Electric Conductivity and Electrophoretic Mobility in Suspensions of

22044

J. Phys. Chem. C 2010, 114, 22044–22054

Electric Conductivity and Electrophoretic Mobility in Suspensions of Charged Porous Spheres Huan J. Keh* and Chun P. Liu Department of Chemical Engineering, National Taiwan UniVersity, Taipei 10617, Taiwan, Republic of China ReceiVed: August 14, 2010; ReVised Manuscript ReceiVed: October 4, 2010

The electric conduction and electrophoresis of a suspension of charged porous spheres in an electrolyte solution with an arbitrary thickness of the electric double layers are analytically studied. The porous particle can be a solvent-permeable and ion-penetrable polyelectrolyte molecule or charged floc with uniformly distributed frictional segments and fixed charges. The effect of particle interactions is taken into account by employing a unit cell model, and the overlap of the electric double layers of adjacent particles is allowed. The electrokinetic equations, which govern the electrostatic potential profile, the ionic concentration (or electrochemical potential energy) distributions, and the fluid velocity field inside and outside the porous particle in a unit cell, are linearized by assuming that the system is only slightly distorted from equilibrium. Through the use of a regular perturbation method, these linearized equations are solved with the dimensionless density of the fixed charges as the small perturbation parameter. Analytical expressions for the electrophoretic mobility of each charged porous sphere and for the effective electric conductivity of the suspension correct to the first and second orders, respectively, of the fixed charge density are obtained in closed forms. The effect of particle interactions on the electrophoresis and electric conduction of the suspension can be significant in typical situations. Comparisons of the results of the cell model with different conditions at the outer boundary of the cell are made. The dependence of the electrophoretic mobility and the electric conductivity on the particle volume fraction and other properties of the particle-solution system is found to be quite complicated. 1. Introduction When an external electric field is imposed to charged colloidal particles suspended in an electrolyte solution, the particles and the surrounding ions are driven to migrate. As a consequence, the fluid is dragged to flow by the motion of the particles and the ions, and there is an electric current through the suspension to produce. To determine the current density distribution and transport properties such as the electrophoretic mobility of the particles and the electric conductivity of the suspension, it is usually necessary to solve a set of coupled electrokinetic equations to obtain the distributions of the electric potential, ionic densities, and fluid velocity in the electrolyte solution. Dukhin and Derjaguin1 derived a simple formula for the effective electric conductivity of a dilute suspension of impermeable charged particles by considering an infinite plane slab of suspension immersed in an infinite homogeneous electrolyte subjected to an electric field perpendicular to the slab. Extending this analysis, O’Brien2 assumed that the particles and their electric double layers occupy only a small fraction of the total volume of the suspension to obtain approximate formulas for the electric conductivity using a perturbation method for particles with low zeta (ζ) potential immersed in a symmetric electrolyte correct to O(ζ2). Their results have some discrepancies with the experimental data reported by Watillon and Stone-Masui,3 who measured the surface conductances of a number of monodisperse polystyrene latices over a range of particle volume fractions. Later, considering the effects of nonspecific adsorption, which alters the concentrations of ions in the solution outside the double layers, and of counterions derived from the * To whom correspondence should be addressed. Telephone: 886-233663048. Fax: +886-2-2362-3040. E-mail: [email protected].

particle charging processes, Saville4 obtained better agreement between theories and experiments. The basic equations governing the electric conductivity of a suspension of charged colloidal particles also describe the electrophoretic phenomena. O’Brien5 derived analytical formulas for the electrophoretic mobility and electric conductivity of a dilute suspension of impermeable dielectric spheres with thin but polarized double layers in a general electrolyte solution. On the other hand, approximate analytical expressions for the electrophoretic mobility and electric conductivity of dilute suspensions of colloidal spheres in symmetric electrolytes were obtained by Ohshima et al.6 These expressions are correct to order (κa)-1, where κ is the Debye-Huckel parameter (defined after eq 18) and a is the particle radius. When the zeta potential of the particles is small, their result is in agreement with O’Brien’s.2 There are many colloidal particles that are porous, that is, permeable to the fluid and ions, such as macromolecules and flocs of fine particles. A theoretical study of the electrokinetic phenomena of charged porous particles was first made by Hermans and Fujita,7 who derived formulas for the electrophoretic mobility of a porous sphere by introducing the Brinkman equation8 for the internal flow field of the particle and assuming that the double layer remains spherically symmetric in the presence of the applied electric field. Later, an analytical expression for the effective electric conductivity of a dilute suspension of charged porous spheres with uniform densities of hydrodynamic frictional segments and fixed charges was derived under the assumption that the density of the fixed charges is low.9 Some general expressions were also derived for the electrophoretic mobility and electric conductivity of a dilute suspension of composite (soft) spherical particles.10-13

