Electrophoresis of a Finite Cylinder Positioned Eccentrically along the

Aug 15, 2006 - of the eccentricity of a particle and its linear size, the radius of the pore, and the thickness of the electrical double layer on the ...
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J. Phys. Chem. B 2006, 110, 17607-17615

17607

Electrophoresis of a Finite Cylinder Positioned Eccentrically along the Axis of a Long Cylindrical Pore Jyh-Ping Hsu* and Chao-Chung Kuo Department of Chemical Engineering, National Taiwan UniVersity, Taipei, Taiwan 10617 ReceiVed: May 29, 2006; In Final Form: July 9, 2006

The electrophoresis of a finite cylindrical particle positioned eccentrically along the axis of a long cylindrical pore is modeled under the conditions of low surface potential and weak applied electric field. The influences of the eccentricity of a particle and its linear size, the radius of the pore, and the thickness of the electrical double layer on the electrophoretic mobility of the particle are investigated. Some interesting results are observed. For instance, for the case of a positively charged particle placed in an uncharged pore, if the double layer is thin and the particle is short, the mobility has a local minimum as the eccentricity varies. Also, for a short particle the mobility at a thinner double layer can be smaller than that at a thicker double layer, which has never been reported for the case of constant surface potential. In general, the mobility increases with the increase in the eccentricity, and this effect is pronounced when the size of a particle is large and/or the radius of a pore is small.

Introduction Electrophoresis, the migration of a charged entity driven by an applied electric field, is one of the most important electrokinetic phenomena. Over the past two decades, capillary electrophoresis has emerged into the forefront of electrophoresis analysis with its advantages of efficiency, sensitivity, and versatile usages.1 For instance, it can be applied to separate biopolymers, metal ions, and inorganic ions; it is also a powerful tool for analyzing pharmaceuticals, monitoring water quality, and studying various drug and drug metabolism. A detailed understanding of a general behavior of the capillary electrophoresis is highly desirable from both a fundamental theory and an applications point of view. This, however, is not an easy task even under idealized conditions. The difficulty arises mainly from the complicated interactions between hydrodynamic and electric effects, which lead to a set of coupled, nonlinear partial differential equations. Also, since the presence of a boundary is usually important in the capillary electrophoresis, the problem becomes even more complicated. The electrophoretic behavior of an entity can be influenced by a boundary in several possible ways. These include, for example, the increase in the viscous drag, that in the induced charge on both the entity surface and on the approaching boundary, the enhancement of the local electric field on the entity surface, and the electroosmotic flow arising from a charged boundary. Many attempts have been made to investigate the boundary effect on electrophoresis including, for instance, a sphere moving parallel2-5 and normal to a planar surface,2,4-9 a sphere moving along the axis of a cylindrical pore,2,4,5,10 a sphere moving at the center11-14 or at an arbitrary position15 in a spherical cavity, and a cylinder moving in a cylindrical pore.16-19 Liu et al.18 analyzed the electrophoresis of a cylindrical particle in a long cylindrical pore for the case when the (radius of particle/radius of pore) ratio is large. Neglecting the end effect of a particle, they were able to solve analytically the case of a concentrically positioned particle, * Address correspondence to this author. Phone: 886-2-23637448. Fax: 886-2-23623040. E-mail: [email protected].

which is a one-dimensional problem. Some numerical data were reported for the case of a finite particle under the condition of thin double layer. Yariv and Brenner24 investigated the electrophoresis of a sphere in a long cylindrical pore under the condition of thin double layer by successive reflection. The same problem was solved numerically by Ye et al.25 Most of the previous studies on the boundary effect of electrophoresis are of one- or two-dimensional nature. Although this simplifies considerably relevant mathematical analysis, a more general treatment is desirable from the practical point of view. In this study, the boundary effect on electrophoresis is investigated theoretically by considering the electrophoresis of a finite cylinder positioned eccentrically in a long, nonconducting cylindrical pore, which simulates, for instance, capillary electrophoresis. The wall of the cylindrical pore can be either uncharged or charged. A fused silica pore with a hydrophobic coating like polyacrylamide or poly(vinyl alcohol), for example, belongs to the former, and a fused silica pore is a typical example for the latter. The influences of the relative eccentricity and the linear sizes of a cylinder, the radius of a pore, and the thickness of the double layer on the electrophoretic mobility of a cylinder are investigated. Analysis Referring to Figure 1, we consider the electrophoresis of a rigid, nonconducting, finite cylindrical particle of radius a and length 2d positioned eccentrically along the axis of a long, nonconducting cylindrical pore of radius b as a response to an applied electric field E0 of strength E0. The liquid phase is an incompressible Newtonian fluid with constant physical properties. Let c be the distance between the axis of the particle and that of the pore and m ) c/(b - a) be the relative eccentricity. The Cartesian coordinates (x,y,z) are adopted with its origin located at the center of the cylindrical pore, and E0 in the z-direction. If E0 is relatively weak than the electric field established by the surface of the particle and/or the pore, and the surface

