Langmuir is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036
Research Article
Diffusiophoresis in a Suspension of Charge-Regulating Colloidal Spheres
Huan J. Keh, and Yu L. Li
Langmuir, 2007, 23 (3), 1061-1072 • DOI: 10.1021/la061517c Downloaded from http://pubs.acs.org on November 19, 2008
More About This Article
Additional resources and features associated with this article are available within the HTML version:
• Supporting Information
• Access to high resolution figures
• Links to articles and content related to this article
Diffusiophoresis in a Suspension of Charge-Regulating Colloidal
Spheres
Huan J. Keh* and Yu L. Li
Department of Chemical Engineering, National Taiwan UniVersity, Taipei 10617, Taiwan, Republic of China
ReceiVed May 29, 2006. In Final Form: August 13, 2006
An analytical study of diffusiophoresis in a homogeneous suspension of identical spherical charge-regulating particles with an arbitrary thickness of the electric double layers in a solution of a symmetrically charged electrolyte with a uniform prescribed concentration gradient is presented. The charge regulation due to association/dissociation reactions of ionogenic functional groups on the particle surface is approximated by a linearized regulation model, which specifies a linear relationship between the surface charge density and the surface potential. The effects of particle-particle electrohydrodynamic interactions are taken into account by employing a unit cell model, and the overlap of the double layers of adjacent particles is allowed. The electrokinetic equations that govern the electric potential profile, the ionic concentration distributions, and the fluid flow field in the electrolyte solution surrounding the particle in a unit cell are linearized assuming that the system is only slightly distorted from equilibrium. Using a regular perturbation method, these linearized equations are solved with the equilibrium surface charge density (or zeta potential) of the particle as the small perturbation parameter. Closed-form formulas for the diffusiophoretic velocity of the charge-regulating sphere correct to the second order of its surface charge density or zeta potential are derived. Our results indicate that the charge regulation effect on the diffusiophoretic mobility is quite sensitive to the boundary condition for the electric potential specified at the outer surface of the unit cell. For the limiting cases of a very dilute suspension and a very thin or very thick electric double layer, the particle velocity is independent of the charge regulation parameter.
1. Introduction
A colloidal particle can be driven to move by the application of a nonuniform solute concentration field that interacts with the surface of the particle. This phenomenon, known as diffusiophoresis,1-3has been demonstrated experimentally for both charged4 and uncharged5 solutes. Diffusiophoresis is of practical importance in some applications to particle analysis or separation and in certain latex-particle coating processes.6,7In a solution of nonionic solute, the solute molecules interact with the particle through the van der Waals and/or dipole forces. For charged particles in an electrolyte solution, the particle-solute interaction is electrostatic in nature, and its range is the Debye screening length κ-1. Particles with zeta potentials on the order of kT/e (∼25 mV; e is the elementary electric charge, k is Boltzmann’s constant, and T is the absolute temperature) in prescribed electrolyte gradients on the order of 100 kmol/m4(1 M/cm) will move by diffusiophoresis at speeds of several micrometers per second.
Using the classical model of the diffuse electric double layer, Prieve8derived a formula for the diffusiophoretic velocity of a nonconducting particle of arbitrary shape in a solution of a symmetrically charged electrolyte with a constant concentration gradient for arbitrary zeta potential (ζ) of the particle in the limit
of thin double layer (κa f∞, where a is the linear dimension
of the particle). This formula agrees with that previously deduced by a more intuitive method.1 Using a method of matched asymptotic expansions with (κa)-1as the small parameter, Prieve et al.2 evaluated the effect of particle curvature on the diffu-siophoretic mobility of a spherical particle accurate to O[(κa)-1]. When the double-layer distortion from equilibrium was taken as a small perturbation, Prieve and Roman9obtained a numerical solution for the diffusiophoresis of a dielectric sphere in concentration gradients of 1:1 electrolytes (KCl or NaCl) which was applicable to a broad range ofζ and κa. Later, analytical
expressions for the diffusiophoretic velocities of a dielectric sphere10,11and cylinder11,12with a “thin but polarized” double layer were derived. Recently, analytical formulas in closed forms for the diffusiophoretic mobility of a colloidal sphere13 and cylinder14in symmetric electrolytes at low surface charge density
σ (valid for ζ up to 50 mV) and arbitrary κa have also been
obtained. All results of the above investigations show that the diffusiophoretic mobility of a particle in general decreases with the increase of (κa)-1. Also, the particle can reverse the direction of its migration when the value of|ζ|, (κa)-1, or the valence of the electrolyte increases.
In some practical applications of the diffusiophoretic mobility, 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 assemblages of particles, unit cell models were often employed to predict the effects of particle interactions on the mean sedimentation rate * To whom correspondence should be addressed. Fax:
+886-2-2362-3040. E-mail: [email protected].
(1) Dukhin, S. S.; Derjaguin, B. V. In Surface and Colloid Science; Matijevic, E., Ed.; Wiley: New York, 1974; Vol. 7.
(2) Prieve, D. C.; Anderson, J. L.; Ebel, J. P.; Lowell, M. E. J. Fluid Mech.
1984, 148, 247.
(3) Anderson, J. L.; Prieve, D. C. Langmuir 1991, 7, 403.
(4) Ebel, J. P.; Anderson, J. L.; Prieve, D. C. Langmuir 1988, 4, 396. (5) Staffeld, P. O.; Quinn, J. A. J. Colloid Interface Sci. 1989, 130, 88. (6) Smith, R. E.; Prieve, D. C. Chem. Eng. Sci. 1982, 37, 1213. (7) Dukhin, S. S. AdV. Colloid Interface Sci. 1993, 44, 1. (8) Prieve, D. C. AdV. Colloid Interface Sci. 1982, 16, 321.
(9) Prieve, D. C.; Roman, R. J. Chem. Soc., Faraday Trans. 2 1987, 83, 1287. (10) Pawar, Y.; Solomentsev, Y. E.; Anderson, J. L. J. Colloid Interface Sci.
1993, 155, 488.
(11) Keh, H. J.; Chen, S. B. Langmuir 1993, 9, 1142.
(12) Keh, H. J.; Wei, Y. K. J. Colloid Interface Sci. 2002, 252, 354. (13) Keh, H. J.; Wei, Y. K. Langmuir 2000, 16, 5289.
(14) Keh, H. J. Wei, Y. K. Langmuir 2002, 18, 10475.
10.1021/la061517c CCC: $37.00 © 2007 American Chemical Society Published on Web 01/03/2007
in a bounded suspension of identical uncharged spheres.15These models 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 velocity at the outer (virtual) surface of a spherical cell are the so-called “free-surface” model of Happel16and “zero-vorticity” model of Kuwabara,17the predictions of which have been tested against the experimental data.
