214 Electrophoresis 2004, 25, 214–228

Review

Sandip Ghosal

Fluid mechanics of electroosmotic flow and its effect

on band broadening in capillary electrophoresis

Department of Mechanical

Engineering,

Northwestern University,

Electroosmotic flow (EOF) usually accompanies electrophoretic migration of charged

Evanston, IL, USA

species in capillary electrophoresis unless special precautions are taken to supress it.

The presence of the EOF provides certain advantages in separations. It is an alterna-

tive to mechanical pumps, which are inefficient and difficult to build at small scales,

for transporting reagents and analytes on microfluidic chips. The downside is that any

imperfection that distorts the EOF profile reduces the separation efficiency. In this

paper, the basic facts about EOF are reviewed from the perspective of fluid mechanics

and its effect on separations in free solution capillary zone electrophoresis is discussed

in the light of recent advances.

Keywords: Band broadening / Capillary electrophoresis / Electroosmosis / Lubrication theory /

Review / Taylor dispersion DOI 10.1002/elps.200305745

observed mobility of water was due to the fact that the

Contents

clay particles (and many other solid substrates such as

1 Introduction ........................... 214

glass, silicon, polymeric materials, minerals of various

2 Electroosmotic flow..................... 215

kinds, etc.) acquire a surface charge when in contact

3 The thin EDL limit....................... 216

with an electrolyte. The immobile surface charge in turn

4 EOF in a uniform cylindrical capillary....... 216

attracts a cloud of free ions of the opposite sign creating

5 Axially inhomogeneous channels.......... 217

a thin (,1–10 nm under typical conditions, e.g., univalent

3

5.1 Exactly solvable models................. 217

electrolyte at a concentration of 1–100 mol per m ) Debye

5.2 Potential flow solution................... 218

layer of mobile charges next to it. The thickness of this

5.3 Lubrication approximation ............... 219

electric double layer (EDL) is determined by a balance be-

6 Dispersion and EOF .................... 220

tween the intensity of thermal (Brownian) fluctuations and

6.1 Dispersion due to finite size of the EDL..... 220

the strength of the electrostatic attraction to the sub-

6.2 Dispersion due to analyte-wall interactions . . 221

strate. In the presence of an external electric field, the

6.3 Thermal broadening .................... 223

fluid in this charged Debye layer acquires a momentum

6.4 Dispersion in curved channels ............ 225

which is then transmitted to adjacent layers of fluid

7 Summary and conclusions............... 226

through the effect of viscosity. If the fluid phase is mobile

8 References............................ 227

(such as in a packed bed of particles or in a narrow capil-

lary), it would cause the fluid to flow (electroosmosis). In a

typical separation in capillary electrophoresis (CE)* both

1 Introduction

electroosmosis (sometimes also called electroendosmo-

sis) and electrophoresis occur simultaneously. Therefore,

Electroosmotic flow (EOF) was first reported by Reuss [1]

the resultant migration velocity of each species ‘i’is

in 1809 in experiments that demonstrated that water

could be made to percolate through porous clay dia-

ðiÞ ðiÞ

u u þ u (1)

eof

phragms through the application of an electric field. The total eph

where the first term is the bulk EOF velocity and the sec-

Correspondence: Dr. Sandip Ghosal, Department of Mechani-

ond term is the migration velocity relative to still fluid of

cal Engineering, Northwestern University, 2145 Sheridan Road,

species i (generally different for each species ). Due to

Evanston, IL 60208, USA

E-mail: s-ghosal@northwestern.edu

Fax:1847-491-3915 * In this paper CE will always refer to ‘free solution capillary zone

elecrophoresis’, which is the only mode considered here,

Abbreviations: EDL, electric double layer; HS, Helmholtz-Smo- though the ideas presented could with appropriate modifica-

luchowski; -TAS, micro total analysis system tion be useful for other separation modes.

2004 WILEY-VCH Verlag GmbH & Co. KGaA, WeinheimElectrophoresis 2004, 25, 214–228 EOF and band broadening 215

tries. In order to handle problems involving inhomoge-

the thinness of the EDL (1–10 nm) compared to typical

channel radii (10–100 mm), the electrical driving forces neous channels of arbitrary cross-sectional shapes, the

are localized in a thin sheath at the solid-fluid interface. potential flow and the lubrication approximation are intro-

duced in Sections 5.2 and 5.3. Finally, the dispersive

EOF can have a number of effects on the efficiency of

effects of EOF in homogeneous and inhomogeneous

ðÞ i

separation. First note that ifjj u 4 u for all i, then all

eof

eph

channels are considered in Section 6. A summary is pre-

species move in the same direction enabling single point

sented in Section 7.

detection of charged species of either sign. If the capillary

is uniformly charged and the inlet and outlet are at the

same pressure, it is well-known that the flow is uniform

2 Electroosmotic flow

throughout the capillary cross-section except for a very

thin EDL near the wall where the flow velocity rapidly

The equations describing the velocity field, u, of the fluid

decreases from its free stream value to zero at the sub-

phase are those of momentum conservation:

strate/fluid interface [2]. This uniform velocity is given by

2

r (q u1 u?Hu)¼2Hp1mH u2r Hf (3)

0 t e

the Helmholtz-Smoluchowski (HS) formula:

and continuity:

eEz

u ¼ (2)

e

4pm

H? u¼ 0 (4)

where e is the dielectric constant of the fluid, E is the

wherer andm are the (constant) density and viscosity of

0

applied electric field,z is the zeta-potential at the electro-

the fluid, p is the fluid pressure,f is the electric potential,

lyte/substrate interface, andm is the fluid viscosity. There-

and the charge density in the EDL, r is related to the

e

fore, except in the thin EDL there is no shear in the flow.

potential by Poisson’s equation

Thus, EOF does not add any significant shear induced

2

eH f¼24pr (5)

axial dispersion (Taylor-Aris dispersion) to the analyte. e

Band broadening in this case is purely due to axial molec-

To close the system, we need an equation for determining

ular diffusion.

f, which is the Poisson-Boltzmann equation

Clearly, Eq. (2) cannot be valid if any of the parameters

2 2

H f¼2k f (6)

that enter into the expression vary in the axial direction,

or, if the cross-section varies along the capillary. This is wherek is a constant determined by the ionic composi-

because any such variation would require the continuity tion of the electrolyte [5]. The Debye length is defined by

condition, that the fluid flux through all cross-sections be l ¼ 2p/k. The form (6) incorporates the Debye-Hückel

D

approximationf k T/e where k is the Boltzmann con-

the same to be violated. Axial variations usually lead to

B B

induced pressure gradients that drive a ‘Poiseuille’ type stant, T is the absolute temperature, and e is the electron-

of flow*. Thus, the flat EOF profile becomes distorted ic charge. At room temperature, k T/e < 25 mV. The elec-

B

resulting in strong band broadening due to Taylor-Aris dis- tric potential at the substrate buffer interface could be as

persion [3, 4]. The effect of such axial variability is dis- high as 100 mV . Thus, the Debye-Hückel approximation

cussed in detail in this review. is not always satisfied, in which case Eq. (6) should be

replaced by the more accurate but nonlinear Gouy-Chap-

The rest of this paper is organized in the following way: in

man form [5]. However, Eq. (6) is still useful for the pur-

the next section, the basic equations describing EOF in

pose of qualitative understanding even in situations

any conduit are presented. In Section 3, these basic

where it may not be strictly valid over the entire width of

equations are simplified by introducing the assumption

the EDL.

