J. Phys. Chem. B 2004, 108, 11871-11875
11871
Approximate Analytical Expressions for the Electrokinetic Flow of a General Electrolyte Solution in a Planar Slit Comprising Dissimilar Surfaces Sung-Hwa Lin and Jyh-Ping Hsu*,† Department of Chemical and Materials Engineering, National Ilan UniVersity, Ilan, Taiwan 26041
Shiojenn Tseng Department of Mathematics, Tamkang UniVersity, Tamsui, Taipei, Taiwan 25137 ReceiVed: April 16, 2004; In Final Form: May 22, 2004
The electroosmotic flow plays an important role in electrokinetic phenomena, in particular, in the flow of an electrolyte solution in a microchannel, where the electrokinetic equations need to be solved. In general, these equations are coupled, nonlinear partial differential equations, and solving them analytically is nontrivial. In this study the electrokinetic equations are solved analytically for the case of a general electrolyte solution flow through a planar slit comprising two planar, parallel surfaces, which can have different charged conditions, and the influences of the key parameters on the flow behavior are discussed. Several interesting phenomena are observed. For example, the absolute value of the mean fluid velocity may exhibit an undulant behavior as the bulk concentration of counterions varies. Also, if both the valence of coions and the bulk concentration of counterions are fixed, then depending upon the level of the latter the absolute value of the mean fluid velocity may have a local maximum or local minimum as the valence of counterions varies.
1. Introduction The fast development in the fabrication techniques for microscaled instruments in the past few years has pioneered new applications of fluid mechanics; relative subjects have drawn the attention of many scientists and engineers in relevant areas. A typical example includes the design of a micro heat sink used in electronic cooling, and a micro fluid pump used in micro-fluidic systems. A microscaled system is capable of exhibiting phenomena that are either inappreciable or negligible in a macroscaled system. One of these is the electroosmotic flow, which is characterized by the flow inside the double layer near a charged surface of a microchannel. Under typical conditions, the thickness of the double layer near a charged surface ranges from several nanometers to several micrometers.1 Since the linear size of a microscaled system is of micrometer order, this implies that the presence of a double layer can play a role in the description of the fluid behavior in such a system, which is summarized by the so-called electrokinetic equations. These include the Navier-Stokes equation for the flow field and the Poisson equation for the electric field. In general, these are coupled, nonlinear partial differential equations, and solving them analytically is nontrivial. In practice, the difficulty of solving the electrokinetic equations is circumvented by considering a simplified problem,2,3 or by solving them numerically.4,5 * Author to whom correspondence should be addressed. Phone: 8863-9357400 ext 285. Fax: 886-3-9353731. E-mail:
[email protected]. † On leave from the Department of Chemical Engineering, National Taiwan University.
Figure 1. Schematic representation of the problem considered (a) and a typical electrical potential distribution, (b).
If the deformation of the double layer is negligible, the electrokinetic equations can be decoupled, that is, the flow field
10.1021/jp0483324 CCC: $27.50 © 2004 American Chemical Society Published on Web 07/07/2004
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Lin et al.
