Langmuir 2003, 19, 1047-1053
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Microfluid Flow in Circular Microchannel with Electrokinetic Effect and Navier’s Slip Condition Jun Yang and Daniel Y. Kwok* Nanoscale Technology and Engineering Laboratory, Department of Mechanical Engineering, University of Alberta, Edmonton, Alberta T6G 2G8, Canada Received July 9, 2002. In Final Form: November 19, 2002 Electrokinetic transport phenomena and slippage of fluid at the surface are often considered independently for description of microchannel flow. Anomalous cases of experimental results disagreeing with theoretical prediction have frequently been reported. By combining the electrokinetic effect and Navier’s slip condition, we illustrate a possible explanation and present an analytical solution for oscillating flow in a circular microchannel. The derived analytical solution is useful for more general time-dependent problems through superposition of time-harmonic solutions weighted by appropriate Fourier coefficients. Our parametric results suggest that these two surface phenomena can be important for a better understanding of microfluid flow with hydrophobic channel walls.
I. Introduction Microelectromechanical system (MEMS) devices with microchannels are widely used in many applications, such as the development of micropumps,1 heat sinks,2 heat exchangers,3 lab-on-a-chip diagnostic devices,4 and intravenous drug delivery systems.5,6 However, many experiments7,8 have shown that the behaviors of fluid in macroflow and microflow are different and are strongly linked to the proportionally large ratio of solid-liquid interfacial area to fluid volume in microchannels. Thus, conventional understanding of fluids cannot be used naively to describe flow in microchannels. In the literature, two major schools of thought for microchannel flow are due to electrokinetic phenomena and slip boundary conditions. As for electrokinetic phenomena in general, electrically neutral liquids have a distribution of electrical charges near a surface because of a charged solid surface9 (Figure 1). This region is known as the electrical double layer (EDL), which induces electrokinetic phenomena. The effect of an EDL during an externally applied pressure gradient is to retard liquid flow, resulting in a streaming potential, whereas, in the absence of an external pressure gradient, an EDL induces fluid flow when an external electric field is applied (electroosmotic pumping). An EDL is primarily a surface phenomenon; its effects tend to appear when * Electronic address:
[email protected]. (1) Song, Y. J.; Zhao, T. S. Modelling and test of a thermally driven phase-change nonmechanical micropump. J. Micromech. Microeng. 2001, 11, 713-719. (2) Pfrahler, J. N.; Bar-Cohen, A.; Kraus, A. D. Advances in Thermal Modeling of Electronic Components and Systems; ASME Press: New York, 1990; Vol. II, Chapter 3. (3) Hunt, C. E.; Desmond, C. A.; Ciarlo, D. R.; Benett, W. J. Direct bonding of micromachined silicon wafers for laser diode heat exchanger applications. J. Micromech. Microeng. 1991, 1, 152-156. (4) Harrison, J. D.; Fluri, K.; Seiler, K.; Fan, Z. H.; Effenhauser, C. S.; Manz, A. Science 1993, 261, 895-897. (5) Blackshear, P. J. Sci. Am. 1979, 241, 52-59. (6) Penn, R. D.; Paice, J. A.; Gottschalk, W.; Ivankovich, A. D. J. Neurosurg. 1984, 61, 302-306. (7) Guncan, A. B.; Peterson, G. P. Appl. Mech. Rev. 1994, 47, 397428. (8) Peiyi, W.; Little, W. A. Measurement of friction factors for the flow of gase in very fine channels used for microminiatures JouleThomson refrigerators. Cryogenics 1983, 23, 273-277. (9) Lyklema, J. Fundamentals of Interface and Colloid Science; Academic Press: 1995; Vol. II.