of thin Debye layers. An exact solution of the EOF prob-

The boundary conditions are those of “no slip” for the fluid

lem in cylindrical capillaries due to Rice and Whitehead is

velocity at the solid-fluid interface:

presented next (Section 4) and compared with the corre-

sponding reduced solution in the case of infinitely thin

uu ¼ 0 (7)

solid surface

Debye layers. In Section 5, the more difficult case of EOF

through capillaries with axial inhomogeneities is consid-

and

ered and solutions are discussed for two special geome-

fu ¼z (8)

solid surface

* Sometimes referred to in the literature as ‘laminar flow’. How-

where the potential at the solid fluid boundary,z, is speci-

ever, this terminology is inconsistent with usage in fluid

fied. Due to the rapid change in the potential at the inter-

mechanics, since all flows of relevance in CE, including the

face the definition of “at the interface” is somewhat

‘pure’ EOF are laminar (as opposed to turbulent) due to the

smallness of the Reynolds numbers involved. ambiguous. It is believed that the solid substrate usually

2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

CE and CEC216 S. Ghosal Electrophoresis 2004, 25, 214–228

2

has a layer of adsorbed immobile ions next to it known as e q f

r ﬃ (10)

e

2

the Stern layer. It is the ions in the fluid phase adjoining

4pqz

the Stern layer that are mobile and the distribution of

On eliminating r between Eqs. (9) and (10), integrating

e

which are governed by the Debye-Hückel or more gener-

the resulting differential equation and using the boundary

ally the Gouy-Chapman theory. The outer edge of the

conditions at the inner and outer edges of the EDL, the

Stern layer is therefore identified as the “slip plane” where

following jump condition across the EDL is derived

the no-slip boundary condition is applied [2]. The “zeta-

potential”z is defined as the electric potential at this slip

eEz

plane and is assumed known for the purpose of this

u u Du¼ (11)

solid

4pm

paper. The applied electric field is assumed to distort the

structure of the EDL by a negligible amount. Clearly, this

where in Eq. (11), u is the velocity of the solid at a point

solid

would be true if V/L z/l where V is the applied voltage

D

on the solid-fluid interface and u is the velocity of the fluid

over a segment L of the capillary. Using typical valuesz

at the corresponding point, just above the (infinitely thin)

5

, 100 mV andl , 10 nm, we find thatz/l , 10 V/cm

D D

Debye layer. A formal asymptotic development in terms of

V/L ,300 V/cm.

the small parameter l /a (where a is a characteristic

D 0 0

radius) has been presented by Anderson [6]. Equation

The ratio of the characteristic magnitude of the left hand

(11) is known as the Helmholtz-Smoluchowski (HS) slip

side to the right hand side of Eq. (3) is measured by

boundary condition after the pioneering work of Helm-

the (dimensionless parameter) Reynolds number, Re =

holtz [7] and Smoluchowski [8]. Thus, in the limit of thin

(a r u /m) where a is a characteristic radius and u is

0 0 e 0 e

Debye layers the term 2r Hf may be dropped from

a characteristic electroosmotic speed. Estimates using e

Eq. (3), instead at the boundary, the no-slip boundary

typical values for microfluidic applications give Re

condition (7) is replaced by (11). Since the external field

, 0.001–1.0, so that the left hand side can often be

E is tangential to the interface, (11) implies that the normal

ignored (in that case Eq. (3) becomes the Stokes equa-

component of the velocity is continuous. Thus, the equa-

tion), or at least treated as a small correction. Unlike ap-

tions of fluid flow inside the capillary become exactly

plications of fluid mechanics to large-scale phenomena

identical to the classical fluid flow equations, the coupling

where the left hand side is dominant leading to instabil-

to the electrical problem is only felt through the boundary

ities and turbulence, microfluidics is always characterized

condition in Eq. (11).

by smooth laminar flow.

3 The thin EDL limit

4 EOF in a uniform cylindrical capillary

The theory of electrophoretic motion of charged particles

In capillary zone electrophoresis (CZE) EOF takes place in

of characteristic size ‘a’ has been well studied in the limit

narrow cylindrical capillaries that to a first approximation

of thick (l a) as well as thin (l a) Debye layers.

D D

can be considered to be infinitely long and uniformly

However, since the characteristic radius of microfluidic

charged. Under those conditions, the Debye-Hückel

channels , 10–100 mm, whereas, the Debye length l

D

form of the Poisson-Boltzmann equation admits an exact

, 1–10 nm, the thin Debye layer approximation is usually

solution [9]:

an excellent one for the purpose of studying EOF, at least

for the majority of current microfluidic applications. In the

IðÞ kr

0

limit of thin EDL, the Navier-Stokes/Poisson-Boltzmann

f¼z (12)

IðÞ ka

0

system described in the last section may be replaced by

a simpler set of equations. The EDL then forms a very

th

where I is the zero order modified Bessel’s function

0

thin boundary layer at the solid fluid interface where the

of the first kind, r is the distance from the axis, and a is

electrical forces are confined.

the internal radius of the capillary. Equation (5) can then

be used to determine the charge density. Further, sub-

At leading order, the dominant balance is between the vis-

stitution ofr in the Navier-Stokes Eq. (3), results in an

cous and the electrical forces in the boundary layer: e

equation that may be integrated exactly to determine

2

q u

m þr Eﬃ 0 (9) the flow profile u(r). In the absence of an imposed pres-

e

2

qz

sure gradient, the solution, after Rice and Whitehead

where E is the external electric field which is in the tan-

[9], is

gential direction. Since rates of change across the bound-

ezE IðÞ kr

ary layer (z-axis) are much larger than along it (x-axis), 0

urðÞ¼ 1 (13)

Poisson’s Eq. (5) may be written as 4pm IðÞ ka

0

2004 WILEY-VCH Verlag GmbH & Co. KGaA, WeinheimElectrophoresis 2004, 25, 214–228 EOF and band broadening 217

satisfy the no-slip boundary condition at r = a. This behav-

Since the fluid flow equation is linear in this limit, clearly a

pressure-driven flow can be added to the solution (super- ior is shown in Fig. 1. Exact solutions within the Debye-

position) in the event that both a pressure gradient and an Hückel approximation are also available for EOF between

electric field are simultaneously applied. parallel plates for any Debye length [11]. They show a

similar ‘boundary layer’ type of behavior.

In the thin EDL limit, the flow problem can be formulated

in terms of the HS slip boundary conditions. Due to sym-

In the case of CE channels etched on substrates for

metry in the axial direction, the Navier-Stokes equation

micrototal analysis system (m-TAS) applications, the

simply reduces to

channels are usually rectangular or trapezoidal rather

than circular in cross-section. However, in the thin EDL

m d du

r ¼ 0 (14)

limit the plug flow profile is valid for channels of arbitrary

r dr dr

cross-sectional shape since it simultaneously solves the

with the ‘slip’ boundary conditions

Stokes flow equation and satisfies the HS slip boundary

condition. This is of course not true in the case of finite

ezE

urðÞ ¼ a¼ (15)

EDL thickness as the detailed structure of the flow within

4pm

the EDL is determined by the cross-sectional geometry of

The only solution, without a singularity at the origin, is the

the capillary.

‘plug flow’ profile

ezE

urðÞ¼urðÞ ¼ a¼ (16)

4pm 5 Axially inhomogeneous channels

Equation (16) may be compared to the Rice-Whitehead

We have seen so far, that except for a very thin sheath

solution (13). Whenka 1 (thin EDL limit) [10]

around the channel walls, EOF has a uniform flow profile.

