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Streaming Potential and Electroosmotic Flow in Heterogeneous Circular Microchannels with Nonuniform Zeta 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 Engineering and 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 commonly found due to fabrication defects, material impurities, and chemical adsorption from solution. Such surface heterogeneity causes a nonuniform surface potential along the microchannel. Other than surface heterogeneity, one could also pattern the various surface potentials along the microchannels. To understand how such variations affect electrokinetic flow, we proposed a model to describe its behavior in circular microchannels with nonuniform surface potentials. Unlike other models, we considered the continuities of flow rate and electric current simultaneously. These requirements cause a nonuniform electric field distribution 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 influence the 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 the solid-liquid interface and its electrokinetic phenomena have been used to develop various chemical and biological instruments.1-3 A common assumption is the uniformity of surface properties during electrokinetic fluid transport in microchannels.4-9 Nevertheless, surface heterogeneity can easily arise from fabrication defects or chemical adsorption onto microchannels. For example, Norde et al.10 studied the relationship between protein adsorption and streaming potentials. Ajdari11,12 presented a theoretical solution for electroosmotic flow through inhomogeneous charged surfaces. Ren and Li13 numerically studied electroosmotic flow in heterogeneous circular microchannels with axial variation of the surface potential. Anderson and Idol14 studied electroosmosis through pores with nonconformed charged walls. They showed that the mean electroosmotic velocity within the capillary was given by * To whom correspondence should be addressed. Phone: (780) 492-2791. Fax: (780) 492-2200. E-mail:
[email protected]. † 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.
the classical Helmholtz equation with the local surface potential replaced by the average surface potential. Keely et al.15 theoretically provided flow profiles inside capillaries with nonuniform surface potentials. Herr et al.16 theoretically and experimentally investigated electroosmotic flow in cylindrical capillaries with nonuniform surface charge distributions. A nonintrusive caged fluorescence imaging technique was used to image the electroosmotic flow; a parabolic velocity profile induced by the pressure gradient due to the heterogeneity of the capillary surfaces was observed. Cohen and Radke studied streaming potentials of a slit with a nonuniform surface charge density.17 Erickson and Li18 studied microchannel flow with patchwise and periodic surface heterogeneity. However, all the above studies10-18 assumed uniform axial electric fields along the microchannel and did not consider continuity of electric current. Phenomenologically, let us consider two independent microchannels with the same geometry, electrolyte, and flow rate. If these two channels have different surface potentials, the electric fields associated with the electric double layer would have to be different. If we assemble these two independent channels in series as sections of a channel, the electric field should be nonuniform along the flow direction and current continuity should also be satisfied. It is the purpose of this paper to study oscillating electrokinetic (streaming potential and electroosmotic) flow in a microchannel with different surface potentials in every section which satisfy both flow rate and current continuity requirements. (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.
10.1021/la035243u CCC: $27.50 © 2004 American Chemical Society Published on Web 04/09/2004
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of the medium. Combining eqs 1 and 2 yields the following Poisson equation in cylindrical coordinates
(
)
Fm 1 d dψm(r) r )r dr dr
(3)
The conditions imposed on ψm(r) are Figure 1. Schematic of a heterogeneous circular microchannel.
