing cell damage and protein denaturing in biological samples, which are often prevalent in DC electrokinetics. Charge buildup at sharp geometries that is due to field singularity (see Chapter 6)isalso minimal with high AC frequencies, thereby reducing aggregation and precipitation of the sample species at corners. A net local AC electric force occurs only if the field- induced polarization (charging) and the electric field are in phase. As the charging is due to the normal field, the phase difference between the field and the polarization is a sensitive function of the AC frequency, the electrode geometry, and the applied voltage. In some interfacial AC electrokinetic phenomena (see, for example, Chapter 9), Faradaic generation of plasma is unavoidable, and the Faradaic polarization of interfaces will be shown to be much more complex than even capacitive polarization of electrode surfaces. Moreover, colloidal manipulation using embedded AC electrokinetic devices requires careful consideration of the electric forces acting on the colloidal particles, be they blood cells, bacteria, immunocolloids, genetic nanobeads, DNA, or proteins. In particular, how an AC field induces a dipole on a colloid to produce dielectrophoretic motion remains poorly understood and is a major topic of this

8 Introduction and Fundamental Concepts

book (see Chapters 5 and 8, for example). How colloids attract each other during pathogen–colloid docking and how a colloid interacts with a DNA molecule, with and without an applied field, are also subjects of interest but are touched on only briefly in this book. It therefore becomes apparent that proper design of electrokinetic devices, particularly those that use AC electric fields, is extremely crucial and requires a fundamental understanding of how the electrode surfaces are polarized because of the electromigration of ions induced by the external AC electric field. This field-induced polarization process is referred to as nonequilibrium electrokinetic phenomena and differs from the classical equilibrium electrokinetics wherein the polarization arises naturally when the solid surface acquires electrical charge because of adsorption during contact with a polar solvent or dissociation or ionization of surface groups. These nonequilibrium electrokinetic phenomena are induced by the applied field, as electromigration is driven by the field. However, the resulting polarization is quite distinct from field-induced dielectric polarization (or the related Maxwell–Wagner polarization) in that the space charge in the double layer is involved. As such, the pertinent nonequilibrium phenomena involve new forces and, in particular, the tan- gential stress within the double layer.

1.1.3 Organization In the next section and the rest of this chapter, we review the underlying fundamental physical and chemical concepts behind electrostatics and electrokinetics that are central to the development of the theories presented in the later parts of this book. In Chapters 2–4, classical equilibrium electrokinetic theories are revisited and applied to new applications for micropumps and micromixers. Subsequently, we introduce the concepts of field-induced dielectric and double layer polarization pertaining to nonequilibrium electrokinetic phenomena in Chapters 5–8. We then conclude the book with two final chapters on electrokinetic phenomena involving free surfaces, namely jets, drops, and bubbles. Readers familiar with the physical concepts of electrostatics and hydrodynamics may prefer to skip the remainder of this introductory chapter and proceed to Chapter 2. More advanced readers already well versed with classical equilibrium electrokinetic theories may choose to omit Chapter 2 altogether and possibly the first sections of Chapters 3 and 4.

1.2 Electrostatics An electric charge is a fundamental property of the elementary particles that make up matter, which allows for a description of how these particles interact. Two classes of electrical charge exist, positive charge and negative charge. Their interaction is governed by the simple law that requires like charges to repel each other and unlike charges to attract each other. This repulsion or attraction gives rise to an interaction force known as the electrical (Maxwell) force, which forms the basis of all electrostatic interactions. In the following subsections, we review the fundamental underlying physics that governs these electrostatic interactions as a foundation for

1.2 Electrostatics 9

Figure 1.2. The electric-ﬁeld vector (dark lines) and electric potential +qi -qj (dashed contours) arising from two point charges qi and qj of opposite rij polarities separated by a distance ri j .

the development of electrokinetic theories in which electrostatic forces give rise to ﬂuid motion. It is not our intention to provide an in-depth treatise of classical electrostatic theories here. For a more rigorous discussion, we refer the reader to the many excellent texts on electrostatics and electromagnetism available. Rather, we brieﬂy touch on important physical concepts that are relevant to the electrokinetic theories discussed in the rest of this book.

1.2.1 Coulomb’s Law It was empirically established that, when two point charges (charges that have spa- tial dimensions that are negligible compared with the length scale of their interaction) are brought together, the static interaction force that arises is inversely proportional to the square of their linear separation. This relationship, known as Coulomb’s Law, in honor of Charles Augustin de Coulomb (1736–1806), can be written in the following manner to describe the electrical force Fi j that arises because of a pair of point charges separated by a scalar linear distance ri j whose algebraic magnitudes are denoted by qi and qj , as shown in Fig. 1.2:

1 q q F =−F = i j e . (1.1) ji i j πε 2 i j 4 0 ri j

Here, F ji denotes the force exerted on charge qj by charge qi ; ei j = ri j /ri j is the unit vector pointing in the direction from qi to qj , ri j being the displacement vec- 2 tor in this direction, and ε0 is the permittivity of free space (ε0 = 1/c µ0, where c is the speed of light and µ0 is the permeability of free space) such that the constant of proportionality 1/4πε0 and hence Eq. (1.1) is universal for any system involving a pair of charges existing in vacuum. If the charge pair is placed within a dielectric medium that is inhomogeneous and isotropically inﬁnite, then additional forces will be exerted because of induced charges in the medium, and the constant of proportionality is modiﬁed accordingly to account for the dielectric properties of the material. When more than two point charges are present, the principle of superposition, which states that the net effect arising from multiple charges can be obtained by a summation of the effects of isolated individual charges, can be used to account for

10 Introduction and Fundamental Concepts

the interaction of all charges on a single point charge such that the total electrostatic force on charge qj is qj qi F j = ei j , (1.2) 4πε r 2 0 i i j

noting that the conservation of charge applies for closed systems: qi = constant. (1.3) i

1.2.2 Electric Field and Potential It is convenient to define an electric-field vector E as the spatially dependent force per unit charge. From Eq. (1.1), the electric field of the point charge qi at point j is then F 1 q 1 q E = ji = i e = r, (1.4) πε 2 i j πε 3 qj 4 0 ri j 4 0 r where r and r are the displacement vector and scalar distance, respectively, from charge q to the point at which the field E is evaluated. In similar fashion to Eq. (1.2), we can write for multiple charges 1 qi 1 qi E = ei j = ri , (1.5) 4πε r 2 4πε 3 0 i i 0 i ri

ri and ri being the displacement vector and scalar distance from charge qi to the point at which E is evaluated, respectively. In Cartesian space, 2 2 2 r = (x − xi ) + (y − yi ) + (z − zi ) . (1.6)

The electric field can perform work on a charge to displace it over a certain distance. Alternatively, energy external to the system in which the field is defined can be imparted to move the charge. The work done W on a unit charge to move it by an infinitesimal distance ds from point a to b is then 1 b b W = ϕ(b) − ϕ(a) =− F · ds =− E · ds, (1.7) q a a where ϕ is a scalar electrostatic potential. When the reference point is taken at infin- ity, the potential at a given point in space (x, y, z) for a single charge at the origin is q 1 ϕ = . (1.8) 4πε0 r Again, for multiple charges, the potential can be written as 1 q ϕ = i . (1.9) 4πε r 0 i i