This is a great advantage in CE applications since it

exp z 1

IðÞ z pﬃﬃﬃﬃﬃﬃﬃﬃ 1þ þ (17)

0

implies that the presence of EOF does not lead to signif-

8z

2pz

icant added dispersion. This conclusion, however, is

therefore the term I (kr)/I (ka) in Eq. (13) is negligible

0 0

valid as long as all of the parameters involved, namely

unless r < a. Near r ¼ a we have I (kr)/I (ka)

0 0

the electric field E, dielectric constante, the zeta-poten-

, exp[2k(a2r)], thus the HS solution (16) is recovered

tial z and viscosity m are constants. Variability in any of

21

except in a thin boundary layer a2r ,k ,l within

D

these parameters could induce axial pressure gradients

which the velocity drops precipitously to zero in order to

which perturb the flow and distort the uniformity of the

flow profile. Calculating the perturbations in the flow

due to such causes and the resultant axial dispersion is

a fundamental fluid mechanics problem of considerable

interest in CE.

5.1 Exactly solvable models

Anderson and Idol [12] considered the problem of EOF

through a uniform, infinite, straight cylindrical capillary

with az-potential that varies solely in the axial direction,

z =z(x). A uniform external electric field and zero imposed

pressure gradient were assumed. An exact solution to the

Stokes flow problem was derived under the assumption

of thin EDL. It was shown that the velocity field u = ux ˆ 1

r ˆv may be expressed in terms of the stream functionc,

1qc

u¼ (18)

r qr

1qc

v¼ (19)

Figure 1. Normalized velocity profile from the Rice-

r qx

Whitehead solution showing the thin boundary layer at

the wall for small Debye lengths. wherec is given by a series expansion

2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim218 S. Ghosal Electrophoresis 2004, 25, 214–228

"

1

2 problem, an exact solution could be obtained by Fourier

X

eE r 2mpx

c

c¼ hi z 2 a ðÞ r cos

m transforming the equations along the planes parallel to

4pm 2 L

m¼1

the plates. In a subsequent paper [14], Ajdari generalized

#

1

X

2mpx the solution to include the effect of small (compared to

s

2 a ðÞ r sin (20)

m

L plate separation) amplitude irregularities on the surface

m¼1

of the plates.

where

Ajdari’s solutions illustrate some interesting features of

c s 2

a a rIðÞ a IðÞ a r r IðÞ a r IðÞ a EOF such as the presence of recirculating regions which

0 m 1 m 0 m 1 m

m m

¼ ¼ (21)

c s 2 2

^ ^

could be useful in the design of microfluidic mixers. In

z z a IðÞ a þ 2IðÞ a IðÞ a a IðÞ a

m m 0 m 1 m m m

m m 1 0

fact, Stroock et al. [15] constructed such mixers using

EOF in a long channel of rectangular cross-section

and

(260 mm6130 mm) with a patterned surface charge of

Z

L

1 2mpx

alternating sign that was fabricated using soft lithographic

c

^

z ¼ zðÞ x cos dx (22)

m

L L

techniques [16]. It is further shown by Ajdari that surface

0

irregularities and variations in the zeta-potential in combi-

Z

L

1 2mpx

s nation could generate net forces on the plates which

^

z ¼ zðÞ x sin dx (23)

m

L L

0 could even be perpendicular to the applied electric field

and need not vanish even if the net charge on either plate

are the cosine and sine transform of thez-potentialz(x),

vanishes. The framework of Ajdari was applied by Long et

a =2mp/L, and78 indicates the average over the length

m

al. [17] to obtain analytical solutions in the neighborhood

of the capillary:

of localized ‘defects’ in the zeta-potential for both the par-

allel plate as well as cylindrical geometries. These solu-

Z

L

1 tions are useful in providing an understanding of the per-

hi f ¼ fdx (24)

L turbations in EOF that may result from various local

0

surface imperfections of the zeta-potential likely in any

practical device.

In Eq. (21), I denotes the modified Bessel function of inte-

n

ger order n.

The above solution implies a remarkably simple formula

5.2 Potential flow solution

for the cross-sectional average of the axial velocity, u (or

Let us consider the problem of EOF in a conduit of arbi-

equivalently, the volume flux per unit cross-sectional

trary geometry but in the thin EDL approximation. Further,

area),

assume that all fluid and material properties are uniform

ezhiE

and that no external pressure gradient has been applied

u¼ (25)

4pm

across the capillary. Let us denote by S the walls of the

conduit and by S and S the equipotential surfacesf =

0 1

which follows on integrating Eq. (18) over the cross-sec-

f andf =f , respectively, near the inlet and outlet sec-

0 1

tion of the capillary. Thus, the flux per unit area at any

tions. These are the potentials of the reservoirs at the

instant over any cross-section is the same and equal to

extremities of the capillary. Then it follows by direct calcu-

that of the flow through a uniform capillary withz = 7z8.

lation that

ez

Ajdari [13] considered the problem of EOF between a pair

u¼ rf (26)

4pm

of parallel plates at z =6 h under the application of a uni-

form external electric field, E, and arbitrary position de- wheref is a solution of the boundary value problem

pendent variations of the zeta-potential on the surface of

2

H f¼ 0; f¼z on S,f on S ,&f on S (27)

0 0 1 1

the platesz =z (x, y). Though the parallel plate geometry

6

is not directly relevant to CE applications, it may serve as

has the following properties

a reasonable approximation to flow in shallow rectangular

channels etched on chips. The analysis is based on the 2

H? u¼ 0, H u¼ 0 (28)

assumptions of low zeta-potentials (z k T, the Debye-

B

Hückel approximation) and low Reynolds numbers Further, u satisfies the HS slip boundary conditions

(Stokes flow). On account of the linearity of the fluid flow (Eq. 11). Thus, Eq. (26) is already the solution of the

equations in the Stokes flow limit, and, since the problem problem of EOF through the conduit in the Stokes flow

for the potential in the EDL is decoupled from the fluid limit.

2004 WILEY-VCH Verlag GmbH & Co. KGaA, WeinheimElectrophoresis 2004, 25, 214–228 EOF and band broadening 219

If the Reynolds number Re is not necessarily negligible If the assumption of slow variations as defined above is

but the flow is known to be steady and irrotational, then satisfied, then a formal asymptotic solution to the prob-

the Navier-Stokes Eqs. (3) and (4) outside the EDL may be lem of EOF in terms of the ratio of characteristic radial dis-

replaced by tance to characteristic axial scale (a small parameter) may

be carried out [22] and the solution may be summarized

H6u¼ 0; H? u¼ 0 (29)

ˆ ˆ

as follows: u ,i u (x, y, z)1 O (E), E ,i E (x)1 O (E),

Once again, we see that Eq. (26) automatically satisfies

u dp eF c

p

u¼ þ (30)

these equations and is therefore the unique solution to

m dx 4pmAxðÞ

the hydrodynamic problem.

u dp eFc

p

Q¼ AxðÞ þ (31)

