Streaming Potential and Electroosmotic Flow inHeterogeneous Circular Microchannels with NonuniformZeta Potentials: Requirements of Flow Rate and Current

Continuities

Jun Yang,† J. H. Masliyah,‡ and Daniel Y. Kwok*,†

Nanoscale Technology and Engineering Laboratory, Department of Mechanical Engineeringand Department of Chemical and Materials Engineering, University of Alberta,

Edmonton, Alberta T6G 2G8, Canada

Received July 9, 2003. In Final Form: December 3, 2003

Real surfaces are typically heterogeneous, and microchannels with heterogeneous surfaces are commonlyfound due to fabrication defects, material impurities, and chemical adsorption from solution. Such surfaceheterogeneity causes a nonuniform surface potential along the microchannel. Other than surfaceheterogeneity, one could also pattern the various surface potentials along the microchannels. To understandhow such variations affect electrokinetic flow, we proposed a model to describe its behavior in circularmicrochannels with nonuniform surface potentials. Unlike other models, we considered the continuitiesof flow rate and electric current simultaneously. These requirements cause a nonuniform electric fielddistribution and pressure gradient along the channel for both pressure-driven flow (streaming potential)and electric-field-driven flow (electroosmosis). The induced nonuniform pressure and electric field influencethe electrokinetic flow in terms of the velocity profile, the flow rate, and the streaming potential.

I. Introduction

The presence of an electric double layer (EDL) at thesolid-liquid interface and its electrokinetic phenomenahave been used to develop various chemical and biologicalinstruments.1-3 A common assumption is the uniformityof surface properties during electrokinetic fluid transportin microchannels.4-9 Nevertheless, surface heterogeneitycan easily arise from fabrication defects or chemicaladsorption onto microchannels. For example, Norde etal.10 studied the relationship between protein adsorptionand streaming potentials. Ajdari11,12 presented a theoreti-cal solution forelectroosmotic flowthrough inhomogeneouscharged surfaces. Ren and Li13 numerically studiedelectroosmotic flow in heterogeneous circular microchan-nels with axial variation of the surface potential. Andersonand Idol14 studied electroosmosis through pores withnonconformed charged walls. They showed that the meanelectroosmotic velocity within the capillary was given by

the classical Helmholtz equation with the local surfacepotential replaced by the average surface potential. Keelyetal.15 theoreticallyprovided flowprofiles insidecapillarieswith nonuniform surface potentials. Herr et al.16 theoreti-cally and experimentally investigated electroosmotic flowin cylindrical capillaries with nonuniform surface chargedistributions. A nonintrusive caged fluorescence imagingtechnique was used to image the electroosmotic flow; aparabolic velocity profile induced by the pressure gradientdue to the heterogeneity of the capillary surfaces wasobserved. Cohen and Radke studied streaming potentialsof a slit with a nonuniform surface charge density.17

Erickson and Li18 studied microchannel flow with patch-wise and periodic surface heterogeneity. However, all theabove studies10-18 assumed uniform axial electric fieldsalong the microchannel and did not consider continuityof electric current.

Phenomenologically, let us consider two independentmicrochannels with the same geometry, electrolyte, andflow rate. If these two channels have different surfacepotentials, the electric fields associated with the electricdouble layer would have to be different. If we assemblethese two independent channels in series as sections ofa channel, the electric field should be nonuniform alongthe flow direction and current continuity should also besatisfied. It is the purpose of this paper to study oscillatingelectrokinetic (streaming potential and electroosmotic)flow in a microchannel with different surface potentialsin every section which satisfy both flow rate and currentcontinuity requirements.

* To whom correspondence should be addressed. Phone: (780)492-2791. Fax: (780) 492-2200. E-mail: daniel.y.kwok@ualberta.ca.

† Department of Mechanical Engineering.‡ Department of Chemical and Materials Engineering.(1) Harrison, J. D.; Fluri, K.; Seiler, K.; Fan, Z. H.; Effenhauser, C.

S.; Manz, A. Science 1993, 261, 895.(2) Blackshear, P. J. Sci. Am. 1979, 241, 52.(3) Penn, R. D.; Paice, J. A.; Gottschalk, W.; Ivankovich, A. D. J.

Neurosurg. 1984, 61, 302.(4) Rice, C. L.; Whitehead, R. J. Phys. Chem. 1965, 69, 4017.(5) Levine, S.; Marriott, J. R.; Robinson, K. J. Chem. Soc., Faraday

Trans. 2 1975, 71, 1.(6) Lyklema, J. Fundamentals of Interface and Colloid Science;

Academic Press: 1995; Vol. II.(7) Yang, J.; Kwok, D. Y. J. Phys. Chem. B 2002, 106, 12851.(8) Yang, J.; Kwok, D. Y. J. Chem. Phys. 2003, 118, 354.(9) Yang, J.; Kwok, D. Y. Langmuir 2003, 19, 1047.(10) Norde, W.; Rouwendal, E. E. J. Colloid Interface Sci. 1990, 139,

169.(11) Ajdari, A. Phys. Rev. Lett. 1995, 75 (4), 755.(12) Ajdari, A. Phys. Rev. E 1996, 53 (4), 4996.(13) Ren, L.; Li, D. J. Colloid Interface Sci. 2001, 243, 255.(14) Anderson, J. L.; Idol, W. K. Chem. Eng. Commun. 1985, 38, 93.

(15) Keely, C. A.; de Goor, T. A. A. M. V.; McManigill, D. Anal. Chem.1994, 66, 4236.

