Two-fluid electroosmotic flow in microchannels

Two-fluid electroosmotic flow in microchannels

Journal of Colloid and Interface Science 284 (2005) 306–314 www.elsevier.com/locate/jcis Two-fluid electroosmotic flow in microchannels Yandong Gao, ...

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Journal of Colloid and Interface Science 284 (2005) 306–314 www.elsevier.com/locate/jcis

Two-fluid electroosmotic flow in microchannels Yandong Gao, Teck Neng Wong ∗ , Chun Yang, Kim Tiow Ooi School of Mechanical and Production Engineering, Nanyang Technological University, 50 Nanyang Avenue, 639798 Singapore Received 10 February 2004; accepted 4 October 2004 Available online 26 January 2005

Abstract This paper presents a mathematical model to describe a two-fluid electroosmotic pumping technique, in which an electrically nonconducting fluid is delivered by the interfacial viscous force of a conducting fluid; the latter is driven by electroosmosis. The electrical potential in the conducting fluid and the analytical solution of the steady two-fluid electroosmotic stratified flow in a rectangular microchannel was presented by assuming a planar interface between the two immiscible fluids. The effects of viscosity ratio, holdup, concentration, and interfacial zeta potential are analyzed to show the potential feasibility of this technique.  2004 Elsevier Inc. All rights reserved. Keywords: Two-fluid stratified flow; Electrical double layer; Electroosmotic pump

1. Introduction It is a well-known phenomenon that most surfaces acquire a finite charge density when in contact with a polar solution. The effect of any charged surface in an electrolyte solution will be to influence the distribution of nearby ions in the solution and lead to the formation of a region close to the charged surface in which there is an excess of counterions over coions to neutralize the surface charge. This high capacitance charged region of ions at the liquid/solid interface, known as the electrical double layer (EDL), induces electrokinetic phenomena [1]. The effect of an EDL on the flow for an externally applied pressure gradient is to retard the liquid flow, resulting in a streaming potential. Instead of imposing an external pressure gradient, when an external electric field is applied, the presence of the EDL induces the fluid flow to form an electroosmotic flow. The thickness of EDL (Debye length) is dependent on the bulk ionic concentration and electrical properties of liquid, usually ranging from several nanometers for high ionic concentration solutions up to the order of micrometers for distilled water. EDL is primarily a surface phenomenon; its effect tends to be sig* Corresponding author. Fax: +65-67911859.

E-mail address: [email protected] (T.N. Wong). 0021-9797/$ – see front matter  2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2004.10.011

nificant when the typical channel dimension is of the order as the EDL thickness. Many research works have been published for single conducting fluid-involved electroosmotic flow and electroosmotic pump. Burgreen and Nakache [2] studied the effect of the surface potential on liquid transport through ultrafine capillary slits under an imposed electrical field. Rice and Whitehead [3] studied the same problem in narrow cylindrical capillaries. Levine et al. [4] extended the Rice and Whitehead model to high zeta potential for the electrokinetic flow. Mala et al. [5] reported microchannel flow and heat transfer on two parallel plates and Yang and co-workers [6,7] studied electroviscous effects in rectangular microchannels. Maynes and Webb [8] analyzed the electroosmotic flow in a parallel-plate microchannel and circular microtube under imposed constant wall heat flux and constant temperature boundary conditions. In application, Zeng et al. [9] fabricated an EOF pump by packing the 3.5-µm-diameter nonporous silica into 500to 700-µm-diameter fused-silica capillaries. The proposed pump can generate maximum pressures in excess of 20 atm and flow rates of 3.6 µl/min for a 2 kV applied potential. Chen and Santiago [10] fabricated an EOF pump on soda-lime glass substrate using standard microlithography and chemical wet-etching techniques. The pump provided