II. Controlling Equations and Boundary Conditions in an Individual Pore We consider a heterogeneous microchannel with N sections shown in Figure 1 where l is the length of each section and the subscript m implies quantities relating to the mth section. We assume each section to be long enough for a fully developed flow and neglect the disturbance across regions between two sections. The effect of entrance length on microchannel flow has been studied.19,20 It is typically found that the entrance length is on the order of microns for microchannel flow and can be neglected. This assumption is consistent with those in refs 15 and 16. In the mth section in Figure 1, we consider a boundary value problem for oscillating electrolyte flow driven by an oscillating pressure gradient and an electric field; a local cylindrical coordinate system (r, θ, z) is used for every section where the z-axis is taken to coincide with the microchannel central axis. All field quantities are taken to depend on the radial coordinate, r, and the time, t. The boundary value problem with the relevant field equations and the boundary conditions is given below. A. Electrical Field. Because each heterogeneous microchannel section has its own physical and chemical properties, the corresponding electric double layer also has its own surface potential and ion distribution. Here, we consider each section to be a different medium. For simplicity, we neglect the differences in the permittivities and 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
um ≡ um(r, z, t) ) ψm(r) + [u0,m - zE′z,m(t)]
(1)
where ψm(r) is the potential due to the double layer at the equilibrium state (i.e., no liquid motion with no applied external field), u0,m is the potential at the beginning of the mth section with z ) 0 [i.e., u0,m ≡ um(r, 0, t)], and E′z,m(t) is the spatially uniform time-dependent electric field strength in the mth layer. The total potential, um, in eq 1 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 flow to be studied here is assumed to be sufficiently slow such that the radial charge distribution is relaxed at its steady state. Further, it is assumed that any induced magnetic fields are sufficiently small and negligible such that the total electric field may still be defined as -∇u bm;22 this definition may then be used to obtain the Poisson equation
Fm ∇ um ) 2
(2)
ψm(a) ) ψs,m and ψm(0) ) a finite number (4) where ψs,m is the surface potential near the microchannel wall in the mth section at r ) a and a is the radius of the microchannel. For brevity, we shall focus on a symmetric, binary electrolyte with univalent charges. The cations and the anions are identified as species 1 and 2, respectively. On the basis of the assumption of thermodynamic equilibrium, the Boltzmann equation provides the local charge density, Fi,m, of the ith species as
[
Fi,m ) zien∞ exp -
(i ) 1, 2)
(5)
where zi is the valence of the ith species, e is the elementary charge, n∞ is the ionic concentration in an equilibrium electrochemical solution at the neutral state where ψm ≈ 0 and can be neglected, k is the Boltzmann constant, and T is the absolute temperature. We employ a Debye-Hu¨ckel approximation for low zeta potentials (zieψm/kT , 1) which provides an acceptable prediction for surface potentials up to 100 mV.23 For simplicity, this approximation is employed to obtain the analytical solution. It should be noted that numerical integration of the Poisson-Boltzmann equation can be found elsewhere.24 From our analytical solutions, we express sinh(z0eψm/kT) ≈ z0eψm/ kT and the total charge density follows from eqs 3 and 5 as
-2n∞e2z20 ψm Fm ) Fi,m ) kT i)1 2
∑
(6)
where we have used z1 ) -z2 ) z0. Finally, the definition of the reciprocal of the double layer thickness for a (z0:z0) electrolyte is given as
κ)
x
2n∞e2z20 kT
(7)
Combining eqs 3 and 6 results in
(
)
1 d dψm(r) r ) κ2ψm r dr dr ψm(a) ) ψs,m and
∂ψm(0) )0 ∂r
(8)
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 NavierStokes equation becomes
where Fm is the free charge density and is the permittivity (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: New York, 1975.
]
zieψm kT
-
( )
1 1 ∂vm 1 ∂pm 1 ∂ ∂vm + r + FmE′z,m ) µ ∂z r ∂r ∂r µ ν ∂t
(9)
where we have taken the pressure gradient [∂pm/∂z ≡ ∂pm/ (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.
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∂z(t)] to be position-independent, µ is the viscosity, and ν is the kinematic viscosity of the liquid. The boundary conditions for the velocity field are
∂vm(0, t) )0 vm(a, t) ) 0 and ∂r
) -2π (10)
The electric current density along the microchannel may be integrated over the channel cross-section to give the electric current
∫0aFmvmr dr +
im ) 2π
2πz0eD E′z,m kT
∫0a(F1,m - F2,m)r dr
2z20e2n∞D (πa2)E′z,m im ) 2π 0 Fmvmr dr + kT
∫
a
∫0avmr dr
(13)
III. Normalized Equations Here, we provide the normalized governing equations and boundary conditions. Using a yet unknown characteristic velocity, 〈v〉, the following normalized quantities are defined as
Vm )
z0e z0e 1 ψm Ψs,m ) ψ R ) r Ψm ) a kT kT s,m z0e z0ea u E′ (14) Um ) Ez,m ) kT m kT z,m
