Electrophoretic Mobility and Electric Conductivity in Dilute

7226

Langmuir 2003, 19, 7226-7239

Electrophoretic Mobility and Electric Conductivity in Dilute Suspensions of Charge-Regulating Composite Spheres Jau M. Ding and Huan J. Keh* Department of Chemical Engineering, National Taiwan University, Taipei 106-17, Taiwan, Republic of China Received March 24, 2003. In Final Form: June 11, 2003 The electrophoresis and electric conduction in a dilute suspension of charged composite particles, each composed of a solid core and a surrounding porous shell, with an arbitrary thickness of the electric double layers are analytically studied. The porous shell of a particle is treated as a solvent-permeable, ionpenetrable, and charge-regulating surface polymer layer of finite thickness with uniformly distributed ionogenic functional groups and frictional segments. The electrokinetic equations that govern the electrostatic potential profile, the electrochemical potential distributions of ionic species, and the fluid flow field inside and outside the surface polymer layer of the particle migrating in an unbounded electrolyte solution are linearized assuming that the system is only slightly distorted from equilibrium. Through the use of a regular perturbation method, these linearized equations are solved for a composite sphere in a uniform applied electric field with the dimensionless fixed charge densities on the surface of the rigid core and of the surface polymer layer as the small perturbation parameters. Analytical expressions for the electrophoretic mobility of the charge-regulating composite sphere and for the effective electric conductivity of the suspension are derived as linear functions of these charge densities. The results demonstrate that the charge regulation phenomenon tends to thin down the electric double layer and to reduce the magnitudes of the electrophoretic mobility and the electric conductivity compared to the case that the fixed charge density in the surface polymer layer is a constant. The intensity of this effect depends on the regulation characteristics such as the association/dissociation equilibrium constants of the ionogenic functional groups in the surface polymer layer and the concentration of the potential-determining ions in the bulk solution.

1. Introduction In the study of phenomena occurring at charged solidliquid interfaces, electrokinetic methods have proved to be useful tools. Although the basic relationships involved in electrokinetic phenomena were derived mainly by using the traditional model of plain distribution of surface charges,1-8 quite a number of investigations have applied these methods to the study of the effects of polyelectrolyte adsorbates. For example, theoretical analyses for the electrophoresis of a colloidal particle coated with an ionpenetrable layer of adsorbed, charged polymers have been made for many years,9-13 in relation to the electrokinetic motion of biocolloids such as cells. These analyses provided formulas for the electrophoretic mobility of such a composite particle by introducing the modified Brinkman equation for the flow field inside the porous surface layer of the particle and assuming that the local radii of * To whom correspondence should be addressed. Fax: +886-22362-3040. E-mail: [email protected]. (1) Henry, D. C. Proc. R. Soc. London, Ser. A 1931, 133, 106. (2) Booth, F. Proc. R. Soc. London, Ser. A 1950, 203, 514. (3) Wiersema, P. H.; Loeb, A. L.; Overbeek, J. Th. G. J. Colloid Interface Sci. 1966, 22, 78. (4) Dukhin, S. S.; Derjaguin, B. V. In Surface and Colloid Science; Matijevic, E., Ed.; Wiley: New York, 1974; Vol. 7. (5) O’Brien, R. W.; White, L. R. J. Chem. Soc., Faraday Trans. 2 1978, 74, 1607. (6) Saville, D. A. J. Colloid Interface Sci. 1979, 71, 477. (7) O’Brien, R. W. J. Colloid Interface Sci. 1981, 81, 234. (8) Ohshima, H.; Healy, T. W.; White, L. R. J. Chem. Soc., Faraday Trans. 2 1983, 79, 1613. (9) Jones, I. S. J. Colloid Interface Sci. 1979, 68, 451. (10) Wunderlich, R. W. J. Colloid Interface Sci. 1982, 88, 385. (11) Levine, S.; Levine, M.; Sharp, K. A.; Brooks, D. E. Biophys. J. 1983, 42, 127. (12) Sharp, K. A.; Brooks, D. E. Biophys. J. 1985, 47, 563. (13) Ohshima, H.; Kondo, T. J. Colloid Interface Sci. 1987, 116, 305.

curvature of the particle are much larger than the thicknesses of the electric double layer and of the porous surface layer (i.e., the particle surface is planar and the applied electric field is parallel to it). Experimental results of the electrophoretic mobility of charged composite particles are also available for human erythrocytes,12 rat lymphocytes,14 and grafted polymer microcapsules.15 On the basis of a formula derived from the theory of a planar particle surface, these experimental results could be used to calculate the fixed charge density and the hydrodynamic resistance parameter of the porous surface layer. On the other hand, Ohshima16 neglected the relaxation effect of the electric double layer and derived a general expression for the electrophoretic mobility of a spherical composite particle which requires complicated numerical calculations. It was found that the electrophoretic mobility of a composite sphere can be quite different from that of a “bare” rigid sphere. Analytical expressions for the electrophoretic mobility and effective electric conductivity in a dilute suspension of charged composite spheres were also obtained by Liu and Keh17 under the Debye-Huckel approximation. The results demonstrated that the presence of the fixed charges in the porous surface layer of the particle can lead to an augmented or a diminished electric conductivity of the suspension relative to that of a corresponding suspension of uncharged composite spheres, depending on the characteristics of the electrolyte solution and the suspending particles. (14) Morita, K.; Muramatsu, N.; Ohshima, H.; Kondo, T. J. Colloid Interface Sci. 1991, 147, 457. (15) Aoyanagi, O.; Muramatsu, N.; Ohshima, H.; Kondo, T. J. Colloid Interface Sci. 1994, 162, 222. (16) Ohshima, H. J. Colloid Interface Sci. 1994, 163, 474. (17) Liu, Y. C.; Keh, H. J. Langmuir 1998, 14, 1560.

10.1021/la030117b CCC: $25.00 © 2003 American Chemical Society Published on Web 08/08/2003

Mobility of Charged Composite Particles

The previous analyses for the electrokinetic phenomena concerning charged composite particles have assumed that the fixed charge density in the surface polymer layer is constant. While this assumption may be convincing under certain circumstances, it only leads to ideal results for limiting cases and can be impractical for some polyelectrolytes. The actual fixed charge density in a polyelectrolyte matrix is usually determined by the dissociation of its ionizable functional groups and/or their association with specific ions. The degree of these dissociation and association reactions will be a function of the local concentration of the charge-determining (and potentialdetermining) ions in the polymer matrix. Thus, the fixed charge density in the surface polymer layer of a composite particle is no longer a uniform distribution but a function of the local electric potential and characteristics of the surface polymer layer such as the equilibrium constants of the association/dissociation reactions and the number density of the ionogenic functional groups. This is the so-called charge regulation phenomenon,18-27 which causes the electric double layer surrounding the particle to become essentially thinner, as will be shown in eq 20b. When a charge-regulating composite particle is subjected to an applied electric field, the fixed charge distribution in the surface polymer layer can be rearranged from its equilibrium state, and electrophoretic movement of the particle according to this rearranged fixed charge distribution occurs. In this work, the electrophoretic mobility and effective electric conductivity of a dilute suspension of charged composite particles with an arbitrary thickness of the electric double layers are analytically examined. The porous shell of each particle is treated as a solventpermeable, ion-penetrable, and charge-regulating surface polyelectrolyte layer with uniformly distributed ionogenic functional groups and frictional segments. The suspension is sufficiently dilute that the suspended particles occupy only a small fraction of the total volume of the suspension, and the double layers surrounding the particles do not overlap with one another. In section 2, we present the fundamental electrokinetic equations and boundary conditions, which govern the electrostatic potential profile, electrochemical potential distributions of ionic species, and the fluid flow field inside and outside the surface polymer layer of a particle migrating in an unbounded electrolyte solution when a constant electric field is applied. These basic equations are linearized assuming that the system is only slightly distorted from equilibrium. In section 3, these linearized equations for a composite sphere are solved by using the regular perturbation method with the charge density on the surface of the rigid core and the leading term of the fixed charge density in the surface polymer layer as the small perturbation parameters. The analytical expression for the electrophoretic mobility of a charge-regulating composite sphere in an unbounded electrolyte solution is obtained. An (18) Ninham, B. W.; Parsegian, V. A. J. Theor. Biol. 1971, 31, 405. (19) Chan, D.; Perram, J. W.; White, L. R.; Healy, T. W. J. Chem. Soc., Faraday Trans. 1 1975, 71, 1046. (20) Prieve, D. C.; Ruckenstein, E.J. Theor. Biol. 1976, 56, 205. (21) Van Riemsdijk, W. H.; Bolt, G. H.; Koopal, L. K.; Blaakmeer, J. J. Colloid Interface Sci. 1986, 109, 219. (22) Krozel, J. W.; Saville, D. A. J. Colloid Interface Sci. 1992, 150, 365. (23) Carnie, S. L.; Chan, D. Y. C. J. Colloid Interface Sci. 1993, 161, 260. (24) Pujar, N. S.; Zydney, A. L. J. Colloid Interface Sci. 1997, 192, 338. (25) Tsao, H. K. J. Colloid Interface Sci. 1998, 205, 538. (26) Tseng, S.; Lin, S. H.; Hsu, J. P. Colloids Surf., B 1999, 13, 277. (27) Dan, N. Langmuir 2002, 18, 3524.

