Diffusioosmosis of electrolyte solutions in a fine capillary slit
Hsien Chen Ma, Huan J. Keh
∗Department of Chemical Engineering, National Taiwan University, Taipei 10617, Taiwan, ROC Received 3 October 2005; accepted 21 November 2005
Available online 20 December 2005
Abstract
The steady diffusioosmotic flows of an electrolyte solution along a charged plane wall and in a capillary channel between two identical parallel charged plates generated by an imposed tangential concentration gradient are theoretically investigated. The plane walls may have either a constant surface potential or a constant surface charge density. The electrical double layers adjacent to the charged walls may have an arbitrary thickness and their electrostatic potential distributions are determined by 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 tangential direction induced by the imposed electrolyte concentration gradient are obtained semianalytically as a function of the lateral position in a self-consistent way. 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 wall, the properties of the electrolyte solution, and other relevant factors. For a given concentration gradient of an electrolyte along a plane wall, the magnitude of fluid velocity at a position in general increases with an increase in its electrokinetic distance from the wall, but there are exceptions. The effect of the lateral distribution of the induced tangential electric field and the relaxation effect in the double layer on the diffusioosmotic flow are found to be very significant.
©2005 Elsevier Inc. All rights reserved.
Keywords: Diffusioosmosis; Plane wall; Capillary slit; Diffusiophoresis
1. Introduction
The flow behavior of fluids in porous media is of fundamen-tal and practical interest in various areas of science and engi-neering. In general, driving forces for the fluid transport through micropores include dynamic pressure differences between the two ends of a capillary pore (convection)[1–6], concentration differences of an impermeable solute between the two bulk so-lutions outside the pores (osmosis)[4,7], and tangential electri-cal fields that interact with the electrielectri-cal double layer adjacent to a charged pore wall (electroosmosis)[2–6,8–13]. Problems of fluid flow induced by these well-known driving forces were treated extensively in the past.
Another driving force for the flow of liquid solutions in a capillary pore, which has commanded less attention, involves concentration gradients of a permeable solute along the
cap-* Corresponding author.
E-mail address:[email protected](H.J. Keh).
illary that interacts with the pore wall. The fluid motion as-sociated with this mechanism is termed diffusioosmosis and has been discussed analytically for solutions of either ionic or nonionic solutes near a plane wall[8,14–19]and inside a capil-lary pore[20–24]. Some experimental results and interesting applications concerning diffusioosmosis are also available in the literature[25]. In a solution of uncharged solute, the solute molecules interact with the adjacent wall through the van der Waals and dipole forces. For an electrolyte solution in contact with a charged wall, the solute-wall interaction is electrosta-tic in nature and its range is the Debye screening length κ−1 (defined right after Eq.(3)). Electrolyte solutions with a con-centration gradient of order 100 kmol/m4 (=1 M/cm) along solid surfaces with a zeta potential of order kT /e (∼25 mV;
eis 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.
A tangential gradient of a dissociating electrolyte produces fluid flow along a charged solid surface by two mechanisms. The first involves the stresses developed by the tangential
gra-0021-9797/$ – see front matter © 2005 Elsevier Inc. All rights reserved.
dient of the excess pressure within the electric double layer (chemiosmotic effect), and the second is based on the macro-scopic 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 con-sidered to some extent in previous investigations for the dif-fusioosmotic flow[8,14–24]. In these studies, however, either the effect of lateral distributions of the counterions and coions (or of the electrostatic potential) on the local electric field in-duced by the imposed electrolyte concentration gradient in the tangential direction inside the double layer or the effect of the ionic convection on it caused by the diffusioosmotic flow was neglected.
In this work we present comprehensive analyses of the diffu-sioosmosis of an electrolyte solution with a constant prescribed concentration gradient along a charged plane wall and in the tangential direction of a capillary channel between two identi-cal parallel charged plates. The zeta potential or surface charge density of the walls is assumed to be uniform, but no assump-tion is made concerning the magnitude of the electric potential or the thickness of the double layer, and both the lateral distri-bution of the induced tangential electric field and the effect of the ionic convection on it are allowed. Semianalytical results for the fluid velocity profile and the bulk-phase diffusioosmotic velocity are obtained for various cases. These results show that the effect of the deviation of the induced tangential electric field in the double layer from its bulk-phase quantity and the effect of the ionic convection on the diffusioosmotic velocity of the fluid are dominantly significant in most practical situations, even for the case of a very thin double layer.