(16) Herr, A. E.; Molho, J. I.; Santiago, J. G.; Mungal, M. G.; Kenny,T. W. Anal. Chem. 2000, 72, 1053.

(17) Cohen, R. R.; Radke, C. J. J. Colloid Interface Sci. 1991, 141,338.

(18) Erickson, D.; Li, D. Langmuir 2002, 18, 8949.

3863Langmuir 2004, 20, 3863-3871

10.1021/la035243u CCC: $27.50 © 2004 American Chemical SocietyPublished on Web 04/09/2004

II. Controlling Equations and BoundaryConditions in an Individual Pore

We consider a heterogeneous microchannel with Nsections shown in Figure 1 where l is the length of eachsection and the subscript m implies quantities relating tothe mth section. We assume each section to be long enoughfor a fully developed flow and neglect the disturbanceacross regions between two sections. The effect of entrancelength on microchannel flow has been studied.19,20 It istypically found that the entrance length is on the orderof microns for microchannel flow and can be neglected.This assumption is consistent with those in refs 15 and16. In the mth section in Figure 1, we consider a boundaryvalue problem for oscillating electrolyte flow driven by anoscillating pressure gradient and an electric field; a localcylindrical coordinate system (r, θ, z) is used for everysection where the z-axis is taken to coincide with themicrochannel central axis. All field quantities are takento depend on the radial coordinate, r, and the time, t. Theboundary value problem with the relevant field equationsand the boundary conditions is given below.

A. Electrical Field. Because each heterogeneousmicrochannel section has its own physical and chemicalproperties, the corresponding electric double layer alsohas its own surface potential and ion distribution. Here,we consider each section to be a different medium. Forsimplicity, we neglect the differences in the permittivitiesand consider only the deviations of electric field strength.The total potential of the mth section, um, at a location (r,z) at a given time, t, is taken to be

where ψm(r) is the potential due to the double layer at theequilibrium state (i.e., no liquid motion with no appliedexternal field), u0,m is the potential at the beginning of themth section with z ) 0 [i.e., u0,m ≡ um(r, 0, t)], and E′z,m(t)is the spatially uniform time-dependent electric fieldstrength in the mth layer. The total potential, um, in eq1 is axisymmetric, and when E′z,m(t) is time-independent,eq 1 is similar to eq 6.1 in ref 21. The time-dependent flowto be studied here is assumed to be sufficiently slow suchthat the radial charge distribution is relaxed at its steadystate. Further, it is assumed that any induced magneticfields are sufficiently small and negligible such that thetotal electric field may still be defined as -∇ubm;22 thisdefinition may then be used to obtain the Poisson equation

where Fm is the free charge density and ε is the permittivity

of the medium. Combining eqs 1 and 2 yields the followingPoisson equation in cylindrical coordinates

The conditions imposed on ψm(r) are

where ψs,m is the surface potential near the microchannelwall in the mth section at r ) a and a is the radius of themicrochannel. For brevity, we shall focus on a symmetric,binary electrolyte with univalent charges. The cations andthe anions are identified as species 1 and 2, respectively.On the basis of the assumption of thermodynamic equi-librium, the Boltzmann equation provides the local chargedensity, Fi,m, of the ith species as

where zi is the valence of the ith species, e is the elementarycharge, n∞ is the ionic concentration in an equilibriumelectrochemical solution at the neutral state where ψm ≈0 and can be neglected, k is the Boltzmann constant, andT is theabsolute temperature.WeemployaDebye-Huckelapproximation for low zeta potentials (zieψm/kT , 1) whichprovides an acceptable prediction for surface potentialsup to 100 mV.23 For simplicity, this approximation isemployed to obtain the analytical solution. It should benoted that numerical integration of the Poisson-Boltz-mann equation can be found elsewhere.24 From ouranalytical solutions, we express sinh(z0eψm/kT) ≈ z0eψm/kT and the total charge density follows from eqs 3 and 5as

where we have used z1 ) -z2 ) z0. Finally, the definitionof the reciprocal of the double layer thickness for a (z0:z0)electrolyte is given as

Combining eqs 3 and 6 results in

where a is radius of the pores.B. Hydrodynamic Field. The axial electric field will

induce a body force of FmE′z,m, and the modified Navier-Stokes equation becomes

where we have taken the pressure gradient [∂pm/∂z ≡ ∂pm/

(19) Werner, C.; Korber, H.; Zimmermann, R.; Dukhin, S.; Jacobasch,H.-J. J. Colloid Interface Sci. 1998, 208, 329.

(20) Ren, L.; Li, D.; Qu, W. J. Colloid Interface Sci. 2001, 233, 12.(21) Masliyah, J. H. Electrokinetic Transport Phenomena; Alberta

Oil Sands Technology and Research Authority: Edmonton, Alberta,Canada, 1994.

(22) Shadowitz, A. The Electromagnetic Field; McGraw-Hill: NewYork, 1975.

(23) Hunter, R. J. Introduction to Modern Colloid Science; Oxford:New York, 1993.

(24) Bowen, W. R.; Jenner, F. J. Colloid Interface Sci. 1995, 173, 388.

