Electrokinetic Flow and Dispersion in Capillary Electrophoresis · 2012-09-28 · bilities is known...

AR266-FL38-12 ARI 21 November 2005 10:11

Electrokinetic Flow and Dispersionin Capillary ElectrophoresisSandip GhosalDepartment of Mechanical Engineering, Northwestern University, Evanston,Illinois 60203; email: [email protected]

Annu. Rev. Fluid Mech.2006. 38:309–38

The Annual Review ofFluid Mechanics is online atfluid.annualreviews.org

doi: 10.1146/annurev.fluid.38.050304.092053

Copyright c© 2006 byAnnual Reviews. All rightsreserved

0066-4189/06/0115-0309$20.00

Key Words

electroosmosis, microfluidics, electrophoresis, Lab-on-a-chip,zeta-potential

AbstractElectrophoretic separation of a mixture of chemical species is a fundamental techniqueof great usefulness in biology, health care, and forensics. In capillary electrophoresis(which has evolved from its predecessor, slab-gel electrophoresis), the sample migratesthrough a single microcapillary instead of through the network of pores in a gel. Afundamental design problem is to minimize dispersion in the separation direction.Molecular diffusion is inevitable and sets a theoretical limit on the best separationthat can be achieved. But in practice, there are a number of effects arising out of theinterplay between fluid flow, chemistry, thermal effects, and electric fields that resultin enhanced dispersion. This paper reviews the subject of fluid flow in such capillarymicrochannels and examines the various causes of enhanced dispersion that limit theefficiency of separation.

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1. INTRODUCTION

Electrophoretic separation of macromolecules such as DNA, RNA, and proteins is anindispensable tool in modern biology, health care, and forensics (Alberts et al. 1994,Watson et al. 2004). Due to the dissociation of molecular groups in the presence ofwater, a macromolecule in aqueous solution often acquires a charge. If an externalelectric field is applied, the molecule migrates in the direction of the electric fieldwith a velocity (v) that is proportional to the field strength (E ). The constant of pro-portionality (µep), which may have either sign, is called the electrophoretic mobilityof the species, thus v = µep E. The separation of a mixture of chemical species intoits components by taking advantage of the differences in their electrophoretic mo-bilities is known as electrophoresis. The calculation of µep from first principles is aninteresting problem in fluid mechanics that has received much attention (Anderson1989, Russel et al. 1989, Saville 1977).

The subject of this review, capillary electrophoresis (CE) was initiated by Hjerten(1967) and developed over the next three decades by various groups. A schematicdiagram of a basic CE apparatus is shown in Figure 1. A microcapillary, usuallymade of fused silica, is stretched between two reservoirs of relatively large capacity.Usually capillaries of 25–75 µm internal diameters and 10–100 cm length are used.The interior of the capillary as well as the two reservoirs are filled with an electrolyte(the buffer). The buffer provides a conducting path for the electric current and mayalso serve other secondary functions; for example, it may contain certain additives toprevent coagulation of the sample or to reduce any tendency of the sample to adsorbto the capillary. The sample to be analyzed (the analyte) is introduced at one end of thecapillary in the form of a plug, which travels down the capillary due to a combinationof the electroosmotic flow (EOF) generated in the capillary as well as due to its ownelectrophoretic mobility. The differences in electrophoretic mobility of the sample

Figure 1A schematic diagram of a basic capillary electrophoresis setup.

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constituents cause them to separate and travel to a detector placed near the outlet inthe form of distinct bands. Most often, the detector is of the UV absorbance type: Thecapillary passes between a small UV light source and a photodetector and the passageof a band is signaled by a drop in the UV light intensity. For specialized applications,various other detection techniques such as laser-induced fluorescence (LIF) may beemployed. Further, the CE output may be interfaced with other analytical devices,such as mass spectrometers, in order to learn about the chemical identity of samplecomponents. The entire set up—capillary, reservoirs, and sample injector—may allbe etched together on a single glass or silicon substrate to make a microfluidic chip.CE has many advantages over the traditional slab-gel electrophoresis techniquesfrom which it evolved. An important advantage is that it can be easily integrated asa component on a microfluidic chip (Auroux et al. 2002, Landers 2003, Reyes et al.2002, Vilkner et al. 2004) to build a Lab-on-a-chip (LOC), a machine capable ofexecuting a complex biochemical task with a high degree of automation. The modeof operation described above is called free solution capillary zone electrophoresis(FSCZE), and it is the only kind that we consider here. Further details about thetechniques of electrophoresis may be found in the many books devoted to the subject(Camilleri 1998, Jorgenson & Phillips 1987, Landers 1996, Weinberger 2000). Barron& Blanch (1995) provide a comprehensive account of DNA separations using bothcapillary and slab-gel electrophoresis. Hu & Dovichi (2002), Quigley & Dovichi(2004), and Karger et al. (1995) review the uses of CE in the analysis of biopolymers.

The resolution of a CE device is controlled by the amount of axial dispersionthat a sample experiences between the time of injection and detection. The axialdispersion also determines the signal strength at the detector. The larger the axialspreading of the sample, the smaller the peak concentration and therefore the weakerthe detected signal. The physical processes that occur within a CE channel involvea complex interplay between fluid mechanics, thermal effects, electrical effects, andchemistry. All of these processes determine the final shape and amount of variance inthe detected signal. This review attempts to summarize our current understandingof the various mechanisms that contribute to axial dispersion of the sample in CEchannels and that also have something to do with the underlying fluid flow. This isonly a subclass of the possible dispersive mechanisms. In particular, dispersion in theabsence of fluid flow such as those associated with electrodiffusion are not covered.The two reviews by Gas & Kendler (2002) and Gas et al. (1997) provide a morecomprehensive discussion of all the different mechanisms of dispersion. The currentreview differs from these earlier accounts in that its focus is on fluid mechanics ratherthan the techniques of analytical chemistry. The recent review by Stone et al. (2004)surveys a much broader area of microfluidics, in particular their section 3.5.2 couldserve as a brief introduction to the subject of this review.

The next section is devoted to a detailed discussion of the fluid flow in CE channels,introducing the basic physics and the essential approximations. The first part ofSection 4 is an elementary discussion of dispersion in an ideal situation in whicha capillary performs at its theoretical best. In the second part, various real-worldeffects that lead to enhanced dispersion are investigated. A summary is provided inSection 5.

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2. FLUID FLOW IN CAPILLARY ELECTROPHORESISCHANNELS—FUNDAMENTALS

2.1. Electroosmosis

Electrophoresis is the motion of a charged particle relative to the surrounding liquiddue to an imposed external electric field. In contrast, electroosmosis is the electricallydriven motion of a fluid relative to the walls of the solid that bounds it. In a typicalCE separation, both electroosmosis (sometimes called electroendosmosis) and elec-trophoresis occur simultaneously. Therefore, the resultant migration speed of eachspecies i is u(i ) = ueof + u(i )

eph, where the first term is the bulk electroosmotic velocity(the same for all analyte components) and the second term is the electrophoreticmigration velocity relative to still fluid of species i (generally different for eachcomponent of the analyte).

