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Langmuir 2004, 20, 7952-7959
Electrophoresis in a Carreau Fluid at Arbitrary Zeta Potentials Eric Lee, Chia-Sheun Tai, Jyh-Ping Hsu,* and Chur-Jen Chen Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan 10617, Department of Chemical and Materials Engineering, National I-Lan University, I-Lan, Taiwan 26041, and Department of Mathematics, Tunghai University, Taichung, Taiwan 403 Received March 29, 2004. In Final Form: June 24, 2004 The electrophoresis of colloidal particles has been studied extensively in the past. Relevant analyses, however, are focused mainly on the electrophoretic behavior of a particle in a Newtonian fluid. Recent advances in science and technology suggest that the electrophoresis conducted in a non-Newtonian fluid can play a role in practice. Here, the electrophoresis of a concentrated colloidal dispersion in a Carreau fluid is investigated under the conditions of arbitrary electrical potential where the effect of double-layer polarization may be significant. A pseudo-spectral method coupled with a Newton-Raphson iteration scheme is used to solve the governing equations, which describe the electric, the flow, and the concentration fields. The results of numerical simulation reveal that, due to the effect of shear thinning, the electrophoretic mobility for the case of a Carreau fluid is greater than for that of a Newtonian fluid. Also, the higher the surface potential of a particle, the more significant the non-Newtonian nature of a Carreau fluid on its electrophoretic mobility.
Introduction 1
Since the pioneer work of Smoluchowski in the early stage of the last century, electrophoresis has been investigated extensively; both experimental and theoretical results are ample in the literature. Today, it is one of the basic analytical tools for the estimation of the surface property of a charged entity in a dispersion medium, and relevant study on this subject is still very active in various areas. Theoretical analyses in the past decades focused mainly on the modification of the assumptions made in the original work of Smoluchowski, which include an isolated rigid entity in an infinite medium, constant and low surface potential, weak applied electric field, thin double layer, and Newtonian medium, so that the problems under consideration were close to reality.2 Among these analyses, that the electrophoresis occurs in a nonNewtonian fluid receives the least attention, which however, can be important in practice. The introduction of surfactant or polymer to a colloidal dispersion to improve its stability, for instance, leads to a shear-thinning fluid.3 In fact, a dispersion will exhibit non-Newtonian behavior if the concentration of particles inside is sufficiently high.4 Apparently, extending the classic analysis of Smoluchowski to the case of non-Newtonian fluids is necessary not only for fundamental study but also for practical applications. If the concentration of particles in a dispersion is appreciable, the interaction between adjacent particles can be significant. In this case, because the analysis involves many-body interactions, the problem becomes complicated. This difficulty can be circumvented by considering a simplified problem where a dispersion is represented by a cluster of cells, each comprises a particle and a concentric liquid shell. The volume fraction * Author to whom correspondence should be addressed. Tel: 8863-9357400 ext.285. Fax: 886-3-9353731. E-mail:
[email protected]. (1) Smoluchowski, M. Z. Phys. Chem. 1917, 92, 129. (2) Hunter, R. J. Zeta Potential in Colloid Science; Academic Press: London, 1981. (3) Hunter, R. J. Foundations of Colloid Science; Oxford University Press: Oxford, 1989; Vol. 1. (4) Hunter, R. J. Foundations of Colloid Science; Oxford University Press: Oxford, 1989; Vol. 2.
