Journal of Colloid and Interface Science 243, 331–341 (2001) doi:10.1006/jcis.2001.7892, available online at http://www.idealibrary.com on
Sedimentation Velocity and Potential in a Suspension of Charge-Regulating Colloidal Spheres
Jau M. Ding and Huan J. Keh1 Department of Chemical Engineering, National Taiwan University, Taipei 106-17, Taiwan, Republic of China
Received February 20, 2001; accepted July 30, 2001; published online October 5, 2001 Dorn in 1878, and this effect is often known by his name (1–3). Body-force-driven migration in a homogeneous suspension of The sedimentation-potential gradient not only alters the veloc- spherical charge-regulating particles with electrical double layers ity and pressure distributions in the fluid due to its action on the of arbitrary thickness is analyzed. The charge regulation due to as- electrolyte ions but also retards the settling of the particles by sociation/dissociation reactions of functional groups on the particle an electrophoretic effect. surface is approximated by a linearized regulation model, which specifies a linear relationship between the surface-charge density Without considering the particle–particle interaction effects, and the surface potential. The effects of particle interactions are Booth (1) solved a set of electrokinetic equations using a pertur- taken into account by employing a unit cell model. The overlap bation method to obtain formulas for the sedimentation velocity of the double layers of adjacent particles is allowed and the relax- and sedimentation potential in a dilute suspension of identi- ation effect in the double layer surrounding each particle is con- cal spherical particles with arbitrary double-layer thickness ex- sidered. The electrokinetic equations which govern the ionic con- pressed as power series in the zeta potential of the particles. centration distributions, the electrostatic potential profile, and the Numerical results relieving the restriction of low surface po- fluid flow field in the electrolyte solution in a unit cell are linearized tential in Booth’s analysis were reported by Stigter (4) using assuming that the system is only slightly distorted from equilib- a modification of the theory of electrophoresis of a dielectric rium. Using a regular perturbation method, these linearized equa- sphere developed by Wiersema et al. (5). It was found that the tions are solved for a symmetrically charged electrolyte with the Onsager reciprocal relation between the sedimentation poten- equilibrium surface potential of the particle as the small perturba- tion parameter. Closed-form formulas for the settling velocity of tial and the electrophoretic mobility derived by de Groot et al. the charge-regulating spheres and for the sedimentation potential (6) is satisfied within good computational accuracy. Taking the in the suspension are derived. Our results show that the charge double-layer distortion from equilibrium as a small perturbation, regulation effects on the sedimentation in a suspension appear Ohshima et al. (7) obtained general expressions and presented starting from the leading order of the equilibrium surface poten- numerical results for the sedimentation velocity and potential tial, which is determined by the regulation characteristics of the in a dilute suspension of identical charged spheres over a broad suspension. C 2001 Academic Press range of zeta potential and double-layer thickness. Recently, Key Words: spherical particle; charge regulation surface; sedi- Booth’s perturbation analysis was extended to the derivation of mentation velocity; sedimentation potential; unit cell model. the sedimentation velocity and potential in a dilute suspension of charged composite spheres with a low density of the fixed charges (8). 1. INTRODUCTION In practical applications of sedimentation, relatively concen- trated suspensions of particles are usually encountered, and ef- The sedimentation or migration of charged colloidal particles fects of particle interactions will be important. To avoid the in electrolyte solutions has received considerable attention in the difficulty of the complex geometry that results when swarms past. This problem is more complex than that of uncharged parti- of particles are used, unit cell models were often employed to cles because the electric double layer surrounding each particle predict the effects of particle interactions on the mean sedimen- is distorted by the fluid flow around the particle. The defor- tation rate in a bounded suspension of identical spheres. These mation of the double layer resulting from the fluid motion is models involve the concept that an assemblage can be divided usually referred to as the polarization or relaxation effect and into a number of identical cells, one sphere occupying each cell gives rise to an induced electric field. The sedimentation po- at its center. The boundary value problem for multiple spheres tential or migration potential, which is set up in a suspension is thus reduced to the consideration of the behavior of a sin- of settling or translating charged particles, was first reported by gle sphere and its bounding envelope. The most acceptable of these models with various boundary conditions at the virtual sur- 1 To whom correspondence should be addressed. Fax: +886-2-2362-3040. face of the cell are the “free-surface” model of Happel (9) and E-mail: [email protected]. the “zero-vorticity” model of Kuwabara (10), the predictions
