Electrophoresis of a charge-regulated sphere at an arbitrary position in a charged spherical cavity

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Journal of Colloid and Interface Science

www.elsevier.com/locate/jcis

Electrophoresis of a charge-regulated sphere at an arbitrary position in

a charged spherical cavity

J.P. Hsu

a,∗

_{, C.Y. Chen}

a

_{, Duu-Jong Lee}

a

_{, Shiojenn Tseng}

b

_{, Ay Su}

c

a_{Department of Chemical Engineering, National Taiwan University, Taipei 10617, Taiwan} b_{Department of Mathematics, Tamkang University, Tamsui, Taipei 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 16 April 2008 Accepted 20 May 2008 Available online 1 July 2008 Keywords:

Electrophoresis Boundary effect Charge-regulated sphere Charged spherical cavity Electroosmotic ﬂow

The electrophoresis of a charge-regulated spherical particle at an arbitrary position in a charged spherical cavity is modeled under conditions of low surface potential (<25 mV) and weak applied electric field (<25 kV/m). The charged cavity allows us to simulate the effect of electroosmotic flow, and the charge-regulated nature of the particle permits us to model various types of surface. The problem studied previously is reanalyzed based on a more rigorous electric force formula. In particular, the influences of various types of charged conditions on the electrophoretic behavior of a particle and the roles of all the relevant forces acting on the particle are examined in detail. Several new results are found. For instance, the mobility of a particle has a local minimum as the thickness of a double layer varies, which is not seen in the cases where the surface of a particle is maintained at a constant potential and at a constant charge density.

1. Introduction

Boundary effect is one of the key factors that are of practical signiﬁcance in electrophoresis operation. This effect can inﬂuence the electrophoretic behavior of a particle both qualitatively and quantitatively[1–3]. Among various types of boundary assumed in theoretical analyses, the spherical cavity proposed by Zydney[4]is simple in structure but is capable of modeling the boundary effect appropriately under certain conditions [5–7]. The geometry of a sphere in a spherical cavity has been adopted by many researchers for studies of boundary effects on the electrophoretic behavior of a particle[4–10].

In a study of the electrophoresis of a charge-regulated particle in a spherical cavity, Yu et al.[6]found that if both the surface of a particle and that of a cavity are kept at constant potential, then the mobility of the particle may have a local minimum as its posi-tion in the cavity varies. A similar result was also observed in Hsu et al.[9]. This behavior was explained by charge reversal as a par-ticle is suﬃciently close to a cavity. In these studies, the evaluation of the electric force acting on a particle was based on the total po-tential, which includes the equilibrium potential and the potential that arises from the applied electric ﬁeld. Although this approach has been adopted by many investigators, we showed recently that

*

Corresponding author. Fax: +886 2 23623040. E-mail address:jphsu@ntu.edu.tw(J.P. Hsu).

for a non-totally-symmetric geometry, it is more realistic to ex-clude the driving force contributed by the equilibrium potential [11,12]. In an attempt to provide more a rigorous explanation for the electrophoretic behavior of a particle near a boundary, the problem considered by Yu et al. [6] is reanalyzed in this study. In particular, the inﬂuence of various types of charged conditions on the electrophoretic behavior of a particle and the roles of all the relevant forces acting on a particle are examined in detail. 2. Materials and methods

Let us consider the problem illustrated inFig. 1, where a non-conductive spherical particle of radius a is at an arbitrary position in a nonconductive spherical cavity of radius b. The surface of the particle bears an acidic function group AH, which is capable of un-dergoing the dissociation reaction expressed by

AH

⇔

A−

+

H+. (1)

The equilibrium constant of this reaction, Ka, is expressed by

Ka

=

[

A−

]

s

(

H+

)

s

[

AH

]

s

,

(2)

where

[

A−

]

s and

[

AH

]

s are the number densities of A− and AH on the particle surface, respectively, and

(

H+

)

sis the surface con-centration of H+. The cavity is ﬁlled with an incompressible New-tonian ﬂuid of constant physical properties containing electrolytes. 0021-9797/$ – see front matter ©2008 Elsevier Inc. All rights reserved.

