Effects of double-layer polarization and electroosmotic flow on the electrophoresis of a finite cylinder along the axis of a cylindrical pore

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Chemical Engineering Science

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Effects of double-layer polarization and electroosmotic flow on the electrophoresis of

a finite cylinder along the axis of a cylindrical pore

Jyh-Ping Hsu

a,∗

_{, Zheng-Syun Chen}

a

_{, Duu-Jong Lee}

a

_{, Shiojenn Tseng}

b

_{, Ay Su}

c a_{Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan 10617, Taiwan}

b_{Department of Mathematics, Tamkang University, Tamsui, Taipei, Taiwan 25137, Taiwan}

c_{Department of Mechanical Engineering & Fuel Cells, Research Center, Yuan Ze University, 135 Yuan-Tung Road, Chung Li, Tao Yuan, 320, Taiwan}

A R T I C L E I N F O A B S T R A C T

Article history:

Received 3 February 2008

Received in revised form 25 June 2008 Accepted 27 June 2008

Available online 2 July 2008 Keywords: Colloidal phenomena Electroosmosis Electrophoresis Mathematical modeling Boundary effect Double-layer polarization

The electrophoresis of a rigid, finite cylindrical particle along the axis of a long cylindrical pore is analyzed theoretically under the conditions of arbitrary surface potential and double-layer thickness. The effects of double-layer polarization and electroosmotic flow on the electrophoretic behavior of the particle are discussed. We show that if both the particle and the pore are positively charged, the mobility of the particle has a local minimum as the thickness of double layer varies. Also, if the level of the surface potential of the particle is sufficiently high and its aspect ratio is sufficiently large, then its mobility may change its sign twice as the thickness of double layer (or the concentration of electrolytes) varies. These findings are of practical significance in designing an electrophoresis operation because it will influence the prediction of the charged conditions on a particle if electrophoresis is used as an analytical tool, and the separation efficiency, if it is adopted as a separation technique.

© 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Electrophoresis, the movement of a charged entity in an elec-trolyte solution as a response to an applied electric field, is one of the most important electrokinetic phenomena. The earliest reported theoretic analysis on electrophoresis was conducted byHelmholtz (1879). Because the dielectric effect of the liquid medium was neglected in his analysis, the applications of the result obtained are limited. The analysis of Helmholtz was extended by Von Smoluchowski (1918) to take that effect into account. Assuming infinitely thin double layer, low surface potential, and steady state, he was able to derive, for a rigid sphere in an infinite medium,



= v/E =



/



, with



, v,



, E,



, and



being the mobility, the velocity, and the surface (zeta) potential of the entity, the strength of the applied electric field, and the dielectric constant and the viscosity of the liquid phase, respectively. The corresponding result for the case of infinitely thick double layer is two-thirds that of Smoluchowski, and in general, for an arbitrarily thick double layer (Henry, 1931),



=



_

f (



a) (1)

∗_{Corresponding author. Tel.: +886 2 23637448; fax: +886 2 23623040.} E-mail address:jphsu@ntu.edu.tw(J.-P. Hsu).

0009-2509/$ - see front matter©2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2008.06.022

where the dependence of the mobility is incorporated into the func-tion f (



a), (1/



) and a being the reciprocal Debye length and the par-ticle radius, respectively. Subsequent studies that considered more general case, such as non-spherical particles, arbitrary level of sur-face potentials, arbitrary thick double layers, and the presence of neighboring particles, are ample in the literature. In general, because the equations governing electrophoresis are coupled and highly non-linear, solving them analytically is almost impossible, and a numeri-cal approach is usually adopted (e.g.,Wiersema et al., 1966;O'Brien and White, 1978). Alternatively, the governing equations are solved by assuming conditions regarding the level of surface potential and the thickness of double layer (e.g., Overbeek, 1943; Dukhin and Semenikhin, 1970;Ohshima et al., 1983). In addition to the factors stated above, the presence of a boundary adds another degree of difficulty to the solution procedure of an electrophoresis problem because the concentration, flow, and electric fields surrounding a particle will all be influenced as it approaches a boundary. In partic-ular, if the boundary is charged, then electroosmotic flow is present, which will also affect the movement of the particle.