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potential is low, the effect of double layer polarization can be neglected. These assumptions are satisfactory if the surface potential is lower than about 25 mV and E0 is weaker than about 25 kV/m. In this case, the electrical potential 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 from E0.20 It can be shown that Ψ1 and Ψ2 satisfy

∇2Ψ1 ) κ2Ψ1

(1)

∇2Ψ2 ) 0

(2)

In these expressions, κ ) [∑jnj0(ezj)2/kT]1/2 is the reciprocal Debye length, nj0 and zj are respectively the bulk number concentration and the valence of the jth ionic species, e is the elementary charge,  is the dielectric constant, and k and T are respectively the Boltzmann constant and the absolute temperature. The following boundary conditions are assumed:

Ψ1 ) ζp and n‚∇Ψ2 ) 0 at the particle surface Ψ1 ) ζw and

∂Ψ2 ) 0 at r ) b ∂r

(3) (4)

I0(κr) and ∇Ψ2 ) -E0ez as |z|f∞, r < b (5) Ψ 1 ) ζw I0(κb) Here, n is the unit normal vector directed into the liquid phase, ez is the unit vector in the z-direction, and I0 is the modified Bessel function of the first kind of order zero. Equations 4 and 5 state that both the particle and the pore are nonconductive and remain at constant surface potentials ζp and ζw, respectively. For electrophoresis, the Reynolds number is very small, and the flow field is in the creeping flow regime and can be described by

∇‚u ) 0

(6)

η∇2u - ∇p ) F∇Ψ

(7)

In these expressions, u is the velocity of the fluid, η and p are respectively the viscosity and the hydrodynamic pressure, and F ) ∑jzjenj is the space charge density. The following boundary conditions are assumed:

u ) Uez at the particle surface

(8)

u ) 0 at r ) b

(9)

u ) u(r)ez ) -

[

]

ζw I0(κr) 1E e as |z|f∞, r < b η I0(κb) 0 z

(10)

In these expressions, U is the velocity of the particle and u(r) is the undisturbed electroosmotic velocity in the absence of the particle when the surface potential of the pore is ζw. Equations 9 and 10 imply that both the surface of the particle and that of the pore are no-slip. The forces acting on a particle include the electrostatic force and the hydrodynamic force. For the present case, only the z-components of these forces, FE and FD, respectively, need be considered. The former can be calculated by

FE )

∫∫SσEz dS

(11)

Figure 1. Electrophoresis of a finite cylindrical particle of radius a and length 2d parallel but eccentric to the axis of a long cylindrical pore of radius b; c is the distance between the axis of the particle and that of the pore. An electric field E0 is applied that is parallel to the axis of the pore, and r and z are respectively the radial and the axial coordinates.

and the latter can be evaluated by21

FD )

∫∫Sη

∂(u‚t) t dS + ∂n z

∫∫S - pnz dS

(12)

where S denotes particle surface, σ ) -n‚∇Ψ1 is the surface charge density, Ez ) - ∂Ψ/∂z is the strength of the local electric field in the z-direction, t is the unit tangential vector on the particle surface, n is the magnitude of n, and tz and nz are the z-components of t and n, respectively. At the steady state, FE + FD ) 0. Following the treatment of O’Brien and White,22 the problem under consideration is decomposed into two subproblems: the particle moves with velocity U in the absence of E0, and it remains fixed when E0 is applied. In the former, a conventional hydrodynamic force FD,1 ) -UD acts on the particle. The drag coefficient D is positive, and depends only upon the geometry of the problem considered. In the latter, an electrostatic force FE and an electric body force FD,2 act on the particle. Although FD,2 is usually a retardation force, as will be shown later, it can be either a drag force or a driving force in the present problem. In our case, both FE and FD,2 are a function of m, κa, (d/a), and λ ()a/b); FD,1 (or D) is a function of (d/a) and λ, but is independent of κa. Since FD ) FD,1 + FD,2, we have

