- Email: [email protected]
Boundary effect on electrophoresis: finite cylinder in a cylindrical pore
Journal of Colloid and Interface Science 283 (2005) 592–600 www.elsevier.com/locate/jcis
Boundary effect on electrophoresis: finite cylinder in a cylindrical pore Jyh-Ping Hsu a,∗ , Ming-Hong Ku b a Department of Chemical and Materials Engineering, National I-Lan University, I-Lan 26041, Taiwan b Department of Chemical Engineering, National Taiwan University, Taipei 10617, Taiwan
Received 21 July 2004; accepted 2 September 2004
Abstract The boundary effect on electrophoresis is investigated by considering a finite cylindrical particle moving along the axis of a long cylindrical pore under conditions of low surface potential and weak applied electric field. The influence of the thickness of the double layer, the aspect ratio of a particle, the ratio particle radius/pore radius, and the charged conditions of the surfaces of the particle and pore on the electrophoretic behavior of a particle are investigated. We show that the effect of the aspect ratio of a particle on its electrophoretic behavior for the case where the particle is charged and the pore is uncharged is larger than that for the case where the particle is uncharged and the pore is charged. Also, depending on the parameters chosen, increasing the aspect ratio of a particle can either promote or hinder its movement, which is not reported in previous studies, and can play a role in electrophoresis measurements. Because both the electric and the flow fields in the gap between the particle and the pore are mediated by those near the top and the end of the particle, the end effect is large when the double layer is thick. 2004 Elsevier Inc. All rights reserved. Keywords: Electrophoresis; Boundary effect; Cylinder in cylindrical pore; Constant surface potential
1. Introduction Electrophoresis has been adopted widely to characterize the surface properties of micrometer- and submicrometersized entities, including both organic and inorganic particles. It is also employed often as a separation tool in both laboratory and industrial operations. Although it has been studied extensively, both theoretically and experimentally, the electrophoretic behavior of an entity under general conditions has not been reported. The difficulty arises mainly from the resolution of the governing equations, the so-called electrokinetic equations, a set of coupled nonlinear partial differential equations. In particular, when the boundary effect comes into play, solving the electrokinetic equations becomes nontrivial even under idealized conditions. In practice, this effect is usually present, since electrophoresis needs * Corresponding author. On leave from Department of Chemical of Engineering, National Taiwan University. Fax: +886-3-9353731. E-mail address: [email protected] (J.-P. Hsu).
0021-9797/$ – see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2004.09.004
to be conducted in a confined space. In electrophoretic separation, for example, electrophoresis is conducted in a porous medium, and the wall of a pore can have a significant influence on the electrophoretic behavior of the entities in the liquid medium. Microelectrophoresis is another example where electrophoresis is conducted in a narrow space. When a boundary is present, the electrophoretic behavior of a particle may be influenced by several factors such as the increase in the viscous drag, the enhancement of the local electric field near the particle surface, and the effect of possible electroosmotic flow, which can be significant if the boundary is charged. Previous efforts regarding the boundary effect on electrophoresis include, for example, a sphere moving parallel to a plane [1–3], a sphere moving normal to a planar surface [1,3–7], a sphere moving along the axis of a cylindrical pore [1,3,8,9], and a sphere moving at the center [10–13] or at an arbitrary position [14] in a spherical cavity. In practice, particles can assume various shapes other than that of a sphere. Many inorganic particles and biological entities are better described, for instance, as a spheroid or
J.-P. Hsu, M.-H. Ku / Journal of Colloid and Interface Science 283 (2005) 592–600
a cylinder. For a simpler mathematical treatment, however, most of the relevant studies on electrophoresis are focused on spherical particles, while those of nonspherical particles are relatively limited. Ye et al. [15], for example, modeled the electrophoresis of a circular cylindrical particle along the axis of a cylindrical microchannel under conditions of thin double layer. The effect of the applied electric field on the flow field is taken into account by incorporating a slip boundary condition into the hydrodynamic equations. Hsu and Kao [16] examined the influence of the charged conditions of a short cylinder on its electrophoresis along the axis of a cylindrical pore. In a recent study, Liu et al. [17] analyzed the electrophoresis of a cylindrical particle in a long cylindrical pore for the case where the ratio radius of particle/radius of pore is large. Neglecting the end effect of a particle, they were able to solve analytically the case of a concentrically positioned particle, which is a onedimensional problem. Some numerical data were reported for the case of a finite particle under the condition of thin double layer. In this study, the boundary effect of electrophoresis is investigated by considering a finite cylinder moving along the axis of a cylindrical pore under the conditions of low surface potential and weak applied electric field. The results obtained in this study have a variety of practical applications such as the electrophoresis of biocolloid, nanorod, or nanofiber in a narrow duct.
2. Theory Referring to Fig. 1, we consider the electrophoresis of a rigid, nonconducting, cylindrical particle of radius a and length 2d along the axis of a long, nonconducting pore of radius b as a response to an applied electric field E0 of strength E0 . The cylindrical coordinates (r, θ , z) are chosen with origin located at the center of the particle, and E0 in the z-direction. The axisymmetric nature of the present problem suggests that only the (r, z) domain needs to be considered. The equation governing the spatial variation of the electrical potential, ψ , can be derived from the Gauss law, and is the Poisson equation, zj enj ρ =− , ε ε N
∇ 2ψ = −
(1)
593
Fig. 1. Schematic representation of the problem considered. A rigid, nonconducting cylindrical particle of radius a and length 2d is moving along the axis of a long, nonconducting cylindrical pore of radius b as a response to a uniform applied electric field E0 . The cylindrical coordinates (r, θ , z) are chosen with origin located at the center of the particle, and E0 is in the z-direction.
from E0 . It can be shown that ψ1 and ψ2 satisfy ∇ 2 ψ1 = κ 2 ψ1 ,
(2)
∇ 2 ψ2 = 0.
(3) In these expressions, κ −1 = [ j nj 0 (ezj )2 /εkT ]−1/2 is the Debye length, nj 0 is the bulk number concentration of the j th ionic species, k is the Boltzmann constant, and T is the absolute temperature. Suppose that the particle and the pore are held at constant surface potentials ζp and ζw , respectively. Then the boundary conditions associated with Eqs. (2) and (3) are
j
where ∇ 2 is the Laplace operator, ε is the permittivity of the liquid phase, ρ is the space charge density, N is the number of ionic species, nj and zj are respectively the number concentration and the valence of the j th ionic species, and e is the elementary charge. If E0 is weak and both the surface potential of the particle and that of the pore are low, the relaxation effect of the double layer can be neglected. In this case [18], ψ can be expressed as a linear superposition of the equilibrium potential ψ1 arising from the presence of the particle and the pore and a perturbed potential ψ2 arising
ψ 1 = ζp
and n · ∇ψ2 = 0 (on particle surface),
ψ 1 = ζw
and
∂ψ2 = 0, ∂r
r = b,
I0 (κr) and ∇ψ2 = −E0 ez I0 (κb) |z| → ∞, r < b.
ψ 1 = ζw
(4) (5)
for (6)
In these expressions, n is the unit normal vector directed into the liquid phase, ez is the unit vector in the z-direction,
594
J.-P. Hsu, M.-H. Ku / Journal of Colloid and Interface Science 283 (2005) 592–600
and I0 is the modified Bessel function of the first kind of order zero. If the liquid phase is an incompressible Newtonian fluid with constant physical properties and the Reynolds number is small, then the flow field is described by ∇ · u = 0,
(7)
η∇ 2 u − ∇p = ρ∇ψ.
(8)
Here, η and u are respectively the viscosity and the velocity of the fluid, p is the hydrodynamic pressure, and ρ∇ψ is the electric body force acting on the fluid. Suppose that both the surface of the particle and that of the pore are no-slip. Then the boundary conditions associated with Eqs. (7) and (8) can be expressed as u = U ez u = 0,
(on particle surface), r = b,
(9) (10)
I0 (κr) εζw 1− E 0 ez , u = u(r)ez = − η I0 (κb) |z| → ∞, r < b.