The presence of boundary effect is common in many applications of electrophoresis. Among the possible geometries that are of prac-tical significance a particle in a cylindrical pore has been considered by many investigators. Often, a spherical particle is assumed, for simplicity. However, because non-spherical particles are encoun-tered frequently, extending relevant analyses to these types of parti-cles is highly desirable. In particular, many inorganic and biological

entities are better described by a cylindrical particle. Typical exam-ple includes clay, protein, and DNA. Previous attempts made on the modeling of the electrophoresis of a cylindrical particle when bound-ary effect can be significant include that ofYe et al. (2002)where a circular cylindrical particle translates along the axis of a cylindri-cal microchannel under the conditions of thin double layer and low surface potential. Assuming low surface potential and weak applied electric field,Hsu and Kao (2002)investigated the influence of the charged conditions of a short cylinder on its electrophoresis along the axis of a cylindrical pore. Under the same conditions,Hsu and Ku (2005)discussed the influence of a charged cylindrical pore on the electrophoretic behavior of a finite cylinder. The analysis ofHsu and Kao (2002)was extended byHsu and Kuo (2006)to the case of an eccentrically positioned finite cylinder translating parallel to the axis of a long cylindrical pore.Liu et al. (2004)considered the electrophoresis of a cylindrical particle in a long cylindrical pore under the conditions of large (radius of a particle/radius of a pore) ratio and low surface potential. For the case of thin double layer and low surface potential,Davison and Sharp (2006)analyzed the eltrophoresis of a cylindrical particle positioned concentrically and ec-centrically in a cylindrical pore.Hsieh and Keh (2007)studied the electrophoresis of a long cylindrical particle of non-uniform zeta po-tential distribution in the angular direction by an imposed electric field in the direction perpendicular or parallel to its axis near a plane wall for the case of a thin double layer and low surface potential.

In this study, our previous analysis on the electrophoresis of a finite cylinder along the axis of a long cylindrical pore (Hsu and Ku, 2005) is extended to the case of arbitrary surface potential and double-layer thickness, taking account of the effects of double-layer polarization and electroosmotic flow. The first effect needs to be con-sidered when the level of surface potential exceeds ca. 25 mV and the thickness of the double layer surrounding a particle is compa-rable to its linear size (O'Brien and White, 1978;Lee et al., 1998). The second effect is important if a boundary is charged, and it can influence both quantitatively and qualitatively the electrophoretic behavior of a particle (Hsu and Ku, 2005;Hsu and Kuo, 2006;Liu et al., 2004). Although the incorporation of these effects into elec-trophoresis modeling makes the analysis much more complicated, it is highly desirable to experimentalists because experiments are usually conducted under conditions where they are significant. In our study, the influences of the thickness of the double layer, the aspect ratio of a particle, the level of surface potential, and the rela-tive size of the pore on the electrophoretic mobility of a particle are examined.

2. Theory

The problem under consideration is illustrated inFig. 1, where a rigid, non-conducting, cylindrical particle of radius a and length 2d translates along the axis of a long cylindrical pore of radius b filled with a Newtonian fluid of constant physical properties as a response to an applied uniform electric field E. The liquid phase contains z₁:

z2electrolytes; z1and z2are the valence of cations and that of anions, respectively, with



= −z2/z1. Let U be the velocity of the particle. The cylindrical coordinates (r,

, z) are adopted, its origin is located at the center of the particle, and E is in the z-direction. Because the present problem is

-symmetric only the (r, z) domain needs to be considered.

Assuming steady state condition, the governing equations of the present problem, including those for the electrical, concentration, and flow fields, can be summarized as follows:

∇2

_{= −}





= − 2  j=1 z_jen_j



(1)

Fig. 1. The electrophoresis of a rigid cylindrical particle of radius a and length 2d

driven by a uniform electric field E along the axis of a long cylindrical pore of radius b. The cylindrical coordinates (r, , z) with its origin located at the center of the particle are adopted; E is in the z-direction.

∇ ·  −Dj  ∇nj+ z_je kBT n_j∇

 + njv  = 0 (2) ∇ · v = 0 (3) −∇p +



∇2_v+



_∇

_{= 0 (4)}

Here, ∇2 _{is the Laplace operator,}

_{the electrical potential,}

_

_the permittivity of the liquid phase,



the space charge density, e the elementary charge, and n_jand D_jare the number concentration and the diffusivity of ionic species j, respectively. kB, T, v,



, and p are the Boltzmann constant, the absolute temperature, the liquid velocity, the viscosity of the liquid, and the hydrodynamic pressure, respec-tively.