1 v 〈v〉
where Vm, R, Ψm, Ψs,m, Um, and Ez,m are the normalized velocity, the radial coordinate, the surface potential, the total potential, and the strength of the electric field of the mth section, respectively. The normalized counterparts of eqs 4 and 8 become, respectively,
(
)
dΨm 1 d R ) K2Ψm and Ψm(1) ) Ψs,m (15) R dR dR and
Ψm(0) ) a finite number
( )
∂Vm ∂Vm ∂Pm 1 ∂ + R - K2ΨmEz,m ) ∂Z R ∂R ∂R ∂τ Vm(1, τ) ) 0
∂Vm(0, τ) )0 ∂R
(16) (17)
(19)
where Pm, Im, and Qm are the normalized pressure, the current, and the flow rate of the mth section, respectively. In deriving eqs 15-19, the following normalized quantities have been identified
Fjm )
a2ez0 F kT m
K ) κa
Pm )
Im )
a p µ〈v〉 m
ez0
im
kT〈v〉
ν t a2 z20e2µD
τ) Σ)
k2T2
(20)
where Fjm, K, τ, and Σ are the normalized charge density, the reciprocal of the double layer thickness, the time, and the conductivity, respectively. The expression for the characteristic velocity can also be identified as
〈v〉 )
k2T2 µae2z20
(21)
We wish to point out that our normalizing schemes in eqs 20 and 21 are different from those of other authors,25-27 as we did not employ the ionic concentration at equilibrium, n∞, to normalize the charge density, Fm. The choice of this selection is important, as we intend to study the effects of electrolyte concentration indirectly through K on the flow properties; otherwise, the results would have been misleading, since it makes no sense to study a quantity that has already been used for normalization. Finally, we define the following normalized quantities
Ω)
qm a2 ω and Qm ) ν 〈v〉a2
(22)
where the parameters ω and qm are the frequency of the external oscillating field and the volumetric flow rate, respectively, and Ω is the normalized frequency. IV. Analytical Solution of the mth Section An analytical solution is sought here for a sinusoidal periodicity in the electrohydrodynamic fields, and this is best addressed by using complex variables. Thus, the general field quantity, X, may be defined as the real part of the complex function (X*ejΩτ) where X* is complex (j ) x-1), Ω is the normalized oscillation frequency oscillation, and t is the time. The general field quantity X is written as
X ) Re[X*ejΩτ]
whereas eqs 9, 10, 12, and 13 become, respectively,
(18)
∫01VmR dR
(12)
and the flow rate, qm, can be written as
qm ) 2π
∫01K2ΨmVmR dR + πΣK2Ez,m Qm ) 2π
(11) where D is the ionic diffusion coefficient. The first term on the right side of eq 11 is due to bulk convection, and the second term is due to charge migration.21 Because of the assumption of an infinitely extended microchannel, the contribution to the current due to concentration gradients vanishes. Using eq 5 for a (z0:z0) electrolyte, we have F1,m - F2,m ) 2ez0n∞ cosh(z0eψm/kT). The DebyeHu¨ckel approximation implies that cosh(z0eψm/kT) ≈ 1 and F1,m - F2,m ) 2z0en∞. With this simplification, eq 11 becomes
2a2z40e4µDn∞ Ez,m 2k3T3
∫01K2ΨmVmR dR + π
Im ) -2π
(23)
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.
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φ ) tan-1
Yang et al.
Im(X*) Re(X*)
(24)
The relevant quantities of an individual mth section in eqs 28-30 are listed below
where Im(X*) and Re(X*) are the imaginary and real parts of X*, respectively. An alternative representation of eq 23 is given as
X ) Re[|X*|ej(Ωτ+φ)]
(25)
/ VE,m (R, Ω) )
where
|X| ) |X*| and |X*| ) xIm2(X*) + Re2(X*)
(26)
∂Pm / ) Re[P/mejΩτ] and Ez,m ) Re[Ez,m ejΩτ] (27) ∂Z
We consider the class of solutions where the amplitude of the pressure gradient and the electric field could be / ≡ frequency-dependent, that is, P/m ≡ P/m(Ω) and Ez,m / Ez,m(Ω). The solution for Ψm will then follow from eq 8 and that for Vm from eq 9. Thus,
K - jΩ J0(jK)
{
≡
[
/ (Ω) ) QP,m
J0(x-jΩ)
/ / / VP,m (R, Ω)P/m(Ω) + VE,m (R, Ω)Ez,m (Ω) (28) / / (R, Ω) and VE,m (R, Ω) will be The expressions for VP,m given at the end of this section. The electric current will follow from eq 11 and may be written as
Im ) Re[I/mejΩτ] where / / / (Ω)P/m(Ω) + IE,m (Ω)Ez,m (Ω) I/m ≡ I/m(Ω) ) IP,m
[ {[
]
]}
[
+ πΣK2 (33)
1 J1(x-jΩ) 2π 1 jΩ 2 x-jΩ J (x-jΩ) 0
[
/ VE,m (R, 0) ) -Ψs,m 1 -
[
Re[Q/mejΩτ]
/ (0) ) -πΨs,m 1 IP,m
where
During pressure-driven flow, the amplitude of the electric / (Ω), is found by field strength in the mth section, Ez,m setting I* ) 0 in eq 29. Thus,
)-
/ IP,m (Ω)
P/m(Ω)
/ IE,m (Ω)
for
I/m
)0
(31)
Equation 31 may be substituted into eqs 28 and 30 to determine the normalized liquid velocity and the volumetric flow rate, respectively. Alternatively, the velocity, the current, and the volumetric flow rate during electroosmotic flow follow from eqs 28-30 by setting P/m(Ω) ) 0.