2. Diffusioosmosis of an electrolyte solution along a charged plane wall
In this section, the diffusioosmotic flow of an infinitely thick solution of a symmetrically charged electrolyte of valence Z (where Z is a positive integer) tangential to a uniformly charged plane wall of length L, as illustrated inFig. 1a, in steady state is
(a)
(b)
Fig. 1. Geometrical sketches for the diffusioosmosis due to an applied concen-tration gradient of an electrolyte: (a) flow tangential to a plane wall; (b) flow in a capillary slit.
examined. The applied electrolyte concentration gradient∇n∞ is a constant along the tangential (z) direction, where n∞(z)
is the linear concentration (number density) distribution of the electrolyte in the bulk solution phase far from the wall (with
y → ∞ or beyond the electrostatic influence of the charged
wall). The end effects are neglected. It is assumed that n∞ is only slightly nonuniform, so that L|∇n∞|/n∞(z= 0) 1,
where z= 0 is set at the midpoint along the plane wall. Thus, the variation of the electrostatic potential and ionic concentra-tions in the electric double layer adjacent to the wall with the tangential position can be neglected in comparison with their corresponding quantities at z= 0.
2.1. Electrostatic potential distribution
We first consider the electrostatic potential distribution in the electrolyte solution near the charged plane wall. If ψ(y) repre-sents the electrostatic potential at a position y from the wall relative to that in the bulk solution and n+(y, z)and n−(y, z)
denote the local concentrations of the cations and anions, re-spectively, then the Poisson equation gives
(1) d2ψ dy2 = − 4π Ze ε n+(y,0)− n−(y,0).
In this equation, ε= 4πε0εr, where εr is the relative
permit-tivity of the electrolyte solution and ε0is the permittivity of a
vacuum.
The local ionic concentrations can also be related to the elec-trostatic potential by the Boltzmann equation,
(2)
n±= n∞exp(∓ψ),
where ψ = Zeψ/kT is the dimensionless potential profile. Substitution of Eq.(2) into Eq.(1) results in the well-known Poisson–Boltzmann equation, (3) d2ψ dy2 = κ 2 sinh ψ,
where κ= [8π(Ze)2n∞(z= 0)/εkT ]1/2is the Debye screen-ing parameter.
For the case of constant surface potential, the boundary con-ditions for ψ are
(4a)
y= 0: ψ= ζ,
(4b)
y→ ∞: ψ= 0,
where the constant ζ is the zeta potential at the shear plane of the wall adjacent to the electrolyte solution having a uniform bulk concentration n∞(z= 0). The solution to Eqs.(3) and (4)
is the well-known Gouy–Chapman result[8,9,11],
(5) ψ= 2 ln 1+ γ exp(−κy) 1− γ exp(−κy) , where γ = tanh(Zeζ/4kT ).
If the constant surface charge density σ , instead of the sur-face potential ζ , is known at the plane wall, the boundary con-dition specified by Eq. (4a)should be replaced by the Gauss
condition, (6) y= 0: dψ dy = − 4π σ ε .
The solution for ψ given by Eq.(5)still holds for this condition, with the relation between ζ and σ as
(7) ζ =2kT Ze sinh −12π Zeσ εκkT .
Equation(7)indicates that σ increases with an increase in κ for the case of constant surface potential and ζ decreases with an increase in κ for the case of constant surface charge density.
