Diffusioosmosis of electrolyte solutions in fine capillaries

Colloids and Surfaces A: Physicochem. Eng. Aspects 233 (2004) 87–95

Diffusioosmosis of electrolyte solutions in fine capillaries Huan J. Keh∗ , Hsien Chen Ma Department of Chemical Engineering, National Taiwan University, Taipei 106-17, Taiwan, ROC Received 14 August 2003; accepted 18 November 2003

Abstract An analytical study is presented for the diffusioosmotic flow of an electrolyte solution in a narrow capillary tube or slit caused by a constant concentration gradient imposed along the axial direction at the steady state. The electric double layer adjacent to the charged capillary wall may have an arbitrary thickness relative to the capillary radius. The electrostatic potential distribution on a cross section of the capillary is obtained by solving the linearized Poisson–Boltzmann equation, which applies to the case of low surface potential at the capillary wall. The macroscopic electric field induced by the imposed electrolyte concentration gradient through the capillary is determined as a function of the radial position. Closed-form formulas for the fluid velocity profile due to the combination of electroosmotic and chemiosmotic contributions are derived as the solution of a modified Navier–Stokes equation. The diffusioosmotic velocity can have more than one reversal in direction over a small range of the surface potential. For a given concentration gradient of electrolyte in a capillary, the fluid-flow rate is not necessarily to increase with an increase in the electrokinetic radius of the capillary, which is the capillary radius divided by the Debye screening length. The effect of the radial distribution of the induced axial electric field in the double layer on the diffusioosmotic flow is found to be of dominant significance in most practical situations. © 2003 Elsevier B.V. All rights reserved. Keywords: Diffusioosmosis; Electroosmosis; Capillary tube; Capillary slit

1. Introduction The flow of electrolyte solutions in a small pore with a charged wall is of much fundamental and practical interest in various areas of science and engineering. In general, driving forces for this electrokinetic flow include dynamic pressure differences between the two ends of the pore (a streaming potential is developed as a result of zero net electric current) and tangential electric fields that interact with the electric double layer adjacent to the pore wall (electroosmosis). Problems of fluid flow in porous media caused by these well-known driving forces were studied extensively in the past century [1–10]. Another driving force for the electrokinetic flow in a micropore, which has commanded less attention, involves tangential concentration gradients of an ionic solute that interacts with the charged pore wall. This solute–wall interaction is electrostatic in nature and its range is the Debye screening length κ−1 (defined byEq. (4)). The ∗ Corresponding author. Tel.: +886-2-2363-5462; fax: +886-2-2362-3040. E-mail address: [email protected] (H.J. Keh).

0927-7757/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.colsurfa.2003.11.014

fluid motion associated with this mechanism, known as “diffusioosmosis”, has been analytically examined for solutions near a plane wall [4,11–14] and inside a capillary tube [15–17]. Electrolyte solutions with a concentration gradient of order 100 kmol/m4 along solid surfaces with a zeta potential of order kT/e (∼25 mV; e is the charge of a proton, k the Boltzmann constant, and T the absolute temperature) can flow by diffusioosmosis at velocities of several micrometers per second. The formula for the diffusioosmotic velocity of electrolyte solutions parallel to a charged plane wall [11,12] can be applied to the corresponding flow in capillary tubes and slits when the thickness of the double layer adjacent to the capillary wall is small compared to the capillary radius. However, in some practical applications involving dilute electrolyte solutions in very fine capillaries, this condition is no longer satisfied and the dependence of the fluid flow on the electrokinetic radius κR or κh, where R is the radius of a capillary tube and h the half thickness of a capillary slit, must be taken into account. Recently, the diffusioosmotic flow of electrolyte solutions in a capillary tube or slit with an arbitrary value of κR or κh was theoretically investigated for the case of small surface potential at the capillary wall, and

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analytical formulas for the fluid velocity profiles on cross sections of the capillary tube and slit were obtained [18]. In this study, however, the effect of radial distributions of the counterions and coions on the local macroscopic electric field induced by the imposed electrolyte concentration gradient in the axial direction inside the double layer, which can be important, was neglected. In the present work, we present a comprehensive analysis of the diffusioosmosis of an electrolyte solution with a constant prescribed concentration gradient through a capillary tube or slit. The electric potential at the capillary wall is assumed to be uniform and low, but no assumption is made about the electrokinetic radius κR or κh and the radial distribution of the induced axial electric field is allowed. Closed-form expressions for the fluid velocity profiles on cross sections of the capillary tube and slit are obtained in Eqs. (20) and (36), respectively. These results show that the effect of the deviation of the induced axial electric field in the double layer from its bulk-phase quantity on the diffusioosmotic velocity of the fluid is dominantly significant in most practical situations.

