Microfluidics

MICROFLUIDICS

Sandip Ghosal

Department of Mechanical Engineering, Northwestern University 2145 Sheridan Road, Evanston, IL 60208

1 Article Outline

Glossary

1. Deﬁnition of the Subject and its Importance. 2. Introduction 3. Physics of Microﬂuidics 4. Future Directions 5. Bibliography

GLOSSARY

1. Reynolds number: A characteristic dimensionless number that determines the nature of fluid flow in a given set up. 2. Stokes approximation: A simplifying approximation often made in fluid mechanics where the terms arising due to the inertia of fluid elements is neglected. This is justified if the Reynolds number is small, a situation that arises for example in the slow flow of viscous liquids; an example is pouring honey from a jar. 3. Ion mobility: Velocity acquired by an ion per unit applied force. 4. Electrophoretic mobility: Velocity acquired by an ion per unit applied electric field. 5. Zeta-potential: The electric potential at the interface of an electrolyte and substrate due to the presence of interfacial charge. Usually indicated by the Greek letter zeta (ζ).

1 6. Debye layer: A thin layer of ions next to charged interfaces (predominantly of the opposite sign to the interfacial charge) due to a balance between electrostatic attraction and random thermal fluctuations. 7. Debye Length: A measure of the thickness of the Debye layer. 8. Debye-Hückel approximation: The process of linearizing the equation for the electric potential; valid if the potential energy of ions is small compared to their average kinetic energy due to thermal motion. 9. Electric Double Layer (EDL): The Debye layer together with the set of fixed charges on the substrate constitute an EDL. It may be thought of as a parallel plate capacitor or a layer of dipoles . 10. Electrophoresis: The motion of charged objects in a fluid due to an imposed electric field. 11. Dielectrophoresis: The phenomenon that results in a polarizable dielectric medium experiencing a force in a non-uniform electric field. 12. Electroosmosis: The motion of an ionic liquid relative to a fixed charged substrate due to an imposed electric field. 13. AC Electroosmosis aka Induced Charge Electroosmosis (ICEO): Fluid flow caused by an electric field in an ionic medium containing embedded polarizable objects (such as metal cylinders). The effect is non-linear in the electric field and can result in unidirectional flow even with an AC voltage. 14. Thermocapillary Effect: Motion of fluid or object in fluid due to spatial inhomogeneities in surface tension caused by variations in the temperature field. 15. Marangoni Effect: Same as the Thermocapillary effect except that the variation of surface tension is caused by inhomogeneities in the concentration of a dissolved chemical species such as a surfactant. 16. Electrowetting: The phenomenon that results in changes of contact angle of a conducting liquid drop placed on a conducting plate when the electric potential of the plate is changed. 17. Electrowetting on Dielectric (EWOD) The same phenomenon as Electrowetting, except a thin dielectric coating is applied on the metal plate so that the liquid and the plate are not in electrical contact.

2 Deﬁnition of the Subject and its Importance

Microﬂuidics is the science of manipulating ﬂuids on spatial scales anywhere between one to a hundred micron. In SI units a micro-meter (µm) or micron refers to a millionth of a meter.

2 To put this in perspective, the average width of a human hair is about 80 microns and the diameter of a hydrogen atom is 0.00005 micron. Thus, microfluidics involves engineering structures for manipulating fluids on scales that are microscopic in comparison to human dimensions but that are nevertheless much larger than atomic dimensions so that the systems can still be treated in the continuum approximation. Microfluidic devices are commonplace in the world of living things; for example, the narrowest capillaries in the human circula- tory system are of the order of 5 − 10 microns and the diameter of a swimming bacteria maybe about 10 microns. However, devices engineered by humans that can be considered microfluidic is of much recent vintage. The methods as well as the motivation for designing microfluidic devices entered an era of rapid development in the last ten years or so, a trend that has continued to accelerate to this date. The primary driving force behind this trend has come from Molecular Biology and the Life Sciences where progress has become dependent upon the ability to perform analytical chemistry at heretofore unprecedented speeds. As of 2006 there were about sixty companies in the US engaged in marketing microfluidic products. Revenues from such products reached 1.7 billion US dollars in 2003 and is projected to reach 2.7 billion US dollars in 2008.

3 Introduction

The idea of a microfluidic chip is derived from the concept of a microelectronic chip. The difference is, channels through which fluids flow take the place of conducting pathways for electrons. The transistors and other micro-electronic elements are replaced by mixers, separators and other micro-fluidic elements designed to perform the basic tasks of analytical chemistry.

3.1 The Case for Microﬂuidics

The technology of microelectronics was invented and developed to fulfill a specific need: pro- cessing of digital information in devices that are smaller, cheaper and faster. The evolution of microfluidics is being driven by exactly the same considerations except that the function that needs to be performed is related to analytical chemistry. Applications in modern biology and medicine such as drug testing, gene sequencing and gene expression studies require the performance of extremely large numbers of repetitive steps. For example, large banks of robots were employed in the Human Genome Project which nevertheless took eleven years and over three billion dollars to complete. An analogy can be drawn with the Manhattan project where large scale numerical calculations were laboriously performed by rooms full of human ‘calculators’. However, large scale automated laboratories for such functions as gene sequencing can only be a temporary measure. The ultimate enabling technology for ‘super- chemistry’ must be minaturization, as it was for ‘super-computing’. This is because (a) as device size shrinks, the time needed to perform a specific operation (such as electrophoretic

3 Figure 1: Left panel: Chipsets for the Agilent 2100 Bioanalyzer for performing various bioanalytical procedures Right panel: an example of a microﬂuidic chip contained within the package. (Image: courtesy of Agilent Technologies). separation) becomes smaller (b) a large number of identical steps can be run in parallel on a small device (c) the amount of sample and reagents consumed becomes much smaller (d) the devices themselves can be mass produced lowering the unit cost.