of particles is determined by the relative magnitudes of the size of a particle and that of a cell.5 This approach was adopted to model the electrophoresis of a concentric dispersion under various conditions.6-15 In an attempt to simulate the electrophoresis conducted in a non-Newtonian fluid, Lee et al.16 considered the electrophoresis of a rigid sphere in a spherical cavity filled with a Carreau fluid for the case of low surface potential and weak applied electric field. The shear-thinning nature of the fluid was found to be advantageous to the movement of a particle, and the difference between the mobility of a particle in a Carreau fluid and that in a Newtonian fluid increases with the decrease in the thickness of the double layer. Also, the mobility correlates with the square of the scaled double-layer thickness. In a recent study, Hsu et al.17 discussed the electrophoresis of a concentrated spherical dispersion in a Carreau fluid for the case of low surface potential and the effect of double-layer polarization is insignificant. The electrophoretic mobility calculated was found to be different, both quantitatively and qualitatively, from that in a Newtonian fluid. The influence of the shear-thinning nature of the fluid on the mobility of a particle is similar to that observed by Lee et al.16 In this study, we extend our previous analysis, where the electrophoresis of a concentrated non-Newtonian dispersion for the case when the surface potential of a particle is low, to the case of arbitrary surface potential. The effect of double-layer polarization, which can be (5) Kuwabara, S. J. Phys. Soc. Jpn. 1959, 14, 527. (6) Levine, S.; Neale, G. H. J. Colloid Interface Sci. 1974, 47, 520. (7) Kozak, M. W.; Davis, E. J. J. Colloid Interface Sci. 1986, 112, 403. (8) Kozak, M. W.; Davis, E. J. J. Colloid Interface Sci. 1989, 127, 497. (9) Kozak, M. W.; Davis, E. J. J. Colloid Interface Sci. 1989, 129, 166. (10) Ohshima, H. J. Colloid Interface Sci. 1997, 188, 481. (11) Happel, J.; Brenner, H. Low-Reynolds Number Hydrodynamics; Martinus Nijhoff: Dordrecht, The Netherlands, 1983. (12) Shilov, V. N.; Zharkikh, N. I.; Borkovskaya, Yu. B. Colloid J. 1981, 43, 434. (13) Lee, E.; Chu, J. W.; Hsu, J. P. J. Colloid Interface Sci. 1999, 209, 240. (14) Hsu, J. P.; Lee, E.; Yen, F. Y. J. Chem. Phys. 2000, 112, 6404. (15) Hsu, J. P.; Lee, E.; Yen, F. Y. J. Phys. Chem. B 2002, 106, 4789. (16) Lee, E.; Huang, Y. F.; Hsu, J. P. J. Colloid Interface Sci. 2003, 258, 283. (17) Hsu, J. P.; Lee, E.; Huang, Y. F. Langmuir 2004, 20, 2149.
10.1021/la0491955 CCC: $27.50 © 2004 American Chemical Society Published on Web 08/06/2004
Electrophoresis in a Carreau Fluid
Langmuir, Vol. 20, No. 19, 2004 7953
constitutive equation can be expressed as
τ ) η0[1 + (λ γ˘ )2](n-1)/2 γ˘
(4)
where η0 is the zero-shear-rate viscosity, λ is the relaxation time, and n is the power-law exponent. According to eq 4, if n ) 1 or λ ) 0, eq 4 reduces to the normally observed Newtonian-type behavior, and if λγ˘ . 1, it reduces to the well-known power-law fluid. Note that the former refers to the corresponding zero-shear-rate viscosity of the dispersion, which in general is a Carreau fluid. It should not be confused with the Newtonian solvent of the dispersion system. Electric, Concentration, and Flow Fields. The spatial variation of the electrical potential, φ, can be derived on the basis of Gauss’s law, which leads to the Poisson equation
∇2φ ) -
Figure 1. Schematic representation of the system under consideration. A concentric dispersion is simulated by a representative cell containing a spherical particle of radius a and a concentric liquid shell of radius b.13-15 An electric field E is applied.