U)

FE + FD,2 D

(13)

The numerator and the denominator on the right-hand side of this expression can be interpreted respectively as the driving force per unit applied electrical field and the drag force per unit velocity of a particle when no external electric field is applied. The value of U can be obtained through the procedure employed previously.19 In subsequent discussions, the scaled electrophoretic mobility ω ) U/U0 is used where U0 ) (kT/e)E0/η. For illustration, we consider two cases: a charged particle is placed in an uncharged pore, and an uncharged particle is placed in a charged pore. Because the present problem is of linear nature, if both a particle and a pore are charged, the result can be obtained by an appropriate linear combination of the results obtained from these two cases. According to eq 13, ω can be expressed as

ω ) (F/E + F/D,2)/D*

(14)

where F/E ) FE/16ηaU0, F/D,2 ) FD,2/16ηaU0, and D* ) D/16ηa. Results and Discussion FlexPDE,23 a differential equation solver based on a finite element method, is adopted to solving the governing equations

Electrophoresis of a Finite Cylindrical Particle

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Figure 2. Variation of the scaled electrophoretic mobility ω as a function of λ for the case when a sphere of constant surface potential ζs ) kBT/e is placed on the axis of an uncharged cylindrical pore at κa ) 4.3. Solid symbols present the numerical result; the doted line shows the numerical result of Shugai and Carnie;5 the solid line shows the results based on the reflection method of Ennis and Anderson.4

and the associated boundary conditions. The applicability of this software is illustrated by considering the electrophoresis of a sphere along the axis of a cylindrical pore. Figure 2 shows the simulated variation of the scaled electrophoretic mobility ω as a function of λ, which measures the significance of the boundary effect. For comparison, the results of Ennis and Anderson4 and that of Shugai and Carnie5 are also presented. The former is unreliable if λ is large, and the latter is inaccurate if λ is small. Figure 2 reveals that the performance of the software adopted in this study is satisfactory for the entire range of λ. We assume that a pore is sufficiently long so that the electroosmotic flow inside is fully developed. Figure 3 illustrates the variations of the scaled transition length of a pore (Ltc/a), where Ltc is the minimum length of a pore necessary to reach fully developed electroosmotic flow, as a function of m and as a function of κa for the case when an uncharged particle is placed in a positively charged pore. Here, Ltc is chosen so that the following conditions are satisfied for Lt > Ltc: (i) The electric potential distribution remains essentially unchanged, and the boundary conditions at infinity are satisfied. (ii) The flow field in the first subproblem remains essentially unchanged, and the velocity at infinity vanishes. (iii) The flow field in the second subproblem remains essentially unchanged, and fully developed electroosmotic flow is achieved. As can be seen in Figure 3a, (Ltc/a) increases with the increase in m. This is because the larger the value of m, although one side of a particle is closer to the wall of a pore the space between the other side of the particle and the wall of the pore becomes larger, and, therefore, it is easier for the flow field to extend itself. Figure 3b indicates that (Ltc/a) decreases with the increase in κa, but approaches a constant value when κa exceeds about unity. This is because the smaller the value of κa the thicker the double layer of the pore, and the more serious it is distorted by the particle. 1. Positively Charged Particle in an Uncharged Pore. Let us consider first the case when a positively charged particle is placed in an uncharged pore. Figure 4a shows the variation of the scaled electrophoretic mobility ω as a function of κa at various relative eccentricity m for the case when both the size of a particle and the radius of a pore are fixed; that as a function of m for various combinations of κa and (d/a) at three levels of λ is presented in Figure 4b-d, where both the radius of a particle

Figure 3. (a) Variation of the scaled transition length (Ltc/a) as a function of m at κa ) 1, λ ) 0.5, d/a ) 1, ζ/p) 0, and ζ/w ) 1. (b) Variation of (Ltc/a) as a function of κa at m ) 0.6, λ ) 0.5, d/a ) 1, ζ/p ) 0, and ζ/w ) 1.

