Diffusioosmosis of Electrolyte Solutions around a Circular Cylinder at

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Ind. Eng. Chem. Res. 2009, 48, 2443–2450

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Diffusioosmosis of Electrolyte Solutions around a Circular Cylinder at Arbitrary Zeta Potential and Double-Layer Thickness Li Y. Hsu and Huan J. Keh* Department of Chemical Engineering, National Taiwan UniVersity, Taipei 10617, Taiwan, Republic of China

The steady diffusioosmotic flow of an electrolyte solution around a charged circular cylinder caused by a constant concentration gradient imposed in the direction along the axis of the cylinder is investigated theoretically. The cylinder can have either a constant surface potential or a constant surface charge density of an arbitrary value. The electric double layer surrounding the cylinder can have an arbitrary thickness, and its electrostatic potential distribution is determined by an analytical approximation to the solution of the Poisson-Boltzmann equation. Solving a modified Navier-Stokes equation with the constraint of no net electric current arising from the cocurrent diffusion, electric migration, and diffusioosmotic convection of the electrolyte ions, the macroscopic electric field and the fluid velocity along the axial direction induced by the imposed electrolyte concentration gradient are obtained semianalytically as functions of the radial position in a selfconsistent manner. The direction of the diffusioosmotic flow relative to the concentration gradient is determined by the combination of the zeta potential (or surface charge density) of the cylinder, the properties of the electrolyte solution, and other relevant factors. For a prescribed concentration gradient of an electrolyte, the magnitude of the fluid velocity generally increases with increasing distance from the surface of the cylinder, but there are exceptions. The effects of the radial distribution of the induced macroscopic electric field and of the ionic convection in the double layer on the diffusioosmotic flow are quite significant in practical situations. 1. Introduction A variety of electrokinetic flows of electrolyte solutions in interstitial channels with charged walls is of much fundamental and practical interest in many areas of science and engineering. Perhaps the most familiar mechanism of electrokinetic flows is electroosmosis, which results from the interaction between the electric double layer adjacent to a charged wall and a tangentially applied electric field. Various aspects of electroosmotic flow have already been studied in the past several decades.1–10 Another mechanism of electrokinetic flows in porous media, termed diffusioosmosis (also known as capillary osmosis),3,11–14 involves a tangential concentration gradient of the electrolyte that interacts with the charged wall. As in the case of electroosmosis, the electrolyte-wall interaction in diffusioosmosis is electrostatic in nature, and its range is the Debye screening length κ-1 (defined after eq 3 in section 2). Recently, the fluid motion caused by diffusioosmosis was analytically studied for flows near a plane wall15,16 and inside a capillary pore.16–21 Some experimental results and interesting applications concerning diffusioosmosis are also available in the literature.22,23 Electrolyte solutions with a concentration gradient on the order of 100 kmol/m4 () 1 M/cm) along solid surfaces with a zeta potential on the order of kT/e (∼25 mV; e is the charge of a proton, k is the Boltzmann constant, and T is the absolute temperature) can flow by diffusioosmosis at a velocity of several micrometers per second. For the case of diffusioosmosis of an ionic fluid solution in a fibrous system constructed of an array of long parallel cylinders, the applied electrolyte concentration gradient can be taken as a combination of its transverse and longitudinal components with respect to the orientation of the cylinders. Hence, the problem can be divided into two problems, if it is linearized, and these component problems can be separately solved. The overall diffusioosmotic velocity of the fluid solution * To whom correspondence should be addressed. Fax: +886-223623040. E-mail: [email protected].

