Electrophoresis of a Charge-Regulated Sphere Normal to a Large

The electrophoresis of a rigid, charge-regulated, spherical particle normal to a large disk is investigated under the conditions of low surface potent...
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Langmuir 2005, 21, 7588-7597

Electrophoresis of a Charge-Regulated Sphere Normal to a Large Disk Jyh-Ping Hsu,* Ming-Hong Ku, and Chao-Chung Kuo Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan 10617 Received March 6, 2005. In Final Form: May 22, 2005 The electrophoresis of a rigid, charge-regulated, spherical particle normal to a large disk is investigated under the conditions of low surface potential and weak applied electric field. We show that, although the presence of a charged disk does not generate an electroosmotic flow, it affects particle motion appreciably through inducing charge on its surface and establishing an osmotic pressure field. The competition between the hydrodynamic force and the electric force may yields a local extremum in mobility; it is also possible that the direction of particle movement is reversed. In general, if a particle remains at constant surface potential, a decrease in the thickness of double layer has the effect of increasing the electrostatic force acting on it so that its mobility increases. However, this might not be the case for a charged-regulated particle because an excess hydrodynamic force is enhanced. For a fixed separation distance, the influence of a charged disk on mobility may reduce to a minimum if the bulk concentration of hydrogen ion is equal to the dissociation constant of the monoprotic acidic functional groups on particle surface.

1. Introduction In most of the applications of electrophoresis, charged entities are not isolated, and usually migrate under the influence of neighboring entities and/or boundaries. If a charged entity is sufficiently close to a charged surface or to another charged entity, the overlapping between and the distortion of the electrical double layers can play a significant role. For a spherical particle, this leads to asymmetric ionic concentration and electrical potential distributions surrounding the particle, which causes asymmetrical stresses of electrical origin on its surface, and the particle experiences an electrical interaction force. The presence of a neighboring entity and/or surface also influences hydrodynamically the movement of an entity through the viscous force arising from a no-slip condition. The evaluation of these forces involves solving a PoissonBoltzmann equation for the electrical field and a NavierStokes equation for the flow field. The degree of difficulty in solving these equations depends largely upon the level of electrical potential, the geometry of the problem, and the types of boundary conditions. Among various types of boundary effects, the electrophoresis of a particle normal to a planar surface is of practical significance. Electrodeposition, for example, involves this phenomenon where the deposition of charged particles onto electrodes is driven by an applied electric field. The particle-boundary interactions under the conditions of thin double layer and low surface potential have been studied intensively in the literature. Analytical solutions in terms of bi-spherical expansions were obtained for the case a sphere moving normal to a planar wall.1,2 Based on a reflection method, Keh and Anderson3 considered various sphere-boundary geometries and approximate expressions for the mobility of a particle were obtained. They concluded that the effect of particle-boundary on electrophoresis is on the order O(λ3), much weaker than that on sedimentation, O(λ), λ being the ratio of the particle radius/distance * Corresponding author. Fax: 886-2-23623040. E-mail: jphsu@ ntu.edu.tw. (1) Morrison, F. A.; Stukel, J. J. J. Colloid Interface Sci. 1970, 33, 88. (2) Keh, H. J.; Lien, L. C. J. Chin. Inst. Chem. Eng. 1989, 20, 283. (3) Keh, H. J.; Anderson, J. L. J. Fluid Mech. 1985, 153, 417.

from boundary. Employing a boundary collocation technique, Keh and Lien4 studied the electrophoresis of a sphere toward a circular hole or a disk. Feng and Wu5 analyzed the electrophoresis of an arbitrary prolate body of revolution toward an infinite conducting wall through using a combined analytical-numerical method. Since all of these analyses are limited to infinitely thin double layers, the surface potential of the planar wall or disk has no contribution to the mobility of a particle when it moves normal to the planar wall or disk. Ennis and Anderson6 extended the analysis of Keh and Anderson3 to the case of a finite double layer thickness, but the double layer surrounding a particle is not allowed to overlap significantly with that near the boundary. For a sphere normal to a planar wall, it was concluded that, if both the surface potential of the sphere and that of the wall are of the same sign, then the presence of the double layer near the latter acts to increase the velocity of the former. Applying Teubner’s method,7,8 Shugai and Carnie9 solved numerically the electrophoresis of a sphere with a thick double layer in bounded flows. They claimed that if the double layer surrounding a sphere is close to the boundary their method is more accurate than the reflection method used by Ennis and Anderson.6 Tang et al.10 examined the electrophoresis of a charge-regulated sphere normal to a planar surface based on a collocation method. Chih et al.11 discussed the effect of polarization on the electrophoresis of a sphere normal to a plane at arbitrary surface potential and double layer thickness. Because the plane was assumed to be free of charge in their analysis, the influence of its charged conditions was neglected. Often, the surface of a particle is assumed to be maintained at either a constant potential or constant (4) Keh, H. J.; Lien, L. C. J. Fluid Mech. 1991, 224, 305. (5) Feng, J. J.; Wu, W. Y. J. Fluid Mech. 1994, 264, 41. (6) Ennis, J.; Anderson, J. L. J. Colloid Interface Sci. 1997, 185, 497. (7) Teubner, M. J. Chem. Phys. 1982, 76, 11. (8) Shugai, A. A.; Carnie, S. L.; Chan, D. Y. C.; Anderson, J. L. J. Colloid Interface Sci. 1997, 191, 357. (9) Shugai, A. A.; Carnie, S. L. J. Colloid Interface Sci. 1999, 213, 298. (10) Tang, Y. P.; Chih, M. H.; Lee, E.; Hsu, J. P. J. Colloid Interface Sci. 2001, 242, 121. (11) Chih, M. H.; Lee, E.; Hsu, J. P. J. Colloid Interface Sci. 2002, 248, 383.