Using the Kuwabara cell model, Levine and Neale18derived an analytical expression for the electrophoretic mobility in a suspension of identical charged spheres with smallζ and arbitrary κa as a function of the volume fraction of the particles. Later,
the Happel and/or Kuwabara cell models were also used by Zharkikh and Shilov,19 Kozak and Davis,20,21 Ohshima,22,23 Dukhin et al.,24Ding and Keh,25and Carrique et al.26,27to predict the electrophoretic mobility of charged spheres in concentrated suspensions and/or the electric conductivity of the suspensions under various conditions. It was found that the Happel model in general shows a better agreement with the available experimental data28,29than the Kuwabara model does.25Recently, the diffu-siophoretic motions in homogeneous suspensions of identical dielectric spheres with thin but polarized double layers30and with small surface potential but arbitrary double-layer thickness31 were analyzed by employing both the Happel and the Kuwabara cell models. Closed-form formulas for the diffusiophoretic mobility as functions of the volume fraction of the particles were obtained.
The previous analyses for the diffusiophoretic mobility of charged particles in either dilute or concentrated suspensions were all based on the assumption that either the surface charge density or the surface potential of the particles remains constant. Although this assumption may be convincing under certain conditions, it only leads to idealized results for limiting cases and can be impractical for some particles. The actual surface charge (and potential) for biological colloids, polymer latices, and particles of metal oxides in electrolyte solutions is usually determined by the dissociation of ionizable surface groups and/ or adsorption (or site-binding) of specific ions. The degree of these dissociation and adsorption reactions will be a function of the local concentrations of the charge-determining (and potential-determining) ions at the particle surfaces. When a relatively concentrated suspension is subjected to an imposed electrolyte concentration gradient (which can cause an induced macroscopic electric field), the neighboring particles will adjust the concen-trations of the potential-determining ions at their surfaces to
minimize the electrostatic energy of interaction among them. Thus, the extent of the surface reactions and the magnitudes of the surface charge density and surface potential for multiple particles undergoing diffusiophoresis will be changed in com-parison with those for a single particle at equilibrium. This is the so-called charge regulation phenomenon.32-42The assumptions of constant surface charge density and constant surface potential provide two limiting cases for the combined electrostatic and hydrodynamic interaction effects on the charge regulation surfaces that exist in these systems.
In this article, the unit cell model is used to obtain analytical expressions for the diffusiophoretic mobility of a suspension of identical, charge-regulating colloidal spheres in a solution of a symmetrically charged electrolyte with a constant imposed concentration gradient. The linearized form of the charge regulation boundary condition proposed by Carnie and Chan38 is employed. No assumption is made about the thickness of the double layers relative to the radius of the particles, and the overlap of adjacent double layers is allowed. 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 electrolyte concentration gradient. Through the use of a regular perturbation method with the equilibrium surface charge density of the particle as the small perturbation parameter, the ion concentration (or electrochemical potential energy), elec-trostatic potential, fluid velocity, and pressure profiles are determined by solving these linearized electrokinetic equations subject to the appropriate boundary conditions. Analytical expressions for the diffusiophoretic mobility of the suspension of charge-regulating spheres are obtained in closed-forms.
2. Basic Electrokinetic Equations
We consider the steady diffusiophoresis of a statistically homogeneous distribution of identical charged spherical particles in a bounded liquid solution of a symmetrically charged electrolyte. The particles can have charge-regulating surfaces on which the chemical equilibrium of ionogenic functional groups is maintained (see the Appendix). The applied electrolyte concentration gradient∇n∞is a constant equal to|∇n∞|ez, and
the diffusiophoretic velocity of the colloidal particles is Uez,
where ezis a unit vector in the positive z direction. Gravitational
effects on the particles are ignored. As shown in Figure 1, we employ a unit cell model in which each particle of radius a is surrounded by a concentric spherical shell of the suspending solution having an outer radius of b such that the particle/cell volume ratio is equal to the particle volume fractionφ throughout
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 polar axis θ ) 0 points toward the positive z direction. Obviously, the
problem for each cell is axially symmetric about the z axis. (15) Happel, J.; Brenner, H. Low Reynolds Number Hydrodynamics; Nijhoff:
The Netherlands, 1983.
(16) Happel, J. AIChE J. 1958, 4, 197.
(17) Kuwabara, S. J. Phys. Soc. Jpn. 1959, 14, 527.
(18) Levine, S.; Neale, G. H. J. Colloid Interface Sci. 1974, 47, 520. (19) Zharkikh, N. I.; Shilov, V. N. Colloid J. USSR 1982, 43, 865 (English translation).
(20) Kozak, M. W.; Davis, E. J. J. Colloid Interface Sci. 1989, 127, 497. (21) Kozak, M. W.; Davis, E. J. J. Colloid Interface Sci. 1989, 129, 166. (22) Ohshima, H. J. Colloid Interface Sci. 1997, 188, 481.
(23) Ohshima, H. J. Colloid Interface Sci. 1999, 212, 443.
(24) Dukhin, A. S.; Shilov, V.; Borkovskaya, Y. Langmuir 1999, 15, 3452. (25) Ding, J. M.; Keh, H. J. J. Colloid Interface Sci. 2001, 236, 180. (26) Carrique, F.; Arroyo, F. J.; Delgado, A. V. J. Colloid Interface Sci. 2001,
243, 351.
(27) Carrique, F.; Arroyo, F. J.; Jimenez, M. L.; Delgado, A. V. J. Phys. Chem.
B 2003, 107, 3199.
(28) Watillon, A.; Stone-Masui, J. J. Electroanal. Chem. 1972, 37, 143. (29) Zukoski, C. F.; Saville, D. A. J. Colloid Interface Sci. 1987, 115, 422. (30) Wei, Y. K.; Keh, H. J. Langmuir 2001, 17, 1437.
(31) Wei, Y. K.; Keh, H. J. J. Colloid Interface Sci. 2002, 248, 76.
(32) Ninham, B. W.; Parsegian, V. A. J. Theor. Biol. 1971, 31, 405. (33) Chan, D.; Perram, J. W.; White, L. R.; Healy, T. W. J. Chem. Soc.,
Faraday Trans. 1 1975, 71, 1046.
(34) Chan, D.; Healy, T. W.; White, L. R. J. Chem. Soc., Faraday Trans. 1
1976, 72, 2844.
(35) Prieve, D. C.; Ruckenstein, E. J. Theor. Biol. 1976, 56, 205. (36) Van Riemsdijk, W. H.; Bolt, G. H.; Koopal, L. K.; Blaakmeer, J. J.