can be obtained by the vectorial addition of the results for the two components. Recently, the problems of the transverse24,25 and longitudinal26,27 diffusioosmotic flows of electrolyte solutions in a homogeneous array of parallel charged circular cylinders were analytically solved using a unit cell model. An imposed concentration gradient of a dissociating electrolyte produces fluid flow along a charged solid surface by two mechanisms. The first involves the stresses developed by the induced tangential gradient of the excess pressure within the electric double layer (chemiosmotic effect), and the second is based on the macroscopic electric field that is generated because the tangential diffusive and convective fluxes of the two electrolyte ions are not equal (electroosmotic effect). Both mechanisms were considered to some extent in previous investigations of diffusioosmotic flow.11–27 In the existing analyses of the longitudinal diffusioosmotic flow of electrolyte solutions in a unit cell with a long dielectric cylinder,26,27 however, neither the effect of radial distributions of the counterions and co-ions (or of the electrostatic potential) on the local electric field induced by the imposed electrolyte concentration gradient in the tangential direction inside the double layer nor the effect of the ionic convection on the local electric field caused by the diffusioosmotic flow was considered. Moreover, this analysis is subject to the severe restriction that the zeta potential is sufficiently low (less than about 25 mV) for the Debye-Huckel approximation to be acceptable. In practical applications, however, zeta potentials as high as 100-200 mV are frequently encountered. In this work, we present a comprehensive analysis of the diffusioosmosis of an electrolyte solution around a long circular cylinder with a constant prescribed concentration gradient in its axial direction. The zeta potential or surface charge density of the cylinder is assumed to be uniform, but no assumption is made concerning the magnitude of the zeta potential or the thickness of the electric double layer, and both the radial distribution of the induced axial electric field and the effect of

10.1021/ie800428p CCC: $40.75  2009 American Chemical Society Published on Web 08/06/2008

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Figure 1. Geometrical sketch for the diffusioosmosis of electrolyte solutions around a long circular cylinder due to an axially applied concentration gradient.

Figure 2. Map showing the relation among the dimensionless parameters j κr*, ζj , and κR and the fundamental subdomains for the solution of ψ (r).

the ionic convection on the local electric field are considered. The Poisson-Boltzmann equation governing the electrostatic potential in the fluid phase is solved by an analytical approximation that has been shown to yield results differing only slightly from the exact numerical solution.4,28 Semianalytical results for the diffusioosmotic velocity profile are obtained for various cases. These results show that the effect of the deviation of the induced axial electric field in the double layer from its bulk-phase value and the effect of the ionic convection on the diffusioosmotic velocity of the fluid are quite significant in practical situations, even for the case of a thin double layer. 2. Electrostatic Potential Distribution In this section, we consider the radial distribution of the electrostatic potential in the fluid solution of a symmetrically charged electrolyte of valence Z (where Z is a positive integer) undergoing diffusioosmosis around a circular cylinder of radius R and length L with R , L, as illustrated in Figure 1, at the steady state. The discrete nature of the surface charges, which are uniformly distributed over the cylinder, is ignored. The applied electrolyte concentration gradient ∇n∞ is constant along the axial (z) direction of the cylinder, where n∞(z) is the linear

Figure 3. Plots of the normalized electric field in the electrolyte solution around a circular cylinder induced by an axially applied concentration gradient versus the dimensionless coordinate r/R for the case of κR ) 1 with various values of the parameter β: (a) ζj ) 1, (b) ζj ) 6. The solid curves represent the case Pe ) 1, and the dashed curves represent the case Pe ) 0.

concentration (number-density) distribution of the electrolyte in the bulk solution phase far from the cylinder (as r f ∞, or beyond the influence of the charged cylinder). The end effects are neglected. It is assumed that n∞ is only slightly nonuniform such that L|∇n∞|/n∞(0) , 1, where z ) 0 is set at the midpoint along the cylinder. Thus, the variations with the axial position of the electrostatic potential (excluding the macroscopic electric field induced by the electrolyte gradient, which is discussed in section 3) and ionic concentrations in the electric double layer adjacent to the surface of the cylinder can be neglected in comparison to their corresponding quantities at z ) 0. That is, the principle of local equilibrium can be used to characterize the polarized double-layer structure.3 If ψ(r) represents the electrostatic potential at a point with distance r from the axis of the cylinder relative to that in the bulk solution and n+(r, z) and n-(r, z) denote the local concentrations of the cations and anions, respectively, then Poisson’s equation gives 1 d dψ Ze r ) - [n+(r,0) - n-(r,0)] r dr dr ε

( )

(1)

where ε is the dielectric permittivity of the electrolyte solution.