Y. Gao et al. / Journal of Colloid and Interface Science 284 (2005) 306–314

a maximum pressure and flow rate performance of 0.33 atm and 1.5 µl/min at 1 kV applied potential. The above studies were focused on the transportation of polar fluid with significant electrical conductivity. Nonpolar fluids, such as oil, cannot be pumped using the EOF due to the low conductivity. Brask et al. [11] proposed an idea to use an EOF as a driving mechanism to drag other liquid. The concept was to use a conducting fluid to pump a nonconducting fluid by viscous shear stress. They analyzed the performance of the pump by equivalent circuit theory and CFD simulations. It marks an interesting development in this field. The aim of this work is to provide a theoretical analysis of the two-fluid electroosmotic flow. Analytical solutions of EDL in the conducting fluid and velocities of two fluids are obtained in the fully developed section in a rectangular channel. In practice, the cross section of microchannel made by modern micromachining technology is close to a rectangular shape. In such a situation, a two-dimensional Poisson-Boltzmann equation is required and the corner of the channel may have particular contribution to the electrical double layer and to the two-fluid flow subsequently [6].

307

Fig. 1. Schematic of a novel two-fluid electroosmotic pump.

Fig. 2. Schematic of coordinate system.

x direction. Due to symmetry, only half of the cross section of the rectangular microchannel is considered. 2. Operational principle of the two-fluid electroosmotic pump The electroosmotic pumping concept [12] for the nonconducting liquid consists of two immiscible fluids: an electrically conducting fluid at the bottom section and a nonconducting fluid at the upper section of the channel as shown in Fig. 1. When an electric field is applied across the conducting fluid, the conducting fluid is driven by electroosmosis, which drags the nonconducting fluid by the hydrodynamic viscous forces at the interface. Through this way, the electrically nonconducting fluid is delivered by electroosmosis. The flow of the nonconducting fluid depends on the viscosity ratio, density, and interface curvature of the two fluids.

3. Theoretical model of the two-fluid electroosmotic flow To analyze this system, a Cartesian orthonormal coordinate system (x, y, z) is used where the origin point, O, is at the intersection of the interface and the symmetric line as shown in Fig. 2. A planar interface is assumed which is satisfied when the contact angle is close to 90◦ . The heights of the conducting fluid and the nonconducting fluid are denoted by h1 and h2 , respectively. EDLs only form near the channel walls in contact with the conducting fluid and at the fluid–fluid interface. Because the bottom wall and the side wall may be made of different materials, we assume that the zeta potential at the bottom wall is ζ1 and at the side walls is ζ2 ; and the zeta potential between the two fluids is ζ3 . The driving electric forces and electroosmotic flow are along the

3.1. Electric double layer in the conducting fluid According to the theory of electrostatics, the relationship between the electrical potential, ψ, in the diffuse layer and the net charge density per unit volume, ρe at any point in the solution is described by the Poisson equation [5,6] ρe , ∇ 2ψ = − (1) εr ε0 where εr is the dielectric constant of the solution and ε0 is the permittivity of vacuum. In a fully developed hydrodynamic state, the net charge density in the diffuse layer variation obeys the Boltzmann distribution [7]. Based on this assumption of thermodynamic equilibrium, for a symmetric electrolyte (e.g., KCl) with univalent charges, the net volumetric charge density, ρe , is connected with the electrical potential, via   z0 eψ , ρe = −2z0 en0 sinh (2) kB T where z0 is the valence of ions, e is the elementary charge, n0 is the bulk concentration of ions, kB is the Boltzmann constant, and T is the absolute temperature. Substituting Eq. (2) into the Poisson equation (1) leads to the well-known Poisson–Boltzmann equation,   2z0 en0 z0 eψ 2 . sinh ∇ ψ= (3) εr ε0 kB T By defining the Debye–Hückel parameter κ = (2z02 e2 n0 / εr ε0 kB T )1/2 and the hydraulic diameter of the microchannel as Dh = 4(h1 + h2 )w/(h1 + h2 + 2w) and introducing

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the dimensionless groups y¯ = y/Dh , z¯ = z/Dh , K = κDh , and ψ¯ = z0 eψ/kB T , the above equation can be nondimensionalized as

exerting on the conducting fluid to modify the conventional Poiseuille flow equation, which will be discussed in the following sections.