2.2. Induced electric field distribution
The ionic concentrations n+and n−in the fluid undergoing diffusioosmosis along the plane wall are not uniform in both tangential (z) and normal (y) directions, and their gradients in the tangential direction can give rise to a “diffusion current” distribution. To prevent a continuous separation of the coun-terions and coions, an electric field distribution along the tan-gential direction arises spontaneously in the electrolyte solution to produce another electric current distribution which exactly balances the diffusion current [15–19]. This induced electric field generates an electroosmotic flow of the fluid parallel to the plane wall, in addition to the chemiosmotic flow caused by the electrolyte gradient directly. Both the chemiosmotic and the electroosmotic flows also generate an electric current distribu-tion by ionic convecdistribu-tion (known as the relaxadistribu-tion 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 (8) J±= −D± ∇n±±ZekTn±(∇ψ − E) + n±u,
where u= u(y)ezis the fluid velocity relative to the plane wall
in the 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(y)ez is the macroscopic electric field induced by the
concentration gradient of the electrolyte, and the principle of superposition for the electric potential is used. To have no net electric current arising from the cocurrent diffusion (conduc-tion), electric migration, and diffusioosmotic convection of the cations and anions, one must require that J+= J−= J (obvi-ously, the normal component of J vanishes and the ionic fluxes induced by ∇ψ in Eq. (8) are balanced by the normal com-ponents of the diffusive ionic fluxes as required by the Boltz-mann distribution given by Eq.(2)). Applying this constraint to Eq.(8), we obtain E=kT Ze ∇n∞ n∞(z= 0) (1+ β)e−ψ− (1 − β)eψ (1+ β)e−ψ+ (1 − β)eψ (9) + Pe sinh ψ (1+ β)e−ψ+ (1 − β)eψ u U∗ , where (10) U∗= ε|∇n ∞| 4π ηn∞(z= 0) kT Ze 2 =2kT ηκ2|∇n ∞|,
which is a characteristic value of the diffusioosmotic velocity,
(11) β=D+− D− D++ D−, (12) Pe= 4n ∞(z= 0)U∗ (D++ D−)|∇n∞|= 8n∞(z= 0)kT (D++ D−)ηκ2,
and η is the fluid viscosity. Evidently,−1 β 1, with the upper and lower bounds occurring as D−/D+→ 0 and ∞,
re-spectively. Typical values of the physical quantities in Eqs.(9)– (12) are U∗ = 10−5 m/s, D± = 10−9 m2/s, n∞(z = 0)/
|∇n∞| = 10−4m, and Pe of order unity.
The induced electric filed E given by Eq. (9) in a self-consistent way depends on the local electrostatic potential ψ and fluid velocity u. It indicates that E is collinear with and pro-portional to the tangentially imposed electrolyte gradient∇n∞. If we consider the situation that κy→ ∞, then ψ → 0 and Eq.(9)for the induced electric field caused by the imposed elec-trolyte concentration gradient reduces to its bulk-phase quan-tity, (13) E∞=kT Ze β∇n∞ n∞(z= 0).
For the special case of an uncharged wall (ζ = 0), E at any location y is also identical to this bulk-phase quantity. Note that
E∞is linearly proportional to the parameter β, but E(y) is not necessarily to vanish if β= 0, as shown in Eq.(9).
2.3. Fluid velocity distribution
For the steady diffusioosmotic flow along a plane wall, the momentum balances on the symmetric electrolyte solution, which is taken to be incompressible and Newtonian, in the y and z directions give
(14a) ∂p ∂y+ Ze(n+− n−) dψ dy = 0, (14b) ηd 2u dy2 = ∂p ∂z− Ze(n+− n−)E,
where p(y, z) is the pressure distribution. The boundary condi-tions for u at the no-slip wall and at infinity are
(15a)
y= 0: u= 0,
(15b)
y→ ∞: du
dy= 0.
After the substitution of Eq. (2) into Eq. (14a) based on the assumption that the equilibrium ionic distributions are not affected by the net electrolyte flux J, which is warranted if |∇n∞|/κn∞(z= 0) 1, the pressure distribution can be
de-termined as
(16)
Here, p∞ is the pressure far away from the wall, which is a constant in the absence of the applied pressure gradient, and the electric potential distribution ψ(y) is given by Eq.(5).
Substituting the ionic concentration distributions of Eq.(2)
and the pressure profile of Eq.(16)into Eq.(14b)and then per-forming the integration with respect to y twice subject to the boundary conditions in Eqs.(15), we obtain
u U∗= κy 0 κy ∞ cosh ψ− 1 (17) +Zen∞(z= 0) kT|∇n∞| Esinh ψ d(κy) d(κy).
After the substitution of Eq.(17)for u and Eq.(5)for ψ into Eq.(9), the induced electric field E can be numerically solved as a function of the dimensionless parameters κy, Zeζ /kT , β, and Pe. With the known results of ψ and E, the diffusioosmotic velocity distribution of the electrolyte solution as a function of Zeζ /kT , β, and Pe can be determined from Eq.(17)with the numerical integrations. Evidently, u/U∗= 0 everywhere if
ζ= 0. It is understood that, for a given value of κy, the quantity u/U∗ with specified values−Zeζ/kT and β is equal to that with the values Zeζ /kT and−β. The bulk-phase diffusioos-motic velocity u∞of the electrolyte solution at a large distance from the plane wall as a function of the relevant parameters can be obtained from the distribution u by taking κy→ ∞.