10.1021/jp107697r  2010 American Chemical Society Published on Web 12/01/2010

Electric Conductivity in Charged Porous Spheres These mobility and conductivity expressions tend to the formulas obtained previously for a suspension of spherical polyelectrolytes7,9 when the hard core of each composite particle vanishes and the electric potentials are low. In some practical applications of electric conduction and electrophoresis, relatively concentrated suspensions of particles are encountered, and effects of particle interactions will be important. To avoid the difficulty of the complex geometry appearing in swarms of particles, unit cell models were often employed to predict the effects of particle interactions on the mean sedimentation velocity in bounded suspensions of identical impermeable or permeable spheres.14-20 These models, which have also been used to evaluate the electrophoretic mobility,21-28 electric conductivity,25-29 and diffusiophoretic mobility30-32 in suspensions of impermeable spheres, involve the concept that an assemblage can be divided into a number of identical cells, one sphere occupying each cell at its center. The boundary value problem for multiple spheres is thus reduced to the consideration of the behavior of a single sphere and its bounding envelope. The most acceptable of these models with various boundary conditions for the fluid flow at the virtual surface of the cell are the so-called “free-surface” model of Happel33 and “zerovorticity” model of Kuwabara,34 the predictions of which for the sedimentation of spherical particles have been tested against the experimental data. Using the Kuwabara cell model, assuming that the overlap of the electric double layers of adjacent particles is negligible on the virtual surface of the cell, and neglecting the polarization (relaxation) effect of each double layer, Ohshima35 obtained a general expression for the electrophoretic mobility in a concentrated suspension of identical charged composite spheres for the case of low electrostatic potentials. In the present Article, the unit cell model is used to obtain analytical expressions for the electrophoretic mobility and effective electric conductivity of a suspension of charged porous spheres with an arbitrary thickness of the electric double layers. The overlap of adjacent double layers is allowed, and for the derivation of the electric conductivity, the polarization effect in the diffuse layer surrounding each particle is included. Both the Happel model and the Kuwabara model are considered. The basic electrokinetic equations are linearized assuming that the electrolyte ion concentrations, the electrostatic potential, and the fluid pressure have only a slight deviation from equilibrium due to the application of the electric field. Through the use of a regular perturbation method with the dimensionless fixed charge density of the porous particle as the small perturbation parameter, the ion concentration (or electrochemical potential energy), electric potential, and fluid velocity profiles are determined by solving these linearized electrokinetic equations subject to appropriate boundary conditions. Analytical expressions for the electrophoretic mobility and effective electric conductivity of the suspension are derived. 2. Basic Electrokinetic Equations We consider a statistically homogeneous distribution of identical charged porous spheres in a bounded liquid solution containing M ionic species. When the suspension is subjected to a constant applied electric field E∞ez, the particles migrate with a velocity equal to Uez at the steady state due to electrophoresis, where ez is a unit vector in the positive z direction. As shown in Figure 1, we employ a unit cell model in which a single particle of radius a is surrounded by a concentric spherical shell of suspending solution having an outer radius of b such that the particle/cell volume ratio is equal to

J. Phys. Chem. C, Vol. 114, No. 50, 2010 22045

Figure 1. Geometrical sketch for a charged porous sphere at the center of a spherical cell under an applied electric field.

the apparent particle volume fraction φ in the entire suspension, viz., φ ) (a/b)3. The cell as a whole is electrically neutral. The origin of the spherical coordinate system (r, θ, φ) is taken at the center of the particle, and the axis θ ) 0 points toward the positive z direction. Obviously, the problem for the cell is axially symmetric about the z-axis. The cell model intends to obtain information on the ensemble by considering the behavior of a single particle in a bubble of solution, together with properly chosen boundary conditions. 2.1. General Governing Equations. Conservation of all ionic species in the electrolyte solution at the steady state requires that

∇ · Jm ) 0, m ) 1, 2, ..., M

(1)

where Jm(r,θ) is the number flux distribution of species m. If the solution is dilute, Jm is given by

(

Jm ) nmu - Dm ∇nm +

)

zme n ∇ψ kT m

(2)

Here, u(r,θ) is the fluid velocity field relative to the particle, ψ(r,θ) is the electric potential distribution, nm(r,θ) and zm are the concentration (number density) distribution and the valence, respectively, of species m, Dm is the diffusion coefficient of species m, which is assumed to be a constant both inside and outside the porous particle, 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(r)fu ) ∇p +

∑ zmenm∇ψ

(3)

m)1

∇·u ) 0

(4)

where η is the viscosity of the fluid (the available evidence36 suggests that it is reasonable to assume the same value of η

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J. Phys. Chem. C, Vol. 114, No. 50, 2010

Keh and Liu

inside and outside the porous particle), f is the hydrodynamic friction coefficient inside the porous particle per unit volume of the fluid (which accounts for the hindrance to the convective transport of the electrolyte solution caused by the frictional segments), p(r,θ) is the dynamic pressure distribution, and h(r) is a unit step function, which equals unity if r e a, and zero otherwise. In eq 3, η and f are assumed to be constant. Note that f can be expressed as 6πηaSNS in the free-draining limit, where NS and aS are the number density and the Stokes radius, respectively, of the hydrodynamic frictional segments of the porous particle. The local electric potential ψ and the space charge density are related by Poisson’s equation: M

∑

1 ∇2ψ ) - [ z en + h(r)Q] ε m)1 m m

(5)

Here, Q is the fixed charge density inside the porous particle, and ε is the dielectric permittivity of the electrolyte solution, which is assumed to be the same inside and outside the porous particle. 2.2. Linearized Governing Equations. 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

p ) p(eq) + δp

(6a)

nm ) nm(eq) + δnm

(6b)

ψ ) ψ(eq) + δψ

(6c)

(eq) (r) are the equilibrium distribuwhere p(eq)(r,θ), n(eq) m (r), and ψ tions of pressure, concentration of species m, and electric potential, respectively, and δp(r,θ), δnm(r,θ), and δψ(r,θ) are the small deviations from the equilibrium state. The quantities p(eq), 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∞ (concentration in the electrolyte solution free from the porous particles) and the equilibrium electric potential by the Boltzmann distribution. Substituting eq 6 and the Boltzmann distribution for n(eq) m into eqs 1, 3, and 5, canceling their equilibrium components, and neglecting the products of the small quantities δnm, δψ, and u, one obtains:

∇2δµm )

[

]

zme kT ∇ψ(eq) · ∇δµm ∇ψ(eq) · u kT Dm m ) 1, 2, ..., M (7)

∇2∇ × u - h(r)λ2∇ × u ) M zme zme ∞ 1 nm exp - ψ(eq) ∇ψ(eq) × ∇δµm η m)1 kT kT

∑

{

[

]

}

(8)

Here, δµm is defined as a linear combination of δnm and δψ on the basis of the concept of the electrochemical potential energy:2,6

kT δnm + zmeδψ nm(eq)