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The local ionic concentrations can also be related to the electrostatic potential by the Boltzmann equation, j) n( ) n∞ exp(-ψ

(2)

j ) Zeψ/kT is the dimensionless potential profile. where ψ Substitution of eq 2 into eq 1 results in the well-known Poisson-Boltzmann equation

( )

j 1 d dψ j r ) κ2 sinh ψ r dr dr

(3)

where κ ) [2Z2e2n∞(0)/εkT]1/2 is the Debye screening parameter. For the case of constant surface potential, the boundary conditions for ψ are j ) ζj r)R:ψ

(4a)

j f0 rf∞:ψ

(4b)

where the constant ζj ) Zeζ/kT is the dimensionless zeta potential at the shear plane of the cylinder surface adjacent to the electrolyte solution having a uniform bulk concentration n∞(0). Because there is no simple analytical solution of eq 3 available for the case of cylindrical symmetry, we follow a previous approach4,28 and use an approximation to provide a j throughout the range ψ j g0 good representation of sinh ψ j )ψ j sinh ψ

j 1 sinh ψ if ψ 2 and replace eq 3 by the pair of equations

( ) ( )

j 1 d dψH 1 j r ) κ2eψH r dr dr 2

if R e r e r*

(5a) (5b)

(6a)

j 1 d dψL j r ) κ2ψ if r* e r (6b) L r dr dr j are considered here without loss Only the positive values of ψ in generality. We have thus divided the fluid phase into two j ) 1 (sinh ψ j ) 1.175 hypothetical concentric regions such that ψ j and eψ/2 ) 1.359) at their junction r ) r*, where the subscripts H and L designate the inner (or high-potential) and outer (or low-potential) regions, respectively, as shown in Figure 1. If ζj e 1, then region L comprises the whole fluid phase. In other cases, eq 6a is subject to eq 4a and the additional boundary conditions j )ψ j )1 r ) r*: ψ H L

(7a)

j j dψ dψ H L ) dr dr

(7b)

j (r) profile, albeit which together ensure that the calculated ψ approximate, will be a smooth continuous function in the neighborhood of r ) r*. The relation among the dimensionless parameters κr*, ζj, and j L(r) κR is displayed in Figure 2, and the analytical solutions for ψ j H(r) are outlined below in terms of several subdomains. and ψ In subdomain I (ζj e 1), the low-potential region fills the fluid phase entirely, and the electrostatic potential distribution is j (r) ) ζj ψ L

K0(κr) K0(κR)

for r g R

(8)

where Kn is the modified Bessel function of the second kind of order n. As expected, Figure 2 illustrates that r* ) R when ζj ) 1. In subdomain II (R < r* < r1*, where r1* is a critical value of r* used to provide ranges for subdomains), the solution of eqs 6a, 7a, and 4a results in j (r) ) ψ L

{

[

K0(κr) K0(κr*)

for r g R

(9a)

( ) ]}

j (r) ) -2ln κr sinh 1 R ln r* + sinh-1 R ψ H R 2 r √eκr* for R e r e r*(9b) where R2 ) C (R is positive) and C is an integration constant that is dependent on the parameter κr* and is given by

[

C ) 2 - κr*

( )

K1(κr*) K0(κr*)

]

2

- (κr*)2e

(10)

The above equation gives 0 < C < 4 (and a meaningful solution in eq 9b) for R < r* < r1* and C ) 0 at r* ) r1* (the junction between subdomains II and III), where κr1* ) 0.601973; in the latter case, eq 9b becomes

{ [ ( ) ( ) ]}

r* 2 j (r) ) -2ln 1 κr ln 1 + ψ H 2 r √eκr*1

for R e r e r* (11)

In subdomain III [r1 < r* < r2, -(κr2) e < C ) -γ < 0, j L is also given by eq 9a, whereas eq 9b is and γ is positive), ψ replaced by *

*

* 2

{ [ ()