2. Diffusioosmosis in a circular capillary In this section, we consider the diffusioosmotic flow of a fluid solution of a symmetrically charged binary electrolyte of valence Z (where Z is a positive integer) in a straight capillary tube of radius R and length L with R  L, as illustrated in Fig. 1(a), at the steady state. The discrete nature of the surface charges, which are uniformly distributed over the capillary wall, is ignored. The applied electrolyte concentration gradient ∇n∞ is a constant along the axial (z) direction in the capillary, where n∞ (z) is the linear concentration distribution of the electrolyte in the bulk solution phase in equilibrium with the fluid inside the capillary. The electrolyte ions can diffuse freely in the capillary, so there

R

r z

∇n ∞

∇n ∞

2.1. Electric potential distribution We first deal with the electrostatic potential distribution in the electrolyte solution on a cross section of the capillary tube. If ψ(ρ) represents the electrostatic potential at a point with distance ρ from the axis of the tube relative to that in the bulk solution, and n+ (ρ, z) and n− (ρ, z) denote the local concentrations of the cations and anions, respectively, then Poisson’s equation has the form:
 1 d dψ 4πZe ρ =− [n+ (ρ, 0) − n− (ρ, 0)]. (1) ρ dρ dρ ε In the above equation, ε = 4πε0 εr , where εr the relative permittivity of the electrolyte solution and ε0 the permittivity of a vacuum. The local ionic concentrations can also be related to the electrostatic potential by the Boltzmann equation,
 Zeψ ∞ n± = n exp ∓ . (2) kT Substitution of Eq. (2) into Eq. (1) leads to the well-known Poisson–Boltzmann equation. For small values of ψ, the Poisson–Boltzmann equation can be linearized (known as the Debye–Huckel approximation), and Eq. (1) becomes:
 1 d dψ ρ = κ2 ψ, (3) ρ dρ dρ where κ is the reciprocal of the Debye screening length defined by:  1/2 8π(Ze)2 ∞ κ= . (4) n (0) εkT The boundary conditions for ψ are

(a)

h

y

exists no regular osmotic flow of the solvent. It is assumed that L|∇n∞ |/n∞ (0)  1, where z = 0 is set at the midpoint through the capillary. Thus, the variation of the electrostatic potential (excluding the macroscopic electric field induced by the electrolyte gradient, which will be discussed in Section 2.2) and ionic concentrations in the electric double layer adjacent to the capillary wall with the axial position can be neglected.

z

(b) Fig. 1. Geometrical sketch for the diffusioosmotic flow in a straight capillary due to a concentration gradient of the electrolyte: (a) a capillary tube; (b) a capillary slit.

ρ=0:

dψ = 0, dρ

(5a)

ρ=R:

ψ = ζ,

(5b)

where ζ is the zeta potential of the capillary wall (at the shear plane). The solution to Eqs. (3) and (5) is [3]: ψ=ζ

I0 (κρ) , I0 (κR)

(6)

where In is the modified Bessel function of the first kind of order n. Due to the linearization of the Poisson–Boltzmann

H.J. Keh, H.C. Ma / Colloids and Surfaces A: Physicochem. Eng. Aspects 233 (2004) 87–95

equation, the normalized potential distribution ψ/ζ is independent of the dimensionless parameter Zeζ/kT. Evidently, ψ/ζ = 1 at any position in the limiting case κR = 0, and ψ/ζ = 0 at an arbitrary position with ρ = R in the limit κR → ∞. If the surface charge density σ, instead of the surface potential ζ, is known at the capillary wall, the boundary condition specified by Eq. (5b) should be replaced by the Gauss condition, ρ=R:

dψ 4πσ = . dρ ε

(7)

The solution for ψ given by Eq. (6) still holds for this condition, with the relation between ζ and σ for an arbitrary value of κR as 4πσ I0 (κR) ζ= . (8) εκ I1 (κR) For a capillary tube with a given radius, Eq. (8) predicts that ζ decreases with an increase in κR for the case of constant surface charge density and σ increases with an increase in κR for the case of constant surface potential.

Since the ionic concentrations n+ and n− in the capillary are not uniform in both axial (z) and radial (ρ) directions, their prescribed gradients in the axial direction (∇n∞ ) can give rise to a “diffusion current” distribution on a cross section of the capillary. To prevent a continuous separation of the counter-ions and co-ions, an electric field distribution E along the axial direction arises spontaneously in the electrolyte solution to produce another electric current distribution which exactly balances the diffusion current [4,12,19]. If the electrolyte solution is dilute, the flux of either ionic species is given by the Nernst–Planck equation,   Ze J± = −D± ∇n± ± (9) n± (∇ψ − E) , kT where D+ and D− are the diffusion coefficients of the cations and anions, respectively, and the principle of superposition for the electric potential is used. In order to have no current arising from the cocurrent diffusion of the cations and anions, one must require that J+ = J− = J (the radial component of J vanishes and the ionic fluxes induced by ∇ψ in Eq. (9) are balanced by the radial components of the diffusive ionic fluxes as required by the Boltzmann distribution given by Eq. (2) for the cations and anions). Applying this constraint to Eq. (9), we obtain 
kT G+ − G− ∇n∞ E= , (10) Ze G+ + G− n∞ (0) where