In previous studies of the diffusioosmosis of electrolyte so-lutions along a plane wall[8,15–18], the constant bulk-phase quantity of the induced electric field E∞given by Eq.(13), tak-ing ψ= 0 everywhere, was used in Eq.(14b)and its solution gives (18) u∞ U∗ = β Zeζ kT + 4 ln cosh Zeζ 4kT .
In this equation, the electroosmotic contribution represented by the first term on the right-hand side and the chemiosmotic con-tribution denoted by the second term are decoupled as a result of the assumption of a constant induced electric field.
2.4. Results and discussion
The distribution of the macroscopic electric field E(y) in-duced by a concentration gradient of a symmetric electrolyte prescribed parallel to a plane wall can be numerically deter-mined after the substitution of Eqs.(17) and (5)into Eq.(9). A simple method of numerical calculation is to make an initial guess of the fluid velocity distribution u1(κy)for a given
com-bination of the dimensionless parameters Zeζ /kT , Pe, and β, and to obtain the resulting induced electric field E1(κy)from
Eq.(9). Then, the next result of the velocity distribution u2(κy)
can be determined from the double integral involving E1(κy)in
Eq.(17). If the difference between u2(κy)and u1(κy)is beyond
the tolerable error, the same procedure will be repeated until an acceptable result of the velocity distribution is obtained.
The induced electric field normalized by its bulk-phase quantity, ZeEn∞(z= 0)/βkT |∇n∞|, as a function of the
elec-trokinetic distance κy from the wall is plotted in Fig. 2 for
(a)
(b)
Fig. 2. Plots of the normalized electric field induced by an electrolyte gradient along a plane wall versus the electrokinetic coordinate κy for various values of the parameter β: (a) Zeζ /kT= 2; (b) Zeζ/kT = 6. The solid curves represent the case Pe= 1 and the dashed curves denote the case Pe = 0.
several values of the parameters Zeζ /kT , Pe, and β. Note that each curve with specified values of −Zeζ/kT and β in this figure would be identical to that with the values Zeζ /kT and −β. As expected, the magnitude of the normalized in-duced electric field in general is a sensitive function of the electrokinetic coordinate κy and becomes infinity for the spe-cial case of β = 0, irrespective of the parameters Zeζ/kT , Pe, and κy. The normalized induced electric field approaches unity (the bulk-phase value) as κy 5 and has a typical value −ζ/|ζ|β at the wall (with κy = 0) if Ze|ζ |/kT 1. For the situation that ζβ > 0, the direction of E(y) can be oppo-site to that of its bulk-phase quantity E∞. When Pe= 0, the magnitude of ZeEn∞(z= 0)/βkT |∇n∞| − 1 (or the
devia-tion of the induced electric field from its bulk-phase quantity) decreases with an increase in κy, increases with an increase in Ze|ζ|/kT , and decreases with an increase in the magni-tude of β (vanishes in the limits of β = ±1), for an other-wise specified condition. When the value of Pe is finite, the magnitude of ZeEn∞(z= 0)/βkT |∇n∞| − 1 may not be a
(a)
(b)
Fig. 3. Plots of the normalized diffusioosmotic velocity along a plane wall versus the electrokinetic coordinate κy for various values of the parameter β: (a) Zeζ /kT= 2; (b) Zeζ/kT = 6. The solid curves represent the case Pe = 1 and the dashed curves denote the case Pe= 0.
monotonic function of κy, and the effect of the electrolyte con-vection on the local induced electric field in the double layer can be quite significant for the case of high zeta potential at the wall.
