1560
Langmuir 1998, 14, 1560-1574
Electric Conductivity of a Dilute Suspension of Charged Composite Spheres Yung C. Liu and Huan J. Keh* Department of Chemical Engineering, National Taiwan University, Taipei 106-17, Taiwan, Republic of China Received August 8, 1997. In Final Form: December 15, 1997 An analytical study of the effective electric conductivity of a dilute suspension of charged composite particles, each composed of a solid core and a surrounding porous shell, in an electrolyte solution is presented. The model used for the porous shell of each composite particle is a solvent-permeable and ion-penetrable surface layer in which the density of hydrodynamic frictional segments, and therefore also that of the fixed charges, is constant. The equations which govern the electrochemical potential distributions of ionic species and the fluid flow field inside and outside the surface layer of a composite particle migrating in an unbounded solution are linearized assuming that the system is only slightly distorted from equilibrium. Using a perturbation method, these linearized equations are solved for a composite sphere in a uniform applied electric field with the charge densities of the rigid core surface and of the porous surface layer as the small perturbation parameters. An analytical expression for the effective conductivity of a dilute suspension of identical charged composite spheres is obtained from the average electric current density calculated using the solution of electrochemical potential distributions of the ions. The results demonstrate that the presence of the fixed charges in the composite particles can lead to an augmented or a diminished electric conductivity of the suspension relative to that of a corresponding suspension of uncharged composite particles, depending on the characteristics of the electrolyte solution and the suspending particles. In the limiting cases, the analytical solutions describing the effective electric conductivity of a dilute suspension of charged composite spheres reduce to those of dilute suspensions of charged solid spheres and of charged porous spheres.
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 determine the current density distribution and transport properties such as the electric conductivity, it is necessary to find out not only the local electric potential but also the local ionic densities and fluid velocity. That is, one must first solve a set of coupled electrokinetic equations, to obtain the distributions of electric potential, ionic concentrations, and fluid velocity in the electrolyte solution, and then compute the average electric current and conductivity in the suspension. 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 plane of the slab. Extending this analysis, Saville2 and O’Brien3 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 conductivity using a perturbation method for particles with low 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,4 who measured the surface * Person to whom correspondence should be addressed. (1) Dukhin, S. S.; Derjaguin, B. V. In Surface and Colloid Science; Matijevic, E., Ed.; Wiley: New York, 1974; Vol. 7. (2) Saville, D. A. J. Colloid Interface Sci. 1979, 71, 477. (3) O’Brien, R. W. J. Colloid Interface Sci. 1981, 81, 234.
conductances of a number of monodisperse polystyrene latices over a range of particle volume fractions. Later, Saville5 considered the effects of nonspecific adsorption, which alters the concentrations of ions in the solution outside the double layers, and of counterions derived from the particle charging processes and obtained better agreement between theories and experiments. Recently, an analytical expression for the effective conductivity of a dilute suspension of charged porous spheres with uniform densities of hydrodynamic frictional segments and fixed charges was derived by the present authors6 under the assumption that the density of the fixed charges is low. The basic equations governing the electric conductivity of a dilute suspension of colloidal particles also describe the electrophoretic phenomena. O’Brien7 derived analytical formulas for the electrophoretic mobility and the electric conductivity of a dilute suspension of dielectric spheres with thin but polarized double layers in a general electrolyte solution. Using a similar analysis, O’Brien and Ward8 also determined the electrophoretic mobility and the effective conductivity of a dilute suspension of randomly oriented spheroids with thin polarized diffuse layers at the particle surfaces. On the other hand, approximate analytical expressions for the electrophoretic mobility and the conductivity of dilute suspensions of colloidal spheres in symmetric electrolytes were obtained by Ohshima et al.9 These expressions are correct to order (κa)-1, where κ is the Debye-Huckel parameter (defined (4) Watillon, A.; Stone-Masui, J. J. Electroanal. Chem. 1972, 37, 143. (5) Saville, D. A. J. Colloid Interface Sci. 1983, 91, 34. (6) Liu, Y. C.; Keh, H. J. J. Colloid Interface Sci. 1997, 192, 375. (7) O’Brien, R. W. J. Colloid Interfrace Sci. 1983, 92, 204. (8) O’Brien, R. W.; Ward, D. N. J. Colloid Interface Sci. 1988, 121, 402. (9) Ohshima, H.; Healy, T. W.; White, L. R. J. Chem. Soc., Faraday Trans. 2 1983, 79, 1613.
S0743-7463(97)00893-7 CCC: $15.00 © 1998 American Chemical Society Published on Web 02/18/1998
Dilute Suspension of Charged Composite Spheres
by eq 34) and a is the particle radius. When the ζ potential of the particles is small, their reduced result is in agreement with O’Brien’s.3 In many practical applications, the electric conductivity of a suspension is known from direct measurement and then the ζ potential of dielectric particles in the suspension can be calculated. Similarly, one can also measure the electrophoretic mobility of a particle in order to get the ζ potential. O’Brien and Perrins10 derived a formula for the electric conductivity of a porous plug composed of closely packed spheres and compared it with the conductivity data for dilute and concentrated dispersions of monodisperse polystyrene particles reported by van der Put and Bijsterbosch.11 They found significant differences between the ζ potentials evaluated from measurements of the electric conductivity and of the electrophoretic mobility. Similar differences were also found by another work12 in which the conductivities and electrophoretic mobilities of polystyrene latex systems were measured. On the other hand, Stigter13 developed a theory based on the concepts used to describe the conductivity of strong electrolyte solutions, in which the specific conductance of the suspension was computed by summing the individual contributions of the particle-ion interactions expressed in terms of equivalent conductances. The differences between the kinetic charges calculated from electrophoresis and from conductance in this theory were found to be small and within the errors of the experiments and the theoretical models.14 Theoretical investigations of the electrokinetic phenomena of colloidal particles covered by charged porous surface layers have been performed for many years.15-20 These investigations 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 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,21 rat lymphocytes,22 and latex particles coated with poly(N-isopropylacrylamide) hydrogel layers.23 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. Later, a general expression for the electrophoretic mobility of a spherical composite particle was derived by Ohshima,24 neglecting the relaxation effect of the double layer. It has been found that the electrophoretic mobility (10) O’Brien, R. W.; Perrins, W. T. J. Colloid Interface Sci. 1984, 99, 20. (11) Van Der Put, A. G.; Bijsterbosch, B. H. J. Colloid Interface Sci. 1980, 75, 512. (12) Zukoski, C. F.; Saville, D. A. J. Colloid Interface Sci. 1985, 107, 322. (13) Stigter, D. J. Phys. Chem. 1979, 83, 1663. (14) Stigter, D. J. Phys. Chem. 1979, 83, 1670. (15) Jones, I. S. J. Colloid Interface Sci. 1979, 68, 451. (16) Wunderlich, R. W. J. Colloid Interface Sci. 1982, 88, 385. (17) Levine, S.; Levine, M.; Sharp, K. A.; Brooks, D. E. Biophys. J. 1983, 42, 127. (18) Sharp, K. A.; Brooks, D. E. Biophys. J. 1985, 47, 563. (19) Ohshima, H.; Kondo, T. J. Colloid Interface Sci. 1989, 130, 281. (20) Hsu, J. P.; Fan, Y. P. J. Colloid Interface Sci. 1995, 172, 230. (21) Kawahata, S.; Ohshima, H.; Muramatsu, N.; Kondo, T. J. Colloid Interface Sci. 1990, 138, 182. (22) Morita, K.; Muramatsu, N.; Ohshima, H.; Kondo, T. J. Colloid Interface Sci. 1991, 147, 457. (23) Makino, K.; Yamamoto, S.; Fujimoto, K.; Kawaguchi, H.; Ohshima, H. J. Colloid Interface Sci. 1994, 166, 251.