The mathematical treatment can be simplified by partitioning

into the equilibrium potential, that is, the potential in the absence of E,

₁, and the potential outside the particle when E is applied,

2,

=

1+

2(O'Brien and White, 1978;Lee et al., 1998). Also, the distortion of the ionic cloud surrounding the particle is expressed as

n_j= nj0exp  −zje(

1+_k

2+ gj) BT  , j= 1, 2 (5)

where n_j0and g_jare the bulk concentration and a perturbed poten-tial associated with ionic species j, respectively. Note that this does not mean that the distribution of n_jis Boltzmann because the actual distribution depends on g_j. In practice, the upper limit of the surface potential of a particle and that of a surface is on the order of 100 mV and the thickness of the double layer on the order of 100 nm. This implies that the strength of the electric field established by the par-ticle and/or the pore is on the order of 103k V/m, which is much

stronger than that of E, E. In this case, all the dependent variables can be partitioned into an equilibrium term and a perturbed term, the later is on the order of E. For the flow field, v and p can ex-pressed as v_{= v}(e)₊



_{v and p= p}(e)₊



_{p, respectively, where} super-script (e) and prefix



denote the equilibrium term and the perturbed term, respectively. Substituting these expressions into Eqs. (3) and (4) and collecting terms on the order of E yield the governing equa-tions for the perturbed flow field. For convenience, the



v and



p in

these equations are replaced by v and p, respectively. Suppose that (

2+gj) (kBT/ezj). Then it can be shown that in scaled symbols the governing equations of the concentration, electrical, and flow fields for the perturbed problem are (Lee et al., 1998; Hsu et al., 2007a)

n∗_{1= exp(−}

_r

∗₁)[1₋

_r(

∗_{2+ g}∗₁)] (6) n∗₂= exp(−

r

∗1)[1−



r(

∗2+ g1∗)] (7) ∇∗2

∗ 1= − 1 (1+



) (



a)2

r [exp(−

r

∗1)− exp(



r

∗1)] (8) ∇∗2

∗2− (



a)2 (1+



)[exp(−

r

∗ 1)+



exp(



r

∗1)]

∗2 = (



a) 2 (1+



)[exp(−

r

∗ 1)g∗1+ exp(



r

∗1)



g2∗] (9) ∇∗2g₁∗−

r∇∗

∗1· ∇∗g∗1=

2rPe1v∗· ∇∗

∗1 (10) ∇∗2g₂∗+



r∇∗

∗1· ∇∗g∗2=

2rPe2v∗· ∇∗

∗1 (11) ∇ · v∗= 0 (12) −∇∗p− ∇∗2v∗+ ∇∗2

∗2∇∗

∗1+ ∇∗2

∗1∇∗

∗2= 0 (13) In these expressions, n∗

j = nj/n10,



a is the surface potential of the particle,

r=



az1e/kBT,

∗j =

j/



a, g∗_j = gj/



a, j= 1, 2, ∇∗2= ∇2/a2,



= [2

j=1nj0(ezj)2/



kBT]1/2is the reciprocal Debye length,∇∗= ∇/a, v∗= v/UE, UE= (



2a/



a), Pej=



(zje/kBT)2/



Dj, j= 1, 2, is the Peclet number of ionic species j, and p∗= p/pref, pref=



2a/a2is a reference pressure.

Suppose that both the particle and the pore are non-conductive, impermeable to ionic species, and remained at constant surface po-tential, the concentration of ionic species away from the particle reaches the corresponding bulk value, and both the surface of the particle and that of the pore are no-slip. Then we have the following boundary conditions:

∗

1=



a/



a on the particle surface (14)

∗1=



b/



a on the pore surface (15)

∗

1= (



b/



a)

I₀(



r)

I₀(



c), z → ∞, r < b (16)

n· ∇

∗2= 0 on the particle surface (17)

n· ∇

∗

2= 0 on the pore surface (18)

∇

∗2= −E∗zez, |z| → ∞, r < b (19)

n_{· ∇}∗g∗_{j = 0, j = 1, 2 on the particle surface} (20)

n· ∇∗g∗_j = 0, j = 1, 2 on the pore surface (21)

g∗_j = −

∗

2, j= 1, 2, z → ∞, r < b (22)

v∗= (v/UE)ez on the particle surface (23)

v∗_{= 0 on the pore surface} (24)

v∗= (v(r)/UE)ez= −(



w/



a)  1− I0(



r) I₀(



b)  ez, z → ∞, r < b (25)

In these expressions, E∗_z= Ez/(



a/a), I0 is the zero-order modified Bessel function of the first kind,



_b is the surface potential of the pore, v is the speed of the particle in the z-direction, and ezis the unit vector in the z-direction. Note that Eq. (15) describes the undisturbed electroosmotic velocity profile, v(r)ez, for a charged cylindrical pore in the absence of the particle (Hsu and Ku, 2005).