δµm )

(9)

and λ ) (f/η)1/2. The reciprocal of the parameter λ is the shielding length characterizing the extent of fluid flow penetration inside the porous particle. For some model porous particle made of steel wool (in glycerin-water solution)37 and plastic foam slab (in silicon oil),38 experimental values of 1/λ can be as high as 0.4 mm, whereas in the surface regions of human erythrocytes,39 rat lymphocytes,40 and grafted polymer microcapsules41 in salt solutions, experimental data of 1/λ range from 1.3 to 3.7 nm. Note that 1/λ2 is the permeability of the porous medium, which is related to its pore size and porosity and characterizes the dynamic behavior of the viscous fluid in it. 2.3. Boundary Conditions. The perturbed electrochemical potential energies and fluid velocity within the porous sphere satisfy the conditions:

r ) 0: δµm and u are finite

(10)

The boundary conditions at the surface of the porous particle are

δµm | r)a+ ) δµm | r)a-

|

(11b)

u|r)a+ ) u|r)a-

(11c)

∂δµm ∂r

er · σ

|

(11a)

) r)a+

|

r)a+

∂δµm ∂r

) er · σ

|

r)a-

r)a-

(11d)

where the superscripts + and - to a represent the external and internal sides, respectively, to the surface of the particle, er is the radial unit vector in spherical coordinates, and σ is the hydrodynamic stress deviation from the equilibrium state given by σ ) -δpI + η[∇u +(∇u)T], in which I is the unit dyadic. Equations 11a and 11b state that the concentration and flux of ionic species m are continuous, whereas eqs 11c and 11d are the continuity requirements of the fluid velocity and stress at the particle surface, which are physically realistic and mathematically consistent boundary conditions for the present problem.36,42,43 Here, the total fluid stress, which consists of the hydrodynamic stress and the Maxwell stress, is continuous at the particle surface. The boundary condition for the continuity of hydrodynamic stress given by eq 11d comes from the fact of continuous Maxwell stress. The boundary conditions at the virtual (outer) surface of the cell, in which the local electric field is compatible with the uniform applied field E∞ez (Levine-Neale model),44 are

r ) b:

∂δµm ) -zmeE∞ cos θ ∂r

(12a)

Electric Conductivity in Charged Porous Spheres

ur ) -µEE∞ cos θ

[ ()

τrθ ) η r


]

∂ uθ 1 ∂ur + ) 0 (for the Happel model) ∂r r r ∂θ (12c)

eφ · (∇ × u) )

ψ(eq)

|

) ψ(eq)

dψ(eq) dr

|

)

(12b)

1∂ 1 ∂ur (ruθ) ) r ∂r r ∂θ 0 (for the Kuwabara model) (12d)

r)a+

r)a+

r ) b:

|

(15a)

r)a-

|

(15b)

dψ(eq) )0 dr

(16)

dψ(eq) dr

r)a-

It can be shown that where µE is the electrophoretic mobility of the charged porous sphere, ur and uθ are the r and θ components, respectively, of u, and eφ is a unit vector in the positive φ direction. Note that the Happel cell model33 assumes that the radial velocity and the shear stress of the fluid on the outer boundary of the cell are zero, whereas the Kuwabara cell model34 assumes that the radial velocity and the vorticity of the fluid are zero there. Because the reference frame is taken to travel with the particle, the porous particle is at rest, and the radial velocity of the fluid given by eq 12b is generated by the particle (electrophoretic) velocity in the opposite direction. The conditions in eqs 12a and 12b imply that there are no net flows of fluid and ionic species between adjacent cells; they are valid because the suspension of the particles is bounded by impermeable and inert walls. For the sedimentation of a suspension of uncharged spherical particles, both the Happel and the Kuwabara models give qualitatively the same flow fields and approximately comparable drag forces on the particle in a cell. However, the Happel model has a significant advantage in that it does not require an exchange of mechanical energy between the cell and the environment.45 The boundary condition of the electric potential at the virtual surface r ) b may be taken as the distribution giving rise to the applied field E∞ez in the cell when the particle does not exist (Zharkikh-Shilov model).46 In this case, eq 12a becomes

r ) b: δµm ) -zmeE∞r cos θ

(13)

Note that the overlap of the electric double layers of adjacent particles is allowed in both of the boundary conditions given by eqs 12a and 13.

j + O(Q j 2) ψ(eq) ) ψeq1(r)Q

(17)

where

ψeq1(r) )

{

kT 1 + [(κa - κb) cosh(κa - κb) + e

(κ2ab - 1) sinh(κa - κb)]

sinh(κr) R(κb)κr

if 0 e r e a

}

(18a) ψeq1(r) )

kT R(κa) 1 [bκ cosh(κb - κr) e R(κb) κr sinh(κb - κr)] if a e r e b (18b)

j ) The function R(x) is defined by eq A8 in the Appendix, Q ZeQ/εκ2kT is the nondimensional charge density of the porous particle, and κ is the Debye screening parameter equal to 2 ∞ 1/2 (eq) as a power (e2∑M m ) 1zmnm/εkT) . The expression in eq 17 for ψ j up to O(Q j ) is the equilibrium solution for the series in Q linearized Poisson-Boltzmann equation that is valid for small values of the electric potential (the Debye-Huckel approximaj 2) to tion). Note that the contribution from the effect of O(Q j 3) for the special case of a solution ψ(eq) in eq 17 reduces to O(Q of symmetrically charged electrolytes. 3.2. Solution to the Electrokinetic Equations and the Electrophoretic Mobility. To solve the small quantities j is small, δµm, u, and µE in eqs 7-13 when the parameter Q these variables can be written as perturbation expansions in j: powers of Q

j + µm2Q j 2 + ... δµm ) -zmeE∞r cos θ + µm1Q 3. Electrophoretic Mobility of Charged Porous Spheres 3.1. Solution for the Equilibrium Electrostatic Potential. Before solving for the problem of electrophoresis of the charged porous sphere in a unit cell filled with the solution of M ionic species with constant bulk concentrations nm∞, we first need to determine the equilibrium electrostatic potential distribution, which is spherically symmetric. The equilibrium potential ψ(eq) satisfies the Poisson-Boltzmann equation and the following boundary conditions:

r ) 0: ψ(eq) is finite

(14)