¯ ∇ 2 ψ¯ = K 2 sinh(ψ),

3.2. Momentum equation of the two-fluid flow

(4) m−1

where κ has the unit of and 1/κ is referred to the characteristic thickness of the EDL. Generally speaking, the nonlinear Poisson–Boltzmann equation (4) cannot be solved analytically. However, in the range of 0 < ψ¯ < 1, the right-hand side of Eq. (4) can be ¯ ≈ K 2 ψ¯ [13]. mathematically approximated as K 2 sinh(ψ) This approximation is termed Debye–Hückel approximation [14], which physically means a small electrical potential as compare to thermal energy of ions, |z0 eψ| < kB T (kB T /e ≈ 25.7 mV at 25 ◦ C). Thus, Eq. (2) can be rewritten approximately as     z0 eψ z0 eψ ≈ −2z0 en0 . ρe = −2z0 en0 sinh (5) kB T kB T Therefore, the Poisson–Boltzmann equation can be linearized as ¯ ∇ 2 ψ¯ = K 2 ψ.

(6)

In the rectangular channel studied in this paper, due to the symmetry of the EDL field, Eq. (6) is subjected to the following boundary conditions in the conducting fluid  ¯ z¯ = 0 at z¯ = 0,  ∂ ψ/∂   ¯ ψ = ζ¯2 at z¯ = w/Dh = w, ¯ (7) ¯ ¯  at y ¯ = 0, ψ = ζ 3   ¯ at y¯ = −h1 /Dh = h¯ 1 , ψ = ζ¯1 where the nondimensional zeta potentials are defined as ζ¯1 = z0 eζ1 /kB T , ζ¯2 = z0 eζ2 /kB T , and ζ¯3 = z0 eζ3 /kB T . Using the separation of variables method, the analytical solution of Eq. (6) with the boundary conditions Eq. (7) is obtained as ∞ √  4ζ¯1 (−1)n+1 sinh K 2 +(2n−1)2 π 2 /4w¯ 2 y¯ ¯ z, y) √ ψ(¯ ¯ =  n=1

K 2 +(2n−1)2 π 2 /4w¯ 2 h¯ 1

(2n−1)π sinh

 × cos (2n − 1)π z¯ /2w¯ ∞ √  4ζ¯3 (−1)m+1 sinh K 2 +(2m−1)2 π 2 /4w¯ 2 (h¯ 1 −y) ¯ + √ m=1

(2m−1)π sinh

K 2 +(2m−1)2 π 2 /4w¯ 2 h¯ 1

 × cos (2m − 1)π z¯ /2w¯



∞ 2ζ¯ [1 + (−1)p+1 ] cosh K 2 + p 2 π 2 /h¯ 21 z¯  2 + 

pπ cosh K 2 + p 2 π 2 /4h¯ 21 w¯ p=1 × sin[pπ y/ ¯ h¯ 1 ].

(8)

Once the electrical potential distribution is known, the ionic net charge density can be obtained from Eq. (5). This local net charge density is required to determine the electrostatic force caused by the presence of an EDL field. Such an electrostatic force is considered as an additional body force

The motion of an incompressible Newtonian fluid is given by the incompressible Navier–Stokes equation [15] as ∂V + ρ (V · ∇) V = −∇p + F + µ∇ 2 V. (9) ∂t In this equation, V is the velocity vector, p is the pressure, ρ and µ are the density and dynamic viscosity of the fluid. When an external electric field (E = −∇φ) is applied, the external electric field interacts with the electric double layers in the conducting fluid and creates the electroosmotic body force, F, on the bulk conducting fluid, whereas there is no such electroosmotic body force in the nonconducting fluid because of the absence of electric double layer. The main simplifying assumptions in the current analysis are as follows:

ρ

(1) The two fluids are simple Newton fluids. (2) The fluids’ properties are independent of local electric field and ion concentration. The electric field strength and ion concentration may affect the properties of the conducting fluid. In the current study, we consider only dilute solutions and these effects are neglected [16]. (3) The fluids’ properties are independent of temperature. The temperature of two fluids will increase because of Joule heating released in the electroosmotic flow. However, the Joule heating can be safely neglected for dilute electrolytes (e.g., C < 10−4 M) and low field strength (e.g., E < 100 V/cm) [17]. (4) The two-fluid flow is steady and fully developed with no-slip boundary condition. Currently, we only consider the fully developed section of the channel, which is away from the entrance and exit. The unsteady term ∂V/∂t and the convection term (V · ∇)V will be vanished. (5) There is no pressure gradient along the channel. There are two sources to generate pressure gradient along the channel. One is the applied pressure between inlet and outlet. Another is the change in the velocity along the channel due to inhomogeneity, such as variance of pH value or nonuniform zeta potential along the channel. Currently, we consider the fully developed flow without external pressure difference between inlet and outlet. Based on these assumptions, the momentum equation for the streamwise velocities of two fluids in the fully developed section are reduced to Poisson equations µ1 ∇ 2 u1 = −Ex ρe

for the conducting fluid

(10)

and µ2 ∇ 2 u2 = 0 for the nonconducting fluid,

(11)

Y. Gao et al. / Journal of Colloid and Interface Science 284 (2005) 306–314

where the subscripts 1 and 2 denote the conducting fluid and the nonconducting fluid, respectively. Ex is external electrical field along the x direction, and ρe is the net charge density obtained by Eqs. (5) and (8). At the interface, matching conditions must be obeyed. They are the continuities of velocity and hydrodynamic shear stress, which are represented by u = u , 1 2 (12) ∂u /∂n = β∂u /∂n at the interface, 1

∇ 2 u¯ 1 = K 2 ψ¯

(13)

and the momentum equation for the nonconducting fluid is ∇ 2 u¯ 2 = 0,

(14)

where u¯ = u/Uref , x¯ = x/Dh , y¯ = y/Dh , and z¯ = z/Dh . In these two equations, the Reynolds number is defined as Re = (ρref Uref Dh )/µref .

u¯ 1 = 0 u¯ 1 = 0

at z¯ = w, ¯ at y¯ = h¯ 1 .

u¯ 2 = 0 u¯ 2 = 0

at z¯ = w, ¯ at y¯ = h2 /Dh = h¯ 2 ,

(16b)

(17a)

Similarly, the hold up of nonconducting fluid is defined as ε2 = 1 − ε1 =

|h¯ 2 | . ¯ 1 + |h¯ 2 | |h|

 Φj (h¯ 1 ) cosh(λj y) ¯ + Φj (y) ¯ cos(λj z¯ ), (18) cosh(λj h¯ 1 )

Φj (y) ¯ =− −

4K 2 ζ¯1 (−1)j +1 sinh(Bj y) ¯

(2j − 1)π(λ2j − Bj2 ) sinh(Bj h¯ 1 )

4K 2 ζ¯3 (−1)j +1 sinh[Bj (h¯ 1 − y)] ¯ 2 2 ¯ (2j − 1)π(λ − B ) sinh(Bj h1 ) j



j

∞  4K 2 ζ¯2 [1 + (−1)p+1 ]λj (−1)j +1 sin(pπ y/ ¯ h¯ 1 )

pπ w(λ ¯ 2j + A2p )[(pπ/h¯ 1 )2 + λ2j ]

p=1

(19) is a parameter function, and

   Ap = K 2 + p 2 π 2 /h¯ 21 ,  = K 2 + (2j − 1)2 π 2 /4w¯ 2 , B j   ¯ λj = (2j − 1)π/2w.

(20)

The nondimensional velocity profile for the nonconducting fluid driven by conducting fluid is u¯ 2 (y, ¯ z¯ ) =

∞    sinh(λj y) ¯ − tanh(λj h¯ 2 ) cosh(λj y) ¯ b2j j =1

× cos(λj z¯ ).