Figure 1. Schematic of an electrical double layer at the channel.
the typical dimensions of the channel are of the same order as the EDL thickness.10 One of the earliest systematic studies was by Rice and Whitehead,11 who studied steady-state liquid flow induced by an electric field in thin circular capillaries. Levine et al.12 studied the electrokinetic steady flow in a narrow parallel-plate microchannel. More recent works are by Mala et al.13,14 on parallelplate microchannels and by Yang et al.15 on rectangular microchannels, where they invoked the classical DebyeHu¨ckel approximation and allowed an analytical solution. Levine et al.16 carried out a semianalytical extension of Rice and Whitehead’s work for high surface potential, (10) Bhattacharyya, A.; Masliyah, J. H.; Yang, J. Oscillating Laminar Electrokinetic Flow in Infinitely Extended Circular Microchannels. J. Colloid Interface Sci., accepted. (11) Rice, C. L.; Whitehead, R. J. Phys. Chem. 1965, 69, 4017-4024. (12) Levine, S.; Marriott, J. R.; Robinson, K. J. Chem. Soc., Faraday Trans. 2 1975, 71, 1-11. (13) Mala, G. M.; Li, D.; Dale, J. D. Int. J. Heat Mass Transfer 1997, 40, 3079-3088. (14) Mala, G. M.; Li, D.; Werner, C.; Jacobasch, H.-J.; Ning, Y. B. Int. J. Heat Liquid Flow 1997, 18, 489-496. (15) Yang, C.; Li, D.; Masliyah, J. H. Int. J. Heat Mass Transfer 1998, 41, 4229-4249. (16) Levine, S.; Marriott, J. R.; Neale, G.; Epstein, N. J. Colloid Interface Sci. 1975, 52, 136-149.
10.1021/la026201t CCC: $25.00 © 2003 American Chemical Society Published on Web 01/14/2003
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whereas Masliyah17 numerically addressed the problem of microchannels in varying cross sections. Bhattacharyya et al.10 derived an analytical solution of oscillating laminar electrokinetic flow in a circular microchannel. With respect to slip boundary conditions, many theoretical and experimental works have contributed to this topic. Watanabe et al.18 observed water slippage in a 16 mm diameter acrylic-resin-coated pipe and obtained a 14% drag reduction. Tretheway and Meinhart measured the velocity profile of water flowing through a 30 × 300 µm2 octadecyltrichlorosilane (OTS)-coated microchannel and observed a 1 µm slip length.19 Zhu and Granick20 measured the hydrodynamic force of water against a methylterminated self-assembled monolayer (SAM) on mica and found that when the flow rate exceeds a critical level, partial slip occurs. Their results could only be explained by consideration of liquid slip. Thompson and Troian21 employed molecular dynamics simulations to probe the fluid behavior at an interface. Anomalous cases of experimental results disagreeing with theoretical prediction have frequently been reported.22,23 By now, the widely accepted slip boundary condition is that from Navier,24,25 which relates the velocity at a solid surface to be proportional to shear stress. However, none of these studies have considered the effects of surface potentials on fluid flow. Clearly, they all suggested that liquid slip occurs for water flow over hydrophobic surfaces on a small scale. The question remains if hydrophobic surfaces have significant ζ potentials with electrolyte solutions. Schweiss et al.16 found that even methyl-terminated SAMs (hydrophobic surfaces) have similar ζ potentials to those of the COOH-terminated SAMs (hydrophilic surfaces) when a 3 mM KCl solution was used, while the water contact angle for the former is 112 ° and that of the latter is 13 °. Therefore, hydrophobic surfaces that induce liquid slip can have significant surface potentials. Consideration of ζ potential and liquid slip for liquid flow in a microchannel is the aim of the present study. We report here analytical solutions for fully developed laminar electrokinetic flow of liquids subjected to a sinusoidal pressure gradient or a sinusoidal external electric field with Navier’s slip condition in an infinitely extended circular microchannel. To the best of our knowledge, this work is the first in depth analysis of microfluid in microchannels combining electrokinetic phenomena with slip boundary condition. The derived analytical solution can be used to study more general timedependent problems through superposition of timeharmonic solutions weighted by appropriate Fourier coefficients. The procedures reported here are presented in a general form so that slip conditions of different types (not necessarily that from Navier) can also be used. II. Controlling Equations and Boundary Conditions We consider the boundary value problem for oscillating liquid flow in an infinitely extended circular microchan(17) Masliyah, J. H. Electrokinetic Transport Phenomena; Alberta Oil Sands Technology and Research Authority: 1994. (18) Watanabe, K.; Udagawa, Y.; Udagawa, H. Drage reduction of Newtonnian fluid in a circular pipe with highly water-repellant wall. J. Fluid Mech. 1999, 381, 225-238. (19) Tretheway, D. C.; Meinhart, C. D. Apparent fluid slip at hydrophobic microchannel walls. Phys. Fluid 2002, 14, 096105. (20) Zhu, Y. X.; Granick, S. Phys. Rev. Lett. 2001, 87, 9, 980. (21) Thompson, P. A.; Troian, S. M. A general boundary condition for liquid flow at solid surfaces. Nature 1997, 389, 360-362. (22) de Gennes, P. G. Langmuir 2002, 18, 3413-3414. (23) Pit, R.; Hervet, H.; Le´ger, L. Phys. Rev. Lett. 2000, 85, 5. (24) Navier, C. L. M. H. Mem. Acad. Sci. Inst. Fr. 1823, 1, 414. (25) Goldstein, S. Modern Developments in Fluid Dynamics; Dover: 1965, 2, 676.