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J.P. Hsu et al. / Journal of Colloid and Interface Science 325 (2008) 516–525 517

Fig. 1. The electrophoresis of a charged, nonconductive spherical particle of radius

aat an arbitrary position in a spherical cavity of radius b, where (r,θ, z) are the cylindrical coordinates with the origin at the center of the cavity, E is an applied uniform electric ﬁeld in the z-direction, and the center of the sphere is at z=m.

The cylindrical coordinates

(

r

, θ,

z

)

are adopted with the origin lo-cated at the center of the cavity. The center of the particle is at

z

=

m. A uniform electric ﬁeld E pointed in the z-direction with

strength E is applied. Because the system is symmetric about

θ

, only the

(

r

,

z

)

domain needs to be considered. Assuming steady state, the governing equations of the present problem can be sum-marized as follows[13]:

∇

2

Ψ

= −

ρ

e

ε

= −

j zjenj

ε

,

(3)

∇ ·

u

=

0

,

(4)

η

∇

2u

− ∇

p

= −

ρ

eE

.

(5) Here,

Ψ

,

∇

2_,

_ε

_,

_ρ

e, nj, zj, and e are the electrical potential, the

Laplacian, the dielectric constant of the liquid phase, the space charge density, the bulk number concentration and the valence of ionic species j, and the elementary charge, respectively. u,

η

, and p are the ﬂuid velocity, the viscosity, and the pressure, re-spectively.

Suppose that E is a weak electric ﬁeld relative to that estab-lished by the particle and/or the cavity, and the surface potential of the particle and/or that of the cavity is low. The former is rea-sonable if E is lower than ca. 25 kV

/

m, and the latter is satisﬁed if the surface potential is lower than ca. 25 mV[11]. Under these conditions, Eq.(3)can be replaced by[14]

∇

2

Ψ

1

=

κ

2

Ψ

1

,

(6)

∇

2

_Ψ

2

=

0

,

(7)

where

Ψ

= Ψ

1

+ Ψ

2,

Ψ

1 is the electrical potential in the absence of E or the equilibrium potential, and

Ψ

2is the electrical potential outside the particle arising from E, E

= −∇Ψ

2.

κ

=

j

n0_j

(

ezj

)

2

/

ε

kBT

1/2

is the reciprocal Debye length, n0_j, kB, and T being the bulk num-ber concentration of ionic species j, the Boltzmann constant, and the absolute temperature. The boundary conditions associated with Eqs.(6) and (7)are assumed to be

n

· ∇Ψ

1

= −

σ

p

ε

and n

· ∇Ψ

2

=

0 on the particle surface

,

(8)

Ψ

1

= ζ

w and n

· ∇Ψ

2

= −

Ezcos

θ

on the cavity surface

,

(9)

where

σ

p is the surface charge density of the particle,

ζ

w is the surface potential of the cavity, n is the unit normal vector pointing to the liquid phase, and Ez is the z-component of E.

Assuming Boltzmann distribution for the spatial variation of the molar concentration of H+, it can be shown that[3]

σ

p

= −

eNs 1

+

C 0 H+ Ka exp

(

−

eζp kBT

)

,

(10)

where Nsis the density of the acidic functional groups on the par-ticle surface,

ζ

p is the surface potential of the particle, and C_H0+ is

the bulk concentration of H+. If

ζ

p is low, this expression can be approximated by

σ

p

∼

=

−

eNs

{

1

+

C0 H+

/

Ka

}

−

(

e 2_N s

/

kBT

)

{

C0_H+

/

Ka

}

{

1

+

C0 H+

/

Ka

}

2

ζ

p

.

(11)

Suppose that both the surface of the particle and that of the cavity are no-slip. Then the boundary conditions associated with Eqs.(4) and (5)are

u

=

U ez on the particle surface

,

(12)

u

=

0 on the cavity surface

.

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Here, U is the z-component of the particle velocity and ez is the

unit vector in the z-direction.