10.1021/la0506022 CCC: $30.25 © 2005 American Chemical Society Published on Web 06/30/2005

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applied in the z direction. Because the disk is infinitely large, the present problem can be treated as the electrophoresis of a sphere normal to an infinite plane.4 In this case, adopting the complicated bispherical coordinates as in previous studies is unnecessary, and the cylindrical coordinates (r,θ, z) can be used to describe the present problem. The origin of the cylindrical coordinates is located at the center of the disk and the infinitely large disk is defined by z ) 0. 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 Figure 1. Schematic representation of the problem considered where a spherical particle of radius a moves normal to an infinitely large disk as a response to a uniform applied electric field E. The cylindrical coordinates (r,θ,z) with origin located at the center of the disk are adopted, and h is the distance between particle and disk.

charge density in electrophoresis analysis. As pointed out by Ninham and Parsegian,12 these are idealized, extreme conditions, and the real situation is somewhere between the two. One of the most important developments in the charge-potential behavior of colloidal systems concerns the dissociation of ionizable surface groups of various types including strong and weak acids, and strong and weak bases. Apparently, a detailed understanding of such a system is essential to various applications such as protein adsorption, mineral preparation, and surface coating, to name a few. Several attempts have been made in previous studies to take this effect into account.10,12-20 In this study, the boundary effect on electrophoresis is investigated by considering a spherical particle moving normal to an infinitely large disk under the conditions of low surface potential and weak applied electric field. The influence of the charged conditions of a particle is examined by considering a charge-regulation model, which simulates bio-colloids such as cells and particles covered by a membrane layer. The results obtained in this study have a variety of practical applications such as electrodeposition of biocolloids. Here, we focus on the classic electrophoresis problem; that is, the movement of a particle is driven solely by an applied electric field. Other possible driving forces, such as the concentration gradient near electrode surface and related gradient of electric field strength arising, for example, from surface reactions or deposition of particles,21 are not within the scope our analysis. 2. Theory Referring to Figure 1, we consider the electrophoresis of a rigid, nonconducting, sphere of radius a normal to an infinite, perfectly conducting disk. Let h be the distance between the centers of the sphere and the disk. Far from the particle, a uniform electric field E of strength E0 is (12) Ninham, B. W.; Parsegian, V. A. J. Theor. Biol. 1971, 31, 405. (13) Chan, D.; Perram, J. W.; White, L. R.; Healy, Y. W. J. Chem. Soc., Faraday Trans. 1 1975, 71, 1046. (14) Chan, D.; Healy, Y. W.; White, L. R. J. Chem. Soc., Faraday Trans. 1 1976, 72, 2844. (15) Krozel, J. W.; Saville, D. A. J. Colloid Interface Sci. 1992, 150, 365. (16) Carnie, S. L.; Chan, D. Y. C. J. Colloid Interface Sci. 1993, 155, 297. (17) Carnie, S. L.; Chan, D. Y. C. J. Colloid Interface Sci. 1993, 161, 260. (18) Carnie, S. L.; Chan, D. Y. C.; Gunning, J. S. Langmuir 1994, 10, 2993. (19) Carnie, S. L.; Chan, D. Y. C.; Stankovich, J. J. Colloid Interface Sci. 1994, 165, 116. (20) Hsu, J. P.; Lee, E.; Yen, F. Y. J. Chem. Phys. 2000, 112, 6404. (21) Ulberg, Z. R.; Dukhin, A. S. Prog. Org. Coat. 1990, 1, 1.