and the electric field can be treated separately. Also, the electric field is described by a Poisson-Boltzmann equation. The solution procedure then involves solving the Poisson-Boltzmann equation for the electric field first, substituting the result obtained into the corresponding Navier-Stokes equation, followed by solving the resultant expression for the flow field. Unfortunately, solving a general Poisson-Boltzmann equation analytically is also not an easy task; the only reported result is that for an infinite planar surface in a symmetric electrolyte solution. Recently, Luo et al.6 proposed using a matching method to solving the Poisson-Boltzmann equation for two dissimilar, planar parallel surfaces in a symmetric electrolyte solution. This method combines the Langmuir approximation for a point near a charged surface7 and the Debye-Huckel approximation for a point far away from the surface. The performance of the approximate analytical result derived was found to be satisfactory. The analysis of Luo et al.6 was extended by Hsu et al.8 to the case of two identical surfaces in a general electrolyte solution, and the result derived used to evaluate the critical coagulation concentration of a colloidal dispersion. In this study, the matching method adopted by Luo et al.6 and Hsu et al.8 is used to solve a general Poisson-Boltzmann equation, and the result derived is used to evaluate the electrokinetic flow in a planar slit comprising two dissimilar surfaces. 2. Theory Referring to Figure 1a, we consider the flow of an a:b electrolyte solution with a bulk anionic concentration C0b in a planar slit comprising two planar, parallel surfaces, which may have different charged conditions. Let x be the distance from surface 1, and h be the separation distance between two surfaces. A uniform electrical field E with strength Ey and a uniform pressure gradient with magnitude dp/dy are applied in the y-direction. Electrical Field. If both the deformation of double layer and the end effect are negligible, then the electrical potential φ can be described by the Poisson-Boltzmann equation
d2φ ) bC0bF(ebFφ/RT - e-aFφ/RT) dx2
(1)
where is the permittivity of electrolyte solution, and F, R, and T are Faraday constant, gas constant, and the absolute temperature, respectively. Let φs1 and φs2 be respectively the electrical potentials on surfaces 1 and 2. Without loss of generality, we assume that both φs1 and φs2 are positive with φs1 e φs2. An approximate analytical solution to eq 1 can be derived on the basis of the matching method used by Luo et al.6 and Hsu et al.8 We consider three possible cases: (a) hmp e h (φm e φmp), (b) hs1 e h e hmp (φmp e φm e φs1), and (c) h e hs1 and φs1 < φs2, where hmp and hs1 are respectively the values of h at which φm ) φmp and φm ) φs1, φm is the value of φ at x ) xm, where dO/dx ) 0, if it exists, and φmp is the electrical potential at the matching point. Figure 1b shows a typical electrical potential distribution. Because it is required that both aFφmp/RT e 1 and bFφmp/RT e 1 at the matching point, we define Fφmp/RT ) 1/max[a,b], where max[a,b] is the larger of a and b. The three cases above cover all the possible conditions
if φmp e φs1 e φs2. In the first case, hmp e h (φm e φmp), it can be shown thatwhere ψm is the root of the equation
{[ {[ [
]
b(a + b) 2 1 ψ ) ln ebψmp (ψmp - ψ2m) × b 2 b2 b ebψmp - (ψ2mp - ψ2m) 4 2(a + b)
sec2 -
tan-1
X+
] }}
0.5 ebψs1 -1 b(a + b) 2 ebψmp (ψmp - ψ2m) 2 for 0 e X e Xmp1 (2)
(
ψ ) ψm cosh X -
[
]
0.5
)
Xmp1 + Xmp2 for Xmp1 e X e Xmp2 2
]
(3)
-0.5 b b2 ebψm - (ψ2mp - ψ2m) × 4 2(a + b) 0.5 ebψs1 tan-1 -1 + b(a + b) 2 (ψmp - ψ2m) ebψmp 2 0.5 ebψs2 -1 tan-1 b(a + b) 2 ebψmp (ψmp - ψ2m) 2 0.5 ebψmp -1 2 tan-1 + b(a + b) 2 bψmp 2 e (ψmp - ψm) 2 2 sech-1(ψm/ψmp) - D ) 0 (4)
{ [ [ [
] ] ]}
In these expressions, ψ ) Fφ/RT, ψs1 ) Fφs1/RT, ψs2 ) Fφs2/ RT, ψmp ) Fφmp/RT, ψm ) Fφm/RT, X ) κx, and Xmp1 and Xmp2 are respectively the values of X at matching points in the intervals [0,Xm] and [Xm,D], where Xm ) kxm, D ) κh, and κ ) (bC0b(a + b)F2/RT)1/2 ) (2IF2/RT)1/2 is the Debye-Huckel parameter, I ) bC0b(a + b)/2 being the ionic strength. The values of Xmp1 and Xmp2 can be estimated by The potential field