Colloid Interface Sci. 1986, 109, 219.
(37) Krozel, J. W.; Saville, D. A. J. Colloid Interface Sci. 1992, 150, 365. (38) Carnie, S. L.; Chan, D. Y. C. J. Colloid Interface Sci. 1993, 161, 260. (39) Reiner, E. S.; Radke, C. J. AdV. Colloid Interface Sci. 1993, 47, 59. (40) Pujar, N. S.; Zydney, A. L. J. Colloid Interface Sci. 1997, 192, 338. (41) Ding, J. M.; Keh, H. J. J. Colloid Interface Sci. 2001, 243, 331. (42) Keh, H. J.; Ding, J. M. Langmuir 2002, 18, 4572.
It is assumed that the magnitude of∇n∞or the particle velocity is not large and hence that the electric double layer surrounding the particle is only slightly distorted from the equilibrium state, where no bulk electrolyte gradient is imposed and the particle and fluid are at rest. Therefore, the ionic concentration (number density) distributions n((r,θ), the electrostatic potential
distribu-tionψ(r,θ), and the dynamic pressure distribution p(r,θ) can be
expressed as
where n((eq)(r),ψ(eq)(r), and p(eq)(r) are the equilibrium distribu-tions of the ionic concentradistribu-tions, electrostatic potential, and dynamic pressure, respectively, and δn((r,θ), δψ(r,θ), and
δp(r,θ) are the small deviations from the equilibrium state.
Here, subscripts + and - refer to the cation and anion, respec-tively. The equilibrium concentration of each ionic species is related to the equilibrium potential by the Boltzmann distribution. It can be shown that the small perturbed quantitiesδn(,δψ, andδp together with the fluid velocity field u(r,θ) satisfy the
following set of linearized electrokinetic equations:31
Here,δµ((r,θ) is defined as a linear combination of δn(andδψ
on the basis of the concept of the electrochemical potential energy
n0∞is the value of n∞at the positions with z ) 0 which can be experimentally taken as the mean bulk concentration of the electrolyte in the vicinity of the diffusiophoretic particle (or in the cell),η is the viscosity of the fluid, D(are the diffusion coefficients of the ionic species, Z is the valence of the symmetric electrolyte which is positive, and ) 4π0r, where ris the relative permittivity of the electrolyte solution and 0 is the permittivity of a vacuum. The use of the bulk concentration n0∞ at the particle center in eq 5 is valid when the solution is only slightly nonuniform in the imposed electrolyte concentration on the length scale of the particle radius (a|∇n∞|/n0∞, 1).
The boundary conditions for u andδµ(at the surface of the particle are
which are obtained from the assumptions that the “shear plane” coincides with the particle surface and no ions can penetrate into the particle. Note that eq 7a takes a reference frame traveling with the particle. To obtain the boundary condition for the small perturbed quantityδψ at the charge-regulating surface, we adopt
the linearized regulation model proposed by Carnie and Chan38 and express the surface charge densityσ as a linear function of
the surface potentialψS
whereσ(eq)andζ are the values of σ and ψ
S, respectively, at equilibrium. The substitution of eqs 1b and 8 into the Gauss condition at the particle surface
results in
where the charge regulation coefficient L is defined by eqs A7 and A8 in the Appendix and can be evaluated in terms of measurable quantities. The constant surface charge density limit corresponds to L ) 0, whereas the constant surface potential limit corresponds to L f∞.
Because the bulk concentration of the electrolyte is not uniform, it is required that the total fluxes of cations and anions are balanced in order to have no current arising from the diffusive fluxes of the electrolyte ions in an electrically neutral solution, and a macroscopic electric field occurs spontaneously due to the difference in ionic mobilities. At the virtual (outer) surface of the cell, the local ionic concentration gradient and electric field are compatible with the prescribed gradient and induced field, respectively. Thus, the boundary conditions there are18,31 Figure 1. Geometric sketch of a spherical particle undergoing
diffusiophoresis at the center of a spherical cell.
n() n((eq)+ δn( (1a) ψ ) ψ(eq)+ δψ (1b) p ) p(eq)+ δp (1c) η∇2u )∇δp - 4π(∇ 2 ψ(eq)∇δψ + ∇2δψ∇ψ(eq)) (2) ∇‚u ) 0 (3) ∇2 δµ() (Ze kT
(
∇ψ (eq)‚∇δµ (- kT D(∇ψ (eq)‚u)
(4) ∇2 δψ )4πZen0 ∞ kT[
exp(
Zeψ(eq) kT)
(δµ- + Zeδψ) -exp(
- Zeψ (eq) kT)
(δµ+ - Zeδψ)]
(5) δµ() kT δn( n((eq) ( Zeδψ (6) r ) a: u ) 0 (7a) ∂µ( ∂r ) 0 (7b) σ ) σ(eq)+(
dσ dψS)
ψS)ζ δψS (8) r ) a: ∂ψ ∂r ) - 4πσ (9) r ) a: ∂δψ ∂r - Lδψ ) 0 (10)where R ) a|∇n∞|/n0∞,β ) (D+- D-)/(D++ D-), U is the diffusiophoretic velocity of the charged sphere to be determined, and urand uθ are the r andθ components, respectively, of u.
Note that the Happel cell model16assumes that the radial velocity and the shear stress of the fluid on the outer boundary of the cell are zero, whereas the Kuwabara cell model17assumes that the radial velocity and the vorticity of the fluid are zero there. Since the reference frame is taken to travel with the particle, the radial velocity given by eq 11a is generated by the particle velocity in the opposite direction. The condition in eq 11a implies that there is no net flow of fluid between adjacent cells; it is valid because the suspension of the particles is bounded by impermeable walls. Thus, the effect of the backflow of fluid occurring in a closed container is included in both cell models.
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.15 The boundary conditions of the ionic concentrations and electric potential at the virtual surface r ) b may be taken as the distributions giving rise to the applied gradient∇n∞in the cell when the particle does not exist. In this case, the Neumann approach given by eqs 11d and 11e becomes the following Dirichlet approach:19,31
Note that the overlap of the electric double layers of adjacent particles is allowed in both of the boundary conditions given by eqs 11d, 11e, and 12.
In the next section, eqs 2-5 subject to eqs 7 and 10-12 are solved using a regular perturbation method for a low equilibrium surface potential or surface charge density of the particles, and the diffusiophoretic velocity is determined with the condition of zero net force acting on the particles or on a unit cell.