∇2Ψ ) -

Fe 

N

)-

∑j

zjenj

(1)



where ∇2 is the Laplace operator,  is the permittivity of the liquid phase, Fe is the space charge density, N is the number of ionic species, nj and zj are respectively the number concentration and the valence of jth ionic species, and e is the elementary charge. If E is weak and the surface potential is low, the polarization of the double layer can be neglected; that is, the equilibrium electrolyte distribution is not influenced by the applied electric field. In this case, Ψ can be expressed as a linear superposition of the equilibrium potential Ψ1 arising from the presence of a charged particle or wall and the external potential Ψ2 arising from E.22 It can be shown that Ψ1 and Ψ2 satisfy

∇2Ψ1 ) κ2Ψ1

(2)

∇2Ψ2 ) 0

(3)

Here, κ ) [∑jn0j (ezj)2/kT]1/2 is the reciprocal Debye length, n0j is the bulk number concentration of j th ionic species, k is the Boltzmann constant, and T is the absolute temperature. The boundary conditions associated with eqs 2 and 3 are assumed as

n‚∇Ψ1 ) -

σp and n‚∇Ψ2 ) 0 on particle surface  (4a)

Ψ1 ) ζw and Ψ2 ) constant at z ) 0

(4b)

Ψ1 ) 0 and ∇Ψ2 ) -E0ez for zf∞

(4c)

n‚∇Ψ1 ) 0 and n‚∇Ψ2 ) 0 for rf∞, z > 0 (4d) In these expressions, σp is the surface charge density of the particle, ζw is the surface potential of the disk, n is the unit normal vector directed into the liquid phase, and ez is the unit vector in the z direction. Suppose that the surface of the particle contains monoprotic acidic functional groups, and the dissociation of them can be expressed by

ΑΗ T A- + H+

(5)

with dissociation constant

Ka )

[A-]s(H+)s [AH]s

(6)

[A-]s and [AH]s are respectively the numbers of A- and (22) Henry, D. C. Proc. R. Soc. London Ser. A 1931, 133, 106.

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Hsu et al.

AH per unit area on particle surface and (H+)s is the concentration of H+ on particle surface. Let Ns be the total number of the acidic functional groups per unit area. Then

Ns ) [A-]s + [AH]s

(7)

that the rotation of the particle needs not be considered. If we let Up be the magnitude of particle velocity in the z direction and assume both the surface of the particle and that of the disk are no-slip, then the boundary conditions associated with eqs 15 and 16 can be expressed as

Suppose that the spatial variation of H+ follows the Boltzmann distribution and Ψ1 ) 0 in bulk solution. Then

( )

(H+)s ) C0H+ exp -

Ns

(9)

( )

eζp 1+ exp Ka kT

eNs C0H+

( )

eζp 1+ exp Ka kT

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

FE )

Therefore, the charge density on particle surface, σp ) -e[A-]s is

σp ) -

(10)

)

|

dσp dζp

-eNs {1 + C0H+/Ka}

(ζp - 0) ζp)0

-

(e2Ns/kT){C0H+/Ka} ζp {1 + C0H+/Ka}2

(11)

eNs/ {1 + C0H+/Ka}

+

(e2Ns/kT){C0H+/Ka} Ψ1 (12) {1 + C0H+/Ka}2

If we let ∇* ) a∇, Ψ1* ) eΨ1/kT, A ) e2Nsa/kT, and B ) 0 CH +/Ka, this expression can be rewritten as

n‚∇*Ψ/1 )

AB A + Ψ/ 1 + B (1 + B)2 1

(13)

Similarly, the scaled charge density on particle surface can be expressed as

σ/p

(18)

FD )

∫S∫η

∂(u‚t) t dS + ∂n z

∫S∫ -pnz dS

(19)

where t is the unit tangential vector on particle surface, n is the magnitude of n, and tz and nz are respectively the z-component of t and that of n. At quasi-steady-state, the net force acting on the particle in the z-direction vanishes, that is

FE + FD ) 0

In this case, the boundary condition associated with Ψ1 on particle surface, eq 4a, needs be replaced by

n‚∇Ψ1 )

∫S∫σpEz dS

where S denotes the surface area of the particle, σp ) -n‚∇Ψ1 is the surface charge density, and Ez ) -∂Ψ2/∂z is the local strength of the external 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 force. FD can be evaluated by23

If ζp is low, this expression can be approximated by

σp ) σp|ζp)0 +

(17b)

(8)

6 through 8 yields

C0H+

(17a)

u ) 0 at z ) 0; for zf∞, rf∞

eζp kT

0 + where CH + is the bulk concentration of H . Combining eqs

[A-]s )

u ) Upez on particle surface

AB A A ≈)ζ/ (14) 1 + B (1 + B)2 p 1 + B exp(-ζ/p)

where σ/p ) eσpa/kT and ζ/p ) eζp/kT. If the liquid phase is an incompressible Newtonian fluid with constant physical properties and the Reynolds number is small, then the flow field can be described by

∇‚u ) 0

(15)

η∇2u - ∇p ) - FeE

(16)

In these expressions, η and u are respectively the viscosity and the velocity of the fluid and p is the fluid pressure and E ()-∇Ψ2) is the applied electric field. Note that the spherical symmetric nature of the present problem implies

(20)