]}
J1(jK) J1(x-jΩ) jK - (x-jΩ) J0(jK) J0(x-jΩ)
] (34)
1 / (R, 0) ) (1 - R2) VP,m 4
(29)
/ / / (Ω)P/m(Ω) + QE,m (Ω)Ez,m (Ω) (30) Q/m ≡ Q/m(Ω) ) QP,m
(32)
where J0 and J1 are the zeroth- and first-order Bessel functions of the first kind. Note that the analysis without the EDL effects follows from eqs 32-34 by setting Ψs,m ) 0. The resulting expressions are identical to those obtained / ) IP,m/ satisfies by Uchida.28 We can also find that QE,m 4 Onsager’s theorem. When Ω f 0, eqs 32-34 reduce to those of the steady state. The steady state response follows from eqs 32-34 by setting Ω ) 0, and such solutions were originally obtained by Rice and Whitehead4 as
The volumetric flow rate, Qm, is defined as Qm ) 2π∫a0RVm dR and can be expressed as
Qm )
]
1 1 J1(jK) × ΩK J0(jK) Ω2 + jΩK2
/ / QE,m (Ω) ) IP,m (Ω)
Ω) )
/ (Ω) Ez,m
J0(Rx-jΩ)
J1(jK) J1(x-jΩ) jK - (x-jΩ) J0(jK) J0(x-jΩ)
where
V/m(R,
-
2 2πK4Ψs,m J21(jK) 1 1 )- 2 1+ 2 × 2 K - jΩ J0(jK) jΩ - K2
Vm ) Re[V/mejΩτ]
V/m
[
2
/ IP,m (Ω) ) 2πK2Ψs,m
/ IE,m (Ω)
]
K2Ψs,m J0(jKR)
With the notation of eq 23, we shall seek the solution of the boundary value problem for the following specific dependencies.
-
[
J0(Rx-jΩ) 1 1jΩ J0(x-jΩ)
/ VP,m (R, Ω) )
/ IE,m (0) )
[
2 K2 1 -πΨs,m
] ]
J0(jKR) J0(jK)
(35)
2 J1(jK) jK J0(jK)
]
2 2 J1(jK) J1(jK) + πΣK2 (36) + 2 jK J0(jK) J (jK) 0
/ QP,m (0) )
π 8
/ (0) ) I/P(0) QE,m
(37)
V. Pressure and Electric Field Distribution Because of the surface heterogeneity of the microchannel, each section has its own normalized parameters: the (28) Uchida, S. ZAMP 1950, 7, 403.
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Langmuir, Vol. 20, No. 10, 2004 3867
pressure gradient, P/m, 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 P/m are uniform field quantities; for the whole microchannel, the total current, Itotal, and the total flow rate, Qtotal, are the same for any section due to the continuities of current and flow rate. The total pressure and the potential drops of the microchannel are the sum of those in each section and can be expressed as
[
the potential drops in each section.
E/z )
N
P*
/ / QE,m (Ω)Ez,m (Ω))ejΩτ] (38) / / ejΩτ] ) Re[(IP,m (Ω)P/m(Ω) + Itotal ) Re[Itotal
(39)
N
∆P ) Re[P*LejΩτ] ) Re[
P/mLmejΩτ] ∑ m)1
(40)
N
∆U ) Re[E/z Lejωt] ) Re[
/ Ez,m LmejΩτ] ∑ m)1
(41)
N Lm is the total normalized length and U where L ) ∑m)1 is the normalized total potential. On the basis of eqs 3841, one can determine the relationships of the pressure gradient and the strength of the electric field in all sections. We will express all quantities as functions of the corresponding 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, eq 31 becomes
/ )Ez,m
/ IP,m
P/m / IE,m
(m ) 1, ... ,N)
(42)
and the continuity of current is satisfied automatically. From eq 38, we have
/ QP,1 -
P/m ) / QP,m -
/ QP,m
-
1
(IP,m)
2
IE,m
/ QP,1
N
∑
-
Lm
m)1 / QP,m -
(IP,1)2 IE,1
]
(IP,m)2 IE,m
(46)
/ / / / / / IP,m (Ω)P/m(Ω) + IE,m Ez,m ) IP,1 (Ω)P/1(Ω) + IE,1 Ez,1 (48) / ) IP,m, and P/m and E/m can be solved from Note that QE,m eqs 47 and 48
P/m
/ / / / / / / / IP,m IP,m IP,1 - IE,m QP,1 IE,1 - IE,m IP,1 / / ) / 2 P1 + / 2 Ez,1 / / / / (IP,m) - IE,mQP,m (IP,m) - IE,mQP,m (49)
/ ) Ez,m
/ / / / / / / / IP,m IP,m QP,1 - QP,m IP,1 IP,1 - QP,m IE,1 / / P + Ez,1 1 / / / / / / (IP,m )2 - IE,m QP,m (IP,m )2 - IE,m QP,m (50)
Following eqs 40 and 41, one can obtain a set of equations.