(11)
In these expressions, U is the magnitude of the particle velocity in the z-direction and u(r) is the undisturbed electroosmotic velocity in the absence of the particle for an infinitely long pore with surface potential ζw . In our case, the forces acting on the particle include the electrostatic force and the hydrodynamic force. The axisymmetric nature of the present problem suggests that only the z-components of these forces need to be considered. The electrostatic force in the z-direction, FE , can be calculated by x Ft = (12) σ Ez dS, S
where S denotes the particle surface, σ = −εn · ∇ψ1 is the surface charge density, and Ez = −∂ψ2 /∂z is the strength of the local electric field in the z-direction. The hydrodynamic force exerted by the fluid on the particle in the z-direction, FD , comprises the viscous force and the pressure gradient. FD can be evaluated by [19] x ∂(u · t) x FD = (13) η −pnz dS, tz dS + ∂n S
S
where t is the unit tangential vector on particle surface, n is the magnitude of n, and tz and nz are, respectively, the zcomponent of t and that of n. At steady state, the net force acting on the particle in the z-direction vanishes; that is, FE + FD = 0.
(14)
Usually, the evaluation of U involves a trial-and-error procedure in which this expression is used as a criterion to see if an assumed U is appropriate. Owing to the linear
nature of the present problem, this difficulty can be circumvented by adopting a superposition method. According to O’Brien and White [20], the problem under consideration can be decomposed into two subproblems. In the first problem, the particle moves with speed U in the absence of the external electric field. In this case, the particle experiences a hydrodynamic force FD,1 = −U D, D(> 0) being the drag per unit velocity, which depends upon the geometry of the particle and the boundary effect. In the second problem, the external electric field is applied, but the particle is remained fixed. In this case, the particle experiences an electrostatic force FE and a hydrodynamic force FD,2 . Here, FD,2 is the hydrodynamic force acting on the particle due to the motion of the mobile ions in the electric double layer when an external electric field is applied. Since this hydrodynamic force always acts against the motion of an isolated, charged particle, it is often called the electrophoretic retardation force. However, as will be shown later that in the present case FD,2 can be either a drag force or a driving force for particle motion, depending on the parameters chosen. Note that both FE and FD,2 are related to the distortion of the electrical double layer and are functions of κa, d/a, and λ (= a/b). FD,1 (or D) is independent of κa and is a function of d/a and λ. Since FD = FD,1 + FD,2 , Eq. (14) yields FE + FD,2 (15) . D The electrophoretic mobility of a particle is evaluated through the following procedure. In the first problem, ψ1 = 0 and ψ2 = 0, and the electric body force ρ∇ψ can be removed from Eq. (8). Assuming an arbitrary U in Eq. (9) and replacing the boundary condition expressed in Eq. (11) by u = 0, we solve the flow filed from Eqs. (7) and (8) and calculate FD,1 (or D = FD,1 /U ) by Eq.(13). In the second problem, ψ1 and ψ2 are evaluated first by solving Eqs. (2) and (3) subject to the boundary conditions expressed in Eqs. (4)–(6), and FE is calculated by Eq. (12). Substituting ψ1 and ψ2 thus obtained into Eq. (8) for ρ∇ψ and replacing the boundary condition expressed in Eq. (9) by u = 0, the flow field is evaluated by solving Eqs. (7) and (8), and FD,2 and FD,1 are calculated by Eq. (13). U is then determined by substituting D, FE , and FD,2 into Eq. (15). U=
3. Results and discussions The electrophoretic behavior of a particle is examined through numerical simulation. The governing equations and the associated boundary conditions are solved numerically by FlexPDE [21], which is based on a finite element approach. This software was justified to be efficient for the resolution of the boundary-valued problem of the present type [16]. For illustration, we consider two cases: (a) the particle is charged with a constant surface potential ζp∗ = eζp /kT and the pore is uncharged; (b) the particle is uncharged and the pore is charged with a constant surface po-
J.-P. Hsu, M.-H. Ku / Journal of Colloid and Interface Science 283 (2005) 592–600
tential ζw∗ = eζw /kT . Note that, due to the linear nature of the present problem, the result for the case when both the particle and the pore are charged can be recovered by an appropriate linear combination of the results obtained from the above two cases. For a more concise presentation, the scaled electrophoretic mobility µE = U/U0 is used in subsequent discussions, where U0 = ε(kT /e)E0 /η is the electrophoretic velocity of an isolated particle with a constant surface potential kT /e predicted by the Smoluchowski’s theory when an electric field of strength E0 is applied. 3.1. Particle positively charged, pore uncharged According to Eq. (15), the scaled electrophoretic mobility can be expressed as ∗ ∗ µE = FE∗ + FD,2 (16) /D , ∗ =F ∗ where FE∗ = FE /16ηaU0 , FD,2 D,2 /16ηaU0 , and D = D/16ηa. The numerator represents the driving force acting on the particle per unit applied electrical field, and the denominator represents the drag force per unit velocity of the particle in the absence of the applied electric field. Fig. 2
Fig. 2. Variation of D ∗ (a) and scaled mobility µE (b) as a function d/a at various λ for the case when particle is positively charged and pore uncharged. Key: ζp∗ = 1, ζw∗ = 0, and κa = 1.
595
shows the simulated variations of D ∗ and µE as a function of d/a at various λ. Fig. 2a reveals that D ∗ increases with an increase in d/a or λ. This is because the hydrodynamic drag on an uncharged particle increases with the increase in its lateral surface area or as the boundary effect becomes important. Note that if d/a → 0 or λ → 0, D ∗ → 1, which must be satisfied because D = 16ηa for an infinitely thin disk of radius a moving axially in an unbounded fluid [22]. As can be seen in Fig. 2b, if λ is small (or κb = κa/λ is large), µE increases with the increase in d/a when κa = 1; however, the reverse is true, if λ is large (or κb is small). This is because as d/a increases, the rate of increase of D ∗ ∗ ) if λ is small, but the reverse is faster than that of (FE∗ + FD,2 is true if λ is large. Note that |µE | depends weakly on d/a, except when λ is sufficiently small (<0.5). The variations of ∗ as a function of d/a at various λ for the case of FE∗ and FD,2 Fig. 2 are illustrated in Fig. 3. According to Eq. (12), FE∗ is related to the product of σ = −εn · ∇ψ1 and Ez = −∂ψ2 /∂z on the particle surface. However, only the electrostatic force arising from the lateral surface of the particle contributes to FE∗ , since Ez must satisfy the boundary condition described by Eq. (4), and Ez = 0 on the basal surfaces of the particle. If the end effect of the particle is neglected, the present
∗ (b) as a function of d/a at various λ Fig. 3. Variation of FE∗ (a) and FD,2 for the case of Fig. 2.
596
J.-P. Hsu, M.-H. Ku / Journal of Colloid and Interface Science 283 (2005) 592–600
problem becomes one-dimensional, and it can be shown that the local electric field across the gap between the particle and the pore is constant and equals to E0 /(1 − λ2 )ez . This indicates that the Ez in the gap increases with the increase in λ, the so-called squeezing of the applied electric field by the boundary. As can be seen in Fig. 3a, FE∗ increases with the increase in d/a or in λ. The former is expected because the area of the lateral surface of a particle is proportional to d/a. The latter is the consequences of two effects as λ increases, namely, the squeeze of the applied electric field becomes more serious, as mentioned above, and the charge density on the lateral surface of a particle becomes higher due to a more serious deformation of the double layer if λ is sufficiently large (>0.3). Therefore, if λ is small, because both Ez and σ on particle surface are insensitive to the variation in λ, so is FE∗ , which is justified in Fig. 3a. According ∗ is the sum of a viscous term and a pressure to Eq. (13), FD,2 ∗ = F∗ ∗ ∗ term, and we write FD,2 D,2(v) + FD,2(p) . Similarly, D ∗ ∗ ∗ can be expressed as D = D(v) + D(p) . In the present study, the electroosmotic flow is in the −z-direction; so is the vis∗ cous force acting on a fixed particle from the fluid, FD,2(v) . However, the pressure force acting on the particle arising ∗ from the pressure gradient, FD,2(p) , can be in the z-direction or in the −z-direction. As illustrated in Fig. 3b, under the ∗ is positive when λ > 0.4, and negconditions assumed, FD,2 ∗ ∗ > |FD,2(v) | when λ > 0.4. ative if λ < 0.4, that is, FD,2(p) Note that the volumetric flow rate of the fluid in the gap between the particle and the pore must vanish; that is, there is no net flow. This implies that there is a recirculation flow in the gap, a counterclockwise vortex is present on the righthand side of the particle and a clockwise vortex on its lefthand side. Therefore, if λ is sufficiently large (or the gap is sufficiently narrow), a negative pressure gradient must be present to counterbalance these recirculation flows. Fig. 4 shows the variation of µE as a function of κa at ∗ various λ, and the corresponding variations of FE∗ and FD,2 are presented in Fig. 5. Here, a is fixed and κ varies; that
is, the particle radius is fixed and the electrolyte concentration varies. Fig. 4 suggests that, for fixed d/a and λ, µE increases monotonically with the increase in κa. This can be explained by showing that the qualitative behavior of ∗ ) follows the same trend when D ∗ = D ∗ (d/a, λ) (FE∗ + FD,2 is fixed. Fig. 4 also suggests that if the double layer is thicker than the width of the gap, κ(b − a) < 1 or κa < λ(1 − λ), µE is insensitive to the variation in κa. This is because if κ(b − a) < 1, the double layer surrounding the particle deforms due to the boundary condition assigned on the pore, ζw = 0 at r = b, and the electrostatic force FE∗ , which is ∗ , is weakly dependent on κa. The much greater than FD,2 assumption that ζw = 0 implies that the surface charge density σ associated with the equilibrium double layer is on the order of εζp /(b − a), which is estimated by the gradient of ψ1 between the particle and the pore, and almost remains constant for a fixed λ or (b − a). This is in contrast to the case when κ(b − a) > 1, where σ is on the order of εζp κ, which is estimated by the gradient of ψ1 within the double layer, and increases drastically with the increase in κ (or κa). Therefore, as can be seen in Fig. 5a, since Ez is independent on κa, FE∗ will increase drastically with the increase in κ
Fig. 4. Variation of scaled mobility µE as a function of κa at various λ. Key: ζp∗ = 1, ζw∗ = 0, and d/a = 1.
