Electrophoresis of a Sphere along the Axis of a Cylindrical Pore: Effects of Double-Layer Polarization and Electroosmotic Flow

Subscriber access provided by NATIONAL TAIWAN UNIV

Langmuir is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036

Research Article

Electrophoresis of a Sphere along the Axis of a Cylindrical Pore:

Effects of Double-Layer Polarization and Electroosmotic Flow

Jyh-Ping Hsu, and Zheng-Syun Chen

Langmuir, 2007, 23 (11), 6198-6204 • DOI: 10.1021/la070079m Downloaded from http://pubs.acs.org on November 18, 2008

Subscriber access provided by NATIONAL TAIWAN UNIV

Langmuir is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036

Subscriber access provided by NATIONAL TAIWAN UNIV

Langmuir is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036

More About This Article

Additional resources and features associated with this article are available within the HTML version:

• Supporting Information

• Access to high resolution figures

• Links to articles and content related to this article

Electrophoresis of a Sphere along the Axis of a Cylindrical Pore:

Effects of Double-Layer Polarization and Electroosmotic Flow

Jyh-Ping Hsu* and Zheng-Syun Chen

Department of Chemical Engineering, National Taiwan UniVersity, Taipei, Taiwan 10617 ReceiVed January 11, 2007. In Final Form: March 9, 2007

The electrophoresis of a rigid sphere along the axis of a cylindrical pore is investigated theoretically. Previous analysis is extended to the case where the effects of double-layer polarization and electroosmotic flow can be significant. The influences of the surface potential, the thickness of the double layer, and the relative size of a pore on the electrophoretic behavior of a sphere are discussed. Some interesting results are observed. For example, if both a sphere and a pore are positively charged, then the mobility of the sphere has a local minimum as the thickness of its double layer varies. Depending upon the level of the surface potential of a sphere and the degree of significance of the boundary effect, the mobility of the sphere may change its sign twice as the thickness of its double layer varies. This result can play a significant role in electrophoresis measurements.

1. Introduction

Electrophoresis has many applications in various fields. It is not only adopted often to characterize the charged conditions of entities of colloidal size but also used widely in both conventional and modern operations. One of the key effects in conducting electrophoresis and electrophoresis-related operations is the presence of a boundary. Electrophoresis conducted in a narrow space such as capillary electrophoresis is a typical example of the former, and electrodeposition where charged particles are driven by an applied electric field toward electrodes is an example of the latter. The charged conditions on a boundary can make things even more complicated because the associated electroos-motic flow will influence not only the magnitude of the electrophoretic velocity of a particle but also its direction.

The boundary effect on electrophoresis has been studied extensively in the last few decades, and various types of problems have been investigated. These include the electrophoresis of a sphere parallel to a plate,1-4_{along the axis of a cylindrical pore,}2-6 positioned eccentrically in a cylindrical pore,7-9 _{through a} converging-diverging pore,10_{along the center line between two} parallel plates,2,4 _{and in a spherical cavity.}11-14 _{Although the} presence of a boundary simply yields an extra viscous force on a particle arising from the nonslip condition on the boundary, the influences of other effects such as the level of surface potential, the thickness of the electrical double layer, and the charged conditions on the boundary surface can be profound. For instance, if a boundary is charged, an electroosmotic flow is generated that will influence both qualitatively and quantitatively the electrophoretic behavior of a particle.2_{The interaction between}

the double layer of a particle with a boundary can also lead to complicated results.6

In this study, previous analysis of the electrophoresis of a sphere along the axis of a cylindrical pore under the conditions of low surface potential and weak applied electric field6 _is extended to the case of arbitrary surface potential and double-layer thickness. Also, the surface of a sphere and that of a pore can be charged. This implies that the effects of double-layer polarization and electroosmostic flow, both of practical significance,15-16 _{can be important and should be taken into} account.

* To whom correspondence should be addressed. E-mail: jphsu@ ntu.edu.tw. Fax: 886-2-23623040.

