Electrophoresis of a Concentrated Spherical Dispersion
at Arbitrary Electrical Potentials
Wen-Hsun Lin, Eric Lee, and Jyh-Ping Hsu1
Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan 10617, Republic of China
Received September 5, 2001; accepted January 12, 2002; published online March 13, 2002
The electrophoretic behavior of a concentrated spherical disper-sion is investigated theoretically. The present analysis extends those in the literature in that both the surface potential of a particle and the strength of the applied electric field are arbitrary and both the effects of double-layer polarization and the overlapping between neighboring double layers are taken into account. Results based on these conditions are highly desirable since they cover essentially all the possible experimental conditions in practice. We show that, for a fixed surface potential and strength of applied electric field, the higher the concentration of particle, the smaller the mobility. Coun-terions are found to accumulate at the downstream side of a particle. Double-layer polarization is inappreciable if either it is thick or the concentration of the particle is high. C2002 Elsevier Science (USA)
Key Words: electrophoresis; spherical dispersion; concentrated dispersion; double-layer polarization; double-layer overlapping.
INTRODUCTION
Electrophoresis, one of the most significant electrokinetic phenomena, has various applications in practice. For example, it is widely used to separate proteins, to measure the stability of a colloidal dispersion, and to characterize the surface prop-erties of biological cells such as bacteria and blood cells, to name a few. The mathematical description of electrophoresis involves the governing equations for the flow, the electrical, and the concentration fields. In general, these equations are cou-pled, nonlinear partial differential equations (1), and solving them simultaneously is nontrivial even if a numerical scheme is adopted. To circumvent this difficulty, assumptions such as low surface potential, thin or thick double layers, and weak applied electric field are often made in previous studies. Un-der these conditions the governing equations can be decoupled and approximated by linear expressions, which are more readily solvable.
The electrophoretic behavior of a colloidal dispersion depends upon various factors such as the nature of colloidal particle, the bulk concentration of electrolytes (or thickness of double layer), the strength of applied electric field, and the presence of a
bound-1To whom correspondence should be addressed. Fax: 886-2-23623040.
E-mail: jphsu@ccms.ntu.edu.tw.
ary. The last factor includes, for instance, the existence of a rigid boundary (2–4) and a concentrated dispersion (5–8) in which the interaction between neighboring particles is important. Previous analyses focused mainly on a dilute dispersion where only the behavior of an isolated entity is considered (9–15). Levine and Neale (5) investigated the electrophoresis of a swarm of particles taking the effect of the interaction between neighboring particles into account. Under the assumption of low surface potential, an analytical expression, which is applicable to an arbitrary thick-ness of double layer, for electrophoretic mobility was derived. Kozak and Davis (6, 7) extended the analysis of Levine and Neale to the case of an arbitrary surface potential and double-layer thickness. The result obtained, however, is limited to the case when the overlapping between neighboring double layers is unimportant. Ohshima (8) was able to derive an approximate expression for the electrophoretic mobility of a concentrated dispersion in which the tedious recursive procedure involved in Levine and Neale can be avoided. The performance of the result obtained was found to be satisfactory with a relative percentage deviation less than 4%. O’Brien and White (14) assumed that the applied electric field is weak and the governing electrokinetic equations could be linearized. The electrical potential was parti-tioned into two parts and the original problem is divided into two subproblems. They concluded that if the double layer is either very thin or very thick, the effect of its polarization is negligible. Also, the variation of electrophoretic mobility as a function of double-layer thickness may have both a local maximum and a local minimum. Following the same procedure as that employed by O’Brien and White (14), Lee et al. (16) considered the case of arbitrary surface potential, double-layer thickness, and con-centration of the particle. In addition, the effects of double-layer polarization and relaxation and the overlapping of neighboring double layers were taken into account.
In this study the analysis of Lee et al. (16) is further extended to the case where the strength of the applied electric field is arbitrary. This is of practical significance since the general elec-trokinetic equations are solved and essentially all the significant factors are taken into account. The electric field used by Pikhitsa
et al. (17) in a study of the drift velocity of particles in water, for
example, ranged from 250 to 1000 V/cm. Apparently, previous results based on a weak applied electric field are inapplicable to correlate their data.
398 0021-9797/02 $35.00
C
2002 Elsevier Science (USA) All rights reserved.
FIG. 1. The cell model adopted. A spherical particle of radius a is enclosed by a concentric liquid shell of radius b. The origin of the spherical coordi-nates (r, θ, ϕ) is located at the center of the particle. E is the applied electric field.