significant if the surface potential of a particle is high and the double layer surrounding it is thick, is taken into account. The electrokinetic equations describing the problem under consideration are solved numerically by adopting a pseudo-spectral method coupled with a Newton-Raphson iteration scheme. Theory. Referring to Figure 1, we consider a spherical dispersion of nonconductive particles of radius a in a nonNewtonian fluid which contains z1:z2 electrolyte with z2)Rz1, z1 and z2 being, respectively, the valence of cations and that of anions. The cell model of Levine and Neale6 assumes that the system under consideration can be described by a representative cell, which comprises a particle and a concentric liquid shell of radius b. The spherical coordinates (r,θ,φ) are chosen with the origin located at the center of the representative cell. Constitutive Equation for Fluid. For a purely viscous non-Newtonian fluid, such as Bingham, power-law, and Carreau fluids, the so-called generalized Newtonian fluid, its constitutive equation can be expressed as18
τ ) -η(γ˘ )γ˘
(1)
where τ is the shear stress tensor of the fluid, γ˘ is the rate of deformation tensor, and γ˘ ) x1/2(γ˘ :γ˘ ) is the second invariant or the magnitude of the rate of deformation tensor, and η is the apparent viscosity, which is a function of γ˘ . We have the following relations:
Π ) -pδ - τ
(2)
γ˘ ) ∇v + (∇v)T
(3)
where Π is the total stress tensor, p is the pressure, δ is the unit tensor, ∇ is the gradient operator, v is the velocity of fluid, and the superscript T denotes matrix transpose. For most concentrated polymer solutions or melts, the (18) Bird, R. B.; Armstrong, R. C.; Hassager, O. Dynamics of Polymer Liquids; Wiley: New York, 1987; Vol. 1.
F
N
∑ j)1
)-
zjenj
(5)
where ∇2 is the Laplace operator, and F are, respectively, the permittivity of the liquid phase and the space charge density, and e is the elementary charge. zj and nj are, respectively, the valence charge and concentration of species j. Here, we assume that is constant. The conservation of ionic species in the liquid phase can be expressed as
∂nj ) -∇‚fj ∂t
(6)
where fj is the concentration flux of ionic species j, which can be written as
[
fj ) -Dj ∇nj +
]
zjenj ∇φ + njv kBT
(7)
Here, Dj is the diffusion coefficient of ionic species j, kB and T are, respectively, the Boltzmann constant and the absolute temperature. Combining eqs 6 and 7 and assuming steady-state condition we obtain
∇2nj +
zje 1 (∇nj‚∇φ + nj∇2φ) - v‚∇nj ) 0 kBT Dj
(8)
Suppose that the liquid phase is incompressible with constant physical properties and the flow of liquid is in the creeping flow regime. Then, the flow field at steady state can be described by
∇‚v ) 0
(9)
∇‚τ + ∇p + F∇φ ) 0
(10)
Solution Procedure. For convenience, the electrical potential is decomposed into the electrical potential in the absence of the applied electric field (or the equilibrium potential due to the presence of particles), φ1, and the perturbed electric field arising from the applied electric field, φ2, that is, φ ) φ1 + φ2. For the present case, because the effect of double-layer polarization may be significant, we assume that nj can be expressed as19
(
nj ) nj0exp -
)
zje(φ1 + φ2 + gj) kBT
(11)
where nj0 is the bulk concentration of ionic species j, and
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Lee et al.