For the present case, the forces acting on the particle include the electric force FE and the hydrodynamic force FD. If we let FE and FDbe the z-components of these forces, then FE

+

FD

=

0 at steady state. FEand FDcan be evaluated by[1–3,14–17]

FE

=

s

σ

pEzdS

,

(14) FD

=

s

η

∂(

u

·

t

)

∂

n tzdS

+

s

−

pnzdS

,

(15)

where S is the surface of the particle, t the unit tangent vector on S

,

n the magnitude of n, and tz and nz are the z-components

of t and n, respectively. For a simpler treatment, the problem un-der consiun-deration is decomposed into two subproblems[18]. In the first subproblem the particle translates with a constant velocity U in the absence of E, and in the second subproblem E is applied but the particle is held fixed. In the first subproblem the particle ex-periences a conventional drag force FD,1

= −

U D, where D is the drag coeﬃcient. In the second subproblem the particle experiences both an electric force FE and a hydrodynamic force FD,2, which arises from the movement of the ionic species in the double layer surrounding the particle. The force balance FE

+

FD

=

0 yields

U

=

FE

+

FD,2

D

.

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For convenience, the following scaled symbols are deﬁned:

P

(

=

100m

/(

b

−

a

)

%

)

is the scaled position of the particle;

κ

a is

the scaled thickness of the double layer;

λ (

=

a

/

b

)

is the scaled size of the cavity; A

=

e2_N

sa

/

ε

kBT is a parameter measuring the density of the acidic functional groups on the particle surface;

B

=

C0_H+

/

Ka is a parameter measuring the bulk concentration of H+ (or pH); U∗

=

U

/

Uref is the scaled electrophoretic mobility,

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Fig. 2. Typical meshes used in the resolution of the electric field (a) and of the flow field (b) for an incompressible Newtonian fluid. Other parameters used areζp∗=1,

ζw∗=0, A=1, B=1,κa=1,λ=0.4, and P=0%.

Fig. 3. Variation of the scaled mobility U∗as a function of P for various values ofλfor the case of a positively charged sphere in an uncharged spherical cavity atζp∗=1

andζw∗=0. Solid curves, present results; discrete symbols, those of Hsu et al.[5].

where Uref

=

ε

(

kBT

/

e

)

E∞

/

η

is a reference velocity;

ζ

_w∗

=

e

ζ

w

/

kBT is the scaled surface potential of the cavity; F_E∗

=

FE

/

6

π η

aUref and

F∗_D_,₂

=

FD,2

/

6

π η

aUref are scaled forces; and D∗

=

D

/

6

π η

a is the scaled drag coeﬃcient.

3. Results and discussion

The governing equations and the associated boundary condi-tions are solved numerically by FlexPDE[19], which is based on a finite element scheme. In the numerical simulation, grid indepen-dence is checked to ensure that the mesh used is appropriate. In Fig. 2, the numbers of nodes necessary for the resolution of the electric field and the flow field are ca. 130,800 and 3669, respec-tively. To illustrate the performance of the software adopted, it is

used to solve the problem considered by Hsu et al. [5], namely, the electrophoresis of a rigid sphere of constant surface potential in an uncharged spherical cavity. Fig. 3 presents the variation of the scaled mobility U∗ as a function of P∗ at various values of

λ

. As seen, the performance of the software adopted is satisfactory.

Note that Eqs.(11) and (8)lead to 1

+

B A d

Ψ

₁∗ dn

−

B 1

+

B

Ψ

∗

1

∼

=

1 on the particle surface

,

(17) where

Ψ

₁∗

=

e

Ψ

1

/

kBT and n is the magnitude of n. This expression implies that the two widely assumed conditions for the description of the surface of a charged particle, namely, constant potential and constant charge density, can be recovered from the present charge-regulation model by letting A

→ ∞

and B

→

0, respectively.

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Fig. 4. Variation of the scaled mobility U∗(a), the electric force F_E∗(b), the scaled excess hydrodynamic force F∗_D_,₂(c), and the scaled hydrodynamic force coeﬃcient D∗(d) as functions of P for various values of B (or pH) at A=1,ζw∗=0,κa=1, andλ=0.4.

Fig. 5. Variation of the scaled mobility U∗as a function of P for various values of A at B=1,ζw∗=0,κa=1, andλ=0.4.