Yang and Kwok
nel: A cylindrical coordinate system (r, θ, z) is used 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 time t. The boundary value problem with relevant field equations and boundary conditions is shown below. A. Electrical Field. The total potential U at location (r, z) at a given time t is taken to be
U ≡ U(r,z,t) ) ψ(r) + [U0 - zE′z(t)]
(1)
where ψ(r) is the potential due to the double layer at the equilibrium state (i.e., no liquid motion with no applied external field); U0 is the potential at z ) 0 (i.e., U0 ≡ U(r,0,t)); and E′z(t) is the spatially uniform, time-dependent electric field strength. The total potential U in eq 1 is axisymmetric, and when E′z(t) is time-independent, eq 1 is identical to eq 6.1 of Masliyah.17 The timedependent 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 B ;27 this definition may then be used to obtain the Poisson equation
∇2U ) -
F
(2)
where F is the free charge density and is the permittivity of the medium. Combining eqs 1 and 2 yields the following Poisson equation in cylindrical coordinates
(
)
F 1 d dψ(r) r )r dr dr
(3)
The conditions imposed on ψ(r) are
ψ(a) ) ψs and ψ(0) is finite
(4)
where ψs is the surface potential at the capillary wall, r ) a. For brevity, we shall focus on a symmetric, binary electrolyte with univalent charges. The cations and anions are identified as species 1 and 2, respectively. On the basis of the assumption of thermodynamic equilibrium, the Boltzmann equation provides a local charge density Fi of the ith species. Thus
[
Fi ) zien∞ exp -
]
zieψ , (i ) 1, 2) kT
(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 ψ ) 0; k is the Boltzmann constant; and T is the absolute temperature. Invoking the Debye-Hu¨ckel approximation for low surface potentials (zieψ/kT , 1), we have sinh(z0eψ/kT) ≈ z0eψ/kT and the total charge density follows from eqs 3 and 5 as
-2n∞e2z20 ψ F ) Fi ) 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) (26) Schweiss, R.; Welzel, P. B.; Werner, C.; Knoll, W. Colloids Surf., A: Physiochem. Eng. Aspects 2001, 195, 97. (27) Shadowitz, A. The Electromagnetic Field; Dover: 1975.