EOF was first reported by Reuss (1809), who demonstrated that water couldpercolate through porous clay diaphragms due to an applied electric field. A porousmedium such as clay provides a complex network of microchannels for the motion ofthe liquid. The basic mechanism is more easily understood in the context of a singlelong, narrow cylindrical fused silica capillary of the kind used in CE. The solid fusedsilica substrate (like various other materials such as most minerals, glass, and certainpolymeric materials) acquires a surface charge when in contact with an electrolyte,due to ionization or ion adsorption or both. The resulting surface charge attractsthe counterions (ions of opposite charge) and repels the co-ions (ions of the samecharge) in the electrolyte, creating a thin (1–10 nm is typical) charged layer of theopposite sign next to it. This is known as the electric double layer (EDL), or the Debyelayer. When an external electric field is applied, the fluid in this charged Debye layeracquires a momentum that is transmitted to adjacent layers of fluid through the effectof viscosity. Thus, if the fluid phase is mobile, it would cause the fluid to flow. Thisis known as electroosmosis or EOF (Probstein 1994).

2.2. The Equations of Electrokinetic Flow

Electrokinetic flows, which include both electroosmosis as well as electrophoresis,are described by the incompressible Navier-Stokes equations with a volume densityof electrical forces fe = −ρe∇φ; ρe is the electric charge density and φ is the electricpotential. In most applications the Stokes-flow approximation is adequate, althoughin some cases the Reynolds number Re ∼ O(1). The potential φ is related to thecharge density through the Poisson equation (Feynmann et al. 1970) of electrostatics

ε∇2φ = −4πρe , (1)

where ε is the dielectric constant of the liquid (in CGS units). If the electrolytecontains N species of ions with charges ezk and concentrations nk (k = 1, . . . , N,where e is the magnitude of the charge on an electron), then ρe = ∑N

k=1 ezknk. Eachion species obeys a conservation equation

∂nk

∂t+ ∇ · jk = 0. (2)

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Here jk, the flux vector for the species k, can be modeled by the Nernst-Planckequation for ion transport (Landau & Lifshitz 2002)

jk = −vkzkenk∇φ − Dk∇nk + nku. (3)

In Equation 3, vk is the ion mobility: the velocity acquired by the ion when acted uponby a unit of external force. It is obviously related to the electrophoretic mobility, thevelocity per unit of electric field as µ

(ep)k = 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 zero ion flux normal to the wall jk · n = 0 (n is the unit normal directed intothe fluid). In the absence of external electric fields, the chemistry at the electrolytesubstrate interface leads to the establishment of a potential, φ = ζ . This so-calledζ -potential at an interface depends on a number of factors, including the nature ofthe substrate and ionic composition of the electrolyte, the presence of impurities,the temperature, and the buffer pH, and can exhibit hysteresis effects with relationto pH (Lambert & Middleton 1990). Methods of determining the ζ -potential andmeasured values for a wide variety of surfaces used in microfluidic technology havebeen reviewed by Kirby & Hasselbrink (2004a,b).

The ion distribution near a planar wall at z = 0 with potential φ(z) is known fromstatistical thermodynamics: nk = nk(∞) exp(−zkeφ/kB T), where kB is the Boltzmannconstant and T is the absolute temperature of the solution. For this expression to bea steady solution of Equation 2 we must have the Einstein relation Dk/vk = kB T.Therefore, Equation 3 can also be written as

jk = −nkvk∇ψk + nku, (4)

where ψk = ezkφ + kB T ln nk is called the chemical potential for the species k.

2.2.1. The Gouy-Chapman model. Suppose that the system is in the steady stateand there is no fluid flow or imposed electric fields. Further suppose that the geometryis such that the electrolyte-substrate interface is a surface of constant ψk. Then, itreadily follows from Equations 2 and 4 and the boundary condition of zero flux intothe wall that ∇ψk = 0 everywhere. Therefore, nk = n(∞)

k exp (−zkeφ/kB T ), wheren(∞)

k is the ion concentration where the potential φ = 0, usually chosen as a point veryfar from the wall. Using the solution for nk in the charge density ρe and substitutingto Equation 1, we get the nonlinear Poisson-Boltzmann equation for the potential,

∇2φ = −4πeε

N∑k=1

n(∞)k zk exp (−zkeφ/kB T ) , (5)

with the boundary condition φ = ζ on walls.Equation 5 was the starting point of a detailed investigation of the structure of the

EDL by Gouy (1910) and Chapman (1913). The description of the EDL in terms ofEquation 5 is therefore known as the Gouy-Chapman model of the EDL.

2.2.2. The Debye-Huckel approximation. Equation 5 is a nonlinear equationand solutions can only be constructed by numerical methods. Debye & Huckel


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linearized it by expanding the exponential terms on the right-hand side in Taylorseries and discarding all terms that are quadratic or of higher order in φ, whichgives

∇2φ − κ2φ = 0, (6)

where

κ =[

N∑k=1

4πz2ke2n(∞)

k

εkB T

]1/2

(7)

is a constant determined by the ionic composition of the electrolyte. In arrivingat Equation 6 the condition

∑Nk=1 n(∞)

k zk = 0 has been used, expressing the factthat the bulk solution (φ = 0) is free of net charge. It is easily verified that κ hasunits such that λD = κ−1 defines a length scale that is called the Debye length. Ifa charged plane at a potential ζ is introduced in an electrolyte where the Debye-Huckel approximation is valid, then the solution to Equation 6 may be written asφ = ζ exp(−κz) = ζ exp(−z/λD), where z is distance normal to the plate. Thus, thepotential due to the charged plate is shielded by the free charges in solution, and theeffect of the charge penetrates a distance of the order of the Debye-length λD, whichgives a physical meaning to this very important quantity. For the buffers commonlyused in CE the Debye length is typically between 1 to 10 nm.

The linearization proposed by Debye-Huckel is justified provided that |zkφ| �kB T/e uniformly in all space and for all k. At room temperature kB T/e ≈ 30 mV.However, for silica substrates −ζ ∼ 50–100 mV in typical applications. Thus, theDebye-Huckel approximation is often not strictly valid. Nevertheless, it is a veryuseful approximation because it enormously simplifies mathematical investigationsrelated to the Debye layer. Further, all deductions from it are usually qualitativelycorrect and even quantitative predictions from it outside its range of formal validitytend not to differ much from the true solution.