gj is a perturbed potential, which simulates the effect of double-layer polarization. For a simpler mathematic treatment, the stream function, ψ, is adopted to describe the flow field. In terms of ψ, the r- and the θ-components of velocity, vr and vθ, can be expressed, respectively, as
vr ) -
1 ∂ψ r sinθ ∂θ
Equations 8 and 11 lead to
∇2gj -
(12)
2
1 ∂ψ vθ ) rsinθ ∂r
(13)
zje 1 1 ∇φ ‚∇gj ) v‚∇φ + v‚∇gj + kBT 1 Dj Dj zje zje ∇φ2‚∇gj + ∇g ‚∇gj (21) kBT kBT j
The boundary conditions associated with this equation are assumed as
fj‚n ) fj‚r ) 0, r ) a
(22)
nj ) nj0, r ) b
(23)
Taking curl on eq 10 and applying eq 9, we obtain 2
2
∇ × (∇η‚γ˘ ) + ∇η × ∇ v + η∇ × ∇ v - ∇ × F∇φ ) 0 (14) Substituting eq 11 into eq 5 yields
∇2φ1 ) -
F1
N
)-
∑ j)1
zjenj0
( )
exp -
zjeφ1 kT
(15)
where F1 represents the charge density at the equilibrium field and in the absence of the applied electric field. Note that the present problem is of a one-dimensional nature, with φ1 ) φ1(r). We assume that the surface of a particle is remained at a constant potential, ζa, and a cell as a whole is at electroneutrality, which implies that there is no net current between adjacent cells. These assumptions lead to the following boundary conditions:
φ1 ) ζa, r ) a
(16)
∂φ1 ) 0, r ) b ∂r
(17)
Since φ2 ) φ - φ1 and ∇2φ2 ) ∇2φ - ∇2φ1, eqs 5, 11, and 15 yield
∇2φ2 ) -
F2
N
)-
∑ j)1
zjenj0
(
exp N
∑ j)1
) ( )
zje(φ1 + φ2 + gj)
+
kBT
zjeφ1 zjenj0 (18) exp kBT
where F2 represents the charge density in the presence of the applied electric field. The boundary conditions associated with this equation are assumed as
∂φ2 ) 0, r ) a ∂r
(19)
∂φ2 ) -Ezcosθ, r ) b ∂r
(20)
The first condition indicates that the particle is nonconductive, and therefore, it is ion-impenetrable. This is realistic because the dielectric constant of a rigid particle is usually much smaller than that of the liquid phase. The second condition was proposed by Levine and Neale.6 (19) O’Brien, R. W.; White, L. R. J. Chem. Soc., Faraday Trans. 2 1978, 74, 1607.
where n and r are, respectively, the unit normal vector and the unit vector in the r direction. The first condition implies that the surface of a particle is ion-impenetrable, and the second condition arises from a cell representing the whole dispersion. It can be shown that, in terms of stream function, the flow field is described by
[( ) ( )] (