(21)

From Eq. (12), the following constants can be obtained b1j = β

Φj (h¯ 1 )/ cosh(λj h¯ 1 )−Φj (0) tanh(λj h¯ 1 )/λj −Φj (0) tanh(λj h¯ 2 )−β tanh(λj h¯ 1 )



Φ  (0) λj

(22)

and

The holdup of the conducting fluid, ε1 , is defined as the ratio of the area occupied by the conducting fluid to the whole area of the cross section of the channel. In this problem, because of planer interface it is simply given as |h¯ 1 | ε1 = . ¯ 1 + |h¯ 2 | |h|

sinh(λj y) ¯ − tanh(λj h¯ 1 ) cosh(λj y) ¯ b1j

where

(16a)

The dimensionless boundary conditions for the nonconducting fluid are ∂ u¯ /∂ z¯ = 0 at z¯ = 0, 2



(15)

In the rectangular channel as shown in Fig. 2, the electric potential in the EDL is give by Eq. (8). The dimensionless boundary conditions which are applied for the conducting fluid are ∂ u¯ /∂ z¯ = 0 at z¯ = 0,

∞   j =1

2

where n is the normal direction along the interface between two fluids; β = µ2 /µ1 is the dynamic viscosity ratio. To nondimensionalize Eqs. (10) and (11), we introduce some reference quantities: Uref = Ex εr ε0 kB T /z0 eµref is the Helmholtz–Smoluchowski electroosmotic velocity and µref = µ1 , ρref = ρ1 . The nondimensional momentum equation for the conducting fluid is thus written as

1

¯ z¯ ) = u¯ 1 (y,

309

(17b)

Using the method of separation of variables, the analytical velocity profile corresponding to the electroosmotic force, u1 , is obtained as

b2j =

Φj (h¯ 1 )/ cosh(λj h¯ 1 )−Φj (0) tanh(λj h¯ 1 )/λj −Φj (0) , tanh(λj h¯ 2 )−β tanh(λj h¯ 1 )

(23)

where Φj (y) ¯ =−

4K 2 ζ¯1 (−1)j +1 Bj cosh(Bj y) ¯ 2 2 (2j − 1)π(λ − B ) sinh(Bj h¯ 1 )

j j 2 j +1 ¯ 4K ζ3 (−1) Bj cosh[Bj (h¯ 1 − y)] ¯ + 2 2 ¯ (2j − 1)π(λj − Bj ) sinh(Bj h1 ) ∞  4K 2 ζ¯ [1+(−1)p+1 ]λ (−1)j +1 pπ cos(pπ y/ ¯ h¯ 1 )/h¯ 1 j 2 − . pπ w(λ ¯ 2j +A2p )[(pπ/h¯ 1 )2 +λ2j ] p=1

(24)

The dimensionless volumetric flow rate through the rectangular microchannel can be defined by q¯1 = q1 /(Dh2 Uref ) and q¯2 = q2 /(Dh2 Uref ) (where q1 , q2 are the volumetric flow rate for the conducting fluid and the nonconducting fluid, respectively). The dimensionless flow rates are given by w¯ 0 q¯1 = 2

u¯ 1 (y, ¯ z¯ ) d y¯ d z¯ 0 h¯ 1

(25)

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and w¯ h¯ 2 q¯2 = 2

u¯ 2 (y, ¯ z¯ ) d y¯ d z¯ .

(26)

0 0

Substituting u¯ 1 (Eq. (18)) and u¯ 2 (Eq. (21)) into Eqs. (25) and (26), the dimensionless volumetric flow rates can be obtained as  ∞   Φj (h¯ 1 ) sinh(λj h¯ 1 ) b1j 1 1− + q¯1 = 2 ¯ cosh(λj h1 ) λj cosh(λj h¯ 1 )λj j =1  sin(λj w) ¯ Φ Φ ¯ + Ij (0) − Ij (h1 ) (27) λj and q¯2 = 2

∞   j =1

Fig. 3. EDL profile at the symmetric line in the conducting fluid.



b2j 1 ¯ − 1 2 sin(λj w), cosh(λj h¯ 2 ) λj

(28)

where IjΦ (y) ¯ =− +

4K 2 ζ¯1 (−1)j +1 cosh(Bj y) ¯ 2 2 (2j − 1)π(λj − Bj ) sinh(Bj h¯ 1 )Bj

4K 2 ζ¯3 (−1)j +1 cosh[Bj (h¯ 1 − y)] ¯ 2 2 (2j − 1)π(λ − B ) sinh(Bj h¯ 1 )Bj j



j

∞  4K 2 ζ¯2 [1 + (−1)p+1 ]λj (−1)j +1 cos(pπ y/ ¯ h¯ 1 ) p=1

pπ w(λ ¯ 2j + A2p )[(pπ/h¯ 1 )2 + λ2j ]pπ/h¯ 1

.