The dimensionless diffusioosmotic velocity distribution
u(κy)/U∗ of an electrolyte solution along a plane wall can be numerically calculated using Eq.(17)with the known distri-butions of ψ and E. The results are plotted inFigs. 3 and 4for several values of the parameters Zeζ /kT , Pe, and β. For fixed values of Zeζ /kT , Pe, and β, the value of u/U∗ approaches a constant (the bulk-phase value u∞/U∗) as κy 5. When Pe= 0 and β = 0, Eq.(9)indicates that E vanishes everywhere andFig. 3shows that the fluid flows toward higher electrolyte concentration (u is negative) which is due to the chemiosmotic contribution only. When Pe= 0 and the product of ζ and β is negative, u is also negative, meaning that the diffusioosmotic flow is in the direction of increasing electrolyte concentration. The magnitude of u/U∗ increases monotonically with an
in-(a)
(b)
Fig. 4. Plots of the normalized diffusioosmotic velocity along a plane wall versus the electrokinetic coordinate κy for various values of the parameter Pe: (a) Zeζ /kT= 2; (b) Zeζ/kT = 6. The solid curves represent the case β= −0.2 and the dashed curves denote the case β = 0.
crease in the electrokinetic distance κy from the wall, with an increase in Ze|ζ|/kT , and with an increase in |β|, for an oth-erwise specified condition. When Pe= 0 and ζβ is positive, the direction of the diffusioosmotic flow depends on the com-bination of parameters Zeζ /kT , β, and κy. If the magnitude of β is sufficiently large, the fluid flows against the electrolyte concentration gradient (u is positive) and u/U∗is a monotonic increasing function of κy and of Ze|ζ |/kT . If the magnitude of β is sufficiently small for a given value of Zeζ /kT , the fluid flows toward the opposite direction (u is negative) and the mag-nitude of u/U∗ is also a monotonic increasing function of κy and of Ze|ζ |/kT . For a specified value of Zeζ/kT with an intermediate magnitude of β, the diffusioosmotic flow may re-verse its direction from along with the concentration gradient to against it as the value of κy increases not much from zero.
When the value of Pe is finite, the dependence of u on κy is similar to that for the case of Pe= 0 if the value of Ze|ζ |/kT is small, but u can be positive for any given value of β if the
(a)
(b)
Fig. 5. The normalized bulk-phase diffusioosmotic velocity along a plane wall: (a) plots versus the dimensionless surface potential Zeζ /kT for various values of the parameter β; (b) plots versus β for various values of Zeζ /kT . The solid curves represent the case Pe= 1 and the dashed curves denote the case Pe = 0.
value of Ze|ζ |/kT is large (the relaxation effect on the diffu-sioosmotic flow is quite significant in this case). In general, the value of u/U∗increases monotonically and remarkably with an increase in the value of Pe for specified values of κy, Zeζ /kT , and β except for the situation that the value of ζ /|ζ |β is close to unity. In the limit of Pe→ ∞, u/U∗is finite. Note that the cases with Pe 10, which are not likely to exist in practice, is exhibited inFig. 4for the sake of numerical comparison.
InFig. 5, the normalized diffusioosmotic velocity u∞/U∗of the electrolyte solution far away from the plane wall is plotted versus the parameters Zeζ /kT and β for the cases of Pe= 0 and 1. Maps showing the direction of this velocity are also drawn inFig. 6. The dependence of u∞/U∗ on Zeζ /kT , β, and Pe is quite similar to that of u/U∗for a given value of κy. When Pe= 0 and the product of ζ and β is negative (in the sec-ond and fourth quadrants inFig. 6a), u∞ is negative and the electrolyte solution flows toward higher concentration. When Pe= 0 and the product of ζ and β is positive (in the first and third quadrants inFig. 6a), the bulk-phase diffusioosmotic
ve-(a)
(b)
Fig. 6. Maps showing the direction of the bulk-phase diffusioosmotic velocity along a plane wall: (a) Pe= 0; (b) Pe = 1.
locity may reverse its direction from against the concentration gradient to along with it as Ze|ζ |/kT increases not much from zero for the case of a small magnitude of β (in addition to a re-versal occurring at ζ= 0), or as |β| decreases from 1 to 0 for a given value of Zeζ /kT . When the value of Pe is finite, as indi-cated inFig. 6b, the probability of the diffusioosmotic flow of the electrolyte solution in the direction toward higher concen-tration for a combination of β and Zeζ /kT is greatly reduced, due to the effect of the electrolyte convection. Evidently, this effect is quite significant, irrespective of the thickness of the electric double layer adjacent to the wall.