3.2 A Brief History of Microﬂuidics

Unlike microelectronics, which by now must be considered a mature technology, microfluidics is still in its infancy. In fact microfluidics draws heavily from many of the same technological breakthroughs that led to the current sophistication in the manufacture of computer chips. It is difficult to say when microfluidics really began or what the first “microfluidic device” was. If one uses the definition of microfluidics given in the introductory paragraph as something that involves a device that manipulates fluid on a 1 − 100 micron scale, then perhaps the ink jet printer should qualify as one of the earliest microfluidic devices. The first inkjet printer was patented by Siemens in 1951 and subsequently went through many stages of innovations and improvement through the efforts of companies such as HP, Xerox, IBM, Cannon, Epson and others. The fluid mechanical principles involved are rich in complexity and there are excellent reviews available on the subject [1]. An early example of a microfluidic chip is due to Jacobson and Ramsey who demonstrated DNA Restriction Fragment Analysis on a single chip under computer control [2]. The first commercial “Lab on a Chip” was the AGILENT 2100 Bioanalyzer introduced in 1999. The number of patents issued per year for inventions related to some aspect of commercial microfluidic applications has grown from a handful per year in the nineties to more than 350 patents a year.

4 Figure 2: Examples of Microﬂuidic systems in the living world (A) the Stenocara beetle from Namibia (left) that harvests water droplets from fog using a textured surface with alternating hydrophobic/hydrophillic areas as seen in the microscope image (middle) (B) the African “Bombardier Beetle” (Stenaptinus insignis) discharging a boiling toxic liquid jet (right) through steerable micro-nozzles (Images reprinted with permission from Nature and Proc. Natl. Acad. Sci. USA – full citations in text).

3.3 Natural Microﬂuidic Systems

Microfluidic systems of beautifully elegant design are commonplace in the natural world. A stunning example is the fog harvesting Stenocara beetle of the Kalahari desert in Namibia. The shell of this beetle is a patterned substrate of alternating hydrophillic and hydrophobic areas [3]. The beetle positions itself on top of sand dunes with its head down and facing the ocean wind that blows in the early morning fog. The surface patterning is designed to catch the water droplets, grow them to a critical size and then detach them. This size selection is important, for the droplets need to be large enough to roll down by gravity against the prevailing headwind and not be blown off the beetle’s back. The African bombardier beetle (Stenaptinus insignis) is armed with a potent microfluidics based weapon [4]. Its abdomen contains a reaction chamber which receives a propellant (hydroquinone) and an oxidizer (hydrogen peroxide) from separate storage organs. These two chemicals combine explosively in the presence of catalysts secreted within the reaction chamber. The result is a boiling mixture of steam and corrosive liquid which the beetle is able to deliver on to the target through a steerable nozzle with an accompanying loud bang! Other examples include fluid flow through the proboscis of insects such as butterflies and mosquitos, the flagellar propulsion of bacteria such as Ecoli, propulsion by a single waving flagellum in spermatozoa, use of cillia by numerous small animals for moving fluids along and the innovative use of surface tension forces by insects and larva that live on the air water interface in stagnant pools [5].

5 3.4 Understanding Microﬂuidics

A rich variety of microfluidic systems have arisen in the natural world through millions of years of evolution and natural selection. However, the history of microfluidics as an area of human endeavor can be measured in decades. It is therefore not surprising that practical microfluidic devices are relatively few, many ideas such as the “Lab On a Chip” exist mostly in concept and the most useful of microfluidic devices are probably yet to be conceived. In order to translate concepts into practical devices, we must learn to control fluids with great precision and speed at ultra small scales. Such control can only be achieved through a deep knowledge of the behavior of fluids on ultra small scales. It is the purpose of this article to give an overview of the mathematical laws that govern such fluid behavior in the small.

4 Physics of Microﬂuidics

The laws of fluid motion for microfluidic systems are not any different from those that govern large scale systems such as the oceanic currents on intercontinental scales. However, the relative importance of different forces and effects change dramatically as we go from macro to micro scales. Thus, for example, surface tension forces and electrostatics play no role at all in the study of ocean currents but are enormously important in the world of microfluidics.

The remainder of this section is presented in three subsections, Stokes Flow which deals with fluid motions that are dominated by viscous resistance, Electrokinetics which deals with the interaction of liquids with electrostatic forces and finally the motion of fluids driven by forces related to Surface Tension. The flow of gases is not considered at all in this article because microfluidic devices based on the flow of a gas is not very common. In the case of gas flow, the most important physical effect to consider is the likely break down of the classical gas dynamic equations as the mean free path approaches the channel diameter. The flow of rarefied gases have been reviewed by Muntz [6].

4.1 Stokes Flow

In microfluidics, relevant physical dimensions are sufficiently large in comparison to atomic scales that it is permissible to treat the fluid as if it were a continuum. Thus, the fluid velocity u and pressure p are regarded as continuous functions of position x and time t, and they obey the incompressible Navier-Stokes equations with a volume density of external forces fe 2 ρ0(∂tu + u · ∇u) = −∇p + µ∇ u + fe. (1)

6 This is supplemented by the continuity equation which takes into account the fact that in a liquid the density changes are slight, even for large changes in pressure : ∇ · u = 0. (2)

In the above, ρ0 is the (constant) density of the ﬂuid, p is the pressure, u is the ﬂow velocity.

The relative size of the term on the left of eq(1) (due to fluid inertia) and the second term on the right (due to viscosity) is characterized by the Reynolds number ULρ Re = 0 (3) µ where U and L denote a characteristic velocity and length for the flow. In most applications of microfluidics, Re << 1. In some applications, Re ∼ 1. By contrast, in large scale flows (aircraft engines, geophysical flows etc.) Re >> 1 is the rule. Because of the smallness of the Reynolds number, Re, in microfluidics, most well known flow instabilities leading to period doubling, chaos and finally turbulence are absent. Furthermore, the left hand side of eq(1) which corresponds to fluid inertia can either be neglected, or treated as a small perturbation. In the former case, we arrive at the Stokes flow equations: 2 −∇p + µ∇ u + fe = 0 (4) which is often referred to as slow, creeping or highly viscous flow. All of these terms mean the same thing, namely Re = (ULρ0)/µ 1. The unknown scalar field p in eq (4) is determined by the constraint provided by eq (2).