In subsequent discussions we consider three representative cases: a charge-regulated sphere in an uncharged cavity, a posi-tively charged cavity, and a negaposi-tively charged cavity. For illustra-tion, the radius of the particle a is ﬁxed.

3.1. Charge-regulated sphere in an uncharged cavity

Let us consider ﬁrst the case where a charge-regulated sphere is placed in an uncharged cavity. Fig. 4 illustrates the varia-tions of the scaled mobility, U∗, and the corresponding scaled forces, F∗_E

,

F∗_D_,₂, and D∗, as functions of the position parameter

P

(

=

100m

/(

b

−

a

)

%

)

at various levels of parameter B

(

=

C0_H+

/

Ka

)

,

or pH. U∗is seen to decline with increasing B because the greater the B the higher the bulk concentration of H+, leading to a lower surface concentration of A− and a smaller electrical driving force

|

F_E∗

|

, as illustrated inFig. 4b[20]. Note that except for P close to 100%, F_E∗ is the dominating force in this case. If P exceeds ca. 90%, D∗

(|

F∗_E

+

F∗_D_,₂

|)

, implying that the retardation arising from the presence of the cavity dominates. In this case,

|

U∗

|

declines rapidly with increasing P , and vanishes at P

=

100%, as required by the nonslip boundary condition assumed for the ﬂow ﬁeld.

Fig. 5 indicates that the absolute value of the scaled mobility,

|

U∗

|

, increases with increased A

(

=

e2_N

sa

/

ε

kT

)

. This is because the greater the A the larger the number of acidic functional groups

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Fig. 6. Variation of scaled mobility U∗(a), the electric force F∗E(b), scaled excess hydrodynamic force FD∗,2(c), and scaled excess hydrodynamic force FD∗,2(v)and FD∗,2(P)(d) as functions ofκa for various P at A=1, B=1,ζw∗=0, andλ=0.4.

Fig. 7. Variations of the average scaled surface potential of the particle ¯ζp∗and the average surface charge densityσp∗as functions ofκa for various values of P at A=1,

B=1,ζw∗=0, andλ=0.4.

on the particle surface, leading to a higher surface charge den-sity[3], and a greater scaled electrical driving force

|

F∗_E

|

. Again, F∗_E dominates except when P is close to 100%. As inFig. 4, the rapid decrease in

|

U∗

|

as P approaches 100% arises from the rapid in-crease in D∗.

The inﬂuence of the thickness of the double layer, measured by

κ

a, on the scaled mobility, U∗, and the corresponding scaled forces, F_E∗ and F_D∗_,₂, is illustrated in Fig. 6. Note that D∗ is inde-pendent of

κ

a.Fig. 6a reveals that if P is not large (0% and 60%),

|

U∗

|

decreases monotonically with increasing

κ

a, which is different

from the behavior of a rigid sphere remaining at constant surface potential.Fig. 7shows the variations of the average surface poten-tial

¯ζ

_p∗and the average surface charge density

σ

p∗ as a function of

κ

a at various levels of P . Note that while

|

σ

p∗

|

increases with in-creasing

κ

a, which is consistent with Fig. 6b,

|¯ζ

_p∗

|

declines at the same time.

It is interesting to note that if P is suﬃciently large (95%) U∗ has a negative local minimum at

κ

a

∼

₌

5. These results can be ex-plained by the trends of F_E∗ and F_D∗_,₂. According to Fig. 6b,

|

F_E∗

|

increases with increasing

κ

a, because the thinner the double layer

the higher the surface charge density, as is pointed out for the electrophoresis of a rigid particle[5]. However, F_D∗_,₂, which is pos-itive except when P is large, also increases with

κ

a, as seen in

Fig. 6c[3]. Note that F_D∗_,₂ comprises a viscous term, F∗_D_,₂₍_v₎, and a pressure term, F_D∗_,₂₍_P₎. The speciﬁc behavior of F_D∗_,₂ at P

=

95%, where it has a negative local minimum as

κ

a varies, arises from

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Fig. 8. Variations of the scaled mobility U∗as a function ofκa for various types of charged conditions on the particle surface atζw∗=0, P=0%, andλ=0.4. Solid curves,

charge-regulated surface at B=1; (- - -) constant surface potential atζp∗=1[5]; (-·-·-·) constant surface charge atσp=2.75×10−15C/m2.