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electrolyte is given as
κ)
x
2n∞e2z20 kT
(7)
where V, R, Ψ, and Ez are the normalized velocity, radius, surface potential, and strength of the electric field, respectively. The normalized counterparts of eqs 8 and 4 become, respectively,
1 d dΨ R ) K2Ψ R dR dR
(
Combining eqs 3 and 6 results in
(
)
(8)
whereas eqs 9, 10, 12, and 13 become, respectively,
-
B. Hydrodynamic Field. The axial electric field will induce a body force FE′z, and the modified Navier-Stokes equation becomes
1 1 ∂v 1 ∂p 1 ∂ ∂v + r + FE′z ) µ ∂z r ∂r ∂r µ ν ∂t
( )
∂v(0,t) ∂v and )0 ∂r ∂r
∂V ∂V ∂P 1 ∂ R - K2ΨEz ) + ∂Z R ∂R ∂R ∂τ
(
V(1,τ) ) B
(9) I ) -2π
where we take the pressure gradient [∂p/∂z ≡ (∂p/∂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
v(a,t) ) β
(15)
Ψ(1) ) Ψs and Ψ(0) is finite
1 d dψ(r) r ) κ2ψ r dr dr
-
)
(10)
where β is the slip coefficient and -β is the slip length. The electric current density along the microchannel may be integrated over the channel cross section to give the electric current
∫
)
(16)
∂V ∂V(0,τ) , )0 ∂R ∂R 2a2z40e4µDn∞ Ez 2K3T3
1
K2ΨVR dR + π 0
∫01K2ΨVR dR + πΣK2Ez
) -2π
∫01VR dR
Q ) 2π
(17)
where P, I, and Q are the normalized pressure, current, and flow rate, respectively. In deriving eqs 15-17, the following normalized quantities have been identified
Fj )
a2ez0 a ν F, K ) κa, P ) p, τ ) 2t, kT µ〈v〉 a z20e2µD β I) , B ) (18) i, Σ ) 2 2 a kT〈v〉 k T ez0
i ) 2π
∫0aFvr dr +
2πz0eD E′z kT
∫0a(F1 - F2)r dr
(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.17 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 - F2 ) 2ez0n∞ cosh(z0eψ/kT). The Debye-Hu¨ckel approximation implies that cosh(z0eψ/kT) ≈ 1 and F1 - F2 ) 2z0en∞. With this simplification, eq 11 becomes
i ) 2π
∫0aFvr dr +
2z20e2n∞D (πa2)E′z kT
(12)
and the flow rate q can be written as
q ) 2π
∫0avr dr
(13)
III. Normalized Equations Here we give the normalized governing equations and boundary conditions. Using a yet unknown characteristic velocity 〈v〉, the following normalized quantities are defined as
V)
e ea 1 1 ψ, and Ez ) E′ (14) v, R ) r, Ψ ) a kT kT z 〈v〉
where Fj, K, τ, B, and Σ are the normalized charge density, reciprocal of the double layer thickness, time, slip coefficient, and conductivity, respectively. As well, the expression for the characteristic velocity can be identified as
〈v〉 )
k2T2 µae2z20
(19)
We wish to point out that our normalizing schemes in eqs 18 and 19 are different from those of other authors,13,14,28 as we did not employ the ionic concentration at equilibrium n∞ to normalize the charge density F. The choice of this selection is important, as we intend to study the effects of electrolyte concentration indirectly through K on 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 four normalized quantities
Ω)
q νj µ a2 ω, Q ) , Ω h ) Ω , and τj ) 2t 2 ν µ j 〈v〉a a
(20)
where the parameters ω and q are the frequency of the external oscillating field and volumetric flow rate, respectively; Ω and Ω h are the normalized frequency and normalized apparent frequency, respectively; the param(28) Hu, L.; Harrison, J. D.; Masliyah, J. H. J. Colloid Interface Sci. 1999, 215, 300-312.
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Yang and Kwok
eters µ j and νj are the apparent viscosity and apparent kinematic viscosity, respectively. They are written with an overbar as opposed to the true viscosity µ and kinematic viscosity ν (without overbars).