∂2η ∂2η ∂η ∂γ˘ rθ ∂η γ˘ rθ + r 2 γ˘ rθ + r + γ˘ + ∂r ∂r ∂r ∂r∂θ θθ ∂r ∂2η ∂η ∂γ˘ rr 1 ∂2η ∂η ∂γ˘ θθ γ˘ rr + + γ˘ + ∂θ ∂r ∂r∂θ ∂r ∂θ r ∂r2 rθ ∂η ∂3ψ cotθ ∂2ψ 1 ∂3ψ 1 ∂η ∂γ˘ rθ + 2 - 2 3 r ∂θ ∂θ ∂r ∂r r ∂r∂θ r ∂r∂θ2
ηD4ψ + sinθ
) (
2 ∂2ψ 2cotθ ∂ψ 1 ∂3ψ ∂η 1 ∂3ψ + 4 3+ + + 3 2 3 2 2 ∂θ ∂θ r ∂r ∂θ r ∂θ r ∂θ r 1 ∂ψ cotθ ∂2ψ - 4 ) r4sin2θ ∂θ r ∂θ2 ∂g2 ∂φ1 - (1 + R)n10e2z12 ∂g1 sinθ (24) n1 + Rn2 kBT ∂θ ∂θ ∂r
[(
)
) ]
where
(( ) ( ) (
D4 ) D2D2 ) D2 )
))
∂2 sinθ ∂ 1 ∂ + 2 ∂r2 r ∂θ sinθ ∂θ
(
)
2
∂2 sinθ ∂ 1 ∂ + 2 ∂r2 r ∂θ sinθ ∂θ
(25)
(26)
Let U be the magnitude of particle velocity in the z direction. The following boundary conditions are assumed for the flow field:
vr ) Ucosθ and vθ ) -Usinθ, r ) a
(27)
∇ × v ) 0, r ) b
(28)
vr ) 0, r ) b
(29)
The first condition arises from the no-slip nature of the particle surface, the second condition was proposed by Kuwabara,5 and the last condition implies that there is no net fluid flow across cell boundary. In terms of stream function, these conditions become
∂ψ 1 ) -Ursin2θ, r ) a ψ ) - Ur2sinθ and 2 ∂r
(30)
Electrophoresis in a Carreau Fluid
ψ ) 0 and
Langmuir, Vol. 20, No. 19, 2004 7955
(
)
1 cosθ ∂ ∂2 1 ∂2 + 3 - 3 2 ψ ) 0, 2 rsinθ ∂r r sin θ ∂θ r sinθ ∂θ2 r ) b (31)
The symmetric nature of the problem under consideration requires that
∂g/1 ) 0, r* ) 1 ∂r*
(42)
g/1 ) -φ/2, r* ) 1/H
(43)
Similarly, for j ) 2 (anions), eq 21 becomes
∂φ1 ∂φ2 ∂g1 ∂g2 ∂ψ ) ) ) )ψ) ) 0, ∂θ ∂θ ∂θ ∂θ ∂θ θ ) 0 and θ ) π (32)
∇*2g/2 + Rφr∇*φ/1‚∇g/2 ) Pe2v*‚∇*φ/1 + Pe2v*‚∇φ/2 +
For a simpler mathematical treatment, all the relevant symbols are expressed in scaled forms in subsequent discussions. To this end, the following characteristic scales are used, the radius of a particle a, the equilibrium surface potential of a particle ζa, the bulk concentration of cations n10, and the electrophoretic velocity based on Smoluchowski’s theory UE ) ζa2/η0a. We define the following scaled quantities: r* ) r/a, n/j ) nj/n10, E/z ) Ez/(ζa/a), U* ) U/UE, v* ) v/UE, φr ) z1eζa/kBT, φ/1 ) φ1/ζa, φ/2 ) φ2/ζa, ψ* ) ψ/UEa2, and g/j ) gj/ζa. In terms of scaled symbols, eq 11 gives
and the corresponding boundary conditions are
n/1 ) exp(-φr(φ/1 + φ/2 + g/1 ))
(33)
n/2 ) exp(Rφr(φ/1 + φ/2 + g/2 ))
(34)
Equation 15 can be rewritten as 2
∇*2φ/1 ) -
(κa) 1 [exp(-φrφ/1) - exp(Rφrφ/1)] (1 + R) φr (35)
where κ-1 ) [kBT/∑ nj0(ezj)2 ]1/2 is the Debye length and φr is the scaled surface potential. The corresponding boundary conditions are