Figure 1. Schematic of a heterogeneous circular microchannel.

um ≡ um(r, z, t) ) ψm(r) + [u0,m - zE′z,m(t)] (1)

∇2um ) -Fm

ε(2)

1r

ddr(rdψm(r)

dr ) ) -Fm

ε(3)

ψm(a) ) ψs,m and ψm(0) ) a finite number (4)

Fi,m ) zien∞ exp[-zieψm

kT ] (i ) 1, 2) (5)

Fm ) ∑i)1

2

Fi,m )-2n∞e2z0

2

kTψm (6)

κ ) x2n∞e2z02

εkT(7)

1r

ddr(r dψm(r)

dr ) ) κ2ψm

ψm(a) ) ψs,m and∂ψm(0)

∂r) 0 (8)

- 1µ

∂pm

∂z+ 1

r∂

∂r(r∂vm

∂r ) + 1µ

FmE′z,m ) 1ν

∂vm

∂t(9)

3864 Langmuir, Vol. 20, No. 10, 2004 Yang et al.

∂z(t)] to be position-independent, µ is the viscosity, and νis the kinematic viscosity of the liquid. The boundaryconditions for the velocity field are

The electric current density along the microchannel maybe integrated over the channel cross-section to give theelectric current

where D is the ionic diffusion coefficient. The first termon the right side of eq 11 is due to bulk convection, andthe second term is due to charge migration.21 Because ofthe assumption of an infinitely extended microchannel,the contribution to the current due to concentrationgradients vanishes. Using eq 5 for a (z0:z0) electrolyte, wehave F1,m - F2,m ) 2ez0n∞ cosh(z0eψm/kT). The Debye-Huckel approximation implies that cosh(z0eψm/kT) ≈ 1and F1,m - F2,m ) 2z0en∞. With this simplification, eq 11becomes

and the flow rate, qm, can be written as

III. Normalized Equations

Here, we provide the normalized governing equationsand boundary conditions. Using a yet unknown charac-teristic velocity, ⟨v⟩, the following normalized quantitiesare defined as

where Vm, R, Ψm, Ψs,m, Um, and Ez,m are the normalizedvelocity, the radial coordinate, the surface potential, thetotal potential, and the strength of the electric field of themth section, respectively. The normalized counterpartsof eqs 4 and 8 become, respectively,

and

whereas eqs 9, 10, 12, and 13 become, respectively,

where Pm, Im, and Qm are the normalized pressure, thecurrent, and the flow rate of the mth section, respectively.In deriving eqs 15-19, the following normalized quantitieshave been identified

where Fjm, K, τ, and Σ are the normalized charge density,the reciprocal of the double layer thickness, the time, andthe conductivity, respectively. The expression for thecharacteristic velocity can also be identified as

We wish to point out that our normalizing schemes in eqs20 and 21 are different from those of other authors,25-27

as we did not employ the ionic concentration at equilib-rium, n∞, to normalize the charge density, Fm. The choiceof this selection is important, as we intend to study theeffects of electrolyte concentration indirectly through Kon the flow properties; otherwise, the results would havebeen misleading, since it makes no sense to study aquantity that has already been used for normalization.Finally, we define the following normalized quantities

where the parameters ω and qm are the frequency of theexternal oscillating field and the volumetric flow rate,respectively, and Ω is the normalized frequency.

IV. Analytical Solution of the mth SectionAn analytical solution is sought here for a sinusoidal

periodicity in the electrohydrodynamic fields, and this isbest addressed by using complex variables. Thus, thegeneral field quantity, X, may be defined as the real partof the complex function (X*ejΩτ) where X* is complex (j )x-1), Ω is the normalized oscillation frequency oscilla-tion, and t is the time. The general field quantity X iswritten as

The phase angle, φ, is defined as

(25) Hu, L.; Harrison, J. D.; Masliyah, J. H. J. Colloid Interface Sci.1999, 215, 300.

(26) Mala, G. M.; Li, D.; Dale, J. D. Int. J. Heat Mass Transfer 1997,40, 3079.

(27) Mala, G. M.; Li, D.; Werner, C.; Jacobasch, H. J.; Ning, Y. B.Int. J. Heat Fluid Flow 1997, 18, 489.

vm(a, t) ) 0 and∂vm(0, t)

∂r) 0 (10)

im ) 2π∫0

aFmvmr dr +

2πz0eDkT

E′z,m∫0

a(F1,m - F2,m)r dr

(11)

im ) 2π∫0

aFmvmr dr +

2z02e2n∞DkT

(πa2)E′z,m (12)

qm ) 2π∫0

avmr dr (13)

Vm ) 1⟨v⟩

v R ) 1a

r Ψm )z0ekT

ψm Ψs,m )z0ekT

ψs,m

Um )z0ekT

um Ez,m )z0eakT

E′z,m (14)

1R

ddR(RdΨm

dR ) ) K2Ψm and Ψm(1) ) Ψs,m (15)

Ψm(0) ) a finite number

-∂Pm

∂Z+ 1

R∂

∂R(R∂Vm

∂R ) - K2ΨmEz,m )∂Vm

∂τ(16)

Vm(1, τ) ) 0∂Vm(0, τ)

∂R) 0 (17)

Im ) -2π∫0

1K2ΨmVmR dR + π

2a2z04e4µDn∞

ε2k3T3

Ez,m

) -2π∫0

1K2ΨmVmR dR + πΣK2Ez,m (18)

Qm ) 2π∫0

1VmR dR (19)

Fjm )a2ez0

εkTFm K ) κa Pm ) a

µ⟨v⟩pm τ ) ν

a2t

Im )ez0

εkT⟨v⟩im Σ )

z02e2µD

εk2T2(20)

⟨v⟩ ) εk2T2

µae2z02

(21)

Ω ) a2

νω and Qm )

qm

⟨v⟩a2(22)

X ) Re[X*ejΩτ] (23)

Heterogeneous Circular Microchannels Langmuir, Vol. 20, No. 10, 2004 3865

where Im(X*) and Re(X*) are the imaginary and real partsof X*, respectively. An alternative representation of eq 23is given as

where

With the notation of eq 23, we shall seek the solution ofthe boundary value problem for the following specificdependencies.