Langmuir, Vol. 14, No. 7, 1998 1561
of a charged composite sphere can be quite different from that of a “bare” rigid sphere. Recently, analytical expressions for the sedimentation velocity and potential in a dilute suspension of charged composite spheres were obtained by the present authors25 under the situation that the electric potentials are low. However, the effects of particle charges on the effective conductivity of a suspension of composite particles have not yet been theoretically examined. In this paper, we analytically study the effective conductivity of a suspension of charged composite particles. The densities of fixed charges and hydrodynamic frictional segments in the porous surface layer of each particle are assumed to be uniform, but no assumption is made as to the thickness of the electric double layer relative to the dimension of the particle. The suspension is sufficiently dilute that the suspended particles occupy only a small fraction of the total volume of the suspension and the double layer surrounding each particle does not overlap with the others. In Section 2, the average electric current density in a dilute suspension of identical charged particles is given as an integral over a large surface enclosing a single particle and its adjacent double layer, and the effective conductivity of the suspension is related to the electrochemical potential energies of the ionic species. In Section 3, we present the fundamental electrokinetic equations and boundary conditions which govern the electrochemical potential distributions, the electrostatic potential profile, and the fluid flow field inside and outside the permeable surface layer of a composite particle migrating in an unbounded solution when a constant electric field is applied. These basic equations are linearized assuming that the ionic concentrations, the electric potential, and therefore the electrochemical potentials have only slight deviations from equilibrium due to the imposed field. The axisymmetric electrophoretic motion of a charged composite sphere in an unbounded electrolyte solution is considered in Section 4. Using the Debye-Huckel approximation, we first get the solution of the equilibrium electric potential distribution. Then the linearized electrokinetic equations are transformed into a set of differential equations by using a regular perturbation method with the fixed charge densities of the surface layer and of the rigid core surface as the small perturbation parameters. The perturbed electrochemical potentials of ions and the fluid velocity are determined by solving this set of differential equations subject to the appropriate boundary conditions. An analytical expression for the electric conductivity of a dilute suspension of identical charged composite spheres is obtained. Finally, analytical expressions in two limiting cases and typical numerical results for the effects of the fixed charges of composite spheres on the effective conductivity of the suspension are presented in Section 5. 2. Average Current Density in a Suspension of Charged Particles We consider a dilute suspension of identical charged particles immersed in a solution containing M ionic species. The particles may be impermeable to the fluid, porous, or composite. It is assumed that the suspension is statistically homogeneous, and all effects of its boundaries are ignored. When the isotropic suspension is subjected to a uniform applied electric field E∞, one has (24) Ohshima, H. J. Colloid Interface Sci. 1994, 163, 474. (25) Keh, H. J.; Liu, Y. C. J. Colloid Interface Sci. 1997, 195, 169.
1562 Langmuir, Vol. 14, No. 7, 1998
E∞ ) -
1 V
∫V∇ψ dx
Liu and Keh
(1)
where ψ(x) is the electric potential field at position x and V denotes a sufficiently large volume of the suspension to contain many particles. There is a resulting volumeaverage current density, which is collinear with E∞, defined by
1 〈i〉 ) V
∫Vi dx
(
∞ n(eq) m ) nm exp -
(2)
where i(x) is the current density distribution. The effective electric conductivity Λ of the suspension can be assigned by the linear relation
〈i〉 ) ΛE∞
distributions of the concentration of species m, the electric potential, and the electrochemical potential energy of species m, respectively, and δnm(x), δψ(x), and δµm(x) are the small perturbations to the equilibrium state (in which no external field is applied). The equilibrium concentration of any species is related to the equilibrium potential by the Boltzmann distribution,
δnm δµm ) kT (eq) + zmeδψ nm
∑
(
M
zmeJm
i)
(4)
Jm ) nmu - nm
∑
zmen(eq) m u -
m)1
m)1
where Jm(x) and zm are the number flux distribution and the valence of species m, respectively, and e is the charge of a proton. If the solution is dilute, the flux Jm is given by
Dm ∇µ kT m
µm ) µ°m + kT ln nm + zmeψ
(6)
Here, u(x) is the fluid velocity field, nm(x) is the concentration (number density) distribution of species m, Dm is the diffusion coefficient of species m, which is assumed to be constant both inside and outside the the particles if they have porous shells, k is the Boltzmann constant, T is the absolute temperature, and µ°m is a constant. The first term on the right-hand side of eq 5 represents the convection of the ionic species by the fluid, and the second term denotes the diffusion and electrically induced migration of the ions. The linear relation (eq 3) between the volume-average current density and the electric field for a suspension will be exact in the limit as E∞ f 0. To calculate the effective conductivity of the suspension, we may assume that the intensity of the applied electric field is not high and hence that the electric double layer surrounding each particle is only slightly distorted from equilibrium by the application of the field. Therefore, the concentration distribution of each ionic species and the electric potential distribution have small deviations from equilibrium, and one can write
nm ) n(eq) m + δnm
(7a)
ψ ) ψ(eq) + δψ
(7b)
µm ) µ(eq) m + δµm
(7c)
(eq)(x), and µ(eq) are the equilibrium where n(eq) m (x), ψ m
)
Dm ∇δµm kT
(10)
Far from any particle (beyond the double layer), n(eq) m f n∞m, and eq 10 becomes
(
)
zmen∞m ∇δψ zmeDm ∇δnm + ifkT m)1 M
∑
(5)
with the electrochemical potential energy field of the mth species µm(x) defined as9
(9)
Substituting eqs 5-9 into eq 4, using the fact that ∇µ(eq) m ) 0, and neglecting products of the small perturbation quantities, u, δnm, and δµm, one has
M
i)
(8)
where n∞m is the constant bulk concentration of type m ions. The perturbation quantity δµm is linearly related to the others by
(3)
Since the measured electric field and current density are equal to E∞ and 〈i〉, respectively, eq 3 reduces to the usual experimental definition of conductivity, provided that the suspension is everywhere homogeneous. The current density i can be written as
)
zmeψ(eq) kT
(11)
By adding and subtracting the current density given by the above equation in the integrand of eq 2, one obtains
(
)
zmen∞m ∇δψ dx + 〈i〉 ) V ∇δnm + V kT m)1 M zmen∞m 1 ∇δψ dx (12) zmeDm ∇δnm + V i + V kT m)1 M
∑
∫
zmeDm
[
∫
∑
(
)]
Note that the magnitude of i and Dm can be taken as zero inside the dielectric rigid core of each composite sphere. In a statistically homogeneous suspension with constant bulk ionic concentrations, the volume average of ∇δnm is zero. According to the definition of eq 1, the first term on the right-hand side of eq 12 equals Λ∞E∞, where ∞