∗ (b) as a function of κa at various λ for Fig. 5. Variation of FE∗ (a) and FD,2 the case of Fig. 4.
J.-P. Hsu, M.-H. Ku / Journal of Colloid and Interface Science 283 (2005) 592–600
(or κa), like that of σ estimated above when κ(b − a) > 1. ∗ shown in Fig. 5b is similar The qualitative behavior of FD,2 to that shown in Fig. 3b. However, if λ is sufficiently large ∗ may have a local maximum as κa varies; that (>0.4), FD,2 ∗ is, FD,2 will become negative if κa is sufficiently large. This ∗ ∗ and FD,2(p) as arises from the competition between FD,2(v) ∗ ∗ κa varies, and |FD,2(v) | > FD,2(p) if κa is sufficiently large. ∗ ∼ 0. This Fig. 5b also suggests that, if κa is small (<1), FD,2 = is because the electroosmotic flow velocity, or the electrical body force, decreases rapidly when κa is small. Figs. 3b and 5b reveal that, regardless of the values of d/a and κa, ∗ is always negative as λ → 0. This is why it is called the FD,2 electrophoretic retardation force in the case of an isolated particle. Fig. 6 shows the variation of µE as a function of κa at various d/a for the case when λ = 0.5. The result of Liu et al. [17] is also presented for comparison. Note that their result is based on the conditions of the end effect of a particle being negligible, that is, a particle being sufficiently long, and LT /LC = 100, LC and LT being respectively the length of particle and that of pore. Fig. 6 indicates that, regardless of the actual length of a particle, the result of Liu et al. [17], is close to the present one for the case when d/a = 10, and the difference between the two is more significant if κa is small. This implies that if the double layer surrounding a particle is thick and comparable to its length, then its end effect is important. For the case of Fig. 6, the end effect is insignificant if d/a exceeds about 10. That is, both the electric field and the flow field in the gap between the particle and the pore are significantly intervened by those near the top and the end of the particle when the double layer is thick. Fig. 6 also suggests that, if κa (or κb) is large, µE increases with the increase in d/a, and the reverse is true if κa (or κb) is small. This is consistent with the result shown in Fig. 2b. Note that the behavior that µE decreases with the increase in d/a (or κd) as κa → 0 (or κ −1 a) is the same as that for
Fig. 6. Variation of scaled mobility µE as a function of κa at various d/a. Discrete symbols: analytical results of Liu et al. [17] for the case when LT /LC = 100, ζp∗ = 1, ζw∗ = 0, and λ = 0.5.
597
the case of a long rod or fiber in an unbounded electrolyte solution [23,24]. 3.2. Particle uncharged, pore positively charged Fig. 7 shows the variation of the scaled electrophoretic mobility of a particle µE as a function of d/a at various λ for the case when the particle is uncharged and the pore positively charged. This figure reveals that |µE | decreases very slightly with the increase in d/a for all values of λ considered. On the other hand, for a fixed d/a, |µE | decreases rapidly with the increase in λ. This can be explained by the fact that the undisturbed electroosmotic velocity u(r) described by Eq. (11) is acting as a driving force for the movement of a particle, and it is fixed if λ (or κb) is fixed. In contrast, due to the overlap of the double layer of the pore, u(r) decreases drastically with the increase in λ (or κb). Note that, in this case, µE is negative for the ranges of the parameters considered, since both the driving forces ∗ are negative, as shown in Fig. 8. Fig. 8a shows FE∗ and FD,2 ∗ that |FE | increases both with the increase in d/a and with that in λ. This can be explained by the same reasoning as that employed in the discussion of Fig. 3a. Note that if FE∗ is negative, a negative charge is induced on the particle surface due to the presence of the positively charged pore. However, if λ is sufficiently small (>0.3), the amount of induced charge on the particle surface decreases rapidly and vanishes as λ → 0, so is |FE∗ |. This is in contrast to the result for the case when the particle is positively charged and the pore is uncharged, where the surface charge density σ is finite and of order εζp κ if λ < 0.3, as can be seen in Fig. 3a. Fig. 8b ∗ | increases with the increase in d/a or in suggests that |FD,2 λ when κa = 1. The former is because the area of the lateral surface of a particle associated with the viscous force is proportioned to d/a. This is similar to the behavior of ∗ here is the drag force D ∗ shown in Fig. 2a, although FD,2 acting as a driving force for the movement of a particle. It
Fig. 7. Variation of scaled mobility µE as a function of d/a at various λ for the case when the sphere is uncharged and the pore positively charged. Key: ζp∗ = 0, ζw∗ = 1, and κa = 1.
598
J.-P. Hsu, M.-H. Ku / Journal of Colloid and Interface Science 283 (2005) 592–600
Fig. 9. Variation of scaled mobility µE as a function of κa at various λ. Key: ζp∗ = 0, ζw∗ = 1, and d/a = 1.
∗ (b) as a function of κa at various λ for Fig. 8. Variation of FE∗ (a) and FD,2 the case of Fig. 7.
can be shown that the volumetric flow rate in the pore is −εζw AC E0 [1 − 2I1 (κb)/κbI0 (κb)]/η, where AC (= πb2 ) is the cross sectional area of the pore, and I1 is the firstorder modified Bessel function of the first kind. This implies that the velocity profile of the fluid in the gap between the particle and the pore is parabolic, and a positive pressure gradient in the z-direction (that is, dp/dz > 0) is necessary ∗ to drive this flow pattern. Therefore, FD,2(p) is always nega∗ tive in this case, and increases with the increase in λ as D(p) does. This is different from the result for the case when the particle is positively charged and the pore uncharged, where ∗ FD,2(p) changes from negative to positive as λ increases. Fig. 9 shows the variation of µE as a function of κa at various λ. This figure indicates that if λ is fixed, |µE | has a local maximum as κa varies. The presence of the local maximum is the net result of two competing driving forces as κa varies, namely, the absolute value of the electrostatic force, |FE∗ |, ∗ | increases with decreases with the increase in κa, but |FD,2 the increase in κa. The value of κa at which |µE | has the maximum increases with the value of λ. As will be shown later, the local maximum occurs at κa ∼ = 10λ, because if ∗ | remains constant and κb = κa/λ > 10 or κa > 10λ, |FD,2 |FE∗ | decreases rapidly as κa varies. Note that in the limit
κa → 0 or κa → ∞; |µE | depends only on λ when d/a is fixed and decreases with the increase in λ due to a more serious viscous drag. It is not surprising, therefore, that the curve for λ = 0.7 crosses that of λ = 0.6 twice at a large κa in Fig. 9, because the |µE | of the former must be smaller than that of the latter as κa → ∞. Moreover, this dependence of |µE | on λ is more appreciable when κa is small than that when it is large, as can be seen in Fig. 9. All of the results of µE described above can be justified by the correspond∗ presented in Fig. 10. Fig. 10a ing results of FE∗ and FD,2 reveals that for a fixed λ, |FE∗ | decreases with the increase in κa. However, if the double layer surrounding the pore surface is sufficiently thick with κ(b − a) < 1 or κa < λ(1 − λ), |FE∗ | becomes insensitive to the variation in κa. This is because if κ(b − a) < 1, the charge induced on the lateral surface of a particle is on the order of −εζw /(b − a). On the other hand, if κ(b − a) > 1, the influence of the double layer surrounding the pore surface on the lateral surface of the particle becomes unimportant, and the amount of charge induced on it decreases rapidly as κa increases; so does |FE∗ |. ∗ | increases with the Fig. 10b indicates that for a fixed λ, |FD,2 increase in κa, but if the electrical double layer is sufficiently ∗ | becomes insensitive to thin with κb = λ/κa > 10, |FD,2 the variation in κa. This is because if κb > 10, the undisturbed electroosmotic velocity u(r) described by Eq. (11) is roughly constant, and so is the drag force acting on the particle by the fluid. On the other hand, if κb < 10, u(r), and ∗ |, increases rapidly with the increase in κa accordingly, |FD,2 (or κb) when λ is fixed. Fig. 10b also suggests that if κa is ∗ | has a local minimum as λ varies. sufficiently small, |FD,2 This arises from the net result of two competing effects as λ varies, namely, u(r) decreases with the increase in λ (or decrease in κb) as is just mentioned, and the gap between the particle and the pore becomes narrower as λ increases so that the viscous drag increases with the increase in λ. Note that if κa is sufficiently large, the later always dominates, as is observed in the discussion of Fig. 8b, where κa = 1.