We consider the class of solutions where the amplitude ofthe pressure gradient and the electric field could befrequency-dependent, that is, Pm

/ ≡ Pm/ (Ω) and Ez,m

/ ≡Ez,m

/ (Ω). The solution for Ψm will then follow from eq 8 andthat for Vm from eq 9. Thus,

where

The expressions for VP,m/ (R, Ω) and VE,m

/ (R, Ω) will begiven at the end of this section. The electric current willfollow from eq 11 and may be written as

where

The volumetric flow rate, Qm, is defined as Qm ) 2π∫0aRVm

dR and can be expressed as

where

During pressure-driven flow, the amplitude of the electricfield strength in the mth section, Ez,m

/ (Ω), is found bysetting I* ) 0 in eq 29. Thus,

Equation 31 may be substituted into eqs 28 and 30 todetermine the normalized liquid velocity and the volu-metric flow rate, respectively. Alternatively, the velocity,the current, and the volumetric flow rate during elec-troosmotic flow follow from eqs 28-30 by setting Pm

/ (Ω)) 0.

The relevant quantities of an individual mth section ineqs 28-30 are listed below

where J0 and J1 are the zeroth- and first-order Besselfunctions of the first kind. Note that the analysis withoutthe EDL effects follows from eqs 32-34 by setting Ψs,m )0. The resulting expressions are identical to those obtainedby Uchida.28 We can also find that QE,m

/ ) IP,m/ satisfies

Onsager’s theorem.4When Ω f 0, eqs 32-34 reduce to those of the steady

state. The steady state response follows from eqs 32-34by setting Ω ) 0, and such solutions were originallyobtained by Rice and Whitehead4 as

V. Pressure and Electric Field DistributionBecause of the surface heterogeneity of the microchan-

nel, each section has its own normalized parameters: the

(28) Uchida, S. ZAMP 1950, 7, 403.

φ ) tan-1 Im(X*)Re(X*)

(24)

X ) Re[|X*|ej(Ωτ+φ)] (25)

|X| ) |X*| and |X*| ) xIm2(X*) + Re2(X*) (26)

-∂Pm

∂Z) Re[Pm

/ ejΩτ] and Ez,m ) Re[Ez,m/ ejΩτ] (27)

Vm ) Re[Vm/ ejΩτ]

Vm/ ≡ Vm

/ (R, Ω) )

VP,m/ (R, Ω)Pm

/ (Ω) + VE,m/ (R, Ω)Ez,m

/ (Ω) (28)

Im ) Re[Im/ ejΩτ]

Im/ ≡ Im

/ (Ω) ) IP,m/ (Ω)Pm

/ (Ω) + IE,m/ (Ω)Ez,m

/ (Ω) (29)

Qm ) Re[Qm/ ejΩτ]

Qm/ ≡ Qm

/ (Ω) ) QP,m/ (Ω)Pm

/ (Ω) + QE,m/ (Ω)Ez,m

/ (Ω) (30)

Ez,m/ (Ω) ) -

IP,m/ (Ω)

IE,m/ (Ω)

Pm/ (Ω) for Im

/ ) 0 (31)

VP,m/ (R, Ω) ) 1

jΩ[1 -J0(Rx-jΩ)

J0(x-jΩ) ]VE,m

/ (R, Ω) )K2Ψs,m

K2 - jΩ[J0(jKR)

J0(jK)-

J0(Rx-jΩ)

J0(x-jΩ) ] (32)

IP,m/ (Ω) ) 2πK2Ψs,m 1

ΩKJ1(jK)

J0(jK)- 1

Ω2 + jΩK2×

[jKJ1(jK)

J0(jK)- (x-jΩ)

J1(x-jΩ)

J0(x-jΩ)]IE,m/ (Ω) ) -

2πK4Ψs,m2

K2 - jΩ 12[1 +

J12(jK)

J02(jK)] - 1

jΩ - K2×

[jKJ1(jK)

J0(jK)- (x-jΩ)

J1(x-jΩ)

J0(x-jΩ)] + πΣK2 (33)

QP,m/ (Ω) ) 2π

jΩ[12 - 1x-jΩ

J1(x-jΩ)

J0(x-jΩ)]QE,m

/ (Ω) ) IP,m/ (Ω) (34)

VP,m/ (R, 0) ) 1

4(1 - R2)

VE,m/ (R, 0) ) -Ψs,m[1 -

J0(jKR)

J0(jK) ] (35)

IP,m/ (0) ) -πΨs,m[1 - 2

jKJ1(jK)

J0(jK)]IE,m/ (0) )

-πΨs,m2 K2[1 - 2

jKJ1(jK)

J0(jK)+

J12(jK)

J02(jK)] + πΣK2 (36)

QP,m/ (0) ) π

8

QE,m/ (0) ) IP

/ (0) (37)

3866 Langmuir, Vol. 20, No. 10, 2004 Yang et al.

pressure gradient, Pm/ , the strength of the electric field,

Ez,m/ , the surface potential, Ψs,m, and the length, Lm,

where Lm ) lm/a. For each section, Ez,m/ and Pm

/ areuniform field quantities; for the whole microchannel, thetotal current, Itotal, and the total flow rate, Qtotal, are thesame for any section due to the continuities of currentand flow rate. The total pressure and the potential dropsof the microchannel are the sum of those in each sectionand can be expressed as

where L ) ∑m)1N Lm is the total normalized length and U

is the normalized total potential. On the basis of eqs 38-41, one can determine the relationships of the pressuregradient and the strength of the electric field in all sections.We will express all quantities as functions of the corre-sponding quantities in the first section (m ) 1).