P/1
N
/ / / / IP,m IP,1 - IE,m QP,1
m)1
/ / / (IP,m )2 - IE,m QP,m
∑ Lm
(44)
/ Ez,1
+
N
/ / / / IP,m IE,1 - IE,m IP,1
m)1
/ / / (IP,m )2 - IE,m QP,m
∑ Lm
(51)
/ / / / IP,m QP,1 - QP,m IP,1 Lm + / / / m)1 (IP,m )2 - IE,m QP,m N
From eq 40, we have
E/z L ) P/1 N
P*L ) P/1
IE,1
/ / / / / / (Ω)P/m(Ω) + QE,m Ez,m ) QP,1 (Ω)P/1(Ω) + QE,1 Ez,1 QP,m (47)
P*L )
/ 2 ) (IP,1 / IE,1 / P1 / )2 (IP,m / IE,m
IE,m
(IP,1)2
B. Electroosmosis (Electric-Field-Driven Flow). In the case of electroosmosis, the known inputs are the normalized total strength of the electric field, E/z , and the normalized pressure gradient, P*. Normally, both sides of an electroosmotic system are opened to the atmosphere to eliminate the effect of hydrostatic pressure on flow pressure such that P* ) 0. The continuities of current and flow rate can be expressed as
/ / / / / / (Ω)P/m(Ω) + QE,m Ez,m ) QP,1 (Ω)P/1(Ω) + QE,1 Ez,1 QP,m (43)
Substituting eq 42 into eq 43, we can relate the pressure gradient in the mth section to that of the first section.
-
∑ -Lm / m)1
/ / Qtotal ) Re[Qtotal ejΩτ] ) Re[(QP,m (Ω)P/m(Ω) +
/ / IE,m (Ω)Ez,m (Ω))ejΩτ]
/ IP,m
/ QP,1
/ QP,1 -
∑ Lm m)1 / QP,m -
(IP,1)2 IE,1 (IP,m)
2
(45)
IE,m
According to the known pressure gradient, one can solve for the pressure gradient in every section. The total streaming potential (potential drop) is then the sum of
∑
/ Ez,1
N
/ / / / IP,m IP,1 - QP,m IE,1
m)1
/ / / (IP,m )2 - IE,m QP,m
∑ Lm
(52)
For brevity, we write eqs 51 and 52, respectively, as / P*L ) A11P/1 + A12Ez,1
(53)
/ E/z L ) A21P/1 + A22Ez,1
(54)
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Yang et al.
Solving the above two equations results in
P/1 )
A22L A12L P* E/ A11A22 - A21A12 A11A22 - A21A12 z
/ )Ez,1
(55)
A21L A11L P* + E/ (56) A11A22 - A21A12 A11A22 - A21A12 z
/ and P/m could be determined from Thus, Ez,m
P/m )
/ / / / IP,m IP,1 - IE,m QP,1
× / / / (IP,m )2 - IE,m QP,m A12L A22L P* E/ + A11A22 - A21A12 A11A22 - A21A12 z
(
/ / / / IE,1 - IE,1 IP,1 IP,m / (IP,m )2
-
/ / IE,m QP,m
)
(
-
A21L P* + A11A22 - A21A12
)
A11L E/ (57) A11A22 - A21A12 z / ) Ez,m
/ / / / IP,m QP,1 - QP,m IP,1
× / / / (IP,m )2 - IE,m QP,m A12L A22L P* E/ + A11A22 - A21A12 A11A22 - A21A12 z
(
(
/ / / / IP,1 - QP,m IE,1 IP,m / / / (IP,m )2 - IE,m QP,m
-
)
A21L P* + A11A22 - A21A12
)
A11L E/ (58) A11A22 - A21A12 z If both sides of an electroosmotic system are opened to the atmosphere, implying that P* ) 0, one can infer from eq 57 that P/m * 0. Because of the heterogeneity of the microchannel, pressure gradients are induced as a result of current and flow rate continuities. The expressions for the experimentally measurable total current, Itotal, and total flow rate, Qtotal, are given by / / / P/m + IE,m Ez,m Itotal ) IP,m
(59)
/ / / P/m + IP,m Ez,m Qtotal ) QP,m
(60)
If we focus on a common electroosmotic experimental system with P* ) 0,
Itotal )
/ / (A11IE,1 - A12IP,1 )L / E A11A22 - A21A12 z
(61)
/ / - A12QP,1 )L / (A11IP,1 Qtotal ) Ez A11A22 - A21A12
(62)
/ / - A12QP,1 Qtotal A11IP,1 ) Itotal A I/ - A I /
(63)
11 E,1
12 P,1
Equation 63 is a complex function of frequency Ω, which includes information on the surface potential and the length of each section. VI. Parametric Study and Discussion We 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 field distributions. We have selected the following values for our calculations: the amplitude of the normalized total pressure gradient P*(Ω) ) 400, the amplitude of the strength of the total electric field E/z (Ω) ) 10, the normalized frequency Ω ) 10, and the normalized conductivity Σ ) 3.85 which represents the diffusion coefficient D ≈ 2 × 10-9 m2/s for a KCl electrolyte.29 We also consider a heterogeneous microchannel consisting of eight 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 dimensional surface potentials of 25, 35, 45, 55, 65, 75, 85, and 95 mV. It is noted that the Debye-Hu¨ckel approximation still provides a good agreement with experiments for the selected zeta potentials.23 A. Pressure and Electric Field Distribution for Pressure-Driven Flow. For pressure-driven flow through a heterogeneous microchannel, we set P ) 0 and Um - Ψm ) 0 as reference values at the beginning of the microchannel. 