The volumetric flow rate q is defined as q ) 2π∫a0 rv dr. Using the definition in eq 20 as well as eqs 24 and 25, the normalized flow rate Q becomes
Q ) Re[Q*ejΩτ] IV. Analytical Solution An analytical solution is sought here for a sinusoidal periodicity in the electrohydrodynamic fields, and this is best addressed by using complex variables. Thus, a 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 (defined in eq 20), and τ is the normalized time (defined in eq 18). The general field quantity X is written as
X ) Re[X*ejΩτ]
Im(X*) Re(X*)
(22)
where Im(X*) and Re(X*) are the imaginary and real parts of X*, respectively. An alternative representation of eq 21 is given as
X ) Re[|X*|ej(Ωτ+φ)] where
|X| ) |X*| and |X*| ) xIm2(X*) + Re2(X*)
Q* ≡ Q*(Ω) ) Q/P(Ω) P*(Ω) + Q/E(Ω) E/z (Ω) (27) During pressure-driven flow, the amplitude of the streaming potential E/z (Ω) is found by setting I* ) 0 in eq 26. Thus
E/z (Ω) ) -
(21)
The phase angle φ is defined as
φ ) tan-1
where
(23)
With the notation of eq 21, we shall seek the solution of the boundary value problem for the following specific dependencies
∂P ) Re[P*ejΩτ], Ez ) Re[EzejΩτ] ∂Z
(24)
[(
)
I/P(Ω) 1 / / P*(Ω)ejΩτ q ) 2n∞a kT Re QP(Ω) - QE(Ω) / µ I (Ω) E
]
The expression for the flow rate without EDL effects and without slippage follows by first setting E/z (Ω) ) 0 and B ) 0 in eq 27. The resulting expression for the dimensional volumetric flow rate (denoted as qapparent) is obtained by replacing µ with µ j:
1 h ,B)0) P*(Ω h )ejΩh τj qapparent ) 2n∞a3kT Re Q/P(Ω µ j
[
V ) Re[V*ejΩτ] where
V* ≡ V*(R,Ω) ) V/P(R,Ω) P*(Ω) + V/E(R,Ω) E/z (Ω) (25) The expressions for V/P(R,Ω) and V/E(R,Ω) will be given at the end of this section. The normalized electric current will follow from eq 17 and may be written as jΩτ
]
]
(30)
where Ω h is the normalized apparent frequency and τj is the normalized apparent time calculated using the apparent kinematic viscosity νj (where νj ) µ j /Fd and Fd is the mass density of the liquid). The normalized quantities Ω h and τj have both been defined in eq 20. Setting q ) qapparent and using eqs 29 and 30, we obtain
Re
[(
)
]
I/P(Ω) 1 / QP(Ω) - Q/E(Ω) / P*(Ω)ejΩτ ) µ I (Ω) E
[
]
1 h ,B)0) P*(Ω h )ejΩh τj (31) Re Q/P(Ω µ j
where
I* ≡ I*(Ω) ) I/P(Ω) P*(Ω) + I/E(Ω) E/z (Ω)
(28)
(29)
We consider the class of solutions where the amplitude of the pressure gradient and the electric field could be frequency-dependent, that is, P* ≡ P*(Ω) and E/z ≡ E/z(Ω). The solution for Ψ will then follow from eq 15, and that for V, from eq 16. Thus
I ) Re[I*e
P*(Ω) for I* ) 0
I/E(Ω)
Equation 28 may be substituted into eqs 25 and 27 to determine the normalized liquid velocity and the volumetric flow rate, respectively. Alternatively, the velocity, current, and volumetric flow rate during electroosmosis flow follow from eqs 25-27 by setting P*(Ω) ) 0. The presence of a streaming potential translates to an increased “drag” on the flow, resulting in an apparent viscosity which is higher than the true liquid viscosity. This is referred to as the electroviscous effect. On the contrary, liquid slip at the solid-liquid interface induces a larger flow rate or smaller apparent viscosity. Both the electroviscous effect and slip condition at the interface are characterized by defining an apparent viscosity µ j . For the steady-state problem, Rice and Whitehead11 defined µ j by proposing that the flow rate (inclusive of EDL effects and the computed true liquid viscosity) be set equal to the flow rate exclusive of an EDL effects and with µ replaced by µ j . Adopting the same approach here, the dimensional flow rate follows from eqs 19 and 20 and 27 and 28 as 3