 Zeψ G± = D± exp ∓ . kT

The positive coefficients G+ and G− defined by the above equation reflect the fact of an increase in the axial diffusive flux of the counter-ions and a decrease in the flux of the co-ions inside the electric double layer. Evidently, the induced electric filed E on a cross section of the capillary given by Eq. (10) depends on the local electrostatic potential ψ. Substitution of Eqs. (10) and (11) into Eq. (9) leads to a net flux distribution of the electrolyte, J = −D∇n∞ ,

(11)

(12)

where the net diffusivity which is a function of the local potential or the radial position in the capillary is: 2G+ G− . (13) D= G+ + G − Eqs. (10) and (12) illustrate clearly that both E and J are collinear with ∇n∞ , which is in the axial direction. For small values of ψ, a Taylor expansion applies to Eq. (11), and Eqs. (10) and (13) can be expressed as:   kT Zeψ ∇n∞ 2 2 E= β + (β − 1) + O(ψ ) ∞ (14) Ze kT n (0) and D=

2.2. Induced electric field distribution

89

  Zeψ 2D+ D− 1+β + O(ψ2 ) , D+ + D − kT

where D+ − D− . β= D+ + D −

(15)

(16)

Obviously, −1 ≤ β ≤ 1. Note that, even if the cation and anion diffusion coefficients are identical (i.e., β = 0, the O(ψ2 ) term in Eq. (14) for the induced electric field E still exists (due to the adsorption of the counterions and depletion of the coions near the capillary wall) and equals −ψ∇n∞ /n∞ . In a previous study of the diffusioosmosis of electrolyte solutions in a capillary [18], only the first term in the parenthesis of Eq. (14) was considered for E (the bulk-phase electrostatic potential ψ = 0 is taken everywhere), and thus, the effect of the radial distribution of the induced electric field on the fluid velocity was not allowed. 2.3. Fluid velocity distribution We now consider the steady flow of an electrolyte solution in a capillary tube under the influence of a constant concentration gradient ∇n∞ of the electrolyte prescribed axially. The momentum balances on the Newtonian fluid in the ρ and z directions give: ∂p dψ + Ze(n+ − n− ) = 0, (17a) ∂ρ dρ
 du η d ∂p ρ = − Ze (n+ − n− ) |E|, (17b) ρ dρ dρ ∂z where u(ρ) is the fluid velocity in the positive z direction (axial direction of increasing electrolyte concentration), p(ρ, z) the pressure, η the viscosity of the fluid, and the macroscopic electric field E(ρ) induced by the imposed concen-

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tration gradient of the electrolyte is given by Eqs. (10) and (14). The boundary conditions for u are ρ=0:

du = 0, dρ

(18a)

ρ=R:

u = 0.

(18b)

After the substitution of Eq. (2) for n± into Eq. (17a) and the application of the Debye–Huckel approximation, the pressure distribution can be determined as: p = p0 +

n∞ (z) (Ze)2 {[ψ(ρ)]2 − [ψ(0)]2 } + O(ζ 3 ), kT

(19)

where p0 is the pressure on the axis of the capillary tube, which is a constant in the absence of applied pressure gradient. Note that the pressure distribution in the ρ and z directions results from the variation of the electrostatic potential ψ with ρ and the variation of the bulk concentration n∞ with z, respectively. Substituting the ionic concentration distributions of Eq. (2), the electrostatic potential distribution of Eq. (6), the pressure profile of Eq. (19), and the electric field profile of Eq. (14) into Eq. (17b), and solving for the fluid velocity subject to the boundary conditions in Eq. (18), we obtain:  2Ze Ze 2 ∞ u = − 2 |∇n | βζΘ1 (κR, ρ/R) + ζ [Θ2 (κR, ρ/R) 8kT ηκ  + 2(β2 − 1)Θ3 (κR, ρ/R)] + O(ζ 3 ) , (20) where Θ1 , Θ2 , and Θ3 are functions given by: Θ1 = 1 −

I0 (κρ) , I0 (κR)

(21a)

Θ2 =

1 [F(κR) − F(κρ) − κ2 (R2 − ρ2 )], [I0 (κR)]2

(21b)

Θ3 =

1 [F(κR) − F(κρ)], [I0 (κR)]2

(21c)

and the function F is defined as: F(x) = x2 I0 (x)[I0 (x) + I2 (x)] − 2[xI1 (x)]2 .

(22)

The average fluid velocity u over a cross section of the capillary tube can be expressed as:  Ze 2 2Ze ∞ ζ [Θ2  u = − 2 |∇n | βζΘ1  + 8kT ηκ  + 2(β2 − 1)Θ3 ] + O(ζ 3 ) , (23) where the definition of the angle brackets is:  2 R u = 2 u(ρ)ρ dρ. R 0

(24)

It is easy to find that Θ1  = 1 −

2I1 (κR) , κRI0 (κR)