J.-P. Hsu, M.-H. Ku / Journal of Colloid and Interface Science 283 (2005) 592–600
599
Fig. 11. Variation of scaled mobility µE as a function of κa at various d/a. Discrete symbols, analytical result of Liu et al. [17] for the case when LT /LC = 100, ζp∗ = 0, ζw∗ = 1, and λ = 0.5.
∗ (b) as a function of κa at various λ Fig. 10. Variation of FE∗ (a) and FD,2 for the case of Fig. 9.
Fig. 11 shows the variation of µE as a function of κa at various d/a when λ = 0.5. For comparison, the analytical result of Liu et al. [17], which is valid for the case when a particle is sufficiently long (LT /LC = 100), or the end effect is negligible, is also presented. As in the case of Fig. 6, the performance of the solution of Liu et al. [17] depends upon the magnitudes of κa and d/a. Fig. 11 also suggests that, if κa is fixed, |µE | decreases with the increase in d/a, which is consistent with the result of Fig. 7 where the drag force D ∗ is more significant than the sum of the driving forces, FE∗ + ∗ . However, the variation of |µ | with d/a is significant FD,2 E only if κa is sufficiently large (>3). This behavior is similar to that for the case when the particle is positively charged and the pore uncharged, where the variation of |µE | with d/a is significant when κa > 2, as shown in Fig. 6. As in the case of Fig. 6, the end effect of the particle becomes unimportant if d/a exceeds about 10. As discussed previously, due to its linear nature, the present problem can be decomposed into two subproblems, and the mobility of a particle can be obtained directly without a trial-and-error procedure as that often adopted in the literature. In our approach, evaluating three independent forces, namely, FD,1 (or D) in the first problem, and FE and
FD,2 in the second problem, is sufficient. In fact, the first problem can be treated as a sedimentation problem and the second one as an electroosmosis problem. This suggests that the present numerical approach is applicable to these problems. In our analysis, we assume that the pore is sufficiently long so that the electroosmotic flow inside is fully developed. Fig. 12a shows the variation of the scaled transition length of a pore Ltc /a, Ltc being the length of a pore necessary to reach fully developed electroosmotic flow, as a function of λ (= a/b) for the case when the particle is uncharged and the pore positively charged; that as a function of κa is presented in Fig. 12b. Here, Ltc is chosen so that the following conditions are satisfied for Lt > Ltc : (a) The electric potential distribution remains essentially unchanged, and the boundary conditions at infinity are satisfied. (b) The flow field in the first problem remains essentially unchanged, and the velocity at infinity vanishes. (c) The flow field in the second problem remains essentially unchanged, and fully developed electroosmotic flow is achieved. Fig. 12a reveals that Ltc /a declines as a/b increases. This is because the larger the a/b, the smaller is the gap between particle and pore, and therefore, the more difficult it is for the flow field to extend itself. Note that Ltc /a is uninfluenced by the scaled length of a particle d/a. Fig. 12b suggests that the smaller the κa, the larger the Ltc /a. This is because the thicker the double layer the more appreciably it is distorted by the particle. As can be seen in Fig. 12b, Ltc /a approaches a constant value as κa → 0. This is because, in this case, the equilibrium electric field is described by a Laplace equation. As κa → ∞, Ltc /a approaches a constant value of 3.5. If κa is large, Ltc /a is mainly determined by the flow field, which is related to a/b. The constant value of 3.5 is close to the value predicted by Liu et al., i.e., 3. In fact, their Fig. 12 suggests that Ltc /a is better estimated by 3.5.
600
J.-P. Hsu, M.-H. Ku / Journal of Colloid and Interface Science 283 (2005) 592–600
ratio of a particle for the case when a particle is charged and the pore uncharged is more important than that for the case when the particle is uncharged and the pore charged. (d) If boundary effect is important, the dependence of the mobility of a particle on its aspect ratio is significant only if the double layer is sufficiently thin. (e) Increasing the aspect ratio of a particle can either promote or hinder its movement, depending upon the parameters chosen. (f) The end effect on electrophoresis is important when the double layer is thick.
Acknowledgment This work is supported by the National Science Council of the Republic of China.
References [1] [2] [3] [4] [5] [6] [7] Fig. 12. Variation of scaled transition length Ltc /a as a function of λ for the case of Fig. 7 (a) and that as a function of κa at d/a = 1 for the case of Fig. 11 (b).
4. Conclusions In summary, the boundary effect on electrophoresis is investigated by considering the electrophoresis of a finite cylindrical particle along the axis of a long cylindrical pore. The results of numerical simulation reveal the followings: (a) Due to the presence of the pore, the mobility of a particle decreases appreciably. (b) The mobility of a particle increases monotonically with the decrease in the thickness of the double layer surrounding it. However, the mobility may have a local maximum as the thickness of the double layer surrounding the pore varies. This local maximum occurs at a thinner double layer when the boundary effect is more important. (c) In general, the influence of the aspect
[8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
H.J. Keh, J.L. Anderson, J. Fluid Mech. 153 (1985) 417. H.J. Keh, S.B. Chen, J. Fluid Mech. 194 (1988) 377. J. Ennis, J.L. Anderson, J. Colloid Interface Sci. 185 (1997) 497. H.J. Keh, L.C. Lien, J. Chin. Inst. Chem. Eng. 20 (1989) 283. H.J. Keh, J.S. Jan, J. Colloid Interface Sci. 183 (1996) 458. M.H. Chih, E. Lee, J.P. Hsu, J. Colloid Interface Sci. 248 (2002) 383. Y.P. Tang, M.H. Chih, E. Lee, J.P. Hsu, J. Colloid Interface Sci. 242 (2001) 121. H.J. Keh, J.Y. Chiou, AIChE J. 42 (1996) 1397. A.A. Shugai, S.L. Carnie, J. Colloid Interface Sci. 213 (1999) 298. A.L. Zydney, J. Colloid Interface Sci. 169 (1995) 476. E. Lee, J.W. Chu, J.P. Hsu, J. Colloid Interface Sci. 196 (1997) 316. E. Lee, J.W. Chu, J.P. Hsu, J. Colloid Interface Sci. 205 (1998) 65. J.W. Chu, W.H. Lin, E. Lee, J.P. Hsu, J. Colloid Interface Sci. 17 (2001) 6289. J.P. Hsu, S.H. Hung, C.Y. Kao, Langmuir 18 (2002) 8897. C. Ye, D. Sinton, D. Erickson, D. Li, Langmuir 18 (2002) 9095. J.P. Hsu, C.Y. Kao, J. Phys. Chem. B 106 (2002) 10605. H. Liu, H.H. Bau, H.H. Hu, Langmuir 20 (2004) 2628. D.C. Henry, Proc. R. Soc. London Ser. A 133 (1931) 106. G. Backstrom, Fluid Dynamics by Finite Element Analysis, Studentlitteratur, 1999. R.W. O’Brien, L.R. White, J. Chem. Soc. Faraday Trans. 2 74 (1978) 1607. PDE Solutions, FlexPDE, Version 2.22. J. Happel, H. Brenner, Low Reynolds Number Hydrodynamics, Nijhoff, The Hague, 1983, chap. 4. J.D. Sherwood, J. Chem. Soc. Faraday Trans. 2 78 (1982) 1091. S.B. Chen, D.L. Koch, J. Colloid Interface Sci. 180 (1996) 466.