Usually, problems of the present type need to be solved through a tedious trial-and-error procedure. This difficulty can be circum-vented by partitioning the original problem into two hypothesized sub-problems (O'Brien and White, 1978). In the first sub-problem, the particle moves with a constant velocity in the absence of E, which is a pure hydrodynamic problem. In the second sub-problem, E is applied but the particle is remained fixed in space. If we let F₁and F₂ be the total forces acting on the particle in these two sub-problems, respectively, then F1=



U∗and F2=



E∗z, where U∗=v/UE, and



and



are proportional constants. Therefore, we must have, at steady state,



=U_E∗∗

z = −





(26)

where



is the scaled electrophoretic mobility of the particle. In our problem, the forces acting on the particle include the electrical and the hydrodynamic forces, and only the z-components of these forces need to be considered. Let Fei and Fdibe the z-components of the electrical and the hydrodynamic forces in sub-problem i, re-spectively. It can be shown that (Hsu et al., 2007a,b;Hsu and Yeh, 2006;Happel and Brenner, 1983)

F_ei∗₌ Fei



2 a =  S

j

∗ 1

j

n

j

∗ 2

j

z∗ −

j

∗ 1

j

t

j

∗ 2

j

t dS ∗ ₍₂₇₎ F_di∗ = Fdz



2 a = S(



H∗_{· n) · e} zdS∗ (28)

In these expressions, n, t, and z∗are the magnitude of the unit normal vector, that of the unit tangential vector, and the z-component of the unit normal vector, respectively. F∗_eiand F_di∗ are the scaled forms of

Feiand Fdi, respectively. S∗= S/a2is the scaled particle surface, and



H∗_{the scaled shear stress tensor.}

3. Results and discussion

The complicated nature of the present boundary-valued problem suggests that it needs to be solved numerically. To this end,FlexPDE (2003), a commercial software, which is based on a finite element method, is adopted. Here, we assume that the pore is sufficiently long so that the end effect of the flow field can be neglected. Our experience reveals that this can be achieved by letting (L/d) > 10, L being the length of the computation domain in the axial direction. Double precision is applied throughout the numerical simulation and grid independence is checked to assure that the mesh used is fine enough. Typically, using a total of 16 000 and 8000 nodes is necessary for the resolution of the electric field and the flow field, respectively. A representative mesh used is shown inFig. 2.

To justify the applicability of the software adopted, it is used to solve the problem considered by Liu et al. (2004)where the elec-trophoresis of a cylindrical particle in a long cylindrical pore is under the conditions of low surface potential and finite double-layer thick-ness was simulated; their analysis was limited to the case where the end effect of a particle is negligible (L_T/L_{C= 100, LC}and L_T are the length of the particle and that of the pore, respectively). As seen

Fig. 2. Typical mesh used where the number of nodes is 8021. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 ω 10 1 0.1 -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 κa 10 1 0.1 κa ω

Fig. 3. Variation of the scaled electrophoretic mobilityas a function ofa at r= 1,= 0.5, and d/a = 10. Solid curves: present numerical results; dashed curves: analytical results of Liu et al. (2004) at LT/Lc= 100. (a)∗a= 1 and

∗ b= 0, (b) ∗ a= 0 and∗b= 1. 0.15 0.20 0.25 0.30 0.35 0.40 0.45 5 4 3 2 φr=1 ω 0.1 0.2 0.3 0.4 0.5 0.6 φr=1 5 4 ω 2 3 10 1 0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 κa 10 1 0.1 κa 10 1 0.1 κa ω 5 4 3 2 φr=1

Fig. 4. Variation of the scaled electrophoretic mobilityas a function ofa for various combinations ofrand (d/a) at

∗

b= 0 and= 0.5. (a) d/a = 0.25, (b) d/a = 1, (c) d/a= 4.

inFig. 3, the performance of the software adopted in our study is satisfactory, in general. Note that if the particle is charged and the pore is uncharged, the result of Liu et al. is less accurate when the double layer is thick (



a small). On the other hand, if the particle is

Fig. 5. Contours of the net scaled ionic concentration CD (=n∗

1− n∗2) for two levels ofrat

∗

b= 0,= 0.5, d/a = 1, anda= 0.5. (a)r= 1, (b)r= 5.

uncharged and the pore is charged, their result becomes less accurate when the double layer is thin.

