Dynamic Electrophoretic Mobility of a Concentrated Dispersion of
Particles with a Charge-Regulated Surface at Arbitrary Potential
Eric Lee, Chi-Hua Fu, and Jyh-Ping Hsu1
Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan 10617, Republic of China
Received December 3, 2001; accepted February 14, 2002; published online May 15, 2002
The dynamic electrophoretic mobility of a concentrated disper-sion of biocolloids such as cells and microorganisms is modeled theoretically. Here, a biological particle is simulated by a particle, the surface of which contains dissociable functional groups. The results derived provide basic theory for the quantification of the surface properties of a biocolloid through an electroacoustic device, which has the merit of making direct measurement on a concen-trated dispersion without dilution. Two key parameters are defined to characterize the phenomenon under consideration: the first, A, is associated with the pH of the dispersion, and the second, B, is associated with the equilibrium constant of the dissociation reac-tion of the funcreac-tional group. We show that if A is large and/or B is small, the surface potential is high, and the effect of double-layer polarization becomes significant. In this case the dynamic electrophoretic mobility may have a local maximum and a phase lead as the frequency of the applied electric field varies. Due to the hydrodynamic interaction between neighboring particles, the dy-namic electrophoretic mobility decreases with the concentration of dispersion. C2002 Elsevier Science (USA)
Key Words: dynamic electrophretic mobility; concentrated dis-persion; charge-regulated surface; arbitrary potential.
I. INTRODUCTION
Electrokinetic theory is one of the basic tools for the descrip-tion of the behavior of charged entities. It also provides the foun-dation for the design of the instruments used to quantify their physicochemical properties. The result of Smoluchoski for elec-trophoresis (1), for example, is often used to estimate the charged conditions of the surface of an entity. This result was also ex-tended by many researchers to more general cases, which take the effects of practical significance into account (2–5). These include, for example, double-layer polarization (2), arbitrary double-layer thickness and arbitrary surface potential (4, 5), and double-layer overlapping and dynamic applied electric field (6–26).
An electroacoustic instrument is a powerful device developed for direct measurement of the properties of a concentrated
1To whom correspondence should be addressed. Fax: 886-2-23623040.
E-mail: jphsu@ccms.ntu.edu.tw.
dispersion (6). In contrast to the static electric field applied in the classic electrophoresis measurement, an alternating electric field is applied, and the electrophoretic behavior observed is of a dynamic nature. The result based on electroacoustic measure-ment is found to be more accurate than that based on the classic method, which involves inevitably some optical limitation (7). The description for the dynamic phenomenon involves solving simultaneously a set of coupled, nonlinear differential equations, which describe electric, flow, and concentration fields. O’Brien (6) was able to derive the dynamic electrophoretic mobility of a dilute dispersion for the case of thin double-layer and low surface potential. His result was extended to more general cases such as arbitrary double-layer thickness and surface potential (8–14). Ohshima (14), for example, investigated the dynamic electro-phoretic behavior of a dilute dispersion under the conditions of low surface potential and arbitrary double-layer thickness.
Most of the available theoretical results for dynamic elec-trophoresis based on an electroacoustic measurement are valid for the case when the volume fraction of the dispersed phase is below 5%. This, however, is unsatisfactory from a practical point of view since dispersion needs to be diluted before its properties can be measured. Apart from a tedious procedure, a diluted dis-persion is easier to be affected by various factors such as the con-centrations of ionic species and solvent. Therefore, direct mea-surement on concentrated dispersions is highly desirable, and the corresponding theoretical analysis becomes imminent. Ohshima (15–17) adopted the cell model of Kuwabara (18), in which a dispersion is simulated by a representative particle enclosed by a concentric spherical liquid shell, to analyze the electrophoretic behavior of a concentrated dispersion. The electric potential on a particle surface was assumed to be low, and the boundary con-dition for electric potential on a cell surface was the same as that adopted by Levine and Neale (19); that is, it is Neumann type. Although the effect of double-layer overlapping was considered, the result obtained still deviated significantly from experimental observations. Dukhin et al. (20) pointed out that the boundary condition used by Shilov and Zharkikh (21), which is Dirichlet type, is more appropriate than that of Levine and Neale (19) for a concentrated dispersion. Lee et al. (22) considered the dy-namic electrophoresis of a concentrated dispersion by adopting the boundary condition of Shilov and Zharkikh (21). The ef-fects of double-layer polarization and the overlapping between
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2002 Elsevier Science (USA) All rights reserved.
