J. Phys. Chem. B 2001, 105, 747-753
747
ARTICLES Conductivity of a Concentrated Spherical Colloidal Dispersion Eric Lee, Ming-Hui Chih, and Jyh-Ping Hsu* Department of Chemical Engineering, National Taiwan UniVersity, Taipei, Taiwan 10617, Republic of China ReceiVed: July 7, 2000; In Final Form: October 24, 2000
The conductivity of a concentrated suspension of charged spherical particles at an arbitrary level of surface potential is estimated. The cell model proposed by Kuwabara is adopted to simulate the present many-body problem by taking the effect of double-layer polarization into account. Several interesting results are found which cannot be observed if the corresponding linearized model is considered. For example, if φr has a medium value, (K*/K∞) has a local minimum as κa varies, φr, K*, K∞, κ, and a being respectively the scaled surface potential, the effective conductivity, the effective conductivity for an infinitely dilute dispersion, the reciprocal Debye length, and the radius of a particle. For both κa f 0 and κa f ∞, (K*/K∞) decreases with the increase in the volume fraction of the dispersed phase, and the reverse is true for a medium κa. For a fixed κa, the variation of (K*/K∞) as a function of φr may have a local minimum as R ()valence of counterions/ valence of co-ions) varies. For a fixed φr, the variation of (K*/K∞) as a function of κa may have a local minimum as R varies.
Introduction If an electric potential is applied to an electrolyte solution containing charged colloid particles, the motion of particles and the flow of liquid-phase yield a net current in the direction of the applied electric field. It is justified that the induced current can be used to estimate the surface properties of a particle. For example, the electric conductivity, defined by the ratio (magnitude of net current)/(magnitude of applied electric field), is found to correlate with the charged conditions of a particle. The prediction of the electric conductivity of a colloidal dispersion involves solving a set of coupled, nonlinear electrokinetic equations including the Navier-Stokes equation describing flow field and the Poisson equation governing the electric field. Solving these equations simultaneously is, in general, a nontrivial task even if a numerical scheme is adopted. If the concentration of particles is appreciable, the interaction between adjacent particles may play a significant role, and the problem becomes even more complicated. Because of the complexity involved in the resolution of the governing equations reported results in the literature are mainly limited to drastically simplified cases. Dukhin and Derjaguin,3 for example, derived a simple formula for the conductivity of planar particles under the conditions that the Maxwell relation is applicable and the double layer is thin. On the basis of this analysis, O’Brien4 analyzed the conductivity of a suspension of spherical particles at a low surface potential. They assumed that the suspension is dilute enough that the interaction between adjacent double layers is negligible and the problem can be simulated by an isolated particle in an infinite electrolyte solution. The result obtained, however, is found to deviate appreciably from the experimental observations of Watillon and * To whom correspondence should be addressed. Fax: 886-2-23623040. E-mail:
[email protected].