THEORY
We consider a concentrated dispersion of nonconducting, monodispersed, positively charged spherical particles in a z1: z2
electrolyte solution, z1and z2being the valences of cations and
anions, respectively. The cell model of Kuwabara (18) is adopted to simulate the system under consideration. As shown in Fig. 1, it is modeled by a representative cell, which comprises a spher-ical particle of radius a surrounded by a concentric spherspher-ical liquid shell of radius b. A uniform electric field E is applied in the Z -direction, and the particle moves in the same direction with velocity U. The spherical coordinates (r, θ, φ) are adopted in the following analysis. The conservation of ions leads to
∂nj ∂t = ∇ · Dj ∇nj+ zjenj kBT ∇φ − njv , [1]
where∇ is the gradient operator, nj is the number density of ionic species j ( j = 1, cations, 2, anions), Dj and zje are the diffusivity and the charge of ionic species j, φ is the electric potential, kB and T are, respectively, the Boltzmann constant and the absolute temperature, and v is the fluid velocity. We assume that the electrical potential is described by the Poisson equation ∇2φ = −ρ ε = − 2 j=1 zjenj ε , [2]
whereε is the permittivity of the liquid phase and ρ is the space charge density.
Suppose that the flow field around the particle in the represen-tative cell can be described by the Navier–Stokes equations in the creeping flow regime with electrical body forces considered. We have
∇ · v = 0 [3]
ρf
∂v
∂t = η∇2v− ∇ p − ρ∇φ, [4]
where p is the pressure andρf andη are the density and the viscosity of the liquid phase, respectively. Here, we assume that the liquid phase is incompressible and has constant physical properties.
For convenience, the electrical potentialφ is decomposed into the electrical potential that would exist in the absence of the applied field,φ1, and that outside the particle which arises from
the applied electrical field,φ2, that is,
φ = φ1+ φ2. [5]
The effect of double-layer relaxation is taken into account by defining the potential gjthrough
nj= nj 0exp −zje(φ1+ φ2+ gj) kBT , [6]
where nj 0is the bulk concentration of ionic species j .
For a more concise presentation, the governing equations are rewritten in scaled forms. In the discussion below, a symbol with an asterisk denotes a scaled quantity. For example, the scaled number concentration of ions, n∗1and n∗2are
n∗1 = exp[−φr(φ1∗+ φ∗2+ g∗1)] [7a]
n∗2 = exp[αφr(φ1∗+ φ2∗+ g2∗)]. [7b]
The scaled governing equation forφ1is
∇∗2φ∗ 1 = − 1 (1+ α) (κa)2 φr [exp(−φrφ1)− exp(αφrφ1)], [8]
where φr = ζaz1e/kBT is the scaled surface potential, α = −z2/z1, and κ = [
2
j=1nj 0(ezj)2/εkBT ]1/2 is the reciprocal Debye length. The scaled boundary conditions associated with Eq. [8] are φ∗ 1 = 1, r∗= 1 [8a] ∂φ∗ 1 ∂r∗ = 0, r∗= 1/λ, [8b]
The scaled governing equation forφ2∗is ∇∗2 φ∗ 2 = − 1 (1+ α) (κa)2 φr {(n∗ 1− n∗2) − [exp(−φrφ1∗)− exp(αφrφ1∗)]}. [9] The associated scaled boundary conditions are
∂φ∗ 2 ∂r∗ = 0, r∗= 1 [9a] ∂φ∗ 2 ∂r∗ = −E∗zcosθ, r∗= 1/λ, [9b] where E∗z = Eza/ζa. At steady state, the left-hand side of Eq. [1] vanishes. In this case, substituting Eqs. [5] and [6] into Eq. [1] yields the following scaled equations for g∗j(g∗j = gj/ζa):
∇∗2 g∗1− φr∇∗φ∗1· ∇∗g∗1 = Pe1u∗· ∇∗φ1∗+ Pe1u∗· ∇∗φ2∗+ Pe1u∗· ∇∗g∗1 + φr∇∗φ∗2· ∇∗g∗1+ φr∇∗g∗1· ∇∗g∗1 [10] ∇∗2 g∗2+ αφr∇∗φ∗1· ∇∗g2∗ = Pe2u∗· ∇∗φ1∗+ Pe2u∗· ∇∗φ2∗+ Pe2u∗·∇∗g∗2 − αφr∇∗φ∗2·∇∗g∗2− αφr∇∗g∗2· ∇∗g2∗. [11] The associated scaled boundary conditions are
∂g∗ 1 ∂r∗ = 0, r∗ = 1 [11a] g1∗= −φ∗2, r∗= 1/λ [11b] ∂g∗ 2 ∂r∗ = 0, r∗ = 1 [11c] g2∗= −φ∗2, r∗= 1/λ. [11d] In these expressions Pej = UEa/Djis the electric Peclet num-bers for ionic species j, UE = ε(z1e/kBT )2/ηa being the elec-trophoretic velocity based on Smoluchowski’s result for an iso-lated particle in an electric field of strengthζa/a.