When the ﬂow is bounded in one or more spatial directions, boundary conditions must be speciﬁed to guarantee unique solutions to eq (4) and (2). Solid boundaries are most commonly encountered where the classical “no slip” boundary condition

u(P ) = usolid(P ) (5) is employed: the fluid velocity at a point P on the solid fluid interface must match the velocity of the solid, usolid(P ) at that interfacial point. The justification is empirical, and based on the intuitive picture that on the micro-scale any smooth surface is actually a rugged terrain of peaks and valleys which effectively randomizes the direction of the velocity vector of a molecule after it collides with the wall. On the scale of macroscopic experiments, these boundary conditions have always provided excellent agreement between theory and experiments except for describing the motion of the contact line for drops moving on a substrate. In these moving contact line problems, the no-slip boundary conditions lead to a singularity of the stress at the contact line [7]. In the flow of low density gases through microchannels, the mean free path of molecules could approach the channel diameter. In such cases deviations from the Navier-Stokes equations with the no-slip conditions have been observed [8, 9]. In microfluidic systems, the validity of the no slip condition for liquid flow have been questioned both from a theoretical as well as an experimental perspective [10]. A consensus is yet to be reached, however it seems that the deviation from the no-slip condition, if there is such a deviation, is quite small. The interpretation of experimental data is often complicated by the presence of adsorbed gas microbubbles on the interface, particularly if the interface is hydrophobic in character [11].

7 4.2 Electrokinetics

Electrokinetics refers to mechanical effects that arise due to the motion of ions in liquids. The working fluid in microfluidic systems is normally water which contain ions of both signs due to dissociated water molecules or other ionic components: acids, salts, and molecules with dissociable charged groups. Normally, a volume element of such a fluid considered “infinitesimal” in the continuum viewpoint still contains a sufficiently large number of ions of either sign for statistical fluctuations to be unimportant and for the fluid element to be considered charge neutral. Therefore, the net algebraic transfer of momentum due to any ambient electric field is also zero (even though a non-zero electric current may exist in the fluid due to the ordered motion of these ions). Electrokinetic effects arise when this balance of positive and negative charges is disturbed due to external factors. For example, at the silica water interface, the hydrated silica often deprotonates resulting in a net negative fixed surface charge on the silica surface. These fixed charges attract a layer of ions of the opposite sign (and repel ions of like sign) resulting in the creation of a fluid layer with a net positive charge next to the interface. Similar effects arise at the surface of large macromolecules, colloidal particles or surfactant micelles. Thus, in the macroscopic description, the fluid in these so called Debye Layers experience an electrical force with volume density fe = −ρe∇φ where ρe is the electric charge density and φ is the electric potential.

4.2.1 Ion Transport

The electric potential φ is related to the charge density through the Poisson equation [12] of electrostatics: 2 ∇ φ = −4πρe, (6) being the dielectric constant of the liquid (in CGS units). If the electrolyte contains N species of ions with charges ezk and concentration nk (k = 1,...,N, and e is the magnitude PN of the electron charge) then ρe = k=1 ezknk. Each ion species obeys a conservation equation ∂n k + ∇ · j = 0. (7) ∂t k

Here jk, the ﬂux vector for the species k can be modeled by the Nernst-Planck equation for ion transport [13] jk = −vkzkenk∇φ − Dk∇nk + nku. (8)

In eq (8), vk is the ion mobility: the velocity acquired by the ion when acted upon by a unit of external force. It is obviously related to the electrophoretic mobility: the velocity per unit (ep) of electric field as µk = ezkvk. The diffusivity of the kth species is Dk and u is the fluid velocity.

The boundary conditions are those of no slip at the wall for the velocity, u = 0, and no ion flux normal to the wall jk · nˆ = 0 (nˆ is the unit normal directed into the fluid). In the absence of external electric fields, the chemistry at the electrolyte substrate interface leads

8 to the establishment of a potential, φ = ζ. This so called ζ-potential at an interface depends on a number of factors including the nature of the substrate and ionic composition of the electrolyte, the presence of impurities, the temperature, the buffer pH and is even known to exhibit hysteresis effects with relation to pH [14]. Methods of determining the ζ-potential and measured values for a wide variety of surfaces used in microfluidic technology have been reviewed in [15] and [16].

4.2.2 Equilibrium Debye Layers

The ion distribution near a planar wall at z = 0 with potential φ(z) is known from statistical thermodynamics: nk = nk(∞) exp(−zkeφ/kBT ) where kB is the Boltzmann constant and T is the absolute temperature of the solution. In order that this expression be a steady solution of eq (7) we must have the Einstein relation Dk/vk = kBT . Therefore eq (8) can also be written as jk = −nkvk∇ψk + nku (9) where ψk = ezkφ+kBT ln nk is called the chemical potential for the species k. To determine φ, one must solve a self-consistent coupled problem because the species concentrations depend upon φ through nk = nk(∞) exp(−zkeφ/kBT ), but φ also depends on the nk through the Poisson equation, eq (6).

The Gouy-Chapman model

Suppose that the system is in the steady state and that there is no fluid flow or imposed electric fields. Further suppose that the geometry is such that the electrolyte-substrate interface is an iso-surface of ψk. Then it readily follows from equations (7) and (9) and the boundary condition of no flux into the wall that ∇ψk = 0 everywhere. Therefore, (∞) (∞) nk = nk exp (−zkeφ/kBT ) where nk is the ion concentration where the potential φ = 0; usually chosen as a point very far from the wall. Using the solution for nk in the charge density ρe and substituting in eq (6), we get the non-linear Poisson-Boltzmann equation for determining the potential N 4πe X (∞) ∇2φ = − n z exp (−z eφ/k T ) . (10) k k k B k=1 with the boundary condition φ = ζ on walls.

Equation (10) was the starting point of a detailed investigation of the structure of the Electric Double Layer (EDL) by Gouy [17] and Chapman [18]. The description of the EDL in terms of eq (10) is therefore known as the Gouy-Chapman model of the EDL.