neighboring double layers have been taken into account, and the result obtained is applicable to arbitrary double-layer thickness, surface potential, and the frequency of the applied electric field. Previous analyses assumed that the surface of a particle re-mains either at constant potential or at constant surface charge density. These assumptions, although they make the mathemati-cal treatment simpler, can be appropriate for biocolloids such as cells and microorganisms. In a study of the adhesion of biologi-cal cells Ninham and Parsegian (27) pointed out that to minimize the total interaction potential, cells are capable of regulating the charged conditions on their surface toward maintaining the inter-action surfaces at an ionic equilibrium with the bulk suspension medium. In this case, neither the potential nor the charge den-sity can remain constant during the period of adhesion. Another example is particles covered by an artificial membrane layer where the charged condition depends upon the properties of the latter. The surface of biocolloids usually carry dissociable func-tional groups. The dissociation of these funcfunc-tional groups yields a charged entity, and the charged condition is a function of the degree of dissociation of the functional groups, which in turn, is affected by factors such as the pH and the ionic strength of the surrounding medium.
In this study the dynamic electrophoretic mobility of a con-centrated dispersion of spherical particles, the surface of which contains dissociable functional groups, is analyzed, taking the effect of double-layer polarization into account. The result de-rived is capable of describing the electrophoretic behavior of bi-ological particles such as cells and microorganisms. The key fac-tors, which characterize the phenomenon under consideration, including pH and the equilibrium constant of the dissociation reaction of the functional group, are discussed. The governing equations are solved numerically by a pseudospectral method based on the Chebyshev polynomial, which is justified to be efficient for the present type of problem (22–26).
II. THEORY
We consider concentrated, monodispersed, spherical parti-cles in a z1: z2 electrolyte solution, z1 and z2 being respec-tively the valences of cations and anions with z1= −αz2. If
n10 and n20 are the bulk concentrations of cations and anions, then n20= (n10/α). The surface of a particle contains dissocia-ble functional groups HA, the dissociation of which yields a negatively charged particle. Referring to Fig. 1, the cell model of Kuwabara (18) is adopted to simulate the system under con-sideration where a representative particle of radius a is enclosed by a concentric liquid shell of radius b. The spherical coordi-nates (r, θ, ϕ) are adopted with the origin located at the center of the representative particle. We define H= λ3, withλ = (a/b). Then H is an estimate for the volume fraction of particles. An electric field is applied which yields an average electric field
E= EZe−iωteZ in the ±Z-direction, where EZ is the magni-tude of E, i=√−1, eZ is the unit vector in the Z -direction, andω and t are respectively the frequency of the applied
elec-U
a b E
ez
FIG. 1. Schematic representation of the system considered, which is rep-resented by a representative particle of radius a and a concentric liquid shell of radius b. E is the average dynamic electric field and U is the electrophoretic velocity.
tric field and time. Let U be the velocity of the representa-tive particle U= U EZe−iωteZ, where U is the magnitude of U,
U= UR+ iUI, UR and UI being respectively the real and the imaginary parts of U .
2.1. Flow Field
We assume that the liquid phase contains incompressible Newtonian fluid with constant physical properties and the flow field can be described by the Navier–Stokes equation in the creeping flow regime
∇ · u = 0 [1]
ρf
∂u
∂t = −∇ p + η∇2u− ρc∇φ. [2]
In these expressions∇ is the gradient operator, u is fluid velocity,
ρf andη are respectively the density and the viscosity of the fluid, p is the pressure,ρcis the space charge density,∇2is the Laplace operator, andφ is the electrical potential.
2.2. Concentration Field
The variation of the concentration of ionic species j can be described by ∂nj ∂t = −∇ · fj [3] fj = −Dj ∇nj+ njˆezj kBT ∇φ + nju, [4] where nj, fj, and Dj are respectively the concentration, the flux, and the diffusivity of ionic species j , kB is the Boltzmann constant, ˆe is the elementary charge, and T is the absolute temperature. Equation [3] represents the conservation of ions,
and Eq. [4] states that the driving force for fj comprises that contributed by the concentration gradient, the electrical poten-tial gradient, and the flow of the liquid phase.