3. Solution of the Electrokinetic Equations and Diffusiophoretic Velocity
Before solving for the problem of diffusiophoresis of the charged sphere in a unit cell filled with the solution of a symmetric electrolyte with a constant bulk concentration gradient∇n∞, we need to determine the equilibrium electrostatic potential first. The equilibrium potentialψ(eq)satisfies the Poisson-Boltzmann equation and the boundary conditions
It can be shown that
where σj ) 4πZeσ(eq)/κkT, which is the nondimensional equilibrium surface charge density of the particle
and
Here κ is the Debye screening parameter equal to [8πZ2e2n 0∞/
kT]1/2. The expression in eq 14 forψ(eq)as a power series in the equilibrium surface charge density of the particle up to O(σj) is
the equilibrium solution to the linearized Poisson-Boltzmann equation that is valid for small values of the electric potential (the Debye-Huckel approximation). Note that the contribution from the effect of O(σj2) toψ(eq)in eq 14 disappears only for the case of a solution of symmetric electrolytes.
Substituting eq 14 together with eq 15 into eq 9, one obtains a relation between the surface charge density and the surface potential of the colloidal sphere in a unit cell at equilibrium
where
andγ ) κ(b - a) ) κa(φ- 1/3- 1). The equilibrium surface
potential (known as the zeta potential)ζ for a charge-regulating
sphere can be found by combining eqs 16 and A4 (withδµS) 0 at equilibrium) and then solving the resulting equation. Thus,
ζ is able to be estimated in terms of measurable quantities. In
the limitφ ) 0, eq 16 reduces to the simple relation σ(eq)) ζ(κa + 1)/4πa for an isolated charged sphere.
To solve for the small quantities u,δp, δµ(, andδψ in terms of the particle velocity U when the parameterσj is small, these
variables can be written as perturbation expansions in powers ofσj
where the functions ui, pi,µi (,ψi, and Uiare not directly dependent
onσj. The zeroth-order terms of U, u, and δp disappear because
a spherical particle withσ(eq)) 0 or ζ ) 0 will not move by
r ) b: ur) -Ucos θ (11a) τrθ) η
[
r ∂ ∂r(
uθ r)
+ 1r ∂ur ∂θ]
) 0(for the Happel model) (11b) (∇ × u)φ) 1 r ∂ ∂r(ruθ) -1 r ∂ur ∂θ ) 0
(for the Kuwabara model) (11c)
∂δµ( ∂r ) kT(1 - β) R acosθ (11d) ∂δψ ∂r ) - kTZeβ R acosθ (11e) r ) b: δµ() kT(1 - β)Rr acosθ (12a) δψ ) -kT ZeβR r acosθ (12b) r ) a: dψ (eq) dr ) - 4πσ (eq) (13a) r ) b: dψ (eq) dr ) 0 (13b)
ψ(eq)(r) )ψeq1σj + O(σj 3 ) (14) ψeq1(r) ) kT Ze
(
κa A)
a r[(κb + 1)e κ(a + r)+ (κb - 1)eκ(a + 2b - r) ] (15a)A ) (κb - 1)(κa + 1)e2κb- (κa - 1)(κb + 1)e2κa
(15b) σj ) WZeζ kT (16a) W )γ cosh γ + (κaγ + κ 2 a2- 1) sinh γ
κa(κa +γ) cosh γ - sinh γ (16b)
u ) u1σj + u2σj 2+ ... (17a) δp ) p1σj + p2σj 2+ ... (17b) δµ() µ0(+ µ1(σj + µ2(σj 2+ ... (17c) δψ ) ψ0+ ψ1σj + ψ2σj2+ ... (17d) U ) U1σj + U2σj2+ ... (17e)
applying an electrolyte concentration gradient [although a “neutral” sphere with ionogenic surface groups in equilibrium with the suspending solution can develop an odd (antisymmetric) distribution of surface charges when an electrolyte gradient is imposed]. It is easy to show that
where the coefficientχ equals 1 - φ when the boundary conditions
in eqs 11d and 11e are used, and equals 1 + φ/2 when the
conditions in eq 12 are used.
Substituting the expansions given by eq 17 andψ(eq)given by eq 14 into the governing equations given by eqs 2-5 and boundary conditions in eqs 7 and 10-12, and equating like powers ofσj
on both sides of the respective equations, one can derive a group of linear differential equations and boundary conditions for each set of functions ui, pi,µi(, andψiwith i equal to 0, 1, 2, etc. After
collecting the zeroth-order and first-order terms in the perturbation procedure, we obtain
with
The solutions forψ0,µ1(, p1, and the r andθ components of u1 subject to eqs 19-23 are
where
and the functions Fir(r), Fiθ(r), Fpi(r), and Fµi(r) with i equal to
0, 1, and 2 were given in ref 31. In eq 27, K and K′are functions of parameters La, κa, and κb
Obviously, K ) K′) 0 as La ) 0. In the limit La f∞, eq 28 reduces to
Among the higher-order terms in the perturbation procedure, the distributions we need in the following calculations areψ1, u2, and p2. The governing equations and boundary conditions for these distributions are given by eqs 19, 20, 22a, 22b, 23c, 23d, and 23e, with the subscript 0 and 1 being replaced by 1 and 2, respectively, and
The solutions of these distributions are
µ0() kT(1 - β)R χ
(
r a+ a 2 2r2)
cosθ (18) ∇2 ψ0) - 4πZekTn0 ∞( µ0+ - µ0-) + κ 2 ψ0 (19) ∇2u 1) 1η∇p1 - 4πη(∇ 2 ψeq1∇ψ0+ ∇2ψ0∇ψeq1) (20a) ∇‚u1) 0 (20b) ∇2 µ1() ( Ze(1 - β)R χa(
1 -a3 r3)
dψeq1 dr cosθ (21) r ) a: ∂ψ0 ∂r - Lψ0) 0 (22a) u1) 0 (22b) ∂µ1( ∂r ) 0 (22c) r ) b: ∂ψ0 ∂r ) - kTZeβ Racosθ (if eq 11e is used) (23a)
ψ0) - kT
ZeβR r
acosθ (if eq 12b is used) (23b)
u1r) -U1cosθ (23c)
τ1rθ) 0 (for the Happel model) (23d)
(∇ × u1)φ) 0 (for the Kuwabara model) (23e)
∂µ1(
∂r ) 0 (if eq 11d is used) (23f)
µ1() 0 (if eq 12a is used) (23g)
ψ0) kT
ZeRβFψ0(r) cosθ (24)
u1r)
[
U1F0r(r) - kTηa2βRF1r(r)
]
cosθ (25a)u1θ)
[
U1F0θ(r) - kT ηa2βRF1θ(r)]
sinθ (25b) p1) η a[
U1Fp0(r) -kT ηa2βRFp1(r) -κ2akT 4πηZeβRψeq1 (r)Fψ0(r)]
cosθ (25c) µ1()-kT(1 - β)RFµ1(r) cosθ (26) Fψ0(r) ) 1 1 -φ[
3a2K 2r2 {[κb(κb - 2) + 2](κr + 1)e κ(b-r)+ [κb(κb + 2) + 2](κr - 1)e-κ(b-r)}-(
a 2 2r2 + r a)
]
(if eq 11e is used) (27a)
Fψ0(r) ) 1 1 +φ/2
{
3a2K' 2r2 [(κb - 1)(κr + 1)e κ(b-r)- (κb + 1)(κr - 1)e-κ(b-r)] -(
a 2 2r2 + r a)}