The Debye-H¨uckel approximation

Equation (10) is a nonlinear equation and solutions can only be constructed by numerical methods. Debye and H¨uckel linearized it by expanding the exponential terms on the right

9 hand side in Taylor series and discarding all terms that are quadratic or of higher order in φ, which gives ∇2φ − κ2φ = 0 (11) where 1/2 " N (∞) # X 4πz2e2n κ = k k (12) kBT k=1 is a constant determined by the ionic composition of the electrolyte. In arriving at eq(11) we PN (∞) used the condition k=1 nk zk = 0 which expresses the fact that the bulk solution (φ = 0) −1 is free of net charge. It is easily verified that κ has units such that λD = κ defines a length scale that is called the Debye-Length. If a charged plane at a potential ζ is introduced in an electrolyte where the Debye-Hückel approximation is valid, then the solution to eq (11) may be written as φ = ζ exp(−κz) = ζ exp(−z/λD) where z is distance normal to the plate. Thus, the potential due to the charged plate is shielded by the free charges in solution and the effect of the charge penetrates a distance of the order of the Debye-length λD; which gives a physical meaning to this very important quantity. In microfluidic applications the Debye length is typically between 1 to 10 nm.

The linearization proposed by Debye and Hückel is justified provided that |zkφ| kBT/e uniformly in all space and for all k. At room temperature kBT/e ≈ 30 mV . However, for silica substrates |ζ| ∼ 50 − 100 mV in typical applications. Thus, the Debye-Hückel approximation is often not strictly valid. Nevertheless, it is a very useful approximation because it enormously simplifies mathematical investigations related to the Debye layer. Further, all deductions from it are usually qualitatively correct and even quantitative predictions from it outside its range of formal validity tend not to differ from the true solution by a large amount.

4.2.3 Thin Debye Layers and Apparent Slip

The characteristic width of microfluidic channels is typically of the order of 10 − 100 µm, whereas the Debye length λD ∼ 1 − 10 nm. Thus, the Debye layer is exceedingly thin compared to characteristic channel diameters. Under those circumstances, the Navier- Stokes/Poisson-Boltzmann system described in the last section may be replaced by a simpler set of equations. Indeed, the EDL then forms a very thin boundary layer at the solid fluid interface where the electrical forces are confined.

In the outer region (that is, outside the EDL) we have a fluid that is electrically conducting but charge neutral. Its motion is therefore described by the Navier-Stokes equations without the electrical force term. Inside the EDL, the problem reduces to that of an electric field parallel to a planar interface, therefore, the use of the Gouy-Chapman model is justified. From the solution (18) ζE u(z → ∞) = − 0 . (13) 4πµ

10 Figure 3: A sketch showing the Debye layers formed at the walls of a parallel channel and the resultant electroosmotic ﬂow in response to an applied Voltage (Image: courtesy of Prof. H. Bruus of the Technical University of Denmark).

In the language of asymptotic theory, this is the “outer limit of the inner solution” [19] that becomes the boundary condition for the outer problem. Thus, in the limit of inﬁnitely thin EDL, the standard no-slip boundary conditions of ﬂuid mechanics is replaced by the following apparent slip condition: Eζ u − u ≡ u = − . (14) solid s 4πµ

In eq (14), usolid is the velocity of the solid, and u is the velocity of the fluid at a point on the solid fluid interface, us is the slip velocity. The electric field E is the field in the buffer due to external sources and can be determined by solving Laplace’s equation within the electrolyte with the boundary condition E · nˆ = 0 on the wall (since the walls are insulating and there can be no current flowing into them). Equation (14) is known as the Helmholtz- Smoluchowski (HS) slip boundary condition after the pioneering work of [20] and [21]. It is the starting point for solutions of more complex problems such as that of flow through axially inhomogeneous channels [22, 23].

4.2.4 Electroosmosis

In the presence of external fields and fluid flow the equilibrium Gouy-Chapman model is generally not applicable and one must proceed from the full electrokinetic equations presented

11 earlier. However, if the external field and fluid velocity are both along the iso-surfaces of the charge density ρe then the presence of the flow or the imposed field does not alter the charge density distribution which may still be obtained from the Gouy-Chapman model. Examples where such a situation holds would be

1. A planar uniformly charged substrate at z = 0 with an applied electric ﬁeld E0 that is tangential to the surface (the x-direction).

2. A uniform inﬁnite cylindrical capillary with an imposed electric ﬁeld E0 along the axis (the x-direction).

3. A narrow slit with uniformly charged walls and an imposed constant electric ﬁeld E0 along the slit (the x-direction).

For any of the above geometries, the fluid flow equations reduce to (assuming steady state and zero imposed pressure gradient) E µ∇2u + ρ E = µ∇2u − 0 ∇2φ(EDL) = 0 (15) e 0 4π where u is the axial velocity and φ(EDL) is the electric potential for the equilibrium problem, that is, without the external field or flow. Therefore, E u = 0 φ(EDL) + χ (16) 4πµ where χ satisfies ∇2χ = 0 (17)

and χ = −(E0ζ)/(4πµ) at the boundaries. Assuming an inﬁnitely long channel, by sym- metry, solutions must be independent of the axial co-ordinate x. The only such solution is χ = −(ζE0)/(4πµ). Therefore, the velocity is determined in terms of the potential distribution in the equilibrium EDL: E u = 0 φ(EDL) − ζ (18) 4πµ If we adopt the Debye-H¨uckel approximation then

 ζ exp(−κz) for (1) infinite plane (EDL)  φ = ζ I0(κr)/I0(κa) for (2) infinite cylindrical capillary (19)  ζ cosh(κz)/ cosh(κb) for (3) narrow slit where a is the capillary radius, r the distance from the axis and I0 is the zero order modified Bessel function of the first kind. In the last formula, 2b is the channel width and z is the wall normal co-ordinate with origin on the plane (in Case 1) or origin at a point equidistant between the two walls (in Case 3). Since the fluid flow equation is linear in this limit, clearly a pressure driven flow can be added to the solution (superposition) in the event that both a

12 pressure gradient and an electric field are simultaneously applied. The solution for an infinite capillary was first obtained by Rice and Whitehead [24]. Solutions in a narrow slit were obtained by Burgreen and Nakache [25] in the context of the Debye-Hückel approximation as well as for a 1 : 1 electrolyte (that is, in our notation N = 2 and z1 = −z2) directly from the full Poisson-Boltzmann equation.