2.3. Electroosmotic Flow—Analytical Solutions

In the presence of external fields and fluid flow the equilibrium Gouy-Chapmanmodel is generally not applicable and one must proceed from the full electrokineticequations presented earlier. However, if the external field and fluid velocity are bothalong the surfaces of constant charge density ρe , then the presence of the flow orthe imposed field does not alter the charge density distribution, which may still beobtained from the Gouy-Chapman model. Examples where such a situation holdswould be

1. A planar uniformly charged substrate at z = 0 with an applied electric field E0

in the x-direction.2. A uniform infinite cylindrical capillary with an imposed electric field E0 in the

axial (x) direction.3. A narrow slit with uniformly charged walls and an imposed constant electric

field E0 along the slit (x direction)

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For any of the above geometries, the fluid flow equations reduce to (assuming steadystate and zero imposed pressure gradient)

µ∇2u + ρe E0 = µ∇2u − εE0

4π∇2φ(E DL) = 0, (8)

where u is the axial velocity and φ(E DL) is the electric potential for the equilibriumproblem (without the external field or flow). Therefore,

u = εE0

4πµφ(E DL) + χ, (9)

where χ satisfies

∇2χ = 0 (10)

and χ = −(εE0ζ )/(4πµ) at the boundaries. The only solution independent of theaxial direction (assuming infinitely long channel) is χ = −(εζ E0)/(4πµ). Therefore,the velocity is determined in terms of the potential distribution in the equilibriumEDL:

u = εE0

4πµ[φ(EDL) − ζ ] (11)

If we adopt the Debye-Huckel approximation then

φ(EDL) =

ζ exp(−κz) for (1) infinite plane

ζ I0(κr)/I0(κa) for (2) infinite cylindrical capillary

ζ cosh(κz)/ cosh(κb) for (3) narrow slit,

(12)

where a is the capillary radius, r the distance from the axis, and I0 is the zero-ordermodified Bessel function of the first kind. In the last formula, 2b is the channelwidth and z is the wall normal coordinate with origin on the plane (in Case 1) ororigin at a point equidistant between the two walls (in Case 3). Because the fluid flowequation is linear in this limit, a pressure-driven flow can be added to the solutionby superposition in the event that both a pressure gradient and an electric field aresimultaneously applied. The solution for an infinite capillary was first obtained byRice & Whitehead (1965). Solutions in a narrow slit were obtained by Burgreen &Nakache (1964) in the context of the Debye-Huckel approximation as well as for a1 : 1 electrolyte (that is, in our notation N = 2 and z1 = −z2) directly from the fullPoisson-Boltzmann equation.

2.4. Helmholtz-Smoluchowski Slip Boundary Conditions

The characteristic radius of microfluidic channels ∼10–100 µm, whereas, in CE ap-plications, the Debye length λD ∼ 1–10 nm. Thus, the Debye layer is exceedinglythin compared to characteristic channel diameters. Under these circumstances, theNavier-Stokes/Poisson-Boltzmann system described in the last section may be re-placed by a simpler set of equations. Indeed, the EDL then forms a very thin boundarylayer at the solid-liquid interface where the electrical forces are confined.

In the outer region (that is, outside the EDL) we have a fluid that is electricallyconducting but charge neutral. Its motion is therefore described by the Navier-Stokes


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equations without the electrical force term. Inside the EDL, the “inner problem”reduces to the problem of the electric field parallel to a planar interface discussed inthe previous section, and, therefore, the use of the Gouy-Chapman model is justified.From the solution (Equation 11)

limz→ ∞

u(z) = − εζ E0

4πµ. (13)

This is the “outer limit of the inner solution” (Van Dyke 1975) that becomes theboundary condition for the outer problem. Thus, in the limit of infinitely thin EDL,the standard no-slip boundary condition of fluid mechanics is replaced by the follow-ing “slip condition”:

u − usolid ≡ us = − εEζ

4πµ. (14)

In Equation 14, usolid is the velocity of the solid, and u is the velocity of the liquid ata point P on the solid-liquid interface, us is called the slip velocity. E is the electricfield in the buffer due to external sources and can be determined by solving Laplace’sequation within the electrolyte with the boundary condition E · n = 0 on the wall(because the walls are insulating and there can be no current across them).

A formal asymptotic derivation of Equation 14 in terms of the small parameterλD/a0 (where a0 is a characteristic radius) was presented by Anderson (1985). Equation14 is known as the Helmholtz-Smoluchowski (HS) slip boundary condition after thepioneering work of Helmholtz (1879) and Smoluchowski (1903).

As an example, for EOF in an infinite cylindrical capillary driven by a constantelectric field E0 and with no applied pressure gradient, we need to solve Stokesequation with the boundary condition u = us at the walls. The solution, in the limit ofinfinitely thin EDL, is simply uniform flow with velocity us . For comparison, Figure 2shows the profile u(r) normalized by us calculated from Rice & Whitehead’s (1965)solution, discussed in the last section. The boundary-layer structure and convergenceto the thin EDL solution is obvious for κa = a/λD � 1. The plug-flow profile wasexperimentally verified by Taylor & Yeung (1993), Tallarek et al. (2000), and Herret al. (2000), among others.

3. FLUID FLOW IN CAPILLARY ELECTROPHORESISCHANNELS—COMPLEX FLOWS

Problems involving nontrivial geometry or other complicating factors becomeamenable to exact analysis as a consequence of the HS slip boundary conditions.In this section we examine some examples of relevance to the dispersion problem inCE.

3.1. Similitude Between Electric Field and Flow Velocity

Consider the problem of EOF in a cylinder of arbitrary cross section but in the thinEDL limit. Further, assume that all fluid and material properties (such as µ, ε, ζ, . . .)are constant. Consider the region bounded by the interface between the liquid and

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Figure 2Velocity profiles(normalized by theHelmholtz-Smoluchowskislip velocity) as a function ofradial distance (r) in acylindrical capillaryaccording to the solution ofRice & Whitehead (1965)for κa = (a/λD) = 1, 10, 30;here a is the capillary radiusand λD is the Debye length.

substrate (Sw) and two equipotential surfaces SA and SB on which the potential φe

equals φA and φB , the inlet and outlet reservoir potentials. Because there can be nocurrent into the walls, n ·∇φe = 0 on Sw, and in the interior of the channel ∇2φe = 0,as there is no net charge. Now it is readily verified that

u = εζ

4πµ∇φe (15)

is both solenoidal and irrotational, and therefore satisfies the incompressible steadyNavier-Stokes equation:

(∇ × u) × u = −∇(

u2

2+ p

ρ

)− ν∇ × ∇ × u, (16)

with the pressure p determined from the Bernoulli equation p/ρ + u2/2 = C , whereρ is a (constant) density of fluid and C is a constant. Further, Equation 15 satisfies theHS slip boundary conditions on Sw, and also the condition of conservation of fluidvolume ∫

SA

u · n d S =∫SB

u · n. (17)

Thus, Equation 15 is a solution of the fluid flow problem through the conduit.However, it is not necessarily the only solution because any solenoidal field that isirrotational and zero on Sw can be added to it, and one would still satisfy Equation 16.


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Indeed, even a solenoidal field that is rotational can be added provided the resultantfield satisfies Equation 16 and vanishes on Sw. Equation 15 is the unique solutiononly when conditions at the lateral boundaries SA and SB are such as to exclude thesealternate solutions.

The existence of this “similitude” between the electroosmotic velocity u and theelectric field −∇φe was first pointed out by Morrison (1970) in the context of elec-trophoresis of small particles of arbitrary shape. In his case, the flow is unbounded,and it was shown that the imposition of the constraints: (a) the net force on the par-ticle is zero and (b) the net torque on the particle is zero, led to a unique solution.A consequence is that the Smoluchowski expression for electrophoretic mobility,

µep = − εζ

4πµ, (18)

is valid for particles of arbitrary shape, not just spheres for which the Smoluchowskiformula was originally derived. Cummings et al. (2000) also discussed the conditionsunder which Equation 15 would be a solution of the problem of EOF in a conduit.