2.3. Electrical Field
Suppose that the electrical field can be described by the Pois-son equation ∇2φ = −ρc ε = − 2 j=1 zjˆenj ε , [5]
whereε is the permittivity of the liquid phase.
We assume that the applied electric field is weak, and each dependent variable can be expressed as the sum of its value in the absence of the applied electric field, i.e., the equilibrium value, and a perturbed term. Also, since the problem under considera-tion isϕ-symmetric, we have
φ(r, θ, t) = φe(r )+ δφ(r, θ)EZe−iωt [6]
nj(r, θ, t) = nej(r, θ) + δnj(r, θ)EZe−iωt [7]
u(r, θ, t) = 0 + δu(r, θ)EZe−iωt [8]
p(r, θ, t) = pe(r, θ) + δp(r, θ)EZe−iωt. [9] These expressions imply that the governing equation for a depen-dent variable can be decomposed into an equation describing its equilibrium state (i.e., state in the absence of the applied electric field) and that describing its perturbed state.
2.4. Equilibrium State
In the absence of the applied electric field u vanishes, and Eqs. [1] and [2] yield
−Dj ∇ne j+ ne jˆezj kBT ∇φe = 0. [10]
Integrating this expression, we obtain
nej = n e j 0 exp −zjˆeφe kBT , [11]
where nej 0is the bulk concentration of ionic species j . Substi-tuting Eq. [5] into Eq. [11] gives
∇∗2φ∗ e(r∗)= − 2 j=1 zjˆe2nej 0a2 εkBT exp(−zjφ∗e), [12]
where r∗ = r/a and φe∗= φe/(kBT/ ˆe).
The present cell model requires that a cell as a whole is elec-trically neutral, which implies that there is no net current across
the virtual surface, r= b. Therefore,
dφ∗
e
dr∗ = 0, r
∗= b/a. [13]
We consider the following dissociation reaction on a particle surface:
AH⇔ A−+ H+. [14]
The equilibrium constant Ka can be expressed as
Ka =
[ A−][H+]s
[ AH] , [15]
where a symbol with a square bracket represents concentration and [H+]s is the concentration of H+on a particle surface. We assume that the spatial variation of [H+] follows the Boltzmann distribution, that is,
[H+]s= [H+]0exp −eφe kBT , [16]
where [H+]0is the bulk concentration of H+. If we let Nsbe the concentration of the dissociable functional groups on a particle surface, then
Ns = [A−]+ [AH]. [17]
Combining Eqs. [15]–[17] gives
[ A−]= Ns
1+ ([H+]0/Ka) exp(−eφe/kBT )
. [18]
If we letσ be the charge density on a particle surface, then
σ = −e[A−]. [19]
Substituting Eq. [18] into Eq. [19] yields
σ = − eNs
1+ ([H+]0/Ka) exp(−eφe/kBT )
. [20]
Suppose that the permittivity of a liquid is much larger than that of a particle. Then, applying Gauss law yields
σ = −ε dφe dr r=a . [21]
Substituting Eq. [20] into Eq. [21] gives
dφ∗ e dr∗ = e2Nsa/εkBT 1+ ([H+]0/Ka) exp(−φ∗e) = A 1+ B exp(−φe∗) , r∗= 1, [22] where A= ˆe2N sa/εkBT and B= [H+]0/Ka.
2.5. Perturbed State
Since u vanishes at equilibrium state, Eqs. [1], [2], [8], and [9] lead to
∇ · (δuEZe−iωt)= 0 [23]
−iωρfδuEZe−iωt = −∇δpEZe−iωt+ η∇2δuEZe−iωt− ρc∇φ. [24] The pressure term in the last expression can be eliminated by in-troducing a stream function,ψ, representation. In terms of ψ the magnitude of the perturbed velocity in the r -direction and that inθ-direction, δur andδuθ, can be represented respectively by
δur = − 1 r2sinθ ∂ψ ∂θ [25] and δuθ =r sin1 θ∂ψ∂r. [26] We have
δuEZe−iωt = (δurˆr+ δuθθ)Eˆ Ze−iωt. [27] Taking curl on both sides of Eq. [24] gives
E4ψ + iωρfE2ψ = 1 η ∂ρ c ∂r ∂φ ∂θ − ∂ρc ∂θ ∂φ ∂r sinθ, [28] where E2 = ∂2 ∂r2 +sinθ r2 ∂ ∂θ 1 sinθ ∂ ∂θ [29] E4 = E2E2= ∂2 ∂r2 +sinθ r2 ∂ ∂θ 1 sinθ ∂ ∂θ 2 . [30]
Note that the concentration field is influenced by the perturbed flow field, which arises from the applied electric field, and there-fore, the first two fields need to be considered simultaneously.