Boundary effect on electrophoresis: finite cylinder in a cylindrical pore Jyh-Ping Hsu a,∗ , Ming-Hong Ku b a Department of Chemical and Materials Engineering, National I-Lan University, I-Lan 26041, Taiwan b Department of Chemical Engineering, National Taiwan University, Taipei 10617, Taiwan
Received 21 July 2004; accepted 2 September 2004
Abstract The boundary effect on electrophoresis is investigated by considering a finite cylindrical particle moving along the axis of a long cylindrical pore under conditions of low surface potential and weak applied electric field. The influence of the thickness of the double layer, the aspect ratio of a particle, the ratio particle radius/pore radius, and the charged conditions of the surfaces of the particle and pore on the electrophoretic behavior of a particle are investigated. We show that the effect of the aspect ratio of a particle on its electrophoretic behavior for the case where the particle is charged and the pore is uncharged is larger than that for the case where the particle is uncharged and the pore is charged. Also, depending on the parameters chosen, increasing the aspect ratio of a particle can either promote or hinder its movement, which is not reported in previous studies, and can play a role in electrophoresis measurements. Because both the electric and the flow fields in the gap between the particle and the pore are mediated by those near the top and the end of the particle, the end effect is large when the double layer is thick. 2004 Elsevier Inc. All rights reserved. Keywords: Electrophoresis; Boundary effect; Cylinder in cylindrical pore; Constant surface potential
1. Introduction Electrophoresis has been adopted widely to characterize the surface properties of micrometer- and submicrometersized entities, including both organic and inorganic particles. It is also employed often as a separation tool in both laboratory and industrial operations. Although it has been studied extensively, both theoretically and experimentally, the electrophoretic behavior of an entity under general conditions has not been reported. The difficulty arises mainly from the resolution of the governing equations, the so-called electrokinetic equations, a set of coupled nonlinear partial differential equations. In particular, when the boundary effect comes into play, solving the electrokinetic equations becomes nontrivial even under idealized conditions. In practice, this effect is usually present, since electrophoresis needs * Corresponding author. On leave from Department of Chemical of Engineering, National Taiwan University. Fax: +886-3-9353731. E-mail address: [email protected] (J.-P. Hsu).
0021-9797/$ – see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2004.09.004
to be conducted in a confined space. In electrophoretic separation, for example, electrophoresis is conducted in a porous medium, and the wall of a pore can have a significant influence on the electrophoretic behavior of the entities in the liquid medium. Microelectrophoresis is another example where electrophoresis is conducted in a narrow space. When a boundary is present, the electrophoretic behavior of a particle may be influenced by several factors such as the increase in the viscous drag, the enhancement of the local electric field near the particle surface, and the effect of possible electroosmotic flow, which can be significant if the boundary is charged. Previous efforts regarding the boundary effect on electrophoresis include, for example, a sphere moving parallel to a plane [1–3], a sphere moving normal to a planar surface [1,3–7], a sphere moving along the axis of a cylindrical pore [1,3,8,9], and a sphere moving at the center [10–13] or at an arbitrary position [14] in a spherical cavity. In practice, particles can assume various shapes other than that of a sphere. Many inorganic particles and biological entities are better described, for instance, as a spheroid or
J.-P. Hsu, M.-H. Ku / Journal of Colloid and Interface Science 283 (2005) 592–600
a cylinder. For a simpler mathematical treatment, however, most of the relevant studies on electrophoresis are focused on spherical particles, while those of nonspherical particles are relatively limited. Ye et al. [15], for example, modeled the electrophoresis of a circular cylindrical particle along the axis of a cylindrical microchannel under conditions of thin double layer. The effect of the applied electric field on the flow field is taken into account by incorporating a slip boundary condition into the hydrodynamic equations. Hsu and Kao [16] examined the influence of the charged conditions of a short cylinder on its electrophoresis along the axis of a cylindrical pore. In a recent study, Liu et al. [17] analyzed the electrophoresis of a cylindrical particle in a long cylindrical pore for the case where the ratio radius of particle/radius of pore is large. Neglecting the end effect of a particle, they were able to solve analytically the case of a concentrically positioned particle, which is a onedimensional problem. Some numerical data were reported for the case of a finite particle under the condition of thin double layer. In this study, the boundary effect of electrophoresis is investigated by considering a finite cylinder moving along the axis of a cylindrical pore under the conditions of low surface potential and weak applied electric field. The results obtained in this study have a variety of practical applications such as the electrophoresis of biocolloid, nanorod, or nanofiber in a narrow duct.
2. Theory Referring to Fig. 1, we consider the electrophoresis of a rigid, nonconducting, cylindrical particle of radius a and length 2d along the axis of a long, nonconducting pore of radius b as a response to an applied electric field E0 of strength E0 . The cylindrical coordinates (r, θ , z) are chosen with origin located at the center of the particle, and E0 in the z-direction. The axisymmetric nature of the present problem suggests that only the (r, z) domain needs to be considered. The equation governing the spatial variation of the electrical potential, ψ , can be derived from the Gauss law, and is the Poisson equation, zj enj ρ =− , ε ε N
∇ 2ψ = −
(1)
593
Fig. 1. Schematic representation of the problem considered. A rigid, nonconducting cylindrical particle of radius a and length 2d is moving along the axis of a long, nonconducting cylindrical pore of radius b as a response to a uniform applied electric field E0 . The cylindrical coordinates (r, θ , z) are chosen with origin located at the center of the particle, and E0 is in the z-direction.
from E0 . It can be shown that ψ1 and ψ2 satisfy ∇ 2 ψ1 = κ 2 ψ1 ,
(2)
∇ 2 ψ2 = 0.
(3) In these expressions, κ −1 = [ j nj 0 (ezj )2 /εkT ]−1/2 is the Debye length, nj 0 is the bulk number concentration of the j th ionic species, k is the Boltzmann constant, and T is the absolute temperature. Suppose that the particle and the pore are held at constant surface potentials ζp and ζw , respectively. Then the boundary conditions associated with Eqs. (2) and (3) are
j
where ∇ 2 is the Laplace operator, ε is the permittivity of the liquid phase, ρ is the space charge density, N is the number of ionic species, nj and zj are respectively the number concentration and the valence of the j th ionic species, and e is the elementary charge. If E0 is weak and both the surface potential of the particle and that of the pore are low, the relaxation effect of the double layer can be neglected. In this case [18], ψ can be expressed as a linear superposition of the equilibrium potential ψ1 arising from the presence of the particle and the pore and a perturbed potential ψ2 arising
ψ 1 = ζp
and n · ∇ψ2 = 0 (on particle surface),
ψ 1 = ζw
and
∂ψ2 = 0, ∂r
r = b,
I0 (κr) and ∇ψ2 = −E0 ez I0 (κb) |z| → ∞, r < b.
ψ 1 = ζw
(4) (5)
for (6)
In these expressions, n is the unit normal vector directed into the liquid phase, ez is the unit vector in the z-direction,
594
J.-P. Hsu, M.-H. Ku / Journal of Colloid and Interface Science 283 (2005) 592–600
and I0 is the modified Bessel function of the first kind of order zero. If the liquid phase is an incompressible Newtonian fluid with constant physical properties and the Reynolds number is small, then the flow field is described by ∇ · u = 0,
(7)
η∇ 2 u − ∇p = ρ∇ψ.
(8)
Here, η and u are respectively the viscosity and the velocity of the fluid, p is the hydrodynamic pressure, and ρ∇ψ is the electric body force acting on the fluid. Suppose that both the surface of the particle and that of the pore are no-slip. Then the boundary conditions associated with Eqs. (7) and (8) can be expressed as u = U ez u = 0,
(on particle surface), r = b,
(9) (10)
I0 (κr) εζw 1− E 0 ez , u = u(r)ez = − η I0 (κb) |z| → ∞, r < b.
(11)
In these expressions, U is the magnitude of the particle velocity in the z-direction and u(r) is the undisturbed electroosmotic velocity in the absence of the particle for an infinitely long pore with surface potential ζw . In our case, the forces acting on the particle include the electrostatic force and the hydrodynamic force. The axisymmetric nature of the present problem suggests that only the z-components of these forces need to be considered. The electrostatic force in the z-direction, FE , can be calculated by x Ft = (12) σ Ez dS, S
where S denotes the particle surface, σ = −εn · ∇ψ1 is the surface charge density, and Ez = −∂ψ2 /∂z is the strength of the local electric field in the z-direction. The hydrodynamic force exerted by the fluid on the particle in the z-direction, FD , comprises the viscous force and the pressure gradient. FD can be evaluated by [19] x ∂(u · t) x FD = (13) η −pnz dS, tz dS + ∂n S
S
where t is the unit tangential vector on particle surface, n is the magnitude of n, and tz and nz are, respectively, the zcomponent of t and that of n. At steady state, the net force acting on the particle in the z-direction vanishes; that is, FE + FD = 0.