and that of a pore are fixed. Figure 4a suggests that, for fixed values of (d/a) and λ, ω increases monotonically with the increase in κa. This is because the thinner the double layer surrounding a particle, the greater the absolute value of the gradient of the electrical potential on the particle surface, the higher the charge density on the surface, and therefore, the greater the electrical driving force. Figure 4a also indicates that if the value of κa is fixed, ω increases with the increase in m, that is, the closer a particle is to the wall of a pore the larger is its mobility. A similar phenomenon was also observed in the electrophoresis of a rigid sphere parallel to a nonconducting planar wall,3,5 and in the electrophoresis of a cylinder along the axis of a long cylindrical pore.18 This is because when a particle is close to a wall, both the increase of the surface charge density and the squeeze of the electric field in the particlewall gap have the effect of increasing the electric force acting on the particle. Panels b-d of Figure 4 show that ω increases with the increase in (d/a), which arises from the fact that the increase in the lateral surface of a particle leads to a greater electric driving force. The behavior of ω in Figure 4c (λ ) 0.5) is similar to that in Figure 4b (λ ) 0.2) except that the effect of m on ω in the former is enhanced. It is interesting to note in Figure 4b that when κa is large and (d/a) is small (0.2 and 1), ω can have a local minimum as m varies. Furthermore, ω at a larger value of κa can be smaller than that at a smaller value of κa. These phenomena are not observed in Figure 4a,c,d, where the boundary effect is important (λ is large) and the

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Figure 4. (a) Variation of the scaled electrophoretic mobility ω as a function of κa for various values of m at λ ) 0.5 and d/a ) 1. (b-d) Variation of ω as a function of m at various combinations of (d/a) and κa: (b) λ ) 0.2, (c) λ ) 0.5, (d) λ ) 0.8. Parameters used are ζ/p ) 1 and ζ/w ) 0. Panels b-d: solid lines, d/a ) 5, dotted lines, d/a ) 1, dash-dotted line, d/a ) 0.2.

influence of the electric driving force dominates. The presence of a local minimum in ω was also reported for the eccentric electrophoresis of a sphere in circular cylindrical microchannels under the condition of a thin double layer.25 The result that ω at a large κa is smaller than that at a small κa has never been reported for the case of constant surface potential. This is because the electrical driving force correlates positively with the side area of a cylinder. When both λ and (d/a) are small, the electrical driving force is small, and so is the induced electroosmotic retardation force; although the latter is smaller than the former, the difference between the two is limited. As κa increases, the rate of increase in the induced electroosmotic retardation force is faster than that in the electrical driving force, yielding a smaller mobility. Panels b-d in Figure 4 suggest that if λ is large, the influences of both κa and (d/a) on the behavior of ω become unimportant. Again, this is due to the role of the electric driving force acting on a particle. The variations of the scaled electrostatic force F/E acting on a particle for the case of Figure 4b-d are illustrated in Figure 5. This figure indicates that F/E increases with the increase in κa; this is because the larger the value of κa the higher the surface charge density of a particle. F/E is found to increases with both (d/a) and λ. The former is expected because the larger the value of (d/a) the greater the surface area of a particle, and therefore, the greater the amount of surface charge. As pointed out by Hsu and Ku,19 the latter is a consequence of the combined effect of the squeeze of the applied electric field between a particle and a pore and the increase in the surface charge density.

The behavior of F/E as m varies is similar to that of ω in Figure 4b-d, implying that F/E is the main driving force. Again, this can be explained by the squeeze of the applied electric field between a particle and a pore and the increase in the surface charge density as the particle approaches the pore. Note that the larger the value of λ (more significant the boundary effect) the faster the rate of increase of F/E as m increases. Also, the larger the value of λ the less significant the influence of κa on the behavior of F/E as m varies. Figure 6 shows the variation of the scaled coefficient D* of the conventional hydrodynamic force acting on a particle as a function of m for the case of Figure 4b-d. As can be seen in this figure D* increases with the increase in (d/a), which is expected since the surface area of a cylinder increases with its (d/a). It is interesting to note in panels a and b of Figure 6 that if λ is sufficiently small, D* has a local minimum as m varies; the presence of the local minimum becomes clearer when (d/a) is large. Figure 6c reveals that the local minimum in D* disappears when λ is sufficiently large, and in that case it declines monotonically with the increase of m. Furthermore, D* becomes smaller when a particle is closer to the wall of a pore. In Figure 6a, only a small part of the cross sectional area of a pore is occupied by a particle, the magnitude of the viscous term in eq 12 is larger than that of the corresponding pressure term. The drag acting on a particle is dominated by the former, especially when (d/a) is large. In addition, D* increases with the increase in λ, as can be seen in Figure 6. As m increases, while one side of a particle approaches the wall of a pore the