Taking∇2on both sides of Eq. [6] yields
∇2
(δφ(r, θ)EZe−iωt)= ∇2φ(r, θ, t) − ∇2φe(r ). [31] Similarly, Eq. [12] leads to
∇2φ = − 2 j=1 zjˆenej 0 ε exp −zjˆe(φ + gjEZe−iωt) kBT , [32] where nj = nej 0exp −zjˆe(φ + gjEZe−iωt) kBT . [33]
This expression suggests that the electrical potential is related to the equilibrium potential, perturbed potential, and the induced potential, which arises from the concentration field. For
con-venience, the concentration field is represented by gj through combining Eqs. [7], [11], and [33]. We have
δnjEZe−iωt = nej 0 exp −zjˆe(φ + gjEZe−iωt) kBT − exp −zjˆeφe kBT . [34]
Combining Eqs. [3] and [4] gives
∂nj ∂t = −Dj ∇2n j+ zjˆe kBT (∇nj· ∇φ + nj∇2φ) + u · ∇nj. [35] Substituting Eqs. [6]–[8] into Eq. [35] yields
−iωδnj = −Dj ∇2(δn j)+ zjˆe kT ∇ne j· ∇(δnj)+ ∇(δnj)· ∇φe + ∇(δnj)· ∇(δφEZe−iωt)+ nej∇2(δφ) + (δnj)∇2φe+ (δnj)∇2(δφEZe−iωt) + (δu) · ∇nej+ δnjEZe−iωt . [36]
For a simpler mathematical treatment, Eqs. [28] and [32]–[36] are linearized to yield the following governing equations for the perturbed variables: ∇∗2δφ∗− (κa)2 1+ α[exp(−φrφ ∗ e)+ α exp(αφrφe∗)]δφ∗ =(κa)2 1+ α[exp(−φrφ ∗ e)g∗1+ α exp(αφrφe∗)g2∗] [37] ∇∗2g∗ 1− φr∇∗φe∗· ∇∗g1∗+ Pe1δu∗· ∇∗φe∗ + iωa2 D1 (δφ∗+ g∗1)= 0 [38] ∇∗2g∗ 2+ αφr∇∗φe∗· ∇∗g2∗+ Pe2δu∗· ∇∗φe∗ + iωa2 D2 (δφ∗+ g∗2)= 0 [39] E∗4ψ∗+ iρfωa 2 η E∗2ψ∗ =(κa)2 1+α ∂g∗ 1 ∂θ exp(−φrφe∗)+α ∂g∗ 2 ∂θ exp(αφrφe∗) ∂φ∗ e ∂r∗ sinθ. [40] The scaled quantities are E∗Z= EZ/(ζ/a), U∗E∗Ze−iωt=
UEZe−iωt/UE, φe∗= φe/ζ , δφ∗E∗Ze−iωt = δφEZe−iωt/ζ ,
g∗jE∗Ze−iωt = gjEZe−iωt/ζ , ψ∗E∗Ze−iωt = ψ EZe−iωt/UEa,
n∗j = nj/ne10, and r∗= r/a. In these expressions, κ = (εkBT/
2 j=1n
e
j 0 (e zj)2)−1/2 is the reciprocal Debye length, Pej=
species j , and (ρfωa2/η) and (ωa2/Dj), j = 1, 2, are scaled frequencies.