Langmuir, Vol. 19, No. 18, 2003 7227

explicit formula for the effective electric conductivity of a dilute suspension of charge-regulating composite spheres is then derived in section 4. Finally, analytical expressions in some limiting cases and typical numerical results for the electrophoretic mobility and electric conductivity of a dilute suspension of charge-regulating composite spheres are presented in section 5. 2. Basic Electrokinetic Equations We consider a charge-regulating composite particle of arbitrary shape in an unbounded liquid solution containing M ionic species when a constant electric field E∞ is applied. The composite particle is composed of a charged solid core and a surrounding layer of adsorbed charge-regulating polymers which is solvent permeable and ion penetrable. The type j ionogenic functional groups in the adsorbed polyelectrolyte layer are assumed to be distributed at a uniform density Nj, and the chemical equilibrium of these groups with the ambient electrolyte solution is maintained (see Appendix A). Experimental values for human erythrocytes,12 rat lymphocytes,14 and grafted polymer microcapsules15 indicate that the thickness of the porous surface layer ranges from 7.8 nm to 3.38 µm and Nj can be as high as 0.03 M, depending on the pH and ionic strength of the electrolyte solution. 2.1. General Governing Equations. Conservation of all species in the ionic solution at the steady state requires that

∇‚Jm ) 0,

m ) 1, 2, ‚‚‚, M

(1)

where Jm is the number flux distribution of species m. If the electrolyte solution is dilute, Jm is given by

(

Jm ) nmu - Dm ∇nm +

)

zmenm ∇ψ kT

(2)

Here, u is the fluid velocity field relative to the particle, ψ is the electric potential distribution, nm and zm are the concentration (number density) distribution and the valence of species m, respectively, Dm is the diffusion coefficient of species m, which is assumed to be constant both inside and outside the porous surface layer, e is the elementary electric charge, k is the Boltzmann constant, and T is the absolute temperature. The first term on the right-hand side of eq 2 represents the convection of the ionic species caused by the fluid flow, and the second term denotes the diffusion and electrically induced migration of the ions. By assuming that the Reynolds number of the fluid motion is very small, the inertial effect on the fluid momentum balance can be neglected. The fluid flow is governed by a combination of the Stokes and Brinkman equations modified with the electrostatic effect M

η∇2u - h(x) fu ) ∇p +

∑ zmenm∇ψ

(3)

m)1

∇‚u ) 0

(4)

where η is the viscosity of the fluid (the available evidence28 suggests that it is reasonable to assume the same value of η inside and outside the porous surface layer), f is the hydrodynamic friction coefficient of the porous surface layer per unit volume of the fluid (which accounts for the hindrance to the convective transport of the electrolyte caused by the polymer segments), p is the dynamic (28) Koplik, J.; Levine, H.; Zee, A. Phys. Fluids 1983, 26, 2864.

7228

Langmuir, Vol. 19, No. 18, 2003

Ding and Keh

pressure distribution, and h(x) is a unit step function which equals unity if the position x is inside the surface polymer layer and zero if x is outside the composite particle. In eq 3, η and f are assumed to be constant. Note that f can be expressed as 6πηaSNS, where NS and aS are the number density and the Stokes radius, respectively, of the hydrodynamic frictional segments in the surface polymer layer. The local electric potential ψ and the charge density are related by Poisson’s equation

∇2ψ ) -

4π M [ zmenm + h(x)Q] m)1

∑

(5)

Here, Q is the fixed charge density distribution inside the surface polymer layer of the composite particle, which is a function of the local electric potential given by eq A4 in Appendix A, and ) 4π0r, where r is the relative permittivity of the electrolyte solution, which is assumed to be the same inside and outside the surface polymer layer, and 0 is the permittivity of a vacuum. For small values of ψ (the Debye-Huckel approximation), eq A4 for Q as a function of ψ can be linearized to become

Q ) A1 - A2ψ

(6)

which represents a straight line tangent to the exact Q versus ψ curve at ψ ) 0 at the equilibrium state (in which no external field is applied), where

A1 )

A2 )

∑j ZjeNj

∑j Zj e Nj 2 2

∞2 nj+ - Kj+Kj∞2 ∞ nj+ + nj+ Kj+ + Kj+Kj-

∞ ∞2 ∞ nj+ Kj+(nj+ + 4nj+ Kj- + Kj+Kj-) ∞2 ∞ kT(nj+ + nj+ Kj+ + Kj+Kj-)2

(7a)

(7b)

In eq 7, Kj+ and Kj- are the dissociation equilibrium constants expressed by eq A2 for the type j functional groups that react with specific potential-determining ∞ cations with valence Zj and bulk concentration nj+ . Obviously, A2 should be a positive value. When A2 ) 0 (Kj+ ) 0 or Kj- f ∞), the surface polymer layer has a uniform fixed charge density Q ) A1 ) (∑ZjeNj with the sign depending on the association or dissociation of the functional groups. 2.2. Linearized Governing Equations and Boundary Conditions. Because the governing equations described above are coupled nonlinear partial differential equations, it is difficult to find a general solution of them. Therefore, we shall assume that the electrokinetic system is only slightly distorted from the equilibrium state, where the particle and fluid are at rest, and replace these nonlinear equations by approximate linear equations. One can write

nm(eq), and ψ(eq) must also satisfy eqs 1-5 under the equilibrium state. The equilibrium concentration of species m is related to its bulk concentration nm∞ and the equilibrium electric potential by the Boltzmann distribution. Substituting eqs 6 and 8 into eqs 1, 3, and 5, canceling their equilibrium components, and neglecting the products of the small quantities δnm, δψ, and u, one obtains

∇2u - h(x)λ2u )

1 ∇δp η

h(x) (∇2ψ(eq)∇δψ + ∇2δψ∇ψ(eq)) + [A2(δψ∇ψ(eq) + 4πη η ψ(eq)∇δψ) - A1∇δψ] (9) ∇2δψ ) ∇2δµm )

4π M [ zmeδnm - h(x)A2δψ] m)1

∑

zme zme ∇ψ(eq) ‚∇δµm ∇ψ(eq)‚u kT Dm

(10) (11)

m ) 1, 2, ...., M Here, δµm is defined as a linear combination of δnm and δψ on the basis of the concept of the electrochemical potential energy7,8