(if eq 12b is used) (27b) K ) La{[κb(κb - 2) + 2][κa(κa + La + 2) + La + 2]eκ(b-a)- [κb(κb + 2) + 2][κa(κa - La - 2) + La +2]e-κ(b-a)}-1 (28a)
K′) La{(κb - 1)[κa(κa + La + 2) + La + 2]eκ(b-a)+ (κb + 1)[κa(κa - La - 2) + La + 2]e-κ(b-a)}-1 (28b)
K∞){[κb(κb - 2) + 2](κa + 1)eκ(b-a)+ [κb(κb + 2) + 2](κa - 1)e-κ(b-a)}-1 (29a)
K′∞){(κb - 1)(κa + 1)eκ(b-a) (κb + 1)(κa
-1)e-κ(b-a)}-1 (29b)
r ) b: ∂ψ1
∂r ) 0 (if eq 11e is used) (30a)
ψ1) 0 (if eq 12b is used) (30b)
ψ1) kTR
where
Since the unit cell as a whole is electrically neutral, the net force exerted on its virtual surface must be zero. Applying this constraint to the above solutions, one can obtain the leading-order terms
U1and U2for the diffusiophoretic velocity of the charge-regulating sphere. When the Neumann-type boundary conditions in eqs 11d and 11e are employed (χ ) 1 - φ), the result is
where W is defined by eq 16b andΘiwith i equal to 1 and 2
are functions of κa, La, andφ given by
for the Happel model, and
for the Kuwabara model. In eq 37
and the finite integrals can be calculated numerically. Note that the effect of charge regulation at the particle surface on the diffusiophoretic velocity is demonstrated in terms of functions
Fψ0(r) and Fψ1(r) given by eqs 27 and 33.
When the Dirichlet-type boundary conditions in eq 12 are chosen, the result for Ui will be independent of the charge
regulation coefficient L, and its analytical expressions for the Happel and Kuwabara models are the same as those obtained for the limiting case of L ) 0 (given by eqs 36-39 with K ) K′) 0 andχ ) 1 + φ/2).
Substituting eqs 16a and 36 into eq 17e, we obtain the expression for the diffusiophoretic velocity of the particle as an expansion in powers of the equilibrium surface potentialζ
Since the solutionsµ1(andψ1given by eqs 26 and 31 are not influenced by the fluid flow, the relaxation effect of the diffuse ions in the electric double layer surrounding the particle is not included in eq 40 up to the orderζ2.
For the reason that all of the governing equations and boundary conditions in this analysis have been linearized, diffusiophoresis of charged particles in an electrolyte solution can be considered as a linear combination of two effects: (i) chemiphoresis due to the nonuniform adsorption of counterions and depletion of co-ions over the surface of the particle and (ii) electrophoresis due to the macroscopic electric field generated by the electrolyte concentration gradient given by eq 11e or 12b. In eq 40 for the diffusiophoretic velocity, the O(ζ) term (involving parameter β)
results from the contribution of electrophoresis, whereas the O(ζ2) term represents the chemiphoretic contribution.
4. Results and Discussion
Before the evaluation of the diffusiophoretic mobility of suspensions of identical charge-regulating spheres from eqs 37 and 40, it is necessary to know how the equilibrium surface potentialζ, equilibrium surface charge density σ(eq), and charge regulation coefficient L depend on the bulk electrolyte concen-tration n∞, surface reaction equilibrium constants K+and K -(defined by eq A2), and particle volume fractionφ. To perform
a typical calculation using eqs 16, A4, and A8, we make the continuous phase an aqueous 1-1 electrolyte solution with relative permittivity r ) 78.54, the particle radius a ) 200 nm, the ionogenic surface group density NS) 5 × 1016site/m2, and the system temperature T ) 298 K. The numerical results of the dimensionless equilibrium surface potentialζh, equilibrium surface
charge densityσj(eq), and charge regulation parameter La calculated as functions of the variables n∞, K+ K-, K-/K+, and φ are plotted in Figures 2 and 3. The value of K+K-is fixed at 10-6 M2in Figure 2 and the value of K-/K+is specified at 10-4in Figure 3. It can be seen that the point of zero charge is given by n∞) (K+K-)1/2. If n∞< (K+K-)1/2, the values ofζ and σ(eq) are negative; the magnitude ofζ decreases monotonically with
u2r)
[
U2F0r(r) + kTηa2RF2r(r)
]
cosθ (32a)u2θ)
[
U2F0θ(r) + kT ηa2RF2θ(r)]
sinθ (32b) p2) η a[
U2Fp0(r) + kT ηa2RFp2(r) + κ2akT 4πηZeRψeq1(r)Fψ1(r)]
× cosθ (32c) Fψ1(r) ) 1 κ3r2[
{[g1+(κa,La) - e 2κag 1-(κa,La)]Aψ1(a,b)-2g1+(κa,La)Bψ1(a,b)}
{
-g2+(κb)[κr cosh(κr) -sinh(κr)] + 12[g2+(κb) - e 2κbg
2-(κb)](κr + 1)e -κr
}
/[e2κag2+(κb)g1-(κa,La) - e2κbg2-(κb)g1+(κa,La)] -[κr cosh(κr) - sinh(κr)]Aψ1(r,b) + (κr + 1)e-κrBψ1(r,b)
]
(33) Aψ 1(x,y) )
∫
x y e-κrκ2(κr + 1)[
Fµ1(r) +Ze kTψeq1(r)Fµ0(r)]
dr (34a) Bψ1(x,y) )∫
x y κ2[κr cosh(κr) -sinh(κr)][
Fµ1(r) +Ze kTψeq1(r)Fµ0(r)]
dr (34b)g1((x,y) ) 2 ( 2x + x2+ y(1 ( x) (35a)
g2((x) ) 2 ( 2x + x2 (if eq 11e is used) (35b)
g2((x) ) 1 ( x (if eq 12b is used) (35c) Ui) (8β)1-iβR 4πηa
(
kT Ze)
2Θi Wi (36) Θi) (-8)i-1 3(3 + 2φ5/3)(κa) 2 WiZe kT{
2 φ2/3ωψeq1(b)Fψ(i - 1)(b)-∫
a b[
1 - 3r 2 a2+ 2 r3 a3- φ 5/3(
2r 2 a2- 3 r3 a3 + r5 a5)
]
Gi(r) dr}
(37a) Θi)(-8) i-1 9 (κa) 2WiZe kT{
2 φ2/3ω'ψeq1(b)Fψ(i-1)(b)-∫
a b[
1 - 3r 2 a2+ 2 r3 a3 - 1 5φ(
2 - 5 r3 a3+ 3 r5 a5)
]
Gi(r) dr}
(37b) Gi(r) ) (-1)i -1a r[
Fµ(i - 1)(r) + (i - 1)Ze kTψeq1(r)Fµ0(r)]
dψeq1 dr (38) ω )(
1 -3 2φ 1/3+ 3 2φ 5/3- φ2)
-1 (39a) ω′)(
1 -9 5φ 1/3+ φ -1 5φ 2)
-1 (39b) U ) R 4πηa[
kT ZeβΘ1ζ + 1 8Θ2ζ 2+ O(ζ3 )]
(40)an increase in n∞for an otherwise specified condition, whereas
σ(eq)may have a maximal magnitude at some values of n∞. If n∞ > (K+K-)1/2, the values ofζ and σ(eq), which are both positive, do not increase monotonically with an increase in n∞and may