Λ )
M
∑
z2me2n∞mDm
m)1
kT
(13)
which is the electric conductivity of the electrolyte solution in the absence of the particles. The integral in the second term on the right-hand side of eq 12 can be calculated by first considering it for a single particle as if the others were absent and then multiplying the result by the particle number N in volume V, since the integrand vanishes beyond the double layers surrounding the particles and the suspension is assumed to be sufficiently dilute that the double layers do not overlap with one another. Also, the volume integral can be transformed into a surface integral over a spherical boundary of infinite radius containing the single particle at its center. With this arrangement, the second term becomes
Dilute Suspension of Charged Composite Spheres
(
Langmuir, Vol. 14, No. 7, 1998 1563
)
zmen∞mDm δµmn dS ) rf∞ V kT m)1 ∞ N M zmenmDm rf∞(n‚∇δµmr - δµmn) dS (14) V m)1 kT
N
∫
M
n‚ir +
∑
∫
∑
where r is the position vector relative to the particle center, r ) |r|, and n is the unit vector outwardly normal to the surface of the boundary. To obtain eq 14, the requirement of the conservation of electric charge (∇‚i ) 0) and eq 11 have been used. Therefore, the average current density given by eq 12 can be expressed as
〈i〉 ) Λ∞E∞ N
M
∑
respectively, of the hydrodynamic frictional segments in the surface layer. The local electric potential ψ and the space charge density are related by Poisson’s equation:
zmen∞mDm
V m)1
kT
∫rf∞(r∇δµm‚n - δµmn) dS
(15)
The determination of δµm in the above equation is concerned with the solution of a set of basic electrokinetic equations for the electrolyte around a single particle. These electrokinetic equations for the case of a charged composite particle are described in the next section, and their analytical solution for a charged composite sphere with low fixed charge densities is presented in Section 4. From this solution we shall derive the effective conductivity of a dilute suspension of identical composite spheres. 3. Basic Electrokinetic Equations for a Charged Composite Particle in an Electric Field In this section we consider a charged composite particle of arbitrary shape in an unbounded liquid solution containing M ionic species when a constant external electric field E∞ is applied. The definition of a charged composite particle is a charged rigid particle core covered by a surface layer of charged porous substance or adsorbed polyelectrolytes in equilibrium with the surrounding electrolyte solution. The porous surface layer is treated as a solvent-permeable and ion-penetrable homogeneous shell in which fixed-charged groups are assumed to distribute at a uniform density. Conservation of all species, which do not react with one another, in the steady state requires that
∇‚Jm ) 0, m ) 1, 2, ..., M
(16)
Here, the species fluxes Jm(x) are defined by eq 5, taking u as the fluid velocity distribution relative to the particle. We assume that the Reynolds number of the fluid motion is very small, so 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
(17)
m)1
∇‚u ) 0
(18)
where η is the viscosity of the fluid, f is the hydrodynamic friction coefficient inside the porous surface layer per unit volume of the fluid, p(x) is the dynamic pressure distribution, and h(x) is a unit step function which equals unity if x is inside the surface layer and zero if x is outside the composite particle. In eq 17, 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,
∇2ψ ) -
4π M [ zmenm + h(x)Q] m)1
∑
(19)
Here, Q is the fixed charge density inside the surface layer of the composite particle and ) 4π0r, where r is the relative permittivity of the electrolyte solution, which is assumed to be the same inside and outside the surface layer, and 0 is the permittivity of free space. Note that the space charge density in the surface layer is the sum of the densities of the mobile ions and the fixed charges. If the strength of the imposed electric field is weak, the deviations in the ionic concentrations and the electric potential from equilibrium are so slight that eqs 16 and 17 can be linearized. It has been found that the perturbed electrochemical potential energies δµm and the fluid velocity u satisfy the following set of electrokinetic equations:7,9 2 (eq) n(eq) m ∇ δµm + ∇nm ‚∇δµm )
kT (eq) ∇n ‚u Dm m
(20)
M
η∇2∇ × u - h(x)f∇ × u )
∑ ∇n(eq) m × ∇δµm m)1
(21)
Substituting eq 8 into eqs 21 and 22, one has
∇2δµm )
(
zme kT (eq) ∇ψ(eq)‚∇δµm ∇ψ ‚u kT Dm
)
(22)
∇2∇ × u - h(x)λ2∇ × u ) ∞ zmeψ(eq) 1 M zmenm exp ∇ψ(eq) × ∇δµm (23) ηm)1 kT kT
∑
[
(
)
]
where λ ) (f/η)1/2 and the solution of ψ(eq) will be given in the following section. Note that the reciprocal of the parameter λ is the shielding length characterizing the extent of flow penetration inside the surface charge layer of the particle. The boundary conditions at the surface of the nonconducting rigid particle core are
n‚∇δµm ) 0
(24a)
u)0
(24b)
Equation 24a results from the fact that ions and fluid cannot penetrate into the rigid core. In eq 24b, we have assumed that the shear plane coincides with the surface of the rigid particle. The boundary conditions at the surface of the composite particle (the boundary between the surface layer and the external solution, S() are
δµm|S+ ) δµm|S-
(25a)
∇δµm|S+ ) ∇δµm|S-
(25b)
u|S+ ) u|S-
(25c)
n‚σ|S+ ) n‚σ|S-
(25d)
1564 Langmuir, Vol. 14, No. 7, 1998
2
∇ψ
(eq)
)-
[∑
(
M
4π
zmen∞m m)1
exp -
Liu and Keh
)
zmeψ(eq) kT
+ h(r)Q
]
(29)
where h(r) equals unity if r0 < r < a and is zero if r > a. The appropriate boundary conditions for the equilibrium potential are
4πσ dψ(eq) | )dr r)r0
(30a)
ψ(eq)|r)a+ ) ψ(eq)|r)a-
(30b)
dψ(eq) dψ(eq) |r)a+ ) | dr dr r)a-
(30c)
ψ(eq)|rf∞ f 0
(30d)
Figure 1. Geometrical sketch for a charged composite sphere under an applied electric field.
Here, σ is the hydrodynamic stress of the fluid given by
σ ) -pI + η[∇u + (∇u)T]
(26)
where I is the unit dyadic. Equations 25a and 25b indicate that the concentration and flux of species m and the electric potential must be continuous. Equations 25c and 25d are the continuity requirement of the fluid velocity and stress tensor at the particle surface. The conditions far from the particle are
u ) -µEE∞
(27a)
where σ is the surface charge density of the “bare” particle core. Equation 30a states that the Gauss condition holds at the surface of the rigid core. The solution to eq 29 satisfying eq 30 is
j + ψeq10(r)Q h + O(σ j 2, σ jQ h, Q h 2) ψ(eq) ) ψeq01(r)σ
(31)
Here,
nm f
n∞m
ψ f -E∞‚x
(27b)
4πeσ κkT
(32a)
Q h )
4πeQ κ2kT
(32b)
(27c) and
Here, µE is the electrophoretic mobility of a charged composite particle, and its expression for a composite sphere will be given in the next section. Because the equilibrium electric potential and concentration of type m ions in the bulk solution have been set equal to zero and n∞m, respectively, from eqs 7, 9, 27b, and 27c, one obtains the boundary condition for δµm at large distances from the particle:
δµm f -zmeE∞‚x
σ j)
which are the nondimensional charge densities of the rigid core surface and of the porous surface layer, respectively, of the composite particle, and
ψeq01 )
(28)
Equations 17, 18, and 27a take a reference frame that the composite particle is at rest and the velocity of the fluid at infinity is the particle (electrophoretic) velocity in the opposite direction.