(25)

but the explicit forms of Θ2  and Θ3  are quite complicated and they can be determined by numerical integration. It can be found from Eq. (21) that all the functions Θ1 , Θ2 , and Θ3 are always positive. Evidently, Θ1 = Θ2 = Θ3 = 0 and u = 0 at the capillary wall (ρ/R = 1) due to the no-slip requirement of Eq. (18b). In the limiting situation that κR = 0 (there is no interaction between the solute ions and the capillary wall), Eq. (21) gives Θ1 = Θ2 = Θ3 = 0 at any position in the tube, which also leads to Θ1  = Θ2  = Θ3  = 0. Note that Θ3 /(κR)2 = 4Θ1 /(κR)2 = 1 − (ρ/R)2 and Θ3 /(κR)2 = 4Θ1 /(κR)2 = 1/2, although Θ2 /(κR)2 = Θ2 /(κR)2 = 0, as κR = 0; thus, the diffusioosmotic flow caused by the induced electric field E still exists (with a parabolic profile) for this case with a finite radius R of the capillary, as shown in Eq. (20). In the other limiting situation that κR → ∞, Eq. (21) yields Θ1 = Θ2 = Θ3 = 1 at an arbitrary position with ρ/R = 1 (so Θ1  = Θ2  = Θ3  = 1, too). It is surprising and interesting that Θ3 , which reflects the effect of the radial distribution of the induced axial electric field in the electric double layer, does not vanish as κR → ∞, where ψ = 0 at any position other than ρ/R = 1 (but the solute–wall interaction is very strong). For all cases of finite κR, Θ3 and Θ3  are comparable with (especially when κR is large) but always greater than Θ2 and Θ2 , respectively. As it is well known, the diffusioosmosis of an electrolyte solution in a capillary pore results from a linear combination of two effects: “chemiosmosis” due to the non-uniform adsorption of counterions and depletion of coions in the electric double layer over the charged surface and “electroosmosis” due to the macroscopic electric field caused by the imposed concentration gradient of the electrolyte and given by Eqs. (10) and (14). The terms in Eqs. (20) and (23) involving the functions Θ1 and Θ3 represent the contribution from electroosmosis, while the remainder terms (containing the function Θ2 ) are the chemiosmotic component. Note that the direction of the local or average fluid velocity caused by electroosmosis is determined by the combination of the parameters κR, Zeζ/kT, and β, while the fluid flow generated by chemiosmosis for small values of ζ is always in the direction of decreasing electrolyte concentration. 2.4. Results and discussion The functions Θ1 , Θ2 , and Θ3 given by Eq. (21) determine the diffusioosmotic velocity of a symmetric electrolyte in a capillary tube correct to O(ζ 2 ). The numerical results of Θ1 and Θ2 have already been given in the literature [18]. In Fig. 2, the function Θ3 is plotted versus the dimensionless coordinate ρ/R for various values of κR. As expected, for a specified value of κR, Θ3 is a monotonically decreasing function of ρ/R from a maximum at the axis of the tube (with ρ = 0 to zero at the tube wall (with ρ = R). For a fixed value of ρ/R, Θ3 is not a monotonic function of κR, due to the competition between the effect of the deviation of the local induced axial electric field from its bulk-phase

H.J. Keh, H.C. Ma / Colloids and Surfaces A: Physicochem. Eng. Aspects 233 (2004) 87–95

1.2

91

Q 3 ( r = 0)

1.2

∞ 10

Q3

5

0.8

0.8

3

Q3

Q 2 ( r = 0)

1

0.4

Q2

0.4

0.5 kR = 0

0

0 0

0.2

0.4

0.6

0.8

1

0.1

1

10

100

kR

r/R Fig. 2. Plots of the function Θ3 as calculated from Eq. (21c) vs. the normalized coordinate r/R for various values of the electrokinetic radius κR.

quantity (strong as κR → 0 and weak as κR → ∞) and the effect of the solute–wall interaction (strong as κR → ∞ and weak as κR → 0), and has a maximal value which is greater than unity at some value of κR. The maximum in Θ3 at the axis of the tube occurs in the vicinity of κR = 2.48 and its location shifts to larger κR as ρ/R increases. When the tube wall is approached (ρ/R → 1), this maximum takes place at ρ/R → ∞ and its value is close to unity. The values of Θ3 at the axis of the tube and of the averaged Θ3  as functions of κR are presented in the first three columns of Table 1 and plotted in Fig. 3. The corresponding values of Θ2 and Θ2  are also illustrated in Fig. 3 for comparison. It can be seen that both functions Θ2  and Θ3  increase monotonically with an increase in κR. As expected, Θ2  and Θ3  are smaller than Θ2 (ρ = 0) and Θ3 (ρ = 0), respectively, for an arbitrary finite value of κR. Although the values of Θ2 (ρ = 0) and Θ3 (ρ = 0) are greater than unity for cases with κR greater than ∼3.2 and 1.35, respectively, the values of Θ2  and Θ3  cannot be greater than unity for any given value of κR. Table 1 Values of Θ3 (ρ = 0), Θ3 , Φ3 (y = 0), and Φ3  for various values of κR or κh κR or κh

Θ3 (ρ = 0)

Θ3 

Φ3 (y = 0)

Φ3 

0 0.2 0.5 1 2 3 4 5 10 20 50 ∞

0 0.0394 0.2281 0.7087 1.3138 1.3307 1.2303 1.1595 1.0597 1.0271 1.0103 1

0 0.0197 0.1152 0.3691 0.7625 0.8810 0.9072 0.9186 0.9531 0.9757 0.9901 1

0 0.0774 0.4101 1.0000 1.2116 1.0789 1.0201 1.0044 1.0000 1.0000 1.0000 1

0 0.0517 0.2757 0.6892 0.9121 0.8884 0.8887 0.9029 0.9500 0.9750 0.9900 1

Fig. 3. Plots of the functions Θ2 (ρ = 0), Θ3 (ρ = 0), Θ2 , and Θ3  vs. the electrokinetic radius κR.