A complex number is introduced to facilitate the analysis of the oscillatory phenomena involved in the electroacoustics. Both the magnitude of the dynamic electrophoretic velocity and the phase angle can be easily represented based on the complex notation (22). Moreover, the geometry of the system under con-sideration justifies the application of the separation of variables technique. As a result, Eqs. [37]–[40] can be rewritten in com-plex form as δφ∗ = ( R+ iI) cosθ [41] g∗1 = (G1R+ iG1I) cosθ [42] g∗2 = (G2R+ iG2I) cosθ [43] ψ∗ = ( R+ iI) sin2θ, [44] where the complex variables in the parentheses are functions of r only. Substituting Eqs. [41]–[44] into Eqs. [37]–[40], we obtain
L21R− (κa)2 1+ α[exp(−φrφ ∗ e)+ α exp(αφrφe∗)]R = (κa)2 1+ α[exp(−φrφ ∗ e)G1R+ α exp(αφrφe∗)G2R] [45] L21I − (κa)2 1+ α[exp(−φrφ ∗ e)+ α exp(αφrφe∗)]I = (κa)2 1+ α[exp(−φrφ ∗ e)G1I + α exp(αφrφ∗e)G2I] [46] L21G1R− φr dφe∗ dr∗ d G1R dr∗ + Pe1 1 r∗2R dφ∗e dr∗ −ωa2 D1 (I + G1I)= 0 [47] L21G1I − φr dφe∗ dr∗ d G1I dr∗ + Pe1 1 r∗2I dφe∗ dr∗ +ωa2 D1 (R+ G1R)= 0 [48] L21G2R+ αφr dφe∗ dr∗ d G2R dr∗ + Pe2 1 r∗2R dφ∗ e dr∗ −ωa2 D2 (I + G2I)= 0 [49] L21G2I + αφr dφe∗ dr∗ d G2I dr∗ + Pe2 1 r∗2I dφe∗ dr∗ +ωa2 D2 (R+ G2R)= 0 [50] L42R− ρfωa2 η L22I = − (κa)2 (1+ α)(G1Rexp(−φrφ ∗ e) + αG2Rexp(αφrφ∗e)) dφe∗ dr∗ [51] L42I +ρ fωa2 η L22R = − (κa)2 (1+ α)(G1Iexp(−φrφ ∗ e) + αG2Iexp(αφrφ∗e)) dφ∗e dr∗, [52] where L42= L22L22 [53] and L21 ≡ d 2 dr∗2 + 2 r∗2 d dr∗ − 2 r∗2 [54] L22 ≡ d 2 dr∗2 − 2 r∗2. [55]
Note that due to the introduction of the complex notation, the number of equations is doubled.
We assume the following: (a) The surface of a particle is im-penetrable to ions. (b) The potential on a cell surface arises from the applied electric field (22). (c) The perturbed concentration vanishes on a cell surface. (d) No-slip conditions exist on a parti-cle surface. (e) The vorticity vanishes on a cell surface. (f) There is no net flow in the radial direction across a cell surface. There-fore, the boundary conditions associated with Eqs. [45]–[52] are
dR dr∗ = 0, r ∗= 1 [56] R = −E∗Z b a, r ∗= b/a [57] dI dr∗ = 0, r ∗= 1 [58] I = −E∗Z b a, r ∗= b/a [59] d G1R dr∗ = 0, r ∗= 1 [60] G1R = −R, r∗= b/a [61] d G1I dr∗ = 0, r ∗= 1 [62] G1I = −I, r∗= b/a [63] d G2R dr∗ = 0, r ∗= 1 [64] G2R = −R, r∗= b/a [65] d G2I dr∗ = 0, r ∗= 1 [66] G2I = −I, r∗= b/a [67] R = 1 2U ∗ Rr∗2, r∗= 1 [68] dR dr∗ = U ∗ Rr∗, r∗ = 1 [69]
L22R = 0, r∗= b/a [70] R = 0, r∗= b/a [71] I = 1 2U ∗ Ir∗2, r∗= 1 [72] dI dr∗ = U ∗ Ir∗, r∗ = 1 [73] L22I = 0, r∗= b/a [74] I = 0, r∗= b/a. [75]
Scaled Frequency , ρfωa2/η
Magnitude of Scaled Dynamic Mobility
10-2 10-1 100 101 0. 5 1 1. 5 2 1000 100 10 A=10000
Scaled Frequency , ρ fωa 2
/η
Phase Angle (deg)
10-2 10-1 100 101 0 10 20 30 40 50 60 1000100 10 A= 10000 a b
FIG. 2. Variation of magnitude of scaled dynamic mobility, (a), and phase angle, (b), as a function of scaled frequency (ρfωa2/η) at various A for the case
B= 1.0, κa = 1.0, and λ = a/b = 0.5.