ψeq10 )
4. Solution for the Conductivity of a Dilute Suspension of Charged Composite Spheres
ψeq10 )
We now consider a charged composite sphere of radius a immersed into an unbounded electrolyte solution with constant bulk ionic concentrations under a uniform applied electric field E∞. As illustrated in Figure 1, the composite sphere has a surface 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. We first seek the solution of ψ(eq) which appears in the governing equations (eqs 22 and 23) for a charged composite sphere. Substituting the Boltzmann distribution (eq 8) into Poisson’s equation (eq 19) at equilibrium, one can get the equilibrium Poisson-Boltzmann equation,
( )
)
kT κr0 r0 -κ(r-r0) e e 1 + κr0 r
{ (
(33a)
kT 1 e-κd 1- 1+ [κr cosh(κr - κr0) + e κa 1 + κr0 0 sinh(κr - κr0)]
{ (
)
}
a , if r0 < r < a (33b) r
kT 1 e-κd 1- 1+ [κr cosh(κd) + e κa 1 + κr0 0 sinh(κd)]
}
a -κ(r-a) e , if r > a (33c) r
where κ is the reciprocal of the Debye screening length, defined by
κ)
(
4πe2
M
∑
)
1/2
z2mn∞m
kT m)1
(34)
Note that ψ(eq) is a function of r only and the solution ψeq01(r) takes the same form in both the regions r0 < r < a and r > a. Expression 31 for ψ(eq) as a power series in the fixed charge densities of the composite sphere up to O(σ j, Q h ) is the equilibrium solution for the linearized Poisson-Boltzmann equation that is valid for small values
Dilute Suspension of Charged Composite Spheres
Langmuir, Vol. 14, No. 7, 1998 1565
of the electric potential (the Debye-Huckel approximation). That is, the charge densities σ and Q of the composite particle must be small enough for the potential to remain small. To solve the small quantities δµm, u, and µE for the case of the small parameters σ j and Q h , these variables can be written as perturbation expansions in powers of σ j and Q h,
( )
δµm ) -zmeE∞ r +
r30 2
cos θ + µm01σ j + µm10Q h + 2r j 2 + µm11σ jQ h + µm20Q h 2 + ... (35a) µm02σ
j + u10Q h + u02σ j 2 + u11σ jQ h + u20Q h 2 + ... u ) u01σ (35b)
where
[ ∫( ) ∫( ∫ )
z2me2 r30 ∞ r30 dψeqij dr + 1 3kT 2r2 r0 r3 dr r30 dψeqij dψeqij 1 r 1- 3 dr + 2 r (r3 - r30) dr (40) dr dr r r 0
Fmij(r) ) r
∞
r
(ar)
∇2µijm ) -
[ ( )]
r0 z2me2E∞ 1kT r
∇2∇ × uij - h(r)λ2∇ × uij )
1
M
∑
3
dψeqij cos θ (36a) dr
z2me2n∞mE∞
ηm)1
[( ) ]
kT r30 cos θ (36b) ∇ r+ 2r2
where (i, j) equal (0, 1) and (1, 0). The boundary conditions for µmij and uij are
r ) r0: n‚∇µmij ) 0 uij ) 0 r ) a: µmij and uij are continuous ∇µmij and n‚σij are continuous r f ∞: µmij f 0 uij f -µEijE∞ez
3
(37a) (37b) (37c) (37d) (37e) (37f)
The solutions for µmij and the r and θ components of uij subject to eqs 36 and 37 are
µmij ) E∞Fmij(r) cos θ
(38)
uijr ) E∞Fijr(r) cos θ
(39a)
uijθ ) E∞Fijθ(r) sin θ
(39b)
3
1
ij4
3
r
2
ij
a
1
3
r
ij
a
6R1(λr)
r β1(λr)Gij(r) dr ∫ (λa) (λr) 2
3 a
6β1(λr)
r R1(λr)Gij(r) dr ∫ (λa) (λr) 2
3 a
(41a)
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 a rr3 Gij(r) dr β1(λr)Gij(r) dr + 2 3 a a λr (λa)2(λr)3 a 3β2(λr) r R1(λr)Gij(r) dr, if r0 < r < a (41b) (λa)2(λr)3 a
()
Fijθ(r) ) -Cij1 +
() ∫
()
∫(
∫
)
∫
(ar) + C ar + C + C (ar) + 1a a r r G (r) dr - ∫ ( ) G (r) dr + 5( r ) ∫ (a) r a ∫ (ar ) G (r) dr - 51(ar ) ∫ G (r) dr (41c) 3
Fijr(r) ) Cij5 3
2
ij6
r
ij7
5
ij8
r
ij
a
∇ψeqij ×
+
(ar) R (λr) + C (ar) β (λr) + a r 2 G (r) dr - ( ) ∫ ( ) G (r) dr] + ∫ [ r a (λa) Cij3
j + µE10Q h + µE02σ j 2 + µE11σ jQ h + µE20Q h 2 + ... µE ) µE01σ (35c) where the functions µmij, uij, and µEij with i and j equal to 0, 1, 2, ... are independent of σ j and Q h . The zeroth-order terms of u and µE disappear because an uncharged particle will not move by applying an electric field. Substituting the expansions given by eq 35 and ψ(eq) given by eq 33 into the linearized governing equations (eqs 22 and 23) and the boundary conditions (eqs 24, 25, 27a, and 28) and equating like powers of σ j and Q h on both sides of the respective equations, we obtain a set of differential equations and boundary conditions for each set of the functions µmij and uij with i and j equal to 0, 1, 2, .... After collecting the first-order (σ j and Q h ) terms in the perturbation procedure, one obtains the following equations:
3
Fijr(r) ) Cij1 + Cij2
]
r
a