The dependence of the diffusioosmotic velocity of an electrolyte solution at the axis of a capillary tube on the dimensionless zeta potential of the tube wall at various values of κR is displayed in Fig. 4. The value of this fluid velocity calculated by Eq. (20) is normalized by a characteristic value given by:
 −ε|∇n∞ | kT 2 2kT ∗ U = = − 2 |∇n∞ | (26) 4πηn∞ (0) Ze ηκ The negative nature of this characteristic velocity, which is dependent not only on the value of |∇n∞ | but also on the value of n∞ (0) or κ, means that its direction is opposite to that of the concentration gradient of the electrolyte. Fig. 4(a) is drawn for the case of an electrolyte that the cation and anion diffusivities are equal (β = 0 , representing the aqueous solution of KCl if Z = 1). Only the results at positive surface potentials are shown because the fluid velocity, which is due to the O(ζ 2 ) contribution entirely as illustrated by Eq. (20), is an even function of the zeta potential. Since our analysis is based on the assumption of small surface potential, the magnitudes of Zeζ/kT considered are <2. It can be seen from the O(ζ 2 ) term of Eq. (20) that the fluid flows contributed from chemiosmosis (involving the function Θ2 ) and electroosmosis (involving Θ3 ) are always in the opposite directions, and for the case of β = 0, the net flow is dominated by the electroosmotic effect (having the direction of increasing electrolyte concentration), even when the electric double layer is very thin or κR → ∞. Fig. 4(a) shows that the magnitude of the reduced diffusioosmotic velocity u(ρ = 0)/U ∗ (or u/U∗ at any other location, or the reduced average fluid velocity u/U ∗ , which are not graphically presented here for conciseness) increases monotonically with the increase of Zeζ/kT for a given value of κR and, similar to the function Θ3 (ρ = 0) illustrated in Figs. 2 and 3, is not a monotonic function of κR for a fixed value of Zeζ/kT. There is no diffusioosmotic motion of the fluid

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negative to positive values. The reversals occurring at the values of Zeζ/kT other than zero result from the combination of the contributions from electroosmosis of O(ζ) and O(ζ 2 ) (which is still dominant in many situations) and chemiosmosis of O(ζ 2 ). Again, for a specified value of Zeζ/kT, u(ρ = 0)/U ∗ (or u/U∗ at any other location, or the average velocity u/U ∗ ) is not necessarily a monotonic function of κR.

kR = 0

0

0.5 ∞

-0.4

5

u ( r = 0) U*

1

3. Diffusioosmosis in a capillary slit

-0.8

2

5

In this section, we consider the diffusioosmotic flow of a symmetric electrolyte solution in a narrow capillary channel between two identical parallel charged plates with a separation distance 2h, as shown in Fig. 1(b), at the steady state. Owing to the planar symmetry of the system, we need consider only the half region 0 ≤ y ≤ h, where y is the distance from the median plane between the slit walls in the normal direction. The analysis for this case is similar to that for the case of diffusioosmosis in a capillary tube presented in the previous section.

1

3.1. Electric potential distribution

-1.2 0

0.4

0.8

(a)

1.2

1.6

2

Zez / kT

∞

0

kR = 0

-0.4

For the electrolyte solution in the capillary slit, let ψ(y) be the electrostatic potential at the position y and n+ (y, z) and n− (y, z) be the local concentrations of the cations and anions, respectively. Now, Poisson’s equation gives

1

2

u ( r = 0) U*

∞

-0.8

5

d2 ψ 4πZe =− [n+ (y, 0) − n− (y, 0)], 2 ε dy

2 -1.2 -2

(b)

-1

0

1

2

Zez / kT

Fig. 4. Plots of the normalized diffusioosmotic velocity at the axis of a capillary tube vs. the dimensionless surface potential Zeζ/kT for various values of κR: (a) β = 0; (b) β = −0.2.

for the special case of Zeζ/kT = 0. In a previous study of the same diffusioosmosis [18], the O(ζ 2 ) contribution from electroosmotic effect was not considered, and the resulted fluid velocity (which is due to chemiosmotic effect only) is in the direction of decreasing electrolyte concentration. In Fig. 4(b), the reduced diffusioosmotic velocity u(ρ = 0)/U ∗ as a function of Zeζ/kT at various values of κR is plotted for the case of a symmetric electrolyte whose cation and anion have different diffusion coefficients (β = −0.2 is chosen, representing the aqueous solution of NaCl if Z = 1). In this case, the diffusioosmotic velocity is neither an even nor an odd function of the zeta potential of the tube wall. It can be seen that the fluid velocity is not necessarily a monotonic function of Zeζ/kT given a constant value of κR. Some of the curves in Fig. 4(b) indicate that the fluid might reverse direction of flow more than once as the zeta potential of the capillary wall varies in a narrow range from

(27)

where z = 0 is set at the midpoint through the capillary. Substitution of Eq. (2) for n+ and n− into the above equation and application of the Debye–Huckel approximation lead to the linearized form d2 ψ = κ2 ψ, dy2

(28)

where the Debye parameter κ is defined by Eq. (4). Now, the boundary conditions for ψ are y=0:

dψ = 0, dy

(29a)

y=h:

ψ = ζ.