(19a) j + u2Q j 2 + ... u ) u1Q

(19b)

j + µE2Q j 2 + ... µE ) µE1Q

(19c)

where the functions µmj, uj, and µEj, with j ) 1, 2, ..., are j . The zeroth-order terms of u and µE disappear independent of Q because an uncharged particle will not move by applying an electric field. Substituting the expansions given by eq 19 and ψ(eq) given by eq 17 into the governing eqs 7 and 8 and boundary conditions

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j on both sides of the 10-13, and equating like powers of Q respective equations, one can derive a group of linear differential equations and boundary conditions for the functions µmj, uj, and j ) terms in the µEj with j ) 1, 2, ... After collecting the O(Q procedure of this regular perturbation, we obtain the following equations:

zm2 e2 ∞ dψeq1 ∇ ∇ × u1 - h(r)λ ∇ × u1 ) n E sin θeφ ηkT m ∞ dr m)1 (20a) M

2

∑

2

∇2µm1 ) -

zm2 e2 dψeq1 E cos θ kT ∞ dr

(20b)

r ) 0: µm1 and u1 are finite

(21)

r ) a: µm1 and u1 are continuous

(22a)

µE1 )

1 × 12(λa) [3(λa) R(λa) + A2φ5/3] 2

2

(κa) [(λa) A { 4εkT ηe 2

2

1

3(λa)4R(λa)φ1/3] + 4(λa)2[3(λa)2R(λa) + a A2φ5/3] J3(a) - J2(a) b b2 a3 J0(a) - 3 J5(a) + + 4(λa)2A3φ2/3 a b 2 2 2/3 7/3 b I (a) + 24(λa) (φ - φ ) a R 4(λa)2[3(λa)2R(λa) - {A2 +

[ [( )

(22b)

(25a)

{

4εkT 1 (κa)2[5A1 2 180(λa) R(λa) ηe A2φ2 + 10A3φ - 18(λa)2R(λa)φ1/3] a3 J (a) + 12(λa)2R(λa) 5J2(a) + b 5 a 4[A2φ5/3 - 5A3φ2/3 + 18(λa)2R(λa)] J3(a) + b b2 120[φ2/3 - φ5/3] I (a) a R b2 4[A2φ5/3 - 5A3] J (a) + a 0 5/3 4[A2φ - 2{5A3 - 15R(λa)}φ2/3 a 10(λa)3 cosh(λa) + 18(λa)2R(λa)] I3(a)} b

[

(23a)

µm1 ) 0 (if eq 13 is used)

(23b)

u1r ) -µE1E∞ cos θ

(23c)

τ1rθ ) 0 (for the Happel model)

(23d)

(23e)

The solutions for µm1 and the r and θ components of u1 subject to eqs 20-23 are

µm1 ) E∞Fm(r) cos θ

(24a)

u1r ) E∞Fu(r) cos θ

(24b)

u1θ ) -E∞

1 d 2 [r Fu(r)] sin θ 2r dr

()

]

for the Happel model, and

∂µm1 r ) b: ) 0 (if eq 12a is used) ∂r

eφ · (∇ × u1) ) 0 (for the Kuwabara model)

]

(λa)2R(λa) - 2(λa)2 sin h(λa)]}φ4/3 a + A2φ5/3 - 2(λa)3 cos h(λa)φ-1/3] I3(a)} b

µE1 ) ∇µm1 and n · σ1 are continuous

+ A4φ5/3 - (λa)2A2φ2 -

(24c)

where the functions Fm(r) and Fu(r) are given by eqs A1 and A2 in the Appendix. Because the unit cell as a whole is electrically neutral, the net force exerted on its virtual surface must be zero. Applying this constraint to eqs 24b and 24c, one can obtain the firstorder term µE1 for the electrophoretic mobility of the charged porous sphere expressed as

()

]

() ()

(25b) for the Kuwabara model, where the constants An and functions In(r), IR(r), Jn(r), and R(x) are defined by eqs A3, A4, A5, A7, and A8. Note that both boundary conditions 12a and 13 for the electric potential (electrochemical potential energies) prescribed at the virtual surface of the unit cell lead to the same result of the first-order mobility µE1, because µm1 is not needed in its calculation. Also, the relaxation effect of the diffuse ions in the electric double layer surrounding the particle is not included in the result of µE1. Among the second-order terms in the perturbation procedure, the only distributions we need in the following calculations are the electrochemical potential energies µm2. If the solution contains only a symmetrically charged electrolyte (M ) 2, z+ ) -z- ) Z, n∞+ ) n∞- ) n∞, where the subscripts + and refer to the cation and anion, respectively), the equation governing µ(2 is

∇2µ(2 ) (

(

Ze kT u ∇ψeq1 · ∇µ(1 kT D( 1

)

(26)

The boundary conditions for µ(2 are given by eqs 21, 22, and 23a or 23b with the subscript 1 being replaced by 2. For a general electrolyte, there is an extra term on the right-hand side j 2) correction to the equilibrium of eq 26 involving the O(Q



potential as expressed by eq 17. This extra term considerably complicates the problem. So we consider here only the case of j 2) term in eq 17 a symmetric electrolyte, in which the O(Q j is O(Q j 3). The vanishes and the leading correction to ψeq1Q solution to eq 26 for µ(2 is