A. Streaming Potential (Pressure-Driven Flow).In this case, the total normalized pressure gradient, P*,is a known input. At equilibrium, for the mth section, eq31 becomes

and the continuity of current is satisfied automatically.From eq 38, we have

Substituting eq 42 into eq 43, we can relate the pressuregradient in the mth section to that of the first section.

From eq 40, we have

According to the known pressure gradient, one can solvefor the pressure gradient in every section. The totalstreaming potential (potential drop) is then the sum of

the potential drops in each section.

B. Electroosmosis (Electric-Field-Driven Flow).In the case of electroosmosis, the known inputs are thenormalized total strength of the electric field, Ez

/, and thenormalized pressure gradient, P*. Normally, both sidesof an electroosmotic system are opened to the atmosphereto eliminate the effect of hydrostatic pressure on flowpressure such that P* ) 0. The continuities of current andflow rate can be expressed as

Note that QE,m/ ) IP,m, and Pm

/ and Em/ can be solved from

eqs 47 and 48

Following eqs 40 and 41, one can obtain a set of equations.

For brevity, we write eqs 51 and 52, respectively, as

Qtotal ) Re[Qtotal/ ejΩτ] ) Re[(QP,m

/ (Ω)Pm/ (Ω) +

QE,m/ (Ω)Ez,m

/ (Ω))ejΩτ] (38)

Itotal ) Re[Itotal/ ejΩτ] ) Re[(IP,m

/ (Ω)Pm/ (Ω) +

IE,m/ (Ω)Ez,m

/ (Ω))ejΩτ] (39)

∆P ) Re[P*LejΩτ] ) Re[∑m)1

N

Pm/ LmejΩτ] (40)

∆U ) Re[Ez/Lejωt] ) Re[∑

m)1

N

Ez,m/ LmejΩτ] (41)

Ez,m/ ) -

IP,m/

IE,m/

Pm/ (m ) 1, ... ,N) (42)

QP,m/ (Ω)Pm

/ (Ω) + QE,m/ Ez,m

/ ) QP,1/ (Ω)P1

/(Ω) + QE,1/ Ez,1

/

(43)

Pm/ )

QP,1/ -

(IP,1/ )2

IE,1/

QP,m/ -

(IP,m/ )2

IE,m/

P1/ (44)

P*L ) P1/ ∑

m)1

N

Lm

QP,1/ -

(IP,1)2

IE,1

QP,m/ -

(IP,m)2

IE,m

(45)

Ez/ )

P*∑m)1

N [-Lm

IP,m/

IE,m/

QP,1/ -

(IP,1)2

IE,1

QP,m/ -

(IP,m)2

IE,m

1

∑m)1

N

Lm

QP,1/ -

(IP,1)2

IE,1

QP,m/ -

(IP,m)2

IE,m

](46)

QP,m/ (Ω)Pm

/ (Ω) + QE,m/ Ez,m

/ ) QP,1/ (Ω)P1

/(Ω) + QE,1/ Ez,1

/

(47)

IP,m/ (Ω)Pm

/ (Ω) + IE,m/ Ez,m

/ ) IP,1/ (Ω)P1

/(Ω) + IE,1/ Ez,1

/ (48)

Pm/ )

IP,m/ IP,1

/ - IE,m/ QP,1

/

(IP,m/ )2 - IE,m

/ QP,m/

P1/ +

IP,m/ IE,1

/ - IE,m/ IP,1

/

(IP,m/ )2 - IE,m

/ QP,m/

Ez,1/

(49)

Ez,m/ )

IP,m/ QP,1

/ - QP,m/ IP,1

/

(IP,m/ )2 - IE,m

/ QP,m/

P1/ +

IP,m/ IP,1

/ - QP,m/ IE,1

/

(IP,m/ )2 - IE,m

/ QP,m/

Ez,1/

(50)

P*L ) P1/ ∑

m)1

N

Lm

IP,m/ IP,1

/ - IE,m/ QP,1

/

(IP,m/ )2 - IE,m

/ QP,m/

+

Ez,1/ ∑

m)1

N

Lm

IP,m/ IE,1

/ - IE,m/ IP,1

/

(IP,m/ )2 - IE,m

/ QP,m/

(51)

Ez/L ) P1

/ ∑m)1

N

Lm

IP,m/ QP,1

/ - QP,m/ IP,1

/

(IP,m/ )2 - IE,m

/ QP,m/

+

Ez,1/ ∑

m)1

N

Lm

IP,m/ IP,1

/ - QP,m/ IE,1

/

(IP,m/ )2 - IE,m

/ QP,m/

(52)

P*L ) A11P1/ + A12Ez,1

/ (53)

Ez/L ) A21P1

/ + A22Ez,1/ (54)

Heterogeneous Circular Microchannels Langmuir, Vol. 20, No. 10, 2004 3867

Solving the above two equations results in

Thus, Ez,m/ and Pm

/ could be determined from

If both sides of an electroosmotic system are opened to theatmosphere, implying that P* ) 0, one can infer from eq57 that Pm

/ * 0. Because of the heterogeneity of themicrochannel, pressure gradients are induced as a resultof current and flow rate continuities. The expressions forthe experimentally measurable total current, Itotal, andtotal flow rate, Qtotal, are given by

If we focus on a common electroosmotic experimentalsystem with P* ) 0,

Equation 63 is a complex function of frequency Ω, whichincludes information on the surface potential and thelength of each section.