Let us consider two representative cases, namely, a positively charged particle in an uncharged pore, and an uncharged particle in a positively charged pore.

3.1. Positively charged particle in an uncharged pore

Fig. 4illustrates the variation of the scaled electrophoretic mo-bility of a particle



as a function of the thickness of the double layer, measured by



a, at various combinations of the aspect ratio

of the particle (d/a) and its scaled surface potential

_rfor the case when a positively charged particle is placed in an uncharged pore. For illustration, the radius of the particle a is fixed and the recipro-cal Debye length



varies, that is, the concentration of electrolytes varies in the simulation. Note that for a fixed



a, although the higher

the

_r, the smaller the



, the actual mobility increases with

_r, as

it should be. As seen inFig. 4, if

_rtakes a small to medium large value,



increases monotonically with the increase in



a, but if

_ris sufficiently high,



has a local minimum as



a varies. The presence

of the local minimum in



as



a varies was also observed in the

electrophoresis of an isolated sphere in an infinite fluid (Wiersema et al., 1966;O'Brien and White, 1978), and in the electrophoresis of a sphere in a cylinder pore (Hsu and Cheng, 2007). Note that for a longer particle, the local minimum occurs at a lower level of surface potential. The presence of the local minimum in



arises from the effect of double-layer polarization, which yields an internal electric field having the direction inverse to that of the applied electric field (Lee et al., 1998;Hsu and Cheng, 2007;Hsu et al., 2007a,b). Because the strength of the induced electric field increases with the increas-ing

_r, the higher the

_r, the easier to observe the local minimum. The effect of double-layer polarization arises from the fact that as a particle moves in an applied electric field, the shape of the ionic cloud near itsfront region becomes asymmetric to that in its rear

0.2 0.3 0.4 0.5 0.6 0.7 0.1 0.180 0.185 0.190 0.195 0.200 κa ω ω d/a=4 2 1 0.5 0.25 10 1 0.1 0.1 0.2 0.3 0.4 0.5 ω κa 10 1 0.1 κa d/a=4 2 1 0.5 0.25 0.2 0.3 0.4 0.5 0.60.7

Fig. 6. Variation of the scaled electrophoretic mobilityas a function ofa for various combinations ofr and (d/a) at= 0.5 and

∗

b= 0. (a)r= 1, (b)r= 5.

region.Fig. 5illustrates the contours for the net scaled ionic con-centration CD (=n∗

2− n∗1) at two levels of

r. For the present case, because the particle is positively charged, CD is positive. As seen in Fig. 5, double-layer polarization is insignificant when

_ris low, but becomes appreciable if

_ris sufficiently high.

The influence of the aspect ratio of a particle (d/a) on its scaled mobility



is illustrated inFig. 6. This figure reveals that if



a is small,



declines with the increase in (d/a), but the reverse is true if



a is

large. These behaviors are the consequence of the competition be-tween the hydrodynamic drag and the electrical driving force acting on a particle as



a varies. For the present case, the larger the (d/a),

the larger the surface area of a particle, the greater the amount of surface charge, and therefore the greater the electrical driving force. The larger the surface area arising from a larger (d/a) also leads to a greater hydrodynamic drag. However, if



a small, the rate of

in-crease in the hydrodynamic drag as (d/a) inin-creases is faster than the rate of increase in the corresponding electrical driving force. If



a is

large, the reverse trend is observed.Fig. 6also reveals that if

_r is sufficiently high,



has a local minimum as



a varies. As mentioned

previously, this arises from the effect of double-layer polarization. Fig. 7illustrates the boundary effect, the significance of which is measured by



(=a/b), on the scaled mobility of a particle



for

0.2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 λ=(a/b) 0.25 0.5 ω 2 1 d/a=8 4 0.1 0.2 0.3 0.4 0.5 1 0.25 0.5 2 ω d/a=8 4 0.3 0.4 0.5 0.6 0.7 0.8 0.2 λ=(a/b) 0.3 0.4 0.5 0.6 0.7 0.8

Fig. 7. Variation of the scaled electrophoretic mobilityas a function of(=a/b) for various combinations ofrand (d/a) at