-
I/P(Ω)
(26)
Note that Ω h τj ) Ωτ. Further, assuming that the amplitude of the imposed pressure gradient is a constant quantity
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P0, we have P*(Ω) ) P*(Ω h ) ) P0. For an arbitrary P0ejΩτ, eq 31 reduces to
µ j (B,E/z ) ) µ
1 B V/P(R,0) ) (1 - R2) 4 2
Q/P(Ω h ,B)0) Q/P(Ω)
-
(32)
I/P(Ω) Q/E(Ω) / IE(Ω)
V/E(R,Ω) )
[
[
]
{
J0(jKR)
2
K - jΩ J0(jK)
{
I/E(0) ) -πΨ2s K2 1 -
J0(jK)
] ]
J1(jK) + jBK J0(jK)
J1(jK) 2 J1(jK) + jBK jK J0(jK) J0(jK)
[
]
J1(jK) 2 J1(jK) 1 + jBK + jK J0(jK) J0(jK)
}
J21(jK)
Q/P(0) )
-
]
J0(Rx-jΩ) J1(jK) 1 + jBK J0(jK) J0(x-jΩ) + Bx-jΩJ1(x-jΩ)
I/P(Ω) ) 2πK2Ψs
[
I/P(0) ) -πΨs 1 -
J0(jKR)
J20(jK)
J0(Rx-jΩ) 1 1jΩ J0(x-jΩ) + Bx-jΩJ1(x-jΩ) K2Ψs
[
V/E(R,0) ) -Ψs 1 -
The resulting apparent viscosity µ j is a complex quantity. At steady state [which is recovered by setting Ω or Ω h ) 0 in eq 32], the imaginary component of µ j /µ vanishes, and the real component for B ) 0 is identical to eq 37 of Rice and Whitehead.11 Equation 32 is an implicit equation in µ j , and hence the ratio µ j /µ has to be determined numerically. The relevant quantities in eqs 25-27 are listed below
V/P(R,Ω) )
When Ω f 0, eq 33 reduces to those of the steady state.
}
{
1 J1(jK) + ΩK J0(jK)
[
+ πΣK2
π π - B 8 2
]
J1(jK) 2 J1(jK) Q/E(0) ) -πΨs 1 + jBK J0(jK) jK J0(jK)
(34)
If the slip coefficient B ) 0, all quantities reduce to those of the electrokinetic flow with no slip condition. If Ψs ) 0 and B ) 0, all quantities reduce to those of macroflow with no EDL effect and no slip condition. V. Parametric Results and Discussion
(33)
We study here the effect of the normalized thickness K-1 in the EDL and that of the normalized slip coefficient B on flow velocity. We have selected the following values for our calculations: the amplitude of the normalized pressure gradient, P*(Ω) ) 2, the amplitude of the electric field, E/z (Ω) ) 39, the normalized slip coefficient, B ) -0.05, the normalized conductivity, Σ ) 3.85, and the normalized surface potential, Ψs ) 3.9. These parameters represent, in the dimensional case, a surface potential of 100 mV (the Debye-Hu¨ckel approximation gives a good agreement with experiments, when the ζ potential is up to 100 mV),29 a diffusion coefficient D ≈ 2 × 10-9 m2/s for an infinitely diluted KCl electrolyte,30 a pressure gradient of 106 Pa/m, an electric field of 0.5 × 105 V/m, and a slip length of 1 µm for microchannel with a radius of 20 µm. A. Pressure Driven Flow. When the pressure gradient drives a flow, a streaming potential will build up by the gathered ions between the two ends of the microchannel. Ions in the conductive current induced by the streaming potential [the second term in eq 12] will bring some water molecules to move opposite to the flow direction because of viscosity. We plot the amplitude of the normalized velocity with respect to the normalized radius in Figure 2 for Ω ) 0. Let us illustrate the effect of the slip condition on flow velocity by comparing the following two cases: (1) no slip without electrokinetic effect [B ) 0 and E/z ) 0 (solid line)] and (2) slip without electrokinetic effect [B * 0 and E/z ) 0 (stars)]. In Figure 2, we see that the flow velocity of the former is smaller than that of the latter. Therefore, if one neglects the slip condition (E/z ) 0 and B ) 0), a smaller velocity would be obtained because B
where J0 and J1 are the zeroth- and first-order Bessel functions of the first kind.
(29) Hunter, R. J. Introduction to Modern Colloid Science; Oxford Science Publications: 1993. (30) Newman, J. S. Electrochemical Systems, 2nd ed.; Prentice Hall: Englewood Cliffs, NJ, 1991.