VI. Parametric Study and DiscussionWe study here the effect of the normalized thickness,

K-1, in the EDL and the normalized frequency, Ω, on the

velocity profile, the pressure, and the electric fielddistributions. We have selected the following values forour calculations: the amplitude of the normalized totalpressure gradient P*(Ω) ) 400, the amplitude of thestrength of the total electric field Ez

/(Ω) ) 10, thenormalized frequency Ω ) 10, and the normalizedconductivity Σ ) 3.85 which represents the diffusioncoefficient D ≈ 2 × 10-9 m2/s for a KCl electrolyte.29 Wealso consider a heterogeneous microchannel consisting ofeight sections with the same normalized length, Lm ) 50.Along the flow direction, the normalized surface potentials,Ψs,m, are set, respectively, to 0.96, 1.37, 1.76, 2.15, 2.54,2.93, 3.32, and 3.71. They correspond to the dimensionalsurface potentials of 25, 35, 45, 55, 65, 75, 85, and 95 mV.It is noted that the Debye-Huckel approximation stillprovides a good agreement with experiments for theselected zeta potentials.23

A. Pressure and Electric Field Distribution forPressure-Driven Flow. For pressure-driven flow througha heterogeneous microchannel, we set P ) 0 and Um - Ψm) 0 as reference values at the beginning of the micro-channel. The results for the magnitude of velocity, |V*|,the pressure drop, P, and the axial potential, Um - Ψm,along the channel for Ω ) 0 and 10 are shown in Figure2. We see that the velocity profile remains parabolic alongthe channel for Ω ) 0. When Ω ) 10 (dashed lines), themagnitude of velocity is smaller, since liquid viscositycauses the flow to lag behind the variation of pressure. Itis noted that the pressure gradient in Figure 2 is indeednonlinear along the channel, as seen in Table 1 from thetabulated results. We see that, as the surface potentialincreases, the pressure gradient increases along the flowdirection. This is due to the fact that a larger pressuregradient is required to compensate for the decrease inflow rate due to the stronger electroviscous effect. WhenΩ ) 10, the pressure distribution is nearly identical tothat of Ω ) 0. The potential distribution along the axisof the microchannel is also nonlinear. As the zeta potentialincreases, the strength of the electric field also increases.The reason for the stronger electric field is to make up anet zero current. The strength of the electric field is smallerwhen Ω ) 10. These differences are shown in Table 1 andFigure 2. In Figure 2, K ) 10 implies a diluted solutionfor a fixed microchannel size. We also plotted the corre-sponding results in Figure 3 for a more concentratedsolution for K ) 1000 with Ω ) 0. When K ) 1000 (athinner EDL), the pressure gradient varies slightly (seeTable 2) as compared to that for K ) 10, since theelectroviscous effect is smaller for a more concentratedsolution. However, the strength of the electric field isseveral orders smaller than that of K ) 10, as a smallerelectric field is sufficient to maintain a net zero current.

B. Pressure and Electric Field of Electric-Field-Driven Flow. For electric-field-driven flow through aheterogeneous microchannel, a nonuniform pressuredistribution will result from the different surface poten-tials and electric fields in each section to maintain theflow rate and current continuities. In actual experiments,the electroosmotic system is often opened to the atmo-sphere so that the total pressure difference between thetwo ends of the microchannel is zero.

The results for the magnitude of the velocity, |V*|, thepressure drop, P, and the axial potential, Um - Ψm, alongthe channel for Ω ) 0 and 10 are shown in Figure 4 forK ) 1000; its tabulated results are given in Table 3. InFigure 4, we find that the velocity profile has a parabolic

(29) Newman, J. S. Electrochemical Systems, 2nd ed.; Prentice Hall:Englewood Cliffs, NJ, 1991.

P1/ )

A22LA11A22 - A21A12

P* -A12L

A11A22 - A21A12Ez

/ (55)

Ez,1/ ) -

A21LA11A22 - A21A12

P* +A11L

A11A22 - A21A12Ez

/ (56)

Pm/ )

IP,m/ IP,1

/ - IE,m/ QP,1

/

(IP,m/ )2 - IE,m

/ QP,m/

×

( A22LA11A22 - A21A12

P* -A12L

A11A22 - A21A12Ez

/) +

IP,m/ IE,1

/ - IE,1/ IP,1

/

(IP,m/ )2 - IE,m

/ QP,m/ (-

A21LA11A22 - A21A12

P* +

A11LA11A22 - A21A12

Ez/) (57)

Ez,m/ )

IP,m/ QP,1

/ - QP,m/ IP,1

/

(IP,m/ )2 - IE,m

/ QP,m/

×

( A22LA11A22 - A21A12

P* -A12L

A11A22 - A21A12Ez

/) +

IP,m/ IP,1

/ - QP,m/ IE,1

/

(IP,m/ )2 - IE,m

/ QP,m/ (-

A21LA11A22 - A21A12

P* +

A11LA11A22 - A21A12

Ez/) (58)

Itotal ) IP,m/ Pm

/ + IE,m/ Ez,m

/ (59)

Qtotal ) QP,m/ Pm

/ + IP,m/ Ez,m

/ (60)

Itotal )(A11IE,1

/ - A12IP,1/ )L

A11A22 - A21A12Ez

/ (61)

Qtotal )(A11IP,1

/ - A12QP,1/ )L

A11A22 - A21A12Ez

/ (62)

Qtotal

Itotal)

A11IP,1/ - A12QP,1

/

A11IE,1/ - A12IP,1

/(63)

3868 Langmuir, Vol. 20, No. 10, 2004 Yang et al.

feature at the two ends of the microchannel when Ω ) 0.In the middle of the microchannel, velocity profiles aremore similar to typical electroosmotic flow. The pressuredistribution corresponds well to the velocity profiles, sincethe pressure gradient in the middle of the channel is much

smaller than those at the two ends. This phenomenon hasbeen experimentally observed by Herr et al.16 andnumerically predicted by Ren and Li.13 When Ω ) 10, aslightly larger pressure drop is found where the flowvelocity is smaller than that for Ω ) 0. Toward the end

Figure 2. The velocity profile, |V*|, the pressure drop, P, and the potential distribution, Um - ψm, in a heterogeneous circularmicrochannel for K ) 10 with Ω ) 0 and 10 for pressure-driven flow. R is a normalized radius where R ) 1 and 0 for the wall andcenter line, respectively.