δµm )

kT nm(eq)

δnm + zmeδψ

(12)

and λ ) (f/η)1/2. The reciprocal of the parameter λ has the dimension of length and characterizes the extent of flow penetration inside the surface polymer layer. For the surface charge layers of human erythrocytes,12 rat lymphocytes,14 and grafted polymer microcapsules,15 experimental data of 1/λ range from 1.35 to 3.7 nm. Because the reference frame is taken to travel with the particle, the composite particle is at rest and the velocity of the fluid at infinity is the particle (electrophoretic) velocity in the opposite direction. Thus, the boundary conditions for u, δψ, and δµm at the surface of the nonconducting rigid particle core are

u)0

(13a)

n‚∇δψ ) 0

(13b)

n‚∇δµm ) 0

(13c)

where n is the unit vector outwardly normal to the surface. Equation 13b represents that the rigid core is insulated, and eqs 13a and 13c result from the fact that no fluid and ions can penetrate into the rigid core. In eq 13a, we have assumed that the shear plane coincides with the surface of the rigid core. The boundary conditions at the surface of the composite particle (the boundary between the surface polymer layer and the external solution, S() are

p ) p(eq) + δp

(8a)

u|S+ ) u|S-

(14a)

nm ) nm(eq) + δnm

(8b)

n‚σ|S+ ) n‚σ|S-

(14b)

ψ ) ψ(eq) + δψ

(8c)

δψ|S+ ) δψ|S-

(14c)

n‚∇δψ|S+ ) n‚∇δψ|S-

(14d)

δµm|S+ ) δµm|S-

(14e)

n‚∇δµm|S+ ) n‚∇δµm|S-

(14f)

where p(eq), nm(eq), and ψ(eq) are the equilibrium distributions of pressure, concentration of species m, and electric potential, respectively, and δp, δnm, and δψ are the small deviations from the equilibrium state. The quantities p(eq),


Langmuir, Vol. 19, No. 18, 2003 7229

|

dψ(eq) 4πσ )dr r)r0

(16a)

ψ(eq)|r)a+ ) ψ(eq)|r)a-

(16b)

dψ(eq) dψ(eq) r)a+ ) dr dr r)a-

(16c)

ψ(eq)|rf∞ f 0

(16d)

|

|

where σ is the surface charge density of the “bare” particle core. Equations 16a and 16d state that the Gauss condition holds at the surface of the rigid core and the equilibrium potential is set to zero in the bulk solution, respectively. It can be shown that

j + ψeq10(r)Q h ψ(eq) ) ψeq01(r)σ Figure 1. Geometric sketch for a charge-regulating composite sphere under an applied electric field.

In eq 14b, σ is the hydrodynamic stress of the fluid given by σ ) -pI + η[∇u + (∇u)T], where I is the unit dyadic. Equations 14a and 14b are the continuity requirements of the fluid velocity and stress tensor at the particle surface, which are physically realistic and mathematically consistent boundary conditions for the present problem.28,29 Equations 14c and 14d indicate that the electric potential and electric field must be continuous. Equations 14e and 14f state that the concentration and flux of species m are also continuous. The conditions far from the particle are

u ) -µEE∞

(15a)

δψ f -E∞‚x

(15b)

δµm f -zmeE∞‚x

(15c)

Here, µE is the electrophoretic mobility of the chargeregulating composite particle, and its expression for a composite sphere will be obtained in the next section. 3. Electrophoretic Mobility of a Charge-Regulating Composite Sphere We now consider a charge-regulating composite sphere of radius a immersed into an unbounded electrolyte solution with constant bulk ionic concentrations nm∞ under a uniform applied electric field E∞. As shown in Figure 1, the composite sphere has a surface polyelectrolyte layer of constant thickness d so that the radius of the rigid core is r0 ) a - d. The electrophoretic velocity of the particle is µEE∞ez, where ez is the unit vector in the axial direction and E∞ ) |E∞|. The origin of the spherical coordinate system (r, θ, φ) is taken to be the center of the particle. 3.1. Solution for the Equilibrium Electrostatic Potential. Before solving for the problem of electrophoresis of the composite sphere, we need to determine the equilibrium electrostatic potential distribution, which is spherically symmetric, first. Under the Debye-Huckel approximation, the equilibrium potential ψ(eq) satisfies the linearized Poisson-Boltzmann equation resulting from eq 5 and the boundary conditions (29) Neale, G.; Epstein, N.; Nader, W. Chem. Eng. Sci. 1973, 28, 1865.

(17)

where

σ j)

4πeσ τkT

(18a)

and

Q h )

4πeA1

(18b)

τ2kT

which are the dimensionless fixed charge density on the rigid core surface and the leading term of that (or the dimensionless Donnan potential) in the charge-regulating polymer layer, respectively, of the composite sphere. In eq 17

kTτr02 {[τ cosh(τd) + κ sinh(τd)] × eνr cosh(τr - τr0) - [κ cosh(τd) + τ sinh(τd)] × sinh(τr - τr0)} (19a)

ψeq01 )

ψeq10 )

{

kT 1 + κa [sinh(τr - τr0) + 1e νr

}

τr0 cosh(τr - τr0)] (19b) when r0 < r < a, and

kT(τr0)2 exp(-κr + κa) ψeq01 ) eνr ψeq10 )

{

(19c)

kT 1 + κa [τr0 cosh(τd) + 1e νa a exp(-κr + κa) (19d) sinh(τd)] r

}

when r > a, where κ is the reciprocal of the Debye screening length defined by

κ)

(

4πe2 kT

(

∑i

τ)κ 1+

)

1/2

zi2ni∞

)

4πA2 κ2

(20a)

1/2

ν ) τ(1 + κr0) cosh(τd) + (κ + τ2r0) sinh(τd)

(20b) (20c)

The reciprocal of the parameter τ is the modified Debye length by taking the charge regulation effect into account.

7230


Ding and Keh

In eq 20b, the value of A2 is positive, and the charge regulation effect of the surface polymer layer increases with an increase in the dimensionless parameter 4πA2/ κ2. Evidently, the charge regulation effect tends to reduce the magnitudes of the effective thickness of the electric double layer and the electric potential distribution. Note that the equilibrium fixed charge density distribution within the surface polymer layer can be determined by the substitution of eq 17 into eq 6. 3.2. Solution to the Electrokinetic Equations. To solve the small quantities u, δp, δµm, δψ, as well as µE in eqs 9-15 when the parameters σ j and Q h are small, these variables can be written as perturbation expansions in powers of σ j and Q h to the first order

(26a)

n‚∇µmij ) 0

(26b)

uij ) 0

(26c)

r ) a: ψ00, µmij, and uij are continuous

(27a)

n‚∇ψ00, n‚∇µmij, and n‚σij are continuous

(27b)

r f ∞: ψ00 f -E∞r cos θ

(28a)

(21a)

µmij f 0

(28b)

j + p10Q h δp ) p01σ

(21b)

uij ) -µEijE∞ez

(28c)

j + µm10Q h δµm ) µm00 + µm01σ

(21c)

δψ ) ψ00 + ψ01σ j + ψ10Q h

(21d)

j + µE10Q h µE ) µE01σ

(21e)

( ) 3

µm00 ) -zmeE∞ r +

r0

2r2

cos θ

(22)

Substituting the expansions given by eq 21 and ψ(eq) given by eq 17 into the governing eqs 9-11 and boundary conditions 13-15, and equating like powers of σ j and Q h on both sides of the respective equations, one can derive a group of linear differential equations and boundary conditions for the functions uij, pij, µmij, and ψij. After collecting the zeroth-order and first-order terms in the perturbation procedure, we obtain