3. Diffusioosmosis of an electrolyte solution in a capillary slit
In this section, we deal with the steady diffusioosmotic flow of a solution of a symmetric electrolyte in a capillary channel between two identical parallel charged plates with a separation distance 2h, as shown inFig. 1b. Again, the end effects are ne-glected. Owing to the planar symmetry of the system, we need consider only the half region 0 y h, where y measures dis-tance from the median plane between the capillary walls in a normal direction. The analysis for this case is similar to that
presented in the previous section for the case of diffusioosmo-sis along a single plane wall.
3.1. Electrostatic potential distribution
For the electrolyte solution in the capillary slit, the electro-static potential distribution ψ(y) is still governed by Eq. (3), but the boundary conditions become
(19a)
y= h: ψ= ζ,
(19b)
y= 0: dψ
dy = 0,
where ζ is the zeta potential at the capillary walls.
Integration of Eq.(3)from y= 0 to y = y using Eq.(19b)
leads to (20) dψ dy = 2κ cosh2 ψ 2 − cosh2 ψc 2 1/2 ,
where ψc= ψ(0) represents the potential at the median plane
between the slit walls. Integrating Eq.(20)again from y= 0 to
y= y, one obtains (21) b F b,π 2 − F (b, φ) = κy, where (22) φ= sin−1cosh(ψc/2) cosh(ψ/2), (23) b=cosh(ψc/2) −1 ,
and F (b, φ) is the elliptic integral of the first kind,
(24) F (b, φ)= φ 0 (1− b2sin2θ )−1/2dθ.
Equation(21)is the solution for the potential profile ψ(y) in an implicit form, and the quantity ψc/ζ can be determined as a
function of the parameters Zeζ /kT and κh by Eq.(21)setting
y= h and using Eq.(19a).
If the surface charge density σ , instead of the surface po-tential ζ , is known at the capillary walls, the boundary condi-tion(19a)should be replaced by the Gauss condition,
(25)
y= h: dψ
dy = 4π σ
ε .
Substitution of this into Eq.(20)yields
(26) coshZeζ kT − cosh Zeψc kT = 1 2 4π Zeσ εκkT 2 ,
which provides the relation between ζ and σ for an arbitrary value of κh.
3.2. Fluid velocity distribution
The momentum equations for the steady flow of the elec-trolyte solution in a capillary slit with an applied concentration gradient of the electrolyte parallel to the slit walls are still given
by Eqs.(14), and the induced electric field distribution can also be expressed as Eq.(9). The boundary conditions for the fluid velocity u are (27a) y= h: u= 0, (27b) y= 0: du dy = 0.
Substituting Eq.(2)into Eq.(14a), we solve for the pressure distribution to obtain
(28)
p= p0+ 2n∞(z)kT (cosh ψ− cosh ψc),
where p0 is the pressure on the mid-plane between the slit
walls. Substituting Eqs.(28) and (2) into Eq.(14b) and then performing the integration with respect to y twice subject to the boundary conditions in Eqs.(27)yields
u U∗= κy κh κy 0 cosh ψ− cosh ψc (29) +Zen∞(z= 0) kT|∇n∞| Esinh ψ d(κy) d(κy).
After the substitution of u expressed by Eq.(29)and ψ calcu-lated from Eq. (21)into Eq. (9), the induced electric field E can be numerically solved. With the results of ψ and E, the diffusioosmotic velocity distribution of the electrolyte solution as a function of the relevant parameters can be calculated from Eq.(29)through the numerical integrations.
3.3. Results and discussion
The distribution of the macroscopic electric field E(y) in-duced by a concentration gradient of a symmetric electrolyte prescribed parallel to the slit walls can be numerically deter-mined after substituting u/U∗ in the form of Eq.(29) and ψ calculated from Eq. (21) into Eq. (9). This induced electric field normalized by its quantity at the midplane between the slit walls, E(0), as a function of the normalized coordinate y/ h is plotted inFigs. 7 and 8for some values of the dimensionless parameters Zeζ /kT , β, Pe, and κh. When Pe= 0, E(y)/E(0) is positive and its value increases with an increase in y/ h from unity at the symmetric plane of the slit (with y= 0) to a maxi-mum at the slit wall (with y= h), increases with an increase in
κh, decreases with an increase in Ze|ζ |/kT , and increases with
an increase in ζ /|ζ|β if it is not too close to unity (vanishes in the limit of ζ /|ζ |β = ±1), for an otherwise specified condition. When the value of Pe is finite, the value of E(y)/E(0) is larger than that for the case of Pe= 0 if the magnitude of Zeζ/kT is small and the value of ζ /|ζ|β is not too close to unity, but it may not be a monotonic function of y/ h and may become neg-ative if the magnitude of Zeζ /kT is large. In general, the effect of the electrolyte convection on the local induced electric field in the slit can be quite significant even for the case of low zeta potential at the walls.