2

( )] }

j (r) ) -2ln κr sin 1 γ ln r* + sin-1 γ ψ H γ 2 r √eκr* for R e r e r*(12) * where r2 is another critical value of r* used to provide ranges for subdomains and κr2* ) 1.552651. At r* ) r2* (the junction of subdomains III and IV), C ) -(κr2*)2e, and in this case, eq 12 becomes

{ [ ( ) ]}

j (r) ) -2ln κr cos 1 γ ln r ψ H γ 2 r*2

for R e r e r* (13)

j L is In subdomain IV [r* > r2* and C ) -γ2 < –(κr2*)2e], ψ still given by eq 9a, whereas eq 9b is replaced by

{( ) [ ( )

( )] }

j (r) ) -2ln κr sin 1 γln r + sin-1 γ ψ H γ 2 r* √eκr* for R e r e r*(14) Usually, an analytical solution of the Poisson-Boltzmann equation in the form of eq 3 is obtained either for a small zeta potential ζj or for a large electrokinetic radius κR. An advantage of the above analysis is the method to find the potential j for any values of ζj and κR. distribution ψ If the constant surface charge density σ, instead of the surface potential ζ, of the cylinder is known, the boundary condition specified by eq 4a should be replaced by the Gauss condition dψ σ )(15) dr ε The solutions for ψ given by eqs 8–14 still hold for this condition, with the relation between ζ and σ for an arbitrary value of κR being r ) R:

σ j K0(κR) ζj ) κR K1(κR)

if ζj e 1

(16a)

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(σ j + 2)2 - C ζj ) ln otherwise (16b) (κR)2 where σ j ) RZeσ/εkT is the dimensionless surface charge density. Equation 16a and 16b indicates that, for a given electrolyte solution around a cylinder with a specified radius, σ increases (almost linearly) with increasing κ or [n∞(0)]1/2 for the case of constant surface potential, and ζ decreases with increasing κ or [n∞(0)]1/2 for the case of constant surface charge density. 3. Induced Electric Field Distribution The ionic concentrations n+ and n- in the fluid undergoing longitudinal diffusioosmosis around a circular cylinder are not uniform in both axial (z) and radial (r) directions; hence, their prescribed gradients in the axial direction can give rise to a “diffusion current” distribution on a cross section of the cylinder. To prevent a continuous separation of the counterions and coions, an electric field distribution along the axial direction arises spontaneously in the electrolyte solution to produce another electric current distribution that exactly balances the diffusion current.11–16 This induced electric field generates an electroosmotic flow of the fluid around the cylinder, in addition to the chemiosmotic flow caused directly by the prescribed electrolyte gradient. Both the chemiosmotic and electroosmotic flows also generate an electric current distribution by ionic convection (known as the relaxation effect), and alternately, this secondary “convection current” again needs to be balanced by the electric current contributed by the induced electric field. The total flux of either ionic species can be expressed in the general form Ze n (∇ψ - E) + n(u (17) kT ( where u ) u(r)ez is the fluid velocity in the axial direction of decreasing electrolyte concentration (i.e., ez is the unit vector in the direction of -∇n∞); D+ and D- are the diffusion coefficients of the cations and anions, respectively; E ) E(r)ez is the macroscopic electric field induced by the prescribed concentration gradient of the electrolyte; ψ(r) is the electrostatic potential profile obtained in eqs 8–14; and the principle of superposition for the electric potential is used. To have zero net electric current arising from the cocurrent diffusion, electric migration, and diffusioosmotic convection of the cations and anions, one must require that J+ ) J- ) J. (Obviously, the radial component of J vanishes, and the ionic fluxes induced by ∇ψ in eq 17 are balanced by the radial components of the diffusive ionic fluxes as required by the Boltzmann distribution given by eq 2.) Applying the constraint J+ ) J- to eq 17, we obtain

[

]

J( ) -D( ∇n( (

E)

[

j

j

kT ∇n∞ (1 + β)e-ψ - (1 - β)eψ + j j Ze n∞(0) (1 + β)e-ψ + (1 - β)eψ j Pe sinh ψ j -ψ