Xmp1 )
[
]
-0.5 b b2 × ebψm - (ψ2mp - ψ2m) 4 2(a + b) 0.5 ebψs1 tan-1 -1 b(a + b) 2 (ψmp - ψ2m) ebψmp 2 0.5 ebψmp -1 tan-1 b(a + b) 2 (ψmp - ψ2m) ebψmp 2
{ [
]
[
Xmp2 ) Xmp1 + 2 sech-1(ψm/ψmp)
]}
(5)
(6)
for the region of Xmp2 e X e D can be recovered by replacing respectively X and ψs1 with (D - X) and ψs2 in eq 2. In the second case hs1 e h e hmp (φmp e φm e φs1), it can be shown that
{
{[
]
b 1 ebψm ψ ) ln ebψm sec2 b 2(a + b)
0.5
tan-1[eb(ψs1-ψm) - 1]0.5
X+
}}
for 0 e X e Xm (7)
Flow of an Electrolyte Solution in a Microchannel
J. Phys. Chem. B, Vol. 108, No. 31, 2004 11873
where ψm is the root of the equation
[
]
b ebψm 2(a + b)
-0.5
velocity, uy, can be described by the one-dimensional NavierStokes equation where η is the viscosity of liquid phase, and
{tan-1[eb(ψs1-ψm) - 1]0.5 + -1
tan [e
b(ψs2-ψm)
d2uy
- 1] } - D ) 0 (8)
The value of Xm can be estimated by
Xm )
[
]
b ebψm 2(a + b)
-0.5
tan-1[eb(ψs1-ψm) - 1]0.5
(9) uy )
In the last case h e hs1 and φs1 < φs2, it can be shown that
{ {[ [
( ) | ]×
b(a + b) dψ 2 1 ψ ) ln ebψs1 b 2 dX sec2 -
tan-1
( )|
( )|
b b2 dψ 2 ebψs1 4 dX 2(a + b)
{ [ [
]
]
[
] }}
×
( )
( )
]}
(10)
[
]
1]
[
where Ca and Cb are respectively the concentrations of cations and anions, uy,a and uy,b are respectively the velocities of cations and anions, ma and mb are respectively the mobilities of cations and anions, and NAV is the Avogadro number. The mean liquid velocity uy can be evaluated by
uy )
]
0.5
-0.5
-1
{tan [e
-1
+ tan [e
]
b ebψs1 2(a + b)
-0.5
b(ψs1-ψmp)
b(ψs2-ψmp)
∫0huy dx
1 h
(17)
Substituting eq 15 into this expression yields
uy ) -
Ey h2 dp Ey (φs1 + φs2) + 12η dy 2η ηh
∫0hφ dx
(18)
Similarly, the mean current density iy can be calculated by
Note that in this case, although surface 1 has a positive potential, it is negatively charged. This is because if two surfaces are sufficiently close to each other, the sign of the charge on the surface having a lower potential will be reversed, a consequence of applying Gauss’s law. The values of hmp and hs1 can be derived from eqs 8 and 11, respectively. We obtain
b Dmp ) κhmp ) 2 ebψmp 2(a + b)
]
for 0 e X e D
-0.5
X)0
FEy (amae-aFφ/RT + NAV
bmbebFφ/RT) (16)
0.5 ebψs2 -1 b(a + b) dψ 2 ebψs1 | 2 dX X)0 0.5 ebψs1 -1 - D ) 0 (11) b(a + b) dψ 2 bψs1 e | 2 dX X)0
Ds1 ) κhs1 )
The electrical current density, iy, can be expressed as
0.5
tan-1
tan-1
]
) bC0bF (e-aFφ/RT - ebFφ/RT)uy +
(D - X) +
where (dψ/dX)2|X)0 is the root of the equation
[
[
1 dp 2 h dp Ey x + + (φ - φs2) x 2η dy 2η dy ηh s1 Ey Ey φs1 + φ (15) η η
) aCa(uy + amaFEy/NAV) - bCb(uy - bmbFEy/NAV)
0.5
X)0
ebψs2 -1 b(a + b) dψ 2 ebψs1 |X)0 2 dX
( )
(14)
iy ) aCauy,a - bCbuy,b
X)0
b2 dψ 2 b ebψs1 4 dX 2(a + b)
dx2
d2φ dp E - )0 2 y dy dx
-
the longitudinal coordinate y is chosen so that Ey g 0. Assuming nonslip conditions on planar surfaces, it can be shown that
The electric field in the region Xm e X e D can be recovered by replacing respectively X and ψs1 with (D - X) and ψs2 in eq 7.
[
η
0.5
iy )
- 1] } (12) 0.5
tan-1[eb(ψs2-ψs1) - 1]0.5 (13)
Flow Field. Suppose that the liquid phase is a Newtonian fluid with constant physical properties, and the flow is in the laminar regime. Therefore, the y-component of the liquid
(19)
Substituting eq 16 into this expression gives
[∫
bC0bF iy ) h
-
∫0hiy dx
1 h
h
0
(e-aFφ/RT - ebFφ/RT)uy dx + FEy NAV
∫0h(amae-aFφ/RT + bmbebFφ/RT) dx
]
(20)
The influence of the nature of the electrical double layer on the electrokinetic flow under consideration is examined through numerical simulation based on the approximate analytical results derived. For two identical surfaces, φmp e φs1 ) φs2, and only two cases need to be considered: (a) hmp e h (φm e φmp) and (b) h e hmp (φmp e φm). In this case, Xmp1 + Xmp2 ) D, Xm ) D/2,
11874 J. Phys. Chem. B, Vol. 108, No. 31, 2004
Lin et al.