have maxima at some values of n∞. The magnitudes ofζ and σ(eq) increase as K-/K+increases, because the concentration of the un-ionized surface group AB decreases with K-/K+, as inferred from eq A2. When the value of K+K-increases, the concentration of the negatively charged surface group AZ-will increase or that
Figure 2. Plots of the dimensionless equilibrium surface potential
ζ
h, equilibrium surface charge density σj(eq), and charge regulation
parameter La versus the bulk concentration n∞of an aqueous 1-1 electrolyte solution under the condition of a ) 200 nm, NS) 5 ×
1016site/m2, and K+K-) 10- 6M2. The solid and dashed curves
represent the cases of the volume fractionφ equal to 0.3 and 0,
respectively.
Figure 3. Plots of the dimensionless equilibrium surface potential
ζ
h, equilibrium surface charge density σj(eq), and charge regulation
parameter La versus the bulk concentration n∞of an aqueous 1-1 electrolyte solution under the condition of a ) 200 nm, NS) 5 ×
1016site/m2, and K-/K+) 10-4. The solid and dashed curves represent the cases of the volume fractionφ equal to 0.3 and 0, respectively,
of the positively charged surface group AB2Z+will decrease according to eq A2; thus, the particles become more negatively charged or less positively charged. The magnitude ofζ increases,
while the magnitude ofσ(eq)decreases, as the volume fraction
φ increases, but these dependencies become negligible when the
value of n∞is relatively high. The regulation parameter La is not a monotonic function of the characteristic variables n∞, K+K-,
K-/K+, andφ of the suspension.
Now we consider several limiting cases of the analytical expressions for the diffusiophoretic velocity. In the limit of an infinitely dilute suspension (φ f 0), eq 37 reduces to
where Enis a function defined by
Interestingly, these reduced results, which are the same as the formulas for the diffusiophoretic velocity obtained previously13 for a single dielectric sphere in an unbounded electrolyte, do not depend on the charge regulation parameter La. However, it is understood that the value ofζ in eq 40 for a charge-regulating
sphere is dependent on the regulation characteristics of the particle and suspending electrolyte solution.
When κa . 1, eq 37 for i ) 1 can be expressed asymptotically as
Again, these asymptotic results are independent of the charge regulation parameter La to the order as listed. When the boundary conditions 11d and 11e are used (i.e.,χ ) 1 - φ), the leading
terms in eq 43 are identical to the formulas of the electrophoretic mobility of a dielectric sphere in a cell derived by Levine and Neale18in the limit κa f∞. Note that, when κa f∞, the value ofΘ1predicted by the Happel model can be as much as 14% greater (occurring at φ = 0.39) than that predicted by the
Kuwabara model.
When κa , 1, eq 37 with i ) 1 for the case that the boundary conditions in eqs 11d and 11e apply (χ ) 1 - φ) can be written
as
where
If the boundary conditions in eq 12 are chosen, eqs 44 and 45 will still be valid as long as one takes L ) 0 andχ ) 1 + φ/2.
The numerical results of the dimensionless coefficientΘ1for a suspension of identical charge-regulating spheres calculated from eq 37 withχ ) 1 - φ as a function of the parameters κa,
La, and φ are plotted in Figures 4 and 5 for the Happel and
Kuwabara models, respectively. The calculations are presented up toφ ) 0.74, which corresponds to the maximum attainable
volume fraction for a swarm of identical spheres.18It can be seen thatΘ1decreases gently and monotonically with an increase in the charge regulation parameter La for specified values of κa and
φ. This coefficient becomes independent of La for the case of
very dilute suspensions and for the situations of very large and very small values of κa. For given values of La and φ, Θ1 decreases monotonically with a decrease in κa (or with an increase in the double layer overlap). When κa ) 0,Θ1) 2/3 as φ ) 0 and Θ1) 0 for all finite values of φ. For the case of the Kuwabara model,Θ1is a monotonic decreasing function ofφ
for fixed values of κa and La and equals unity as κa f∞, whereas for the case of the Happel model,Θ1is a monotonic function ofφ only for a given value of κa less than about unity. For any
combination of κa, La, andφ, the Kuwabara model predicts a
smaller value ofΘ1than the Happel model does. This occurs because the zero-vorticity model yields larger energy dissipation in the cell than that due to the particle drag alone for the free-surface model, owing to the additional work done by the stresses at the outer boundary.15
In Figures 6 and 7, the coefficientΘ2 for a suspension of identical charge-regulating spheres calculated from eq 37 with
χ ) 1 - φ is plotted for the Happel and Kuwabara models,
respectively, as a function of the parameters κa, La, and φ.
Analogous to the case ofΘ1,Θ2decreases monotonically with an increase in the charge regulation parameter La for specified values of κa andφ and becomes independent of La for the case
of very dilute suspensions and for the situations of very large and very small values of κa. However,Θ2 is not monotonic functions of either κa orφ, and a local maximum and/or minimum
of each of these functions would appear, for an otherwise unchanged condition. For a combination of κa, La, andφ not
too close to these minima, the Kuwabara model predicts a somewhat smaller value ofΘ2 than the Happel model does. When κa ) 0, both models predict thatΘ2) 0 for all values ofφ and La.