(1 + β)e

u j U* ψ

+ (1 - β)e

]

(18)

Pe )

8n∞(0)kT 4n∞(0)U* ) (D+ + D-)| ∇ n∞| (D+ + D-)ηκ2

(21)

and η is the fluid viscosity. From the definition in eq 20, -1 e β e 1, with the upper and lower bounds occurring as D-/D+f0 and ∞, respectively. Typical values of the physical quantities in eqs 18–21 are U* ) 10-5 m/s, D( ) 10-9 m2/s, n∞(0)/|∇n∞| ) 10-4 m, and Pe ≈ 1. The induced electric field E given by eq 18 depends, in a self-consistent manner, on the local electrostatic potential ψ and fluid velocity u. This equation indicates that E is collinear with and proportional to the axially imposed electrolyte gradient ∇n∞. If one considers conditions such that κR . 1, then, at a position r . R, ψ f 0, and eq 18 for the induced electric field caused by the imposed electrolyte concentration gradient reduces to its bulk-phase value E∞ )

kT β ∇ n∞ Ze n∞(0)

(22)

For the special case of an uncharged wall (ζ ) 0), E at any location r is also identical to this bulk-phase value. Note that E∞ is linearly proportional to the parameter β, but E(r) does not necessarily vanish if β ) 0, even for Pe ) 0, as shown in eq 18. 4. Fluid Velocity Distribution We now consider the steady diffusioosmotic flow of a symmetric electrolyte solution around a circular cylinder under the influence of a constant axial concentration gradient of the electrolyte. Momentum balances on the incompressible and Newtonian fluid in the r and z directions give dψ ∂p + Ze(n+ - n-) )0 ∂r dr

(23a)

η d du ∂p - Ze(n+ - n-)E r ) r dr dr ∂z

(23b)

( )

where p(r,z) is the dynamic pressure distribution. The boundary conditions for u at the no-slip surface of the cylinder and at infinity are r ) R: u ) 0

(24a)

du )0 (24b) dr After substitution of eq 2 into eq 23a based on the assumption that the equilibrium ionic distributions are not affected by the net electrolyte flux J, which is warranted if |∇n∞|/κn∞(0) , 1, the pressure distribution can be determined as r f ∞:

j - 1) p ) p0 + 2kTn∞(z)(cosh ψ

(25)

where ε|∇n∞| kT 2 2kT (19) ) 2 |∇n∞| ηn∞(0) Ze ηκ which is a characteristic value of the diffusioosmotic velocity

( )

U* )

β)

D+ - DD+ + D-

(20)

Here, p0 is the pressure at infinity, which is a constant in the absence of the applied pressure gradient, and the electric j (r) is given by eqs 8–14. potential distribution ψ Substituting the ionic concentration distributions of eq 2 and the pressure profile of eq 25 into eq 23b and then performing the integration with respect to r twice, subject to the boundary conditions in eq 24a, we obtain

u ) (κR)2 U*

∫

r/R

1

R r

∫

r/R

∞

[

r j -1+ cosh ψ R

Ind. Eng. Chem. Res., Vol. 48, No. 5, 2009 2447

]()()

Zen∞(0) j d r d r (26) Esinh ψ ∞ R R kT|∇n | j into eq 18, After substitution of eq 26 for u and eqs 8–14 for ψ the induced electric field distribution E can be numerically solved as a function of the dimensionless parameters κR, ζj, β, j and E, the diffusioosmotic and Pe. With the known results of ψ velocity distribution of the electrolyte solution as a function of κR, ζj, β, and Pe can be determined from eq 26 by numerical integration. Clearly, u/U* ) 0 everywhere if ζ ) 0. It is understood that, for given values of r/R, κR, and Pe, the quantity u/U* with specified values -ζj and β is equal to that with the values ζj and -β. 5. Results and Discussion The distribution of the macroscopic electric field E(r) in a fluid solution of a symmetric electrolyte around a circular cylinder induced by a concentration gradient prescribed axially can be determined numerically after substituting the fluid

Figure 5. Plots of the normalized diffusioosmotic velocity of the electrolyte solution around a circular cylinder versus the dimensionless coordinate r/R for the case of κR ) 1 with various values of the parameter β: (a) ζj ) 1, (b) ζj ) 6. The solid curves represent the case Pe ) 1, and the dashed curves represent the case Pe ) 0.