(14)
Usually, the evaluation of U involves a trial-and-error procedure in which this expression is used as a criterion to see if an assumed U is appropriate. Owing to the linear
nature of the present problem, this difficulty can be circumvented by adopting a superposition method. According to O’Brien and White [20], the problem under consideration can be decomposed into two subproblems. In the first problem, the particle moves with speed U in the absence of the external electric field. In this case, the particle experiences a hydrodynamic force FD,1 = −U D, D(> 0) being the drag per unit velocity, which depends upon the geometry of the particle and the boundary effect. In the second problem, the external electric field is applied, but the particle is remained fixed. In this case, the particle experiences an electrostatic force FE and a hydrodynamic force FD,2 . Here, FD,2 is the hydrodynamic force acting on the particle due to the motion of the mobile ions in the electric double layer when an external electric field is applied. Since this hydrodynamic force always acts against the motion of an isolated, charged particle, it is often called the electrophoretic retardation force. However, as will be shown later that in the present case FD,2 can be either a drag force or a driving force for particle motion, depending on the parameters chosen. Note that both FE and FD,2 are related to the distortion of the electrical double layer and are functions of κa, d/a, and λ (= a/b). FD,1 (or D) is independent of κa and is a function of d/a and λ. Since FD = FD,1 + FD,2 , Eq. (14) yields FE + FD,2 (15) . D The electrophoretic mobility of a particle is evaluated through the following procedure. In the first problem, ψ1 = 0 and ψ2 = 0, and the electric body force ρ∇ψ can be removed from Eq. (8). Assuming an arbitrary U in Eq. (9) and replacing the boundary condition expressed in Eq. (11) by u = 0, we solve the flow filed from Eqs. (7) and (8) and calculate FD,1 (or D = FD,1 /U ) by Eq.(13). In the second problem, ψ1 and ψ2 are evaluated first by solving Eqs. (2) and (3) subject to the boundary conditions expressed in Eqs. (4)–(6), and FE is calculated by Eq. (12). Substituting ψ1 and ψ2 thus obtained into Eq. (8) for ρ∇ψ and replacing the boundary condition expressed in Eq. (9) by u = 0, the flow field is evaluated by solving Eqs. (7) and (8), and FD,2 and FD,1 are calculated by Eq. (13). U is then determined by substituting D, FE , and FD,2 into Eq. (15). U=
3. Results and discussions The electrophoretic behavior of a particle is examined through numerical simulation. The governing equations and the associated boundary conditions are solved numerically by FlexPDE [21], which is based on a finite element approach. This software was justified to be efficient for the resolution of the boundary-valued problem of the present type [16]. For illustration, we consider two cases: (a) the particle is charged with a constant surface potential ζp∗ = eζp /kT and the pore is uncharged; (b) the particle is uncharged and the pore is charged with a constant surface po-
J.-P. Hsu, M.-H. Ku / Journal of Colloid and Interface Science 283 (2005) 592–600
tential ζw∗ = eζw /kT . Note that, due to the linear nature of the present problem, the result for the case when both the particle and the pore are charged can be recovered by an appropriate linear combination of the results obtained from the above two cases. For a more concise presentation, the scaled electrophoretic mobility µE = U/U0 is used in subsequent discussions, where U0 = ε(kT /e)E0 /η is the electrophoretic velocity of an isolated particle with a constant surface potential kT /e predicted by the Smoluchowski’s theory when an electric field of strength E0 is applied. 3.1. Particle positively charged, pore uncharged According to Eq. (15), the scaled electrophoretic mobility can be expressed as ∗ ∗ µE = FE∗ + FD,2 (16) /D , ∗ =F ∗ where FE∗ = FE /16ηaU0 , FD,2 D,2 /16ηaU0 , and D = D/16ηa. The numerator represents the driving force acting on the particle per unit applied electrical field, and the denominator represents the drag force per unit velocity of the particle in the absence of the applied electric field. Fig. 2
Fig. 2. Variation of D ∗ (a) and scaled mobility µE (b) as a function d/a at various λ for the case when particle is positively charged and pore uncharged. Key: ζp∗ = 1, ζw∗ = 0, and κa = 1.
595
shows the simulated variations of D ∗ and µE as a function of d/a at various λ. Fig. 2a reveals that D ∗ increases with an increase in d/a or λ. This is because the hydrodynamic drag on an uncharged particle increases with the increase in its lateral surface area or as the boundary effect becomes important. Note that if d/a → 0 or λ → 0, D ∗ → 1, which must be satisfied because D = 16ηa for an infinitely thin disk of radius a moving axially in an unbounded fluid [22]. As can be seen in Fig. 2b, if λ is small (or κb = κa/λ is large), µE increases with the increase in d/a when κa = 1; however, the reverse is true, if λ is large (or κb is small). This is because as d/a increases, the rate of increase of D ∗ ∗ ) if λ is small, but the reverse is faster than that of (FE∗ + FD,2 is true if λ is large. Note that |µE | depends weakly on d/a, except when λ is sufficiently small (<0.5). The variations of ∗ as a function of d/a at various λ for the case of FE∗ and FD,2 Fig. 2 are illustrated in Fig. 3. According to Eq. (12), FE∗ is related to the product of σ = −εn · ∇ψ1 and Ez = −∂ψ2 /∂z on the particle surface. However, only the electrostatic force arising from the lateral surface of the particle contributes to FE∗ , since Ez must satisfy the boundary condition described by Eq. (4), and Ez = 0 on the basal surfaces of the particle. If the end effect of the particle is neglected, the present
∗ (b) as a function of d/a at various λ Fig. 3. Variation of FE∗ (a) and FD,2 for the case of Fig. 2.
596
J.-P. Hsu, M.-H. Ku / Journal of Colloid and Interface Science 283 (2005) 592–600
problem becomes one-dimensional, and it can be shown that the local electric field across the gap between the particle and the pore is constant and equals to E0 /(1 − λ2 )ez . This indicates that the Ez in the gap increases with the increase in λ, the so-called squeezing of the applied electric field by the boundary. As can be seen in Fig. 3a, FE∗ increases with the increase in d/a or in λ. The former is expected because the area of the lateral surface of a particle is proportional to d/a. The latter is the consequences of two effects as λ increases, namely, the squeeze of the applied electric field becomes more serious, as mentioned above, and the charge density on the lateral surface of a particle becomes higher due to a more serious deformation of the double layer if λ is sufficiently large (>0.3). Therefore, if λ is small, because both Ez and σ on particle surface are insensitive to the variation in λ, so is FE∗ , which is justified in Fig. 3a. According ∗ is the sum of a viscous term and a pressure to Eq. (13), FD,2 ∗ = F∗ ∗ ∗ term, and we write FD,2 D,2(v) + FD,2(p) . Similarly, D ∗ ∗ ∗ can be expressed as D = D(v) + D(p) . In the present study, the electroosmotic flow is in the −z-direction; so is the vis∗ cous force acting on a fixed particle from the fluid, FD,2(v) . However, the pressure force acting on the particle arising ∗ from the pressure gradient, FD,2(p) , can be in the z-direction or in the −z-direction. As illustrated in Fig. 3b, under the ∗ is positive when λ > 0.4, and negconditions assumed, FD,2 ∗ ∗ > |FD,2(v) | when λ > 0.4. ative if λ < 0.4, that is, FD,2(p) Note that the volumetric flow rate of the fluid in the gap between the particle and the pore must vanish; that is, there is no net flow. This implies that there is a recirculation flow in the gap, a counterclockwise vortex is present on the righthand side of the particle and a clockwise vortex on its lefthand side. Therefore, if λ is sufficiently large (or the gap is sufficiently narrow), a negative pressure gradient must be present to counterbalance these recirculation flows. Fig. 4 shows the variation of µE as a function of κa at ∗ various λ, and the corresponding variations of FE∗ and FD,2 are presented in Fig. 5. Here, a is fixed and κ varies; that
is, the particle radius is fixed and the electrolyte concentration varies. Fig. 4 suggests that, for fixed d/a and λ, µE increases monotonically with the increase in κa. This can be explained by showing that the qualitative behavior of ∗ ) follows the same trend when D ∗ = D ∗ (d/a, λ) (FE∗ + FD,2 is fixed. Fig. 4 also suggests that if the double layer is thicker than the width of the gap, κ(b − a) < 1 or κa < λ(1 − λ), µE is insensitive to the variation in κa. This is because if κ(b − a) < 1, the double layer surrounding the particle deforms due to the boundary condition assigned on the pore, ζw = 0 at r = b, and the electrostatic force FE∗ , which is ∗ , is weakly dependent on κa. The much greater than FD,2 assumption that ζw = 0 implies that the surface charge density σ associated with the equilibrium double layer is on the order of εζp /(b − a), which is estimated by the gradient of ψ1 between the particle and the pore, and almost remains constant for a fixed λ or (b − a). This is in contrast to the case when κ(b − a) > 1, where σ is on the order of εζp κ, which is estimated by the gradient of ψ1 within the double layer, and increases drastically with the increase in κ (or κa). Therefore, as can be seen in Fig. 5a, since Ez is independent on κa, FE∗ will increase drastically with the increase in κ
Fig. 4. Variation of scaled mobility µE as a function of κa at various λ. Key: ζp∗ = 1, ζw∗ = 0, and d/a = 1.