In Figures 8 and 9, the dependence of the diffusiophoretic velocity U in a suspension of identical charge-regulating spheres on their dimensionless zeta potential (Zeζ/kT) at various values
of the parameters κa and La calculated from eq 40 withχ ) 1
- φ and φ ) 0.1 is plotted for the Happel and Kuwabara models, respectively. The magnitude of the diffusiophoretic velocity is normalized by a characteristic value given by
The case that the cation and anion diffusivities are equal (β )
0) is displayed in Figures 8a and 9a. Only the results at positive zeta potentials are shown since the particle velocity, which is due to the chemiphoretic effect entirely, is an even function of the zeta potential as illustrated by eq 40. Because our analysis is based on the assumption of small surface charge density or zeta potential, the magnitudes of Zeζ/kT considered are less than
ν ) 2 + φ - 9Laφ 2(1 -φ) + La(1 + 2φ)(κa) 2+ O[(κa)4 ] (45) U* ) R 4πηa
(
kT Ze)
2 (46) Θ1) 1 - e κa[5E7(κa) - 2E5(κa)] (41a) Θ2) 1 +
1 3e
2κa
[10E6(2κa) + 7E8(2κa)] - 40e2κaE7(κa) [E3(κa) - E5(κa)] +4
3e
κa
[3E3(κa) + 9E4(κa) - 7E5(κa) -15E6(κa)] (41b) En(x) )
∫
1 ∞ t-ne-xtdt (42) Θ1) 3 χ{
1 -φ5/3 3 + 2φ5/3- (κa) -1+ O[(κa)-2 ]}
(for the Happel model) (43a) Θ1) 1χ{1 -φ - 3(κa)
-1+ O[(κa)-2]}
(for the Kuwabara model) (43b)
Θ1) ν 6χ
(
3 + 2φ-5/3 3 + 2φ5/3φ 2/3- φ-2/3)
(κa)2+ O[(κa)4](for the Happel model) (44a) Θ1) ν45χ(5φ
-1- 9φ-2/3+ 5 - φ)(κa)2+ O[(κa)4
] (for the Kuwabara model) (44b)
2. It can be seen that, in this range of Zeζ/kT, the reduced
diffusiophoretic velocity U/U* increases monotonically with an increase in the magnitude of Zeζ/kT for fixed values of κa and
La and with a decrease in La for given values of κa and Zeζ/kT.
For the case of very large and very small values of κa, U/U* becomes independent of La. On the other hand, consistent with the result ofΘ2, U/U* is not a monotonic function of κa for an otherwise specified condition. There is no chemiphoretic motion of the particles for the special cases of Zeζ/kT ) 0 or κa ) 0.
Figures 8b and 9b are drawn for the reduced diffusiophoretic mobility U/U* of the particles for a case that the cation and anion have different diffusion coefficients (β ) -0.2). In this case,
both the chemiphoretic and the electrophoretic effects contribute to the particle’s movement, and the net diffusiophoretic velocity is neither an even nor an odd function of the zeta potential. For constant values of κa and La, the reduced particle velocity U/U* is not necessarily a monotonic function of Zeζ/kT. When the
product ofβ and Zeζ/kT is negative, U/U* is not a monotonic
function of κa for given values of Zeζ/kT and La. The reduced
diffusiophoretic velocity U/U*, which is not necessarily to decrease with an increase in La for given values of κa and Zeζ/
kT, becomes independent of La for the case of very large and
very small values of κa. Some of the curves in Figures 8b and
9b indicate that particles might reverse direction of movement more than once as their zeta potential varies from negative to positive values. The reversals occurring at the values of Zeζ/kT
other than zero result from the competition between the contributions from chemiphoresis and electrophoresis. In the limit κa ) 0, the diffusiophoretic velocity vanishes for any finite value ofφ, irrespective of the value of La. Note that the situations
associated with Figures 8a or 9a (β ) 0) and 8b or 9b (β )
-0.2) taking Z ) 1 are close to the diffusiophoresis in the aqueous solutions of KCl and NaCl, respectively.
When the boundary conditions in eq 12 are chosen for the electrostatic and electrochemical potentials at the virtual surface of the unit cell, the diffusiophoretic mobility of a suspension of identical charge-regulating spheres is independent of the charge regulation parameter La. Graphical results of the dimensionless coefficientsΘ1andΘ2as well as the reduced diffusiophoretic mobility U/U* as functions of the parameters κa andφ for this
case are the same as those presented in a previous paper31for the limit La ) 0. For a given suspension of identical charge-regulating spheres, the diffusiophoretic mobility obtained by each cell model can be evaluated as functions of the regulation characteristics of the suspension (such as n∞, K+, K-, NS, a,φ, etc.) from eqs 37 and 40 incorporating with eqs 16, A4, and A8. Figure 4. Plots of the dimensionless coefficientΘ1for a suspension
of identical spheres calculated from eq 37a withχ ) 1 - φ (using
boundary conditions 11d and 11e) for the Happel model versus the parameters κa andφ. The solid, dotted-dashed, and dashed curves
represent the cases of the charge regulation parameter La equal to 0, 5, and∞, respectively.
Figure 5. Plots of the dimensionless coefficientΘ1for a suspension
of identical spheres calculated from eq 37b withχ ) 1 - φ (using
boundary conditions 11d and 11e) for the Kuwabara model versus the parameters κa andφ. The solid, dotted-dashed, and dashed curves
represent the cases of the charge regulation parameter La equal to 0, 5, and∞, respectively.
These functions are quite complicated for most situations and cannot be predicted systematically by simple general rules.
5. Concluding Remarks
In this work, the steady-state diffusiophoresis in a homogeneous suspension of identical charge-regulating spheres in an electrolyte solution with arbitrary values of the parameters κa, La, andφ
(and of the regulation characteristics of the suspension) is 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 electrochemical potential energy distributions of the electrolyte ions, the electrostatic potential profile, and the fluid flow field through the use of a linearized charge regulation model. The requirement that the net force exerted on the unit cell is zero leads to eqs 37 and 40 for the diffusiophoretic velocity of the charged sphere correct to the orderζ2. With the use of the Neumann-type boundary conditions in eqs 11d and 11e at the virtual surface of the unit cell, the normalized diffusiophoretic mobility is a monotonic decreasing function of La for fixed values of κa andφ.
Equation 40 with eq 37 are derived on the basis of the Debye-Huckel approximation for the equilibrium potential distribution
around the charge-regulating sphere in a unit cell. Comparing our results of the coefficientΘ1with the numerical solution for the electrophoretic mobility of a charged sphere in unbounded KCl solutions obtained by O’Brien and White43valid for an arbitrary value of zeta potential, one can find that eq 41a for a charged sphere with a low zeta potential in an unbounded electrolyte solution is also quite accurate for reasonably high zeta potentials (with errors less than 4% for|ζ|e/kT e2). Therefore, our results in eqs 37 and 40 might be used tentatively for the situation of reasonably high electric potentials.