4.2.5 Electrophoresis

Electrophoresis refers to the transport of small charged objects in a fluid due to an applied electric field. Many macromolecules contain dissociable charge groups on its surface, and therefore, spontaneously acquire a charge in aqueous solution. They therefore move in response to an applied electric field, and this motion is the basis for separating macromolecules from solution using such bioanalytical techniques as Capillary Electrophoresis (CE), Slab Gel Electrophoresis (SGE) and Isoelectric Focussing (IEF). As far as the physics is concerned, electroosmosis and electrophoresis are essentially identical, in the former case the reference frame is fixed to the substrate and in the latter case the reference frame is fixed to the fluid at infinitely large distances from the object.

Uniformly charged sphere:

The simplest ‘classic’ problem in electrophoresis involves determining the velocity of a uniformly charged dielectric sphere placed in an ionic ﬂuid (usually water with dissolved salts such as KCl). An exact analytical solution to even this reduced problem is unavailable except in certain special limits. These are the limits of extremely large Debye length (corresponds to very low ionic strengths) and extremely short Debye length (corresponds to very high ionic strength).

Indeed, for low ionic strengths, the velocity v may be obtained by simply equating the electric force qE to the viscous force 6πµav where E is the applied electric ﬁeld, q is the charge on the sphere, µ is the dynamic viscosity of the ﬂuid and a is the particle radius, thus, v q ζ µ(∞) = µ (low salt) = = = (20) ep ep E 6πµa 6πµ where the second part of the equality follows on expressing the charge q in terms of the electrostatic potential ζ on the surface of the sphere. The ratio v/E = µep is called the electrophoretic mobility.

For high ionic strengths, we may invoke the Helmholtz Smoluchowski apparent slip boundary conditions. Then the problem reduces to solving a Stokes Flow problem for a sphere with a pseudo ‘slip’ boundary condition. It is easy to verify that since the electric potential φ is irrotational as well as solenoidal, the quantity ζ∇φ u = (21) 4πµ

13 satisfies the Stokes flow equations with zero pressure gradient. Further, by construction, the apparent slip boundary conditions hold on the surface of the sphere. It can be easily shown that the flow defined by eq (21) corresponds to zero net force and torque on the particle. Therefore, it corresponds to the actual flow field in a reference frame fixed to the particle. The flow speed at distant points gives the electrophoretic speed of the particle, thus, v ζ µ(0) = µ (high salt) = = (22) ep ep E 4πµ Therefore, 2 µ(∞) = µ(0) (23) ep 3 ep

Approximate solutions for intermediate Debye lengths as well as situations where the nonlinear convective term in the Navier-Stokes and species transport equations are small but not negligible have been addresses by various researchers. A careful review is provided by Saville [26].

Non-spherical shapes and non-uniform charge:

For a long time, eq (22) had been the basis for determining experimentally the zeta potentials of particles, though the particles were not always spherical in shape. A formal proof of the validity of eq (22) for particles of any shape was provided by Morrison [27]. Morrison showed that in the limit λD/R → 0, where R is the smallest radius of curvature on the surface of the particle, eq (22) remains valid irrespective of the particle shape provided that ζ, and µ are uniform, the particle is non-conducting and polarization eﬀects are neglected.

The surface charge on small objects such as macomolecules, cells and crystalline colloidal objects like particles of clay are often distributed in a non-uniform manner. The effect of non-uniformity of the ζ potential on the movement of the particle was first investigated by Anderson [28]. Anderson considered a spherical particle in the limit of infinitely thin Debye layers. The distribution of the zeta-potential on the particle surface was assumed arbitrary but known. Anderson showed that in general the particle would both translate and rotate. The translational velocity depended upon the average value of ζ (the monopole) as well as on its quadrapole moment. The angular velocity depended upon the dipole and quadrapole moments and thus vanished in the case of a uniformly charged particle.

4.2.6 Dielectrophoresis

Particles that are uncharged but polarizable experience a force in a non-uniform electric ﬁeld, the resulting motion is known as di-electrophoresis. The force density in a polarizable medium is fe = (P · ∇)E (24)

14 Figure 4: Dielectrophoretic trapping of fluorescently labeled particles on a microchip with an etched pattern consisting of a periodic array of cylindrical obstacles. The overlayed picture shows the simulated isopotentials experienced by particles. (Reproduced with permission from [29]). where P is the dipole moment per unit volume and E is the electric field. In an isotropic and linear medium, P = χE where χ is the susceptibility. Since χ is usually positive, the dielectrophoretic force is normally directed from regions of weak electric field to that of strong fields. For particles that are polarizable and also carry a net charge, electrophoretic and dielectrophoretic effects act in conjunction. The interplay of the two effects can be exploited for the design of elegant microfluidic elements. For example, Cummings et al. have designed a device that is able to separate live and dead bacterial cells in this way [29]. Figure 4 shows the trapping of cells labelled with a fluorescent dye in a 2D array of cylindrical posts. Another important aspect of dielectrophoresis is that unlike electrophoresis, the effect does not dissappear if the constant electric field is replaced by an oscillating one. Thus, the dielectrophoretic force can be fine tuned by adjusting the frequency. Since a conducting object would develop a dipole moment in an external field, conducting regions inside a dielectric fluid (such as water droplets in oil) also experience the dielectrophoretic force.