The existence of this “similitude” between the electroos- m dx 4pm

motic velocity u and the electric field 2Hf was first

E(x)¼ F/A(x) (32)

pointed out by Cummings et al. [18]. When the conditions

needed for such similitude are satisfied, Eq. (26) provides

Here, F is a constant representing the electric flux through

a remarkably simple solution to the problem of determin-

any cross-section, A(x) is the cross-sectional area and the

ing the EOF since one only needs to solve the Dirichlet

21

overbar indicates average over the cross-section, f = A

problem for the potential and a wide variety of analytical

$f dydz. The constant Q represents the volume flux of fluid

and numerical techniques exist for this task. It should also

through any cross-section. The functions u , defined by

p

be clear that if the prefactor multiplyingHf in Eq. (26) fails

2

H u = 0 and u | = 0 and the function c defined by

p pqD(x)

to be a constant, then u does not satisfy either the conti-

2

H c = 0 andc| =2z are properties of the channel ge-

qD(x)

nuity equation or the equation for the conservation of

ometry and charge distribution alone. They are defined on

momentum and Eq. (26) therefore is no longer a solution.

the domain D(x) representing the cross-section of the

It is this class of problems that we consider next.

channel with boundary qD at axial location ‘x’. Both of

these functions u andc may be evaluated by quadrature

p

from a knowledge of the Green function, G, of the Laplace

5.3 Lubrication approximation

operator with zero boundary condition corresponding to

the domain D(x):

When the HS slip velocity is variable over the capillary sur-

Z

face, an analytical solution for the flow field is difficult 1

u ¼ GxðÞ ;y;z;y ;z dy dz

p

except for the special geometries discussed in the last

4p

DxðÞ

section. Generally one may need to resort to the more I

1 qG qG

expensive process of full numerical simulation. However, c¼ zðÞ x;y ;z m þ n ds (33)

4p qy qz

qDxðÞ

if the variations are ‘slow’ in the axial direction; a term that

will be made more precise later, the technique of lubrica-

where (m, n) are the direction cosines of the unit normal

tion theory [19] permits analytical solutions to be obtained

onqD(x).

even for channels with complicated geometrical shapes.

According to Eqs. (30)–(32), the flow velocity in the axial

Lubrication theory was originally developed to analyze

direction in a slowly varying channel is a linear superposi-

the motion of lubricants in the narrow gap between

tion of a purely pressure-driven flow, and a pure EOF. The

machine parts (and hence the name). It has since found

axial pressure pressure gradient and electric field are cal-

wide application in various areas of fluid mechanics,

culated by using the dual conditions: (a) the fluid is incom-

such as in the analysis of blood flow in very narrow capil-

pressible, (b) the electric flux must obey the Gauss law.

laries [20, 21]. It has recently been applied to the prob-

The solution is completely specified by two independent

lem of EOF by Ghosal [22] in the context of the thin EDL

physical constants, the volume flux of fluid, Q, and, the

limit.

electric flux, F. These constants may be expressed, if

The theory is based on the following assumptions: (i) The

desired, in terms of the total pressure drop, and, the total

characteristic length scale for the variation of the cross-

voltage drop, respectively, between the inlet and outlet

sectional shape and area in the axial (x) direction of the

sections, which yields the following generalization of Poi-

channel is very much larger than a characteristic radius

seuille’s law:

(a ). (ii) The characteristic length scale for the variation

0

p p ez V V

a b a b

4 2

of the slip velocity in the x direction is very much larger

Q¼ pa pa (34)

8mL 4pm L

than a . (iii) The characteristic time-scale (T) for any tem-

0

poral variations is very much larger than the diffusion Here, p 2 p is an applied pressure drop and V 2 V is an

a b a b

2

scale (t : a /n)(n being the kinematic viscosity of the applied voltage drop along a length L of the capillary. The

d 0

fluid). constants a andz may be regarded as “effective” values

* *

2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim220 S. Ghosal Electrophoresis 2004, 25, 214–228

for the radius and thez-potential, respectively, and they

are defined as

2 3

1=4

8

4 5

DE

a ¼

1

1

p u A

p

DE

1

1

cu A

p

1

z ¼ pﬃﬃﬃﬃﬃﬃ (35)

DE

1=2

8p

1 1

1

A u A

p

Here, the parameter a is determined solely by channel

*

geometry andz is determined solely by the channel ge-

*

ometry and the nature of the distribution of thez-poten-

tial.

It is instructive to compare the lubrication theory result

with the exact solution of the problem of EOF in a uniform

cylindrical capillary with inhomogeneous charge distribu-

tion solved by Anderson and Idol [12], also in the thin EDL

Figure 2. Comparison of the asymptotic solution in the

limit. For a uniform cylindrical capillary (radius a ) with

0

lubrication limit (symbols) with the exact solution (dashed

z varying only in the x direction Eq. (35) implies z = 7z8

lines) according to Anderson and Idol for a /l = 10.0, 2.0,

0

*

which is the same as the result derived by Anderson and

1.0, 0.5, 0.1, and 0.01. The asymptotic solution corre-

Idol and discussed earlier. For a sinusoidally varying wall sponds to a /l? 0. An infinitely thin EDL is assumed for

0

both solutions.

charge,

z(x)¼z 1Dz sin(2px/l) (36)

0

Then Anderson and Idol’s result implies:

6 Dispersion and EOF

u Dz

The resolving power in CE is limited by axial dispersion of

¼ 1þ FðÞ r sinðÞ aX (37)

u z

0 0

the analyte. The minimum resolvable difference in mobility

Dm ,mN wherem is the average mobility of the two spe-

where cies and ‘N’ is the number of theoretical plates (square of

r

aIðÞ a

1 0

the separation length divided by the variance of the con-

a IðÞ ar 1 þ IðÞ ar

0 1

2IðÞ a

2

1

centration peak). Unlike a classical pressure-driven flow,

FðÞ r ¼ (38)

2

1 IðÞ a

0 which has a parabolic profile, EOF has an essentially flat

1

a IðÞ a þ IðÞ a

0 1

2 2IðÞ a

1

profile (except in a thin EDL near the walls). Thus, under

ideal circumstances, EOF should not contribute signifi-

r = r/a , X = x/a , u =2eEz /(4pm) anda =2p(a /l). This

0 0 0 0 0

cantly to shear-induced axial dispersion (Taylor-Aris dis-

may be compared with our solution which is Eq. (37) but

persion) so that essentially “diffusion limited” separation,

with F (r) instead of F(r) where

0

where resolution is limited only by molecular diffusion is

2

potentially realizable. However, in practice, various inho-

F (r)=2r 2 1 (39)

0

mogeneities can perturb the ideal EOF leading to signifi-

Clearly F(r) ,F (r) in the limita 1 (the lubrication limit).