(1) Keh, H. J.; Chen, S. B. J. Fluid Mech. 1988, 194, 377. (2) Keh, H. J.; Anderson, J. L. J. Fluid Mech. 1985, 153, 417. (3) Shugai, A. A.; Carnie, S. L. J. Colloid Interface Sci. 1999, 213, 298. (4) Ennis, J.; Anderson, J. L. J. Colloid Interface Sci. 1997, 185, 497. (5) Keh, H. J.; Chiou, L. C. Am. Inst. Chem. Eng. J. 1996, 42, 1397. (6) Hsu, J. P.; Ku, M. H. J. Colloid Interface Sci. 2005, 283, 592. (7) Yariv, E.; Brenner, H. Phys. Fluids 2002, 14, 3354. (8) Ye, C.; Xuan, X.; Li, D. Microfluid Nanofluid 2005, 1, 234. (9) Hsu, J. P.; Kuo, C. C. J. Phys. Chem. B 2006, 110, 17607.

(10) Davison, S. M.; Sharp, K. V. J. Colloid Interface Sci. 2006, 303, 288. (11) Zydney, A. L. J. Colloid Interface Sci. 1995, 169, 476.

(12) Lee, E.; Chu, J. W.; Hsu, J. P. J. Colloid Interface Sci. 1997, 196, 316. (13) Lee, E.; Chu, J. W.; Hsu, J. P. J. Colloid Interface Sci. 1998, 205, 65. (14) Hsu, J. P.; Hung, S. H.; Kao, C. Y. Langmuir 2002, 18, 8897.

Figure 1. Schematic representation of the problem considered where a sphere of radius a is placed on the axis of a long cylindrical pore of radius b. A uniform electric field E parallel to the axis of the pore is applied. The cylindrical coordinates (r,θ, z) with the origin located at the center of the sphere are adopted.

6198 Langmuir 2007, 23, 6198-6204

10.1021/la070079m CCC: $37.00 © 2007 American Chemical Society Published on Web 05/01/2007

2. Theory

Let us consider the electrophoresis of a rigid sphere of radius a along the axis of a long cylindrical pore of radius b shown in Figure 1. The liquid phase is an incompressible Newtonian fluid with constant physical properties. The cylindrical coordinates (r,θ, z) are chosen with the origin located at the center of the

sphere. A uniform electric field E is applied in the z direction, and U is the corresponding electrophoretic velocity of the sphere. The symmetric nature of the system under consideration suggests that only the (r, z) domain has to be considered. For the present case, the electrical potential φ can be described by the Poisson equation

Here, ∇2_{is the Laplace operator,  is the permittivity of the} liquid phase, F is the space charge density, e is the elemen-tary charge, and nj and zj are the number concentration

and the valence of ionic species j, respectively. If we let Dj, kB,

T, and v be the diffusivity of ionic species j, the Boltzmann constant, the absolute temperature, and the liquid velocity, respectively, then the conservation of ionic species at steady state leads to

For convenience, φ is decomposed into an equilibrium potential or the potential in the absence of E, φ1, and a perturbed potential

(15) Wong, P. K.; Wang, T. H.; Deval, J. H.; Ho, C. M. IEEE-ASME Trans.

Mech. 2004, 9, 366.

(16) Stone, H. A.; Stroock, A. D.; Ajdari, A. Annu. ReV. Fluid Mech. 2004,

36, 381.

(17) Wiersema, P. H.; Loeb, A. L.; Overbeek, J. Th. G. J. Colloid Interface

Sci. 1966, 22, 78.

Figure 2. Variation of the scaled transition length (Ltc/a)

(a) as a function of κa at φr) ζaz1e/kBT ) 1,ζ*b) 0.2ζ*a, andλ

) 0.5 and (b) as a function of λ () a/b) at φr) 1, ζ*_b) 0.2ζ*_a,

and κa ) 1. ∇2 φ ) -F  ) -

∑

Figure 3. Variation of the scaled electrophoretic mobilityµE(defined

in eq 28) as a function of κa at various levels of φr()ζaz1e/kBT)

for the case ofζ*_b) 0. (a) λ ) 0.3, (b) λ ) 0.5, and (c) λ ) 0.7.