∗

b= 0 anda= 1. (a)r= 1, (b)r= 5.

various combinations of the aspect ratio (d/a) of the particle and the level of the scaled surface potential of the particle

_r. As seen in Fig. 7, the larger the



, the smaller the



, that is, the presence of the pore has the effect of retarding the translation of the particle. If



is small,



increases with the increasing (d/a), but the reverse trend is observed if



is large. This can be explained by the fact that if



is small, the rate of increase in the electrical driving force acting on a particle as (d/a) increases is faster than that of the hydrodynamic drag, but this trend becomes reversed if



is large. Note that for the case where

ris low, the boundary effect becomes unimportant if



is sufficiently large.

3.2. Both particle and pore are positively charged

For the present case, two additional effects that are absent in the case where a positively charged particle is placed in an un-charged pore, namely, a negative charge is induced on the parti-cle surface as it approaches the pore, and an electroosmotic flow developed by the charged pore. Because both of these effects will reduce the electrophoretic velocity of the particle, its mobility is ex-pected to be smaller than that of the previous case, as is justified by comparing the results presented inFigs. 8 and 4. Except that, the

0.00 0.05 0.10 0.15 0.20 0.25 0.30 5 4 3 2 φr=1 ω 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 5 4 2 3 φr=1 ω 10 1 0.1 -0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 5 4 2 3 φr=1 ω κa 10 1 0.1 κa 10 1 0.1 κa

Fig. 8. Variation of the scaled electrophoretic mobilityas a function ofa for various combinations ofr and (d/a) at

∗

b= 0.2

∗

a and= 0.5. (a) d/a = 0.25, (b) d/a= 1, (c) d/a = 4.

general qualitative behaviors of



inFig. 8are similar to those of Fig. 4.

Fig. 9illustrates the variation of the scaled electrophoretic mobil-ity of a particle



as a function of the thickness of the double layer



a for various aspect ratios (d/a) at two levels of the scaled surface

potential of the particle

_r. A comparison betweenFigs. 9 and 6 re-veals that although the general behaviors of



in these two figures

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 1 ω d/a=4 2 0.5 0.25 10 1 0.1 -0.05 0.00 0.05 0.10 0.15 0.20 0.25 ω κa 10 1 0.1 κa d/a=4 2 1 0.5 0.25

Fig. 9. Variation of the scaled electrophoretic mobilityas a function ofa for various combinations ofrand (d/a) at= 0.5 and

∗

b= 0.2

∗

a. (a)r= 1, (b)r= 5.

seem to be similar to each other, there exist two basic differences. Firstly, if

_r is low, for a small to medium large



a,



increases monotonically with the increasing



a inFig. 6, but the reverse trend is observed inFig. 9. Secondly, while



is always positive inFig. 6, it may change its sign from positive to negative as



a increases in

Fig. 9(b), where

_ris high. In addition,



may change its sign from negative to positive again if



a is further increased. The variations in

the sign of



as



a varies arises mainly from the effect of double-layer

polarization, the negative charge induced on the particle surface as it approaches the positively charged pore, and the electroosmotic flow generated by the charged pore. In the present case, the polarized double layer yields a local electric field, the direction of which is op-posite to that of the applied electric field. In practice, attention needs be paid to the changes in the direction of electrophoresis because it will influence the prediction of the charged conditions on a parti-cle if electrophoresis is used as an analytical tool, or the separation efficiency if it is adopted as a separation technique. The results pre-sented inFig. 9suggest that the concentration of electrolytes plays the key role in designing an electrophoresis operation.

A comparison betweenFigs. 10 and 7indicates that if the scaled surface potential of a particle

_r is low, the influence of the pore on the scaled electrophoretic mobility of the particle



for the case where both the particle and the pore are positively charged is similar

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.25 1 2 4 d/a=8 ω 0.5 0.2 -0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 1 d/a=8 ω λ=(a/b) 4 2 0.5 0.25 0.3 0.4 0.5 0.6 0.7 0.8 0.2 λ=(a/b) 0.3 0.4 0.5 0.6 0.7 0.8

Fig. 10. Variation of the scaled electrophoretic mobilityas a function of(=a/b) for various combinations of (d/a) andrat

∗

b=0.2

∗

aanda=1. (a)r=1, (b)r=5.

to that for the case where the particle is positively charged and the pore is uncharged, except that the value of



in the former is smaller than that in the latter. However, if

_ris high, they are different both quantitatively and qualitatively. In particular, the



inFig. 10(b) can have a negative local minimum as



varies if the aspect ratio of the particle (d/a) is sufficiently large. Again, this arises from the effects of double-layer polarization, the negative charge induced on the particle surface, and the electroosmotic flow.