Usually, the evaluation of Up involves a trial-and-error procedure in which eq 20 is used as a criterion to see if an assumed Up is appropriate. However, because the electric field and the flow field of the present problem are of a linear nature, a superposition method can be adopted to circumvent this difficulty by considering two subproblems.24 In the first problem, a particle moves with speed Up in the absence of the external electric field. In this case, the particle experiences a (conventional) hydrodynamic force FD,1 ) -UpD, where D presents the drag force per unit velocity which depends on the geometry of the particle and the type of boundary. For this problem, Brenner25 and Maude26 derived the formula

D* ) )

[

D 6πηa 4

∞

sinh(R)

m(m + 1)

× ∑ m)1(2m - 1)(2m + 3)

3 2 sinh(2m + 1)R + (2m + 1) sinh(2R)

4 sinh2(m + 1/2)R - (2m + 1)2 sinh2(R)

]

- 1 (21)

where D* (g 1) is a wall correction factor and R ) cosh-1(h/a). In the second problem, the external electric field is (23) Backstrom, G. Fluid Dynamics by Finite Element Analysis; Studentlitteratur: Sweden, 1999. (24) O′Brien, R. W.; White, L. R. J. Chem. Soc., Faraday Trans. 2 1978, 74, 1607. (25) Brenner, H. Chem. Eng. Sci. 1961, 16, 242. (26) Maude, A. D. Brit. J. Appl. Phys. 1961, 12, 293.

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applied, but the particle is held stationary. In this case, the particle experiences an electrostatic force FE and a (excess) hydrodynamic force FD,2, which is the hydrodynamic force acting on the particle due to the motion of the mobile ions in the equilibrium double layer when the external electric field E 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 when a charged boundary is present near the particle, FD,2 can be either a retarding force or a driving force. Since FD ) FD,1 + FD,2, eq 20 yields

Up )

FE + FD,2 D

(22)

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 -FeE can be removed from eq 16. Assuming an arbitrary Up in eq 17a, we solve the flow filed from eqs 15 and 16, and calculate FD,1 (or D ) -FD,1/Up) by eq 19. In the second problem, Ψ1 and Ψ2 are evaluated first by solving eqs 2 and 3 subject to the boundary conditions expressed in eqs 4a-4d, and FE is calculated by eq 18. Substituting Ψ1 and Ψ2 thus obtained into eq 16 for -FeE, the flow field is evaluated by solving eqs 15 and 16, and FD,2 is calculated by eq 19. Up is then determined by substituting D, FE, and FD,2 into eq 22. 3. Results and Discussion 27 which is based on a finite element approach,

FlexPDE, is adopted to solve the governing equations subject to the associated boundary conditions. For a more concise presentation, the scaled electrophoretic mobility µE ) Up/ U0 is used in subsequent discussions, where U0 ) (kT/ e)E0/η is the electrophoretic velocity of an isolated particle with a constant surface potential kT/e predicted by the classic Smoluchowski’s theory when an external electric field of strength E0 is applied. According to eq 22, if we let F /E ) FE/6πηaU0 and F /D,2 ) FD,2/6πηaU0, then µE can be expressed as

µE )

F /E + F /D,2 D*

(23)

The numerator represents the net driving force acting on a particle per unit external electrical field, and the denominator represents the (conventional) drag force per unit velocity of the particle in the absence of the external electric field. The behavior of µE can be explained by the variations of these forces as the key parameters vary. 3.1. Isolated, Charge-Regulated Sphere. In this section, the dependence of the surface potential and/or the surface charge density of an isolated, charge-regulated particle on the key parameters of the present problem is discussed first, followed by an analysis on the ranges of these parameters. This provides necessary information for subsequent discussions. The scaled surface potential of an isolated, chargeregulated sphere, ζ /p,iso ) eζp,iso/kT, can be determined from the equation

2κa sinh

( )

( )

/ / ζp,iso ζp,iso + 4 tanh ) 2 4

A (24) / 1 + B exp(-ζp,iso )

/ Figure 2. Variation of scaled surface potential ζ p,iso as a function of parameter A (a), B (b), and κa (c), for an isolated, charge-regulated sphere. Solid lines, approximate result based on eq 25, discrete symbols, exact result based on eq 24. Key: B ) 1, κa ) 1 in (a), A ) 1, κa ) 1 in (b), and A ) 1, B ) 1 in (c).

The left-hand side of this expression is a semiempirical formula proposed by Loeb et al.28 for symmetric electrolytes, and its right-hand side comes from eq 14; both represent the scaled surface charge density of the sphere, / / σ p,iso ) eσp,isoa/kT. If ζ p,iso is low, it can be shown that / ζ p,iso )-

(

A

(1 + B) 1 + κa +

)

AB (1 + B)2

(25)

/ based on eqs 24 Figure 2 illustrates the variations of ζ p,iso and 25 for various A, B, and κa. This figure reveals that the low potential approximation, eq 25, is close to the exact result, eq 24, for the ranges of the parameters

(27) FlexPDE version 2.22, PDE Solutions Inc., USA. (28) Loeb, A. L.; Wiersema, P. H.; Overbeek, J. Th. G. The Electric Double-Layer Around a Spherical Colloid Particle; MIT Press: Boston, 1961.