φ/1 ) 1, r* ) 1 ∂φ/1
(36)
) 0, r* ) 1/H
/
(37)
∂r
2
(κa) 1 ((n/1 - n/2) - (exp(-φrφ/1) (1 + R) φr exp(Rφrφ/1)))
∂φ/2
(38)
(46)
[(
∂2η* ∂η* ∂γ˘ rθ* ∂η* γ˘ rθ* + r* 2 γ˘ rθ + r* + ∂r* ∂r* ∂r* ∂r* ∂2η* ∂η* ∂ γ˘ θθ* ∂η* ∂γ˘ rr* ∂2η* γ˘ θθ* + γrr* + + ∂r*∂θ ∂θ ∂r* ∂r*∂θ ∂r* ∂θ ∂η* ∂3ψ* 1 ∂2η* 1 ∂η* ∂γ˘ rθ* γ˘ rθ* + 2 r* ∂r* r* ∂θ ∂θ ∂r* ∂r*3
η *D*4ψ* + sinθ
) (
)] (
)
1 ∂3ψ* 2 ∂2ψ* 2cotθ ∂ψ* cotθ ∂2ψ* + 2 - 3 + + 2 ∂r*∂θ 2 r* r* ∂r* ∂θ r* ∂θ2 r*3 ∂θ 1 1 ∂3ψ* ∂η* 1 ∂3ψ* ∂ψ* + 4 2 + 4 2 2 ∂θ r* ∂r* ∂θ r* ∂θ3 r* sin θ ∂θ / / / (κa)2 ∂g1 / ∂g2 / ∂φ1 cotθ ∂2ψ* n + Rn sinθ (47) ) ∂θ 2 ∂r* (1 + R) ∂θ 1 r*4 ∂θ2
(
[(
)
)]
∂ψ* 1 ) -U*r*sin2θ, r* ) 1 ψ* ) - U*r*2sin2θ and 2 ∂r* (48) ψ* ) 0 and
(39)
) -E/z cosθ, r* ) 1/H
(40)
In terms of scaled symbols, eq 21 becomes, for j ) 1 (cations)
∇*2g/1 - φr∇*φ/1‚∇g/1 ) Pe1v* ‚∇φ/1 + Pe1v* ‚∇φ/2 + Pe1v* ‚∇g/1 + φr∇*φ/2 ‚∇*g/1 + φr∇*g/1‚∇*g/1 (41) The corresponding boundary conditions are
(
∂2 1 cosθ ∂ + r*sinθ ∂r*2 r*3sin2θ ∂θ
)
1 ∂2 ψ* ) 0, r* ) 1/H (49) r* sinθ ∂θ2
) 0, r* ) 1
/
∂r
g/2 ) -φ/2, r* ) 1/H
3
∂r
/
(45)
where Pej ) UEa/Dj is the Peclet number of ionic species j. We define the scaled stream function, ψ*, and the scaled space charge density, F*, as ψ* ) ψ/UEa2 and F* ) F/F0 ) n/1 - n/2, where F0 ) κ2ζa/(1 + R)φr. In terms of scaled symbols, eq 24 can be rewritten as
The corresponding boundary conditions become
∂φ/2
∂g/2 ) 0, r* ) 1 ∂r*
The corresponding boundary conditions are
where H ) a/b, which is a measure for the concentration of particles. Similarly, eq 18 can be rewritten as
∇*2φ/2 ) -
Pe2v*‚∇g/2 - Rφr∇*φ/2‚∇*g/2 - Rφr∇*g/2‚∇*g/2 (44)
Electrophoretic Mobility. The electrophoretic mobility of a particle can be determined through a force balance. At steady state, the total force acting on a particle in the z direction, which includes the electric force FEz and the drag force FDz, vanishes, that is, FEz + FDz ) 0. The electric force acting on a particle per unit area is σ(-∇φ)s, σ and (-∇φ)s being, respectively, the surface charge density and the electric field on a particle surface. According to Gauss’s law, we have σ ) - (∇φ‚n)S. Since the applied electric field is in the z direction, so is the electric force acting on a particle. Therefore, based on the Maxwell stress tensor, we have (Appendix)
FEz )
∫σ(-∇φ)s‚izdA ) ∫(∇φ‚n)s(∇φ‚iz)sdA
(50)
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Langmuir, Vol. 20, No. 19, 2004
Lee et al.