The scaled Navier–Stokes equation, in terms of scaled stream function, and the associated scaled boundary conditions are
E∗4ψ∗= − (κa) 2 (1+ α) ∂g∗ 1 ∂r∗n∗1+ ∂g∗ 2 ∂r∗αn∗2 ∂φ∗ ∂θ − ∂g∗ 1 ∂θn∗1+ ∂g∗ 2 ∂θ αn∗2 ∂φ∗ ∂r∗ sin θ [12] ψ∗ = −1 2U ∗r∗2sin2θ, r∗= 1 [12a] ∂ψ∗ ∂r∗ = −U∗r∗sin2θ, r∗ = 1 [12b] ψ∗= 0, r∗= 1/λ [12c] 1 r∗sinθ ∂2 ∂r∗2− cosθ r∗3sin2θ ∂ ∂θ + 1 r∗3sinθ ∂2 ∂θ2 ψ∗= 0, r∗= 1/λ. [12d]
In these expressions,ψ∗= ψ/UEa2, U∗= U/UE, and E∗4=
E∗2E∗2, where E∗2= ∂ 2 ∂r∗2 + sinθ r∗2 ∂ ∂θ 1 sinθ ∂ ∂θ . [12e]
The physical meaning of each boundary condition can be found in Lee et al. (16).
The external force acting on a particle in the Z -direction in-cludes the electrical force and the hydrodynamic force. The scaled form of the former, FE z, is
FE z= 2πεζa2 π 0 ∂φ∗ 1 ∂r∗ r∗=1 ∂(φ∗ 1+ φ2∗) ∂r∗ cosθ − 1 r∗ ∂(φ∗ 1+ φ∗2) ∂θ sinθ r∗=1 r∗2sinθ dθ = 2πεζ2 aE∗zKE, [13] where Ez∗KE = π 0 ∂φ∗ 1 ∂r∗ r∗=1 ∂(φ∗ 1+ φ2∗) ∂r∗ cosθ − 1 r∗ ∂(φ∗ 1 + φ2∗) ∂θ sinθ r∗=1 r∗2sinθ dθ. [13a]
The scaled form of the hydrodynamic force, FDz, can be ex-pressed by FDz = πεζa2 π 0 r∗4sin3θ ∂ ∂r∗ E∗2ψ∗ r∗2sin2θ r∗=1 dθ − πεζ2 a (κa)2 (1+ α)φr π 0 r∗2sin2θ(n∗1− n∗2)∂φ ∗ 2 ∂θ r∗=1 dθ = πεζ2 a U∗KD f − (κa)2 (1+ α)φr E∗zKDe = F∗ D f + FDe∗ , [14] where U∗KD f = π 0 r∗2sin3θ ∂ ∂r∗ E∗2ψ∗ r∗2 sin2θ r∗=1 dθ [14a] Ez∗KDe= π 0 r∗2sin2θ(n∗1− n∗2)∂φ ∗ 2 ∂θ r∗=1 dθ. [14b]
FD f∗ and FDe∗ are the scaled viscous force and the scaled electric body force exerted on the particle. At steady state the net force acting on the particle vanishes, that is,
FDz+ FE z= 0 [15]
It can be shown that the scaled electrophoretic mobility of the particle, Um∗, is Um∗ = ηU εζaE = U∗ Ez∗ = (κa)2[KDe/(1 + α)φr]− 2KE KD f [16]
Note that since KE, KDe, and KD fare all functions of the applied electric field and the terminal velocity of the particle, Um∗needed to be calculated through an iterative procedure.
RESULTS AND DISCUSSION
The governing equations and the associated boundary con-ditions are solved by the pseudo-spectral method based on the Chebyshev polynomials (4). Although this method is appropri-ate forκa smaller than about 10, it is sufficient for our purpose since we are mainly interested in the behavior of the system un-der consiun-deration for the case of medium to thick double layers. Figure 2 shows the variation of the scaled mobility Um∗ as a function ofκa at various scaled applied field E∗zfor two different levels of scaled surface potential. For comparison, the results of Lee et al. (16) in which the applied electric field is weak and the governing equations, Eqs. [9]–[12], are linearized are also pre-sented. Figure 2 reveals that ifκa is small (thick double-layer), both models lead to essentially the same result. However, ifκa has a medium value, the linear result of Lee et al. (16) is ap-propriate for Ez∗∼= 1 at φr = 3 and for Ez∗ 1 at φr = 5. A comparison between Fig. 2 suggests that, forκa in the range [1, 10], the behavior of Um∗ againstκa curve at a higher φr is more complicated than that at a lowerφr. As pointed out by Lee
et al. (16), this is due to the fact that asκa increases (double
layer becomes thin), the electric field induced by a charged par-ticle, which is in the opposite direction as the applied electric field, becomes more important. Therefore, if the applied electric field is weak, it will be dominated by the electric field induced by the particle. Apparently, for a fixed surface potential, the weaker the applied electric field, the more complicated the be-havior of the Um∗ againstκa curve, as is justified in Fig. 2b. On the other hand, if the applied electric field is sufficiently strong, it will dominate the electric field induced by a particle and the
Um∗ againstκa curve is more smooth.