∇‚

[

(

z_je

k_BTnj∇φ

)

]

arising from E, φ2, that is, φ ) φ1+ φ2.18Also, njis expressed

as18

where gj is a perturbed potential that takes account of the

deformation of the double layer surrounding a particle as it moves. Note that the form of eq 3 is assumed for convenience. Until now, gjhas been considered to be an arbitrary function. Under

conditions of practical significance, the applied electric field is weak compared with that established by the sphere and/or by the cavity so that the expressions for the distortion of the double layer, the electric potential, and the flow field near the sphere can be linearized. For example, the scaled number concentrations of ions, n*₁and n*₂, can be approximated respectively by

where n*_j) nj/n10, φj

/

) φj/ζa, gj

/

) gj/ζa, j ) 1, 2, R ) -z2/z1, φr) ζaz1e/kBT, andζais the surface potential of the sphere. In scaled quantities, the equilibrium electrical potential can be

described by

where∇*2_{) ∇}2_/a2_{is the scaled Laplace operator and κ ) [}∑ j)1 2 nj0(ezj)2/kBT]1/2is the reciprocal Debye length. It can be shown that eqs 1, 3, 4, 5, and 6 lead to

after neglecting terms involving products of small quantities such as g₁/, g₂/, and φ₂/. Similarly, the variation of g_j/ can be approximated by employing eqs 3-5 to obtain

Here,∇* ) ∇/a is the scaled gradient operator, v* ) v/UE, UE ) ζa2/ηa is the magnitude of the velocity based on Smolu-chowski’s theory when an electric field of strengthζa/a is applied, η is the viscosity of the liquid phase, and Pej) (zje/kBT)2/ηDj

for j )1, 2 is the electric Peclet number of ionic species j. We assume that both the sphere and the pore are nonconductive, impermeable to ionic species, and remain at constant surface potential. Also, the concentrations of ionic species reach the bulk values at both the inlet and the outlet of a pore. These lead to the following boundary conditions:

In these expressions, E*_z ) Ez/(ζa/a), I0 is the zeroth-order modified Bessel function of the first kind, andζbis the surface potential of the pore.

If we let p be the hydrodynamic pressure, then the flow field can be described by

(18) O’Brien, R. W.; White, L. R. J. Chem. Soc., Faraday Trans. 2 1978, 74, 1607.

Figure 4. (a) Variation ofβ (defined in eq 27) as a function of κa at various levels of φr()ζaz1e/kBT) and (b) that of F*_ez(defined

in eq 31) as a function of κa for the case whenζ*_b) 0 and λ ) 0.5. n_j) n_j0exp

(

)

ζ_aon the sphere surface (10)

φ*₁)ζb

ζ_aon the pore surface (11)

φ*₁)ζb ζa

I₀(κr)

I₀(κb) |z| f∞ r < b (12)

n‚∇*φ*₂) 0 on the sphere surface (13)

n‚∇*φ*₂) 0 on the pore surface (14)

n‚∇*φ*₂) -E*_z |z| f∞ r < b (15)

n‚∇*g*_j) 0 j ) 1, 2 on the sphere surface (16)

n‚∇*g*_j) 0 j ) 1, 2 on the pore surface (17) g*_j) -φ*₂ j ) 1, 2 |z| f∞ r < b (18)

∇‚v ) 0 (19)

-∇p + η∇2

v + F∇φ ) 0 (20)

These expressions can be rewritten in scaled forms as

where p* ) p/prefand pref) ξa2/a2.