where iz is the unit vector in the z direction. It can be shown that
∫0
π
FEz ) 2πζa2 1 r*
∂(φ/1
+ ∂θ
( ) ( ) ∂φ/1 ∂r*
sinθ
2
r* sinθdθ )
r*)1
2πζa E/z KE 2
(51)
(
r*2sin2θφ/1
( ) ( ∂φ/
∂(φ/1 + φ/2) cosθ ∂r* / / 1 ∂(φ1 + φ2) sinθ r*2sinθdθ (52) r* ∂θ r*)1
∫0π ∂r*1 r*)1
E/z KE )
)
Note that, because FEz is independent of fluid viscosity since the applied electric field is held constant, eq 51 is the same as that for the case of a Newtonian fluid. The drag force arising from the flow of fluid can be evaluated by
∫sΠz‚dS ) ∫sΠz‚ndS ) ∫sΠnzdS
Fn ) Fz )
(53)
where S is a unit tangent vector to a meridian curve on the surface, Πnz is the component of the stress vector in the z direction. According to Happel and Brenner,11
Πnz ) -
{
}
∂p 1 ∂ 2 1 ∂ ∂ψ (w ˜ p) - w ˜2 - 2η 2w ˜ ∂s ∂s w ˜ ∂z ∂s 1 ∂w ˜ 2 D ψ (54) η w ˜ ∂n
( )
where w ˜ is the distance from the axis of revolution. Substituting this expression into eq 53 yields
∫s ΠnzdS ) 2π ∫Πnzw˜ds ) -π ∫∂s∂ (w˜2p)ds + ∂p ∂ ∂w ˜ ˜ vw ˜ )ds - 2π ∫η D2ψds π ∫w ˜ 2 ds - 4π ∫η (w ∂s ∂s ∂n
Fz )
(55) ˜ direction. It can where vw˜ represents the velocity in the w be shown that this expression yields
Fn ) Fz )
[
1 ∂ (sinθτθθ) + ∫0π r3sin2θ(r12 ∂r∂ r2τrθ) + rsinθ ∂θ cotθ τφ r φ 4π
[
]
dθ - π
r)a
∫0π
[
]
τrθ r
∂φ r2sin2θF dθ ∂θ
]
]
/
sinθ ∂ψ dθ ∫0π µ∂θ∂ (cosθ∂ψ ∂r r ∂θ ) r)a π 2π∫0 µ(rsinθ)D2ψdθ
(56)
In scaled symbols, this expression can be rewritten as
)
∂φ/2 ∂θ
sinθ ∂ψ* r* ∂θ
where
-π
[
∫0π r*3sin2θ(r*12 ∂r*∂ r*2τrθ/ ) +
τrθ cotθ ∂ 1 (sinθτ/θθ) + τφ r*sinθ ∂θ r* r* φ
∂(φ/1 + φ/2) cosθ ∂r*
r*)1
φ/2)
F/n) -πφ20
)]
r*
r*
) 1dθ - 4πφ20
) 1dθ - 2πφ20
r*
∫0π
) 1dθ + πφ20
[
∫0π
∂ ∂ψ* µ* cosθ ∂θ ∂r*
(
∫0πµ*(r*sinθ)D*2ψ*dθ (57)
The governing equations for the concentration, the flow, and the electric fields are solved numerically subject to the corresponding boundary conditions. The pseudospectrum method based on Chebyshev polynomials, coupled with a Newton-Raphson iteration scheme, used in our previous studies13-15 is adopted. Results and Discussions In this study, the cell model of Levine and Neale6 is adopted to simulate a concentrated spherical dispersion. Applying this model has the advantage that a many-body problem is reduced to a one-body problem, thereby simplifying dramatically the mathematical analysis. In particular, the geometry involved in a cell model is of onedimensional nature, which makes both analytical and numerical treatments simpler. On the other hand, a cell model provides only a primary prediction for a concentrated dispersion. For example, as can be seen in Figure 1, the volume fraction of particles is overestimated by the cell model because some of the liquid volume is neglected. Another problem is the boundary conditions that should be assumed on the virtual surface of a cell.20 Levine and Neale,6 for example, assumed that the boundary condition for the electrical field is of Neumann type, but Shilov and Zharkikh12 pointed out that a Dirichlet-type condition is more appropriate. Despite all the potential drawbacks, the performance of the cell model was found to be satisfactory qualitatively, in general.13 The influence of the key parameters of the system under consideration on its electrophoretic behavior is investigated through numerical simulation. These parameters include the concentration of particles measured by H ) a/b, the surface potential of a particle represented by φr, the scaled thickness of double layer surrounding a particle, κa, and n and λUE/a, which describe the shear-thinning nature of a fluid. The influence of the surface potential of a particle on its electrophoretic behavior is illustrated in Figure 2, where the variation of the scaled mobility, U/m, as a function of κa at various scaled surface potentials, φr and λUE/a. This figure reveals that if the surface potential of a particle is low (φr