Electrophoresis of a Finite Cylindrical Particle

Figure 5. Variation of the scaled electrostatic force acting on a particle F/E as a function of m for the case of Figure 4b-d: (a) λ ) 0.2, (b) λ ) 0.5, (c) λ ) 0.8.

other side of the particle moves away from the other side of the wall of the pore, leading to a decrease in the drag acting on the particle. However, if m is sufficiently large, D* increases with m because the former effect dominates. Figure 6a also reveals that the value of D* at a large m (close to unity) can be larger than that at m ) 0, implying that if λ is small, the drag acting on a particle is relatively insignificant until it is sufficiently close to the wall. In Figure 6b, a particle takes a moderately large part of the cross sectional area of a pore. The behavior of D* in this figure is similar to that in Figure 6a except that its value at a large m is smaller than that at m ) 0, which means that the drag is significant even if m is small. In Figure 6c, a large portion of the cross sectional area of a pore is

J. Phys. Chem. B, Vol. 110, No. 35, 2006 17611

Figure 6. Variation of the scaled conventional hydrodynamic force coefficient D* as a function of m for the case of Figure 4b-d except that κa ) 0.5, 1, and 3: (a) λ ) 0.2, (b) λ ) 0.5, (c) λ ) 0.8.

occupied by a particle, the wall effect is significant, and the drag is dominated by the pressure term. In this case the increase in the viscous term as m increases is not important compared with the corresponding decrease in the pressure term, which arises from the increase in the space between the other side of the particle and the other side of the pore. Here, D* decreases monotonically with the increase in m, and the local minimum in Figure 6a,b disappears. Figure 7 illustrates the variation of the scaled electric body force F/D,2 as a function of m for the case of Figure 5. F/D,2 is known as the electroosmotic retardation force because it always has the opposite sign as that of the driving force F/E when a boundary effect is unimportant, as Figure 7a shows. However,

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Figure 8. Variation of the scaled electrophoretic mobility ω as a function of κa at various values of m for the case when particle is uncharged and pore positively charged. Parameters used are ζ/p ) 0, ζ/w ) 1, λ ) 0.5, and d/a ) 1.

Figure 7. Variation of the scaled electric body force acting on a particle F/D,2 as a function of m for the case of Figure 5: (a) λ ) 0.2, (b) λ ) 0.5, (c) λ ) 0.8.

as shown in Figure 7c, if a boundary effect is significant, F/D,2 may become a driving force for the movement of a particle. This occurs when the pressure term on the right-hand side of eq 12 dominates.19 Figure 7a indicates that F/D,2 has a local minimum as m varies, which is more pronounced when κa or (d/a) is large. This local minimum is not seen in Figure 7c, where F/D,2 declines monotonically with the increase in m. The different behavior of F/D,2 in panels a and c of Figure 7 arises from the competition between the viscous term and the pressure term in eq 12. In the former, the boundary effect is less important, F/D,2 is dominated by the viscous term, especially when (d/a) is large. As m increases, a particle begins to