4.2.7 AC Electroosmosis

AC Electroosmosis [31, 32, 30] or more generally Induced Charge Electroosmosis (ICEO) refers to the flow of fluid due to the interaction of an applied electric field with Debye layers,

15 (a) (b) (c) + +_ _

u E ______+ _ + + + _ _ ++ + _ + + + + + + E J E _ _ J _ _ _ _ _ + _ _ _ + + _ _ _ + + + + + + +

+ _ +_ u

Figure 5: Illustration of the operational principle of ICEO: Lines of electric ﬁeld E around a cylindrical metal wire in an electrolyte before (left) and after (middle) double-layer charging in response to a suddenly applied dc ﬁeld and the resulting ICEO streamlines (right). (Image: courtesy of Prof. T.M. Squires of UCSB, also see reference [30]).

except that the charges that give rise to the Debye layers are the polarization charges due to the original electric field. An example is shown in Figure 5 which shows an insulator coated metal cylinder placed in an ionic medium. Switching on the electric field polarizes the cylinder causing the formation of the Debye layers. The tangential electric field acting on the unbalanced charges in the Debye layer drives a flow in the indicated direction. Since the polarization charges themselves are due to the applied field, ICEO is quadratic in the applied field. This nonlinearity opens up new possibilities, for example with cleverly shaped electrodes flow rectification can be achieved, that is, an oscillating field can drive a flow with a DC component. ICEO has a further advantage that electrolysis and the formation of gas bubbles, a disturbing side effect in direct current electroosmosis is eliminated if only oscillating fields are employed. Through careful design of metal on glass electrodes fluids can be effectively driven using the dielectrophoretic force by applying voltages that are much less than those required to drive flows by electroosmosis.

4.3 Surface Tension

2 The surface energy of a spherical drop of liquid of radius R is Es = 4πR γ where γ is the surface tension coefficient. If the drop rests on a surface which is taken as the zero level of potential, then the gravitational potential energy is (ρ is the fluid density) 3 4 2 Eg = (4/3)πR ρg(R/2) = (2/3)πρgR . Therefore, Eg/Es = (ρgR )/(6γ) = Bo/6. The dimensionless number, Bo = ρgR2/γ is called the Bond number and it measures the relative p importance of gravity and surface tension effects. The value R = Rc = γ/ρg that makes the Bond number unity is called the capillary length. For water droplets, Rc = 1 − 3 mm. Thus, droplets of water smaller than one millimeter are dominated by surface tension forces, a fact that is readily apparent in the everyday world from the nearly spherical shape of small water droplets. The forces of surface tension can be used to manipulate small droplets on a

16 30

20 90 10 80 70 60 60 50 40 20 40 80 30 60 40 20 20 80 10

Figure 6: Computed equilibrium shape of a liquid droplet on a patterned substrate consisting of alternating hydrophobic and hydrophilic stripes (Image: reproduced with permission from [33]).

surface to create a variety of useful microﬂuidic devices.

4.3.1 The Static Meniscus

The shape of a liquid drop on a ﬂat substrate may be described by some function h(x, y) giving the height of the free surface above the substrate at each location (x, y). The function h(x, y) has a ﬁnite support, the region D, which is the wetted area on the substrate. The equilibrium shape is the function h(x, y) that minimizes the total energy of the droplet

E[h(x, y)] = Elv + Els + Esv + Eg (25)

subject to the constraint of a ﬁxed liquid volume, V Z h dS = V. (26) D

p 2 2 Here dS = 1 + hx + hy dxdy is a surface element on the free surface (suﬃxes denote partial derivatives). The interaction energies per unit area at the liquid-vapor, liquid-solid and solid- vapor interfaces are denoted by Elv, Els and Esv and Eg is the gravitational potential energy. These quantities are functionals of the shape function h(x, y) and may be written as follows: Z Z Elv = γlv dS, Els = γls dxdy (27) D D Z ρg Z Esv = γsv dxdy, Eg = h dS (28) Dc 2 D

17 where Dc denotes the region outside of D. The constraint equation (26) is enforced by introducing the Lagrange multiplier, p, and minimizing the auxilliary functional Z F [h(x, y)] = E[h(x, y)] − p hdS (29) D The problem of determining the equilibrium shape then reduces to ﬁnding the shape for the domain D and the function h(x, y) that minimizes F [h(x, y)]. The Lagrange multiplier actually corresponds to the pressure within the drop and F is the free energy. Usually, γlv is a constant, for a patterned substrate γls is a periodic function of x and y. The Euler Lagrange equation for the variational problem formulated above can be written down but analytical solutions cannot be found easily. Numerical solutions can be readily constructed with tools such as the Surface Evolver [34]. Figure 6 shows the drop shape on a substrate with a striped pattern of γls determined from a Lattice Boltzmann simulation [33].

An alternative to imprinting a periodic pattern of surface energy on a substrate is to change its texture by sculpting it into a series of peaks and valleys as shown in Figure 8. The liquid drop in this case has a true minimum energy state corresponding to complete wetting of the region between the grooves as well as a metastable state where the drop only makes contact with the tops of the ridges (the analogy is sometimes made to a“fakir” walking on a bed of nails). In order to go from the metastable to the true energy minimum the drop will need to go through intermediate states of greatly increased surface area (energy). If an external disturbance large enough to overcome this energy barrier is not available, the drop remains in the metastable state where the “eﬀective” contact angle is much larger than what it would be in the absence of the texture. Contact angles very close to 180 degrees can be generated in this way, and such surfaces are called “super-hydrophobic”. The leaves of certain plants (such as the lotus) naturally have a micro-textured surface and exhibit super-hydrophobicity (Figure 8). Super-hydrophobic surfaces have many applications due to its water repellant properties.