0

cant band broadening. The important ones among such

Figure 2 compares F(r) and F (r) for several values of the

0

processes are discussed in the subsequent sections

ratio a /l. It is seen, that, for a /l 1 (in practice 0.1 or

0 0

(also see the two reviews by Gas ˇ and Kenndler [23] and

less), the prediction of the lubrication analysis is in excel-

ˇ

by Gas ˇ,Ste ˇ dry ´, and Kenndler [24]).

lent accord with the exact solution as expected. For a /l

0

, 1, the exact solution deviates significantly from the

lubrication solution. In the opposite limit of a /l 1, the

0

6.1 Dispersion due to finite size of the EDL

frequent reversal of the electric force results in no net

transfer of momentum to the interior of the fluid, and the We have seen that in the thin EDL limit and in the absence

lubrication limit solution is qualitatively incorrect. The of axial inhomogeneities, the dispersion is essentially due

latter situation may describe random fine scale inhomo- to molecular diffusion alone. However, the contribution of

geneities in the wall charge of the type considered by the zone of shear in the EDL can easily be estimated by

Ajdari [13]. actually calculating the Taylor-Aris dispersion coefficient

2004 WILEY-VCH Verlag GmbH & Co. KGaA, WeinheimElectrophoresis 2004, 25, 214–228 EOF and band broadening 221

from u

for the EOF profile derived by Rice and Whitehead pre- by 0.1% or less. In typical CZE applications it is

e

sented in Section 4. This has been done by Datta and easily estimated that Pe , 102 100 for small to moder-

210 26

Kotamarthi [25] for a uniform infinitely long capillary. The ate sized molecules. From Eq. (43), X , 10 2 10 ,so

e

same result was arrived at by Griffiths and Nilson [26] by a thatd/D21 , 0.01 or smaller. Thus, for such analytes

different method and also by McEldoon and Datta [27] in the finiteness of the EDL usually has an insignificant

the more general case where the solute could interact effect and the dispersion is essentially limited by molec-

with the wall. Subsequently, Griffiths and Nilson [28] ular diffusion, unless axial inhomogenieties drastically

extended the analysis to the case where the z-potential reduce performance. This conclusion may not be true

need not be small, and therefore the Debye-Hückel if in the future channel radii become very much nar-

approximation cannot be made. However, in that case rower (in the sub-mm range) for microfluidic appli-

ˇ

most of the calculations must be done numerically. Ste ˇ dry ´ cations. Further, for very large macromolecules, such

et al. [29] replaced the actual velocity profile by the com- as proteins, the diffusion coefficient can be , 102 100

2 4

bination of a plug flow region in the core and a stationary times smaller so that the Pe , 10 2 10 , in which case

annular region near the wall and obtained a simplified d/D2 1 1 and the finiteness of the EDL could be

expression for the plate height which depends on the important.

thickness of the stationary zone. The evaluation of this

thickness requires a numerical integration of the Pois-

son-Boltzmann equation.

6.2 Dispersion due to analyte-wall interactions

Within the Debye-Hückel theory, the effective axial diffu-

sion coefficient is given by [25]

An important source of dispersion in the analysis of cati-

onic proteins is the variation in wall zeta-potential that

d

2

2

¼ 1þ Pe XðÞ fðÞ 1 n (40)

e

results from the tendency of charged species to stick to

D

the capillary walls [30]. This alters thez-potential in a non-

homogeneous and time-dependent manner inducing a

where Pe = ua /D is a Peclet number based on the molec-

0

pressure gradient that alters the flow profile as well as

ular diffusion coefficient, D, capillary radius a and bulk

0

the bulk flow rate [31]. Highly asymmetric and broadened

flow velocity (volume flux per unit cross-section) u;

peaks are symptomatic of the presence of wall adsorp-

u ¼ u (12Z) (41)

e

tion. The combination of loss of sample and enhanced

dispersion can lead to reduction of the peak concentra-

where u =2(ezE)/(4pm) and the parameterZ is a measure

e

tion in the sample below the detection threshold, in which

of the reduction of the flow due to the presence of the

case no peak at all is discernible in the detector response.

double layer. It can be shown that

Various strategies have been explored to overcome the

2 IðÞ f

problem of adsorption [32], however, in this paper we

1

Z¼ (42)

fIðÞ f only review work related to the fluid mechanical pro-

0

cesses at play.

where I and I are modified Bessel’s functions of order

0 1

zero and one, andf = a /l (l is the Debye length). The

0 D D

The basic equations are those of fluid flow and scalar dis-

function X is defined as

e

persion with a loss term, f (c , s) which depends on the

w

concentration at the wall, c = c(a, x, t) and the concentra-

w

2

Z 3 2 1 1

tion of adsorbed solute per unit area of the wall, s. A sim-

XðÞ f ¼ þ (43)

e

2 2 2 2

2

8

f Zf Z f

ðÞ 1 Z

ple and commonly used wall interaction model is the

Langmuir law:

The parametern is zero for a pure EOF. However, if a pres-

sure drop is applied across the capillary in addition to an

f (c , s)¼ k c (s 2 s)2 k s (44)

w a w m d

electric voltage, n is defined as the ratio of volume flux

due to the pressure drop alone to that of the volume flux

where k and k , respectively, are adsorption and desorp-

a d

due to the voltage drop alone. It could have a negative

tion coefficients and s = s is the saturation concen-

m

sign if these two external influences on their own would

tration, the maximum solute concentration that the wall

drive a flux in opposite directions.

can hold. The z-potential in general is a function of

Typical Debye lengths in microfluidic applications are the adsorbed concentration and could potentially also

l , 12 10 nm and typical radii are a , 102 100mm. depend explicitly on x and t:

D 0

3 5

Thus,f , 10 2 10 . Using the asymptotic form for large

25 23

f, we haveZ < 2/f , 10 2 10 . Therefore, u differs z¼ g (s (x, t); x, t) (45)

2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim222 S. Ghosal Electrophoresis 2004, 25, 214–228

with z’= z 2 7z8 and given pressure drop Dp across the

The form of g is determined by the structure of the EDL. If

adsorption is assumed to simply alter the density of fixed capillary. The effective axial diffusivity and source terms

charges at the wall and the Debye-Hückel approximation are

is assumed it is readily shown that

2 2

a u

0

d¼ Dþ (50)

48D

g¼z 2 (2pl zF/e) s (46)

0 D

hi

a

o

wherel is the Debye length, F is the Faraday constant, s¼ f 1 f (51)

D 1

4D

z is the ionization state (number of fundamental charges

per molecule), and s is assumed to be in units of moles

where

per unit area.

qf

Early works on this problem are due to Gas ˇ et al. [33] and

f ¼ fðÞ c; s f ¼ (52)

1

ˇ

qc

Ste ˇ dry ´ et al. [34]. Though their analyses take account of w

cw¼c

the fact that analyte is lost to the wall, the consequent

modification of thez-potential and therefore the hydrody-

The sample concentration, c(r, x, t) itself may be ex-

namic flow field were neglected. Indeed, the hydrody-

pressed as

namics was restricted to the trivial case of uniform flow

2 2 4 2 2 4

at constant velocity independent of the adsorption pro-

a 2r qs 2a 6a r þ3r qc

0 0 0

crðÞ ;x;t¼cxðÞ ;tþ u (53)

2

cess. Analysis of similar ‘purely kinematic’ models that

4a D qt qx

24a D

0

0

neglect the perturbation in the hydrodynamic field have

also been presented by other authors (see [35–37]). A

where the second and third terms are small corrections to

detailed experimental study of wall adsorption and its

the first.

consequences for the underlying EOF and dispersion is

According to Eq. (47) the mean concentration profile is

due to Towns and Regnier [38]. These experiments as

advected at the bulk flow speed plus any electrophoretic

well as others [39] together with their interpretation using

migration speed if present, while simultaneously diffusing

simple fluid mechanical models [40, 41] indicate, that, the

with an ‘effective’ axial diffusion coefficient shown in

modification of the z-potential by adsorption, and the

consequent perturbation of the hydrodynamic field, is an Eq. (50). This axial dispersion coefficient is precisely the

important, if not the principal cause of dispersion in these classical Taylor-Aris dispersion coefficient calculated

systems. with that part of the flow field that is proportional to the

induced pressure gradient. The last term represents

At distances large compared to a Pe (Pe is the Peclet

0

losses to the wall after accounting for a small correction

number based on the EOF speed in the unmodified capil-

caused by the fact that the concentration at the wall

lary) from the injection point, axial diffusion ensures that

differs by a small amount from the mean concentration c.