331 0021-9797/01 $35.00 Copyright C 2001 by Academic Press All rights of reproduction in any form reserved. 332 DING AND KEH of which for uncharged spherical particles have been tested which the surface contains single ionizable functional groups to against the experimental data. Using the Kuwabara cell model illustrate the charge regulation behavior of cellular surfaces. This and assuming that the overlap of the double layers of adjacent site-binding model was extended by Prieve and Ruckenstein particles is negligible on the virtual surface of the cell, Levine (17) to a surface bearing multiple ionizable (zwitterionic) func- et al. (11) derived analytical expressions for the sedimentation tional groups and by Chan et al. (15, 16) to a general amphoteric velocity and sedimentation potential in a homogeneous suspen- surface involving surface equilibria that are controlled by the sion of identical charged spheres with small surface potential concentration of the potential-determining ion in the bulk solu- as functions of the fractional volume concentration of the parti- tion. On the basis of the law of mass action for the dissociation cles. The Kuwabara model with nonoverlapping double layers reactions and the Boltzmann distribution for the mobile ions, in was also used by Ohshima (12) to demonstrate the Onsager rela- general, the relation between the surface-charge density and the tion between the sedimentation potential and the electrophoretic surface potential is nonlinear. Carnie and Chan (20) proposed a mobility of charged spheres with low zeta potential in concen- model to linearize this surface-charge-potential relation which is trated suspensions. Recently, the migration phenomena in ho- sufficiently accurate and permits analytical solutions for various mogeneous suspensions of identical charged spheres with small charge regulation systems of interest. surface potential and arbitrary double-layer thickness were an- In this article, the unit cell model is used to study the sed- alyzed by the present authors (13), who employed both the imentation phenomena in a suspension of identical, charge- Happel and the Kuwabara cell models and allowed the overlap regulating, colloidal spheres. The linearized form of the charge of adjacent double layers. Closed-form formulas for the sedi- regulation boundary condition proposed by Carnie and Chan mentation velocity and potential expressed as power series in is employed. The overlap of adjacent double layers is allowed the surface-charge density or surface potential of the particles and the polarization effect in the diffuse layer surrounding each were obtained, and these results demonstrate that the effects of particle is included. No assumption is made about the thickness overlapping double layers are quite significant even for the case of the double layer relative to the dimension of the particle. Both of thin double layers. the Happel model and the Kuwabara model are considered. The The previous analyses for the sedimentation of charged parti- basic electrokinetic equations are linearized assuming that the cles in either dilute or concentrated suspensions were all based electrolyte ion concentrations, the electrostatic potential, and on the assumption that either the surface-charge density or the the fluid pressure have only a slight deviation from equilibrium surface potential of the particles remains constant. While this due to the motion of the particle. Through the use of a regular assumption may be convincing under certain conditions, it only perturbation method with the surface potential of the particle leads to idealized results for a limiting case and can be imprac- as the small perturbation parameter, the ion concentration (or tical for some particles. The actual surface charge for biological electrochemical potential), electric potential, fluid velocity, and colloids, polymer latices, and particles of metal oxides in elec- pressure profiles are determined by solving these linearized elec- trolyte solutions is usually determined by the dissociation of trokinetic equations subject to the appropriate boundary con- ionizable surface groups and/or adsorption (or site-binding) of ditions. Analytical expressions for the settling velocity of the specific ions. The degree of these dissociation and adsorption