experience a greater F/D,2 arising from the higher charge density on its surface, which enhances the electroosmotic flow surrounding the particle and raises the magnitude of the viscous term. If m is sufficiently large, |F/D,2| begins to decline because the electroosmotic flow is hindered by the narrow gap between the particle and the wall of the pore. The presence of the local minimum of the mobility of a particle observed in Figure 4b is related to these behaviors of F/D,2. In Figure 7c, F/D,2 is dominated by the pressure term, and the observation that F/D,2 decreases monotonically as m increases can be explained by the same reason as that used to elaborate the behavior of D* in Figure 6c. Panels a and c of Figure 7 indicate that for both the case when a boundary effect is insignificant (λ is very small) and when it is important (λ is very large), |F/D,2| increases with the increase in (d/a) and in κa. If κa is large, the charge density on the surface of a particle is high, the electroosmotic flow surrounding it is enhanced, leading to a large value of |F/D,2|. For the case when λ takes a medium value as in Figure 7b, if (d/a) is small (0.2 or 1), the behavior of F/D,2 as m varies is similar to that in Figure 7a, that is, the larger the value of κa the larger the negative value of F/D,2. This is because the pressure term in this case is as small as that in Figure 7a where λ is small. Note that F/D,2 may take a small positive value when κa is small. As in Figure 7a, F/D,2 has a local minimum as m varies when (d/a) is large (d/a ) 5). The behavior of F/D,2 as m varies is similar to that in Figure 7c, especially when m is close to zero, that is, the larger the value of κa the larger the positive value of F/D,2. This is because the pressure term in this case is as large as that in Figure 7c where λ is large. 2. Uncharged Particle in a Positively Charged Pore. Consider the case when an uncharged particle is placed in a positively charged pore. Figure 8 shows the variation of the scaled electrophoretic mobility ω as a function of κa at various values of m. This figure indicates that for a fixed value of m, |ω| increases with the increase in κa, which is similar to that observed when a particle is positively charged and a pore uncharged except that ω is negative for the ranges of the parameters considered. Note that |ω| has a local maximum as κa varies, and this phenomenon is more pronounced when m is small. Similar behavior was also observed in the corresponding concentric case,19 and was explained by the net result of the competition of the electrostatic force and the electric body force.

Electrophoresis of a Finite Cylindrical Particle

Figure 9. Variation of the scaled electrophoretic mobility ω as a function of m at various combinations of (d/a) and κa for the case when ζ/p ) 0 and ζ/w ) 1: (a) λ ) 0.2, (b) λ ) 0.5, (c) λ ) 0.8. Solid lines, d/a ) 5; dotted lines, d/a ) 1; dash-dotted lines, d/a ) 0.2.

This will be discussed latter. Figure 8 also illustrates that for a fixed value of κa, |ω| increases with the increase in m, which is similar to the result shown in Figure 4a where a particle is positively charged and a pore uncharged except that the particle is moving in the opposite direction. This phenomenon was also reported by Liu et al.18 in an analysis of the electrophoresis of a cylindrical particle without the end effect eccentrically positioned in a long cylindrical pore. According to eq 14, this is because when m increases F/E increases but at the same time D* decreases. Figure 9 illustrates the variation of |ω| as a function of m for various combinations of κa and (d/a) at three levels of λ. In

J. Phys. Chem. B, Vol. 110, No. 35, 2006 17613 general, if the boundary effect is important, the larger the value of m the larger the value of |ω| is. This is because when m increases, D* declines but F/E increases. Figure 9c reveals that if λ is sufficiently large, |ω| becomes insensitive to the variation of (d/a), or the length of a particle, when the value of κa is fixed. Similar behavior was also reported for the case when a cylinder is positioned concentrically in a cylindrical pore.19 The rationale explanation is that the undisturbed electroosmotic velocity u(r) described by eq 10 is the main driving force for the movement of a particle, especially when both κa and λ are large, and the behavior of u(r) as m varies correlates mainly with κa, and the role of (d/a) is insignificant. Panels a and b of Figure 9 indicate that if the boundary effect is relatively unimportant, |ω| has a local minimum as m varies, which is more pronounced when κa is large and (d/a) is small. The presence of the local minimum arises from that if m is small, F/D,2 is the main driving force and the decrease in D* as m increases can hardly affect the decline of |ω|. On the other hand, if m is large, F/E increases rapidly with the increase in m and can exceed the influence of F/D,2. Panels a and b of Figure 9 also show that if κa is constant, |ω| is insensitive to the variation of (d/a) when m is small; it becomes sensitive to the variation of (d/a) if m is sufficiently large. As mentioned previously, this is because as m increases the magnitude of F/E can exceed that of F/D,2, and the magnitude of F/E is closely related to the value of (d/a). Figure 10 shows the variations of the negative scaled electrostatic force -F/E acting on a particle as a function of m for the case of Figure 9. This figure indicates that the qualitative behaviors of -F/E are similar to those of F/E shown in Figure 5 for the case when a particle is positively charged and a pore is uncharged, that is, -F/E increases with the increase in (d/a) or m, and can be explained by the same reasoning. Note that -F/E decreases with the increase in κa, which is contrary to the behavior of F/E seen in Figure 5. This is because negative charge is induced on the surface of a particle as it approaches a positively charged pore, leading to a negative electrostatic force acting on the former. If κa is large, the influence of the double layer of a pore on the surface of a particle becomes insignificant, and the amount of charge induced is small, so is |F/E|. The degree of variation of -F/E as m varies is more appreciable in Figure 10 than that of F/E in Figure 5, especially in Figure 10a where λ is small. This is because the particle in the former is free of charge originally, and as it approaches the pore |F/E| increases rapidly and the order of its magnitude is comparable to or even greater than that of |-F/D,2|, especially when κa is small. This as will be discussed later. Figure 10 also reveals that |F/E| increases with the increase in λ, which can also be explained by the same reasoning as that employed in the discussions of Figure 5. The variations of the negative scaled electric body force F/D,2 acting on a particle as a function of m for the case of Figure 9 are illustrated in Figure 11. This figure indicates that -F/D,2 declines with the increase in m and increases with the increase in κa or (d/a). The former is because the u(r) decreases with the increase in m, so is the drag acting on a particle by the surrounding fluid. The later is because when κa (or κb) is large, the u(r) is roughly constant over the cross section of a pore, that is, the velocity gradient is almost flat at the center region and is steep near the pore wall, yielding a large average u(r) across the cross sectional area of a pore. The increase in -F/D,2 with (d/a) arises from the increase in the lateral surface of a particle. Figure 11 also suggests that -F/D,2 increases with the