3.2. Axially Varying Flows

It is clear that if the prefactor multiplying ∇φe in Equation 15 fails to be a constant,then u does not satisfy either the continuity equation or the Navier-Stokes equationand therefore is not a solution to the flow problem. Such inhomogeneous problemsfrequently arise in CE. For example, adsorption of small amounts of analyte to thewall could alter the ζ -potential, and adsorption occurs in an axially inhomogeneousmanner with greater contamination near the capillary inlet. This could cause vari-ations of the ζ -potential in the axial direction of the CE channel. Apart from theseaccidental inhomogeneities, ideas about the possibility of building useful microflu-idic devices with engineered axial variations in ζ have emerged in recent years. Forexample, Ghowsi & Gale (1991), Hayes & Ewing (1992), and Lee et al. (1990) in-vestigated the possibility of a dynamically modulated ζ -potential using an externalvoltage source. Barker et al. (2000) considered channels chemically imprinted withthe desired variation of ζ by using polymeric coatings. The possibility of microfluidicmixers based on engineered variations in ζ was suggested by Stroock et al. (2001).

3.2.1. Exactly solvable models. Anderson & Idol (1985) considered a simple modelwith axially varying ζ -potential in the thin EDL limit. They considered the problemof a very long (length, L, much larger than its radius, a0) uniform, straight cylindricalcapillary with a ζ -potential that varies only in the axial direction, ζ = ζ (x). A potentialdrop �V = EL is imposed between inlet and outlet, and the reservoir pressures at theinlet and outlet are considered equal (�p = 0). Using the HS slip condition, Anderson& Idol (1985) derived the following exact solution to the Stokes flow problem incylindrical coordinates: u = ux + rv = −r−1(∂rψ)x + r−1(∂xψ)r, where x and rdenote the axial and radial coordinates and the stream function

ψ = εE4πµ

[r2

2〈ζ 〉 − 2

∞∑m=1

acm(r) cos

(2mπx

L

)− 2

∞∑m=1

a sm(r) sin

(2mπx

L

)], (19)

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whereac

m

ζ cm

= a sm

ζ sm

= a0r I0(αm)I1(αmr/a0) − r2 I0(αmr/a0)I1(αm)αm I1

2(αm) + 2I0(αm)I1(αm) − αm I02(αm)

, (20)

and ζ cm, ζ s

m are the cosine and sine transform of the ζ -potential. Here 〈〉 indicates theaverage over the length of the capillary 〈 f 〉 = L−1

∫ L0 f dx. In Equation 20 In denotes

the modified Bessel function of integer order n, and αm = 2πa0m/L.The above solution implies a remarkably simple formula for the cross-sectional

average u of the axial velocity (or equivalently, the volume flux per unit cross-sectionalarea),

u = − ε〈ζ 〉E4πµ

. (21)

Thus, the flux per unit area over any cross section is the same as that of a uniformcapillary with ζ = 〈ζ 〉. This result can be generalized to a channel of any cross-sectional shape and arbitrary variations of ζ provided that any such variations occuron a length scale that is much larger than a characteristic channel width (Ghosal2002c).

One consequence of Equation 21 is the phenomenon of elution-time delays insituations where the analyte has an affinity for the wall and can attach to it and alterits ζ -potential (Ghosal 2002b). This lack of reproducibility in migration times hasreceived considerable attention as it makes it difficult to calibrate CE systems (Cohen& Grushka 1994, Lee et al. 1994, Towns & Regnier 1992). If X(t) denotes the positionof an analyte plug at time t in a capillary of length L, then, using Equation 21, onehas

dXdt

= ueo〈ζ/ζ0〉 + uep, (22)

where ueo = −εζ0 E/(4πµ) is the electroosmotic speed in the unmodified capillary(for which ζ = ζ0), and uep is the electrophoretic migration velocity. To calculate theaverage 〈〉, we need to know exactly what the modified ζ -potential is behind the plug,which of course cannot be known without solving the detailed transport equations forthe analyte discussed in Section 4. However, the special situation where the adsorbedanalyte completely neutralizes the charge on the capillary allows an exact solution tobe obtained, because in that case, ζ (x, t) = 0 if x < X(t) and ζ (x, t) = ζ0 otherwise.Thus,

dXdt

= ueo〈ζ/ζ0〉 + uep = ueo(1 − X/L) + uep, (23)

which can be solved to give the arrival time t(X) at location X down the capillary:

t(X) = Lue

ln[

ueo + uep

ueo(1 − X/L) + uep

]. (24)

t(X) is a monotonically increasing function of X, as seen in the experimental datadue to Towns & Regnier (1992) shown in Figure 3. In the experiment a small fixedamount of protein mixed with a neutral marker was injected into the capillary andthe arrival time of the marker at two different downstream points was noted. t(X)increases monotonically with X as in Equation 24, with larger delays corresponding


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Figure 3Elution time delays due to interaction of protein samples with walls of a fused silica capillary.Experimental data replotted from Towns & Regnier (1992); lines are fit to a model (see text).

to more cationic (positively charged) proteins that adsorb more strongly to the walls.The lines show a model similar to Equation 22, but assuming a prescribed exponentialform for the ζ -potential behind the plug to account for the fact that the ζ -potentialbehind the moving plug is only partially neutralized (Ghosal 2002b).

Further examples of exactly solved models related to flow over surfaces with vari-ations in the ζ -potential were presented by Ajdari (1995, 1996). There the flow isconfined between a pair of parallel plates at z = ±h under the application of a uni-form external electric field, E, and arbitrary position-dependent variations of thezeta-potential on the surface. Long et al. (1999) obtained analytical solutions in theneighborhood of localized defects in the zeta-potential for both the parallel plate aswell as cylindrical geometries, including a step-wise change in the ζ -potential in aninfinitely long uniform cylindrical capillary.

3.2.2. The lubrication approximation. When the HS slip velocity is variable overthe capillary surface analytical solutions for the flow field are usually difficult to

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find except for the simple geometries discussed in the last section. However, if thevariations are slow in the axial direction, that is, the length scale for characteristicaxial variations is much larger than the channel width, lubrication theory (Batchelor2000, Lighthill 1968) permits analytical solutions to be obtained even for channelswith complicated geometrical shapes.