charge-regulating spheres in the solution of a symmetrically reactions will be a function of the local concentrations of the charged electrolyte and for the sedimentation potential in the charge- (and potential- ) determining ions at the particle surface. suspension are obtained in closed forms. Since the electric double layer surrounding a particle is distorted during its sedimentation, the concentrations of both positively 2. BASIC ELECTROKINETIC EQUATIONS and negatively charged ions at the particle surface are differ- ent from their corresponding concentrations at the equilibrium We consider the sedimentation (or any other body-force- state. Also, in a relatively concentrated suspension, the neigh- driven motion) of a statistically homogeneous distribution of boring particles will adjust the concentrations of the potential- identical charged spherical particles in a bounded liquid solu- determining ions at their surfaces to minimize the electrostatic tion containing M ionic species at the steady state. The parti- energy of interaction among them. Thus, the extent of the sur- cles can have charge-regulating surfaces on which the chemical face reactions and the magnitudes of the surface-charge density equilibrium of ionizable functional groups is maintained (see and surface potential for multiple particles undergoing sedimen- Appendix A). The acceleration of gravity (or the uniformly im- tation will be changed in comparison with those for a single par- posed body force field) equals gez and the sedimentation (or ticle at equilibrium. This is the so-called charge regulation phe- migration) velocity of the colloidal particles is Uez, where ez is nomenon (14–22). The assumptions of constant surface-charge the unit vector in the positive z direction. As shown in Fig. 1, density and constant surface potential provide two limiting cases we employ a unit cell model in which each particle of radius for the combined electrostatic and hydrodynamic interaction ef- a is surrounded by a concentric spherical shell of suspending fects on the charge regulation surface that occur in these systems. solution having an outer radius of b such that the particle/cell The condition of surface-charge regulation was first pointed volume ratio is equal to the particle volume fraction ϕ through- out by Ninham and Parsegian (14), who proposed a model in out the entire suspension, viz., ϕ = (a/b)3. The cell as a whole is SEDIMENTATION OF CHARGE-REGULATING SPHERES 333 2 zme (eq) kT (eq) ∇ m = ∇ ∇ m ∇ u , kT Dm m = 1, 2,...,M, [4] 4 XM z en∞ z e (eq) ∇2 = m m m . ε exp ( m zme ) m=1 kT kT [5]
Here, m(r, ) is defined as a linear combination of nm and on the basis of the concept of the electrochemical potential energy (7),
= kT + , m (eq) nm zm e [6] nm
∞ nm is the concentration of the type m ions in the bulk (electrically neutral) solution where the equilibrium potential is set equal to zero, is the viscosity of the fluid, Dm and zm are the diffu- sion coefficient and valence, respectively, of species m, e is the elementary electric charge, k is Boltzmann’s constant, T is the absolute temperature, and ε = 4 ε0εr, where εr is the relative FIG. 1. Geometrical sketch for the sedimentation of a spherical particle at ε the center of a spherical cell. permittivity of the electrolyte solution and 0 is the permittivity of a vacuum. electrically neutral. The origin of the spherical coordinate system The boundary conditions for u and m at the surface of the (r, , ) is taken at the center of the particle and the axis = 0 particle are points toward the positive z direction. Obviously, the problem for each cell is axially symmetric about the z-axis. r = a, u = 0, [7a] It is assumed that the magnitude of the particle velocity is ∂ not large and hence that the electric double layer surrounding m = , ∂ 0 [7b] the particle is only slightly distorted from the equilibrium state, r where the particle and fluid are at rest. Therefore, the concen- which are obtained from the assumptions that the “shear plane” , tration (number density) distribution nm (r ) of species m, the coincides with the particle surface and no ions can penetrate into , electric potential distribution (r ), and the pressure distribu- the particle. Note that Eq. [7a] takes a reference frame traveling , tion p(r ) can be expressed as with the particle. To obtain the boundary condition for the small perturbed quantity at the charge-regulating surface, we adopt n = n(eq) + n , [1a] m m m the linearized regulation model proposed by Carnie and Chan = (eq) + , [1b] (20) and express the surface-charge density as a linear function of the surface potential , = (eq) + , S p p p [1c] (eq) d (eq) (eq) (eq) = + , [8] where nm (r), (r), and p (r, ) are the equilibrium distri- S d S = butions of the concentration of species m, electric potential, and S pressure, respectively, and n (r, ), (r, ), and p(r, ) are m where (eq) and are the values of and , respectively, at the small deviations from the equilibrium state. The equilibrium S equilibrium. The substitution of Eqs. [1b] and [8] into the Gauss concentration of each ionic species is related to the equilibrium condition at the particle surface, potential by the Boltzmann distribution. It can be shown that the small perturbed quantities nm , , ∂ 4 and p together with the fluid velocity fluid u(r, ) satisfy the r = a, = , [9] ∂r ε following set of linearized electrokinetic equations (13): results in ∇ u=0, [2] ε ∂ ∇2u =∇ p ∇2 (eq)∇ +∇2 ∇ (eq) , [3] r = a, L = 0, [10] 4 ∂r 334 DING AND KEH where the charge regulation coefficient L is defined by Eqs. [A7] the boundary conditions and [A8] in Appendix A and can be evaluated in terms of measur- able quantities. The constant surface-charge limit corresponds r = a, (eq) = , [12a] to L = 0; while the constant surface potential limit corresponds d (eq) to L →∞. r = b, = 0. [12b] The boundary conditions at the virtual surface of the cell, dr in which the overlap of the electric double layers of adjacent It can be shown that particles is allowed, are (eq) 3 = eq1(r) ¯ + O( ¯ ), [13] r = b, ur = Ucos , [11a] ¯ = / ∂ u 1 ∂ur where Ze kT, which is the nondimensional equilibrium = r + =0 (for Happel model), [11b] r ∂r r r ∂ surface potential of the particle, 1 ∂ 1 ∂ur kT a bcosh( r b) + sinh( r b) (∇ u) = (ru ) =0 (for Kuwabara model), (r) = , [14] r∂r r ∂ eq1 Ze r b cosh( a b) + sinh( a b) [11c] = ∂ and is the Debye–Huckel parameter defined by m = , (8 Z 2e2n∞/εkT)1/2. Expression [13] for (eq) as a power se- ∂ 0 [11d] r ries in the surface potential of the particle up to O( ¯) is the equi- ∂ librium solution for the linearized Poisson–Boltzmann equation = 0, [11e] r that is valid for small values of the electric potential (the Debye– Huckel approximation). Note that the contribution from the ef- ¯ 2 (eq) where ur and u are the r and components, respectively, of fect of O( )to in Eq. [13] disappears only for the case of u. Note that the Happel cell model (9) assumes that the radial a solution of symmetric electrolytes. velocity and the shear stress of the fluid on the outer boundary of Substituting Eq. [13] together with Eq. [14] into Eq. [9], one the cell are zero, while the Kuwabara cell model (10) assumes obtains a relation between the surface-charge density and the that the radial velocity and the vorticity of the fluid are zero surface potential of the colloidal sphere at equilibrium, there. Because the reference frame is taken to travel with the particle, the radial velocity given by Eq. [11a] is generated by ε cosh + ( 2a2 + a 1) sinh (eq) = , [15] the particle velocity in the opposite direction. The conditions 4 a ( a + ) cosh sinh [11a], [11d], and [11e] imply that there are no net flows of fluid, ionic species, or electric current between adjacent cells. They where = (b a) = a(ϕ 1/3 1). The equilibrium surface are valid because the suspension of the particles is bounded by potential for a charge-regulating sphere can be found by com- impermeable, inert, and nonconducting walls. Thus, the effect of bining Eqs. [15] and [A4] (with S = 0 at equilibrium) and the backflow of fluid occurring in a closed container is included then solving the resulting equation. Thus, can be estimated in in both cell models. terms of measurable quantities. In the limiting case of ϕ = 0, For the sedimentation of a suspension of uncharged spher- Eq. [15] reduces to the relation (eq) = ε ( a + 1)/4 a for an ical particles, both the Happel and the Kuwabara models give isolated charged sphere. qualitatively the same flow fields and approximately comparable To solve the small quantities u, p, , and in terms drag forces on the particle in a cell. However, the Happel model of the particle velocity U when the parameter ¯ is small, these has a significant advantage in that it does not require an ex- variables can be written as perturbation expansions in powers change of mechanical energy between the cell and the environ- of ¯, ment (23). 2 u = u0 + u1 ¯ + u2 ¯ + , [16a] 2 3. SOLUTION OF THE ELECTROKINETIC EQUATIONS p = p0 + p1 ¯ + p2 ¯ + , [16b] FOR SYMMETRIC ELECTROLYTES 2 = 1 ¯ + 2 ¯ + , [16c] We now consider the sedimentation of a charged sphere in a 2 = 1 ¯ + 2 ¯ + , [16d] unit cell filled with the solution of a symmetrically charged ∞ 2 binary electrolyte with a constant bulk concentration n (M = 2, U = U0 + U1 ¯ + U2 ¯ + , [16e] ∞ ∞ ∞ z+ = z =Z, n+ =n =n , where subscripts + and refer to the cation and anion, respectively). The equilibrium elec- where the functions ui , pi , i , i, and Ui are not directly (eq) tric potential satisfies the Poisson–Boltzmann equation and dependent on ¯. The zeroth-order terms of both and SEDIMENTATION OF CHARGE-REGULATING SPHERES 335