4.3.2 Moving Contact Lines

The mathematical description of a drop moving on a substrate is considerably more complicated. Figure 7 shows a static drop on a level surface and the same drop sliding down the surface when it is tilted. Unlike the static drop which has a symmetric shape and a contact angle θs given by the Young-Dupr´econdition

γlv cos θs = γsv − γls, (30) the moving drop is asymmetric. The contact angle at the front of the drop, θa, which is called the advancing contact angle exceeds θs whereas the contact angle at the rear, θr, called the receding contact angle, is less than the equilibrium contact angle θs. When U is very close to zero, the contact line moves in a ‘stick-slip’ fashion so that smooth curves of θa and θr as a function of the speed U can no longer be drawn. This phenomenon is called contact

18 Figure 7: A static drop (left) and a drop sliding down (right) an inclined plane. In the latter case the advancing contact angle (θa) at the front of the drop exceeds the receeding contact angle (θr) at the rear. (Image: courtesy of Prof. A. Amirfazli of University of Alberta). line pinning. A consequence of contact line pinning is the phenomenon of “contact angle hysteresis”. Contact angle hysteresis can be demonstrated by the following experiment with the inclined plane. If one measures the sliding speed of the drop (U) at different inclination angles (α) of the inclined plane one obtains a graph of the function α(U). Extrapolating to U → 0 we can obtain the critical angle α0 = limU→0 α(U). One might expect then that α0 is the angle of the incline at which the drop would first start to slide down. However, if a drop is first placed on a level surface and then the surface is gradually tilted, the drop does not start to slide when α = α0. Instead, sliding first starts at some larger inclination α = α∗ > α0. This is known as contact angle hysteresis. Contact angle hysterisis and stick slip can be eliminated for certain molecularly smooth substrates such as those with a monolayer coating. However, for most solid liquid interfaces, they are the norm.

4.3.3 Propulsion Mechanisms

The well known phenomenon of capillary action is an example of how a liquid with a free surface can be made to flow by manipulating the contact angle. When a capillary is dipped into the horizontal free surface of a liquid pool, the interfacial energies cause the free surface to develop a curvature. Such curvature immediately causes a local pressure drop by virtue of the Young Laplace equation 1 1 ∆p = γ + (31) R1 R2 where ∆p is the pressure jump across an interfacial point and R1 and R2 are the principal radii of curvature at that point. This sets up a pressure gradient which then causes the liquid to flow and the contact line to move. The same principle is involved in all cases of droplet motion, the difference lies only in the physical mechanism responsible for changes in

19 Figure 8: (a) Lily leaf showing a rugose coating. SEM, scale bar = 3 µm. (b) Water droplet on the top of leaves from the South American plant Setcreasea. (c) Industrial rugose surface of silica. SEM, scale bar =1 µm. (d) Water droplet on industrial hydrophobic coatings. Reprinted with permission from Nature Materials 4, 277 - 288 (2005). 20 the local contact angle.

If the contact angle is shallow, very often the system admits an asymptotic reduction where the ﬂuid motion is described in the lubrication limit. In such cases, an evolution equation for the height of the free surface h(x, y, t) can be constructed: ∂h = L[h] (32) ∂t where L is some (possibly nonlinear) diﬀerential operator in the variables x and y.

Wettability gradients

If a drop is placed on a chemically treated surface where the static contact angle θs varies, pressure differences are created between the fore and aft which drives a flow. An equation of the form (32) was derived by Greenspan [35], who in addition to the smallness of θs assumed that the contact line moves with a velocity vcl = κ(θ − θs)nˆ where nˆ is the unit normal to the contact line on the plane of the substrate. He further modified the no slip boundary condition as follows: α ∂v v − = 0. (33) 3h ∂z z=0 The constant α is a dimensional constant chosen to be of the order of 10−10 cm2. Equa- tion (33) reduces to the classical non-slip boundary condition a short distance away from the contact line while reducing to a zero shear stress condition at the contact line itself. This is a way to work around the difficulty of the shear stress singularity on the contact line. Neglecting contact angle hysteresis and disjoining pressure effects Greenspan derived the following equation for h(x, y, t) ∂h γ + ∇ · h(h2 + α)∇(∇2h) = 0 (34) ∂t 3µ Greenspan applied equation (34) to some special situations with insightful results, for example, he showed that if θs varies slowly in the x direction, so that the fractional change over the drop length is small, then droplets with a circular wetted area of radius R0 move with a velocity U = −κR0(dθs/dx), thus it migrates towards the more hydrophillic regions.

Thermocapillary Eﬀects

Spatial variations in temperature cause corresponding variations in the coefficient of surface tension γ, which could drive a drop on a substrate. The effect is often seen in the kitchen, heating a little oil in a non-stick pan causes a dry spot to appear at the center with the contact line moving radially out towards the cooler edges of the fry pan. An equation similar to (34) can be developed in this case as well. Since electrical heating wires can be embedded in a substrate, microfluidic devices based on controlled motion of droplets using thermocapillary effects is possible [36].

Marangoni stresses

21 Figure 9: Illustrating the physical phenomenon of Electrowetting on a dielectric or EWOD (Image: courtesy of Prof. J. Loo, UCLA).

Marangoni stresses arise when spatial variations in surface tension is set up due to the presence of a trace chemical such as a surfactant. The classic example of motion generated by Marangoni stresses is the toy called the camphor boat. A receptacle at the rear of the tiny ‘boat’ holds the camphor. The difference in concentration of the camphor between the fore and aft of the boat creates a surface tension gradient which propels the boat. Marangoni stresses can be exploited in microfluidics by using a chemical coating on the substrate so that its wettability can be modulated by shining UV light. Droplets can then be directed to move in a specific pattern by spatial modulation of the UV light on a substrate.

Electrowetting

The contact angle of a liquid drop can be modiﬁed by applying an electric voltage. In early applications the drop was placed directly on top of the electrode. This however has the disadvantage of causing electrolysis and the formation of gas bubbles within the liquid drop. In modern applications, the drop is placed on a metal electrode coated with a thin layer of a dielectric material. This set up is called Electrowetting on Dielectric (EWOD). When a voltage V is applied to the plate, the equilibrium shape of the droplet will change. The new shape can be obtained by repeating the calculation sketched in section 4.3.1, except that the electrostatic energy must be included in the minimization of the total energy. Such a calculation shows that the new contact angle θ can be related to the old one θ0 through the relation 2 cos θ = cos θ0 + V . (35) 8πdγlv Here is the dielectric constant of the insulator and d is its thickness.