4πe

M

∑

kT m)1

∇2µmij ) -

[

zmnm∞µm00 + κ2 + h(r)

]

4πA2

ψ00 (23)

( )

zm2e2E∞ r03 dψeqij cos θ 1- 3 kT dr r

(24)

[

ψeqij -

(

{

2

)] }

ikT κ 1+ ∇ψ00 e 4πA2

∇‚uij ) 0

where (i, j) equals (0, 1) or (1, 0) and ψeqij(r) is given by eq 19. The solutions for ψ00, µmij, pij, and the r and θ components of uij subject to eqs 23-28 are

pij )

ψ00 ) E∞Fψ00(r) cos θ

(29)

µmij ) E∞Fmij(r) cos θ

(30)

uijr ) E∞Fijr(r) cos θ

(31a)

uijθ ) E∞Fijθ(r) sin θ

(31b)

[

η E F (r) + µEijFp00(r) + a ∞ pij κ2a ψ (r)Fψ00(r) cos θ (31c) 4πη eqij

]

where

[

Fp00(r) ) λ2ar -C001 +

( )]

C002 a 2 r

3

(32a)

Fψ00(r) ) W-1(τ2 - κ2)(6a + 6κa2 + 2κ2a3 +

{ ( )

1 + τr exp[τ(a + 2r0 - r)] + r2 2 r03 1 - τr κ exp[τ(a + r)] r + (32b) γ1 τ r2 2r2

κ2r03) -γ2

(

)

} ( )( )

if r0 < r < a, and

Fp00(r) ) C006(a/r)2

1 ∇pij (∇2ψeqij∇ψ00 + η 4πη A2 ψ00∇ψeqij + ∇2ψ00∇ψeqij) + h(r) η

∇2uij - h(r)λ2uij )

with

n‚∇ψ00 ) 0

u ) u01σ j + u10Q h

where the functions uij, pij, µmij, ψij, and µEij, with i and j equal to 0 or 1, are not directly dependent on σ j and Q h . The zeroth-order terms of µE, u, and δp disappear because a spherical particle with σ ) 0 and A1 ) 0 will not move by applying an electric field [although a “neutral” composite sphere with ionogenic groups in its porous surface layer in equilibrium with the suspending solution can develop an odd (antisymmetric) distribution of fixed charges when an electric field is imposed]. It is easy to show that

∇2ψ00 ) -

r ) r0:

(32c)

Fψ00(r) ) W-1(τ2 - κ2)[γ1(6a - 6τa2 + 2τ2a3 + τ2r03) exp(2τa) - γ2(6a + 6τa2 + 2τ2a3 + τ2r03) × (25a)

( )

r03 1 + κr exp[κ(a - r)] - r + 2 (32d) exp(2τr0)] r2 2r

(25b) if r > a, and the functions Fmij(r), Fijr(r), Fijθ(r), and Fpij(r) are given by eqs B1-B3 in Appendix B. In eq 32


W ) 2τ2[γ1(τ + κ)(τ - κ + τκR) exp(2τR) γ2(τ - κ) (τ + κ + τκR) exp(2τr0)] (33a) γ1 ) 2 + 2τr0 + (τr0)2

(33b)

2

(33c)

γ2 ) 2 - 2τr0 + (τr0)

{

Langmuir, Vol. 19, No. 18, 2003 7231

Λ Λ∞

)1-

1 - γ c p 1 - p

φ

M

()

3 r0

3

+

2 a

M

[

∫a

∞

r3 G (r) dr a ij

()

]

∑ zm2nm∞Dm

(34) Xij )

4. Electric Conductivity of a Dilute Suspension of Charge-Regulating Composite Spheres For a homogeneous dilute suspension of identical spherical particles subjected to a uniform applied electric field E∞, the effective electric conductivity can be determined from the solution for the fluid velocity, electrostatic potential, and ionic electrochemical potentials obtained in the previous section. One can write

1 V

∫V ∇δψ dV

(35)

where V denotes a sufficiently large volume of the suspension to contain many particles. To obtain eq 35, we have used eq 8c and the fact that the volume average of the gradient of the equilibrium electric potential is zero. There is a resulting volume-average current density, which is collinear with E∞, defined by

〈i〉 )

1 V

∫V i(x) dV

(36)

where i(x) is the electric current density at position x. The effective electric conductivity Λ of the suspension can be assigned by the linear relation

〈i〉 ) ΛE∞

() () 1 τa

1+i

r0 a

3

e kT

(37)

Λ Λ∞

(38)

M Here Λ∞ ) ∑m)1 zm2e2nm∞Dm/kT, which is the electric conductivity of the electrolyte solution containing M ionic species in the absence of the particles, and r is the position vector relative to the particle center. Substituting eqs 21c, 22, and 30 into eq 38, making relevant calculations, and using the relation given by eq 37, we obtain the electric conductivity of a suspension of identical charge-regulating composite spheres to the first order of σ j and Q h

0

[ ( ) ( )] 1 r 2 r0

3

+

1 r0 2 r

3

dψeqij dr dr (40)

5.1. Reduced Expressions for the Electrophoretic Mobility and Electric Conductivity. Before presenting the numerical results for the equilibrium characteristics and the electrophoretic mobility of a charge-regulating composite sphere as well as the effective electric conductivity of a dilute suspension of charge-regulating composite spheres, we first give some reduced expressions of eqs 21e and 39 for the electrophoretic mobility and electric conductivity. When there is no polymer layer on the surface of each rigid particle core, one has d ) 0, r0 ) a, γc ) 1, Q ) 0, λ ) 0, and τ ) κ. Then, eqs 21e and 39 reduce to

M

∑ zmnm∞Dm ∫rf∞ (δµmn - r∇δµm‚n) dS

∫r ∞

5. Results and Discussion

〈i〉 ) Λ∞E∞ + VkT m)1

(39)

Here, p is the porosity of the surface polymer layer of the composite particle, γc ()4πr03/3Vt, where Vt ) 4πa3[1 - p + p(r0/a)3]/3 is the dry volume of a composite particle) is the volume fraction of the rigid core in a composite sphere, and φ ()NVt/V) is the true volume fraction of the dry composite particles. Note that eq 39 is correct to order φ, and τaσ j and (τa)2Q h are independent of τ (or A2 and nm∞). The coefficients Xij are independent of the shielding parameter, λ, and disappear for a symmetric electrolyte when the diffusion coefficients of the cation and anion take the same value. In the following section, typical numerical results of the normalized electric conductivity Λ/Λ∞ calculated from eqs 39 and 40 will be presented.

The average current density can be expressed as17

Ne

}

j + X10(τa)2Q h] [X01τaσ

where

where the constant Cij6 and the functions Gij(r) are defined by eqs B4f and B5. Note that the relaxation effect of the diffuse ions in the electric double layer surrounding the particle is not included in the result of the first-order mobility µEij.