Figure 4. Plots of the normalized electric field in the electrolyte solution around a circular cylinder induced by an axially applied concentration gradient versus the dimensionless coordinate r/R for the case of β ) -0.2 with various values of the parameter κR: (a) Pe ) 0, (b) Pe ) 1. The solid curves represent the case ζj ) 1, and the dashed curves represent the case ζj ) 6.

velocity u(r)/U* in the form of eq 26 and the electric potential j (r) calculated from eqs 8–14 into eq 18. A simple method for ψ this numerical calculation is to make an initial guess of the fluid velocity distribution u1(r/R)/U* for a given combination of the dimensionless parameters ζj, β, Pe, and κR and obtain the resulting induced electric field E1(r/R) from eq 18. Then, the next result of the velocity distribution u2(r/R)/U* can be determined from the double integral involving E1(r/R) in eq 26. If the difference between u2(r/R) and u1(r/R) is beyond the tolerable error, then u2(r/R)/U* is used in eq 18 to obtain E2(r/ R), and the same procedure is repeated until an acceptable result for the velocity distribution is obtained. The induced electric field caused by the axially prescribed electrolyte gradient along the cylinder normalized by its value at infinity, E∞, as a function of the normalized coordinate r/R is plotted in Figures 3 and 4 for several values of the parameters ζj, β, Pe, and κR. Note that each curve with specified values of -ζj and β in the figures would be identical to that with the values ζj and -β. The situation associated with β ) -0.2 (taking Z ) 1) is close to the case of an aqueous solution of NaCl. As expected, the normalized induced electric field E(r)/E∞ approaches its bulk-phase value as r/R g 5, but can deviate significantly from unity as the value of r/R becomes smaller.

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Figure 6. Plots of the normalized diffusioosmotic velocity of the electrolyte solution around a circular cylinder versus the dimensionless coordinate r/R for the case of ζj ) 6: (a) κR ) 1, (b) Pe ) 1. The solid curves represent the case β ) -0.2, and the dashed curves represent the case β ) 0.

When Pe ) 0, E(r)/E∞ equals unity in the limit of β ) (1, and its magnitude becomes infinity for the special case of β ) 0, irrespective of the parameters ζj and κR, as expected from eqs 18 and 22. The magnitude of [E(r)/E∞] - 1 (or the deviation of the induced electric field from its bulk-phase value) decreases with increasing κR, increases with increasing ζj, and decreases with increasing magnitude of β, for an otherwise specified set of conditions. When the value of Pe is finite, the value of E(r)/ E∞ is generally smaller than that for the case of Pe ) 0, and when the magnitude of β is large, it might not be a monotonic function of r/R. The effect of the electrolyte convection on the local induced electric field in the fluid around the cylinder can be significant, especially for the case of high zeta potential at the surface of the cylinder. The dimensionless diffusioosmotic velocity distribution u(r)/ U* of an electrolyte solution around a circular cylinder as calculated numerically using eq 26 is plotted in Figures 5 and 6 for several values of the parameters ζj, β, Pe, and κR. This diffusioosmotic velocity can be either positive or negative and is generally a decreasing function of βζ/|ζ|. The magnitude of u/U* is not necessarily a monotonic function of the normalized coordinate r/R or of the parameter κR, for an otherwise specified set of conditions. When βζ/|ζ| e 0, u is positive, meaning that the diffusioosmotic flow is in the direction of decreasing

Figure 7. Normalized bulk-phase diffusioosmotic velocity of the electrolyte solution around a circular cylinder for various values of the parameter β: (a) plots versus κR for the case of ζj ) 6, (b) plots versus ζj for the case of κR ) 1. The solid curves represent the case Pe ) 1, and the dashed curves represent the case Pe ) 0.