1 1 × jΩJ0(jK) J0(x-jΩ) + Bx-jΩJ1(x-jΩ) jKJ1(jK) J0(x-jΩ) - x-jΩJ0(jK) J1(x-jΩ) jΩ - K2 I/E(Ω)
{[
]
[
]
}
2πK4Ψ2s 1 J21(jK) 1 )- 2 1+ 2 × 2 J (jK) K - jΩ J0(jK) 0 1 × J0(x-jΩ) + Bx-jΩJ1(x-jΩ) J1(jK) 1 + jBK × J0(jK)
jKJ1(jK) J0(x-jΩ) - x-jΩJ0(jK) J1(x-jΩ) jΩ - K2 Q/P(Ω) )
{
}
+ πΣK2
J1(x-jΩ) 2π 1 1 jΩ 2 x-jΩ J (x-jΩ) + Bx-jΩJ (x-jΩ) 0 1 Q/E(Ω) )
{
[
]
}
2πK2Ψs 1 J1(jK) J1(jK) × - 1 + jBK 2 J0(jK) K -jΩ jK J0(jK)
J1(x-jΩ)
x-jΩ
1 J0(x-jΩ) + Bx-jΩJ1(x-jΩ)
}
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Figure 2. Effects of electrokinetic and slip conditions on the velocity profile in the steady state (Ω ) 0).
Figure 3. Effects of electrokinetic and slip conditions on the normalized flow rate in the steady state (Ω ) 0).
might not necessarily be zero. The slip length indeed changes and becomes larger on hydrophobic solid surfaces. A similar comparison can be made for the following two cases, with similar results: (1) no slip with electrokinetic effect [B ) 0 and E/z * 0 (dotted line)] and (2) slip with electrokinetic effect [B * 0 and E/z * 0 (circles)]. Further, if no EDL effect is considered, the predicted velocity would always be larger than that with the EDL effect, by comparing the following two cases: (1) slip without electrokinetic effect [B * 0 and E/z ) 0 (stars)] and (2) slip with electrokinetic effect [B * 0 and E/z * 0 (circles)]. A similar comparison can be made, with similar results, for (1) no slip without electrokinetic effect [B ) 0 and E/z ) 0 (solid line)] and (2) no slip with electrokinetic effect [B ) 0 and E/z * 0 (dotted line)]. It should be noted that the solid line (B ) 0 and E/z ) 0) corresponds to the conventional fluid velocity profile with no slip condition and no electrokinetic effect; the line with circles (B * 0 and E/z * 0) represents the real velocity profile with the slip condition and electrokinetic effect. We see that there is a large deviation in velocity prediction if both effects have not been taken into account. We also plot the normalized flow rate with respect to the normalized reciprocal thickness K in Figure 3. Direct comparison of (1) no slip without electrokinetic effect [B ) 0 and E/z ) 0 (solid line)] and (2) no slip with electrokinetic effect [B ) 0 and E/z * 0 (dotted line)] suggests that an EDL affects flow rate over a large range
Yang and Kwok
Figure 4. Effects of the slip condition on the normalized streaming potential in the steady state (Ω ) 0).