3
ij
a
2
2
ij
r
a
ij
Cij5 a 3 Cij6 a r2 - Cij7 - 2Cij8 + 2 r 2 r a 3 5 3 r r 1 a a rr G (r) dr + G (r) dr 10 r a a ij 2r a a ij r r 2 2r2 r Gij(r) dr + G (r) dr, if r > a (41d) a a 5 a a ij
() ( )∫( ) ∫( )
Fijθ(r) )
()
()
∫(
)
( )∫
In eq 41,
Gij(r) ) -
[ ( )]
r0 (κa)2 2+ 24πη r
3
dψeqij dr
(42)
R1(x) ) x cosh x - sinh x
(43a)
R2(x) ) (x2 + 1) sinh x - x cosh x
(43b)
β1(x) ) x sinh x - cosh x
(43c)
β2(x) ) (x2 + 1) cosh x - x sinh x
(43d)
and the constants Cij1, Cij2, ..., Cij8 are given by eqs A1-A8 in the Appendix. Note that the functions Fmij(r) in eq 40 take the same form in the regions r0 < r < a and r > a. According to a characteristic of electrophoretic motion, the velocity field far from the particle (beyond the double
1566 Langmuir, Vol. 14, No. 7, 1998
Liu and Keh
layer) has the form9,26
Km11(r) )
u f -µEE∞ez + O(r-2)
(44)
Equation 44 satisfies the requirement that the net force on a large surface enclosing the particle and its adjacent double layer must be zero. Using eqs 35c, 39, 41, 44, and A6, one can get the first-order term for the electrophoretic mobility of a charged composite sphere expressed as
∫a (
1 C µEij ) C006 001
[
C003
∫a Gij(r) dr +
r3 G (r) dr + C002 a ij
r0
)
r0
∫ar R1(λr)Gij(r) dr + C004∫ar β1(λr)Gij(r) dr ∞ ∞ r 2 C005∫a Gij(r) dr - C006∫a ( ) Gij(r) dr + a 3 ∫a∞(ar ) Gij(r) dr] (45) 0
0
where the constants C001, C002, ..., C005, and C006 are given by eqs A11-A16 in the Appendix. Among the second-order terms in the perturbation procedure, the only distributions we need in the following calculations are the electrochemical potential energies µm02, µm11, and µm20. If the solution contains only a symmetrically charged, binary electrolyte, the equations governing µm02, µm11, and µm20 are
∇2µm02 )
∇2µm11 )
(
)
zme kT ∇ψeq01‚ ∇µm01 u kT Dm 01
[
(
)
)]
(
)
zme kT ∇ψeq10‚ ∇µm10 ∇2µm20 ) u kT Dm 10
[
0
∫
1 r2
r 3 r Kmij(r) r0
)
dFm10 kT dψeq10 F10r dr Dm dr
(48c)
[( )
1 - γcp 3 r0 φ 1 - p 2 a
3
+ Λ01κaσ j+
h + Λ02(κa)2σ j 2 + Λ11(κa)3σ jQ h + Λ20(κa)4Q h2 + Λ10(κa)2Q j 2Q h, σ jQ h 2, Q h 3) O(σ j 3, σ
]}
(49)
Here, p is the porosity of the surface layer of the composite particle, γc ()4πr30/3Vt, where Vt is the dry volume of a composite particle) is the volume fraction of the rigid core in a composite sphere, φ ()NVt/V) is the true volume fraction of the dry composite particles, M
z3mn∞mDm ∑ m)1 M
Xij
(50)
∑ z2mn∞mDm
m)1
(46c)
]
dr cos θ (47)
dFm01 kT dψeq01 F01r dr Dm dr
)
)
with (i, j) equal to (0, 1) and (1, 0) for a general electrolyte, and
where
(
{
Λ ) Λ∞ 1 -
Λij ) Λij ) -
z(D+ - D-) X , if (i, j) ) (0, 1) or (1, 0) (51a) D+ + D- ij k2T2 Yij - z2Zij, 2πη(D+ + D-)e2 if (i, j) ) (0, 2), (1, 1), or (2, 0) (51b)
for a symmetric electrolyte (M ) 2, z+ ) -z- ) z, n∞+ ) n∞- ) n∞, Λ∞ ) (D+ + D-)z2e2n∞/kT, where the subscripts + and - refer to the cation and anion, respectively). In eqs 50 and 51,
(
X01 ) Φ κr0,
∞
∞
(
(
In eq 48, the functions Fmij(r), Fijr(r), and ψeqij(r) with (i, j) equal to (0, 1) and (1, 0) are given by eqs 40, 41, and 33, respectively. By substituting eqs 35a, 38, and 47 into eq 15, making relevant calculations, and then comparing the result with eqs 3, the effective conductivity in a dilute suspension of identical charged composite spheres is obtained as
(46b)
∫r Kmij(r) dr + r∫r Kmij(r) dr +
Km02(r) )
Km20(r) )
Λij )
The boundary conditions for µmij with (i, j) equal to (0, 2), (1, 1), and (2, 0) are also given by eq 37. For a general electrolyte there are extra terms on the right-hand side jQ h, Q h 2) corrections to the of eq 46 involving the O(σ j 2, σ equilibrium potential as expressed by eq 31. These extra terms considerably complicates the problem. So we consider here only the case of a symmetric electrolyte, in jQ h, Q h 2) terms in eq 31 vanish and the which the O(σ j 2, σ j + ψeq10Q h and O(σ j 3, σ j 2Q h, σ jQ h 2, leading corrections to ψeq01σ Q h 3). The solution for the second-order perturbation µmij is
zmeE∞ r30 µmij ) 3kT 2r2
)
dFm01 kT dψeq10 + F01r dr Dm dr dFm10 kT dψeq01 (48b) F10r dr Dm dr
(46a)
zme kT ∇ψeq01‚ ∇µm10 u + kT Dm 10 kT ∇ψeq10‚ ∇µm01 u Dm 01
(
(
(48a)
(26) O’Brien, R. W.; Hunter, R. J. Can. J. Chem. 1981, 59, 1878.