Note again that the number of boundary conditions is doubled due to the introduction of the complex notation. The details of these derivations can be found elsewhere (22).
2.6. Electrophoretic Mobility
For convenience, the problem under consideration is decom-posed into two subproblems: a particle moves with velocity (UR∗+ iUI∗)E∗Ze−iωt in the absence of the applied electric field and the particle remains fixed when an electric field E∗Ze−iωt
is applied. The force experienced by the particle is the sum of the hydrodynamic force, Fh, and the electric force Fe. In scaled
form, we have Fh∗+ Fe∗ = −i 4 3πεζ 2 ρfωa2 η ρp− ρf ρf × (U∗ R+ iUI∗)E∗Ze−iωt, [76] where Fh∗and Fe∗are respectively the magnitudes of the scaled hydrodynamic force, Fh∗, and the electrical force, Fe∗, exerted on a particle. Fh∗and Fe∗can be evaluated respectively by
Fe∗= 8 3πεζ 2 r∗dφ ∗ e dr∗(R+ iI)E ∗ Ze−iωt r∗=1 [77]
FIG. 3. Variation of magnitude of scaled dynamic mobility, (a), and phase angle, (b), as a function of scaled frequency (ρfωa2/η) at various B for the case
A= 100, κa = 1.0, and λ = a/b = 0.5.
and Fh∗ = 4 3πεζ 2 r∗4 ∂ ∂r∗ L2 2(R+ iI) r∗2 Ez∗e−iωt r∗=1 +4 3πεζ 2[r∗2(exp(−φ rφ∗e)− exp(αφrφe∗)) × (R+ iI)E∗ze−iωt]r∗=1. [78] The electrophoretic velocity of a particle can be calculated based on Eqs. [76]–[78].
III. RESULTS AND DISCUSSION
Figure 2 shows both the variations of the magnitude and the phase angle of the scaled dynamic mobility as a function of
the scaled frequency of the average electric field (ρfωa2/η) at various A (=ˆe2N
sa/εkBT ). Figure 2a reveals that for a fixed (ρfωa2/η) the larger the A, the greater the magnitude of the scaled dynamic mobility. This is because a large A implies a high concentration of the dissociable functional groups on a particle surface, which leads to a high surface potential. Figure 2a also indicates that if A is small, the magnitude of the scaled dynamic mobility decreases with the increase in (ρfωa2/η). However, if
A is sufficiently large, the magnitude of the scaled dynamic
mo-bility exhibits a local maximum at a medium value of (ρfωa2/η). This is mainly due to the phenomenon of double-layer
polariza-FIG. 4. Variation of magnitude of scaled dynamic mobility, (a), and phase angle, (b), as a function of scaled frequency (ρfωa2/η) at various H for the case
A= 100, B = 1.0, and κa = 1.0.
tion, which is significant if the surface potential is sufficiently high. Double-layer polarization has the effect of inducing an internal electric field, which is in the inverse direction as that of the applied electric field. In this case, as the direction of the average electric field varies, since the induced electric field can-not respond instantaneously, it has the effect of accelerating the movement of a particle. However, if the frequency of the average electric field is high, the rate of change in the direction of the movement of the particle is fast. In this case it moves back and forth too fast, and increasing (ρfωa2/η) has a negative effect on the dynamic mobility. Note that (ρfωa2/η) can be interpreted
as the ratio (dynamic inertia effect/viscous effect), which is a measure for the rate of response of the induced electric field to that of the average electric field. Figure 2b reveals that if A is small, the phase angle of the dynamic mobility is positive; that is, the average electric field leads the dynamic mobility, and the greater the (ρfωa2/η), the larger the phase angle. However, if A is sufficiently large, the phase angle is negative (i.e., the dynamic mobility leads the average electric field) and has a negative local minimum as (ρfωa2/η) varies. Again, this is due to the effect of double-layer polarization.