We note that the unit cell models with various boundary conditions at the virtual surface of the cell lead to somewhat different results of the effect of charge regulation at the particle surfaces on the diffusiophoretic velocity. Neither of these boundary conditions is rigorously correct, for the surrounding spheres affect the ionic concentrations, electrostatic potential, and fluid velocity at this surface. Mathematically, the concentra-tion, potential, and flow fields in the surrounding matrix should be coupled to the corresponding fields in the unit cell through appropriate compatibility conditions (as opposed to boundary conditions), and specification of boundary conditions at the virtual (43) O’Brien, R. W.; White, L. R. J. Chem. Soc., Faraday Trans. 2 1978, 74, 1607.
Figure 6. Plots of the dimensionless coefficientΘ2for a suspension
of identical spheres calculated from eq 37a withχ ) 1 - φ (using
boundary conditions 11d and 11e) for the Happel model versus the parameters κa andφ. The solid, dotted-dashed, and dashed curves
represent the cases of the charge regulation parameter La equal to 0, 5, and∞, respectively.
Figure 7. Plots of the dimensionless coefficientΘ2for a suspension
of identical spheres calculated from eq 37b withχ ) 1 - φ (using
boundary conditions 11d and 11e) for the Kuwabara model versus the parameters κa andφ. The solid, dotted-dashed, and dashed curves
represent the cases of the charge regulation parameter La equal to 0, 5, and∞, respectively.
surface is, at best, an approximation. Nonetheless, the analysis presented here provides meaningful information for the volume fraction effects on the diffusiophoretic mobility in concentrated suspensions of particles. In many other electrokinetic phenomena, the Dirichlet approach given by eq 12 has been shown to be better than Neumann’s (better agreement with the ensemble-averaged predictions and experiments as regards electrophoretic mobility and electric conductivity, compatibility with Onsager reciprocity laws connecting different phenomena, etc., for instance), probably due to the fact that the angular components of the concentration and potential gradients at the virtual surface of the unit cell are not specified in eqs 11d and 11e. The results of our analysis show that the regulation mechanism for the surface charge is of minor importance if the Dirichlet option is chosen. The relevant experimental data (such as if the diffusiophoretic mobility in a concentrated suspension is enough sensitive to the changes of the charge regulation parameter), which are not available in the literature yet, would be needed for confirming the validity of each approach of the cell model at various situations.
Acknowledgment. This research was supported by the National Science Council of the Republic of China.
Appendix: Model for a Charge-Regulating Surface Following the previous studies,33,34we consider a general model for the charge-regulating surface which develops surface charges via association/dissociation equilibrium of ionogenic surface groups. The surface reactions may be expressed as
where AB represents the associable/dissociable functional group on the surface, BZ+denotes the ion to determine the status of
charges on the surface groups (the potential-determining ion), and the positive integer Z is the valence of ionization. For the case of an amphoteric surface, BZ+is usually the hydrogen ion
H+. The equilibrium constants for the reactions in eq A1 are given by
Figure 8. Plots of the reduced diffusiophoretic mobility in a suspension of identical spheres versus the dimensionless zeta potential at fixed values of κa calculated from eq 40 withχ ) 1 - φ (using
boundary conditions 11d and 11e) andφ ) 0.1 for the Happel
model: (a)β ) 0; (b) β ) -0.2. The solid, dotted-dashed, and
dashed curves represent the cases of the charge regulation parameter La equal to 0, 5, and∞, respectively.
Figure 9. Plots of the reduced diffusiophoretic mobility in a suspension of identical spheres versus the dimensionless zeta potential at fixed values of κa calculated from eq 40 withχ ) 1 - φ (using
boundary conditions 11d and 11e) andφ ) 0.1 for the Kuwabara
model: (a)β ) 0; (b) β ) -0.2. The solid, dotted-dashed, and
dashed curves represent the cases of the charge regulation parameter La equal to 0, 5, and∞, respectively.
AB2Z+ S AB + BZ+ (A1a)
where [BZ+]
Sis the concentration of BZ+next to the surface. The surface dissociation constants K+and K-are taken to be functions of temperature only.
For NSionizable surface groups per unit area, the net surface charge density is
By the substitution of the Boltzmann distribution for the equilibrium concentration of BZ+and the utilization of the concept
of electrochemical potential energy, eq A3 forσ can be expressed
in terms of the surface potentialψSas
where
δµSis the deviation in electrochemical potential of BZ+next to the surface from the equilibrium state defined by eq 6, and n∞ is the concentration of BZ+ in the bulk solution where the
equilibrium potential is set equal to zero. Equation A6 is the Nernst equation relating the Nernst potentialψNto the isoelectric point [with n∞) (K+K-)1/2]. It can be seen from eq A4, which acts as an equation of the electric state of the surface, that the sign ofσ is opposite to that of ψS- ψNat equilibrium (withδµS ) 0). The surface charge density approaches the saturation values (ZeNSwhen the difference between the surface potential and its Nernst value becomes large [e.g., when the value of n∞/(K+
K-)1/2approaches zero or infinity and the value ofψ
Sis finite]. With the relationship betweenσ and ψSgiven by eq A4, the charge regulation capacitance of the surface at equilibrium can be written as
whereζ is the value of ψSat equilibrium and
The reciprocal of the positive quantity L can be regarded as the characteristic length controlling the charge regulation condition at the surface. The limiting values of L ) 0 and L f∞correspond to the cases of constant surface charge density and constant surface potential, respectively. Note that L is small when the difference between the equilibrium surface potential and its Nernst value is large. LA061517C -
(
dσ dψS)
ψS)ζ ) 4πL (A7) L )4πZ 2 e2NSδ{δ + cosh[Ze(ψN (eq)- ζ)/kT]}kT{1 +δ cosh[Ze(ψN(eq)- ζ)/kT]}2 (A8)
K+ ) [AB][BZ+]S/[AB2 Z+ ] (A2a) K- ) [AZ-][BZ+]S/[AB] (A2b) σ ) ZeNS [AB2Z+] - [AZ-] [AB] + [AB2Z+] + [AZ-] ) ZeNS [B Z+] S 2- K +K -K+[BZ+]S+ [B Z+ ]S2+ K+K -(A3)
σ ) ZeNS δ sinh{[Ze(ψN- ψS) +δµS]/kT}
1 +δ cosh{[Ze(ψN- ψS) +δµS]/kT} (A4) δ ) 2(K-/K+)1/2 (A5) ψN) kTZeln n∞ (K+K-)1/2 (A6)