electrolyte concentration, and the magnitude of u/U* increases with increasing magnitude of ζj. When the magnitude of ζj is large and the product of ζ and β is positive, the diffusioosmotic velocity can reverse its direction from against the concentration gradient to along it as r/R increases from unity. In general, the magnitude of u/U* decreases with an increase in the value of Pe (and vanishes in the limit of Pe f ∞) for specified values of r/R, κR, ζj, and β. This relaxation effect on the diffusioosmotic flow is quite significant if the magnitude of ζj is large. Note that the cases with Pe g 10, which are not likely to exist in practice, are exhibited in Figure 6a for the sake of numerical comparison. In Figure 7, the normalized bulk-phase diffusioosmotic velocity u∞/U* [) u(rf∞)/U*] of the electrolyte solution around the circular cylinder is plotted versus the parameters κR and ζj at specified values of Pe and β. The dependence of u∞/U* on ζj, β, κR, and Pe is quite similar to that of u/U* for a given value of r/R, and it is not necessarily a monotonic function of κR for given values of ζj, β, and Pe. When the product of ζ and β is positive, u∞ can reverse its direction from along the concentration gradient to against it as the magnitude of ζj increases not much from zero for some practical cases of β (in addition to a reversal occurring at ζj ) 0). Again, the effect of

Ind. Eng. Chem. Res., Vol. 48, No. 5, 2009 2449

Figure 8. Normalized diffusioosmotic velocity distribution in the electrolyte solution around a circular cylinder for the case of ζj ) 6, κR ) 1, and Pe ) 1. The dashed curves represent the results obtained by using the j (r) in section 2, and mathematical approximation for the evaluation of ψ the solid curves represent the numerical exact solution.

the electrolyte convection is significant, irrespective of the thickness of the electric double layer adjacent to the surface of the cylinder. Throughout this work we have adopted the mathematical approximation presented in section 2 for the solution of the electrostatic potential distribution on a cross-sectional plane of the circular cylinder. To check the accuracy of this approximation for the resulting diffusioosmotic velocity profile, we j (r) and then numerically solved eqs 3, 4a, and 4b for ψ substituted its values into eqs 18 and 26 to numerically determine the “exact” solution for the diffusioosmotic velocity u(r) in several cases. Figure 8 shows a typical comparison between the approximate solution and the exact solution. It can be seen that the difference between the two solutions for u(r) is less than 1%. This outcome means that the mathematical j (r) is approximation presented in section 2 for the solution of ψ generally acceptable in the evaluation of the longitudinal diffusioosmotic velocity of electrolyte solutions around a circular cylinder when compared with the relevant experimental data. 6. Concluding Remarks A theoretical study of the steady diffusioosmotic flow of solutions of symmetric electrolytes around a long circular cylinder is presented in this work. It is assumed that the fluid is only slightly nonuniform in the electrolyte concentration along the axial direction, but no assumption is made about the thickness of the electric double layer adjacent to the surface of the cylinder. Both the effect of the radial distribution of the electrolyte ions (or of the electrostatic potential) and the effect of ionic convection caused by the diffusioosmotic flow itself (relaxation effect) on the axial electric field induced by the applied electrolyte concentration gradient are taken into account. The cylinder can have either a constant surface potential or a constant surface charge density of an arbitrary value. When the Poisson-Boltzmann equation in an approximate form and the modified Navier-Stokes equation applicable to the system are solved, the distributions of the electrostatic potential, induced electric field, and dynamic pressure under the influence of the imposed electrolyte gradient are determined either analytically or semianalytically. Numerical results for the