Stone-Masui.5 An approach similar to that adopted by O’Brien4 was used by Saville6 to estimate the conductivity of a dilute suspension of charged particles, similar result as that derived by the former was obtained. The analysis of Saville6 was also extended to take the effect of nonspecific adsorption of into account.7 The result derived was found to be more consistent with the experimental data than that obtained by O’Brien.4 Under the conditions of O’Brien,4 the conductivity of a concentrated colloidal suspension was discussed by Ohshima.8 Based on Kuwabara’s cell model,2 which assumes that a concentrated dispersion can be simulated by a swarm of cells, each comprises a particle and a concentric spherical shell of liquid phase, a simple expression for the conductivity was derived which has a better performance than that of O’Brien in the limiting case of dilute dispersion. In this study, the problem discussed by Ohshima8 is generalized to the case the surface potential of a particle is not necessarily low. The effect of double-layer polarization, a phenomenon that arises from the relative motion between a particle and the surrounding ion cloud, is taken into account. Theory We consider identical, nonconducting, spherical particles of radius a in a z1:z2 electrolyte solution, z1 and z2 being the valences of cations and anions, respectively. Let R ) -z2/z1. Then the electroneutrality in the bulk liquid phase requires that n20 ) (n10/R), n10 and n20 are respectively the bulk concentrations of cations and anions. Referring to Figure 1, the cell model of Kuwabara2 is used where the system under consideration is simulated by a representative spherical particle surrounded by a concentric spherical shell of liquid phase of radius b. The spherical coordinates (r,θ,φ) with the origin located at the center of the representative particle. A uniform electric field B E of
10.1021/jp002459l CCC: $20.00 © 2001 American Chemical Society Published on Web 01/06/2001
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(
nj ) nj0 exp -
)
zje(φ1 + φ2 + gj) kBT
j ) 1, 2
(6)
where gj is a function which accounts for the effect of fluid flow on ionic concentration, that is, a measure for the degree of double-layer polarization. If φ2 and gj are small, then eq 6 can be approximated by
(
nj* ) exp -
Figure 1. Schematic representation of the cell model considered. A representative particle of radius a is surrounded by a spherical shell of liquid of radius b. U B is the terminal velocity of the representative particle, and E is the induced electric field. The spherical coordinates (r,θ,φ) with its origin located at the center of the particle is used.
magnitude E is in the Z-direction (θ ) 0) is applied. We assume that the liquid phase has constant physical properties. Suppose that the migration of the particle is slow so that the system under consideration is at a quasi-steady state. Then, the conservation of ions leads to where ∇ B denotes the gradient operator, e and φ
[[
∇ B ‚ Dj ∇ B nj +
] ]
zjenj ∇ B φ - n jb ν )0 kBT
j ) 1, 2
(1)
are respectively the elementary charge and the electrical potential, b ν represents the fluid velocity, kB and T are the Boltzmann constant and the absolute temperature respectively, and nj, Dj, and zj are the number concentration, the diffusivity, and the valence of ion species j respectively. We assume that the spatial variation of the electrical potential can be described by the Poisson-Boltzmann equation 2
∇ φ)2
∑
∇ B ‚ν b)0
(3)
ν-∇ B p - F∇ Bφ ) 0 η∇ b
(4)
2
In these expressions p, η, and F are the pressure, the viscosity, and the space charge density, respectively. Following the treatment of Lee et al.,9 φ is decomposed as