The normalized diffusioosmotic velocity distribution u(y)/
U∗of an electrolyte solution in a slit can be numerically calcu-lated using Eq.(29)with the known results of ψ and E. This
(a)
(b)
Fig. 7. Plots of the normalized electric field induced by an electrolyte gradi-ent parallel to the walls of a slit versus the dimensionless coordinate y/ h for the case of κh= 1 with various values of the parameter β: (a) Zeζ/kT = 2; (b) Zeζ /kT= 6. The solid curves represent the case Pe = 1 and the dashed curves denote the case Pe= 0.
distribution is plotted inFigs. 9 and 10for several values of the parameters Zeζ /kT , β, Pe, and κh. The diffusioosmotic ve-locity u/U∗is a monotonic increasing function of ζ /|ζ|β. In general, the magnitude of u/U∗decreases monotonically with an increase in the normalized coordinate y/ h. When Pe= 0 and
β= 0, the fluid flows toward higher electrolyte concentration
(u is negative) which is due to the chemiosmotic contribution only. When Pe= 0 and ζ/|ζ|β is not too close to unity, u is negative, meaning that the diffusioosmotic flow is in the direc-tion of increasing electrolyte concentradirec-tion. The magnitude of
u/U∗ increases with an increase in Ze|ζ |/kT and with a
de-crease in ζ /|ζ |β, for an otherwise specified condition. When Pe= 0 and ζ/|ζ|β approaches unity, the fluid flows against the electrolyte concentration gradient (u is positive) and u/U∗is a monotonic increasing function of Ze|ζ |/kT . When the value of Pe is finite, the dependence of u on y/ h is similar to that for
(a)
(b)
Fig. 8. Plots of the normalized electric field induced by an electrolyte gradient parallel to the walls of a slit versus the dimensionless coordinate y/ h for the case of β= 0 with various values of the parameter κh: (a) Pe = 0; (b) Pe = 1. The solid curves represent the case Zeζ /kT= 2 and the dashed curves denote the case Zeζ /kT= 6.
the case of Pe= 0 if the value of Ze|ζ |/kT is small, but u can be positive for any given value of β if the value of Ze|ζ |/kT is large. In general, the value of u/U∗increases monotonically and strikingly with an increase in the value of κh and with an increase in the value of Pe (the relaxation effect on the dif-fusioosmotic flow is quite significant) for specified values of
y/ h, Zeζ /kT , and β except for the situation that the value of
ζ /|ζ|β is close to unity. In the limit of Pe → ∞, u/U∗ is fi-nite.
InFig. 11, the normalized diffusioosmotic velocity u(0)/U∗ of the electrolyte solution at the central plane of the slit is plotted versus the parameters κh and Zeζ /kT at specified values of Pe and β. Maps showing the direction of this ve-locity for a typical value of κh are also drawn in Fig. 12. The dependence of u(0)/U∗ on Zeζ /kT , β, κh, and Pe is quite similar to that of u/U∗for a given value of y/ h. When
(a)
(b)
Fig. 9. Plots of the normalized diffusioosmotic velocity in a slit versus the di-mensionless coordinate y/ h for the case of κh= 1 with various values of the parameter β: (a) Zeζ /kT= 2; (b) Zeζ/kT = 6. The solid curves represent the case Pe= 1 and the dashed curves denote the case Pe = 0.
Pe= 0 and the product of ζ and β is negative (in the sec-ond and fourth quadrants inFig. 12a), u∞is negative and the electrolyte solution flows toward higher concentration, analo-gous to the case of diffusioosmosis along a plane wall dis-cussed in the previous section. When Pe= 0 and the prod-uct of ζ and β is positive (in the first and third quadrants inFig. 12a), the bulk-phase diffusioosmotic velocity may re-verse its direction from against the concentration gradient to along with it as Ze|ζ |/kT increases not much from zero for all practical cases of β (in addition to a reversal occur-ring at ζ = 0), or as |β| decreases from 1 to 0 for a small magnitude of Zeζ /kT . When the value of Pe is finite, as indicated in Fig. 12b, the probability of the diffusioosmotic flow of the electrolyte solution in the direction toward higher concentration for a combination of β and Zeζ /kT is again greatly reduced, due to the effect of the electrolyte convec-tion.