∗ (b) as a function of κa at various λ for Fig. 5. Variation of FE∗ (a) and FD,2 the case of Fig. 4.
J.-P. Hsu, M.-H. Ku / Journal of Colloid and Interface Science 283 (2005) 592–600
(or κa), like that of σ estimated above when κ(b − a) > 1. ∗ shown in Fig. 5b is similar The qualitative behavior of FD,2 to that shown in Fig. 3b. However, if λ is sufficiently large ∗ may have a local maximum as κa varies; that (>0.4), FD,2 ∗ is, FD,2 will become negative if κa is sufficiently large. This ∗ ∗ and FD,2(p) as arises from the competition between FD,2(v) ∗ ∗ κa varies, and |FD,2(v) | > FD,2(p) if κa is sufficiently large. ∗ ∼ 0. This Fig. 5b also suggests that, if κa is small (<1), FD,2 = is because the electroosmotic flow velocity, or the electrical body force, decreases rapidly when κa is small. Figs. 3b and 5b reveal that, regardless of the values of d/a and κa, ∗ is always negative as λ → 0. This is why it is called the FD,2 electrophoretic retardation force in the case of an isolated particle. Fig. 6 shows the variation of µE as a function of κa at various d/a for the case when λ = 0.5. The result of Liu et al. [17] is also presented for comparison. Note that their result is based on the conditions of the end effect of a particle being negligible, that is, a particle being sufficiently long, and LT /LC = 100, LC and LT being respectively the length of particle and that of pore. Fig. 6 indicates that, regardless of the actual length of a particle, the result of Liu et al. [17], is close to the present one for the case when d/a = 10, and the difference between the two is more significant if κa is small. This implies that if the double layer surrounding a particle is thick and comparable to its length, then its end effect is important. For the case of Fig. 6, the end effect is insignificant if d/a exceeds about 10. That is, both the electric field and the flow field in the gap between the particle and the pore are significantly intervened by those near the top and the end of the particle when the double layer is thick. Fig. 6 also suggests that, if κa (or κb) is large, µE increases with the increase in d/a, and the reverse is true if κa (or κb) is small. This is consistent with the result shown in Fig. 2b. Note that the behavior that µE decreases with the increase in d/a (or κd) as κa → 0 (or κ −1 a) is the same as that for
Fig. 6. Variation of scaled mobility µE as a function of κa at various d/a. Discrete symbols: analytical results of Liu et al. [17] for the case when LT /LC = 100, ζp∗ = 1, ζw∗ = 0, and λ = 0.5.
597
the case of a long rod or fiber in an unbounded electrolyte solution [23,24]. 3.2. Particle uncharged, pore positively charged Fig. 7 shows the variation of the scaled electrophoretic mobility of a particle µE as a function of d/a at various λ for the case when the particle is uncharged and the pore positively charged. This figure reveals that |µE | decreases very slightly with the increase in d/a for all values of λ considered. On the other hand, for a fixed d/a, |µE | decreases rapidly with the increase in λ. This can be explained by the fact that the undisturbed electroosmotic velocity u(r) described by Eq. (11) is acting as a driving force for the movement of a particle, and it is fixed if λ (or κb) is fixed. In contrast, due to the overlap of the double layer of the pore, u(r) decreases drastically with the increase in λ (or κb). Note that, in this case, µE is negative for the ranges of the parameters considered, since both the driving forces ∗ are negative, as shown in Fig. 8. Fig. 8a shows FE∗ and FD,2 ∗ that |FE | increases both with the increase in d/a and with that in λ. This can be explained by the same reasoning as that employed in the discussion of Fig. 3a. Note that if FE∗ is negative, a negative charge is induced on the particle surface due to the presence of the positively charged pore. However, if λ is sufficiently small (>0.3), the amount of induced charge on the particle surface decreases rapidly and vanishes as λ → 0, so is |FE∗ |. This is in contrast to the result for the case when the particle is positively charged and the pore is uncharged, where the surface charge density σ is finite and of order εζp κ if λ < 0.3, as can be seen in Fig. 3a. Fig. 8b ∗ | increases with the increase in d/a or in suggests that |FD,2 λ when κa = 1. The former is because the area of the lateral surface of a particle associated with the viscous force is proportioned to d/a. This is similar to the behavior of ∗ here is the drag force D ∗ shown in Fig. 2a, although FD,2 acting as a driving force for the movement of a particle. It
Fig. 7. Variation of scaled mobility µE as a function of d/a at various λ for the case when the sphere is uncharged and the pore positively charged. Key: ζp∗ = 0, ζw∗ = 1, and κa = 1.
598
J.-P. Hsu, M.-H. Ku / Journal of Colloid and Interface Science 283 (2005) 592–600
Fig. 9. Variation of scaled mobility µE as a function of κa at various λ. Key: ζp∗ = 0, ζw∗ = 1, and d/a = 1.
∗ (b) as a function of κa at various λ for Fig. 8. Variation of FE∗ (a) and FD,2 the case of Fig. 7.
can be shown that the volumetric flow rate in the pore is −εζw AC E0 [1 − 2I1 (κb)/κbI0 (κb)]/η, where AC (= πb2 ) is the cross sectional area of the pore, and I1 is the firstorder modified Bessel function of the first kind. This implies that the velocity profile of the fluid in the gap between the particle and the pore is parabolic, and a positive pressure gradient in the z-direction (that is, dp/dz > 0) is necessary ∗ to drive this flow pattern. Therefore, FD,2(p) is always nega∗ tive in this case, and increases with the increase in λ as D(p) does. This is different from the result for the case when the particle is positively charged and the pore uncharged, where ∗ FD,2(p) changes from negative to positive as λ increases. Fig. 9 shows the variation of µE as a function of κa at various λ. This figure indicates that if λ is fixed, |µE | has a local maximum as κa varies. The presence of the local maximum is the net result of two competing driving forces as κa varies, namely, the absolute value of the electrostatic force, |FE∗ |, ∗ | increases with decreases with the increase in κa, but |FD,2 the increase in κa. The value of κa at which |µE | has the maximum increases with the value of λ. As will be shown later, the local maximum occurs at κa ∼ = 10λ, because if ∗ | remains constant and κb = κa/λ > 10 or κa > 10λ, |FD,2 |FE∗ | decreases rapidly as κa varies. Note that in the limit
κa → 0 or κa → ∞; |µE | depends only on λ when d/a is fixed and decreases with the increase in λ due to a more serious viscous drag. It is not surprising, therefore, that the curve for λ = 0.7 crosses that of λ = 0.6 twice at a large κa in Fig. 9, because the |µE | of the former must be smaller than that of the latter as κa → ∞. Moreover, this dependence of |µE | on λ is more appreciable when κa is small than that when it is large, as can be seen in Fig. 9. All of the results of µE described above can be justified by the correspond∗ presented in Fig. 10. Fig. 10a ing results of FE∗ and FD,2 reveals that for a fixed λ, |FE∗ | decreases with the increase in κa. However, if the double layer surrounding the pore surface is sufficiently thick with κ(b − a) < 1 or κa < λ(1 − λ), |FE∗ | becomes insensitive to the variation in κa. This is because if κ(b − a) < 1, the charge induced on the lateral surface of a particle is on the order of −εζw /(b − a). On the other hand, if κ(b − a) > 1, the influence of the double layer surrounding the pore surface on the lateral surface of the particle becomes unimportant, and the amount of charge induced on it decreases rapidly as κa increases; so does |FE∗ |. ∗ | increases with the Fig. 10b indicates that for a fixed λ, |FD,2 increase in κa, but if the electrical double layer is sufficiently ∗ | becomes insensitive to thin with κb = λ/κa > 10, |FD,2 the variation in κa. This is because if κb > 10, the undisturbed electroosmotic velocity u(r) described by Eq. (11) is roughly constant, and so is the drag force acting on the particle by the fluid. On the other hand, if κb < 10, u(r), and ∗ |, increases rapidly with the increase in κa accordingly, |FD,2 (or κb) when λ is fixed. Fig. 10b also suggests that if κa is ∗ | has a local minimum as λ varies. sufficiently small, |FD,2 This arises from the net result of two competing effects as λ varies, namely, u(r) decreases with the increase in λ (or decrease in κb) as is just mentioned, and the gap between the particle and the pore becomes narrower as λ increases so that the viscous drag increases with the increase in λ. Note that if κa is sufficiently large, the later always dominates, as is observed in the discussion of Fig. 8b, where κa = 1.