φ ) φ 1 + φ2
(5)
where φ1 and φ2 represent respectively the electrical potential that would exist in the absence of the applied electric field, and that outside a particle arising from the applied electric field. According to O’Brien and White,10 the effect of double-layer polarization can be taken into account by expressing nj as
(7)
(κa) 1 [exp(-φrφ1*) - exp(Rφrφ1*)] (8) (1 + R) φr
with
φr ) ζazle/kBT
(8a)
2
(2)
where is the permittivity of the liquid phase. Suppose that the flow field can be described by the Navier-Stokes equation in the creeping flow region
j ) 1, 2
2
∇2φ1* ) -
κ)[
]
where nj* ) nj/nj0, φj* ) φj/ζa and gj* ) gj/ζa. Here, the surface potential of a particle is characterized by the zeta potential ζa. It should be pointed out that φ2 and gj are small does not imply that their influences are insignificant.10 For a simpler mathematical treatment, scaled variables are used in the subsequent analysis. The particle radius a is chosen as the characteristic length. The electrophoretic velocity of an isolated particle predicted by the Smoluchowski’s theory when an electric field of strength (ζa/a) is applied, and UE ) (ζa2/ηa) is chosen as the scaled factor for velocity. In solving the flow field, the Navier-Stokes equation is first transformed by taking curl of eq 4 under the constraint expressed by eq 3 to obtain an expression describing the variation of stream function, ψ. For the present coordinate system, ι is related to the r- and θ-component of particle velocity, Vr, and νθ, by νr ) -(1/r2 sin θ)(∂ψ/∂θ) and νθ ) (1/r sin θ)(∂ψ/∂r). We define the scaled stream potential ψ* as ψ* ) ψ/UEa2. It can be shown that10 φ2*, g1*, g2*, and ψ* can be expressed respectively as φ2* ) Φ2(r) cos θ, g1* ) G1(r) cos θ, g2* ) G2(r) cos θ, and ψ* ) Ψ(r) sin2 θ. Equations 1-4 can be simplified accordingly. The scaled electric potential at equilibrium, φ1*, can be described by9
zjenj
j)1
)[
zjeζa zjeζa φ1* 1 (φ * + gj*) kBT kBT 2
nj0(ezj)2/kBT]1/2 ∑ j)1
(8b)
In these expressions, r* ) r/a, φr is the scaled surface potential, L is a linear operator, and 1/κ is the reciprocal Debye length. We assume that the boundary conditions associated with eq 8 are
φ 1* ) 1
r* ) 1
(8d)
dφ1* ) 0, r* ) b/a dr*
(8e)
The last condition implies that the unit cell as a whole is electrically neutral. Equation 8 needs to be solved simultaneously with the following equations (Appendix):
[
L-
]
(κa)2 [exp(-φrφ1*) + R exp(Rφrφ1*)] Φ2 ) 1+R (κa)2 [G exp(-φrφ1*) + RG2 exp(Rφrφ1*)] (9) 1+R 1
Conductivity of a Spherical Colloidal Dispersion
LG1 - φr
dφ1* φ1* dG1 ) Pe1φr2νr* dr* dr* dr*
dφ1* dφ1* dG2 ) Pe1φr2νr* LG2 + Rφr dr* dr* dr* D4Ψ ) -
J. Phys. Chem. B, Vol. 105, No. 4, 2001 749
[
(10) (11)
]
dφ1* (κa)2 (n *G + Rn2*G2) 1+R 1 1 dr*
(12)
Here, Pej ) (Z1e/kBT)/ηDj, j ) 1, 2, denotes the electric Peclet number of ion species j, n1* ) exp(-φrφ1*), n2* ) exp(Rφrφ1*), and the operators D4 and L are defined as
)
(12a)
2 2 d d2 - 2 + 2 r* dr* dr* r*
(12b)
D4 ) (D2)2 ) L≡
(
d2 2 - 2 2 dr* r*
2
Figure 2. Variation of effective conductivity (K*/K∞) as a function of κa at various scaled surface potential of a particle φr. Key: λ ) 0.125, R ) 1, Pe1 ) 0.01, and Pe2 ) 0.01.
Since the particle is nonconductive we have
dΦ2 )0 dr*
r* ) 1
(13a)
Suppose that the electric field at the virtual surface r ) b equals the applied electric field, that is,
dΦ2 ) -Ez* dr*
r* ) b/a
(13b)
The surface of the particle is impenetrable to liquid phase. Therefore,
dGj )0 dr*
r* ) 1, j ) 1, 2
r* ) b/a
j ) 1, 2
dΨ ) U*r* dr*
r* ) 1
[
]
∫VBi dV
(14)
-1 V
∫V∇Bφ2 dV
(15)