µ(2

ZeE∞ )[r 3kT

(

∫r

b

(

∫

(

j Λ ) Λ∞[1 - βΛ1(κa, φ)(κa)2Q j 2 + O(Q j 3)] (32) Λ2(κa, λa, φ)(κa)4Q

dF( dψeq1 kT Fu dr + dr D( dr

1 r 3 dF( r dr r2 0 2r b 3 dF( kT r Fu 3 0 dr D b (

∫

)

the relation given by eq 30, we obtain the electric conductivity of a suspension of identical charged porous spheres as a power j: series in Q

)

)

In this expression:

dψeq1 kT F dr + D( u dr dψeq1 dr] cos θ dr

M

β)

(27)

〈E〉

)-

1 V

∑

)

1 V

∫

(29)

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〉

) Λ〈E〉

(30)

On the basis of the mathematical analysis given in a previous article,25 the average current density is obtained as

〈i〉

) Λ∞〈E〉 +

M

∑

3e zmnm∞Dm 3 4πb kT m)1

∫r)b (δµm r

∂δµm )e dS (31) ∂r r

Here, Λ∞ ) e2∑mM ) 1z2mn∞mDm/kT, which is the electric conductivity of the electrolyte solution containing M ionic species in the absence of the particles. 4.2. Solution for the Electric Conductivity. Substituting eqs 24 and 27 into eq 31, making relevant calculations, and using

φR(κb) - R(κa) (κa)2R(κb)

(33b)

D+ - DD+ + D-

(34a)

-2εk2T2 A - Z2B ηe2(D+ + D-)

(34b)

β)Z

for a symmetric electrolyte, where

A)

B)

i(x) dV V

(33a) zm2 nm∞Dm

for a general electrolyte, and

(28)

where V denotes a sufficiently large volume of the suspension to contain many particles. To obtain eq 28, we have used eq 6c 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〉

Λ1 )

Λ2 )

∫V ∇δψ dV

m)1 M m)1

where the functions F((r) and Fu(r) are given by eqs A1 and A2, and ψeq1 has been expressed by eq 18. 4. Electric Conductivity of a Suspension of Charged Porous Spheres 4.1. Formulation for the Electric Conductivity. For a homogeneous suspension of identical spherical porous particles subjected to a uniformly applied electric field E∞ez, 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. The average of the local electric field (E ) -∇ψ) can be expressed as

∑ zm3 nm∞Dm

ηe2 εk2T2(κa)4b3 -1 Z kT(κa)4b3 2

∫0b r3Fu

dψeq1 dr dr

dF dψeq1 dr dr

∫0b r3 dr(

(35a)

(35b)

j ) term Note that the dimensionless coefficient Λ1 of the O(Q given by eq 33b for the electric conductivity of the suspension does not depend on the boundary conditions at the virtual surface j 2 term in eq 32 is limited to of the unit cell, and the result of Q a symmetric electrolyte. As expected, the electric conductivity of a suspension of uncharged porous particles penetrable to electrolyte ions is the same as that in the absence of the particles (Λ1 ) Λ2 ) 0 and Λ ) Λ∞ if Q ) 0 or φ ) 0). The coefficient j term, which is independent of the shielding βΛ1 of the Q parameter λ and the boundary conditions at the virtual surface of the unit cell, disappears for a symmetric electrolyte when the diffusivities of the cation and anion take the same value. j 2 term for a symmetric As to the coefficient Λ2 of the Q electrolyte, the first term of the right-hand side of eq 34b (which depends on parameters κa, λa, and φ) represents the effect due to the convection of the fluid, while the second term (which is a function of κa and φ only) denotes the effect due to the deviations of the electrochemical potential distributions from their combined equilibrium and applied values. Equation 32 indicates that the value of the electric conductivity of a suspension of charged porous particles (Λ) can be smaller or larger than that of the suspension in which the porous particles

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Figure 2. Plots of the dimensionless electrophoretic mobility ηκ2µE/Q in a suspension of identical charged porous spheres versus κa for various values of λa and φ: (a) λa ) 10; (b) φ ) 0.3. The solid and dashed curves represent the calculations for the Happel and Kuwabara cell models, respectively.

Figure 3. Plots of the dimensionless electrophoretic mobility ηκ2µE/Q in a suspension of identical charged porous spheres versus λa for various values of κa and φ: (a) κa ) 1; (b) φ ) 0.3. The solid and dashed curves represent the calculations for the Happily and Kuwabara cell models, respectively.

Λ1 )

are uncharged or in the absence of the particles (Λ∞), depending on the signs of βΛ1 and Q.

5.1. Reduced Expressions for the Electrophoretic Mobility and Electric Conductivity. Before presenting the numerical results for the electrophoretic mobility and effective electric conductivity of a concentrated suspension of charged porous spheres, we first give the reduced expressions of eqs 25, 33b, and 35 for the electrophoretic mobility and electric conductivity in the limiting case of a dilute suspension. For an infinitely dilute suspension of charged porous spheres [φ ) (a/b)3 f 0], the parameters µE1 in expression 25 for the electrophoretic mobility and Λ1, A, and B in eqs 33b and 35 for the electric conductivity reduce to

µE1

{()

(

)

κ2 1 - e-2κa + 2 + 1 + e-2κa × κa λ - κ2 -2κa -2κa λ 2 κa(1 + e )-1+e - 1 + e-2κa κ λa coth(λa) - 1

κ εkT ) 3 3ηe λ

(1 + κa1 )[( )

2

]}

(36)

(

{

(37a)

1 2 + κa + [R(κa)]2e-2κa (λa)2(κa)2 (κa)8 κ2 (1 + κa)2 -2κa e [R(κa) sinh(κa) - sinh2(κa) + 2 2 λ -κ (κa)8 2λ2κ2 (1 + κa)2 R(κa)e-2κa × (κa)2] + 2 2 2 8 (λ - κ ) (κa) λ 2 R(κa) sinh(λa) - sinh(κa) (37b) κ R(λa)

A)φ

5. Results and Discussion

φ (κa)2

)

[( )

]}

{

2(κa)2 + 3κa + 3 κa + 1 -κa R(κa)e-κa e sinh(κa) 3(κa)7 3(κa)5 3 (κa + 1)2 -2κa 3 κa + 2 2 -2κa [R(κa)] e e [R(κa) sinh(κa) 4 (κa)8 4 (κa)8

B)φ

sinh2(κa) + (κa)2]

}

(37c)

Equations 36 and 37 are the same as the corresponding formulas obtained by Hermans and Fujita7 and Liu and Keh,9 respectively, for a single porous sphere in an unbounded electrolyte.