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considered except that when the value of A exceeds about 10, and the larger the value of A the greater the deviation of the approximate result. According to the definition of A ( ) e2Nsa/kT), a large A implies a high concentration of the acidic functional groups on particle surface, which / / . Since ζ p,iso ) yields a large surface charge density, σp,iso / σp,iso/(1+ κa), the higher the surface charge density the higher the surface potential, which is justified in Figure 2a. Note that the approximate result based on eq 25 does not have this nature; it levels off when A exceeds about 100. Figure 2b indicates that |ζ /p,iso| increases with the 0 decrease in B ()CH +/Ka). This is because for a fixed Ka, a small B implies that the bulk concentration of H+ is low, and it is relatively easy for the acidic functional groups on particle surface to dissociate, and therefore, the surface / is high so is the absolute value of the charge density σp,iso surface potential. Figure 2c shows that |ζ /p,iso| decreases with the increase in κa. This is because the larger the κa the thinner the equilibrium double layer, and the stronger the equilibrium electric field set up by the particle, which causes the acidic functional groups on its surface to dissociate more easily and leads to a higher σ/p,iso. How/ / is dominated by the denominator of ζ p,iso ever, since ζ p,iso / / ) σp,iso/(1+ κa), which makes |ζ p,iso| decreasing as κa increases. Note that, according to the definition of κ () [∑jn0j (ezj)2/kT]1/2), for an aqueous solution at 25 C we 0 have κ = 3.288xI (nm-1), where I ) ∑N j Cj zj/2 is the ionic 0 strength and Cj is the bulk molar concentration of jth ionic species. For the special case of a mono-protic 0 electrolyte solution, we have I ) CH +, and therefore, κ = 0 0 3.288xCH+ (nm-1). This suggests that if CH + is fixed, both 0 κa and B ()CH+/Ka) are also fixed for given a and Ka. For the case of a multiple electrolyte solution containing H+, 0 N 0 since CH + < 2I ) ∑j Cj zj is a natural condition, the values of κa must satisfy κa > 6.576a(BKa)1/2, where a is in nm-1. That is, for fixed values of a, B, and Ka, κa should be chosen so that this inequality is satisfied. Similarly, if other parameters are given, B should be chosen so that the inequality is satisfied. In subsequent discussions, we consider two special cases

(a) σ/p ) -

A AB ζ/ and ζ/w ) 0 1 + B (1 + B)2 p

(b) σ/p ) -

AB ζ/p and ζ/w ) 1 (1 + B)2

where ζ /w ) eζw/kT. The first case represents a chargeregulated particle and a neutral disk, and the second case / ) 0) and denotes an initially neutral particle (σ/p,iso ) ζ p,iso a charged disk. The linear nature of the present system implies that the results for the case when both particle and disk are charged can be obtained by a linear combination of those of the above two cases. Note that based on the boundary condition of the second case it can be deduced that the influence of a nonzero ζw on Ψ1 is linear, so is that on the space charge density Fe, and therefore, the influence of ζw on fluid velocity is also linear. We conclude that ζ /p, σ*, F /E, F /D,2, and µE are all linearly dependent on ζ /w. 3.2. Charge-Regulated Particle and Neutral Disk. In this case, σ/p ) -A/(1 + B) - AB/(1 + B)2 ζ /p and ζ /w ) 0, the excess hydrodynamic force acts always against the electrostatic force; that is, F /E is negative and F /D,2 is positive. Also, since F /E is always the dominant driving

Hsu et al.

Figure 3. Variations of µE (a) and F /E and F /D,2 (b) as a function of κa at various h/a for the case of a charge-regulated particle near a neutral disk. Key: A ) 1 and B ) 1. / force, that is, |F /E| > F /D,2, µE is negative and F D,2 is a resistant force. Figure 3 shows the variations of µE, F /E, and F /D,2 as a function of κa at various h/a when A ) 1 and B ) 1. Figure 3a indicates that |µE| declines monotonically with the increase in κa, which is consistent with the observation of Tang et al.10 when A is small ()10). It should be pointed out, however, that the reduction in |µE| at a small A as κa increases arises from a substantial increase in the excess hydrodynamic force, not due to a decrease in the electrostatic force or due to variations in surface charge density and surface potential. In contrast, if A is large ()100), the electrostatic force is strongly dependent on the double layer thickness and competes with the excess hydrodynamic force. This is why µE may have a local maximum as κa varies, as reported by Tang et al.10 However, as discussed previously, if the low potential approximation, eq 25, is adopted, A should be limited to a small enough value. Figure 3b indicates that |F /E| increases slightly as κa increases. This is because, for A j /p = σ/p,iso, ) B ) 1, the average surface charge density, σ varies only from -0.42 to -0.49 as κa varies from 0.1 to 10. Although the surface charge density of a chargedregulated particle changes relatively little, a stronger body force in a thinner double layer is still expected as κa becomes larger. This yields a stronger electroosmotic flow, and therefore, a greater F /D,2 and a smaller |µE|, as observed in Figure 3a. Note that if κa ) 0, the space charge density Fe vanishes and there are no ions available to drive the electroosmotic flow. This leads to F /D,2 ) 0 and F /E ) constant for a fixed h/a, and the equilibrium electric field is governed by a Laplace equation, ∇2Ψ1 ) 0.