and it can be shown that for the first case the electrical field is described by
{[ {[ [
]
b(a + b) 2 1 (ψmp - ψ2m) × ψ ) ln ebψmp b 2 sec2 -
tan-1
b b2 ebψmp - (ψ2mp - ψ2m) 4 2(a + b)
]
0.5
X+
] }}
0.5 ebψs -1 b(a + b) 2 (ψmp - ψ2m) ebψmp 2 for 0 e X e Xmp1 (21)
ψ ) ψm cosh(X - D/2) for Xmp1 e X e D/2
(22)
where Xmp1 ) (D/2) - sech-1(ψm/ψmp), and ψm is the root of the equation
[
]
-0.5 b b2 ebψm - (ψ2mp - ψ2m) 4 2(a + b) 0.5 ebψs tan-1 -1 b(a + b) 2 (ψmp - ψ2m) ebψmp 2 0.5 ebψmp tan-1 + -1 b(a + b) 2 bψmp 2 (ψmp - ψm) e 2 D sech-1(ψm/ψmp) - ) 0 (23) 2
{ [ [
]
]}
In the second case, it can be shown that
{[
{
] }}
b 1 ebψm ψ ) ln ebψm sec2 b 2(a + b) tan-1[eb(ψs-ψm) - 1]0.5
0.5
X+
for 0 e X e D/2 (24)
where ψm is the root of the equation
[
]
b ebψm 2(a + b)
-0.5
tan-1[eb(ψs-ψm) - 1]0.5 -
D )0 2
(25)
The value of Dmp can be determined from this expression as
Dmp ) κdmp ) 2
[
]
b ebψmp 2(a + b)
-0.5
tan-1[eb(ψs-ψmp) - 1]0.5 (26)
Also, eqs 15 and 18 can be simplified respectively to
uy )
Ey Ey 1 dp 2 h dp x xφ + φ 2η dy 2η dy η s1 η
(27)
and
uy ) -
h2 dp Eyφs1 Ey + 12η dy η ηh
∫0hφ dx
(28)
3. Results and Discussions The applicability of the present approach is justified in Figure 2, where the simulated spatial variations in the scaled electrical potential Fφ/RT and the magnitude of fluid velocity uy at various separation distances between two surfaces are illustrated. The
Figure 2. Simulated spatial variation in scaled electrical potential Fφ/ RT (a) and magnitude of fluid velocity uy (b) at various κh for the case when Fφs1/RT ) 1, Fφs2/RT ) 10, a:b ) 2:3, C0b ) 1 × 10-4 M, and κ-1 ) 1.122 × 10-8 m. Curve 1: κh ) 0.5, 2, κh ) 2, 3, κh ) 4. Solid curves show present model, while discrete symbols show exact numerical results. Key: Ey ) 1 × 103 V m-1, T ) 298.15 K, η ) 1 × 10-3 kg m-1 s-1, ) 7.1 × 10-10 F m-1.
corresponding results based on the exact numerical solutions are also presented for comparison. This figure reveals that the performance of the present approach is satisfactory. Figure 2b suggests that an increase in the separation distance between two surfaces has the effect of increasing the maximal value of |uy|. This is expected because the friction force acting on the bulk liquid decreases with the increase in the separation distance between two surfaces. The influence of the bulk concentration of anions (counterions) C0b on the absolute value of the mean velocity of liquid, | uy|, is illustrated in Figure 3. In general, |uy| increases with the increase in C0b, reaches a maximum value, and then decreases with a further increase in C0b. It is interesting to note that |uy| exhibits an undulant behavior when C0b exceeds about 1 × 10-3 M. As C0b increases, the electrical force acting on the electrolyte solution from the applied electrical field increases. However, because the thickness of the double layer decreases, the region influenced by the applied electrical field decreases accordingly. The specific behavior observed in Figure 3 is the net result of these two competing factors. Figure 3 also shows that |uy| increases with the increase in the surface potential, which is expected because the higher the surface potential the higher the absolute value of the space charge density in the double layer, |aCa - bCb|. This leads to a greater electrical force acting on the electrolyte solution, and, therefore, a higher mean velocity.