disappear due to not imposing the concentration gradient and coefficients as electric field. 2 Substituting the expansions given by Eq. [16] and (eq) given a ( p )g U0 = , [20a] by Eq. [13] into the governing equations [2]–[5] and boundary 3 C02 conditions [7], [10], and [11], and equating like powers of ¯ U1 = 0, [20b] on both sides of the respective equations, we obtain a group of ε 2 2 ε 2 linear differential equations and boundary conditions for each = U0 b + , U2 C22 eq1(b)F 1(b) G2(b) set of functions ui , pi , i , and i , with i equal to 0, 1, 2, .... C02 12 a 24 a After solution of these perturbation equations, the results for the [20c] 2 r and components of u, p (to the order ¯ ), , and (to the order ¯, which will be sufficient for the calculation of where the sedimentation velocity and potential to the order ¯ 2) can be Z x written as n d eq1 Gn(x) = r [F1+(r) F1 (r)] dr, [21] a dr
ur ={U0F0r(r)+U1F0r(r) ¯ p and are the densities of the particle and fluid, respectively, 2 3 +[U0F2r(r)+U2F0r(r)] ¯ + O( ¯ )} cos , [17a] and coefficients C02 and C22 are given by Eq. [B2b] for the Happel cell model and by Eq. [B3b] for the Kuwabara cell model. u ={U0F0 (r)+U1F0 (r) ¯+[U0F2 (r) In Eq. [20], U0 is the settling velocity of an uncharged sphere 2 3 = ϕ 1/3 ϕ +U2F0 (r)] ¯ + O( ¯ )} sin , [17b] in the cell, b a , and C02 is a function of parameter only and equals 3/2asϕ=0. The definite integrals of Gn(b) p = {U0 Fp0(r) + U1 Fp0(r) ¯ + [U0 Fp2(r) + U2 Fp0(r) in Eq. [20c] as well as in the coefficient C can be performed a 22 numerically. Note that the correction for the effect of surface ε 2 a 2 3 charges to the particle velocity starts from the second order, + U0 eql(r)F 1(r)] ¯ + O( ¯ )} cos , [17c] 4 2, instead of the first order, . The reason is that this effect is due to the interaction between the particle charges and the local = ¯ + ¯ 2 , Ze[U0F1 (r) O( )] cos [18] induced sedimentation-potential gradient; both are on the order 2 2 of and thus the correction is on the order of . = [U F (r) ¯ + O( ¯ )] cos . [19] 0 1 Substitution of Eq. [20] into Eq. [16e] results in an expression for the sedimentation velocity as a perturbation expansion in Here, the function eq1(r) has been given by Eq. [14], and the powers of , , , functions Fir(r) Fi (r) Fpi (r) (with i equal to 0 and 2), F1 (r), ε 2 and F 1(r) for both the Happel and the Kuwabara cell models 1 1 3 U = U0 1 H + +O( ) . [22] are defined by Eqs. [B1], [B7], and [B8] in Appendix B. Since 8 D+ D F1 (r) and F 1(r) are influenced by the fluid flow via F0r (r), the leading order of the effect of the relaxation (or polarization) Here, the dimensionless coefficient H is a function of parameters of the diffuse ions in the electrical double layer surrounding the a, La, and ϕ, particle is included in the solutions for and up to the order ¯. Note that the perturbation parameter ¯ in expansions 2 1 = U2 8 Ze 1 + 1 . [13] and [16]–[19] can be replaced by the dimensionless equi- H [23] U0 ε kT D+ D librium surface-charge density a¯ (eq)[=4 aZe (eq)/εkT] using the relation given by Eq. [15]. In Eqs. [22] and [23], H is defined such that it does not depend on the diffusion coefficients of the electrolyte ions. The numerical result of H calculated by using Eqs. [20a] and [20c] is presented 4. SEDIMENTATION VELOCITY in Section 6.
The total force exerted on the charged sphere settling in the 5. SEDIMENTATION POTENTIAL electrolyte solution within a unit cell can be expressed as the sum of the gravitational force (and buoyant force), the electric The electric fields around the individual charged particles un- force, and the hydrodynamic drag force acting on the particle. dergoing sedimentation in a suspension superimpose to result At the steady state, the total force acting on the settling par- in a sedimentation potential. For a homogeneous suspension of ticle (or the unit cell) is zero. Applying this constraint to the identical spherical particles, the sedimentation potential field is perturbation solution given by Eqs. [17]–[19] for a symmetric uniform and can be regarded as the average of the gradient of electrolyte, we obtain the sedimentation velocity of the charged electric potential over a sufficiently large volume of the suspen- sphere in the expansion form of Eq. [16e], with the first three sion to contain many particles. In order to calculate this field, 336 DING AND KEH the requirement that there exist no net electric current in the an aqueous 1 : 1 electrolyte solution with relative permittivity suspension must be satisfied. εr = 78.54, the particle radius a = 50 nm, the ionogenic sur- 18 2 For identical charged spheres suspended in a symmetric elec- face group density NS = 1 10 site/m , and the system tem- trolyte with the absolute value of valence Z, the sedimentation- perature T = 298 K. The numerical results of the dimensionless potential field obtained from the cell model is (13) equilibrium surface potential ¯, equilibrium surface-charge den- (eq) sity ¯ , and charge