Fig. 9. Variation of the scaled mobility U∗as a function ofλfor various values of P at A=1, B=1,ζw∗=0, andκa=1.

the competition of these two terms, shown inFig. 6d. If

κ

a is small

and P is large (i.e., thick double layer and boundary effect impor-tant), then F∗_D_,₂₍_P₎dominates. On the other hand, if

κ

a is large and P is small, F∗_D_,₂₍_v₎ dominates[5,15,16].

A comparison between the electrophoretic behavior under var-ious types of charged conditions is illustrated inFig. 8. This ﬁgure reveals that if the surface of a particle is kept at constant po-tential, U∗ increases monotonically with decreasing thickness of the double layer (increasing

κ

a) [5]. On the other hand, if the surface of a particle is kept at constant charge density,

|

U∗

|

de-creases monotonically with decreasing thickness of the double layer. It is interesting to note that if the surface of a particle is of a charge-regulated nature,

|

U∗

|

has a local maximum as

κ

a

varies.

Fig. 9 shows the inﬂuence of the scaled size of the cavity

λ

(

=

a

/

b

)

on the scaled mobility of the particle, U∗. This ﬁgure in-dicates that U∗

→

0 as P

→

100% and/or

λ

→

1. This is expected

because the retardation due to the presence of the cavity, which is of a nonslip nature, becomes signiﬁcant as P and/or

λ

increase.

3.2. Charge-regulated sphere in a positively charged cavity

Consider next the case where a charge-regulated sphere is in a positively charged cavity. In this case, the presence of the charged cavity yields a clockwise (counterclockwise) electroosmotic flow on the right-(left-)hand side of the particle. Because if the cavity is uncharged, the flow of the fluid near the particle is counter-clockwise (counter-clockwise) on its right-(left-)hand side, the presence of the electroosmotic flow has the effect of reducing the particle velocity. A comparison between Figs. 4c and 10breveals that the

|

F∗_D_,₂

|

for the present case is greater than that for the case where the cavity is uncharged, leading to a greater hydrodynamic force acting on the particle [4]. The competition between the hydrody-namic force and the electric force acting on the particle makes its

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Fig. 10. Variation of the scaled mobility U∗(a), and the scaled electric force F∗D,2(b), as functions of P for various values of B (or pH) at A=1,ζw∗=1,κa=1, andλ=0.4.

electrophoretic behavior more complicated than that for the case where the cavity is uncharged. The variations of U∗ and F_D∗_,₂ as a function of P at various levels of B are presented inFig. 10. Note that in the present case F∗_E and D∗ are the same as those shown in Figs. 4b and 4d, respectively. Fig. 10a reveals that if B is not large, U∗is negative and

|

U∗

|

declines monotonically with increas-ing P . On the other hand, if B is suﬃciently large, U∗ is positive and U∗ decreases monotonically with increasing P . The difference between the behavior of U∗ shown in Fig. 10a and that inFig. 4a arises mainly from the difference of the behavior of F_D∗_,₂ in these two case. InFig. 10b, F_D∗_,₂ is always positive, and it increases with decreasing B and/or increasing P .

The behavior of U∗ as A and/or P vary is similar to that ob-served in Fig. 5, that is,

|

U∗

|

increases with increasing A and declines with increasing P , and can be explained by the same reasoning. Note that although negative charge is induced on the particle surface as it approaches the positively charged cavity, lead-ing to a higher surface charge density and electrical drivlead-ing force, the electroosmotic ﬂow arising from the presence of the charged

cavity has the effect of reducing the mobility of the particle. There-fore, the behavior of A is roughly the same as that shown inFig. 5. Fig. 11reveals that if

κ

a is small, U∗ is negative and

|

U∗

|

de-clines with increasing

κ

a. On the other hand, if

κ

a exceeds ca.

unity, U∗ becomes positive and U∗ increases with increasing

κ

a.