Flow of an Electrolyte Solution in a Microchannel
Figure 3. Variation of |uy| as a function of C0b at various Fφs1/RT for the case when φs2 ) φs1, h ) 2 × 10-7 m, and a:b ) 1:1. Curve 1: Fφs1/RT ) 1. Curve 2: Fφs1/RT ) 2. Curve 3: Fφs1/RT ) 3. Key: same as in Figure 2.
J. Phys. Chem. B, Vol. 108, No. 31, 2004 11875
Figure 5. Variation of |uy| as a function of Fφs2/RT at various C0b for the case when Fφs1/RT ) 1, h ) 2 × 10-7 m, and a:b ) 1:1. Curve 1: C0b ) 1 × 10-4 M. Curve 2: C0b ) 2 × 10-4 M. Curve 3: C0b ) 5 × 10-4 M. Key: same as in Figure 2.
the greater the difference between the surface potentials of two surfaces, the larger the absolute value of the mean velocity. This is because raising the potential of a surface has the effect of increasing the mean velocity, as is discussed in Figure 4. 4. Conclusion
Figure 4. Variation of |uy| as a function of b for various Fφs1/RT for the case when φs2 ) φs1, h ) 2 × 10-7 m, and a ) 1. Curves 1 and 1′: Fφs1/RT ) 1. Curves 2 and 2′, Fφs1/RT ) 2. C0b ) 1 × 10-4 M in curves 1 and 2, C0b ) 1 × 10-3 M in curves 1′ and 2′. Key: same as in Figure 2.
The influence of the valence of anions (counterions) on the absolute value of the mean velocity |uy| at fixed a and C0b is illustrated in Figure 4. It is interesting to observe that depending upon the level of C0b, |uy| may have a local maximum or local minimum as b varies. This is because if both the valence of cations (coions) a and C0b are fixed, the concentration of cations increases with b. The increase in C0a has the effect of decreasing the mean velocity, and the increase in b has the effect of increasing the mean velocity. Also, because the thickness of the double layer decreases with the increase of b, the region influenced by the applied electrical field decreases. The behavior of |uy| shown in Figure 4 reflects the net result of these effects. Figure 5 shows the variation of the absolute value of the mean velocity |uy| as a function of Fφs2/RT for various levels of C0b. Note that since Fφs1/RT is fixed, this figure shows the influence of the difference between the surface potentials of two surfaces on the flow rate of electrolyte solution. Figure 5 suggests that
In summary, the electrokinetic flow of a general electrolyte solution in a planar slit comprising two planar, parallel surfaces, which can have different charged conditions, is analyzed. Adopting a matching method, analytical expressions for the electric field and the velocity field are derived. The results of numerical simulation reveal the following: (a) An increase in the separation distance between two surfaces has the effect of increasing the maximal fluid velocity. (b) In general, the absolute value of the mean fluid velocity increases with the increase in the bulk concentration of counterions, reaches a maximum value, and then decreases with a further increase in that concentration. (c) The absolute value of the mean fluid velocity may exhibit an undulant behavior as the bulk concentration of counterions varies. (d) The absolute value of the mean fluid velocity increases with the level of surface potential. (e) If both the valence of coions and the bulk concentration of counterions are fixed, then depending upon the level of the latter the absolute value of the mean fluid velocity may have a local maximum or local minimum as the valence of counterions varies. Acknowledgment. This work is supported by the Department of Economics of the Republic of China under grant 92EC-17-A-09-S1-019 and the National Science Council of the Republic of China. References and Notes (1) Hunter, R. J. Foundations of Colloid Science; Oxford University Press: New York, 1992; Vol. I. (2) Tseng, S.; Kao, C. Y.; Hsu, J. P. Electrophoresis 2000, 21, 3541. (3) Ohshima, H.; Kondo, T. J. Colloid Interface Sci. 1990, 135, 443. (4) Hsu, J. P.; Kao, C. Y. J. Phys. Chem. B 2001, 105, 8135. (5) Hsu, J. P.; Kao, C. Y.; Tseng, S.; Chen, C. J. J. Colloid Interface Sci. 2002, 248, 176. (6) Luo, G.; Feng, R.; Jin, J.; Wang, H. P. J. Colloid Interface Sci. 2001, 241, 81. (7) Langmuir, I. J. Chem. Phys. 1938, 6, 873. (8) Hsu, J. P.; Lin, S. H.; Tseng, S. J. Phys. Chem. B 2004, 108, 4495.