local diffusioosmotic velocity on a cross-sectional plane of the cylinder as a function of relevant parameters are presented in detail. The results show that the effect of the deviation of the local induced axial electric field inside the double layer from its bulk-phase value and the relaxation effect cannot be neglected in the evaluation of the diffusioosmotic velocity of electrolyte solutions in the axial direction of the cylinder, even for the case of a thin double layer. The bulk-phase diffusioosmotic velocity u∞ obtained in this work can also be used as the diffusiophoretic velocity (in the opposite direction) of a charged circular cylinder along its axis. As for the diffusiophoretic/diffusioosmotic motion about a circular cylinder with an arbitrary electric double layer generated by a transverse electrolyte concentration gradient, the relative velocity of the cylinder was already obtained in a previous work.25 For the diffusiophoresis/diffusioosmosis about a dielectric cylinder oriented arbitrarily with respect to the imposed electrolyte gradient, the relative cylinder velocity is the vectorial sum of its transverse and longitudinal contributions. For an ensemble of circular cylinders with random orientations, the average diffusiophoretic velocity (aligned with the direction of ∇n∞) can be obtained by one-third of the longitudinal diffusiophoretic velocity of a single cylinder plus two-thirds of its transverse counterpart. Acknowledgment This research was supported by the National Science Council of the Republic of China under Grant NSC96-2221-E-002-078. Literature Cited (1) Burgreen, D.; Nakache, F. R. Electrokinetic flow in ultrafine capillary slits. J. Phys. Chem. 1964, 68, 1084. (2) Rice, C. L.; Whitehead, R. Electrokinetic flow in a narrow cylindrical capillary. J. Phys. Chem. 1965, 69, 4017. (3) Dukhin, S. S.; Derjaguin, B. V. In Surface and Colloid Science; Matijevic, E., Ed.; Wiley: New York, 1974; Vol. 7. (4) Levine, S.; Marriott, J. R.; Neale, G.; Epstein, N. Theory of electrokinetic flow in fine cylindrical capillaries at high zeta-potentials. J. Colloid Interface Sci. 1975, 52, 136. (5) Anderson, J. L.; Idol, W. K. Electroosmosis through pores with nonuniformly charged walls. Chem. Eng. Commun. 1985, 38, 93. (6) Keh, H. J.; Liu, Y. C. Electrokinetic flow in a circular capillary with a surface charge layer. J. Colloid Interface Sci. 1995, 172, 222. (7) Ohshima, H. Electroosmotic velocity in fibrous porous media. J. Colloid Interface Sci. 1999, 210, 397. (8) Keh, H. J.; Tseng, H. C. Transient electrokinetic flow in fine capillaries. J. Colloid Interface Sci. 2001, 242, 450. (9) Keh, H. J.; Ding, J. M. Electrokinetic flow in a capillary with a charge-regulating surface polymer layer. J. Colloid Interface Sci. 2003, 263, 645. (10) Park, H. M.; Lee, J. S.; Kim, T. W. Comparison of the NernstPlanck model and the Poisson-Boltzmann model for the electroosmotic flows in microchannels. J. Colloid Interface Sci. 2007, 315, 731. (11) Prieve, D. C.; Anderson, J. L.; Ebel, J. P.; Lowell, M. E. Motion of a particle generated by chemical gradients. Part 2. Electrolytes. J. Fluid Mech. 1984, 148, 247. (12) Anderson, J. L. Colloid transport by interfacial forces. Annu. ReV. Fluid Mech. 1989, 21, 61. (13) Pawar, Y.; Solomentsev, Y. E.; Anderson, J. L. Polarization effects on diffusiophoresis in electrolyte gradients. J. Colloid Interface Sci. 1993, 155, 488. (14) Keh, H. J.; Chen, S. B. Diffusiophoresis and electrophoresis of colloidal cylinders. Langmuir 1993, 9, 1142. (15) Keh, H. J.; Ma, H. C. Diffusioosmosis of electrolyte solutions along a charged plane wall. Langmuir 2005, 21, 5461. (16) Ma, H. C.; Keh, H. J. Diffusioosmosis of electrolyte solutions in a fine capillary slit. J. Colloid Interface Sci. 2006, 298, 476. (17) Keh, H. J.; Wu, J. H. Electrokinetic flow in fine capillaries caused by gradients of electrolyte concentration. Langmuir 2001, 17, 4216.

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ReceiVed for reView March 16, 2008 ReVised manuscript receiVed May 31, 2008 Accepted June 16, 2008 IE800428P