regulation parameter Lacalculated as func- 2 2 ∞ ∞ Z e n tions of the variables n , K+ K , K /K+, and ϕ are plotted in E = U [2b (b)F (b) + D F (b) 4 2 SED bkT 3∞ 0 eq1 0r 1 Figs. 2 and 3. The value of K+ K is fixed at 10 M in Fig. 2 6 and the value of K /K+ is specified at 10 in Fig. 3. It can be Ze ∞ = 1/2 D+ F1+(b)] ¯ (D+ D ) eq1(b) seen that the point of zero charge is given by n (K+ K ) . kT ∞ 1/2 (eq) If n < (K+ K ) , the values of and are negative; the ∞ 2 3 magnitude of decreases monotonically with an increase in n F (b) ¯ + O( ¯ ) e . [24] 1 z for an otherwise specified condition, while (eq) has a maximal ∞ ∞ 1/2 magnitude at some values of n .Ifn >(K+K ) , and (eq) ∞ Substituting Eq. [20a] for U0 into the above equation, we can are positive and both increase with an increase in n until express this field in the form the value of n∞ is very high. The magnitudes of and (eq) increase (so does the value of La)asK /K+ increases, because ε = ( p )gϕ + Ze D+ D the concentration of the unionized surface group AB decreases ESED ∞ E 4 3 kT D D+ with K /K+, as inferred from Eq. [A2]. When the value of K+ K increases, the concentration of negatively charged sur- 2 3 Z E ez + O( ¯ ), [25] face groups A will increase or that of the positively charged Z+ surface group AB2 will decrease according to Eq. [A2]; thus, the particles become more negatively charged or less positively where charged. The magnitude of increases, while the magnitudes of (eq) ϕ 2 and La decrease, as the volume fraction increases, but ( a) Ze ∞ E = [2b eq1(b)F0r (b) these dependencies become negligible when the value of n or 6bϕC02 kT K /K+ is relatively high. + D F1 (b) D+ F1+(b)], [26] For the limiting case of an infinitely dilute suspension of 2 2 1 charged spheres (ϕ = 0), the dimensionless coefficients in ( a) Ze 1 1 = + , Eqs. [22] and [25] for the sedimentation velocity and sedimen- E ϕ eq1(b)F 1(b) [27] 6b C02 kT D+ D tation potential reduce to and ϕ = (a/b)3 is the volume fraction of the particles in the 1 2 a 2 a suspension. The definite integrals in F (b) and F (b)in H = {e [5E6( a) 3E4( a)] 8e [E5( a) E3( a)] 1 1 12 Eqs. [26] and [27] can be performed numerically. Similar to + 2 a }, the definition of H in Eqs. [22] and [23], the dimensionless co- e [7E8(2 a) 3E4(2 a) 4E3(2 a)] [28] efficients E and E defined by Eqs. [25]–[27] are independent = a , E 1 e [5E7( a) 2E5( a)] [29] of the ionic diffusion coefficients. The first-order coefficient E = , is a function of parameters a and ϕ only and does not depend E 0 [30] on the charge regulation parameter La, while the second-order coefficient E is a function of parameters a, La, and ϕ. Note where En is a function defined by that the sign of the O( 2) term in Eq. [25] is decided by that of Z ∞ D+ D . n xt En(x) = t e dt. [31] 1 6. RESULTS AND DISCUSSION It is interesting that these reduced results, which are the same Before the evaluation of the settling velocity of the charge- as the formulas for H and E obtained by Ohshima et al. (7) regulating spheres and the sedimentation potential in the sus- for a single dielectric sphere in an unbounded electrolyte, do not pension from Eqs. [22] and [25], respectively, it is necessary depend on the charge regulation parameter La.However,itis to know how the equilibrium surface potential , equilibrium understood that the value of in Eqs. [22] and [25] for a charge- surface charge density (eq), and charge regulation coefficient L regulating sphere is dependent on the regulation characteristics depend on the bulk electrolyte concentration n∞, surface reac- of the particle and ambient electrolyte solution. tion equilibrium constants K+ and K (defined by Eq. [A2]), The numerical results of the dimensionless coefficient H in and particle volume fraction ϕ. To perform a typical calcula- Eqs. [22] and [23] for the sedimentation velocity of charged tion using Eqs. [15], [A4], and [A8], we make continuous phase spheres in electrolyte solutions calculated as a function of SEDIMENTATION OF CHARGE-REGULATING SPHERES 337
FIG. 2. Plots of the dimensionless equilibrium surface potential ¯, equilib- FIG. 3. Plots of the dimensionless equilibrium surface potential ¯, equilib- rium surface-charge density ¯ (eq), and charge regulation parameter La versus rium surface-charge density ¯ (eq), and charge regulation parameter La versus the bulk concentration n∞ of an aqueous 1 : 1 electrolyte solution under the con- the bulk concentration n∞ of an aqueous 1 : 1 electrolyte solution under the 18 2 4 2 18 2 6 ditions a = 50 nm, NS = 1 10 site/m , and K+ K = 10 M . The solid and condition of a = 50 nm, NS = 1 10 site/m , and K /K+ = 10 . The solid dashed curves represent the cases of the volume fraction ϕ equal to 0.2 and 0, and dashed curves represent the cases of volume fraction ϕ equal to 0.2 and 0, respectively. respectively. 338 DING AND KEH