of K. A similar conclusion can be made by comparing (1) slip without electrokinetic effect [B * 0 and E/z ) 0 (stars)] and (2) slip with electrokinetic effect [B * 0 and E/z * 0 (circles)]. In general, an EDL causes a large reduction in flow rate, whereas the slip condition counteracts the effect of an EDL and induces a larger flow rate. However, in Figure 3, when K f 0 or K f ∞, the EDL effect disappears since K f 0 implies a uniform EDL in the microchannel and K f ∞ implies a very thin double layer thickness.10 Closer scrutiny of Figure 3 reveals that when there is no slip (B ) 0, dotted line), the electrokinetic effect on flow rate diminishes as K f ∞. When there is liquid slip (circles), however, the electrokinetic phenomenon has a larger effect on flow rate. This is apparent by comparing the dotted line and circles at K ) 100. This result suggests that liquid slip can enlarge the EDL effect on fluid flow. Since the slip condition induces higher velocity and hence increases the convection current (first term in eq 12), a larger streaming potential is needed to reach a balance, I ) 0. This fact is confirmed in Figure 4, where we see the streaming potential is larger than that with no slip. We wish to emphasis that the presence of an EDL retards flow rate, while slip conditions induce a higher flow velocity. Both phenomena cause the observed flow rate to be different from the traditional one (without the EDL effect and slip conditions) and can result in an apparent viscosity µ j . The ratio of the apparent viscosity to the liquid viscosity |µ j /µ| with respect to K is shown in Figure 5. In this figure, if we consider only the slip condition and no electrokinetic effect (B * 0 and E/z ) 0, stars), the viscosity ratio |µ j /µ| would be smaller than 1, suggesting that the observed apparent viscosity would be smaller than the true liquid viscosity when the slip condition is considered. Thus, liquid slip enhances fluid movement. With the presence of electrokinetic effect, the effect of slip condition on viscosity can be obtained by comparing the following two cases: (1) slip with electrokinetic effect [B * 0 and E/z * 0 (circles)] and (2) no slip with electrokinetic effect [B ) 0 and E/z * 0 (dotted line)]. That is, a higher apparent viscosity would be predicted if the slip condition is neglected. B. Electric Field Driven Flow. Other than a pressure gradient, an electric field can also be used to move the ions in the electric double layer, and such ions will carry water molecules with them because of viscosity, resulting in fluid flow. We normally call such electric field driven phenomena “electroosmosis pumps”. This phenomenon has wide applications for lab-on-a-chip technology. A plot
Microfluid Flow in Circular Microchannel
Figure 5. Effects of the slip condition on the normalized streaming potential in the steady state (Ω ) 0).
Langmuir, Vol. 19, No. 4, 2003 1053
Figure 7. Effects of the slip condition with electrokinetics on the amplitude of the normalized current with respect to |µ j /µ| in the steady state (Ω ) 0).
without consideration of the EDL effect. Thus, a smaller slip length would have obtained. On the other hand, in EDL experiments, liquid slip could be an important condition to be considered, especially for hydrophobic solid surfaces. We shall point out that our solution is based on Navier’s assumption on the slip motion. A recent experiment, however, observed a rate-dependent slip length.20 Therefore, slip length might vary with the flow velocity. Our oscillatory solution can be modified for variable slip coefficient if a slip condition other than that proposed by Navier is known. We are in the process of setting up microchannel flow experiments to determine if Navier’s assumption holds. VI. Conclusions Figure 6. Effects of the slip condition with electrokinetics on the amplitude of the normalized flow rate with respect to |µ j /µ| in the steady state (Ω ) 0).
of the normalized flow rate with respect to K is given in Figure 6. In this figure, we find that liquid slip can greatly enhance the flow. When K f 0 (a very thick EDL), the flow rate quickly decreases because the amount of movable ions has been reduced. When K f ∞ (a very thin EDL), the flow rate without slip (dotted line) approaches a constant, since the velocity becomes more uniform everywhere in the channel. We also plot the normalized current with respect to K in Figure 7. It is apparent that the slip condition has a very small effect on the current because the conduction current is much larger than the convention one. When K is in the range between 10 and 100 (upper Figure 7), there is very little difference between the currents for the slip and nonslip cases. From the results presented here, we have shown that fluid flow in circular microchannels can be significanlty different by consideration of EDL effects with liquid slip. In some experiments,18 attempts have been make to measure slip length from the velocity profile or flow rate
We have presented, to the best of our knowledge, the first in depth analysis and analytical solution of an oscillating microfluid in a circular microchannel by combining the electrokinetic effect with the slip condition. In general, an electrokinetic double layer (EDL) causes a large reduction in flow rate, whereas the slip condition based on Navier’s assumption counteracts the effect of the EDL and induces a larger flow rate. We have also shown that, in the steady state, both slip condition and EDL effects can be important phenomena to be considered simultaneously for a better description of fluid flow, especially with hydrophobic channel walls. Our results provide guidelines to the design and operations of microfluidic devices. Acknowledgment. We gratefully acknowledge financial support from the Alberta Ingenuity Establishment Fund, Canada Research Chair Program, and Natural Science and Engineering Research Council of Canada (NSERC) in partial support of this research. J.Y. acknowledges financial support from a Studentship awarded by the Alberta Ingenuity in the Province of Alberta. LA026201T