)
r0 a
{ () () ( )( ) ∫ () ( )
(52a)
1 1 r0 3 1 r0 6 1 + 2 a 2 a (κa)2 4 aa 1 e-κd 3 r0 6 1+ [κr0 cosh(κr - κr0) + 5 r 2 a κa 1 + κr0 0 r 3 r0 6 κa sinh(κr - κr0)] dr e E5(κa) 1 2 a
X10 )
1+
[
]}
1 e-κd (κr cosh(κd) + sinh(κd)) κa 1 + κr0 0
(52b)
Dilute Suspension of Charged Composite Spheres
Y02 ) Y11 )
4πηe2 1 k2T2 (κa)2
∫r∞W(r)F01r(r)
4πηe2 1 k2T2 (κa)3
0
[
dψeq01 dr (53a) dr
dψeq10 + dr dψeq01 dr (53b) F10r(r) dr
∫r∞W(r) F01r(r)
Y20 ) Z02 ) -
0
4πηe2 1 k2T2 (κa)4
]
∫r∞W(r)F10r(r) 0
∫r∞W(r)
1 2 z kT(κa)2
Z11 )
0
∞
dF+01 dψeq01 dr dr dr
(
(53d)
)
{ [ (
Λ ) Λ∞ 1 - φ
Φ(x, y) )
4
() ( ) [ r a
3
+
1 r0 2 a
3
3y 1 1 1 x + + e E (x) 1 + x x x2 2 5
En(x) )
∫
∞ -n -xt t e 1
dt
(54a)
]
{ [ ][ ( ) 3+
[
(54c)
j jQ hi Note that eq 49 is correct to order φ, that (κa)2i+jσ with i and j equal to 0, 1, and 2 is indepdndent of κ or n∞, jQ h , and Q h 2 terms are limited and that the results of the σ j 2, σ to a symmetric electrolyte. The coefficients Λ01 and Λ10, which are independent of the shielding parameter λ, disappear for a symmetry electrolyte when the diffusivities of the cation and anion take the same value. As to the coefficients Λ02, Λ11, and Λ20 for a symmetric electrolyte, the first term of the right-hand side of eq 51b (which is a function of the parameters κa and λa) denotes the effect due to the convection of the fluid, while the second term (which is independent of λa) represents the effect due to the deviations of the electrochemical potential distributions from their combined equilibrium and applied values. It is understood that the result given by eqs 49-53 is only valid with the requirements that φ , 1 and κaφ-1/3 . 1. 5. Results and Discussion In this section, we first consider two limiting cases of expression 49 for the effective conductivity in a dilute suspension of identical charged composite spheres. The correctness of this expression may be confirmed by examining these limiting cases for which analytical solutions are already known. Then, numerical results of the parameters Xij (with i + j ) 1), Yij, and Zij (with i + j ) 2) in association with the coefficients Λij by eqs 50 and 51 for the general cases will be presented. Note that Xij and Zij depend on the parameters κa and r0/a only, while Yij are functions of κa, λa, and r0/a. Although the parameters Xij, Yij, and Zij are positive values, the coefficients Λ01 and Λ10 can be either positive or negative depending on the diffusion coefficients of the ionic species
]
15 κa e E6(κa) + 2κa
1 3κa E (κa)(E4(κa) - E6(κa)) + E6(2κa) 4 5 2
]}
(56a)
and
{
3 1 5 + (1 + κa)2 4κa 2(κa)2 6 1 6 3 + eκaE5(κa) - e2κa (E5(κa))2 + E6(2κa) κa (κa)2 2 4 (56b)
Z02 ) (54b)
(55)
1 2 1 + + κa (1 + κa)2 2κa (κa)2 κa 2 κa e (E3(κa) - E5(κa)) + e E (κa) 1 + 5 4 (κa)2
Y02 )
e2κa W(r) )
) ]}
Here, the function Φ is defined by eq 54a;
∞
where
3 z(D+ - D-) + κaΦ(κa, 1)σ j2 D+ + D-
k2T2 Y + z2Z02 (κa)2σ j2 2 02 2πη(D+ + D-)e
dF+10 dψeq10 dr (53f) W(r) r0 dr dr
∫
1 Z20 ) - 2 z kT(κa)4
in the electrolyte solution, and the coefficients Λ02, Λ11, and Λ20 are always negative. When there is no permeable layer on the surface of each rigid particle core in the suspension, one has d ) 0, r0 ) a, γc ) 1, Q ) 0, and λ ) 0. Then, eq 49 for a symmetric electrolyte reduces to
dψeq10 dr (53c) dr
dF+01 dψeq10 dF+10 dψeq01 + W(r) dr r0 dr dr dr dr (53e)
∫
1 - 2 z kT(κa)3
Langmuir, Vol. 14, No. 7, 1998 1567
[
]
[
]}
where the function En(x) is given by eq 54b. From eq 31 by letting r0 ) a, one can get the following relation between the surface potential and the surface charge density of a rigid sphere at equilibrium:
σ)
(κa + 1) (eq) ψ (a) 4πa
(57)
Substituting the above equation into eq 55, this degenerated result is the same as that of a dilute suspension of identical rigid spheres with low surface potential obtained by O’Brien.3 Note that there is a typographic error in eq 5.34 of O’Brien’s paper. When the composite particles are homogeneous porous spheres, one has r0 ) 0, d ) a, γc ) 0, and σ ) 0. In this limiting case, eq 49 for a symmetric electrolyte becomes
{
Λ ) Λ∞ 1 -
(
[
φ z(D+ - D-) Q h 1 - p D+ + D2
) ]}
k T2 Y20 + z2Z20 (κa)4Q h2 2πη(D+ + D-)e2
(58)
where
2(κa)2 + 3κa + 3 κa + 1 -κa R1(κa)e-κa e × 7 3(κa) 3(κa)5 2 3 (1 + κa) 3 κa + 2 2 -2κa [R (κa)] e × sinh(κa) 4 (κa)8 1 4 (κa)8
Z20 )
e-2κa[R1(κa) sinh(κa) - sinh2(κa) + (κa)2] (59a)
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Liu and Keh
Figure 2. Plot of the parameters X01 and X10 in eqs 50 and 51a for the dimensionless coefficients Λ01 and Λ10 versus κa at fixed values of r0/a.
1 2 + κa + [R1(κa)]2e-2κa 2 (λa) (κa) (κa)8 (1 + κa)2 -2κa κ2 e [R1(κa) sinh(κa) λ2 - κ2 (κa)8 2 2λ2κ2 (1 + κa) sinh2(κa) + (κa)2] + 2 R1(κa)e-2κa × (λ - κ2)2 (κa)8 2 λ R1(κa) sinh(λa) - sinh(κa) (59b) κ R1(λa)
Y20 )
(
2
)
[( )
]
and the function R1(x) is defined by eq 43a. Equation 58 is identical to the formula for the effective conductivity in a dilute suspension of charged porous spheres previously derived.6 To use the general expression (eq 49) for the effective conductivity in a dilute suspension of identical charged composite spheres and its simplified formulas (eqs 55 and j, 58) in the limiting cases, the parameters κa, λa, r0/a, σ and Q h of the colloidal system have to be determined. Experimental data for the surface layers of human erythrocytes21 and rat lymphocytes22 in electrolyte solutions indicated that the shield length 1/λ has values about 3 nm and the magnitude of Q ranges from quite low to about 1.6 × 106 C/m3, depending on the pH value and ionic strength of the electrolyte solution. For some
Figure 3. Plot of the parameters X01 and X10 in eqs 50 and 51a for the coefficients Λ01 and Λ10 versus r0/a at fixed values of κa.
temperature-sensitive poly(N-isopropylacrylamide) hydrogel layers on latex particles in salt solutions,23 the values of 1/λ were found to be about 1-50 nm and the magnitude of Q could be as high as 8.7 × 106 C/m3. As to the surface charge density, an experimental study for the adsorption of poly(vinyl alcohol) onto AgI reported that the value of σ changes from 0 to -0.035 C/m2 upon increasing the pAg from 5.6 to 11, while experimental data for a positively charged polystyrene latex used as the adsorbent for the polyelectrolyte poly(acrylic acid) showed that σ can have a value as high as 0.16 C/m2.27 It is widely understood that the Debye length 1/κ is in the range from angstroms to about a micron, depending on the ionic strength of the solution. For a composite particle with σ ) 2 × 10-3 C/m2 and Q ) 2 × 106 C/m3 in an aqueous solution with 1/κ ) 10-9 m, one obtains the dimensionless charge densities σ j ≈ 0.1 and Q h ≈ 0.1. The parameters X01 and X10 in association with the coefficients Λ01 and Λ10 by eqs 50 and 51a for the effective conductivity of a dilute suspension can be evaluated for given values of the parameters κa and r0/a using eq 52, and their results are plotted in Figures 2 and 3. Figure 2 indicates that both X01 and X10 (or the magnitudes of Λ01 and Λ10) decrease monotonically with increasing κa for a given value of r0/a. Figure 3 shows that X01 is a (27) Blaakmeer, J.; Bohmer, M. R.; Cohen Stuart, M. A.; Fleer, G. J. Macromolecules 1990, 23, 2301.