(a)
(b)
Fig. 10. Plots of the normalized diffusioosmotic velocity in a slit versus the dimensionless coordinate y/ h for the case of Zeζ /kT = 4: (a) κh = 1; (b) Pe= 1. The solid curves represent the case β = −0.2 and the dashed curves denote the case β= 0.
4. Concluding remarks
A theoretical study of the steady diffusioosmotic flows of solutions of symmetric electrolytes parallel to a charged plane wall and in a capillary slit is presented in this work. It is as-sumed that the fluid is only slightly nonuniform in the elec-trolyte concentration along the plane walls, but no assumption is made about the thickness of the electric double layers ad-jacent to the walls. The effect of lateral distributions of the electrolyte ions (or of the electrostatic potential) and the effect of ionic convection caused by the diffusioosmotic flow (relax-ation effect) on the tangential electric field induced by the ap-plied concentration gradient are taken into account. The plane walls may have either a constant surface potential or a constant surface charge density of an arbitrary quantity. By solving the Poisson–Boltzmann equation and the modified Navier–Stokes equation applicable to the system, the electrostatic potential distribution, the induced electric field distribution, and the
pres-(a)
(b)
Fig. 11. The normalized diffusioosmotic velocity at the midplane in a slit for various values of the parameter β: (a) plots versus κh for the case of Zeζ /kT= 4; (b) plots versus Zeζ/kT for the case of κh = 1. The solid curves represent the case Pe= 1 and the dashed curves denote the case Pe = 0.
sure distribution under the influence of the imposed electrolyte gradient are determined either analytically or semianalytically. Numerical results for the local and bulk-phase diffusioosmotic velocities as functions of relevant parameters are presented in detail. The results show that the effect of the deviation of the local induced tangential electric field inside the double layer from its bulk-phase quantity and the relaxation effect can not be neglected in the evaluation of the diffusioosmotic velocity of electrolyte solutions along the walls, even for the case of very thin double layer.
Our analysis also indicates that the velocity variation in the lateral direction for the diffusioosmotic flow of an electrolyte solution can be significant only inside the electric double layer. Therefore, this variation is important for pore size about hun-dreds of nanometers or smaller as there is an upper limit for the double layer thickness. Some membranes made from thin
(a)
(b)
Fig. 12. Maps showing the direction of the diffusioosmotic velocity at the mid-plane in a slit for the case of κh= 1: (a) Pe = 0; (b) Pe = 1.
(about 7 µm) sheets of muscovite mica by the track-etch process
[22]provide a realistic example with such small pores (the pore radius, defined as the radius of a circle of equivalent area, ranges from several to hundreds nanometers), and a slit might represent such a pore reasonably well (the pores are parallel, cylindrical capillaries of rhombic (60◦) cross section).
In contrast to diffusioosmosis, diffusiophoresis refers to the migration of colloidal particles in response to a macroscopic gradient of solute concentration in a solution. Without consid-ering the variation of the induced tangential electric field with the normal position and the effect of ionic convection inside the electric double layer, u∞ given by Eq.(18)has been used as a slip velocity at the surface of a nonconductive charged par-ticle of arbitrary shape with a very thin double layer to derive the steady diffusiophoretic velocity of the particle[15,16]. This terminal velocity turned out to be equal in magnitude but oppo-site in direction to u∞expressed by Eq.(18). When the effect of the lateral distribution of the induced electric field and the relaxation effect in the double layer are taken into account, the correct diffusiophoretic velocity of the dielectric particle should be equal in magnitude but opposite in direction to the result of
It is worth repeating that all the results in this study are ob-tained on the basis of a small external gradient of the electrolyte concentration along the plane wall. If the imposed concentra-tion gradient|∇n∞| is relatively large, then the effect of varia-tion of the electrostatic potential ψ in the double layer adjacent to the wall with the tangential position may not be neglected. However, it is reasonable for one to expect that this effect will lead to quantitatively rather than qualitatively different results.
Acknowledgment
This research was supported by the National Science Coun-cil of the Republic of China under Grant NSC94-2214-E-002-019.
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