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Figure 10. Variation of the negative scaled electrostatic force acting on a particle -F/E as a function of m for the case of Figure 9.

increase in λ, which is similar to the behavior of D* shown in Figure 6, although F/D,2 is acting as a driving force for the movement of a particle. For the present case, F/D,2 is negative and decreases with the increase in m. This is different from the result for the case when a positively charged particle is placed in an uncharged pore, where F/D,2 may change from negative to positive as m increases. Conclusions The boundary effect of electrophoresis is investigated by considering the electrophoresis of a finite cylindrical particle eccentrically placed in a long cylindrical pore under the

Hsu and Kuo

Figure 11. Variation of the negative scaled electric body force acting on a particle -F/D,2 as a function of m for the case of Figure 9.

conditions of low surface potential and applied weak electric field. The boundary-valued problem is solved numerically and the behavior of a particle is examined through numerical simulation. The following conclusions are drawn: (i) The more eccentrically a particle is positioned in a pore the larger its mobility. If the boundary effect is insignificant, the mobility may have a local minimum as the eccentricity increases, which is not observed in the electrophoresis of a sphere. (ii) Similar to the case of a cylinder on the axis of a cylindrical pore, the mobility of a cylinder increases with the decrease in the thickness of the double layer surrounding it, and the mobility may have a local maximum as the thickness of the double layer varies. (iii) The electrostatic force acting on a particle increases with the increases in its eccentricity. (iv) If the boundary effect

Electrophoresis of a Finite Cylindrical Particle is less significant, the conventional hydrodynamic force may have a local minimum as the eccentricity varies; if it is significant, the conventional hydrodynamic force declines monotonically as the eccentricity increases. (v) For the case when the boundary effect is significant and the double layer is thick, if a particle is positively charged and a pore uncharged, the electric body force (or the electroosmotic retardation force) may become a driving force for the movement of the particle. On the other hand, if a particle is uncharged and a pore positively charged, the electric driving force may be greater than the electric body force. (vi) In general, the influence of the aspect ratio of a particle on its mobility for the case when the boundary effect is insignificant is more important than that when it is significant. (vii) For the case of an uncharged particle placed in a positively charged pore, if the boundary effect is significant, the mobility of a particle as a function of its size is insensitive to the variation of the eccentricity. If the boundary effect is less significant, the mobility becomes sensitive to the size of a particle especially when the eccentricity is large, which is not observed in the case when a positively charged particle is placed in an uncharged pore. Acknowledgment. This work is supported by the National Science Council and the Industrial Development Bureau, Ministry of Economic Affairs of the Republic of China. References and Notes (1) Kemp, G. Biotechnol. Appl. Biochem. 1998, 27, 9-17. (2) Keh, H. J.; Anderson, J. J. Fluid Mech. 1985, 153, 417-439. (3) Keh, H. J.; Chen, S. B. J. Fluid Mech. 1988, 194, 377-390.

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