(29b)

The solution to Eqs. (28) and (29) is: ψ=ζ

cosh(κy) . cosh(κh)

(30)

Analogous to the case of a capillary tube, ψ/ζ = 1 at any position in the limit κh = 0, and ψ/ζ = 0 at any position other than y = h in the limit κh → ∞. If the Gauss condition at the capillary wall, y=h:

4πσ dψ = , dy ε

(31)

H.J. Keh, H.C. Ma / Colloids and Surfaces A: Physicochem. Eng. Aspects 233 (2004) 87–95

is used to replace Eq. (29b), the solution for ψ given by Eq. (30) is still valid, with ζ=

4πσ cosh(κh) εκ sinh(κh)

(32)

For a constant value of σ, ζ is a decreasing function of κh; for a constant value of ζ, σ is an increasing function of κh. 3.2. Fluid velocity distribution The momentum equations for the steady flow of an electrolyte solution in a capillary slit with an applied concentration gradient of the electrolyte parallel to the slit walls are: dψ ∂p + Ze(n+ − n− ) =0 ∂y dy η

∂p d2 u = − Ze(n+ − n− ) |E| ∂z dy2

(33a)

(33b)

where u(y) is the fluid velocity profile in the direction of the electrolyte concentration gradient and p(y, z) is the pressure distribution. The induced electric field profile E(y) is still given by Eqs. (10) and (14) with G± and β defined by Eqs. (11) and (16). Here, the boundary conditions for u are: y=0

du = 0, dy

(34a)

u = 0.

(34b)

y=h:

Substituting the linearized form of Eq. (2) into Eq. (33a), we solve for the pressure distribution to obtain: p = p0 +

n∞ (z) (Ze)2 {[ψ(y)]2 − [ψ(0)]2 } + O(ζ 3 ) kT

(35)

where p0 is the pressure on the midplane between the slit walls. The fluid velocity u can be solved from Eqs. (33b) and (34) by the substitution of Eqs. (2), (14), (30) and (35), with the result  Ze 2 2Ze ∞ ζ [Φ2 (κh, y/ h) u = − 2 |∇n | βζΦ1 (κh, y/ h) + 8kT ηκ  + 2(β2 − 1)Φ3 (κh, y/ h)] + O(ζ 3 ) , (36) where the functions Φ1 , Φ2 , and Φ3 are: Φ1 = 1 −

cosh(κy) , cosh(κh)

Φ2 =

1 2cosh (κh) 2

(37a)

[cosh(2κh) − cosh(2κy) − 2κ2 (h2 − y2 )] (37b)

Φ3 =

1 2cosh2 (κh)

[cosh(2κh) − cosh(2κy) + 2κ2 (h2 − y2 )]. (37c)

93

The average fluid velocity u over a cross section of the slit is:  Ze 2 2Ze ∞ ζ [Φ2  u = − 2 |∇n | βζΦ1  + 8kT ηκ  (38) + 2(β2 − 1)Φ3 ] + O(ζ 3 ) , where the definition of the angle brackets becomes  1 h u = u(y) dy. h 0

(39)

Substitution of Eq. (37) into Eq. (39) leads to sinh(κh) , κh cosh(κh)  1 cosh(2κh) Φ2  = 2cosh2 (κh)  1 4 − sinh(2κh) − (κh)2 , 2κh 3  1 cosh(2κh) Φ3  = 2cosh2 (κh)  1 4 2 − sinh(2κh) + (κh) . 2κh 3

Φ1  = 1 −

(40a)

(40b)

(40c)

Similar to the functions Θ1 , Θ2 , and Θ3 discussed in the previous section, the functions Φ1 , Φ2 , and Φ3 are all positive and equal to zero at the capillary walls (y/ h = 1). In the limit κh = 0, Eqs. (37) and (40) result in Φ1 = Φ2 = Φ3 = Φ2 /(κh)2 = 0 at any position in the capillary and Φ1  = Φ2  = Φ3  = Φ2 /(κh)2 = 0, respectively. However, the diffusioosmotic velocity due to the contribution of the induced electric field E given by Eq. (36) does not vanish for this case since Φ3 /(κh)2 = 4Φ1 /(κh)2 = 2[1−(y/ h)2 ] and Φ3 /(κh)2 = 4Φ1 /(κh)2 = 4/3. Under the situation of κh → ∞, Eqs. (37) and (40) give that Φ1 = Φ2 = Φ3 = 1 at positions with y/ h = 1 and Φ1  = Φ2  = Φ3  = 1. For all other cases, Φ3 and Φ3  are comparable with (especially when κh is large) but greater than Φ2 and Φ2 , respectively. Again, the effect of the lateral distribution of the induced axial electric field always exists and does not vanish even as κh → ∞. Eq. (36) shows that the O(ζ 2 ) contributions from chemiosmosis (involving Φ2 ) and electroosmosis (involving Φ3 ) are always in opposite directions, and the net diffusioosmotic flow is dominated by the electroosmotic effect in most practical situations. 3.3. Results and discussion The functions Φ1 , Φ2 , and Φ3 given by Eq. (36) and (37) for the diffusioosmotic velocity of a symmetric electrolyte in a capillary slit have the same physical meanings as those of the functions Θ1 , Θ2 , and Θ3 for the corresponding flow in a capillary tube, and the numerical results of Φ1 and Φ2 could be found in the literature [18]. In Fig. 5, the function