Assuming no-slip conditions on the surfaces of the sphere and the pore, the boundary conditions associated with eqs 21 and 22 can be expressed as

Here, V is the speed of the sphere in the z direction, and ez is the unit vector in the z direction. Equation 25

de-scribes the scaled undisturbed electroosmotic velocity pro-file for a charged cylindrical pore in the absence of the sphere.6

The problem under consideration is decomposed into two subproblems.18_{In the first subproblem, a sphere moves} with a constant velocity in the absence of E, and in the second subproblem, a sphere is fixed in space when E is applied. The total forces acting on a sphere in these two subproblems,

Figure 5. Contours of the net scaled ionic concentration CD () n*₁- n*₂) on the semiplaneθ ) π/2 at two levels of φr()ζaz1e/kBT)

for the case whenζb

/

) 0, λ ) 0.5, and κa ) 0.8. (a) φr) 1 and (b) φr) 5.

∇‚v* ) 0 (21)

-∇p* + ∇2_{v* +∇}2

φ*∇φ* ) 0 (22)

v* )

(

U_E

)

v* ) 0 on the pore surface (24)

v* )

(

)

(

)

[

]

F1and F2, are

whereχ and β are constant and independent of U* and E*_z, respectively. Because F1+ F2) 0 is at steady electrophoretic mobility,µEis given by

In our case, the force acting on a sphere in subproblem i, Fi,

comprises the electrical force Feand the hydrodynamic force Fd.

Let Fzi, Fez, and Fdzbe the z components of F, Fe, and Fd in

subproblem i, respectively. Then

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, Fez, can be calculated by integrating the Maxwell stress tensor over the particle surface,

Here, S denotes the surface of a sphere, E ) -∇φ, and σE≡ (EE - (1/2) E2_{I) is the Maxwell stress tensor. It can be shown} that this expression leads to19,20

where F*_ezand S* are the scaled form of Fezand S, respectively (Appendix). In our case, the hydrodynamic force acting on a sphere in the z direction, Fdz, comprises the viscous force and the pressure and can be evaluated by21

whereσH≡ -pI + η[∇v + (∇v)T_{]. This expression can be}

rewritten in terms of scaled symbols as

where F*_dzis the scaled form of Fdz.

The governing equations and the associated boundary condi-tions are solved numerically by FlexPDE,22_{a differential equation} solver based on a finite element method. The applicability of this software to the resolution of the electrokinetic phenomena of the present type was justified by Hsu and Ku.6_{Double precision is} used throughout the computation, and grid independence is checked to ensure that the mesh used is fine enough. In general, a convergent result can be obtained by setting the error limit of 10-6 for the electric field and 10-3 for the flow field. Two represented cases are considered in the numerical simulation, namely, a positively charged sphere is placed in an uncharged

pore, and a positively charged sphere is placed in a positively charged pore.

3. Results and Discussion

The influences of the key parameters of the system under consideration on the electrophoretic behaviors of a sphere are examined through numerical simulation. For illustration, we assume that a ) 100 nm and T ) 298 K. For an aqueous solution of a common electrolyte such as KCl, we have Dj) 2 × 10-9

m2_{/s,  ) 8.854}× 10-12× 78.54688 F/m, η ) 1 cP, and R ) 1. These lead to Pe1) Pe2) 0.1. Also, we assume that a pore is sufficiently long so that the end effects of the flow field can be neglected. To make sure that this is appropriate, the transition length of a pore Ltc, the shortest length to achieve fully developed electroosmotic flow, is estimated under conditions of interest. The estimation of Ltcis based on the criteria proposed by Hsu and Kuo.9_{Figure 2a illustrates the variation of the scaled transition} length of a pore (Ltc/a) as a function of κa for the case when both a sphere and a pore are positively charged, and that as a function ofλ () a/b) is shown in Figure 2b. According to this Figure,

assuming a pore length on the order of 10a is sufficient to assuming a fully developed flow field.