(29) 4. Results and discussion In the previous section, general equations were derived for the EDL distribution in the conducting fluid and velocity profiles for the two-fluid electroosmotic flow through a rectangular microchannel. In analysis, the dimension of microchannel is chosen as 20-µm width and 40-µm height. The conducting fluid holds the bottom half of the channel and the nonconducting fluid holds the upper half. The default holdup in this paper is 0.5. There are many methods to determine the zeta potential at the channel wall and interface [14]. The zeta potentials at the channel wall ζ1 and ζ2 depend on the material properties of the wall and the fluid ionic properties [18]. The zeta potential between two immiscible liquids not only depends on the ionic properties of two fluids but also depends on the pH value and the concentration of electrolyte [19,20]. We choose ζ1 = ζ2 = ζ3 = −25 mV as the default values. The concentration of the conducting fluid is chosen in the range from 10−6 to 10−5 M, and the applied electrical potential Ex = 10 kV/m. Therefore, the reference velocity is 1.82 × 10−4 m/s and Reynolds number is about 0.005. 4.1. EDL potential in the conducting fluid The nondimensional parameter K is presented to evaluate parameters affecting EDL profiles, which is defined

as K = κDh . 1/κ is referred to the characteristic thickness of the EDL. As the Debye–Hückel parameter κ = (2z02 e2 n0 /εr ε0 kB T )1/2 is proportional to the square root of the bulk ionic concentration n0 , the variation of the ionic concentration will alter the EDL thickness. In the analysis, the concentration of the conducting fluid is chosen within the range of 10−6 –10−5 M, therefore the bulk concentration n0 = 6.022 × 1020 –6.022 × 1021 m−3 and the EDL dimensionless parameter K = 87–275. The zeta potentials are kept at the same value as −25 mV. The EDL profiles are plotted in Fig. 3 where K = 87 (1/κ ≈ 300 nm) and K = 275 (1/κ ≈ 97 nm). It can be seen from the figure that the value of K controls the dimensionless EDL thickness: a larger value of K corresponds a thinner EDL. 4.2. Effect of viscosity ratio, β, on velocity of the two fluids The flow characteristics depend on the coupling effect between the two fluids which involve the electrokinetic phenomenon of the conducting fluid and the interfacial stresses at the interface of two fluids. The velocity at the fluid–fluid interface must match; i.e., the conducting and the nonconducting velocities must be the same and the forces must be balanced. The coupled two-fluid velocities are presented by Eqs. (18)–(24). To investigate the effect of viscosity ratio between the two fluids, the values of β are chosen as different values. Figs. 4–6 show the dimensionless velocity profiles at the symmetric line when the viscosity ratio β is equal to 1, 10, and 100. The electrical body force results from the interaction of the electric field and the net charge density. This driving force exists only within the nonneutral charges region— the electrical double layer in the conducting fluid. Liquid outside the EDL region is set in motion passively due to the hydrodynamic shear stress. The velocity profile of the nonconducting fluid is also passive very much like the Coutte flow. It is purely due to the interfacial shear stress dragged by the conducting fluid on the nonconducting fluid.

Y. Gao et al. / Journal of Colloid and Interface Science 284 (2005) 306–314

Fig. 4. Dimensionless velocity distribution at the symmetric line for β = 1 (K = 87).

Fig. 5. Dimensionless velocity distribution at the symmetric line for β = 10 (K = 87).