the analyte concentration is slowly varying in the axial

direction relative to a length scale defined by the capillary

In order to check the accuracy of the asymptotic theory,

radius a . In this limit, the lubrication theory discussed

the complete set of coupled fluid flow and scalar equation

0

earlier can be extended to include a diffusing solute. The

for concentration was solved numerically assuming a thin

following one-dimensional coupled partial differential

EDL (via the HS slip boundary conditions) and Stokes

equations for the cross-sectionally averaged concen-

flow. Periodic boundary conditions were assumed in the

tration c(x, t) and adsorbed concentration, s(x, t) have

x (axial) direction. As initial conditions we used a trapezoi-

been derived by Ghosal using asymptotic theory [31]:

dal profile for c(r, x,0)= c (x) approximately 10 radii wide

0

and used a reasonable [42] set of parameter values

qc qc a q qc

0

þðuþ u Þ þ ðu fÞþ u f ¼

ep 1

shown in Table 1. Note that all parameters are in dimen-

qt qx 12D qx qx

sionless units. Figure 3 shows the distribution of c/c

m

q qc 2

(where c is the maximum concentration) andz/z at the

m 0

¼ d s (47)

qx qx a

o

and

Table 1. Parameter values for simulation

qs

¼s (48)

qt

Parameter L/a Re Pe k s /u a k /u s

0 a m e 0 d e m

where for convenience

Definition a u /n a u /D

0 e 0 e

0

2

eEz Dpa ez E

0 Value 1000 0 100 0.1 0.0005 0.01

u uþ ¼ þ (49)

4pm 8mL 4pm

2004 WILEY-VCH Verlag GmbH & Co. KGaA, WeinheimElectrophoresis 2004, 25, 214–228 EOF and band broadening 223

The effect is due to the fact that the ‘tail’ of the distribution

of c is primarily from analyte that is closer to the wall

which is lost at a faster rate thereby displacing the cen-

troid forward. Thus, the centroid moves with a velocity

that is higher than the mean velocity, u , but lower than

the maximum velociy u =2u . If the adsorption rate

max

k s is small (which corresponds to our ‘slow variations’

a m

assumption), the expression for the velocity of the cen-

troid [46, 47] can be linearized with respect to k s .It is

a m

then found that the resulting expression is identical to

Eq. (55). The same physical effect also brings about a

reduction in the effective axial diffusion coefficient. How-

ever, the effect is of second order in k s and is thus not

a m

reflected in the expression (50) ford.

The problem of CZE in the presence of wall adsorption is

Figure 3. Comparison of asymptotic theory (symbols)

very similar to the problem of open-tubular capillary elec-

with numerical simulation (lines) of the cross-sectionally

trochromatography (CEC) (see, e.g., [48]). However, in

averaged analyte concentration (lower curves) and

CEC the interaction with the wall is an important compo-

z-potential (upper curves) at two time instants. The

nent of the mechanism of separation rather than an unde-

parameters are as in Table 1.

sirable source of band broadening. In the case of a pres-

sure-driven (Poiseuille) flow, the correction to the migration

instant when the concentration peak arrives at a hypo-

velocity due to wall interactions, as well as the effective

thetical detector placed at a distance x from the inlet.

d

axial diffusion coefficient have been worked out by Golay

The figure shows two independent sets of results for x =

d

[44] and later, with greater generality by Aris [45]. In CEC,

450a and x = 900a . It is seen that as the sample moves

0 d 0

the flow is electroosmotic rather than pressure-driven.

down the capillary, the peak height decreases, the peak

McEldoon and Datta [27] replaced the Poiseuille profile in

width increases and the peak shape becomes markedly

the Golay-Aris theory by the the Rice-Whitehead flow pro-

asymmetric. The peak shapes have a striking qualitative

file discussed in Section 4 to derive an expression for the

similarity with observed CZE signals in an uncoated cap-

dispersion in CEC due to wall interactions. A similar

illary for cationic proteins (see, e.g., Fig. 8 of [38] and

approach has been used earlier by Martin and Guichon

Fig. 10 of [43].) The z-potential is reduced behind the

[49] and by Martinetal. [50], exceptadhoc approximations

peak, however, with passage of time, thez-potential at a

to the electrokinetic flow profile were used. The analyte

fixed position undergoes a gradual recovery. This is due

concentration peak profile itself has been calculated

to desorption from the capillary walls. For both c and z

numerically recently [51] by integrating a set of one-dimen-

the simulation is seen to be in excellent agreement with

sional model equations. The asymptotic theory presented

the theoretical calculation using the 1-D equations.

in this section should also apply to the CEC problem.

In the case of a pure pressure-driven flow (E = 0 butDp

4 0), our problem reduces to the classical chromato-

6.3 Thermal broadening

graphic problem first studied by Golay [44] and later

refined by Aris [45] and others [46, 47]. If we linearize f by

The flow of electric current through the buffer in CE sys-

assuming s s and neglect desorption, k = 0 then we

m d

tems produces a significant amount of Joule heat which

can rewrite Eq. (47) as

results in temperature variations in the microcapillary. In

fact, the problem of Joule heat was a major impediment

qc qc q qc 2

e to the development of CE as an analytical tool. Excessive

þðuþ u Þ ¼ D s (54)

ep

qt qx qx qx a

0

heating could cause convective overturning in the fluid

that would result in obliteration of all signals. The fluid

where

could even vaporize leading to a “vapor lock” and cata-

strophic failure. These problems were overcome only

a k s

0 a m

e

u¼ u 1þ (55)

when microcapillaries (less than 200mm internal diame-

6D

ter) became available. In modern capillaries or microflui-

Thus, in this case, the profile of c is advected somewhat dic channels, Joule heating does not have such disas-

faster than the mean flow speed, a result derived by var- trous effects, nevertheless it can be a significant source

ious authors under slightly different assumptions [44–47]. of band broadening.

2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim224 S. Ghosal Electrophoresis 2004, 25, 214–228

of the thermal band broadening phenomenon. Knox also

The transport problem for temperature in a cylindrical

capillary carrying a current of uniform density can be calculated an explicit expression for the plate height (the

easily solved [52] yielding a radially varying temperature variance developed by an initially sharp concentration

field, T(r). The electrophoretic migration speed of mole- peak per unit length of capillary traversed by the sample).

cules varies inversely with the fluid viscositym. For exam-

A similar analysis was published later by Grushka et al.

ple, for a spherical particle with a thin EDL (EDL thickness

[54] who also provided an expression* for the plate height:

much less than the particle radius) the Stern-Gouy-Chap-

man theory [5] gives

6 4 2 2 2

2D R E C B L u

1 b

H¼ þ (61)

2

2 2 2

u

2ezE

24D 8k T E LC R B

1 b

1 1

u ¼ (56)

ep

3 4pm

where D is the molecular diffusivity, u is the cross-section-

(in cgs units) wheree andm are the dielectric constant and

ally averaged value for the migration velocity, R is the

1

viscosity of the buffer, E is the applied electric field, and

internal capillary diameter, E is the applied electric field,

z is the zeta-potential at the surface of the migrating par-

C andL are, respectively, the buffer electrolyte concen-

b

ticle. Since the viscosity m varies with temperature, T,

tration and equivalent conductance and k is the buffer

1

which in turn varies with r, analyte molecules near the

thermal conductivity. Finally, the coefficients A and B

wall migrate with a slightly different speed than those at

characterize the temperature dependance of the viscosity

the center. This, clearly would lead to band broadening in

of the buffer:m = A exp (B/T), T being the absolute temper-

the sample plug.