This behavior can be explained by the behavior of F∗_E and F∗_D_,₂ as

κ

a varies, presented in Figs. 11b and 11c. As seen inFig. 9b, the smaller the

κ

a the larger the

|

F_E∗

|

because the smaller the

κ

a the

more important the electrical interaction between the particle and the cavity, and therefore, the greater the amount of negative charge induced on the particle surface. If

κ

a exceeds ca. 4.5, the inﬂuence

of the cavity on the particle is unimportant and the situation be-comes similar to that for the case when the cavity is uncharged (Fig. 6b), where

|

F_E∗

|

κ

a. The trend of F_D∗_,₂ seen inFig. 11c arises from the competition between F∗_D_,₂₍_P₎ and F_D∗_,₂₍_v₎. If the boundary effect is important (small

κ

a and/or

large P ), F∗_D_,₂ is dominated by F∗_D_,₂₍_P₎; on the other hand, it is dominated by F_D∗_,₂₍_v₎.

The variations of the scaled mobility, U∗, and the correspond-ing scaled forces, F∗_E, F∗_D_,₂, and D∗, as a function of the scaled size

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Fig. 11. Variation of the scaled mobility U∗(a), the scaled electric force F_E∗(b), and the scaled excess hydrodynamic force F_D∗_,₂(c) as functions ofκa for various values of P at A=1, B=1,ζw∗=1, andλ=0.4.

Fig. 12. Variation of the scaled mobility U∗(a), the scaled electric force F∗E(b), the scaled excess hydrodynamic force F∗D,2(c), and the scaled hydrodynamic force coeﬃcient

D∗(d) as functions ofλfor various values of P at A=1, B=1,ζw∗=1, andκa=1.

of the cavity

λ (

=

a

/

b

)

are illustrated in Fig. 12. The behavior of

U∗ shown in Fig. 12a is more complicated than that in Fig. 9. In the former, the sign of U∗ depends on the levels of P and

λ

. In ad-dition, U∗ may have a negative local minimum as

λ

varies. These can be explained by the behaviors of F_E∗, F_D∗_,₂, and D∗ shown in

Figs. 12b and 12c. As seen, F∗_E

<

0, F∗_D_,₂

>

0, and D∗

>

0; if

λ

is small, F∗_E and F_D∗_,₂dominate, and if

λ

is large, D∗dominates. Here,

|

F∗_E

|

λ

and/or increasing P . This is be-cause in these cases the compression of the electric ﬁeld between the particle and the cavity leads to an increase in the charge

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den-Fig. 13. Variation of the scaled mobility U∗(a), the scaled electric force F_E∗(b), and the scaled excess hydrodynamic force F_D∗_,₂(c) as functions of P for various values of B (or pH) at A=1,ζw∗= −1,κa=1, andλ=0.4.

sity on the particle surface, and therefore an increase in

|

F∗_E

|

[5,15]. As seen inFig. 12c, the qualitative behavior of F_D∗_,₂ at P

=

0% and 60% is dissimilar to that at P

=

95%. This arises from the competi-tion between F_D∗_,₂₍_v₎and F_D∗_,₂₍_P₎[5,15,16]. If the boundary effect is important (

λ

and/or P are large), F∗_D_,₂ is dominated by F_D∗_,₂₍_P₎; on the other hand, if it is unimportant, F_D∗_,₂ is dominated by F_D∗_,₂₍_v₎ [5,15,16]. As mentioned previously, the rapid increase in D∗ at large

λ

and P seen inFig. 12d arises from the nonslip condition on the cavity surface.

3.3. Charge-regulated sphere in a negatively charged cavity

Let us consider next the case where a charge-regulated sphere is in a negatively charged cavity.Fig. 13a indicates that the qual-itative behavior U∗ as P and B vary is similar to that observed inFig. 4a, and can be explained by the same reasoning. Note that the variation of D∗ as P and B vary is the same as that shown in Fig. 4d. In the present case, because the negatively charged cavity yields a counterclockwise (clockwise) electroosmotic ﬂow on the right- (left-)hand side of the particle, the velocity of the particle is raised. A comparison betweenFigs. 4c and 13cindicates that

|

F_D∗_,₂

|

for the present case is greater than for the case where the cavity is uncharged; that is, the hydrodynamic force experienced by the

Fig. 14. Variation of the scaled mobility U∗(a), the scaled electric force F∗E(b), and

the scaled excess hydrodynamic force (c) as functions ofκa for various values of P at A=1, B=1,ζw∗= −1, andλ=0.4.

particle is greater than that for the case where the cavity in un-charged. As seen inFigs. 13b and 13c, although the magnitudes of

F_D∗_,₂ and F_E∗ are comparable, they have the same sign, and there-fore, the behavior of U∗ is not as complicated as that in the case where the cavity is positively charged.