22 Figure 10: The liquid lens based on electrowetting by Varioptic: a water drop is deposited on a substrate made of metal, covered by a thin insulating layer. The voltage applied to the substrate modiﬁes the contact angle of the liquid drop. The liquid lens uses two isodensity liquids, one is an insulator while the other is a conductor. The variation of voltage leads to a change of curvature of the liquid-liquid interface, which in turn leads to a change of the focal length of the lens. (Image courtesy of Dr. B. Berge of Varioptic)

An interesting application of the EWOD is the variable focus liquid lens [37] shown in Figure 10. Application of a small voltage (a few volts) can alter the contact angle and thereby change the lens shape. Thus, variable focal length and focussing control can be achieved with no moving parts. If the lens is suﬃciently small (Bond number much less than unity) shape changes due to changes in orientation is negligible because gravity becomes insigniﬁcant. The liquid lens is of course ubiquitous in the natural world; our eyes are made of soft materials (liquids and gels) though shape changes are brought about by pressure rather than electrostatic actuation. It is interesting to observe that one of the earliest microscopes invented by the Dutch inventor Anton van Leeuwenhoek (1623-1723) used a small water drop placed inside a hole in a brass plate as the lens.

The EWOD set up is being applied to implement early versions of the “lab on a chip”. An array of individually addressable electrodes under a dielectric layer is used to move water drops along prescribed paths. By controlling the electrode voltages with electronics, a drop can be divided into daughter droplets or two drops (perhaps containing two diﬀerent chemicals) can be merged and the contents stirred to create a microscopic reaction chamber. A great advantage of the EWOD set up, is that large voltages (and the consequent bulky power supplies) are not needed. Thus, commercial products using EWOD are likely to be more portable. One shortcoming is that evaporative loss from small drops can be signiﬁcant. Figure 11 shows shape oscillations of a drop in response to an ac voltage. Such oscillations can be used to homogenize the contents of the liquid drop.

23 Figure 11: Sequence of images showing a 7-µL KCl droplet oscillating in response to a 60 V AC forcing at 180 Hz. Images are 1 ms apart and scale bar corresponds to 0.5 mm. Such oscillations may be used to mix the contents after reagent A and B are brought together by a pair of coalescing daughter droplets (Image: courtesy of R. Miraghaie, J. D. Sterling and A. Nadim, the Keck Graduate Institute of Applied Life Sciences. See also reference [38]).

5 Future Directions

Since “necessity is the mother of invention” there is no doubt that the field of microfluidics will see rapid growth in the near future fueled by the demands of the biosciences. The direction this growth might take is however much more difficult to predict.

The field of microelectronics has grown from the 1947 Bell Labs Transistor to chips that run the Blackberrys and iPods of today. Nevertheless, the basic elements of the technology such as the use of silicon wafers, thin films, photoresist and etching have not changed but have been progressively refined and improved. The situation is very different in the case of microfluidics. Let us first take the case of materials. Initially glass and silicon wafers were the only platforms being explored. But then soft materials such as PDMS came into the picture [39, 40]. Soft materials have many advantages, for example chips can be mass produced by stamping from a master copy, the flexibility of the material can be exploited for pneumatic pumping [41] etc. However, they do have some drawbacks, for example the zeta potential is hard to control and electrokinetic pumping is not very effiicient. It is unclear at this time whether a unique platform will emerge for microfluidics and if so, which one it is going to be.

Consider next the methods of moving fluids around. Electrokinetic methods have the advantage that the voltage needed to achieve a certain flow speed is independent of the channel diameter. By contrast, for pressure driven flows, the pressure head needed scales inversely as the cross-sectional area. Furthermore, electrokinetic flows have a uniform rather than a

24 parabolic profile and therefore cause less Taylor dispersion. On the other hand, electrokinetic flows are very sensitive to surface contamination, a serious difficulty when dealing with proteins and peptides. Currently both pressure driven and electrokinetic approaches are being explored. One disadvantage of both of these approaches is that while the microfluidic chip itself is small and portable, the power source (compressed air for pressure driven flows or kilovolt power sources for electrokinetic ones) is usually quite bulky. Two approaches that seek to overcome this difficulty are pumping based on ICEO [42] and the approach of EWOD which only require a few volts to move droplets around on a substrate. These approaches have other drawbacks of their own, for example small droplets tend to evaporate and ICEO involves the added complexity of patterned electrodes on one or more of the substrates. It still remains to be seen whether one among these very different approaches would emerge as the winner or if they would continue to co-exist as the subject matures.

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27 6 Books and Reviews

Books:

1. Introduction to Microﬂuidics by Patrick Tabeling (Translated by Suelin Chen), Cam- bridge University Press, New York

2. Microﬂows and Nanoﬂows: Fundamentals and Simulation (Interdisciplinary Applied Mathematics) by George Karniadakis, Ali Beskok and Narayan Aluru, Springer-Verlag, Heidelberg

3. Fundamentals of Microfabrication: The Science of Miniaturization by Anthony Ghiotto, CRC Press, Boca Raton, Florida

4. Microﬂuidics for Biotechnology (Microelectromechanical Systems) by Jean Berthier and Pascal Silberzan, Artech Press, Norwood, MA

5. Low Reynolds number hydrodynamics by John Happel and Howard Brenner, Springer- Verlag, Heidelberg, Germany

6. Boundary Integral and Singularity Methods for Linearized Viscous Flow (Cambridge Texts in Applied Mathematics) by Costas Pozrikidis, Cambridge University Press, Cambridge, UK

Reviews:

1. Microﬂuidics: Fluid physics at the nanoliter scale by T.M. Squires and S. R. Quake (2005) Rev. Mod. Phys. 77, 977-1026.

2. Engineering ﬂows in small devices: Microﬂuidics toward a Lab-on-a-Chip by H.A. Stone, A.D. Stroock and A. Ajdari (2004) Annu. Rev. Fluid Mech. 36, 381-411.

3. Principles of microﬂuidic actuation by modulation of surface stresses by Troian, S. (2005) Annu. Rev. Fluid Mech., 37, 425-455.

4. Electrokinetic ﬂow and dispersion in Capillary Electrophoresis by S. Ghosal (2006) Annu. Rev. Fluid Mech., 38, 309-338.

5. Electrowetting: from basics to applications by Mugele, F. and Baret, J. (2005) J. Phys.: Condens. Matter 17, R705 - R774.