E∞ ) -

×

m)1

and the coefficients C001, C002, and C006 are defined by eq B7. Applying the constraint that the net force exerted on a large surface enclosing the particle and its adjacent double layer must be zero to eq 31, one can obtain the first-order term for the electrophoretic mobility of the charge-regulating composite sphere expressed as

1 -Cij6 + µEij ) C006

∑ zm3nm∞Dm m)1

)1-φ

{

3

2

j µE ) µE01σ M

+

∑ zm3nm∞Dm m)1 M

∑ zm2nm∞Dm

m)1

}

(41)

X01κaσ j

(42)

where

µE01 )

kTκa {1 - exp(κa)[5E7(κa) - 2E5(κa)]} 4πeη(1 + κa) (43)

X01 )

[

]

1 3 1 1 + exp(κa)E5(κa) + 1 + κa (κa)2 κa 2

(44)

7232


Ding and Keh

with

En(x) )

∫1∞ t-n exp(-xt) dt

(45)

From eq 17, one can obtain the following relation between the surface potential and the surface charge density of a rigid sphere at equilibrium

σ)

(1 + κa) (eq) ψ (a) 4πa

(46)

Substituting the above equation into eqs 41 and 42, these degenerated results are the same as the formulas for the electrophoretic mobility and electric conductivity obtained by Henry1 and O’Brien,7 respectively, for a single dielectric sphere with constant surface charge density in an unbounded electrolyte solution. When the composite particles are homogeneous porous spheres, one has r0 ) 0, d ) a, γc ) 0, and σ ) 0. The expressions for electrophoretic mobility and electric conductivity given by eqs 21e and 39 become

h µE ) µE10Q

(47)

M

Λ Λ

∞

)1-

∑ zm3nm∞Dm

φ

m)1

1 - p

M

X10(τa)2Q h

(48)

∑ zm nm Dm 2

∞

m)1

where

kT {3 sinh(λa)[τa(τ4(-2λ2κa3 + 3κ2a2 + 36πeη 3κa + 3) + τ2(2λ4κa3 + 2λ2κ3a3 - λ2κ2a2 - 3λ2κa 3λ2 - 3κ3a - 3κ2) + 3λ2κ2(1 + κa)) cosh(τa) (λ2 - τ2)(τ2(2λ2κa3 + 3κ3a3 - 3κa - 3) + 3κ2(1 + κa)) sinh(τa)] + 3a cosh(λa)[λτa(λ2 τ2)(τ2(2λ2a2 + 3κ2a2 + 3κa + 3) - 3κ2(1 + κa)) cosh(τa) + λ(τ4(2λ2a2 - 3κ3a3 + 3κa + 3) τ2(2λ4a2 - λ2κ3a3 + 2λ2κ2a2 + 3λ2κa + 3λ2 + 3κ3a + 3κ2) + 3λ2κ2(1 + κa)) sinh(τa)]}{ λ2τ2(λ2 τ2)a3[λa cosh(λa) - sinh(λa)][τ cosh(τa) + κ sinh(τa)]}-1 (49)

µE10 )

{

X10 ) (1 + κa)[(3 + τ2a2) sinh(τa) - 3τa cosh(τa)] + τ2 (3 + 3κa + κ2a2)[τa cosh(τa) κ2

}

sinh(τa)] {(τa)4[τa cosh(τa) + κa sinh(τa)]}-1 (50) When the fixed charge density Q in the porous particle is assumed to be a constant (A2 ) 0 and τ ) κ), the expressions given by eqs 47 and 48 further reduce to the corresponding results obtained by Liu and Keh30 for the electrophoretic mobility of a porous sphere and the electric conductivity of a suspension of porous spheres having constant Q (with no charge regulation effect). (30) Liu, Y. C.; Keh, H. J. J. Colloid Interface Sci. 1997, 192, 375.

Figure 2. Plots of the leading term of the normalized fixed charge density in a charge-regulating polymer matrix (τa)2Q h versus the bulk concentration n∞ for an aqueous 1:1 electrolyte solution under the condition of a ) 100 nm, N ) 1023 m-3, and T ) 298 K: (a) K+K- ) 10-8 M2; (b) K+/K- ) 1.

To perform typical calculations, we make the continuous phase an aqueous 1:1 electrolyte solution (z+ ) -z- ) Z ) 1 and n+∞ ) n-∞ ) n∞) with the relative permittivity r ) 78.54, the radius of the composite particle a ) 100 nm, the number density of the ionogenic functional groups inside the surface polymer layer N ) 1023 m-3, and the system temperature T ) 298 K in the following section. 5.2. Equilibrium Characteristics of a ChargeRegulating Porous Particle. The numerical results of the leading term of the normalized fixed charge density h ) 4πea2A1/ in a charge-regulating polymer matrix, (τa)2Q kT, calculated as a function of the variables n∞, K+, and K- are plotted in Figure 2. The value of K+ K- is fixed at 10-8 M2 in Figure 2a, and the value of K+/K- is specified at 1 in Figure 2b. It can be seen that the point of zero charge (with ψ(eq) ) 0 and Q ) 0) is given by n∞ ) (K+ K)1/2, as expected from examining eq A3. If n∞ < (K+ K- )1/2, the value of (τa)2Q h is negative, and its magnitude increases monotonically and approaches a saturated value gradually with a decrease in n∞ for an otherwise specified condition. If n∞ > (K+ K-)1/2, the value of (τa)2Q h is positive and increases with an increase in n∞ until the other saturation point is reached. The magnitude of (τa)2Q h decreases as K+/K- increases in that the concentration of the un-ionized functional groups (AB) increases with K+/K- as inferred from eq A2. When the value of K+ K- increases, the


Figure 3. Plots of the dimensionless parameter 4πA2/κ2 for a charge-regulating polymer matrix versus the bulk concentration n∞ for an aqueous 1:1 electrolyte solution under the condition of N ) 1023 m-3 and T ) 298 K: (a) K+K- ) 10-8 M2; (b) K+/K- ) 1.

concentration of the negatively charged groups AZ- will increase or that of the positively charged groups AB2Z+ will decrease according to eq A2; thus, the surface polymer layer becomes more negatively charged or less positively charged. The numerical results of the dimensionless parameter 4πA2/κ2 in eq 20b, which identifies the strength of the charge regulation effect in a polymer matrix, are plotted versus the bulk ionic concentration n∞ in Figure 3. Again, the value of K+ K- is fixed at 10-8 M2 in Figure 3a and the value of K+/K- is specified at 1 in Figure 3b. To clarify the dependence of 4πA2/κ2 on variables K+, K-, and n∞, some reduced expressions of eq 7b should be derived. When K+/K- f 0 with K+ K- being kept constant, the value of 4πA2/κ2 is in direct proportion to n∞K+ K-/(n∞2 + K+ K-)2 with a maximum appearing at n∞ ) (K+ K-/3)1/2; if K+/Kf ∞, then the value of 4πA2/κ2 is proportional to (n∞2 + K+ K-)/n∞2K+, which is a monotonic decreasing function of n∞. So, one can see a bulgy curve turn to a monotonic curve gradually as K+/K- increases in Figure 3a. When K+ K- f ∞ but K+/K- remains constant, eq 7b leads to 4πA2/κ2 f 0; if K+ K- f 0, then 4πA2/κ2 is in direct proportion to K+/n∞2 which is also a monotonic decreasing function of n∞. In Figure 3b, one can see how the curves of 4πA2/κ2 versus n∞ change as the value of K+ K- changes in the intermediate range.

Langmuir, Vol. 19, No. 18, 2003 7233

Figure 4. Plots of the normalized equilibrium potential ψ(eq)/ 4πeNa2 and fixed charge density Q(eq)/eN around a porous sphere (r0/a ) 0) versus the relative position r/a for an aqueous 1:1 electrolyte solution under the condition of a ) 100 nm, N ) 1023 m-3, T ) 298 K, and K+ ) K- ) 10-4 M for various values of n∞.

The numerical results of the dimensionless equilibrium potential ψ(eq)/4πeNa2 around a charge-regulating porous sphere (with r0 ) 0) calculated as a function of the relative position r/a from the particle center for several values of n∞ are plotted in Figure 4a with K+ ) K- ) 10-4 M. As expected, the magnitude of the electric potential decreases monotonically with an increase in r/a and changes sharply near the surface of the porous particle for a constant value of n∞. For a fixed value of r/a, ψ(eq)/4πeNa2 is negative and a monotonic increasing function of n∞ when n∞ < (K+ K-)1/2, equals zero at the point of zero charge and becomes a positive function which first reaches a maximum (that cannot be seen clearly in Figure 4a due to the large scale of the coordinate) and then approaches zero as n∞ increases when n∞ > (K+ K- )1/2. The dimensionless fixed charge density Q(eq)/eN within the porous sphere at the equilibrium state for the case given in Figure 4a is plotted versus the relative position r/a in Figure 4b. It can be seen that the magnitude of the fixed charge density becomes smaller, while the magnitude of the electric potential becomes greater, as the position is closer to the center of the porous particle. This can be explained by examining eq A3 as well as the relation between the concentration distribution of the potentialdetermining ions and the electric potential distribution.