Table 1. Pressure Gradient, -DPm/DZ, Electric Field Strength, Ez,m, Magnitude of the Flow Rate, |Q*|, and Strength of theTotal Streaming Potential, Ez

/, in a Heterogeneous Circular Microchannel for K ) 10 with Ω ) 0 and 10 forPressure-Driven Flow

section 1 section 2 section 3 section 4 section 5

Ω ) 0 ∂P1/∂Z ) 187.82 ∂P2/∂Z ) 190.06 ∂P3/∂Z ) 192.98 ∂P4/∂Z ) 196.55 ∂P5/∂Z ) 200.7Ez,1 ) -0.376 Ez,2 ) -0.522 Ez,3 ) -0.664 Ez,4 ) -0.8 Ez,5 ) -0.93

Ω ) 10 ∂P1/∂Z ) 193.66 ∂P2/∂Z ) 194.81 ∂P3/∂Z ) 196.32 ∂P4/∂Z ) 198.17 ∂P5/∂Z ) 200.33Ez,1 ) -0.153 Ez,2 ) -0.214 Ez,3 ) -0.272 Ez,4 ) -0.329 Ez,5 ) -0.384

section 6 section 7 section 8 |Q*| Ez/

Ω ) 0 ∂P6/∂Z ) 205.37 ∂P7/∂Z ) 210.5 ∂P8/∂Z ) 216.02 72.82 -0.85Ez,6 ) -1.052 Ez,7 ) -1.168 Ez,8 ) -1.275

Ω ) 10 ∂P6/∂Z ) 202.78 ∂P7/∂Z ) 205.49 ∂P8/∂Z ) 208.43 38.5 -0.35Ez,6 ) -0.437 Ez,7 ) -0.487 Ez,8 ) -0.534

Figure 3. The velocity profile, |V*|, the pressure drop, P, and the potential distribution, Um - ψm, in a heterogeneous circularmicrochannel for K ) 10 and 1000 with Ω ) 0 for pressure-driven flow. R is a normalized radius where R ) 1 and 0 for the walland center line, respectively.

Heterogeneous Circular Microchannels Langmuir, Vol. 20, No. 10, 2004 3869

of the channel with a larger zeta potential, the charac-teristic oscillation becomes more noticeable from thevelocity profile. Similar velocity profiles of oscillating flowwithout the EDL effect can be found in ref 30.

A direct comparison of the results for K ) 10 and 1000are shown in Figure 5, and its tabulated results are givenin Table 4. In Table 4, we see that the validity of theassumption of a uniform electric field13-16 depends on thevalues of K. When K ) 1000, the strength of the electricfield changes only slightly; when K ) 10, a larger decreasein Ez is more apparent for electroosmotic flow. When K )10, which corresponds to the case of a diluted solution,the velocity profile in the thicker diffusion layer of the

EDL is parabolic (see Figure 5). Because of a thicker EDL,two inflection points are found in the velocity profile atthe end of the channel with a larger zeta potential. Theelectroosmotic flow for K ) 10 is slower due to the smallerionic concentration. As the flow rate for K ) 10 is smaller,less pressure drop is induced.

VII. Conclusions

In this paper, a model of oscillating electrokinetic flowthrough heterogeneous microchannels is proposed. Ourmodel provides more details of electrokinetic flow due toheterogeneity of microchannels and reasonably predictsa nonuniform electric field. We considered the continuitiesof current and flow rate simultaneously for both pressure-driven flow (streaming potential) and electric-field-driven

(30) Rott, N. Theory of Laminar Flow, Section D; Princeton UniversityPress: Princeton, NJ, 1964.

Table 2. Pressure Gradient, -DPm/DZ, Electric Field Strength, Ez,m, Magnitude of the Flow Rate, |Q*|, and Strength of theTotal Streaming Potential, Ez

/, in a Heterogeneous Circular Microchannel for K ) 10 and 1000 with Ω ) 0 forPressure-Driven Flow

section 1 section 2 section 3 section 4 section 5

K ) 10 ∂P1/∂Z ) 187.82 ∂P2/∂Z ) 190.06 ∂P3/∂Z ) 192.98 ∂P4/∂Z ) 196.55 ∂P5/∂Z ) 200.7Ez,1 ) -0.376 Ez,2 ) -0.522 Ez,3 ) -0.664 Ez,4 ) -0.8 Ez,5 ) -0.93

K ) 1000 ∂P1/∂Z ) 199.998 ∂P2/∂Z ) 199.998 ∂P3/∂Z ) 199.999 ∂P4/∂Z ) 199.998 ∂P5/∂Z ) 200Ez,1 ) -5 × 10-5 Ez,2 ) -7 × 10-5 Ez,3 ) -9 × 10-5 Ez,4 ) -1.1 × 10-4 Ez,5 ) -1.3 × 10-4

section 6 section 7 section 8 |Q*| Ez/

K ) 10 ∂P6/∂Z ) 205.37 ∂P7/∂Z ) 210.5 ∂P8/∂Z ) 216.02 72.82 -0.85Ez,6- ) -1.052 Ez,7 ) -1.168 Ez,8 ) -1.275