for very large and very small values of a. For fixed values of La and ϕ, the coefficient H has a maximum at some finite value of a and vanishes in the limits a → 0 and a →∞. The location of the maximum shifts to greater a as La or ϕ increases. For specified values of a and La in a broad range, H is not a monotonic function of ϕ and has a maximal value. The location of this maximum shifts to greater ϕ as a increases or La decreases. The first-order coefficient E in Eq. [25] for the sedimentation potential in suspensions of identical charged spheres is not a function of the charge regulation parameter La, and its relation
FIG. 4. Plots of the dimensionless coefficient H in Eq. [22] for sediment- ing spheres versus parameters a and ϕ. The solid, dotted, and dashed curves represent the cases of the charge regulation parameter La equal to 0, 5, and ∞, respectively. the parameters a, La, and ϕ are plotted in Fig. 4 for the Happel cell model. The calculations are presented up to ϕ = 0.74, which corresponds to the maximum attainable volume fraction for a swarm of identical spheres (24). The fact that H is always positive demonstrates that the presence of the surface charges reduces the sedimentation rate for any volume fraction of particles in the suspension. It can be seen that the effect of the surface charges on the sedimentation rate decreases gently FIG. 5. Plots of the dimensionless coefficient E in Eq. [25] for the sed- and monotonically with an increase in the charge regulation pa- imentation potential in a suspension of identical spheres versus parameters a rameter La for given values of a and ϕ. This effect becomes and ϕ. The solid, dotted, and dashed curves represent the cases of the charge independent of La in the case of very dilute suspensions and regulation parameter La being equal to 0, 5, and ∞, respectively. SEDIMENTATION OF CHARGE-REGULATING SPHERES 339 to the parameters a and ϕ has already been discussed in a was shown to give an excellent approximation for the case of previous article (13; the term ϕ 1/3( a)3 in its Eqs. [52a] and reasonably high zeta potential (with an error less than 0.1% for [52b] should be omitted). Figure 5 shows plots of numerical | |e/kT 2 in a KCl solution) (7). Therefore, our results might results for the second-order coefficient E as a function of the be used tentatively for the situation of reasonably high electric parameters a, La, and ϕ for the Happel model. Analogous to potentials. the coefficient H illustrated in Fig. 4, E is always a positive value, decreases monotonically with an increase in the charge APPENDIX A regulation parameter La for fixed values of a and ϕ, has a maximum at some finite value of a for constant values of La Model for a Charge-Regulating Surface ϕ ϕ and , and may be maximal at a value of for given values of a Following the previous studies (15, 16, 22), we consider a and La. For the limiting cases a → 0, a→∞,orϕ→0, general model for the charge-regulating surface which develops the value of E equals zero. surface charges via association/dissociation equilibrium of iono- For conciseness, the numerical results of the dimensionless genic surface groups. The surface reactions may by expressed , coefficients H and E as functions of the parameters a La, as and ϕ for the Kuwabara cell model are not presented here. How- Z+ ⇔ + Z+, ever, it can be shown that these results are qualitatively similar AB2 AB B [A1a] and quantitatively close to those for the Happel cell model. For ⇔ Z + Z+, a given suspension of identical charge-regulating spheres, the AB A B [A1b] sedimentation velocity and potential for each model can be eval- where AB represents the associable/dissociable functional uated as functions of the regulation characteristics of the suspen- + ∞ group on the surface, B Z denotes the ion that determines the sta- sion (such as n , K+, K , NS, a, and ϕ, etc.) from Eqs. [22] and [25] incorporating Eqs. [15], [A4], and [A8]. These func- tus of charges on the surface groups (the potential-determining tions are quite complicated for most situations, and cannot be ion), and the positive integer Z is the valence of ionization. For Z+ predicted systematically by simple general rules. the case of an amphoteric surface, B is usually the hydrogen ion H+. The equilibrium constants for the reactions in Eq. [A1] are given by 7. SUMMARY = Z+ Z+ , K+ [AB][B ]S AB2 [A2a] In this work, the steady-state sedimentation phenomena in ho- = Z Z+ / , mogeneous suspensions of identical charge-regulating spheres K [A ][B ]S [AB] [A2b] in electrolyte solutions with arbitrary values of a, La, and ϕ Z+ Z+ are analyzed by employing the Happel and Kuwabara cell mod- where [B ]S is the concentration of B next to the surface. els. Solving the linearized electrokinetic equations applicable to The surface dissociation constants K+ and K are taken to be the system of a sphere in a unit cell by a regular perturbation functions of temperature