7234


Ding and Keh

Figure 5. Plots of the dimensionless coefficients 4πeηµE01/ kTτa and 4πeηµE10/kTτ2a2 versus the bulk concentration n∞ for a charge-regulating composite sphere in an aqueous 1:1 electrolyte solution under the condition of a ) 100 nm, N ) 1023m-3, r0/a ) 0.5, and T ) 298 K for various values of λa. The solid and dashed curves represent the cases that the equilibrium constants K+ ) K- ) 10-4 M and that the fixed charge density Q within the surface polymer layer is constant, respectively.

n∞

If the electric potential decreases for fixed values of , K+ , and K- (so the concentration of the potentialdetermining ions increases exponentially according to the Boltzmann equation), then the equilibrium of the ionogenic functional groups shifts to a lower negatively charged state (or higher positively charged state), and vice versa. In general, the fixed charge density within the porous particle is not a sensitive function of position. The equilibrium electric potential profile around an arbitrary composite sphere and the fixed charge density distribution inside the surface polymer layer of the particle can also be calculated and graphically exhibited. For conciseness we do not present these results here, but they are qualitatively similar to those shown in Figure 4. 5.3. Electrophoretic Mobility of a Particle. In Figures 5 and 6, the dimensionless coefficients 4πeηµE01/ kTτa and 4πeηµE10/kTτ2a2 in association with eq 21e for the electrophoretic mobility of a charge-regulating composite sphere calculated from eq 34 are plotted as a function of n∞, λa, and r0/a. The solid and dashed curves represent respectively the cases that the equilibrium constants K+ ) K- ) 10-4 M and that the fixed charge density Q within the surface polymer layer is constant

Figure 6. Plots of the dimensionless coefficients 4πeηµE01/ kTτa and 4πeηµE10/kTτ2a2 versus the dimensionless parameter r0/a for a charge-regulating composite sphere in an aqueous 1:1 electrolyte solution under the condition of a ) 100 nm, N ) 1023m-3, λa ) 10, and T ) 298 K for various values of n∞. The solid and dashed curves represent the cases that the equilibrium constants K+ ) K- ) 10-4 M and that the fixed charge density Q within the surface polymer layer is constant, respectively.

(i.e., no charge regulation effect). The value of r0/a is fixed at 0.5 in Figure 5, and the value of λa is fixed at 10 in Figure 6. For the simple case of constant Q, as expected, when λa and r0/a remain unchanged, both the first-order coefficients 4πeηµE01/kTτa and 4πeηµE10/kTτ2a2 are monotonic decreasing functions of n∞; when n∞ and r0/a remain unchanged, they are monotonic decreasing functions of λa; and when n∞ and λa remain unchanged, 4πeηµE01/kTτa is a monotonic increasing function of r0/a and equals zero as r0/a ) 0 (the composite sphere reduces to a homogeneous porous sphere). On the other hand, 4πeηµE10/kTτ2a2 is a monotonic decreasing function of r0/a and equals zero as r0/a ) 1 (the composite sphere degenerates to a rigid sphere with no surface layer). By taking the charge regulation effect into account, Q is no longer a constant but a function of the local electric potential. Evidently, the charge regulation effect tends to reduce the values of both the coefficients 4πeηµE01/kTτa and 4πeηµE10/kTτ2a2 and can even cause these two quantities to increase as n∞ increases under some specific conditions. It can be found that the deviations between the solid and dashed curves in Figures 5 and 6 increase with an increase in the value of 4πA2/κ2 shown in Figure 3 and disappear when r0/a ) 1.


Figure 7. Plots of the normalized electrophoretic mobility 4πeηµE10Q h /kT versus the bulk concentration n∞ for a chargeregulating composite sphere in an aqueous 1:1 electrolyte solution under the condition of a ) 100 nm, N ) 1023 m-3, σ ) 0, λa ) 1, r0/a ) 0.5, and T ) 298 K: (a) K+K- ) 10-8 M2; (b) K+/K- ) 1.

When the surface charge density σ on the solid core of the composite sphere is zero, the normalized electrophoretic mobility of the particle is given by 4πeηµE10Q h /kT (see eq 21e) and its numerical results are shown in Figure 7 with λa ) 1 and r0/a ) 0.5. The value of K+ K- is fixed at 10-8 M2 in Figure 7a, and the value of K+/K- is specified at 1 in Figure 7b. Note that the value of 4πeηµE10Q h /kT is the product of the dimensionless coefficient 4πeηµE10/ kTτ2a2 (plotted in Figures 5b and 6b) and the leading term of the normalized fixed charge density in the surface polymer layer of the composite sphere (τa)2Q h (depicted in Figure 2). It can be seen in Figure 7 that, for a given chargeregulating composite sphere in an electrolyte with specified values of K+ and K-, the electrophoretic velocity may not be a monotonic function of the bulk ionic concentration n∞. Similar to the tendency for (τa)2Q, the value of 4πeηµE10Q h /kT is a monotonic decreasing function of K+ K- and the magnitude of the electrophoretic mobility of the particle decreases as the value of K+/K- increases. 5.4. Effective Electric Conductivity of a Suspension. The dimensionless parameters X01 and X10 in association with eq 39 for the effective electric conductivity of a dilute suspension of charge-regulating composite spheres can be evaluated by using eq 40, and the results are plotted in Figures 8 and 9. The solid and dashed curves represent respectively the cases that the equilibrium

Langmuir, Vol. 19, No. 18, 2003 7235

Figure 8. Plots of the parameters X01 and X10 in eq 39 versus the bulk concentration n∞ for a suspension of charge-regulating composite spheres in an aqueous 1:1 electrolyte solution under the condition of a ) 100 nm, N ) 1023 m-3, and T ) 298 K for various values of r0/a. The solid and dashed curves represent the cases that the equilibrium constants K+ ) K- ) 10-4 M and that the fixed charge density Q within the surface polymer layer is constant, respectively.

constants K+ ) K- ) 10-4 and that the fixed charge density Q inside the surface polymer layer is constant. When Q is constant, the trends of the dependence of the first-order parameters X01 and X10 on n∞ and r0/a are quite similar to those of the dimensionless coefficients 4πeηµE01/ kTτa and 4πeηµE10/kTτ2a2 illustrated in Figures 5 and 6, except that X01 and X10 are not functions of λa. Note that the magnitude of X10 can be quite large even for a particle with a relatively thin porous surface layer (say, with r0/a ≈ 0.95). When the charge regulation effect is considered, the magnitudes of both X01 and X10 decrease. The deviations between the solid and dashed curves in Figures 8 and 9 again increase with an increase in the value of 4πA2/κ2 but are not as evident as the results of the dimensionless coefficients for the electrophoretic mobility shown in Figures 5 and 6. Figure 10 shows the numerical results of the normalized electric conductivity of a dilute suspension of chargeregulating composite spheres in HCl aqueous solutions (with D+ > D-) versus n∞ calculated by using eq 39 with σ ) 0, p ) 0.99, φ ) 0.001, and r0/a ) 0.5. The solid curves are the results obtained from different combinations of the equilibrium constants K+ and K-, and the dashed line

7236


Figure 9. Plots of the parameters X01 and X10 in eq 39 versus r0/a for a suspension of charge-regulating composite spheres in an aqueous 1:1 electrolyte solution under the condition of a ) 100 nm, N ) 1023 m-3, and T ) 298 K for various values of n∞. The solid and dashed curves represent the cases that the equilibrium constants K+ ) K- ) 10-4 M and that the fixed charge density Q within the surface polymer layer is constant, respectively.