K ) 1000 ∂P6/∂Z ) 200 ∂P7/∂Z ) 200.002 ∂P8/∂Z ) 200.003 78.54 -1.2 × 10-4

Ez,6 ) -1.5 × 10-4 Ez,7- ) -1.7 × 10-4 Ez,8 ) -1.9 × 10-4

Figure 4. The velocity profile, |V*|, the pressure drop, P, and the potential distribution, Um - ψm, in a heterogeneous circularmicrochannel for K ) 1000 with Ω ) 0 and 10 for electric-field-driven flow. R is a normalized radius where R ) 1 and 0 for thewall and center line, respectively.

Table 3. Pressure Gradient, -DPm/DZ, Electric Field Strength, EZ,m, and Magnitude of the Flow Rate, |Q*|, in aHeterogeneous Circular Microchannel for K ) 1000 with Ω ) 0 and 10 for Electric-Field-Driven Flow

section 1 section 2 section 3 section 4 section 5

Ω ) 0 ∂P1/∂Z ) 108.64 ∂P2/∂Z ) 77.53 ∂P3/∂Z ) 46.45 ∂P4/∂Z ) 15.4 ∂P5/∂Z ) -15.61Ez,1 ) 10.014 Ez,2 ) 10.011 Ez,3 ) 10.008 Ez,4 ) 10.004 Ez,5 ) 10

Ω ) 10 ∂P1/∂Z ) 117.07 ∂P2/∂Z ) 83.54 ∂P3/∂Z ) 50.05 ∂P4/∂Z ) 16.59 ∂P5/∂Z ) - 16.83Ez,1 ) 10.014 Ez,2 ) 10.011 Ez,3 ) 10.008 Ez,4 ) 10.004 Ez,5 ) 10

section 6 section 7 section 8 |Q*|Ω ) 0 ∂P6/∂Z ) -46.58 ∂P7/∂Z ) -77.49 ∂P8/∂Z ) - 108.34 73.23

Ez,6 ) 9.994 Ez,7 ) 9.988 Ez,8 ) 9.981Ω ) 10 ∂P6/∂Z ) -50.19 ∂P7/∂Z ) -83.5 ∂P8/∂Z ) -116.74 34.28

Ez,6 ) 9.994 Ez,7 ) 9.988 Ez,8 ) 9.8

3870 Langmuir, Vol. 20, No. 10, 2004 Yang et al.

flow (electroosmosis). From our solutions, pressure andelectric field distributions in each layer can be obtained.To maintain current and flow rate continuities, a non-uniform induced electric field (pressure-driven flow) anda nonuniform induced pressure distribution are predictedby our model. We have also shown that the validity of theassumption of a uniform electric field in electroosmosisdepends very much on the values of the nondimensionalreciprocal of the double layer thickness, K.

Acknowledgment. D.Y.K. gratefully acknowledgesfinancial support from the Alberta Ingenuity Establish-ment Fund, the Canada Research Chair (CRC) Program,the Canada Foundation for Innovation (CFI), and theNatural Sciences and Engineering Research Council ofCanada (NSERC). J.Y. acknowledges financial supportfrom a studentship award by the Alberta Ingenuity Fundin the Province of Alberta.

LA035243U

Figure 5. The velocity profile, |V*|, the pressure drop, P, and the potential distribution, Um - ψm, in a heterogeneous circularmicrochannel for K ) 10 and 1000 with Ω ) 0 for electric-field-driven flow. R is a normalized radius where R ) 1 and 0 for thewall and center line, respectively.

Table 4. Pressure Gradient, -DPm/DZ, Electric Field Strength, EZ,m, and Magnitude of the Flow Rate, |Q*|, in aHeterogeneous Circular Microchannel for K ) 1000 and 10 with Ω ) 0 for Electric-Field-Driven Flow

section 1 section 2 section 3 section 4 section 5

K ) 1000 ∂P1/∂Z ) 108.64 P2/∂Z ) 77.53 ∂P3/∂Z ) 46.45 ∂P4/∂Z ) 15.4 ∂P5/∂Z ) - 15.61Ez,1 ) 10.014 Ez,2 ) 10.011 Ez,3 ) 10.008 Ez,4 ) 10.004 Ez,5 ) 10

K ) 10 ∂P1/∂Z ) 79.01 ∂P2/∂Z ) 53.37 ∂P3/∂Z ) 29.1 ∂P4/∂Z ) 6.36 ∂P5/∂Z ) - 14.74Ez,1 ) 10.9 Ez,2 ) 10.69 Ez,3 ) 10.45 Ez,4 ) 10.19 Ez,5 ) 9.91

section 6 section 7 section 8 |Q*|K ) 1000 ∂P6/∂Z ) - 46.58 ∂P7/∂Z ) - 77.49 ∂P8/∂Z ) - 108.34 73.23

Ez,6 ) 9.994 Ez,7 ) 9.988 Ez,8 ) 9.981K ) 10 ∂P6/∂Z ) - 34.09 ∂P7/∂Z ) - 51.64 ∂P8/∂Z ) - 67.37 58.04

Ez,6 ) 9.61 Ez,7 ) 9.3 Ez,8 ) 8.97

Heterogeneous Circular Microchannels Langmuir, Vol. 20, No. 10, 2004 3871