4. Conclusions

The influences of the presence of a boundary, the effect of double-layer polarization, and that of electroosmotic flow on the electrophoretic behavior of a particle are discussed by considering the electrophoresis of a rigid, finite cylindrical particle along the axis of a long, charged cylindrical pore. The results of numerical simulation can be summarized as follows. (a) The presence of a positively charged pore leads to two effects that are absent in the case where it is uncharged, namely, negative charge is induced on the particle surface as it approaches the pore, and an electroosmotic flow is developed by the charged pore. Both of these effects will reduce the electrophoretic velocity of a positively charged particle. (b) For the case of a positively charged particle and an uncharged

pore if the surface potential of the particle is low and the thickness of the double layer takes a small to medium large value, the mo-bility of the particle increases monotonically with the decreasing double-layer thickness, but the reverse trend is observed for the case of a positively charged particle and a positively charged pore. On the other hand, if the surface potential of the particle is high, the mobility in the former case is always positive as the double-layer thickness declines, but that in the latter case may change its sign from positive to negative and from negative to positive again. These findings have practical significance because the change in the direc-tion of electrophoresis will influence the predicdirec-tion of the charged conditions on a particle if electrophoresis is used as an analytical tool, or the separation efficiency if it is adopted as a separation technique. (c) For a positively charged particle, if its surface poten-tial is low, the behavior of its mobility for the case of a positively charged pore is similar to that of an uncharged pore, except that the mobility in the former is smaller than that in the latter. However, if the surface potential is high, they are different both quantitatively and qualitatively. In particular, the mobility in the former can have a negative local minimum as the degree of boundary effect varies.

Notation

a radius of the particle, m

b radius of the pore, m

CD =n∗ 2− n∗1

d length of the particle, m

D_j diffusivity of ionic species j, m2/s

e elementary charge, C

ez unit vector in the z-direction

E strength of the applied electric field, m2/V/s

E applied electric field, V/m

EZ strength of the applied electric field in the z-direction, m2_/V/s

E∗_Z scaled strength of the applied electric field in the

z-direction

Fi total force acting on the particle in sub-problem i, N/m

F_di z-component of the hydrodynamic force in

sub-problem i, N/m

Fei z-component of the electrical force in sub-problem i, N/m

F∗_di scaled z-component of the hydrodynamic force in sub-problem i

F∗_ei scaled z-component of the electrical force in sub-problem i

g_j perturbed potential associated with ionic species j, V

g∗

j scaled perturbed potential

I₀ zero-order modified Bessel function of the first kind

kB Boltzmann constant, J/K

Lc length of the particle, m

L_T length of the pore, m

n magnitude of a unit normal vector

n_j number concentration of ionic species j, 1/m3

n∗_j scaled number concentration of ionic species j, 1/m3

n_j0 bulk number concentration associated with ionic

species j, 1/m3

p hydrodynamic pressure, Pa

p∗ scaled hydrodynamic pressure

Pe_j Peclet number of ionic species j

p_ref reference pressure, Pa

S particle surface

S∗ scaled particle surface

t unit tangential vector

U velocity of the particle, m/s

UE = (



2a/



a), m/s

v speed of the particle in the z-direction, m/s

v liquid velocity, m/s

v∗ scaled liquid velocity

z∗ z-component of a unit normal vector

z1 valence of cations z₂ valence of anions Greek letters



= − z2/z1



proportional constant ∇ gradient operator, 1/m

∇∗ scaled gradient operator

∇2 _{Laplace operator, 1/m}2

∇∗2 scaled Laplace operator



dielectric constant, C2/N/m2



surface (zeta) potential, V



a surface potential of the particle, V



b surface potential of the pore, V



viscosity, kg/m/s2



reciprocal Debye length, m



=a/b



mobility, m2/V/s



space charge density, C/m3



H∗ _{scaled shear stress tensor}

1 equilibrium potential, V

2 perturbed potential, V

r scaled surface potential of the particle

∗

j scaled potential



proportional constant



scaled electrophoretic mobility

Acknowledgment

This work is supported by the National Science Council of the Republic of China.

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