is the corresponding results for a suspension of neutral composite spheres (Q ) 0). The value of K+ K- is fixed at 10-10 M2 in Figure 10a and the value of K+/K- is specified at 1 in Figure 10b. It can be seen in Figure 10 that all the solid curves intersect with the dashed curve at the point with n∞ ) (K+ K-)1/2 (the point of zero charge). When n∞ < (K+K-)1/2, Λ/Λ∞ is a monotonic decreasing function of n∞ and its value is larger than that of the dashed line. When n∞ > (K+K-)1/2, the value of Λ/Λ∞ is less than that of the dashed line and may have a minimum at some values of n∞. (On the contrary, for a suspension in an electrolyte with D+ < Dsuch as particles in NaOH aqueous solutions, Λ/Λ∞ will be a monotonic increasing function of n∞ with its value smaller than that of the dashed line as n∞ < (K+K-)1/2 and have a greater value than that of the dashed line as n∞ > (K+ K-)1/2.) If the value of n∞ is relatively large, Λ/Λ∞ will converge to the result of the dashed line due to the h in eq 39 approaches a saturated value and fact that (τa)2Q X10 diminishes. 6. Concluding Remarks In this paper, the electrophoresis and electric conduction in a dilute suspension of charge-regulating composite

Ding and Keh

Figure 10. Plots of the normalized electric conductivity of a suspension of charge-regulating composite spheres in HCl aqueous solution versus n∞ calculated by using eq 39 with a ) 100 nm, N ) 1023 m-3, σ ) 0, p ) 0.99, φ ) 0.001, r0/a ) 0.5, and T ) 298 K: (a) K+K- ) 10-10 M2; (b) K+/K- ) 1. The solid and dashed curves represent the results obtained from different combinations of the equilibrium constants K+ and K- and from a suspension of neutral composite spheres (Q ) 0), respectively.

spheres with an arbitrary thickness of the electric double layers are analyzed. The surface polymer layer of each composite sphere is treated as a solvent-permeable and ion-penetrable shell of constant thickness in which ionogenic functional groups and frictional segments are distributed at uniform densities. The fixed charge density in the surface polymer layer depends on the local electric potential, the number density and dissociation equilibrium constants of the functional groups, and the concentration of the potential-determining ions in the bulk solution. By taking the charge regulation effect into account, the linearized electrokinetic equations applicable to the system of a charge-regulating composite sphere in an unbounded electrolyte solution are solved by using the Debye-Huckel approximation to obtain the electrostatic potential profile, the ionic electrochemical potential energy distributions, and the fluid velocity field under the influence of a uniform applied electric field. Analytical expressions for the electrophoretic mobility of a charge-regulating composite sphere and for the effective electric conductivity of the suspension (eqs 21e and 39) are obtained as linear functions of the fixed charge densities on the surface of the rigid core and in the surface


Langmuir, Vol. 19, No. 18, 2003 7237

polymer layer of the particle. These expressions can reduce to the limiting cases of charged solid spheres with constant surface charge density (eqs 41 and 42) and of chargeregulating porous spheres (eqs 47 and 48). Numerical results indicate that the charge regulation phenomena tend to decrease the effective thickness of the electric double layer (which can be inferred from eq 20b), and the magnitudes of the dimensionless coefficients 4πeηµE01/ kTτa, 4πeηµE10/kTτ2a2, X01, and X10 decrease accordingly compared to the case that the fixed charge density in the surface polymer layer is a constant. The intensity of this effect can be evaluated by examining the dimensionless parameter 4πA2/κ2 that is a function of the regulation characteristics (such as the equilibrium constants K+ and K- as well as the number density of the ionogenic functional groups within the surface polymer layer) and the concentration of the potential-determining ions in the bulk solution. It is worth repeating that the above-mentioned analytical solutions are obtained on the basis of the DebyeHuckel approximation. This means that these results are satisfactory when used for low values of ψ(eq) and may be used tentatively for slightly higher values of ψ(eq). For example, comparing with the numerical results for the electrophoretic mobility of a charged rigid sphere in KCl solutions obtained by O’Brien and White5 valid for an arbitrary value of zeta potential, one can find that eq 41 for a charged rigid sphere with a low zeta potential in an unbounded electrolyte solution is quite accurate for reasonably high zeta potentials (with errors less than 4% for |ζ|e/kT e 2). On the other hand, the decay of the density distributions of the polymer segments and ionogenic functional groups in the porous surface layer of the composite sphere with the distance from the rigid core has not been considered in our calculations. To see whether our theory can be reasonably extended to the higher values of ψ or to the nonuniform density distributions of the polymer segments and ionogenic functional groups, we propose to obtain a numerical solution to the electrokinetic differential equations with no assumption on the magnitude of electric potential, allowing the use of arbitrary distributions of the fixed charge and fluid drag components in the radial direction within the surface polymer layer of the composite sphere, and compare it with the approximate solution. Appendix A: Model for the Charge Regulation in Porous Media We consider a general model for the charge regulation in a porous medium (such as a polymer matrix) which develops fixed charges via association/dissociation equilibrium of uniformly distributed ionizable functional groups in contact with an electrolyte solution. The ionogenic reactions may be expressed as

AB2Z+ S AB + BZ+

(A1a)

AB S AZ- + BZ+

(A1b)

where AB represents the stationary associable/dissociable functional group inside the porous medium, BZ+ denotes the cation to determine the status of charges on the functional groups (the potential-determining ion), and the positive integer Z is the valence of ionization. For the case of an amphoteric functional group, BZ+ is usually the hydrogen ion H+. The equilibrium constants for the reactions in eq A1 are given by

K+ ) [AB][BZ+]/[AB2Z+]

(A2a)

K- ) [AZ-][BZ+]/[AB]

(A2b)

where [BZ+] represents the local concentration of BZ+. The dissociation constants K+ and K- are taken to be functions of temperature only. For N ionizable functional groups per unit volume, the net fixed charge density Q in the porous medium is proportional to the total number of positively charged sites less the total number of negatively charged sites and can be expressed as

Q ) ZeN

) ZeN

[AB2Z+] - [AZ-] [AB] + [AB2Z+] + [AZ-] [BZ+]2 - K+KK+ [BZ+] + [BZ+]2 + K+K-

(A3)

where e is the charge of a proton. In eq A3 we have ignored the discrete nature of the fixed charges and assumed that the ionic equilibria are always applicable. By the substitution of the Boltzmann distribution for the equilibrium concentration of BZ+ and the utilization of the concept of electrochemical potential energy, eq A3 for Q can be expressed in terms of the local electric potential ψ as

Q ) ZeN

δ sinh{[Ze(ψN - ψ) + δµ]/kT} 1 + δ cosh{[Ze(ψN - ψ) + δµ]/kT}

(A4)

In this equation

δ ) 2(K-/K+)1/2

(A5)

∞

ψN )

n+ kT ln Ze (K+K-)1/2

(A6)

δµ is the deviation in the local electrochemical potential energy of BZ+ in the porous medium from the equilibrium state defined by eq 12 and n+∞ is the concentration of BZ+ in the bulk solution where the equilibrium potential is set equal to zero. Equation A6 is the Nernst equation relating the Nernst potential ψN to the isoelectric point (with n+∞ ) (K+K-)1/2). Note that, although the number density of ionogenic functional groups, N, is a constant inside the porous medium, the fixed charge density distribution Q is not uniform but a function of the local electric potential, due to interactions between the functional groups and the potential-determining ions in the electrolyte solution. Appendix B: Some Functions in Section 3 For conciseness the functions Fmij(r), Fijr(r), Fijθ(r), and Fpij(r) in eqs 30 and 31 are defined here.