ature and T its value at the wall. Andreev and Lisin [55]

1

analyzed the problem by numerically solving the coupled

How is the EOF modified and what role does it play in the

equations for Stokes flow, the Poisson-Boltzmann equa-

band broadening process? At first sight one might con-

tion for the electric potential, and the advection diffusion

clude that since the electroosmotic velocity

equation for the analyte concentration taking into account

ezE the dependence of the transport coefficients on concen-

u ¼ (57)

eo

tration and temperature. The thermal equation was inte-

4pm

grated analytically assuming a small temperature drop

the EOF too should have a profile similar to the electro-

between the axis and the wall of the capillary, which

phoretic velocity. This, however, is not true. The reason

implies a parabolic distribution of the temperature profile.

is, the formula (57) is valid only for a constantm not ifm =

It was found that depending on the parameter regime, the

m(r). If the viscosity depends on r, we must go back to the

effect of the nonuniform temperature on the EOF could

Stokes equation for determining the correct velocity pro-

sometimes dominate effects due to variations of the elec-

file. In the absence of a pressure gradient Stokes equation

trophoretic velocity. This is due to the finite Debye layer

becomes

effects which disappear in the thin EDL limit considered

by Knox [53] and by Grushka et al. [54].

1 d du

rmðÞ r ¼ 0 (58)

r dr dr

The wall temperature can be calculated by solving the

steady state diffusion equation with a source. This has

with the HS slip boundary conditions (in the thin EDL

been done by various authors under different assump-

limit):

tions. Grushka et al. [54] assume a polyimide-coated cap-

illary and constant electrical and thermal conductivity of

ezE

urðÞ ¼ a¼ (59)

0

the buffer, in which case

4pmðÞ a

0

2

a being the capillary radius. Assuming that u is not singu-

0 GR 1 R 1 R 1

2 c

1

T ¼ T þ ln þ ln þ (62)

1 a

lar at r = 0, the unique solution

2 k R k R R h

2 1 c 2 c

ezE

urðÞ¼urðÞ ¼ a¼ (60)

0 where T is the ambient temperature, h is the heat transfer

a

4pmðÞ a

0

coefficient to the surroundings (power radiated per unit

area per unit temperature difference between the outer

is determined. Thus, the velocity still has a flat profile and

the result (57) may still be used, provided that the ‘m’is

* The author’s expression for what they call the “electrophoretic

interpreted as them at the wall, r = a . Thus, it is the mod-

0

migration velocity” appears to be missing a pre-factor (2/3 for

ification of the electrophoretic velocity, not the electroos-

spherical particles with thin EDL), however, this should not

motic velocity that causes band broadening, as pointed

affect the final result, Eq. (61) which depends only on the fact

out by Knox [53] who provided the first correct treatment that this velocity is inversely proportional to the viscosity.

2004 WILEY-VCH Verlag GmbH & Co. KGaA, WeinheimElectrophoresis 2004, 25, 214–228 EOF and band broadening 225

sion due to the Taylor-Aris mechanism. Minimizing such

wall and the environment), R and R are the outer capil-

c 2

lary diameters with and without the polyimide coating, k turn induced “geometric dispersion” is a subject of active

2

and k are the thermal conductivities of fused-silica and research interest (see, e.g., the short report by Zubritsky

c

polyimide coating, respectively, and G is the (constant) [57] for an overview of the approaches being pursued).

heat generation rate. The temperature distribution within

The simplest geometry is that of a rectangular channel of

the buffer itself has a parabolic profile. A more accurate

width ‘a’ and infinite depth that has a curved region with a

model for calculating the temperature distribution must

turn angle ofy (in radians) connecting infinitely long inlet

account for the variation of buffer electrical conductivity

and outlet sections. The channel is assumed to lie in a

with temperature. In that case the heat generation rate is

single plane. The following formula has been proposed

no longer constant but to a good approximation a linear

by Griffiths and Nilson [28] for the turn induced axial var-

function of temperature. The resulting equation for ther-

2

iances :

mal transport is nevertheless still linear, and may be read-

ily solved in terms of the zeroth order Bessel function [56]. 2

2

s y Pe 2r y

¼ þ (63)

However, as shown by Jones and Grushka [56], under

a 15r þ 3Pe Pe

typical CE operating conditions the correction to the

parabolic temperature profile obtained assuming con-

Here, r is the mean of the radii of curvature of the inner

*

stant conductivity is very small.

and outer walls normalized by the channel width, and Pe

= aU/D is the Peclet number based on the (uniform) flow

In addition to the obvious radial dependence on tempera-

speed far upstream of the bend (U) and the molecular dif-

ture, axial variations in temperature could also occur due

fusion coefficient (D).

to various inhomogeneities in the capillary. Such varia-

tions could induce pressure gradients and lead to band

There are two qualitatively different regimes of interest.

broadening. It is not clear whether such axial variations

The distinction is based on whether the cross stream dif-

are present and if so whether they do cause significant

fusion of species is dominant or if it is a small correction.

dispersion. Convective motion of fluid in the capillary is

The characteristic time scale required for any concentra-

also possible. These are open areas for investigation.

tion variation across the channel to be homogenized is t

D

2

,a /D. This is to be compared with the characteristic

residence time of the sample in the curved section of

6.4 Dispersion in curved channels

the capillary, t ,ar y/U. Clearly if t t diffusion is un-

R A D

*

important and the dispersion may be calculated from

In a typical laboratory CE unit the capillary is usually

purely geometrical considerations. The ratio

straight, or has a radius of curvature very much larger

t /t ,ry/Pe (64)

than the capillary diameter (in which case it may be con-

A D

*

sidered essentially straight). However, for CE systems,

Thus, we distinguish between the (i) Low Peclet number

the requirement of a long analysis section (in order to sup-

regime: t /t 1 which corresponds to turns of relatively

A D

port a larger voltage drop, the number of theoretical

large radius of curvature or large diffusion coefficients. In

plates being proportional to the voltage drop) forces one

this case, cross-stream diffusion is dominant so that the

to consider sinuous channels in order to fit it on a chip of

concentration is almost constant across the capillary.

modest footprint.