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H.J. Keh, H.C. Ma / Colloids and Surfaces A: Physicochem. Eng. Aspects 233 (2004) 87–95

1.2

∞ 10 0.8

3

F3

2 1

0.4

0.5 kh = 0

0 0

0.2

0.4

0.6

0.8

1

y/h

this maximum arises at κh → ∞ and its value is close to unity. Different from the monotonic increase of Θ3  with an increase in κR for the fluid flow in a capillary tube, the function Φ3  for the flow in a slit does not increase monotonically with an increase in κh (although Φ3  is smaller than Φ3 (y = 0) and can not be greater than unity for any given value of κh). It can be found that, under the situation of κh = κR and 0 ≤ y/ h = ρ/R < 1, Φ3 (κh, y/ h) < Θ3 (κR, ρ/R) if < κh < 1.83 and Φ3 (κh, y/ h) < Θ3 (κR, ρ/R) if 1.84 < κh < ∞. On the other hand, as κh = κR, Φ3  is greater than Θ3  for cases with 0 < κh < 3.14, and Φ3  is smaller than Θ3  otherwise. In Fig. 7, the diffusioosmotic velocities of electrolyte solutions at the midplane of a capillary slit normalized by their characteristic value given by Eq. (26) are plotted as functions of the dimensionless surface potential Zeζ/kT (with its

Fig. 5. Plots of the function Θ3 as calculated from Eq. (37c) vs. the normalized coordinate y/h for various values of κh.

Φ3 is plotted versus the normalized coordinate y/h for various values of κh. The values of Φ3 at the midplane of the capillary slit (with y = 0) and its averaged results over a cross section of the slit as functions of κh are also presented in the last two columns of Table 1 and plotted in Fig. 6. The corresponding values of Φ2 and Φ2  are also shown in Fig. 6 for comparison. Similar to the case of fluid flow in a capillary tube, both Φ2 and Φ3 decrease monotonically with an increase in y/h for a constant value of κh, from maxima at the midplane of the capillary slit to zero at the slit walls (with y = h). For a given value of y/h, different from the trend of Φ2 as an increasing function of κh, Φ3 is not necessarily a monotonic function of κh and may have a maximal value greater than unity at some value of κh. The maximum of Φ3 at the midplane of the slit, which is greater than unity for cases with κh greater than about 1, occurs in the vicinity of κh = 1.67. When the slit wall is approached (y/ h → 1),

kh = 0

0

0.2 ∞

-0.4

3

u ( y = 0) U*

2

-0.8

1.4 -1.2 0

0.4

(a)

0.8

1.2

1.6

Zez / kT

∞

kh = 0

0

F 3 ( y = 0)

1.2

2

2 -0.4

F3 0.8

u ( y = 0) U*

F 2 ( y = 0)

1

∞

-0.8

F2

0.4

2 -1.2 -2

(b)

0 0.1

1

kh

10

100

Fig. 6. Plots of the function Φ2 (y = 0), Φ3 (y = 0), Φ2 , and Φ3  vs. κh.

-1

0

1

2

Zez / kT

Fig. 7. Plots of the normalized diffusioosmotic velocity at the midplane of a capillary slit vs. the dimensionless surface potential Zeζ/kT for various values of ␬h: (a) β = 0; (b) β = −0.2.

H.J. Keh, H.C. Ma / Colloids and Surfaces A: Physicochem. Eng. Aspects 233 (2004) 87–95

magnitude <2) for various values of κh. For the case of a symmetric electrolyte that the cation and anion diffusivities are equal (β = 0) shown in Fig. 7(a), the magnitude of the normalized velocity u(y = 0)/U ∗ (or u/U∗ at any other location, or u/U ∗ , which are not illustrated here) increases monotonically with an increase in Zeζ/kT for a fixed value of κh, but is not a monotonic function of κh for a constant value of Zeζ/kT. This behavior is similar to the corresponding results for a capillary tube. For the case of a symmetric electrolyte whose cation and anion have different diffusion coefficients (β = 0) as illustrated in Fig. 7(b), the fluid velocity is not necessarily a monotonic function of Zeζ/kT for a fixed value of κh and the fluid flow can reverse its direction more than once as the surface potential of the capillary wall varies from negative to positive values, similar to the diffusioosmosis in a capillary tube. In general, the local or average fluid velocity is not a monotonic function of κh for a constant value of Zeζ/kT. Both Fig. 7(a) and (b) demonstrate that the electroosmotic effect of O(ζ 2 ) (involving the function Φ3 ) gives a dominant contribution to the net diffusioosmotic flow.