Dilute Suspension of Charged Composite Spheres
Figure 4. Plot of the parameters Y02, Y11, and Y20 in eq 51b for the dimensionless coefficients Λ02, Λ11, and Λ20 with r0/a ) 0.5 versus κa at fixed values of λa.
monotonically increasing function of r0/a and X10 is a monotonically decreasing function of r0/a for a fixed value of κa. In the limiting case of r0/a ) 0, the composite particle degenerates to a homogeneous porous sphere and X01 equals zero. In the special case of r0/a ) 1, the composite particle reduces to a rigid sphere with no surface layer and X10 vanishes. 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).
Langmuir, Vol. 14, No. 7, 1998 1569
Figure 5. Plot of the parameters Y02, Y11, and Y20 in eq 51b for the coefficients Λ02, Λ11, and Λ20 with r0/a ) 0.5 versus λa at fixed values of κa.
The parameters Y02, Y11, and Y20 defined by eq 51b are calculated using eqs 53a-c for various values of the parameters κa, λa, and r0/a, and the results are presented in Figures 4-7. Figure 4 illustrates that the parameters Y02, Y11, and Y20 are monotonically decreasing functions of κa for given values of λa and r0/a, while Figure 5 shows that they decrease monotonically with increasing λa for given values of κa and r0/a. When λ f ∞, the resistance
1570 Langmuir, Vol. 14, No. 7, 1998
Liu and Keh
Figure 6. Plot of the parameters Y02, Y11, and Y20 in eq 51b for the coefficients Λ02, Λ11, and Λ20 with κa ) 1 versus r0/a at fixed values of λa.
Figure 7. Plot of the parameters Y02, Y11, and Y20 in eq 51b for the coefficients Λ02, Λ11, and Λ20 with λa ) 10 versus r0/a at fixed values of κa.
to the fluid motion inside the surface layer of the composite particle is infinitely large and the velocity profile in the surface layer disappears. The ions can still penetrate the surface layer, and the equilibrium potential distribution ψ(eq) is the same as eq 31. Therefore, for given values of κa and r0/a with λ f ∞, Y02, Y11, and Y20 approach constant values, as shown in Figure 5. When λ f 0, the surface layer does not exert resistance to the fluid motion, and
Y02, Y11, and Y20 for given values of κa and r0/a approach other constant values in Figure 5. In Figures 6 and 7, the parameters Y02, Y11, and Y20 are plotted as functions of the parameter r0/a for given values of κa and λa. It can be seen in Figures 6a and 7a that Y02 is a monotonically increasing function of r0/a for fixed values of κa and λa. In the special case of r0/a ) 0, the composite particle degenerates to a homogeneous porous
Dilute Suspension of Charged Composite Spheres
Figure 8. Plot of the parameters Z02, Z11, and Z20 in eq 51b for the coefficients Λ02, Λ11, and Λ20 versus κa at fixed values of r0/a.
sphere and Y02 must be equal to zero. Obviously, Y02 is independent of λa in the limiting case of r0/a ) 1, in which the composite particle reduces to a rigid sphere with no surface layer. Conversely, as illustrated in Figures 6c and 7c, Y20 decreases monotonically with increasing r0/a and vanishes when r0/a ) 1. As expected, Y11 equals zero in both limiting cases of r0/a ) 0 and r0/a ) 1 and there exists a maximal value of Y11 for given values of κa and λa, as shown in Figures 6b and 7b. Note that the location
Langmuir, Vol. 14, No. 7, 1998 1571
Figure 9. Plot of the parameters Z02, Z11, and Z20 in eq 51b for the coefficients Λ02, Λ11, and Λ20 versus r0/a at fixed values of κa.
of the maximum shifts to larger values of r0/a when κa or λa increases. Figures 8 and 9 show plots of the parameters Z02, Z11, and Z20 defined by eq 51b for various values of κa and r0/a calculated using eqs 53d-f. It can be seen that these parameters decrease monotonically with increasing κa for a given value of r0/a. As are the cases for X01 and Y02, Z02 is a monotonically increasing function of r0/a for a
1572 Langmuir, Vol. 14, No. 7, 1998
Liu and Keh
fixed value of κa. In the special case of r0/a ) 0, Z02 equals zero. However, the variation of Z20 with r0/a for a given value of κa is not always monotonic. As shown in Figures 8c and 9c, Z20 decreases monotonically with increasing r0/a when κa j 5 but has a maximum at a value of r0/a other than zero when κa J 5. For any given value of κa, Z20 vanishes as r0/a ) 1. On the other hand, similar to the case of Y11, Z11 equals zero in both limiting cases of r0/a ) 0 and r0/a ) 1 and there exists a maximum of Z11 for a given value of κa, as shown in Figure 9b. The locations of the maximal values of Z20 (if it exists) and Z11 also shift to larger values of r0/a when κa increases. An analytical study25 has predicted that composite spheres with zero net charge (4πr20σ + (4π/3)(a3 - r30)Q ) 0) can undergo electrophoresis in an electric field and produce sedimentation potential under gravity. It would be of interest to know whether the electric conductivity of a dilute suspension of such “neutral” composite spheres differs from that of a corresponding suspension of composite spheres with σ ) 0 and Q ) 0. For such “neutral” h , and eq composite spheres, σ j ) -(κa/3)[(a/r0)2 - (r0/a)]Q 49 for a symmetric electrolyte becomes
{
Λ ) Λ∞ 1 -
[( )
1 - γcp 3 r0 φ 1 - p 2 a
3
-
z(D+ - D-) h X (κa)2Q (D+ + D-) 0
(
)
k2T2 Y0 + z2Z0 (κa)4Q h 2 + O(Q h 3) 2πη(D+ + D-)e2
]}
(60)
where
[( ) ( )] [( ) ( )] [( ) ( )]
X0 ) -X10 + 1 a Y0 ) Y20 3 r0
Z0 ) Z20 -
2
1 a 3 r0
-
r0 X01 a
r0 1 a Y11 + a 9 r0
[( ) ( )]
1 a 3 r0
2
2
-
2
r0 a
2
[( ) ( )]
r0 1 a Z11 + a 9 r0
2
-
r0 a
(61a)
Y02 (61b)
2
Z02 (61c)
Figures 10-12 display the numerical results for the dimensionless parameters X0, Y0, and Z0 calculated using eq 61 as functions of the parameters κa, λa, and r0/a. It can be seen that X0, Y0, and Z0 are all positive and the presence of the fixed-charge distribution in the “neutral” particles would influence the effective conductivity of the suspension. The direction of the influence is decided by the fixed charges in the porous surface layers (rather than the surface charges of the rigid cores) of the particles. The parameters X0 and Z0 decrease monotonically with the increase of κa for a given value of r0/a, while Y0 is not necessarily a monotonic function of κa for fixed values of λa and r0/a. The parameters Y0 and Z0 decrease monotonically with increasing r0/a for fixed values of κa and λa, while X0 approaches zero in both limiting cases of r0/a ) 0 and r0/a ) 1 and has a maximum for a given value of κa. The parameter Y0 is a monotonic decreasing function of λa for constant values of κa and r0/a. In general, the trends of the dependence of X0, Y0, and Z0 on κa and r0/a are quite different from those of X10, Y20, and Z20 presented in Figures 2-9.