Figure 4. Plots of the dimensionless electrophoretic mobility ηκ2µE/Q in a suspension of identical charged porous spheres versus φ for various values of κa and λa: (a) λa ) 10; (b) κa ) 1. The solid and dashed curves represent the calculations for the Happel and Kuwabara cell models, respectively.

5.2. Electrophoretic Mobility of the Particles. In Figures 2-4, the numerical values of the first-order dimensionless electrophoretic mobility ηκ2µE/Q ()ηeµE1/εkT) in a suspension of charged porous spheres calculated from eq 25 incorporating with eq 19c are plotted as a function of the parameters κa, λa, and φ for both the Happel and the Kuwabara models. The results are presented up to φ ) 0.74, which corresponds to the maximum attainable volume fraction for a swarm of identical spheres.44 As expected, ηκ2µE/Q is always a positive value. For given values of λa and φ, ηκ2µE/Q decreases monotonically with a decrease in κa (or with an increase in the electric-doublelayer overlap),and is proportional to (κa)2 in wide ranges. For fixed values of κa and φ, ηκ2µE/Q decreases monotonically with an increase in λa (or with a decrease in fluid permeability inside the particles), approaches a constant value as λa is large, and is proportional to (λa)-2 in the limit λa f 0. For specified values of κa and λa in a broad range, ηκ2µE/Q is a monotonic decreasing function of φ. When the double layers are thicker (κa is smaller) or the fluid flow penetration inside the particle is thinner (λa is greater), the particle concentration effect on the electrophoretic mobility is more significant, as expected. Figures 2-4 indicate that, for any combination of κa, λa, and φ, the Kuwabara model predicts a smaller value (or a stronger particle concentration dependence) for the dimensionless electrophoretic mobility than does the Happel model (but the


j ) term Figure 5. Plots of the dimensionless coefficient Λ1 of the O(Q in eq 33b for the effective electric conductivity of a suspension of identical charged porous spheres versus the parameters κa and φ.

difference in general is small). This occurs because the zerovorticity model yields a larger energy dissipation in the cell than that due to particle drag alone, due to the additional work done by the stress at the outer boundary.45 Using the Levine-Neale and Kuwabara cell models (with boundary conditions given by eqs 12a, b, and d) and neglecting the relaxation effect of the electric double layer, Ohshima35 obtained an approximate expression for the electrophoretic mobility of charged porous spheres in concentrated suspensions for the case of low electric potentials. This expression, which is linear in the fixed charge density Q inside the porous particle, yields results almost identical to those shown by dashed curves in Figures 2-4. 5.3. Effective Electric Conductivity of the Suspension. The j ) term in eq 33b for dimensionless coefficient Λ1 of the O(Q the effective electric conductivity of a suspension of charged porous spheres is plotted in Figure 5 as a function of the parameters κa and φ. It can be seen that Λ1 is always a positive value, and thus the presence of the particle charges reduces the magnitude of the effective conductivity for any volume fraction of particles in the suspension if the product of β and Q is positive and increases this magnitude if βQ < 0. For a fixed value of φ, Λ1 is a monotonic decreasing function of κa from a constant value as κa f 0 as κa f ∞, where the fixed charges of the porous particles are screened out by the counterions in the infinitely thin diffuse layers. For a given value of κa, the effect of the particle charges on the effective conductivity is not a

22052


Figure 6. Plots of the dimensionless parameter A in eq 35a for the effective electric conductivity of a suspension of identical charged porous spheres versus κa for various values of λa and φ: (a) λa ) 10; (b) φ ) 0.3. The solid and dashed curves represent the calculations for the Happel and Kuwabara cell models, respectively.

monotonic function of φ (has a maximum) and, as expected, vanishes as φ ) 0. The location of the maximum in Λ1 shifts to greater φ as κa increases. In general, the particle interaction effect on the electric conduction of the suspension is significant. The dimensionless parameter A in association with the j 2) term in eq 34b can be evaluated for coefficient Λ2 of the O(Q given values of parameters κa, λa, and φ using eq 35a, and its results are plotted in Figures 6-8. Unlike the coefficient Λ1 (and parameter B discussed later), the value of A depends on the boundary condition for the fluid velocity at the virtual surface of the unit cell. It can be found that the value of A predicted by the Happel model is larger than that predicted by the Kuwabara model, but, in general, the difference is small. For given values of λa and φ, A is a monotonic decreasing function of κa, approaches a constant value as κa is small, and is proportional to (κa)-2 in the limit κa f ∞. For specified values of κa and φ, A decreases monotonically with an increase in λa, approaches a constant value as λa is large, and is proportional to (λa)-2 in the limit λa f 0. For fixed values of κa and λa, A is maximal at some value of φ and, as expected, vanishes as φ ) 0. The location of the maximum shifts to greater φ as κa increases, but is not sensitive to the variation of λa. Figure 9 shows plots of the dimensionless parameter B in j 2) term calculated using eq 35b the coefficient Λ2 of the O(Q for various values of parameters κa and φ. In general, the value

Keh and Liu

Figure 7. Plots of the dimensionless parameter A in eq 35a for the effective electric conductivity of a suspension of identical charged porous spheres versus λa for various values of κa and φ: (a) κa ) 1; (b) φ ) 0.3. The solid and dashed curves represent the calculations for the Happel and Kuwabara cell models, respectively.