{

zm2e2 r03 Fmij(r) ) 3kT 2r2 1 r2

∫r r (r3 - r03) 0

∫r∞ 0

[ ( )] ∫ [ ( )] 1-

dψeqij dr + r dr

r0 r

3

∞

r

where ψeqij(r) is given by eq 19.

dψeqij dr + dr

1-

r0 r

3

}

dψeqij dr dr (B1)

7238


(ar)

Ding and Keh

(ar) R (λr) + a 2 C ( ) β (λr) + [∫ G (r) dr r (λa) 6R (λr) (ar) ∫ (ar) G (r) dr] + (λa) (λr) ∫ β (λr)G (r) dr -

Fijr(r) ) Cij1 + Cij2

3

1

3

ij4

3

r

1

2

3

r

(λa) (λr)

3

()

a λr

∫ar R1(λr)Gij(r) dr

(B2a)

∫

∫ar (

∫

)

∫

[

Fpij(r) ) λ2ar -Cij1 + 2r a

( ) ]-

Cij2 a 2 r

3

()

∫

() ()∫

Cij5 a 3 Cij6 a r2 - Cij7 - 2Cij8 + 2 r 2 r a 3 5 3 r r 1 a a r r Gij(r) dr + G (r) dr a a 10 r 2r a a ij r r 2 2r2 r Gij(r) dr + Gij(r) dr (B3b) a a a 5a

() () ()∫() ∫()

()

()

()

∫

a2 r 2r + 10Cij8 r a a a2 r

()

Fpij(r) ) Cij6

∫

()

()

2λa[λr0 sinh(λa) - cosh(λr0)]

Cij4 )

∫a Gij(r) dr 3 ∫ar (ar ) Gij(r) dr r

(B3c)

λa [C sinh(λr0) + Cij4 cosh(λr0)] Cij1 ) 3r0 ij3 r0 r0 2 2 Gij(r) dr + 3 2 [cosh(λr0) a × 2 a (λa) λ a r0

∫ar

0

0

R1(λr)Gij(r) dr +

0

(ar) G (r) dr} (B4c) 3

ij

{

1 3λr0Cij7 + (λa) [λr0 sinh(λa) - cosh(λr0)]

6r0

∫ar

0

2

λa

Gij(r) dr +

6 [sinh(λr0) (λa)2

∫ar

0

∫ar

0

×

}

R1(λr)Gij(r) dr]

(B4d)

Cij5 ) Cij1 + Cij2 + R1(λa)Cij3 + β1(λa)Cij4 Cij6 - Cij7 - Cij8 (B4e)

∫a∞ Gij(r) dr + C006 ∫a∞ (ar ) Gij(r) dr 2

∫ar

0

C001 C003

∫ar

0

(ar) G (r) dr - C 3

ij

002

R1(λr)Gij(r) dr - C004

Cij7 ) -

2 3

R1(λr)Gij(r) dr - sinh(λr0)

∫ar

3

Cij8 )

if r > a. In eqs B2 and B3

∫

∫ar

3(λr0)2 sinh(λa) cosh(λr0) + 3λr0]

Cij6 ) C005 Fijθ(r) )

Gij(r) dr -

∫

(B2c)

a3 a r2 + Cij6 + Cij7 + Cij8 + r r a r r 5 a r r3 Gij(r) dr G (r) dr + a a r a a ij 2 2 r r r 1r Gij(r) dr Gij(r) dr (B3a) a a a 5a

3

0

2 [B cosh(λr0) - 3(λr0)2 sinh(λa) sinh(λr0)] × (λa)2 r0 2 R1(λr)Gij(r) dr + [B sinh(λr0) a (λa)2

β1(λr)Gij(r) dr - cosh(λr0)

() () ()∫() ∫()

1a 5r

∫ar

(λr0)2 cosh(λr0) + 3β1(λa) - 3β1(λr0)]

3

if r0 < r < a, and

Fijr(r) ) Cij5

[2(λa)3 sinh(λa) + λa2

15λr0Cij8 - (λa)3[λr0 cosh(λa) - sinh(λr0)]Cij3 +

∫ar Gij(r) dr - (ar) ∫ar (ar ) Gij(r) dr 2

2r0

6λa cosh(λr0)]Cij8 +

ij

()

()

2 3

1

a

Cij2 a 3 Cij3 a 3 R (λr) 2 r 2 r 2 Cij4 a 3 r 2 β2(λr) Gij(r) dr 2 a 2 r (λa) 3R2(λr) r r3 Gij(r) dr β1(λr)Gij(r) dr + 2 3 a a (λa) (λr) 3β2(λr) r R1(λr)Gij(r) dr (B2b) 2 3 a (λa) (λr)

Fijθ(r) ) -Cij1 +

3 [A + 3λr0(β1(λa) - β1(λr0))]Cij7 + (λa)3Ω 5[A - 3λr0(λa sinh(λa) + cosh (λa) + β1(λr0)) +

Cij3 )

r

3

6β1(λr) 2

ij

a

1 2

ij

a

{

3

+ Cij3

Gij(r) ) -

∫ar ∫ar

0

0

Gij(r) dr β1(λr)Gij(r) dr (B4f)

∫a∞ (ar ) Gij(r) dr

(B4g)

∫a∞ Gij(r) dr

(B4h)

2

1 5

[ ( )]

r0 (κa)2 2+ 24πη r

3

dψeqij dr

(B5)

∫

R1(x) ) x cosh(x) - sinh(x)

(B6a)

β1(λr)Gij(r) dr] (B4a)

β1(x) ) xsinh(x) - cosh(x)

(B6b)

R2(x) ) (x2 + 1) sinh(x) - xcosh(x)

(B6c)

β2(x) ) (x2 + 1) cosh(x) - x sinh(x)

(B6d)

2 (Cij6 + 10Cij8) Cij2 ) 2Cij1 + (λa)2

(B4b)


In eq B4

C001 )

3 [λr0 - λa cosh(λd) + sinh(λd)] (B7a) Ω

C002 )

3r0 2 3

λaΩ

Langmuir, Vol. 19, No. 18, 2003 7239

C005 )

1 {6λar0 - λ[3(4a - 3r0)(a + r0) + 2aΩ 2 λ a(2a3 + r03)] cosh(λd) + 3[4a + 3r0 + λ2(2a3 - ar02 + r03)] sinh(λd)} (B7e)

{2λ3a3 + λ[-3r0 + a(3 +

λ2r02)] cosh(λd) - [3 + r0(-3a + r0)λ2] sinh(λd)} (B7b) 3 C003 ) - 2 3 {-3r0 cosh(λa) + [3r0 + (2a3 + λaΩ r03)λ2] cosh(λr0) + 3λr0[a sinh(λa) - r0 sinh(λr0)]} (B7c) 3 {-3r0[-λa cosh(λa) + λr0 cosh(λr0) + 2 3 λaΩ sinh(λa)] + [3r0 + λ2(2a3 + r03)] sinh(λr0)} (B7d)

C006 )

3 {λ[3dr0 + λ2a(2a3 + r03)] cosh(λd) 2aΩ [3r0 + λ2(2a3 - 3ar02 + r03)] sinh(λd)} (B7f) A ) [2(λa)3 + (λr0)3] cosh(λr0)

(B8)

B ) [2(λa)3 + (λr0)3 + 3λa + 3λr0] sinh(λa) 3 cosh(λa) (B9) Ω ) λ{-6r0 + [3(a + r0) + λ2(2a3 + r03)] cosh(λd)} + 3(λ2r02 - 1) sinh(λd) (B10)

C004 )

LA030117B

Electrophoretic Mobility and Electric Conductivity in Dilute

Recommend Documents