3.1. Positively Charged Sphere in an Uncharged Pore. Let us first consider the case when a positively charged sphere is placed in an uncharged pore. The influence of the surface potential of a sphere on its mobilityµEat various levels of double-layer thickness, measured by κa, and the boundary effect, measured byλ () a/b), is illustrated in Figure 3. Here, the value of a is

fixed, and that of κ varies, that is, we fix the radius of a sphere and let the concentration of electrolytes vary. In general, for a fixed value of κa, the higher the surface potential, the smaller the mobility; the more important the boundary effectλ, the smaller

the mobility. The latter is expected because the presence of the pore wall has the effect of retarding the movement of a sphere. The former can be explained by the effect of double-layer polarization, which induces an internal electric field, the direction of which is opposite that of the applied electric field. Because the strength of the induced electric field increases with the increase in the surface potential of a sphere,µEdecreases with the increase in the scaled surface potential of a sphere φr. Figure 3 also indicates that for a low to medium value of φr,µE increases with the increase in κa, and if φris sufficiently high, thenµEhas a local minimum as κa varies. Similar phenomena were also observed by Wiersema17_{and O’Brien}18_{for the electrophoresis of an isolated}

(19) Hsu, J. P.; Yeh, L. H.; Ku, M. H. J. Colloid Interface Sci. 2007, 305, 324. (20) Hsu, J. P.; Yeh, L. H. J. Chin. Inst. Chem. Eng. 2006, 37, 601. (21) Happel, J.; Brenner, H. Low-Reynolds Number Hydrodynamics; Nijhoff: Boston, 1983.

(22) FlexPDE, version 2.22; PDE Solutions Inc.: Sunol, CA.

F₁) χU* (26) F₂) βE_z* (27) µE) U* E_z/ ) - β χ) -F₂ F₁ (28) F_zi) F_ezi+ F_dzi) F_i i ) 1, 2 (29) F_ez)

∫

∫

∫

∫

Figure 6. Variation of the scaled electrophoretic mobilityµE(defined

in eq 28) as a function ofλ () a/b) at various levels of φr()

ζaz1e/kBT) for the case whenζb /

) 0 and κa ) 1.

sphere in an infinite fluid and by Lee et al.13_{for the electrophoresis} of a sphere in a spherical cavity. Note that if κa is sufficiently small thenµEbecomes independent of φr.

Figure 4 shows the variations ofβ and F*_ezas a function of the double-layer thickness measured by κa for the case presented in Figure 3 atλ ) 0.5. According to the definitions, β and F*_ez

are measures of the net driving force and the electric force acting on a sphere in the z direction in the second subproblem, respectively. As can be seen in Figure 4, the qualitative trends in bothβ and F*_ezare the same as that of the mobility illustrated in Figure 3.

Figure 5 illustrates the contours for the net scaled ionic concentration CD () n*₁- n*₂) on semiplaneθ ) π/2 for two

levels of the scaled surface potential of a sphere φr. This Figure

reveals that if φris low then the contours are essentially symmetric around a sphere, implying that double-layer polarization is insignificant. On the other hand, if φris sufficiently high, then it becomes significant. For the present case, a sphere moves upward, and the concentration of anions (counterions) near its bottom is higher than that near its top. An internal electric field is induced, the direction of which is opposite to that of the applied electric field.

The boundary effect on the mobility of a sphere for the case presented in Figure 3 at κa ) 1 is summarized in Figure 6. As mentioned previously, the more important the boundary effect, the smaller the mobility. Note that the influence of the boundary effect is threefold: the squeezing of the applied electric field between a sphere and a pore, the increase in the surface charge of a sphere due to double-layer deformation, and the increase in the viscous drag on a sphere.

3.2. Both the Sphere and Pore are Positively Charged. Consider next the case when a positively charged sphere is placed in a positively charged pore. Figure 7 shows the variation of the mobilityµEas a function of the double-layer thickness measured by κa at various combinations of the scaled surface potential of a sphere φrand the boundary effect measured byλ () a/b), and the corresponding variations of the net driving force measured byβ and the scaled electric force F*_ezfor the case in whichλ ) 0.7 in the second subproblem are illustrated in Figure 8. A comparison between Figures 3 and 7 reveals that the mobility in the former is larger than that in the latter. This is because the direction of electroosmotic flow arising from the charged pore Figure 7. Variation of the scaled electrophoretic mobilityµE(defined

in eq 28) as a function of κa at various levels of φr()ζaz1e/kBT)

for the case whenζ*_b) 0.2ζ*_a. (a)λ ) 0.3, (b) λ ) 0.5, and (c) λ ) 0.7.