of B calculated using the Dirichlet-type boundary condition 13 (Zharkikh-Shilov model), which is always positive, is greater than that calculated using the Neumann-type boundary condition 12a (Levine-Neale model), which can be either positive or negative depending on the combination of κa and φ. Both boundary conditions predict that B vanishes in the limits κa f ∞ and φ f 0 as expected. Note that the trend of the dependence of B on κa and φ calculated using the boundary condition 13 is similar to that for Λ1, but the magnitude of B is much smaller. In a study of the effect of cell-model boundary conditions on the electric conductivity and electrophoretic mobility of a concentrated suspension of impermeable spheres,28 both the Levine-Neale model and the Zharkikh-Shilov model lead to identical results if each of them is associated with its corresponding evaluation of the macroscopic electric field. 6. Concluding Remarks In this Article, the electrophoresis and electric conduction in a homogeneous suspension of identical charged porous spheres with an arbitrary thickness of the electric double layers are analyzed by employing the Happel and Kuwabara cell models. Solving the linearized electrokinetic equations applicable to the system of a sphere in a unit cell by a regular perturbation method, we have determined the electrophoretic mobility of the charged porous sphere correct to the order of the fixed charge density Q in eq 25 and the effective electric conductivity of the suspension as a power


Figure 8. Plots of the dimensionless parameter A in eq 35a for the effective electric conductivity of a suspension of identical charged porous spheres versus φ for various values of κa and λa: (a) λa ) 1; (b) κa ) 1. The solid and dashed curves represent the calculations for the Happel and Kuwabara cell models, respectively.

series in Q up to O(Q2) in eq 32. The particle charges can result in an increase or a decrease in the effective conductivity relative to that of a corresponding suspension of uncharged particles, depending on the diffusion coefficients of the electrolyte ions and the sign of the particle charges. Comparisons of the results of the electrophoretic mobility and the effective conductivity between the Happel model and the Kuwabara model with different conditions for the electric potential at the virtual boundary of the unit cell have been made. These results from using the Happel and Kuwabara boundary conditions in general are indistinguishable and adhere to just either model. In typical situations, the effect of particle interactions on the electrophoresis and electric conduction in a suspension of charged porous spheres can be significant. Although experimental results on the motion of porous particles in the presence of electric fields are available in the literature for dilute suspensions,47,48 the relevant experimental data for relatively concentrated suspensions would be needed to confirm the validity of the unit cell model at various ranges of κa, λa, and φ. Equations 25 and 32-35 are obtained on the basis of the Debye-Huckel approximation for the equilibrium potential distribution around the charged porous sphere in a unit cell. Similar formulas for the electrophoretic mobility and electric conductivity of a dilute suspension of identical impermeable spheres with low ζ potential were shown to give good


Figure 9. Plots of the dimensionless parameter B in eq 35b for the effective electric conductivity of a suspension of identical charged porous spheres versus the parameters κa and φ. The solid and dashed curves represent the calculations using the boundary conditions 12a and 13, respectively.

approximations for the case of reasonably high ζ potential (with an error of about 5% in a KCl solution and less than 2% in a HClO4 solution for the case of |ζ|e/kT ) 2).2,49 Therefore, our results might be used tentatively for the situation of reasonably high electric potentials. Acknowledgment. This research was partly supported by the National Science Council of the Republic of China. Supporting Information Available: Coefficients Cn in eq A2. This material is available free of charge via the Internet at http://pubs.acs.org.

Appendix: Some Functions in Section 3 For conciseness, the functions Fm(r) and Fu(r) in eq 24 and the variables An in eq 25 are defined here:

Fm(r) )

[

]

2 2 a3 -η zme a3 I (r) + χ rI3(b) + rJ0(r) 2 3kT 2 3 b ε(κa) r

()

(A1)

22054


Keh and Liu

( ar ) + C ( ar ) + C ( ar ) + 1 1 a 1 r J (r) - J (r) + ( )J (r) 15 ( a ) 3 3 r 1 a J (r) for a < r < b 15 ( r ) 3

Fu(r) ) C1 + C2

2

3

References and Notes

4

2

0

2

3

3

5

(A2a) a 2 [I (r) - ( ) I (r) + ( ar ) R(λr) + 3(λa) r

Fu(r) ) C5 + C6

3

3

2

0

3

3R(λr) 3β(λr) Iβ(r) - 3 3 IR(r)] for 0 < r < a λ3r3 λr

(A2b) A1 ) 3R(λa) + 2(λa)3 cosh(λa)

(A3a)

A2 ) 30R(λa) + (λa)2[2λa cosh(λa) - 12 sinh(λa)] (A3b) A3 ) 6R(λa) + (λa)2[λa cosh(λa) - 3 sinh(λa)] (A3c) A4 ) 90R(λa) + 6(λa)2[7λa cosh(λa) - 12 sinh(λa)] + 3(λa)4[λa cosh(λa) - 5 sinh(λa)] (A3d) where χ ) 2 when the condition eq 12a (Levine-Neale model) is used, and χ ) -1 when the condition eq 13 (Zharkikh-Shilov model) is used:

In(r) ) -

ε(κa)2 η

n dψeq1

∫0r ( ar )

dr

dr

(A4)

IR(r) ) -

ε(κa)2 η

∫0r R(λr)

dψeq1 dr dr

(A5)

Iβ(r) ) -

ε(κa)2 η

∫0r β(λr)

dψeq1 dr dr

(A6)

Jn(r) ) -

ε(κa)2 η

∫rb ( ar )

n dψeq1

dr

dr

(A7)

R(x) ) x cosh x - sinh x

(A8)

β(x) ) x sinh x - cosh x

(A9)

The expressions for the coefficients Cn are available in the Supporting Information.