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 Figure 2. We see that the velocity profile remains parabolic along the channel for Ω ) 0. When Ω ) 10 (dashed lines), the magnitude of velocity is smaller, since liquid viscosity causes the flow to lag behind the variation of pressure. It is noted that the pressure gradient in Figure 2 is indeed nonlinear along the channel, as seen in Table 1 from the tabulated results. We see that, as the surface potential increases, the pressure gradient increases along the flow direction. This is due to the fact that a larger pressure gradient is required to compensate for the decrease in flow rate due to the stronger electroviscous effect. When Ω ) 10, the pressure distribution is nearly identical to that of Ω ) 0. The potential distribution along the axis of the microchannel is also nonlinear. As the zeta potential increases, the strength of the electric field also increases. The reason for the stronger electric field is to make up a net zero current. The strength of the electric field is smaller when Ω ) 10. These differences are shown in Table 1 and Figure 2. In Figure 2, K ) 10 implies a diluted solution for a fixed microchannel size. We also plotted the corresponding results in Figure 3 for a more concentrated solution for K ) 1000 with Ω ) 0. When K ) 1000 (a thinner EDL), the pressure gradient varies slightly (see Table 2) as compared to that for K ) 10, since the electroviscous effect is smaller for a more concentrated solution. However, the strength of the electric field is several orders smaller than that of K ) 10, as a smaller electric field is sufficient to maintain a net zero current. B. Pressure and Electric Field of Electric-FieldDriven Flow. For electric-field-driven flow through a heterogeneous microchannel, a nonuniform pressure distribution will result from the different surface potentials and electric fields in each section to maintain the flow rate and current continuities. In actual experiments, the electroosmotic system is often opened to the atmosphere so that the total pressure difference between the two ends of the microchannel is zero. The results for the magnitude of the velocity, |V*|, the pressure drop, P, and the axial potential, Um - Ψm, along the channel for Ω ) 0 and 10 are shown in Figure 4 for K ) 1000; its tabulated results are given in Table 3. In Figure 4, we find that the velocity profile has a parabolic (29) Newman, J. S. Electrochemical Systems, 2nd ed.; Prentice Hall: Englewood Cliffs, NJ, 1991.
Heterogeneous Circular Microchannels
Langmuir, Vol. 20, No. 10, 2004 3869
Figure 2. The velocity profile, |V*|, the pressure drop, P, and the potential distribution, Um - ψm, in a heterogeneous circular microchannel 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 and center line, respectively. Table 1. Pressure Gradient, -DPm/DZ, Electric Field Strength, Ez,m, Magnitude of the Flow Rate, |Q*|, and Strength of the Total Streaming Potential, E/z, in a Heterogeneous Circular Microchannel for K ) 10 with Ω ) 0 and 10 for Pressure-Driven Flow Ω)0 Ω ) 10
Ω)0 Ω ) 10
section 1
section 2
section 3
section 4
section 5
∂P1/∂Z ) 187.82 Ez,1 ) -0.376 ∂P1/∂Z ) 193.66 Ez,1 ) -0.153
∂P2/∂Z ) 190.06 Ez,2 ) -0.522 ∂P2/∂Z ) 194.81 Ez,2 ) -0.214
∂P3/∂Z ) 192.98 Ez,3 ) -0.664 ∂P3/∂Z ) 196.32 Ez,3 ) -0.272
∂P4/∂Z ) 196.55 Ez,4 ) -0.8 ∂P4/∂Z ) 198.17 Ez,4 ) -0.329
∂P5/∂Z ) 200.7 Ez,5 ) -0.93 ∂P5/∂Z ) 200.33 Ez,5 ) -0.384
section 6
section 7
section 8
|Q*|
E/z
∂P6/∂Z ) 205.37 Ez,6 ) -1.052 ∂P6/∂Z ) 202.78 Ez,6 ) -0.437
∂P7/∂Z ) 210.5 Ez,7 ) -1.168 ∂P7/∂Z ) 205.49 Ez,7 ) -0.487
∂P8/∂Z ) 216.02 Ez,8 ) -1.275 ∂P8/∂Z ) 208.43 Ez,8 ) -0.534
72.82
-0.85
38.5
-0.35
Figure 3. The velocity profile, |V*|, the pressure drop, P, and the potential distribution, Um - ψm, in a heterogeneous circular microchannel for K ) 10 and 1000 with Ω ) 0 for pressure-driven flow. R is a normalized radius where R ) 1 and 0 for the wall and center line, respectively.