J.-P. Hsu, M.-H. Ku / Journal of Colloid and Interface Science 283 (2005) 592–600
599
Fig. 11. Variation of scaled mobility µE as a function of κa at various d/a. Discrete symbols, analytical result of Liu et al. [17] for the case when LT /LC = 100, ζp∗ = 0, ζw∗ = 1, and λ = 0.5.
∗ (b) as a function of κa at various λ Fig. 10. Variation of FE∗ (a) and FD,2 for the case of Fig. 9.
Fig. 11 shows the variation of µE as a function of κa at various d/a when λ = 0.5. For comparison, the analytical result of Liu et al. [17], which is valid for the case when a particle is sufficiently long (LT /LC = 100), or the end effect is negligible, is also presented. As in the case of Fig. 6, the performance of the solution of Liu et al. [17] depends upon the magnitudes of κa and d/a. Fig. 11 also suggests that, if κa is fixed, |µE | decreases with the increase in d/a, which is consistent with the result of Fig. 7 where the drag force D ∗ is more significant than the sum of the driving forces, FE∗ + ∗ . However, the variation of |µ | with d/a is significant FD,2 E only if κa is sufficiently large (>3). This behavior is similar to that for the case when the particle is positively charged and the pore uncharged, where the variation of |µE | with d/a is significant when κa > 2, as shown in Fig. 6. As in the case of Fig. 6, the end effect of the particle becomes unimportant if d/a exceeds about 10. As discussed previously, due to its linear nature, the present problem can be decomposed into two subproblems, and the mobility of a particle can be obtained directly without a trial-and-error procedure as that often adopted in the literature. In our approach, evaluating three independent forces, namely, FD,1 (or D) in the first problem, and FE and
FD,2 in the second problem, is sufficient. In fact, the first problem can be treated as a sedimentation problem and the second one as an electroosmosis problem. This suggests that the present numerical approach is applicable to these problems. In our analysis, we assume that the pore is sufficiently long so that the electroosmotic flow inside is fully developed. Fig. 12a shows the variation of the scaled transition length of a pore Ltc /a, Ltc being the length of a pore necessary to reach fully developed electroosmotic flow, as a function of λ (= a/b) for the case when the particle is uncharged and the pore positively charged; that as a function of κa is presented in Fig. 12b. Here, Ltc is chosen so that the following conditions are satisfied for Lt > Ltc : (a) The electric potential distribution remains essentially unchanged, and the boundary conditions at infinity are satisfied. (b) The flow field in the first problem remains essentially unchanged, and the velocity at infinity vanishes. (c) The flow field in the second problem remains essentially unchanged, and fully developed electroosmotic flow is achieved. Fig. 12a reveals that Ltc /a declines as a/b increases. This is because the larger the a/b, the smaller is the gap between particle and pore, and therefore, the more difficult it is for the flow field to extend itself. Note that Ltc /a is uninfluenced by the scaled length of a particle d/a. Fig. 12b suggests that the smaller the κa, the larger the Ltc /a. This is because the thicker the double layer the more appreciably it is distorted by the particle. As can be seen in Fig. 12b, Ltc /a approaches a constant value as κa → 0. This is because, in this case, the equilibrium electric field is described by a Laplace equation. As κa → ∞, Ltc /a approaches a constant value of 3.5. If κa is large, Ltc /a is mainly determined by the flow field, which is related to a/b. The constant value of 3.5 is close to the value predicted by Liu et al., i.e., 3. In fact, their Fig. 12 suggests that Ltc /a is better estimated by 3.5.
600
J.-P. Hsu, M.-H. Ku / Journal of Colloid and Interface Science 283 (2005) 592–600
ratio of a particle for the case when a particle is charged and the pore uncharged is more important than that for the case when the particle is uncharged and the pore charged. (d) If boundary effect is important, the dependence of the mobility of a particle on its aspect ratio is significant only if the double layer is sufficiently thin. (e) Increasing the aspect ratio of a particle can either promote or hinder its movement, depending upon the parameters chosen. (f) The end effect on electrophoresis is important when the double layer is thick.
Acknowledgment This work is supported by the National Science Council of the Republic of China.
References [1] [2] [3] [4] [5] [6] [7] Fig. 12. Variation of scaled transition length Ltc /a as a function of λ for the case of Fig. 7 (a) and that as a function of κa at d/a = 1 for the case of Fig. 11 (b).
4. Conclusions In summary, the boundary effect on electrophoresis is investigated by considering the electrophoresis of a finite cylindrical particle along the axis of a long cylindrical pore. The results of numerical simulation reveal the followings: (a) Due to the presence of the pore, the mobility of a particle decreases appreciably. (b) The mobility of a particle increases monotonically with the decrease in the thickness of the double layer surrounding it. However, the mobility may have a local maximum as the thickness of the double layer surrounding the pore varies. This local maximum occurs at a thinner double layer when the boundary effect is more important. (c) In general, the influence of the aspect
[8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
H.J. Keh, J.L. Anderson, J. Fluid Mech. 153 (1985) 417. H.J. Keh, S.B. Chen, J. Fluid Mech. 194 (1988) 377. J. Ennis, J.L. Anderson, J. Colloid Interface Sci. 185 (1997) 497. H.J. Keh, L.C. Lien, J. Chin. Inst. Chem. Eng. 20 (1989) 283. H.J. Keh, J.S. Jan, J. Colloid Interface Sci. 183 (1996) 458. M.H. Chih, E. Lee, J.P. Hsu, J. Colloid Interface Sci. 248 (2002) 383. Y.P. Tang, M.H. Chih, E. Lee, J.P. Hsu, J. Colloid Interface Sci. 242 (2001) 121. H.J. Keh, J.Y. Chiou, AIChE J. 42 (1996) 1397. A.A. Shugai, S.L. Carnie, J. Colloid Interface Sci. 213 (1999) 298. A.L. Zydney, J. Colloid Interface Sci. 169 (1995) 476. E. Lee, J.W. Chu, J.P. Hsu, J. Colloid Interface Sci. 196 (1997) 316. E. Lee, J.W. Chu, J.P. Hsu, J. Colloid Interface Sci. 205 (1998) 65. J.W. Chu, W.H. Lin, E. Lee, J.P. Hsu, J. Colloid Interface Sci. 17 (2001) 6289. J.P. Hsu, S.H. Hung, C.Y. Kao, Langmuir 18 (2002) 8897. C. Ye, D. Sinton, D. Erickson, D. Li, Langmuir 18 (2002) 9095. J.P. Hsu, C.Y. Kao, J. Phys. Chem. B 106 (2002) 10605. H. Liu, H.H. Bau, H.H. Hu, Langmuir 20 (2004) 2628. D.C. Henry, Proc. R. Soc. London Ser. A 133 (1931) 106. G. Backstrom, Fluid Dynamics by Finite Element Analysis, Studentlitteratur, 1999. R.W. O’Brien, L.R. White, J. Chem. Soc. Faraday Trans. 2 74 (1978) 1607. PDE Solutions, FlexPDE, Version 2.22. J. Happel, H. Brenner, Low Reynolds Number Hydrodynamics, Nijhoff, The Hague, 1983, chap. 4. J.D. Sherwood, J. Chem. Soc. Faraday Trans. 2 78 (1982) 1091. S.B. Chen, D.L. Koch, J. Colloid Interface Sci. 180 (1996) 466.