Figure 8. Variation ofβ (defined in eq 27) as a function of κa (a) at various levels of φr()ζaz1e/kBT) and that of (b) F*_ez(defined

in eq 31) as a function of κa for the case whenζ*_b) 0.2ζ*_aand λ ) 0.7.

is in the opposite direction to that of the applied electric field. The general trends inµEin Figure 7 are similar to that in Figure 3. However, the level of φrat whichµEhas a local minimum as κa varies in the former is lower than that in the latter. It is interesting that in Figure 7c the presence of a local minimum in

µEmay lead to two changes in sign as κa varies. The reverse direction of the electrophoresis of a sphere is interesting and can play an important role in electrophoretic measurements. The behaviors ofβ and F*_ezshown in Figure 8 are similar to those observed in Figure 4 and can be explained by the same reasoning. Figure 9 shows the variation in mobilityµEas a function of the boundary effect measured byλ () a/b) at various levels of

the scaled surface potential of a sphere φr. A comparison between this Figure and Figure 6 reveals that althoughµEdeclines with the increase inλ in both cases as a result of the hydrodynamic

retardation of the pore,µEis smaller in the former. This is because the movement of a sphere in the former is influenced by both the electroosmotic flow and the charge induced on its surface, and theses effects dominate its electrophoretic behavior as λ

varies.

4. Conclusions

The boundary effect on electrophoresis is examined by considering the movement of a rigid sphere along the axis of a cylindrical pore at an arbitrary level of electrical potential. The effects of both double-layer polarization and electroosmotic flow arising from a charged pore are taken into account. The results of numerical simulation can be summarized as follows. For a positively charged sphere in an uncharged pore, if the surface potential of the sphere is high, then its mobility has a local minimum as the thickness of its double layer varies, which is of practical significance, For example, if capillary electrophoresis is used as a separation tool, then the bulk electrolyte concentration needs to be selected carefully to prevent minimum efficiency. For a fixed double-layer thickness, the mobility decreases with the increases in the surface potential. In general, the presence

of the pore wall has the effect of reducing the mobility of a sphere. For the case when both a sphere and a pore are positively charged, the mobility of the sphere may also have a local minimum as the thickness of its double layer varies even if its surface potential is low. Depending upon the level of the surface potential of a sphere, the direction of its movement may change twice as the thickness of its double layer varies, which is a result of practical significance, for example, when capillary electrophoresis is chosen as a separation tool. In this case, the separation efficiency depends largely upon the bulk concentration of electrolytes.

Appendix: Derivation of Equation 3119,20

For convenience, eq 30 is written as

whereσE_{≡ (EE - (1/2) E}2_{I) and E ) -∇φ ) n(∂φ/∂n) +} t(∂φ/∂t). Here, I is the unit tensor, t is the unit tangential vector on the particle surface, n and t are the magnitudes of n and t, respectively, and E2) E‚E. Substituting σE_{into eq A1 yields}

where nzis the z component of n. Because φ ) φ1+ φ2, eq A2 becomes, after eliminating some minor terms,

For the present case, because the sphere is nonconducting,

vanishes, and eq A3 reduces to

If the surface of a particle is maintained at a constant potential, then ∂φ1/∂t ) 0, and this expression can be further simplified to

Rewriting this equation with scaled symbols leads to eq 31 in the text.

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

LA070079M

Figure 9. Variation of the scaled electrophoretic mobilityµE(defined

in eq 28) as a function ofλ () a/b) at various levels of φr()

ζaz1e/kBT) for the case whenζ*_b) 0.2ζ*_aand κa ) 1.

F_ez)

∫∫

∫∫

(

[

(

)

(

)

]

)

∫∫

(

[

]

)

∫∫

[

(

)

(

)

]

∫∫

(

[

]

[

]

)

∫∫