Ghosal (2002c) obtained the following solution for the flow in a channel of anyshape, and arbitrary distribution of ζ -potential provided that variations of channelgeometry and ζ in the x-direction are slow and that there is no curvature of thechannel centerline. If ε is a small parameter (not to be confused with the dielectricconstant, ε) representing the ratio of a characteristic width to a characteristic lengthscale for axial variations, then u ∼ iu(x, y, z) + O(ε), E ∼ iE(x) + O(ε), and

u = −u p

µ

d pd x

+ εF4πµ

ψ

A(x), (25)

Q = − u p

µA(x)

d pd x

+ εF ψ

4πµ, (26)

E(x) = F/A(x). (27)

Here F is a constant representing the electric flux through any cross section, A(x) isthe cross-sectional area, and the overbar indicates average over the cross section, f =A−1

∫f dy dz. The constant Q represents the volume flux of fluid through any cross

section. The functions u p , defined by ∇2u p = −1 and u p∣∣∂ D(x) = 0, and the function

ψp defined by ∇2ψp = 0 and ψp∣∣∂ D(x) = −ζ , are properties of the channel geometry

and charge distribution alone. They are defined on the domain D(x) representingthe cross section of the channel with boundary ∂ D at axial location ‘x’. According toEquations 25–27, the flow velocity in the axial direction in a slowly varying channelis a linear superposition of a purely pressure-driven flow and a purely EOF. Theaxial pressure gradient and electric field are calculated by using the two conditions:Equations 26 and 27. These conditions express the physical facts that (a) the fluid isincompressible, and (b) the electric flux must obey the Gauss law of electrostatics. Thesolution is completely specified by two independent physical constants, the volumeflux of fluid, Q, and the electric flux, F. If desired, these constants may be expressedin terms of the total pressure drop and the total voltage drop, respectively, betweenthe inlet and outlet sections, yielding the following generalization of Poiseuille’s law:Q = QP + QE , where QP is proportional to the applied pressure drop and equalsthe flow rate through a channel of “effective radius” a∗ that can be expressed purelyin terms of channel geometry. QE is proportional to the potential drop and equalsthe electroosmotic flux through a cylindrical channel of radius a∗ and some effectiveζ -potential, ζ∗, that can be calculated from the known distribution of the actual ζ -potential.

Figure 4 illustrates the accuracy of the lubrication approximation by comparingAnderson & Idol’s exact solution in Section 3.2.1 with the prediction of Equation 25–27 for the example of an infinite cylindrical capillary of radius a0 with a sinusoidallyvarying ζ -potential ζ (x) = ζ0 + �ζ sin(2πx/λ). Using dimensionless variables


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Figure 4Comparison of theAnderson & Idol (1985)exact solution (lines) withthe result of the lubricationapproximation (circles) for aninfinite cylindrical capillary(radius a0) with sinusoidalvariation (wavelength λ) ofζ along its length fora0/λ = 0.01, 0.1, 0.5, 1.0,2.0, and 10.0.

ρ = r/a0 and X = x/a0, the axial velocity has the form

uu0

= 1 + �ζ

ζ0F (ρ) sin(αX), (28)

where u0 = −εEζ0/(4πµ), α = 2πa0/λ and

F (ρ) =α−1 I0(αρ)

[1 − α I0(α)

2I1(α)

]+ ρ

2 I1(αρ)

α−1 I0(α) + 12 I1(α) − I2

0 (α)2I1(α)

, (29)

from Anderson & Idol’s solution, and

F (ρ) = 2ρ2 − 1, (30)

according to lubrication theory. For small values of α, the two expressions are asymp-totically equivalent. Figure 4 compares Equation 29 with Equation 30 for severalvalues of a0/λ. For λ ≈ 10a0 or greater, the prediction of the lubrication analysis isin excellent accord with the exact solution, as expected. For λ = a0 or less, the exactsolution departs widely and is even qualitatively different from the lubrication theoryresult. At such short wavelengths the frequent reversal of the electric force resultsin no net transfer of momentum to the interior of the fluid, and u ≈ u0. Thus, itis the long wavelength variations in ζ that are effective in causing significant flowmodification and associated Taylor dispersion in the channel.

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4. DISPERSION IN CAPILLARY ELECTROPHORESISCHANNELS

4.1. Elementary Considerations

We have seen that EOF in a uniform narrow cylindrical capillary takes the form ofa plug flow (except within a thin Debye layer), u = ueo (a constant). Therefore, acomponent that has electrophoretic velocity uep and molecular diffusivity D has itsconcentration c evolve according to a one-dimensional advection-diffusion equationwith a constant advective velocity uep + ueo and diffusivity D. If the initial peak widthis considered negligible, the variance after time t is σ 2 ≈ 2Dt. The time betweeninjection and detection is t = L/(ueo + uep), where L is the capillary length. Therefore,

σ 2 = 2Dt = 2DL(ueo + uep)

= 2DL(µeo + µep)E0

= 2DL2

(µeo + µep)V, (31)

where µeo and µep are the electroosmotic and electrophoretic mobilities, respectively,and V is the voltage applied across the capillary. The last equation can be written indimensionless form as

N ≡ L2

σ 2≡ L

H= (µeo + µep)V

2D= Nmax . (32)

The dimensionless quantity N is called “the number of theoretical plates” and isthe most important performance measure in CE separations. Alternatively, the plateheight H = σ 2/L: The variance collected per unit length traversed may be used. Hhas dimensions of length and N = L/H. An interesting aspect of Equation 32 is thatN is independent of L and proportional to the voltage V applied across the capillary.In practice, N ∼ 105 − 106 and to achieve these high plate counts V ∼ 30 kV arecommon. In principle, Equation 32 predicts that N is independent of the capillarylength L. However, there are practical limitations on how small one could take L.Because the resistance R is proportional to L, and the heat generated per unit time∼V2/R ∼ V2/L, L cannot be too small. Due to the high voltages involved in CE,Joule heating is a significant concern.

The resolution (Rs) in CE is defined as the distance at the detector between twopeaks divided by four times the standard deviation of each peak. It is easily shown(Giddings 1969, Jorgenson 1987) that

Rs =√

N4

�µ

µ, (33)

where N is the number of theoretical plates, �µ is the difference in mobility betweentwo species of similar mobilities, and µ is the average of the mobilities. Thus, twopeaks can be resolved only if �µ > 4µ/

√N. A concise discussion on dispersion in

CE is provided by Jorgenson & Lukacs (1981).

4.2. Anomalous Dispersion

Because EOF has an essentially flat profile (except in a thin EDL near the walls), unlikea classical pressure-driven flow, an analyte plug introduced in it is not subjected to any


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Taylor dispersion. Even under these conditions, some axial spreading is unavoidablesimply due to molecular diffusion, and Equation 32 represents this ideal diffusion-limited separation. However, in practice, various physical effects may arise that causeN to be smaller than Nmax . Frequently, the cause of such enhanced dispersion is someinhomogeneity in the capillary that causes the EOF to deviate from the ideal plugflow profile, resulting in significant Taylor dispersion.

4.2.1. Dispersion due to Joule heating. Typical CE systems operate at voltagesof the order of tens of kilovolts. At such high voltages, the flow of electric currentthrough the buffer produces a significant amount of Joule heat, which results intemperature variations in the microcapillary. At worst Joule heating could result incatastrophic failure: convective overturning that obliterates all bands or productionof microbubbles, resulting in complete blockage of EOF and current (“vapor lock”).In modern CE systems with capillary diameters less than 100 µm, such failures arerare; however, Joule heating can be a significant source of band broadening.

In Joule heating, the source density for heat is j · E = σ E2, where j is the currentdensity, E is the electric field, and σ is the electric conductivity. Thus, if variationsof conductivity with temperature are neglected, the heat source is uniform withinthe capillary. The transport problem for temperature in a cylindrical capillary withuniformly distributed heat sources can readily be solved (Burgi et al. 1991, Davis1990), yielding a radially varying temperature field, T(r). The electrophoretic mi-gration speed of molecules varies inversely with the fluid viscosity µ. Because theviscosity µ varies with temperature T, which in turn varies with r, analyte moleculesnear the wall migrate with a slightly different speed than those at the center. Thiswould lead to band broadening in the sample plug. However, the radial variationof viscosity does not cause any distortion in the EOF. This is because u(r) = u(a0)still satisfies the Stokes equation with µ = µ(r), and the HS slip boundary condi-tion is satisfied if u(a0) = −(εζ E)/(4πµ(a0)), because in Equation 14, µ refers toµ in the EDL, which in this case would be µ(a0). Thus, it is the modification ofthe electrophoretic velocity, not the electroosmotic velocity, that causes band broad-ening, as pointed out by Knox (1988), who studied the thermal band-broadeningphenomenon. Knox, like Grushka et al. (1989), provided an expression for the plateheight.