The results indicate that the velocity profiles of the conducting fluid are strongly dependent on the viscosity ratio, β. While the viscosity ratio is small, the flow resistance of the nonconducting fluid is small, the nonconducting fluid can be driven with less flow resistance as shown in Fig. 4. When the viscosity ratio is higher, the flow resistance of the nonconducting fluid is higher, resulting in a steeper velocity gradient at the interface of the conducting fluid as shown in Fig. 6. For an extreme case, the infinite viscosity of the nonconducting fluid makes the flow of the nonconducting fluid resemble that of the channel wall. Hence the flow of the conducting fluid resembles the single-fluid EOF, through which we can compare the results of the steady-state analytical model with those of the previous works. If the channel width is much larger than the channel height, in the fully developed state, the flow profile at the symmetric section in this type channel is identical with the flow between two infinite parallel flat plats. For microchannel between two infinite parallel flat plates, if the zeta potentials are assumed as the same at two plates, Patankar and Hu [21] presented an analytical re-

311

Fig. 6. Dimensionless velocity distribution at the symmetric line for β = 100 (K = 87).

Fig. 7. Comparison the analytical solution between two-fluid model and single-fluid model [20]. The viscosity ratio of the two fluids, β = 104 .

sult of the streamwise velocity,   cosh(K y¯ − K h¯ 1 /2) ¯ u( ¯ y) ¯ = ζ1 −1 . cosh(K h¯ 1 /2)

(30)

Fig. 7 shows the comparison between the present exact velocity profile of the conducting fluid from the two-fluid model when β = 104 , and the exact solution from Eq. (30). In this figure, the channel is chosen as 60-µm width and 20-µm height. From this figure, it is clearly seen that the two results from different models are identical. Fig. 8 shows the viscosity ratio on the volume flow rate. It is seen from the figure that the volumetric flow rate increases with decrease in the viscosity ratio. The rapid increase in flow rate occurs when the viscosity ratio, β, decreases from 10 to 10−1 . 4.3. Effect of the value of K Fig. 9 shows the nondimensional velocity profile when K = 87 and 275. If the net charge density in the EDL region is known, the electrical force can be obtained, which is expressed by −2Ex z0 en0 ψ¯ .

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Fig. 8. Dimensionless volumetric flow rates with viscosity ratio, β.

Fig. 10. Dimensionless velocity distributions at the symmetric line with the holdup of the nonconducting fluid, ε2 .

Fig. 9. Dimensionless velocity distributions at the symmetric line for different values of K.

The Debye length, 1/κ, is referred to the characteristic thickness of the EDL. It can be seen that 1/κ decreases inversely as the square root of the concentration. But the increasing of the electroosmotic force is proportional to the bulk ionic concentration. As the result, though the EDL is thinner when the concentration is 10−5 M (K = 275), the overall velocities are larger than those when the concentration is 10−6 M (K = 87). 4.4. Effect of the holdup of nonconducting fluid, ε2 The effect of the holdup, ε2 , of the nonconducting fluid on the velocity profile for β = 100 is shown in Fig. 10. From Eq. (17b), ε2 = h¯ 2 /(h¯ 1 + h¯ 2 ), the interface location of the fluid–fluid interface varies with ε2 . The shapes of velocity profile remain similar for different values of holdup. Fig. 11 shows the variation of volume flow rate with holdup for β = 1 and 100. For a large viscosity ratio (β = 100), the volumetric flow rate of nonconducting fluid increases with increasing ε2 as expected. For a small viscosity ratio (β = 1), the result indicates an optimum holdup ε2 for the maximum flow rate.

Fig. 11. Volumetric flow rates with the holdup of the nonconducting fluid, ε2 .