95

induced axial electric field inside the electric double layer from its bulk-phase quantity can not be neglected in the evaluation of the diffusioosmotic flow rate of electrolyte solutions in a capillary, even for the case of very thin double layer. The macroscopic electric field E arising spontaneously due to the imposed concentration gradient of the electrolyte in the axial direction of the capillary is provided by Eqs. (10) and (14) and the diffusioosmotic velocity u of the electrolyte solution is obtained in Eqs. (20) and (36). In addition to the ionic fluxes due to electric migration (given by the second term on the right-hand side of Eq. (9)), the induced electric field E can generate an electric current by electric conduction. Moreover, the diffusioosmotic fluid flow leads to another electric current by ionic convection. These two electric currents are not included in the current balance for the determination of E. Thus, a secondary induced electric field must build-up through the capillary, which is just sufficient to prevent the net electric current flow. This secondary electric field and its contribution to the fluid flow can be calculated via a similar approach in the calculation of the streaming potential induced across the capillary in the presence of an applied pressure gradient [2,3,10].

4. Concluding remarks Under the Debye–Huckel approximation, the steady diffusioosmosis of a symmetric electrolyte solution in a fine capillary tube or slit is analyzed in this work. It is assumed that the fluid is only slightly non-uniform in the electrolyte concentration along the capillary axis, but no assumption is made about the thickness of the electric double layer over the capillary wall relative to the radius of the capillary and the effect of radial distributions of the counterions and coions on the induced axial electric field due to the electrolyte concentration gradient is taken into account. The capillary wall may have either a constant surface potential or a constant surface charge density. Solving the linearized Poisson–Boltzmann equation and the modified Navier–Stokes equation applicable to the system, the electrostatic potential distribution, the induced electric field distribution, and the pressure distribution under the influence of a constant imposed gradient of the electrolyte concentration are obtained analytically. Closed-form formulas and numerical results for the local and average diffusioosmotic velocities over a cross section of the capillary correct to O(ζ 2 ) are also presented. It is shown in Section 2.2 that the macroscopic electric field induced by the prescribed electrolyte concentration gradient through the capillary is a function of the radial position rather than a constant bulk-phase quantity. The contribution to the diffusioosmotic flow made by the position dependence of the induced electric field is of the same order [O(ζ 2 )] as, but has an opposite direction to, that made by the chemiosmotic effect, and the former is dominant in most practical situations, as indicated by Eqs. (20) and (36). Therefore, the effect of the deviation of the

Acknowledgements This research was supported by the National Science Council of the Republic of China. References [1] M. Smoluchowski, in: I. Graetz (Ed.), Handbuch der Electrizitat und des Magnetismus, vol. II, Barth, Leipzig, 1921, p. 336. [2] D. Burgreen, F.R. Nakache, J. Phys. Chem. 68 (1964) 1084. [3] C.L. Rice, R. Whitehead, J. Phys. Chem. 69 (1965) 4017. [4] S.S. Dukhin, B.V. Derjaguin, in: E. Matijevic (Ed.), Surface and Colloid Science, vol. 7, Wiley, New York, 1974. [5] J.L. Anderson, K. Idol, Chem. Eng. Commun. 38 (1985) 93. [6] J.H. Masliyah, Electrokinetic Transport Phenomena, AOSTRA, Edmonton, Alberta, Canada, 1994. [7] H.J. Keh, Y.C. Liu, J. Colloid Interface Sci. 172 (1995) 222. [8] C. Yang, D. Li, J. Colloid Interface Sci. 194 (1997) 95. [9] A. Szymczyk, B. Aoubiza, P. Fievet, J. Pagetti, J. Colloid Interface Sci. 216 (1999) 285. [10] H.J. Keh, H.C. Tseng, J. Colloid Interface Sci. 242 (2001) 450. [11] D.C. Prieve, Adv. Colloid Interface Sci. 16 (1982) 321. [12] D.C. Prieve, J.L. Anderson, J.P. Ebel, M.E. Lowell, J. Fluid Mech. 148 (1984) 247. [13] J.L. Anderson, Annu. Rev. Fluid Mech. 21 (1989) 61. [14] H.J. Keh, S.B. Chen, Langmuir 9 (1993) 1142. [15] J.C. Fair, J.F. Osterle, J. Chem. Phys. 54 (1971) 3007. [16] V. Sasidhar, E. Ruckenstein, J. Colloid Interface Sci. 85 (1982) 332. [17] G.B. Westermann-Clark, J.L. Anderson, J. Electrochem. Soc. 130 (1983) 839. [18] H.J. Keh, J.H. Wu, Langmuir 17 (2001) 4216. [19] V.G. Levich, Physicochemical Hydrodynamics, Prentice Hall, Englewood Cliffs, NJ, 1962.