Andreev & Lisin (1992) analyzed the problem by numerically solving the coupledequations for Stokes flow, the Poisson Boltzmann equation for the electric potential,and the advection-diffusion equation for the analyte concentration, taking into ac-count the dependence of the transport coefficients on concentration and temperature.The thermal equation was integrated analytically assuming a small temperature dropbetween the axis and the wall of the capillary, which implies a parabolic distributionof the temperature profile. It was found that, depending on the parameter regime,the effect of the nonuniform temperature on the electroosmotic flow could some-times dominate effects due to variations of the electrophoretic velocity. This is dueto the finite Debye-layer effects. Numerical solutions of the thermal transport equa-tions have been presented by various authors incorporating various levels of detail intheir models. Thus, Grushka et al. (1989) assumed a polyimide-coated capillary and

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constant electrical and thermal conductivity of the buffer. Jones & Grushka (1989)accounted for the temperature dependence of the buffer electrical conductivity. Inthat case, the thermal source term has a radial dependence. Approximating the sourceterm with a linear function of temperature results in a temperature profile in termsof the zeroth-order Bessel function instead of the parabolic radial dependence. How-ever, Jones & Grushka (1989) showed that, assuming constant conductivity, undertypical CE operating conditions, the correction to the parabolic temperature profileobtained is very small.

In addition to the obvious radial dependence on temperature, axial variations intemperature could also occur due to various inhomogeneities in the capillary. Suchvariations could induce pressure gradients and lead to band broadening. It is not clearwhether such axial variations are present and, if so, whether they cause significant dis-persion. Convective motion of fluid in the capillary, like thermally driven instabilitiesof various kinds, seem possible, but appear not to have been investigated.

4.2.2. Finite Debye-layer effects. In the thin EDL limit, most of the cross-sectionalarea of the capillary is free of shear, so that N ≈ Nmax . However, there is a zone ofshear in a region of thickness ∼λD from the wall. The contribution of this zoneof shear in the EDL can easily be estimated by actually calculating the Taylor-Arisdispersion coefficient for the EOF profile derived by Rice & Whitehead (1965), pre-sented in Section 2.3. This was done by Datta & Kotamarthi (1990) for a uniforminfinitely long capillary, and independently by Griffiths & Nilson (1999) by a differ-ent method, and also by McEldoon & Datta (1992) in the more general case wherethe solute could interact with the wall. Griffiths & Nilson (2000) later extended theanalysis to the case where the ζ -potential need not be small and, therefore, the Debye-Huckel approximation cannot be made: In that case most of the calculations mustbe done numerically. Stedry et al. (1995) and Gas et al. (1995a) replaced the actualvelocity profile by the combination of a plug flow region in the core and a stationaryannular region near the wall and obtained a simplified expression for N that dependson the thickness of the stationary zone. The evaluation of this thickness requires anumerical integration of the Poisson-Boltzmann equation. Zholkovskij et al. (2003)considered the EOF through a cylindrical capillary of arbitrary cross-sectional shape.The Debye length is small but finite; however, the Debye-Huckel approximationis not invoked. The Taylor-Aris dispersion formula for an arbitrary cross-sectionalshape is used to express the shear-induced dispersion in terms of the velocity distribu-tion over the cross section. An approximation for this velocity distribution is obtainedby assuming that the potential in the EDL can be replaced by a thin “ribbon” withinwhich the solution of the one-dimensional Poisson-Boltzmann equation is applica-ble. Such an approximation requires the radius of curvature at every point on theboundary curve to be much larger than the Debye length. For sharp corners, such aspolygonal shapes, the contribution from the overlapping regions of the EDL must besmall.

Typical Debye lengths in microfluidic applications are λD ∼ 1–10 nm and typicalradii are a0 ∼ 10–100 µm, so the effect of the finite EDL on axial dispersion is usuallysmall.


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4.2.3. Dispersion due to channel curvature. In tabletop CE units employing cylin-drical microcapillaries, the capillary is usually straight or has a radius of curvature thatis much larger than the capillary diameter, so it can be considered essentially straight.However, a CE channel of the same length cannot be etched on a microfluidic chipof modest footprint without introducing turns. It may appear from Equation 32 thatone could perhaps scale down the CE channel because the separation efficiency isindependent of the channel length. However, the resolution depends on the appliedvoltage and a short capillary could lead to unacceptable levels of Joule heating becauseof the very high voltages employed in CE. Therefore, turns appear to be inevitablein any microfluidic-based CE system. One way in which fluid mechanics could con-tribute to microfluidic technology is by helping one understand the role of channelcurvature in the axial dispersion of analytes and how channels could be designed tominimize such effects.

The basic mechanism of dispersion in curved channels may be understood withreference to Figure 5. In the thin EDL limit, because the external electric field istangential to the wall, the isopotential surfaces must intersect the channel boundariesat right angles, as shown by the dashed lines. Thus, in Figure 5 the same poten-tial drop, VA − VB , occurs over a shorter distance (AB) on the inner side of a curvethan on the outer side (A′ B ′). As a result, the applied voltage creates a stronger fieldon the inside edge of the channel and, therefore, by the slip boundary condition,

Figure 5Illustrating themechanism of dispersioncaused by channelcurvature.

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drives a faster flow. Thus, a shear is created in the EOF with the fluid flowing fasteron the inner edge, as shown in Figure 5. Solute particles near the inner edge ofthe turn traverse a shorter distance at this higher speed than particles at the outeredge, the so-called race track effect. Thus, as a band goes round the bend it issheared out of shape as shown in the figure. Cross-stream molecular diffusion actson this shear, resulting in an enhanced effective axial dispersion due to the Taylor-Aris mechanism. Minimizing such turn-induced “geometric dispersion” is a sub-ject of great current interest in which rigorous fluid mechanical insight can bringforth advancements in design. The short report by Zubritsky (2000) provides an in-teresting overview of the various approaches being pursued in trying to solve thisproblem.

In addition to the time needed to diffuse across the capillary, tD ∼ a2/D, wherea is a characteristic width, there is a second timescale in this problem. This is thecharacteristic residence time of the sample in the curved section of the capillary,tR ∼ Rθ/ue , where R is the radius of curvature, ue is a characteristic electroosmoticspeed, and θ is the angle of the turn. Therefore,

tR

tD= Pe−1 R

aθ, (34)

where Pe = aue/D is a Peclet number. Two qualitatively different regimes exist basedon the ratio of these two times.