4.5. Effect of the interfacial zeta potential The fluid–fluid interfacial zeta potential depends on the properties of two fluids and varies with the pH value, the concentration of the conducting fluid and ionic surfactants [20]. The interfacial zeta potential influences the potential distribution in the EDL regions, hence the electroosmotic force distribution and then the flow. To analyze the effects of the interfacial zeta potential, ζ3 are chosen as 0, ζ1 /2, and ζ1 , respectively. Figs. 12 and 13 show the velocity profiles for different interface zeta potential when β = 1 and 100, respectively. From these figures, it is noted that the effects of interfacial zeta potential are more significant than the effects of the wall zeta potential. This is because the electroosmotic force at the interface will not be constrained as the force near the wall. Higher interfacial zeta potential resulted in a higher fluid velocity. The lower the value of β, the more significant these effects are. When the viscosity of nonconducting fluid is low (β = 1), the maximum velocities of two fluids with low interface zeta potential (ζ¯3 /ζ¯1 = 0) are smaller by one order than high interfacial zeta potential

Y. Gao et al. / Journal of Colloid and Interface Science 284 (2005) 306–314

Fig. 12. Velocity profiles at the symmetric line for different interface zeta potentials for β = 1.

313

Fig. 14. Dimensionless volumetric flow rates for different interface zeta potentials.

In this paper, an analytical scheme was used to solve the steady-state two-fluid electroosmotic flow in a rectangular channel. The results showed that the viscosity ratio is an important parameter to the two-fluid flow. Higher viscosity ratio leads to a larger interface velocity gradient of the conducting fluid. The results indicated that for a given electrical field, the volume flow rate decreases with the viscosity ratio. It is noted that the interface location of the fluid–fluid interface varies for different holdups; however, the shapes of velocity profile remain similar. The interfacial zeta potential depends on the properties of two fluids. The results indicate that the effects of interfacial zeta potential are significant.

Fig. 13. Velocity profiles at the symmetric line for different interface zeta potentials for β = 100.

Appendix A. Nomenclature Ap

(ζ¯3 /ζ¯1 = 1); while β = 100, the maximum velocities are in the same order. The velocity profile also varies for different interfacial zeta potentials. For low interfacial zeta potential (ζ¯3 /ζ¯1 = 0), the maximum velocity occurs near the walls, where the electroosmotic force centralized. For a high zeta potential (ζ¯3 /ζ¯1 = 1), the maximum velocity occurs near the wall and the interface. The flow rates for different interfacial zeta potentials are shown in Fig. 14. The results show that the flow rate has a linear relationship with the interfacial zeta potential.

5. Conclusions In most situations, the electrical double layer is very thin and close to the channel wall and the interface. Therefore, the electric body force is concentrated in the very narrow regions. This makes the numerical simulations highly challenging. Thus analytical analysis provides a detailed understanding of the characteristics of two-fluid electroosmotic flow.

parameter in the analytical solution of the velocity defined in Eq. (20) b1j , b2j parameter in the analytical solution of the velocities defined in Eqs. (22) and (23), respectively parameter in the analytical solution of the velocity Bj defined in Eq. (20) C concentration of the solution channel hydraulic diameter Dh e elementary charge induced electrical field in x direction Ex F force vector F electroosmotic force h height of the channel h1 , h2 height of the conducting fluid and the nonconducting fluid φ ¯ parameter function defined as Eq. (33) Ij (y) K dimensionless Debye–Hückel parameter Boltzmann constant kB ionic concentration in equilibrium electrochemical n0 solution at the neutral state concentration of the type i ions ni Re Reynolds number T absolute temperature

314

u Uref V w x, y, z zi

Y. Gao et al. / Journal of Colloid and Interface Science 284 (2005) 306–314

fluid velocity reference velocity fluid velocity vector half-width of the channel Cartesian coordinates valence of type i ions

Greek symbols β viscosity ratio defined as µ2 /µ1 ε holdup of the fluid dielectric constant of the solution εr permittivity of vacuum ε0 ζ1 , ζ2 , ζ3 zeta potential on the bottom wall, side wall, and interface, respectively parameter in the analytical solution of the velocity λj used in Eq. (20) µ fluid viscosity ρ fluid density net volume charge density ρe φ applied electric potential ¯ Φj (y) ¯ parameter function defined as Eqs. (19) and Φj (y), (24), respectively χ aspect ratio defined as Eq. (30) Subscripts 1 2

conducting fluid nonconducting fluid

Superscripts –

dimensionless parameter

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