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Figure 4. Variations of µE (a) and F /E and F /D,2 (b) as a function of h/a at various κa for the case of a charge-regulated particle near a neutral disk. Key: same as in Figure 3.

Figure 5. Variations of µE (a) and F /E and F /D,2 (b) as a function of A at h/a ) 2 for the case of a charge-regulated particle near a neutral disk. Key: B ) 1 and κa ) 1.

Figure 4 shows the variations of µE, F /E, and F /D,2 as a function of h/a at various κa when A ) 1 and B ) 1. Figure 4a indicates that |µE| decreases monotonically with the decrease in h/a, and µE f 0 as h/a f 1, that is, as the particle touches the disk. For the present case (a chargeregulated particle near a neutral disk), the rapid reduction in |µE| as h/a decreases arising from a substantial increase in the conventional drag force D*, especially when h/a < 2. It is interesting to note that, |F /E| decrease as h/a decreases, as shown in Figure 4b. This is because when a charged particle approaches a natural disk, in contrast to the slight increase of the charge density on particle surface, the reduction of the local strength of the applied electric field on particle surface is more appreciable so is |F /E|. Although the presence of a neutral disk only slightly affects the electrostatic force, it substantially retards the electroosmotic flow surrounding the particle by the viscous force through a nonslip surface boundary condition. This is why F /D,2 is more sensitive to the variation in h/a than F /E is, and its magnitude decreases with the decrease in h/a, as shown in Figure 4b. Figure 5 shows the variations of µE, F /E, and F /D,2 as a function of A when B ) 1, κa ) 1, and h/a ) 2. Figure 5a indicates that |µE| increases monotonically with the increase in A. This is because the dominant driving force, F /E, is strongly dependent on A through the electrostatic boundary condition on particle surface, and its magnitude increases rapidly with the increase in A. Similar to the result for the case of an isolated particle illustrated in Figure 2a, a large A yields a large surface charge density on the particle, and therefore, F /E is large. On the other

hand, due to electroneutrality, a large amount of (positive) space charge density is needed to balance the surface charge, which yields a strong electroosmotic flow. In this case, the movement of particle is retarded, and therefore, F /D,2 is large. Note that, if A ) 0, σ/p ) σ/p,iso and Fe ) 0, which leads to F /E ) F /D,2 ) 0 and µE ) 0. Figure 6 shows the variations of µE, F /E, and F /D,2 as a function of B at various h/a when A ) 1 and κa ) 1. Figure 6a indicates that |µE| initially increases with the decrease in B, and then it approaches to a constant if B < 0.01. The former is because the dominant driving force, F /E, is strongly dependent on B through the electrostatic boundary condition on particle surface, and its magnitude increases rapidly as B decreases, as in the case when A increases. The latter is due to the fact that the driving forces, F /E and F /D,2, are fixed by constant σ/p or ζ /p if B is relatively small. Similar to the results for an isolated particle illustrated in Figure 2b, a small B represents a high surface charge density on particle, which yields a large F /E and F /D,2, as in the case when A is large. If B < 0.01, the dissociation of the acidic functional groups on particle surface is complete, and it becomes constant / = -A ) -1), and therefore, charged density (σ/p ) σp,iso / / both F E and F D,2 approach constants. In contrast, if B > 100, the dissociation of the acidic functional groups on / = particle surface is negligible, which leads to σ/p ) σp,iso / / 0 and F E ) F D,2 = 0. 3.3. Neutral Particle and Charged Disk. Let us discuss next the case when σ/p) AB/(1 + B)2ζ /pand ζ /w ) 1. In this case, an initially neutral particle becomes charged as it approaches a charged disk. Also, an osmotic

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Figure 6. Variations of µE (a) and F /E and F /D,2 (b) as a function of B at various h/a for the case of a charge-regulated particle near a neutral disk. Key: A ) 1 and κa ) 1.