only. method, we have obtained the ion concentration (or electrochem- For NS ionizable surface groups per unit area, the net surface ical potential energy) distributions, the electric potential profile, charge density is and the fluid flow field through the use of a linearized charge Z+ Z AB2 [A ] regulation model. The requirement that the total force exerted = ZeNS + Z+ + Z on the particle be zero leads to an explicit formula, Eq. [22], for [AB] AB2 [A ] Z+ 2 the settling velocity of the charged spheres. The correction for [B ] K+K = S . the effect of the particle charges to the settling velocity begins ZeNS + + 2 [A3] K+[BZ ] +[BZ ] + K+K at the second order, 2. Numerical results indicate that this ef- S S fect has a maximum at some finite values of a and disappears By the substitution of the Boltzmann distribution for the equi- when a approaches zero and infinity. Another explicit for- librium concentration of B Z+ and the utilization of the concept mula, Eq. [25], for the sedimentation potential is also derived to of electrochemical potential energy (7), Eq. [A3] for can be order 2 by letting the net electric current in the suspension be expressed in terms of the surface potential S as zero. It is found that the effects of charge regulation at the par- { + / } ticle surface on the sedimentation velocity and potential occur = sinh [Ze( N S) S] kT , ZeNS [A4] in the leading order of , which is dependent on the regulation 1 + cosh{[Ze( N S)+ S]/kT} characteristics of the suspension. Equations [22]–[27] are obtained on the basis of the Debye– where Huckel approximation for the equilibrium potential distribution 1/2 = 2(K /K+) , [A5] around the charged sphere in a unit cell. The reduced formula of ∞ Eq. [23] for the sedimentation velocity of a charged sphere with = kT n , N ln / [A6] low zeta potential in an unbounded electrolyte solution, Eq. [28], Ze (K+ K )1 2 340 DING AND KEH Z+ 2 2 S is the deviation in electrochemical potential of B next to a r ε a F (r) = C + 10C + the surface from the equilibrium state defined by Eq. [6], and pi i2 r i4 a i2 12 n∞ is the concentration of B Z+ in the bulk solutions, where the 1 equilibrium potential is set equal to zero. Equation [A6] is the G (r) rG (r) , [B1c] 2 2 1 Nernst equation relating the Nernst potential N to the isoelec- 2r ∞ 1/2 tric point [with n = (K+ K ) ]. It can be seen from Eq. [A4], which acts as an equation of the electric state of the surface, that for i = 0 and 2. In the above equations, the functions Gn(r) are the sign of is opposite that of S N at equilibrium (with defined by Eq. [21], ij is the Kronecker delta which equals unity S = 0). The equilibrium surface-charge density approaches if i = j but vanishes otherwise, the saturation values ZeNS when the difference between the equilibrium surface potential and its Nernst value becomes large ∞ 1/2 1 5/3 (e.g., when the value of n /(K+ K ) approaches zero or in- C = 2 + 3ϕ A + 5ϕB ω, [B2a] i1 2 i i finity and the value of S is finite). 1 With the relationship between and S given by Eq. [A4], 5/3 2/3 Ci2 = 3 + 2ϕ Ai + 5ϕ Bi ω, [B2b] the charge regulation capacitance of the surface at equilibrium 2 can be written as 1 C = A + 3ϕ2/3 2ϕ B ω, [B2c] ! i3 2 i i d ε = L, [A7] 1 d 4 C = ϕ5/3 A 2ϕ2/3 3ϕ B ω [B2d] S = i4 i i S 2 where is the value of S at equilibrium and for the Happel model, and 2 2 + (eq) 4 Z e NS cosh Ze N kT = . 1 0 / 0 0 L 2 [A8] = + ϕ + ϕ + ϕ5 3 ω , ε + (eq) Ci1 (2 )Ai 2 5 3 Bi [B3a] kT 1 cosh Ze N kT 2 1 = 0 + ϕ2/3 + ϕ5/3 0 ω0, The reciprocal of the positive quantity L can be regarded as the Ci2 3Ai 3 5 2 Bi [B3b] characteristic length controlling the charge regulation condition 2 at the surface. The limiting values of L = 0 and L →∞corre- 1 0 / 0 0 C = (5 2ϕ)A + 5 1 3ϕ2 3 + 2ϕ B ω , [B3c] spond to the cases of constant surface-charge density and con- i3 10 i i stant surface potential, respectively. Note that L is small when 1 = ϕ 0 + ϕ2/3 ϕ + ϕ5/3 0 ω0 the difference between the equilibrium surface potential and its Ci4 3 A 5 2 3 B [B3d] 10 i i Nernst value is large. for the Kuwabara model, where APPENDIX B Definitions of Some Functions in Section 3 ε 2 1 A = + G (b) G (b) , [B4a] i i0 i2 24 1 b 2 For conciseness the definitions of some functions in Section 3 are listed here. In Eq. [17], ε 2 1 2 B = G (b) b G (b) , [B4b] i i2 120 b3 4 1 3 2 ε 2 = + a + a + r + Fir(r) Ci1 Ci2 Ci3 Ci4 i2 ε 2 1 r r a 24 A0 = + G (b) G (b) i i0 i2 24 1 b 2 1 1 r2 G (r) G (r)+ G (r) G (r) , 2 1 r 2 5r3 4 5 1 1 b + G (b) G (b) , [B5a] 3 4 1 [B1a] 5b 5 3 2 ε 2 C a C a r 0 1 2 i2 i3 B = i2 G2(b)+b G 1(b) , [B5b] Fi (r) = Ci1 + 2Ci4 i 2 r 2 r a 120 b 2 1 ε 1 1 3 / 3 / + G (r)+ G (r) ω = 1 ϕ1 3 + ϕ5 3 ϕ2 , [B6a] i212 2 1 4r 2 2 2 2 1 1 r 0 9 1/3 1 2 + G (r) + G (r) , [B1b] ω = 1 ϕ + ϕ ϕ . [B6b] 20r 3 4 5 1 5 5 SEDIMENTATION OF CHARGE-REGULATING SPHERES 341 + In Eqs. 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