Thus, the variance may be obtained through a straight for-

ward substitution of the analytical expression for the ve-

Such curved channels contribute to axial dispersion due

locity profile in the Taylor-Aris formalism for calculating

to the following mechanism: Since the isopotential sur-

the variance. (ii) High Peclet number regime: t /t 1

faces intersect the channel boundaries at right angles A D

which corresponds to small radius of curvature and low

(ignoring the finite thickness of the EDL) the same poten-

molecular diffusion coefficients. In this case, the disper-

tial drop occurs over a shorter distance on the inner side

sion may be calculated by purely geometrical means

of a curve than on the outer side. As a result, the applied

since the solute is simply advected along streamlines.

voltage creates a stronger field on the inside edge of the

channel. Therefore, by the HS slip boundary condition,

The expression (63) is an empirically constructed “com-

the fluid velocity is higher at the inner edge than at the

posite expression” that reduces to the correct limits in

outer one. Further, solute particles near the inner edge of

the low and high Peclet number regimes. In these two lim-

the turn traverse a shorter distance at this higher speed

its the expression (63) for the variance take the following

than particles at the outer edge (the so-called “race track

forms

effect”). Thus, as a band goes round the bend it is sheared

2 4

u a y

out of shape. Cross-stream molecular diffusion acts on

2

s 2Dþ tðÞ Pe! 0 (65)

A

2

this shear resulting in an enhanced effective axial disper- 15r D

2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim226 S. Ghosal Electrophoresis 2004, 25, 214–228

2

2

bine several of these ideas. For example, Dutta and Leig-

y a

2

s þ 2DtðÞ Pe!1 (66)

A

ton have proposed spirals with inner walls that are wavy

3

to compensate for the shorter path [66] thereby both

where r = ar is the dimensional mean radius of curvature increasing r as well as reducing the geometric prefactor

* *

and t = ry/U is the transit time across the bend. The for- in (63). Johnson et al. [67] achieve the same effect by

A

mer expression clearly has the form of the variance due to modifying the wall z-potential through laser ablation,

molecular diffusion added to a Taylor-like term. In the sec- which will also change the pattern of EOF in the channel.

ond case we have molecular diffusion plus a purely geo- Fiechtner and Cummings [68] have proposed a ‘faceted’

metrical quantity that is independent of the diffusion coef- design which is a polygonal shape approximating a

ficient D but depends solely on the difference of path smooth spiral. Griffiths and Nilson [69] have investigated

lengths ay between the inner and outer edges of the ‘pleated channels’ where some of the turn induced dis-

bend. In microfluidic applications both limits are relevant. persion is ‘undone’ at the following turn in the opposite

For example, if we take a , 100mm, r , 1 cm andy =p direction.

1 2

then ry , 300. Since Pe , 10 2 10 for small molecules

*

3 4

but Pe , 10 2 10 for macromolecules it is clear from

Eq. (64) that the ‘high’ the ‘low’ as well as the ‘intermedi-

7 Summary and conclusions

ate’ Peclet number regimes are relevant for microfluidic

Electroosmosis and electrophoresis are closely related

applications. Culbertson et al. [58] also considered the

phenomena which are often present together in CZE.

problem of geometric dispersion in a rectangular channel

EOF is an effect of the action of the applied electric field

undergoing a 1807 turn. They used an ad hoc modification

on the Debye layer of free charges adjoining the walls of a

of the expression for the axial stretching of an initially

microfluidic channel. Electroosmosis is both an ally and

sharp bend in the absence of diffusion to incorporate dif-

an enemy of the researcher interested in achieving effi-

fusive effects. The model was fitted to experimental data.

cient separation. The main advantage of EOF is that it is

However, their model differs from Eq. (63) by a numerical

a useful means for transporting analytes and buffer in a

factor at low Peclet numbers and the apparent agreement

microfluidic circuit. Unlike pressure-driven flows in which

with experimental data in this regime has been attributed

the pressure drop needed to maintain a certain fixed

by Griffiths and Nilson [28] to uncertainties in the meas-

mean flow speed increases inversely as the square of the

urement of the channel width.

capillary radius, in EOF, the voltage needed is independ-

ent of the capillary radius. Thus, EOF is very efficient for

The first term in Eq. (63) increases linearly with Pe for

transporting infinitesimal volumes of fluid through very

small values of Pe and saturates at large Pe. Clearly the

narrow capillaries. The presence of EOF in the microchan-

dispersion contribution from this term can be reduced if

nels enables single point detection (species of either sign

either Pe can be reduced or else if r could be increased.

*

elute at the same end), reduces analysis times and

Two main ideas for designing low dispersion bends have

enables operation of the microdevice in a continuous

evolved out of these two possibilities. The channel can be

mode. The disadvantage is that any effect that causes

‘pinched’ so that it becomes very narrow in the curved

the EOF flow profile to deviate from the classical “plug

sections. This effectively reduces the Pe locally by

flow” shape would cause band broadening.

decreasing a. This approach has been investigated by

Paegel et al. [59]. Alternatively one could design the

The state of current knowlege of the fluid mechanics of

separation channel in the form of spiral turns of large radii

EOF was reviewed with special attention to the role of

of curvature. This approach, which relies on increasing

EOF on dispersion in CZE systems. EOF is fully described

the effective radius of curvature r , has been followed, for

*

by the Poisson-Boltzmann equation coupled to the in-

example, by Culbertson and others [60–62]. Yet a third

compressible Navier-Stokes and continuity equations

possibility is to redesign the channel geometry at the

describing fluid flow. This set of equations is, however,

bend so as to compensate for both the higher electric

quite complex and nonlinear, exact analytical solutions

field and the ‘race track’ effect. Effectively this is equiva-

can be found only for certain highly idealized systems.

lent to attempting to reduce the prefactor multiplying the

Fortunately, however, a series of simplifications can be

Pe in the first term of Eq. (63) by altering the velocity dis-

made to these equations, at each step exploiting a certain

tribution. This approach has been adopted for example

disparity in scales inherent in the problem.

by Molho et al. [63, 64] who tried to come up with optimal

shapes using computer simulation. The results were com- The first level of simplification comes about through the

pared to experimental data and appeared to show rea- assumption of thin Debye layers. This is justified, since in

sonable agreement. Similar designs were proposed by most current applications to EOF, characteristic channel

Griffiths and Nilson [65]. Others have attempted to com- radii are of the order of 10–100mm whereas for the normal

2004 WILEY-VCH Verlag GmbH & Co. KGaA, WeinheimElectrophoresis 2004, 25, 214–228 EOF and band broadening 227

incompressibility constraint. Such pressure fluctuations

range of buffer concentrations used, the Debye length is

1–10 nm. This disparity in scales allows us to drop the give rise to a “Poiseuille” type of flow with a parabolic pro-

term representing the electrical force in the Navier-Stokes file, which through the mechanism of Taylor-Aris disper-

equations and instead, to replace the classical “no slip” sion leads to greatly enhanced effective axial dispersion.

boundary conditions at the solid fluid interface by the Radial variations are often caused by nonuniform temper-

“HS slip boundary conditions”. Thus, within the realm of ature distributions inside the capillaries and cause band

this approximation, the electrical forces are described by broadening due to differential rates of electromigration

a single parameter, ‘the z-potential’ that enters the fluid over the capillary cross-section. Channel curvature

flow description solely through the new ‘slip’ boundary results in ‘geometric dispersion’ and is relevant at the

conditions and the Poisson-Boltzmann equation may be low, intermediate as well as high Peclet number regimes.

replaced by the Laplace equation for a charge free region.

Though the basic analytical machinery for treating EOF in

If fluid properties and thez-potential are uniform, no exter- a wide variety of situations of interest appear to be avail-

nal pressure gradient is applied, the substrate is a poor able, there are many areas where theoretical understand-

ing is in a relatively primitive state. Advances in the study

electrical conductor and the Reynolds number is negligi-

of this new area of fluid mechanics should facilitate the

bly small, then the fluid velocity through a channel of any

development of software and numerical tools that could

geometry is simply proportional to the electric field which

be very useful in the quest to developm-TAS technology

may be determined by solving a Dirichlet problem for the

and more efficient separation methods.

electric potential. This ‘similitude’ between the electric

field and the hydrodynamic flow also applies to the finite

Reynolds number situation provided the flow is irrota-

Received August 18, 2003

tional. However, the assumption of uniform fluid proper-

ties andz-potential is not valid in all problems of interest,

such as in the problem of flow modification due to wall

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