The general behavior of U∗ as A and/or P varies is the same as that observed inFig. 5. For the present case, although the posi-tive charge is induced on the particle surface as it approaches the negatively charged cavity surface, thereby reducing the net surface charge density and the electrical driving force, the electroosmotic ﬂow arising from the presence of the charged cavity has the effect of raising the mobility of the particle. Therefore, the behavior of A is roughly the same as that shown inFig. 5.

The variations of the scaled mobility, U∗, and the correspond-ing scaled forces, F_E∗ and F_D∗_,₂, as functions of

κ

a are illustrated

in Fig. 14. Because the positive charge is induced on the parti-cle surface as it approaches the negatively charged cavity, and the smaller the

κ

a the more the amount of induced charge,

|

F_E∗

|

, and therefore U∗, declines accordingly. As expected, the larger the P the more signiﬁcant that effect is. Note that the general trend of

U∗ illustrated inFig. 14a is different from that of U∗ observed in Figs. 6a and 12a. This is because the order of magnitude of F_D∗_,₂

(10)

(positive) inFig. 6a is roughly the same as that of F_E∗(negative), in Fig. 12a F_D∗_,₂ dominates (except for small

κ

a), and both F_D∗_,₂ and

F_E∗ are negative inFig. 12a. The trend of F_D∗_,₂ seen inFig. 14c can be explained by the result of the competition between F∗_D_,₂₍_P₎ and

F_D∗_,₂₍_v₎. As mentioned previously, if the boundary effect is signiﬁ-cant (small

κ

a and/or large P ), F∗_D_,₂ is dominated by F∗_D_,₂₍_P₎. On the other hand, if the boundary effect is insigniﬁcant, F_D∗_,₂is dom-inated by F_D∗_,₂₍_v₎. The dependence of U∗ on P and

λ

is the same as that shown inFig. 9.

4. Summary

In summary, the influence of a charged boundary on the elec-trophoretic behavior of a charge-regulated particle is modeled by considering the electrophoresis of a spherical particle, the sur-face of which contains dissociable functional groups, in a spherical cavity. We show that, due to the presence of an electroosmotic flow, the charged boundary can influence the mobility of the par-ticle significantly, both qualitatively and quantitatively. For the case where the cavity is uncharged we conclude the following: (a) The absolute value of the mobility of the particle increases with in-creasing pH and with the density of the acidic functional groups on the particle surface. (b) If the particle is not too close to the cavity, the absolute value of the mobility decreases with decreasing thickness of a double layer, that is different from the correspond-ing behavior of a rigid particle. (c) If the particle is sufficiently close to the cavity, the mobility has a negative local minimum as the thickness of a double layer varies. If the cavity is positively charged, then the following conclusions can be drawn: (a) If pH is sufficiently high, the mobility is negative and the closer the parti-cle to the cavity the smaller its absolute value. On the other hand, if the pH is sufficiently low, the mobility is positive, and the closer the particle is to the cavity the smaller the mobility. (b) If a double layer is thick, the mobility is negative, and its absolute value de-clines with decreasing double layer thickness. On the other hand, if the double layer is sufficiently thin, the mobility is positive and

it increases with decreasing double layer thickness. (c) The sign of the mobility depends on the relative position in the cavity and its relative size with respect to that of the cavity. In addition, the mo-bility may have a negative local minimum as its relative size varies. If the cavity is negatively charged, then (a) The qualitative behavior of the mobility as its relative position in the cavity and pH vary is similar to that in the case where the cavity is uncharged. (b) The thicker a double layer the smaller the absolute value of the mo-bility. (c) The general trend of the mobility as the thickness of a double layer varies is different from that for the case where the cavity is uncharged or positively charged.

Acknowledgment

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

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