feature at the two ends of the microchannel when Ω ) 0. In the middle of the microchannel, velocity profiles are more similar to typical electroosmotic flow. The pressure distribution corresponds well to the velocity profiles, since the pressure gradient in the middle of the channel is much
smaller than those at the two ends. This phenomenon has been experimentally observed by Herr et al.16 and numerically predicted by Ren and Li.13 When Ω ) 10, a slightly larger pressure drop is found where the flow velocity is smaller than that for Ω ) 0. Toward the end
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Yang et al.
Table 2. Pressure Gradient, -DPm/DZ, Electric Field Strength, Ez,m, Magnitude of the Flow Rate, |Q*|, and Strength of the Total Streaming Potential, E/z, in a Heterogeneous Circular Microchannel for K ) 10 and 1000 with Ω ) 0 for Pressure-Driven Flow K ) 10 K ) 1000
K ) 10 K ) 1000
section 1
section 2
section 3
section 4
section 5
∂P1/∂Z ) 187.82 Ez,1 ) -0.376 ∂P1/∂Z ) 199.998 Ez,1 ) -5 × 10-5
∂P2/∂Z ) 190.06 Ez,2 ) -0.522 ∂P2/∂Z ) 199.998 Ez,2 ) -7 × 10-5
∂P3/∂Z ) 192.98 Ez,3 ) -0.664 ∂P3/∂Z ) 199.999 Ez,3 ) -9 × 10-5
∂P4/∂Z ) 196.55 Ez,4 ) -0.8 ∂P4/∂Z ) 199.998 Ez,4 ) -1.1 × 10-4
∂P5/∂Z ) 200.7 Ez,5 ) -0.93 ∂P5/∂Z ) 200 Ez,5 ) -1.3 × 10-4 E/z
section 6
section 7
section 8
|Q*|
∂P6/∂Z ) 205.37 Ez,6- ) -1.052 ∂P6/∂Z ) 200 Ez,6 ) -1.5 × 10-4
∂P7/∂Z ) 210.5 Ez,7 ) -1.168 ∂P7/∂Z ) 200.002 Ez,7- ) -1.7 × 10-4
∂P8/∂Z ) 216.02 Ez,8 ) -1.275 ∂P8/∂Z ) 200.003 Ez,8 ) -1.9 × 10-4
72.82
-0.85
78.54
-1.2 × 10-4
Figure 4. The velocity profile, |V*|, the pressure drop, P, and the potential distribution, Um - ψm, in a heterogeneous circular microchannel for K ) 1000 with Ω ) 0 and 10 for electric-field-driven flow. R is a normalized radius where R ) 1 and 0 for the wall and center line, respectively. Table 3. Pressure Gradient, -DPm/DZ, Electric Field Strength, EZ,m, and Magnitude of the Flow Rate, |Q*|, in a Heterogeneous Circular Microchannel for K ) 1000 with Ω ) 0 and 10 for Electric-Field-Driven Flow Ω)0 Ω ) 10
Ω)0 Ω ) 10
section 1
section 2
section 3
section 4
section 5
∂P1/∂Z ) 108.64 Ez,1 ) 10.014 ∂P1/∂Z ) 117.07 Ez,1 ) 10.014
∂P2/∂Z ) 77.53 Ez,2 ) 10.011 ∂P2/∂Z ) 83.54 Ez,2 ) 10.011
∂P3/∂Z ) 46.45 Ez,3 ) 10.008 ∂P3/∂Z ) 50.05 Ez,3 ) 10.008
∂P4/∂Z ) 15.4 Ez,4 ) 10.004 ∂P4/∂Z ) 16.59 Ez,4 ) 10.004
∂P5/∂Z ) -15.61 Ez,5 ) 10 ∂P5/∂Z ) - 16.83 Ez,5 ) 10
section 6
section 7
section 8
|Q*|
∂P6/∂Z ) -46.58 Ez,6 ) 9.994 ∂P6/∂Z ) -50.19 Ez,6 ) 9.994
∂P7/∂Z ) -77.49 Ez,7 ) 9.988 ∂P7/∂Z ) -83.5 Ez,7 ) 9.988
∂P8/∂Z ) - 108.34 Ez,8 ) 9.981 ∂P8/∂Z ) -116.74 Ez,8 ) 9.8
73.23
of the channel with a larger zeta potential, the characteristic oscillation becomes more noticeable from the velocity profile. Similar velocity profiles of oscillating flow without the EDL effect can be found in ref 30. A direct comparison of the results for K ) 10 and 1000 are shown in Figure 5, and its tabulated results are given in Table 4. In Table 4, we see that the validity of the assumption of a uniform electric field13-16 depends on the values of K. When K ) 1000, the strength of the electric field changes only slightly; when K ) 10, a larger decrease in 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 (30) Rott, N. Theory of Laminar Flow, Section D; Princeton University Press: Princeton, NJ, 1964.