Figure 3 illustrates the variation of the scaled mobility Um∗ as a function ofκa at various λ (=a/b), for the case φr = 1, and that for the caseφr = 5 is presented in Fig. 4. Note that λ is a measure for the concentration of the particle; the larger its value, the higher the concentration. These figures reveal that, for a fixed surface potential and applied electric strength, the higher the concentration of the particle, the smaller the mobility. This
FIG. 2. Variation of scaled mobility Um∗as a function ofκa at various scaled
applied field E∗z . φr = 2 (a), φr= 5 (b). The linear case is based on Lee et al.
(16). Key:λ = 0.5, α = 1, and Pe1= Pe2= 0.01.
is mainly due to the electrical interaction between neighboring particles. Figure 5 shows the variation of 2KE as a function of
κa at various λ for the case of Fig. 3. It can be shown that the KE in Eq. [16] can be rewritten as
KEE∗z = − π 0 ∂φ∗ 1 ∂r∗ r∗=1 ∂φ∗ 2 ∂θ r∗=1 sin2θ dθ. [17]
According to Eq. [16], the larger the KE, the smaller the mo-bility. If E∗z is constant, then −(∂φ1∗/∂r∗)r∗=1 increases with the increase inκa. Due to the interaction between neighboring particles, this effect is pronounced if the concentration of the
FIG. 3. Variation of scaled mobility Um∗ as a function ofκa at various
λ (=a/b) for the case φr = 1, E∗z = 3, Pe1= Pe2= 0.01, and α = 1.
particle is high. Ifκa is large, this leads to a drastic increase in KE.
Figures 6–9 show the contours of the scaled net ion concen-trations CD (=n∗
1− n∗2) at various conditions. In our case since
a particle is positively charged, the concentration of anions near it is higher than that of cations and CD is negative. Also, the ap-plied electric field is in the positive Z direction, and the particle moves in this direction too. In Fig. 6, the applied electric field is weak, and the polarization of the double layer is inapprecia-ble. As the strength of the applied electric field increases, the distortion of the double layer surrounding a particle becomes significant, as illustrated in Fig. 7. This figure shows that
an-FIG. 4. Variation of scaled mobility Um∗ as a function ofκa at various
λ (=a/b) for the case for Fig. 3 except that φr= 5.
FIG. 5. Variation of 2KE as a function ofκa at various λ for the case of
Fig. 3.
ions will accumulate at the rear of a particle. If the double layer surrounding a particle is thin, its polarization becomes inappre-ciable, as can be seen in Fig. 8. This is because that for a fixed surface potential the thinner the double layer (larger κa), the stronger the electric field induced by a particle, which is pro-portional to (∂φ1∗/∂r∗)r∗=1. Since this induced electric field is in the opposite direction as that of the applied electric field, the
FIG. 6. Contours of net scaled ion concentrations CD (=n∗1− n∗2) for the case φr= 3, Ez∗= 0.1, κa = 0.1. Key: λ = 0.5, Pe1= Pe2= 0.01, and
FIG. 7. Contours of net scaled ion concentrations CD (=n∗1− n∗2) for the caseφr = 3, Ez∗= 3, κa = 0.1. Key: same as Fig. 8.
extent of double-layer polarization is lowered. A comparison between Figs. 7 and 9 reveals that double-layer polarization is less significant if the concentration of the particle is high. This is because the interaction between neighboring particles becomes significant, which has the effect of compressing the double layer surrounding a particle.
FIG. 8. Contours of net scaled ion concentrations CD (=n∗1− n∗2) for the
caseφr = 3, Ez∗= 3, κa = 5.6. Key: same as Fig. 8.
FIG. 9. Contours of net scaled ion concentrations CD (=n∗1− n∗2) for the
caseφr= 3, Ez∗= 3, κa = 0.1, λ = 0.8, Pe1= Pe2= 0.01, and α = 1.
ACKNOWLEDGMENT
This work is supported by the National Science Council of the Republic of China.
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