Figure 10. Plot of the parameter X0 in eq 61a for a dilute suspension of identical composite spheres with zero net charge versus κa and r0/a.
6. Concluding Remarks The effective electric conductivity of a dilute suspension of charged composite particles in an electrolyte solution is analytically studied in this work. The surface layer of each composite particle is treated as a solvent-permeable and ion-penetrable object in which fixed-charged groups and hydrodynamic frictional segments are distributed at uniform densities. The electric double layer surrounding each particle is assumed not to overlap with the others. The average electric current density passing through the suspension is given by eq 15 as an integral over a large spherical surface surrounding a single particle plus its double layer and is related to the electrochemical potential energies of the electrolyte ions. Solving the linearized electrokinetic equations applicable to the system of an isolated composite sphere by a regular perturbation method, we have derived the electrochemical potential distributions of ionic species and the fluid velocity field under the application of a uniform electric field. An analytical expression (eq 49) is obtained as a power series in the two fixed charge densities of the composite particles jQ h, Q h 2) for the effective electric conductivity up to O(σ j 2, σ of a dilute suspension of identical charged composite spheres. According to this formula, the presence of the fixed charges in the composite particles can result in an increase or a decrease in the effective conductivity relative
Dilute Suspension of Charged Composite Spheres
Langmuir, Vol. 14, No. 7, 1998 1573
Figure 11. Plot of the parameter Y0 in eq 61b for a dilute suspension of identical composite spheres with zero net charge versus κa, λa, and r0/a.
to that of a corresponding suspension of uncharged particles, depending on the diffusion coefficients of the electrolyte ions and the fixed-charge densities of the particles. Expressions 55 and 58 for the electric conductivity of a dilute suspension of identical charged composite spheres in a symmetric electrolyte reduce to the corresponding formulas for the charged solid spheres and the charged porous spheres, respectively, in the limiting cases of r0/a ) 1 and r0/a ) 0. It is worth repeating that eq 49 with eqs 50 and 51 is obtained on the basis of the Debye-Huckel approximation for the equilibrium potential distribution around a composite sphere. A similar formula for the electric conductivity of a dilute suspension of identical impermeable spheres with low ζ potential was shown to give a good approximation 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 ζe/kT ) -2).3 Therefore, our results might be used tentatively for the situation of reasonably high electric potentials. On the other hand, the decay of the density distributions of hydrodynamic frictional segments and fixed charges in the surface layer of the composite particle with the distance from the particle center has not been considered in our calculations. In order to see whether our theory can be reasonably extended to the higher values of electric potential or to the nonuniform density distributions of
segments and fixed charges, we propose to obtain a numerical solution of the electrokinetic differential equations with no assumption on the magnitude of electric potential, allowing the use of arbitrary distributions of charge and fluid drag components in the surface layer of the composite particle, and compare it with the approximate solution. Acknowledgment. This research was supported by the National Science Council of the Republic of China under Grant NSC86-2214-E-002-013. Appendix The constants in eq 41 for the functions Fijr(r) and Fijθ(r) are listed here:
λ2a3 [C sinh(λr0) + Cij4 cosh(λr0)] 3r0 ij3 r0 r0 2 2 Gij(r) dr + 3 2 [cosh(λr0) a R1(λr)Gij(r) dr 2 a (λa) λ a r0
Cij1 ) -
∫
∫
sinh(λr0) Cij2 ) 2Cij1 +
∫ar β1(λr)Gij(r) dr] 0
2 (Cij6 + 10Cij8) (λa)2
(A1) (A2)
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Liu and Keh
{
3 [A + 3λr0(β1(λa) - β1(λr0))]Cij7 + (λa)3Ω 5[A + 6λa cosh(λr0) - 3λr0(λa sinh(λa) + 2r0 cosh(λa) + β1(λr0))]Cij8 + 2[2(λa)3 sinh(λa) + λa
Cij3 )
(λr0)2 cosh(λr0) + 3β1(λa) - 3β1(λr0)]
∫ar Gij(r) dr 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
∫
∫ar β1(λr)Gij(r) dr + 3 r r 2λa∆∫a Gij(r) dr (A3) a
3(λr0)2 sinh(λa) cosh(λr0) + 3λr0]
0
0
{
()
}
1 3λr0Cij7 + 15λr0Cij8 (λa)3∆ (λa)3[λr0 cosh(λa) - sinh(λr0)]Cij3 + r0 6 [sinh(λr0) a β1(λr)Gij(r) dr 2 (λa) 6r0 r0 r0 cosh(λr0) a R1(λr)Gij(r) dr] + 2 a Gij(r) dr λa
Cij4 )
∫
∫
∫
}
(A4)
Cij5 ) Cij1 + Cij2 + R1(λa)Cij3 + β1(λa)Cij4 Cij6 - Cij7 - Cij8 (A5)
∫a∞Gij(r) dr + C006∫a∞(ar ) Gij(r) dr r r 3 r C001∫a ( ) Gij(r) dr - C002∫a Gij(r) dr a r r C003∫a R1(λr)Gij(r) dr - C004∫a β1(λr)Gij(r) dr + 2
Cij6 ) C005
0
0
0
0
C006µEij (A6)
Cij7 ) -µEij Cij8 )
∫a∞(ar ) Gij(r) dr 2
∫a∞Gij(r) dr
1 5
(A7)
Figure 12. Plot of the parameter Z0 in eq 61c for a dilute suspension of identical composite spheres with zero net charge versus κa and r0/a.
C005 ) 1 + C001 + C002 + R1(λa)C003 + β1(λa)C004 -
(A8) C006 )
where
A ) [2(λa)3 + (λr0)3] cosh(λr0)
C001 )
3 [λr - λa cosh(λd) + sinh(λd)] (A11) Ω 0 C002 )
C003 ) C004 )
Γ + 2C001 (λa)3Ω∆
-3 {A + 3λr0[β1(λa) - β1(λr0)]} (λa)3Ω
(A12) (A13)
3 {[2(λa)3 + (λr0)3] sinh(λr0) + 3 (λa) Ω 3λr0[R1(λa) - R1(λr0)]} (A14)
Γ 2λaΩ∆
∆ ) λr0 sinh(λa) - cosh(λr0)
(A9)
B ) [2(λa)3 + (λr0)3 + 3λa + 3λr0] sinh(λa) 3 cosh(λa) (A10)
Γ (A15) 2λaΩ∆ (A16) (A17)
Ω ) -6λr0 + [2(λa)3 + (λr0)3 + 3λa + 3λr0] × cosh(λd) + [3(λr0)2 - 3] sinh(λd) (A18) with
Γ ) 3λr0A - 9(λr0)2[β1(λa) + β1(λr0)] + 3{-λaA + λr0[2(λa)3 + (λr0)3 + 3λr0]β1(λa) + 3λ2ar0β1(λr0)} cosh(λd) + 3{A + 3λr0[(λr0)2β1(λa) β1(λr0)]} sinh(λd) (A19) In the above equations, Gij(r), R1(x), and β1(x) are defined by eqs 42, 43a, and 43c, respectively, and the definite integrals can be performed numerically. LA970893H