(1) Dukhin, S. S.; Derjaguin, B. V. In Surface and Colloid Science; Matijevic, E., Ed.; Wiley: New York, 1974; Vol. 7. (2) O’Brien, R. W. J. Colloid Interface Sci. 1981, 81, 234. (3) Watillon, A.; Stone-Masui, J. J. Electroanal. Chem. 1972, 37, 143. (4) Saville, D. A. J. Colloid Interface Sci. 1983, 91, 34. (5) O’Brien, R. W. J. Colloid Interface Sci. 1983, 92, 204. (6) Ohshima, H.; Healy, T. W.; White, L. R. J. Chem. Soc., Faraday Trans. 2 1983, 79, 1613. (7) Hermans, J. J.; Fujita, H. Proc. K. Ned. Akad. Wet., Ser. B 1955, 58, 182. (8) Brinkman, H. C. Appl. Sci. Res., Sect. A 1947, 1, 27. (9) Liu, Y. C.; Keh, H. J. J. Colloid Interface Sci. 1997, 192, 375. (10) Levine, S.; Levine, M.; Sharp, K. A.; Brooks, D. E. Biophys. J. 1983, 42, 127. (11) Ohshima, H. J. Colloid Interface Sci. 1994, 163, 474. (12) Liu, Y. C.; Keh, H. J. Langmuir 1998, 14, 1560. (13) Ding, J. M.; Keh, H. J. Langmuir 2003, 19, 7226. (14) Levine, S.; Neale, G.; Epstein, N. J. Colloid Interface Sci. 1976, 57, 424. (15) Ohshima, H. J. Colloid Interface Sci. 1998, 208, 295. (16) Ohshima, H. J. Colloid Interface Sci. 2000, 229, 140. (17) Ding, J. M.; Keh, H. J. J. Colloid Interface Sci. 2001, 243, 331. (18) Carrique, F.; Arroyo, F. J.; Delgado, A. V. Colloids Surf., A 2001, 195, 157. (19) Lee, E.; Chou, K. T.; Hsu, J. P. J. Colloid Interface Sci. 2006, 295, 279. (20) Keh, H. J.; Chen, W. C. J. Colloid Interface Sci. 2006, 296, 710. (21) Masliyah, J. H. Electrokinetic Transport Phenomena; AOSTRA: Edmonton, Alberta, Canada, 1994. (22) Ohshima, H. J. Colloid Interface Sci. 1997, 188, 481. (23) Dukhin, A. S.; Shilov, V.; Borkovskaya, Yu. Langmuir 1999, 15, 3452. (24) Carrique, F.; Ruiz-Reina, E.; Arroyo, F. J.; Jimenez, M. L.; Delgado, A. V. Langmuir 2008, 24, 2395. (25) Ding, J. M.; Keh, H. J. J. Colloid Interface Sci. 2001, 236, 180. (26) Keh, H. J.; Ding, J. M. Langmuir 2002, 18, 4572. (27) Carrique, F.; Arroyo, F. J.; Jimenez, M. L.; Delgado, A. V. J. Phys. Chem. B 2003, 107, 3199. (28) Carrique, F.; Cuquejo, J.; Arroyo, F. J.; Jimenez, M. L.; Delgado, A. V. AdV. Colloid Interface Sci. 2005, 118, 43. (29) Cuquejo, J.; Jimenez, M. L.; Delgado, A. V.; Arroyo, F. J.; Carrique, F. J. Phys. Chem. B 2006, 110, 6179. (30) Wei, Y. K.; Keh, H. J. J. Colloid Interface Sci. 2002, 248, 76. (31) Hsu, J. P.; Lou, J.; He, Y. Y.; Lee, E. J. Phys. Chem. B 2007, 111, 2533. (32) Lou, J.; Lee, E. J. Phys. Chem. C 2008, 112, 12455. (33) Happel, J. AIChE J. 1958, 4, 197. (34) Kuwabara, S. J. Phys. Soc. Jpn. 1959, 14, 527. (35) Ohshima, H. J. Colloid Interface Sci. 2000, 225, 233. (36) Koplik, J.; Levine, H.; Zee, A. Phys. Fluids 1983, 26, 2864. (37) Matsumoto, K.; Suganuma, A. Chem. Eng. Sci. 1977, 32, 445. (38) Masliyah, J. H.; Polikar, M. Can. J. Chem. Eng. 1980, 58, 299. (39) Kawahata, S.; Ohshima, H.; Muramatsu, N.; Kondo, T. J. Colloid Interface Sci. 1990, 138, 182. (40) Morita, K.; Muramatsu, N.; Ohshima, H.; Kondo, T. J. Colloid Interface Sci. 1991, 147, 457. (41) Aoyanagi, O.; Muramatsu, N.; Ohshima, H.; Kondo, T. J. Colloid Interface Sci. 1994, 162, 222. (42) Neale, G.; Epstein, N.; Nader, W. Chem. Eng. Sci. 1973, 28, 1865. (43) Natraj, V.; Chen, S. B. J. Colloid Interface Sci. 2002, 251, 200. (44) Levine, S.; Neale, G. H. J. Colloid Interface Sci. 1974, 47, 520. (45) Happel, J.; Brenner, H. Low Reynolds Number Hydrodynamics; Nijhoff: Dordrecht, The Netherlands, 1983. (46) Zharkikh, N. I.; Shilov, V. N. Colloid J. USSR 1982, 43, 465. (47) Garcia-Salinas, M. J.; Romero-Cano, M. S.; de las Nieves, F. J. J. Colloid Interface Sci. 2001, 241, 280. (48) Lietor-Santos, J. J.; Fernandez-Nieves, A. AdV. Colloid Interface Sci. 2009, 147, 178. (49) O’Brien, R. W.; White, L. R. J. Chem. Soc., Faraday Trans. 2 1978, 74, 1607.

JP107697R

Electric Conductivity and Electrophoretic Mobility in Suspensions of

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