Diffusion-dominated regime: If Pe is sufficiently small, or the turns have relativelylarge radius of curvature, tR � tD. In this case the cross-stream diffusion is dominantso that the concentration is almost constant across the capillary. Thus, we are in thelong time limit of the Taylor-Aris theory.Advection-dominated regime: If Pe is sufficiently large, or the turns have rel-atively small radius of curvature, tR � tD. This is usually the case with analytescontaining very large molecules such as proteins. In this case the dispersion may becalculated by purely geometrical means because the solute is simply advected alongstreamlines.

The following formula was proposed by Griffiths & Nilson (2002) for the turn-induced axial variance σ 2 in a two-dimensional planar channel of width ‘a’ :(σ

a

)2= θ2Pe

15r∗θ + 3Pe+ 2r∗θ

Pe. (35)

Here r∗ is the mean of the radii of curvature of the inner and outer walls normalizedby the channel width, and Pe = aU/D is the Peclet number based on the (uniform)flow speed far upstream of the bend (U) and the molecular diffusion coefficient (D).Equation 35 is an empirically constructed “composite expression” that reduces to thecorrect limits in the low and high Peclet number regimes. In these two limits theexpression for the variance take the following forms

σ 2 ∼ 2DtR (Pe → 0), (36)

σ 2 ∼ θ2a2

3, (Pe → ∞), (37)


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where r = ar∗ is the dimensional mean radius of curvature and tR = rθ/U is the transittime across the bend. The former expression clearly has the form of the variance dueto molecular diffusion. In the second case we have a purely geometrical quantity thatis independent of the diffusion coefficient D but depends solely on the difference ofpath lengths aθ between the inner and outer edges of the bend.

In microfluidic applications, Pe ∼ 101−102 for small molecules but Pe ∼ 103−104

for large macromolecules, so the high, low, and intermediate Peclet number regimesare all relevant in practice. Culbertson et al. (1998) also considered the problem ofgeometric dispersion in a rectangular channel undergoing a 180-degree turn. Theyused an ad hoc modification of the expression for the axial stretching of an initiallysharp band in the absence of diffusion to incorporate diffusive effects. The model wasfitted to experimental data.

The first term in Equation 35 increases linearly with Pe for small values of Pe andsaturates at large Pe. The dispersion contribution from this term can be reduced if Pecan be reduced or if r∗ can be increased. Two main ideas for designing low dispersionbends have evolved out of these two possibilities. The channel can be pinched so thatit becomes very narrow in the curved sections. This effectively reduces the Pe locallyby decreasing a. This approach was investigated by Paegel et al. (2000). Alternatively,one could design the separation channel in the form of spiral turns of large radii ofcurvature. This approach, which relies on increasing the effective radius of curvature,r∗, has been followed, for example, by Culbertson and others (Culbertson et al. 2000,Gottschlich et al. 2001, Miyahara et al. 1990). A third possibility is to redesign thechannel geometry at the bend to compensate for both the higher electric field and therace track effect. This is equivalent to attempting to reduce the prefactor multiplyingthe Pe in the first term of Equation 35 by altering the velocity distribution. Thisapproach has been adopted, for example, by Molho et al. (2001) and Mohammadiet al. (2000), who tried to come up with optimal shapes using computer simulation.The results were compared to experimental data and showed reasonable agreement.Griffiths & Nilson (2001) proposed similar designs. Others have attempted to com-bine several of these ideas. For example, Dutta & Leighton (2002) proposed spiralswith inner walls that are wavy to compensate for the shorter path, thereby bothincreasing r∗ as well as reducing the geometric prefactor in Equation 35. Johnsonet al. (2001) achieved the same effect by modifying the wall ζ -potential throughlaser ablation, which also changes the pattern of EOF in the channel. Fiechtner &Cummings (2003) proposed a faceted design, which is a polygonal shape approximat-ing a smooth spiral. Griffiths & Nilson (2002) investigated pleated channels wheresome of the turn-induced dispersion is undone at the following turn in the oppositedirection.

4.2.4. Dispersion due to inhomogeneities in the ζ -potential. The nonuniformadsorption of some analytes onto the capillary wall alters the ζ -potential, creatinga variation of ζ in the axial direction. As discussed earlier, such variations inducea pressure gradient and secondary flow. The resultant flow has shear- and cause-increased spreading due to Taylor dispersion. The effect is particularly severe with

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proteins because many proteins are positively charged (Bonvent et al. 1996) in thephysiological pH range and have a strong affinity for the negatively charged capillarywalls.

Early work on the wall interaction problem is due to Gas et al. (1995b) and Stedryet al. (1995). Although their analyses consider that analyte is lost to the wall, the con-sequent modification of the ζ -potential, and therefore the hydrodynamic flow field,are neglected. Numerical simulation of similar purely kinematic models that neglectthe perturbation in the hydrodynamic field have been presented by other authors aswell (Ermakov et al. 1995, Schure & Lenhoff 1993, Zhukov et al. 1997). Minariket al. (1995) use experimental data to fit various empirical models for predicting plateheight in the presence of wall interactions. A detailed experimental study of wall ad-sorption and its consequences (Towns & Regnier 1992) show unambiguously that it isthe axial variation of the ζ -potential and consequent pressure-driven secondary flowthat cause the observed enhancement of axial dispersion. Potocek et al. (1995) shownumerical solutions to the advection-diffusion equation for a neutral scalar field in aflow field generated by solving the steady Stokes equation in a section of the capillarywith a specified axially varying slip velocity. The results agree qualitatively with realobservations related to peak broadening.

Figure 6 represents one of the experiments conducted by Towns & Regnier (1992)to quantify the extent of dispersion generated by variability in the ζ -potential. In thisexperiment, the first 15 cm of a 100-cm long CE microcapillary was coated with acertain polymeric material that altered the ζ -potential. Thus, the distribution of ζ

becomes a step function, ζ = ζ1 in the coated section and ζ = ζ0 in the rest of thecapillary. A neutral marker (does not adsorb to the wall and has no electrophoretic ve-locity) is introduced at the inlet and its dispersion observed at the detector is recorded.The experiment is repeated after snipping off a 3-cm length of capillary from the inlet(“cut 1” in Figure 6) and adjusting down the potential to keep the electric field thesame. This results in an altered ratio of coated to uncoated lengths. After recordingthe data the procedure is repeated until only 70 cm of the original capillary is left.Figure 6 shows data from this experiment. As a control, the exact same experimentwas repeated with an ordinary uncoated 100-cm long capillary and the resulting datais shown (square symbols) alongside the experiment with the partly coated capillary(circle symbols). When the capillary length is less than 85 cm (that is, all of the coatedsection has been removed), the two sets of data coincide, showing that coating a sec-tion of the capillary did not inadvertently change the ζ -potential on the remainingsection.

The flow field may be determined by representing the flow in each section of thecapillary as a linear superposition of a pure electroosmotic and a pure Poiseuille profile(Ghosal 2002b, Herr et al. 2000). The unknown constants can then be determinedusing the conditions of mass continuity and the requirement that the total pressuredrop between inlet and outlet be zero. The Taylor dispersion coefficient in any sectioncan be calculated using the amplitude of just the Poiseuille part of the velocity profile.The solid line in Figure 6 shows N determined in this way as a function of capillarylength in the case of the partially coated capillary. The dashed lines on either siderepresent the uncertainty in the prediction and the dotted line is a fit to the data


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Figure 6Dispersion caused by a step change in the ζ -potential. Experimental data replotted fromTowns & Regnier (1992); lines represent theoretical calculation (see text).

for the uncoated capillary that determines some of the experimental parameters (seeGhosal 2002a for details).