The variations of the magnitude and the phase angle of the scaled dynamic mobility and as a function of the scaled frequency of the average electric field (ρfωa2/η) at various
B (=[H+]0/Ka) are presented in Fig. 3. Figure 3a suggests that for a fixed (ρfωa2/η) the magnitude of the scaled dynamic mobility increases with the decrease in B. This is because the smaller the B, the larger the equilibrium constant Ka, which im-plies a higher charge density on a particle surface. Figure 3a also reveals that if B is large, the magnitude of the scaled dynamic mobility decreases with the increase in (ρfωa2/η). However, if it is small, the magnitude of the scaled dynamic mobility has a local maximum as (ρfωa2/η) varies. These can be explained by the same reasoning as that in the discussion of Fig. 2a. Similarly, the qualitative behaviors of the phase angle shown in Fig. 3b can be explained by the same reasoning as that in the discussion of Fig. 2b.
Figure 4 illustrates both the variations of the magnitude and the phase angle of the scaled dynamic mobility as a function of the scaled frequency of the average electric field (ρfωa2/η) at various H (=(a/b)3). Figure 4a shows that for a fixed (ρ
fωa2/η) the larger the H , the smaller the magnitude of the scaled dynamic mobility. This is because the larger the H , the higher the con-centration of particle, and the more significant the hindrance of neighboring particles, which leads to a smaller dynamic mo-bility. Figure 4a indicates that if H is small, the magnitude of the scaled dynamic mobility has a local maximum as (ρfωa2/η) varies; the phase angle has a negative (i.e., phase lead) local min-imum, as shown in Fig. 4b. Again, these are due to the effect of double-layer polarization. As can be seen in Fig. 4 if H is large, both the local maximum in the magnitude of the scaled mobility and the local minimum in the phase angle become inappreciable. This is because if H is large, the polarization of the double layer surrounding a particle is confined by the neighboring particles, and therefore its effect becomes relatively insignificant. For the same reason, the lag in the phase angle for the case where H is large is smaller than the lag for the case where H is small, as can be seen in Fig. 4b, since it is relatively easy for the particle to respond to the variation in the applied electric field.
Figure 5 shows the variation of the magnitude of the scaled dynamic mobility as a function of double-layer thicknessκa at various scaled frequency of the average electric field (ρfωa2/η). It was found that, for the case of a static electric field, the mo-bility has a local maximum forκa in the range [1,10] (26). This is because if the double layer is thick, the resistant force
expe-κa
Magnitude of Scaled Dynamic Mobility
10-2 10-1 100 101 0. 25 0. 5 0.75 1 1.25 1. 5 1.75 2 5 10 ρfωa 2 /η=2 0 0 0. 5 1. 0
FIG. 5. Variation of magnitude of scaled dynamic mobility as a function of
κa at various scaled frequency (ρfωa2/η) for the case A = 1000, B = 1.0, and
λ = a/b = 0.5.
rienced by a particle is large, which yields a small mobility. On the other hand, a thin double layer has the effect of confining the degree of dissociation of functional groups, which leads to a low surface charge density, and therefore, a small mobility. As can be seen in Fig. 5, a similar result to that observed in the case of a static electric field is obtained for the present case of a dynamic electric field. This figure also reveals that the local maximum of the scaled dynamic mobility increases first with the increase in (ρfωa2/η), reaches a maximum, and then decreases with a further increase in (ρfωa2/η). As mentioned previously, the effect of double-layer polarization can be advantageous to the movement of a particle if a dynamic electric field of appro-priate frequency is applied. However, if (ρfωa2/η) is high, the polarization of a double layer is insignificant, and therefore, the acceleration on the movement of the particle which arises from the induced electric field becomes unimportant. The observation that the maximum in the maximal scaled dynamic mobility oc-curs at (ρfωa2/η) ∼= 0.5 is consistent with previous discussion.
IV. CONCLUSION
The dynamic electrophoretic mobility of a concentrated dis-persion of charge-regulated particles is investigated. The dy-namic mobility of a particle is found to depend on two key parameters, A and B, which characterize respectively the con-centration and the degree of dissociation of the functional groups on a particle surface. If A is large and/or B is small, the absolute surface potential of the particle is high, and the effect of double-layer polarization becomes significant. In this case the dynamic mobility may have a local maximum, and the phase angle may have a negative (phase lead) local minimum as the thickness of double layer varies. Due the hindrance of neighboring particles, the dynamic mobility decreases with the increase in the volume fraction of a particle.
ACKNOWLEDGMENT
This work is supported by the National Science Council of the Republic of China.
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