34.28
EDL is parabolic (see Figure 5). Because of a thicker EDL, two inflection points are found in the velocity profile at the end of the channel with a larger zeta potential. The electroosmotic flow for K ) 10 is slower due to the smaller ionic 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 flow through heterogeneous microchannels is proposed. Our model provides more details of electrokinetic flow due to heterogeneity of microchannels and reasonably predicts a nonuniform electric field. We considered the continuities of current and flow rate simultaneously for both pressuredriven flow (streaming potential) and electric-field-driven
Heterogeneous Circular Microchannels
Langmuir, Vol. 20, No. 10, 2004 3871
Figure 5. The velocity profile, |V*|, the pressure drop, P, and the potential distribution, Um - ψm, in a heterogeneous circular microchannel for K ) 10 and 1000 with Ω ) 0 for electric-field-driven flow. R is a normalized radius where R ) 1 and 0 for the wall and center line, respectively. Table 4. Pressure Gradient, -DPm/DZ, Electric Field Strength, EZ,m, and Magnitude of the Flow Rate, |Q*|, in a Heterogeneous Circular Microchannel for K ) 1000 and 10 with Ω ) 0 for Electric-Field-Driven Flow K ) 1000 K ) 10
K ) 1000 K ) 10
section 1
section 2
section 3
section 4
section 5
∂P1/∂Z ) 108.64 Ez,1 ) 10.014 ∂P1/∂Z ) 79.01 Ez,1 ) 10.9
P2/∂Z ) 77.53 Ez,2 ) 10.011 ∂P2/∂Z ) 53.37 Ez,2 ) 10.69
∂P3/∂Z ) 46.45 Ez,3 ) 10.008 ∂P3/∂Z ) 29.1 Ez,3 ) 10.45
∂P4/∂Z ) 15.4 Ez,4 ) 10.004 ∂P4/∂Z ) 6.36 Ez,4 ) 10.19
∂P5/∂Z ) - 15.61 Ez,5 ) 10 ∂P5/∂Z ) - 14.74 Ez,5 ) 9.91
section 6
section 7
section 8
|Q*|
∂P6/∂Z ) - 46.58 Ez,6 ) 9.994 ∂P6/∂Z ) - 34.09 Ez,6 ) 9.61
∂P7/∂Z ) - 77.49 Ez,7 ) 9.988 ∂P7/∂Z ) - 51.64 Ez,7 ) 9.3
∂P8/∂Z ) - 108.34 Ez,8 ) 9.981 ∂P8/∂Z ) - 67.37 Ez,8 ) 8.97
73.23
flow (electroosmosis). From our solutions, pressure and electric field distributions in each layer can be obtained. To maintain current and flow rate continuities, a nonuniform induced electric field (pressure-driven flow) and a nonuniform induced pressure distribution are predicted by our model. We have also shown that the validity of the assumption of a uniform electric field in electroosmosis depends very much on the values of the nondimensional reciprocal of the double layer thickness, K.
58.04
Acknowledgment. D.Y.K. gratefully acknowledges financial support from the Alberta Ingenuity Establishment Fund, the Canada Research Chair (CRC) Program, the Canada Foundation for Innovation (CFI), and the Natural Sciences and Engineering Research Council of Canada (NSERC). J.Y. acknowledges financial support from a studentship award by the Alberta Ingenuity Fund in the Province of Alberta. LA035243U