Herr et al. (2000) performed a similar experiment in which they neutralized theζ -potential in a section of the capillary and then used the caged laser fluorescencetechnique to visualize the analyte plug. As expected, enhanced shear-induced spread-ing occurs when a step function in the ζ -potential is created in this manner. However,

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unlike Towns & Regnier, Herr et al. do not provide quantitative data on the amountof added dispersion. Chien & Helmer (1991) also measured peak variances in a sys-tem with a step function in the HS slip velocity. They created such a step changethrough the rather simple arrangement of partially filling the capillary with a secondbuffer of different conductivity. Constancy of the current density ( J ) requires thatJ = σ1 E1 = σ2 E2, where σ denotes the conductivity and E the electric field strengthsin each of the buffers 1 and 2 indicated by the suffixes. Because σ1 �= σ2, E1 and E2

are different. Because the flow is determined solely by the slip velocity (Equation 14),a step change in the ζ -potential is indistinguishable from a step change in the electricfield.

In general, wall adsorption of analytes and the consequent alteration of ζ lead-ing to flow modification happen simultaneously and therefore ought to be treatedtogether as a coupled nonlinear system. This was done by Ghosal (2003), whoused the assumption of slow variations to derive a pair of coupled one-dimensionalpartial differential equations for the cross-sectionally averaged analyte concentra-tion in the buffer, c (x, t), and the concentration of analyte adsorbed on the wall,s (x, t). Shariff & Ghosal (2003) performed a numerical simulation of the fullthree-dimensional Stokes equations coupled with the advection-diffusion equationfor the analyte concentration c (r, x, t) with a Langmuir adsorption law for wallinteractions:

−D∂c∂r

∣∣∣∣r=a0

= ka c (a0, x, t)(sm − s ) − kd s , (38)

where ka and kd are adsorption and desorption coefficients and sm is the maximumconcentration that the wall can hold. The geometry was that of a cylindrical mi-crocapillary of radius a0. The ζ -potential was assumed to decrease linearly withadsorbed concentration, s. Because in the Stokes limit the flow is exactly knownfor any ζ = ζ (x, t) from Anderson & Idol’s solution discussed in Section 3.2.1,the velocity field can be determined at each time step. Thus, only the advection-diffusion equation for c needed to be solved and this was done using a finitevolume approach. The correspondence between the one-dimensional asymptoticsystem and the full three-dimensional simulations was excellent for the parameterstested.

Various strategies have been explored to overcome the problem of adsorption.A common approach involves chemically coating the capillary interior to mask thesurface charge. Generally, coatings have the disadvantage of neutralizing the EOFas well. Furthermore, the chemical stability of the coating itself, and its potential forinteracting with or contaminating the analyte, are relevant concerns. The relativemerits of various kinds of coatings have been reviewed by Doherty et al. (2003).Nashabeh & Rassi (1992) proposed the use of tandem capillaries: a coated and anuncoated capillary joined together end to end. The proteins are restricted to thecoated section, whereas the uncoated section generates EOF through both capillaries.Lauer & McManigill (1986) proposed raising the pH of the buffer above the isoelectricpoint so that the sample proteins are no longer cationic.


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5. SUMMARY

EOF in CE channels and the resulting axial dispersion of analytes was reviewedin this paper. Electroosmosis is an interfacial phenomenon that is observed when-ever charge separation occurs at the interface of the substrate and buffer. Generally,EOF is present together with electrophoretic migration of individual species in aCE channel, even though it is possible to suppress it by applying special chemicalcoatings to the substrate. EOF presents both an advantage as well as a hindrance toefficient electrophoretic separation. The presence of EOF in the CE microchannelenables single-point detection (species of either charge elutes at the same end), re-duces analysis times, and enables operation of the microdevice in a continuous mode.On the other hand, the disadvantage is that any effect that causes the EOF to deviatefrom its classical flat profile would lead to Taylor dispersion and consequent bandbroadening.

Electrokinetic flows are described by the incompressible Navier-Stokes equationswith an electric body force term together with the equation of continuity. Theseequations are coupled to Poisson’s equation relating the potential to the charge dis-tribution and drift-diffusion–type conservation equations for each of the ionic species.This system needs to be supplemented by an advection-diffusion equation for the an-alyte. The resulting system of equations is quite complex and nonlinear. Fortunately,a series of simplifications can be made to these equations, at each step exploiting acertain disparity in scales inherent in the problem.

The first of these is the thin Debye-layer approximation, which is justified becausechannel widths ∼10–100 µm are many orders of magnitude larger than the typicalDebye-layer thickness ∼1–10 nm. This makes it possible to treat the EDL as aboundary layer; thus, the body force term in the Navier-Stokes equation may bedropped and the classical no-slip boundary conditions at the solid-fluid interface isreplaced by the HS slip boundary conditions. Within the realm of this approximation,the electrical forces are described by a single parameter, the ζ -potential that entersthe fluid flow description solely through the new slip boundary conditions.

If fluid properties and the ζ -potential are uniform, no external pressure gradientis applied, and the substrate is a poor electrical conductor, then the fluid velocitythrough a channel of any geometry is simply proportional to the electric field, whichmay be determined by solving Laplace’s equation for the electric potential. Thissimilitude between the electric field and the hydrodynamic flow can be very useful.However, the assumption of uniform fluid properties and ζ -potential is not valid inall problems of interest, and for such problems similitude does not apply.

In such cases an alternate approximation becomes possible provided that any axialinhomogeneity has a characteristic length scale that is much larger than the channeldiameter (10–100 µm). This is the method of lubrication theory for slowly varyingchannels. Axial inhomogeneity can arise due to various reasons, in particular due toadsorption of charged sample components to the wall, which alters the ζ -potential,variations in temperature due to nonuniform heating or cooling, alteration of theelectrical conductivity of the buffer by the sample, or axial variation in buffer pH, asin sample stacking or isoelectric focusing.

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One of the most important results of flow modification in CE is its effect onthe axial dispersion of the analyte through the mechanism of Taylor dispersion. Ax-ial dispersion or band broadening arises out of three main sources: (a) axial inho-mogeneities, (b) radial inhomogeneities, and (c) channel curvature. In general, axialinhomogeneities of any kind lead to an induced axial pressure gradient due to the in-compressibility constraint. Such pressure fluctuations give rise to a Poiseuille flow,1

which, through the Taylor-Aris mechanism, leads to greatly enhanced effective axialdispersion. Radial variations are often caused by nonuniform temperature distribu-tions inside the capillaries and cause band broadening due to differential rates of elec-tromigration over the capillary cross section. Channel curvature results in geometricdispersion and is relevant at the low, intermediate, and high Peclet number regimes.

ACKNOWLEDGMENTS

The author’s work in electroosmotic flow is supported by the NSF under grant NSFCTS-0330604.

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Electrokinetic Flow and Dispersion in Capillary Electrophoresis · 2012-09-28 · bilities is known...

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