Figure 7. Variations of µE (a) and F /E and F /D,2 (b) as a function of κa at various h/a for the case of a natural particle near a charged disk. Key: same as in Figure 3.

pressure field is established. These affect appreciably the behavior of the particle. Especially, since the excess hydrodynamic force is enhanced directly by the extra pressure field, it is capable of competing with the electrostatic force, and therefore, the behavior of the particle is more complicated than that for the case of a charge-regulated particle near a neutral disk. The influence of the extra pressure field is discussed as follows. For an isolated, charged disk, the spatial variation of mobile ions is described by Fe,iso(z) ) -κ2ζw exp(-κz), and an equilibrium osmotic or hydrostatic pressure field, piso(z) ) κζwE0 exp(-κz), is set up to balance the body force. If both ζw and E0 are positive, the local pressure drop of the pressure field is positive, -dpiso/dz ) -Fe,isoE0 > 0, and both the particle and the fluid feel a pressure in the z direction when the particle disturbs the equilibrium pressure field. If we neglect the influence on the equilibrium system due to the presence of the particle, the z component of the hydrostatic pressure force acting on the sphere can be shown as Fp ) 4πaζwE0 exp(-κh)[cosh(κa) - sinh(κa)/κa], which is the last term of eq 12 of Shugai and Carnie.9 Note that this hydrostatic pressure force Fp acting on an imaginary sphere is unequal to the real excess hydrodynamic force FD,2 calculated from the full (electroosmotic) flow field and (hydrodynamic) pressure field. However, it suffices to use the former to describe the qualitative behavior of FD,2 because the constructions of both Fp and FD,2 are directly related to the presence of a charged disk; that is, they are directly related to the equilibrium pressure piso (or the pressure drop -dpiso/dz). Also, although the body force acting on the fluid is in the -z-direction, its movement is influenced seriously by the

positive pressure drop, which is in the z direction, and therefore, F /D,2 are always positive for the present case. Figure 7 shows the variations of µE, F /E, and F /D,2 as a function of κa at various h/a when A ) 1 and B ) 1 for the case of a neutral particle near a charged disk. Figure 7a indicates that µE may change its sign from negative to positive as κa increases, and it may have a local maximum. These phenomena arise from a sudden raise and fall in F /D,2 for κ(h - a) in the range [1,2], and are a consequence of the competition between the driving forces F /E and F /D,2, as is justified by Figure 7b. This figure reveals that, for a smaller κa (thicker double layer), a more serious distortion of the double layer near the disk arising from the approach of the neutral sphere, leading to a greater electrostatic force acting on it. Note that, this effect is significant only for κ(h - a) < 3, where double layer distortion or overlap is insignificant. However, in contrast to the proportional relation between F /E and κa, F /D,2 shows a local maximum as κa varies. For a fixed h/a, although a large κa yields a stronger pressure field near the disk, it also causes a rapid exponential decay in pressure. This leads to a local maximum of FD,2 (or Fp) as κa varies. Figure 8 shows the variations of µE, F /E, and F /D,2 as a function of h/a at various κa when A ) 1 and B ) 1. Figure 8a indicates that |µE| may have a local maximum as h/a varies. The occurrence of this local maximum is a consequence of the competition between the conventional / ) as h/a drag force D* and the net driving force (F /E + F D,2 varies, and its sign depends on which of two driving forces,

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Figure 8. Variations of µE (a) and F /E and F /D,2 (b) as a function of h/a at various κa for the case of a natural particle near a charged disk. Key: same as in Figure 3.

Figure 9. Variations of µE (a) and F /E and F /D,2 (b) as a function of A at h/a ) 2 for the case of a natural particle near a charged disk. Key: same as in Figure 5.

F /E or F /D,2, dominates. For example, if κa ) 0.5, the electrostatic force always dominates, and an increase in its magnitude leads to a more negative µE as h/a decreases from infinite. Further decrease in h/a ( 3 or h/a>(1 + 3/κa), where double layer distortion or overlap is insignificant. Unlike the case of a charge-regulated sphere near a neutral disk, F /E varies appreciably with h/a because of a rapid increase in the negative induced charge on an initially neutral particle when it approaches a charged disk. On the other hand, F /D,2 increases with the decrease in h/a because both the particle and the fluid are subjected to a larger positive pressure drop near the disk. Figure 9 shows the variations of µE, F /E, and F /D,2 as a function of A when B ) 1, κa ) 1, and h/a ) 2. Figure 9a indicates that |µE| may vanish as A varies, which implies that the effect of a nonzero ζ /w on particle mobility may be neglected. This is a consequence of the competition between the two driving forces in (F /E + F /D,2). If A > 3.7, negative F /E dominates, and an increase in its magnitude leads to a more negative µE as A increases. If A < 3.7,

although positive F /D,2 dominates, the fast increase of electrostatic force causes a smaller positive µE as A increases. All of the above results can be justified by Figure 9b. This figure indicates that both magnitudes of F /E and F /D,2 increase with the increase in A, as the illustration in Figure 5b. However, due to the driving of the extra pressure field, F /D,2 is sufficiently large to compete with F /E, and F /D,2 * 0 when A ) 0. Note that if A f ∞, the linearized charge-regulated model reduces to the case of constant surface potential, as shown in Figure 2a. This implies that the induced surface potential of a chargeregulated particle due to the presence of a charged disk must satisfy ζ /p ) -σ/p(1 + B)2/ABf0 as A f ∞ and B is finite. In our case, if A ) 1000, B ) 1, h/a ) 2, κa ) 1, and ζ /w ) 1, the average induced potential on particle surface is about 0.002 and the maximum electrostatic force is obtained as F /E = -0.21, which is identical to the induced electrostatic force for the case of constant surface potential. That is, if ζ /p ) 0, ζ /w ) 1, h/a ) 2, and κa ) 1, F /E = -0.21. The above statements indicate that a finite A (or Ns) acts to reduce the amount of induced charge on an initially neutral particle when it approaches a charged disk. Figure 10 shows the variations of µE, F /E, and F /D,2 as a function of B at various h/a when A ) 1 and κa ) 1. Figure 10a indicates that µE is always positive and may have a local minimum as B varies. Recall the boundary condition in the present case, σ/p ) -AB/(1 + κa)ζ /p, that the (induced) surface charge density σ/p is expected to be proportional to A. Because the value of A chosen ()1) in the numerical simulation is small, so is F /E. Therefore,

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Figure 10. Variations of µE (a) and F /E and F /D,2 (b) as a function of B at various h/a for the case of a natural particle near a charged disk. Key: same as in Figure 6.

positive F /D,2 is always the dominant driving force and leads to positive µE. However, because the extra pressure field is only sensitive to the variations of ζw, κa, and h/a, F /D,2 is insensitive to the variation of B, as the case when A is varied. It is interesting to note that, both the magnitude of F /E and that of F /D,2 have local maxima at B ) 1. As stated previously, this is because that both F /E and F /D,2 are sensitive to the electrostatic boundary condition, σ/p ) -AB/(1 + B)2ζ /p, which is expected to have a maximal value at B ) 1. However, due to a stronger or more direct influence of B on F /E, F /E is more sensitive to the variation of B than F /D,2 does, which leads to a local minimum in µE at B ) 1. On the other hand, if B < 0.01 or B > 100, since σ/p) σ/p,iso= 0, F /E = 0 and F /D,2 constant, both F /E and F /D,2 become insensitive to the variation of B; that is, a finite F /D,2 or µE arises only from the contribution of the extra pressure field, as in the case when A ) 0. Finally, the limiting case of A ) 0 or B f ∞, where the charge-regulated model reduces to a constant charge model with σ/p ) σ/p,iso ) 0 deserves further discussion. As can be seen in Figure 11a, the simulated results for µE agree well with the numerical results of Shugai and Carnie9 for the case of constant charge at κa ) 1 where the boundary conditions σ/p ) 0 and σ/w ) 1 are used. The small difference between the two results for κ(h - a) < 3 comes from the occurrence of σ/w = ζ /w)1 in our case when the distortion of double layer is significant. Since electrostatic force is absent in the present limiting case, the existence of a local maximum in Figure 11a is a consequence of the

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Figure 11. Variation of µE (a) and F /D,2 (b) as a function of h/a when κa ) 1. Discrete symbols, present results with A ) 0 (or B ) ∞) and ζ /w ) 1 (σ/p ) 0 and σ/w = 1), solid curves, numerical result of Shugai and Carnie9 for the case of constant charge (σ/p ) 0 and σ/w ) 1).

competition between the conventional drag force D* and the driving force F /D,2 as h/a varies. If h/a < 1.7, the former dominates, which is justified in Figure 11b. This figure also shows that log(F /D,2) is linearly dependent on h/a, which implies that if κa is fixed F /D,2 decays exponentially as κh (or h) increases. This behavior is the same as that of the equilibrium osmotic pressure or the pressure drop. Again, the direct relationship between FD,2 and Fp is justified. 4. Conclusions In summary, the boundary effect on electrophoresis is investigated by considering the electrophoresis of a spherical particle normal to an infinitely large disk. On the basis of the results of numerical simulations, we conclude that if a charge-regulated particle approaches a neutral disk the latter retards the movement of the former through the conventional hydrodynamic resistance arising from the nonslip condition on the latter. In this case, the electrostatic force acting on a particle is the dominant driving force for its movement. If the concentration of acidic functional groups is low, the increase in the electrostatic force due to a decrease in double layer thickness is relatively limited, but the excess hydrodynamic force, which retards particle movement, is enhanced. Increasing the concentration of the acidic functional groups or decreasing the bulk concentration of the hydrogen ion

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can effectively increase the surface charge density and accelerate the movement of the particle. The presence of a charged disk, although does not cause an electroosmotic flow parallel to the applied electric field, influences appreciably the movement of a particle through inducing charge on its surface and establishing an osmotic pressure field. The competition of these two factors can lead to a local maximum in the variation of the mobility of a particle as the double layer thickness or the separation distance between particle and disk varies, or even change the direction of its movement. Even if the particle is free of

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charge, its mobility driven by the charged disk can be appreciable. For a fixed separation distance, the influence of a charged disk on mobility may reduce to a minimum if the bulk concentration of hydrogen ion is equal to the dissociation constant of the monoprotic acidic functional groups on particle surface. Acknowledgment. This work is supported by the National Science Council of the Republic of China. LA0506022