Employing Gauss’s divergence theorem, eq 14 gives
〈B〉 i )
1 V
r Bn b dS ∫Sb‚i(r)
(16)
where b r is the position vector, b n is the unit outer normal and S represents the virtual surface r* ) b/a. Similarly, applying gradient theorem, eq 15 becomes
(13e) 〈E B〉 )
In addition, we require that the virtual surface is steady and satisfies the Kuwabara’s model of zero vorticity. As pointed out by Levine and Neale,1 this boundary condition is capable of recovering the result of Smoluchowski for the case of an isolated particle. We have
2 1 d2Ψ Ψ)0 Ψ ) 0 and r* dr*2 r*3
〈E B〉 )
(13d)
Let U be the terminal velocity of the particle. Then
1 Ψ ) U*r*2 2
1 V
〈B〉 i )
(13c)
The basic property of the present unit cell model requires that the number density of each ionic species reaches the corresponding equilibrium value at r ) b, that is,
Gj ) -Φ2
(Φ2,G1,G2,Ψ) at the (k + 1)th stage, (Φ2,G1,G2,Ψ)k+1. (3) Compare (Φ2,G1,G2,Ψ)k with (Φ2,G1,G2,Ψ)k+1. If the difference between any two corresponding component is larger than a prespecified value, then return to step (2) with k replaced by k + 1 and (Φ2,G1,G2,Ψ)k replaced by (Φ2,G1,G2,Ψ)k+1. The effective conductivity of a homogeneous suspension can be expressed as K* ) 〈ı˘ 〉, 〈E B〉, 〈ı˘ 〉 and 〈E B〉 being the volume average of current density and that of electric field defined respectively by
∫Sφ2bn dS
(17)
The current density can be expressed by bı ) ∑zjenjb νj. The velocity of ion species j, b νj, and that of the bulk solution, b ν, are related by
(
ν - Dj b νj ) b
r* ) b/a (13f)
where Ez* and Eza/ζa and U* ) U/UE, Ez being the z-component of the applied electric field. The following iterative procedure is suggested for the resolution of eqs 8-12: (1) Solve eq 8 subject to eqs 8d and 8e for φ1*. (2) The values of (Φ2,G1,G2,Ψ) at the kth stage, (Φ2,G1,G2,Ψ)k are guessed. Solve eq 9 for Φ2 subject to eqs 13a and 13b, solve eqs 10 and 11 for G1 and G2, respectively, subject to eqs 13c and 13d. Also, solve eq 12 for Ψ subject to eqs 13e and 13f. This yields the values of
1 V
)
zje ∇ B nj ∇ Bφ + kBT nj
(18)
Since R ) -z2/z1, and n20 ) n10/R, we have
Bi )
( )
{
2φr2 kBT 3 (κa)2 [exp(-φrφ1*) + exp(Rφrφ1*)]ν b* + ηa3 z1e (1 + R)
[
]}
R 1 1 exp(-φrφ1*)∇ B *g1* + exp(Rφrφ1*)∇ B *g2* φr Pe1 Pe2
(19)
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Figure 4. Variation of effective conductivity (K*/K∞) as a function of κa at various radius ratio λ ()a3/b3) for the case φr ) 3.0, R ) 1, Pe1 ) 0.01, and Pe2 ) 0.01.
Evaluating the integral of eq 17 gives
〈E B〉 ) -
4πa2 b 2 (Φ2)r*)b/aB δz φ 3V a a
()
(22)
If we define (K*/K∞) as the effective conductivity, then K* K∞ -
)
(
) ( )[
1 R + Pe1 Pe2
-1
]
b 1 dG1 R dG2 exp(-φrφ1) + exp(Rφrφ1) a Pe1 dr* Pe2 dr*
r*)b/a
(Φ2)r*)b/a
(23) Discussion
Figure 3. Variation of effective conductivity (K*/K∞) as a function of κa. φr ) 0.01, (a), φr ) 1, (b), φr ) 3, (c). The dashed line in (a) denotes the result of Ohshima.8 Key: same as Figure 2.
where b ν* ) b ν/UE. Substitute eq 19 into eq 14 yields
()
4πa2 b 3 × R 3V a 1 + Pe1 Pe2 dG R dG2 1 1 exp(-φrφ1) + exp(Rφrφ1) Pe1 dr* Pe2 dr*
〈B〉 i ) K∞
φa
[
]
r* ) b/a
B δ z (20)
where B δz is the unit vector in the positive z-direction, and the conductivity of an infinitely dilute dispersion, K∞, is defined as ∞
K )
2
∑ i)1
Dizi2e2ni∞ kBT
)
(
2 (κa) UE 1
a 1 + R φ 2 Pe1 r
+
R
)
Pe2
(21)
Figure 2 shows the variation of the effective conductivity (K*/K∞) as a function of κa at various scaled surface potential φr. This figure reveals that (K*/K∞) approaches a constant as κa f 0; this constant increases with φr. This is because if κa is small, the double layer surrounding a particle becomes thick, and the contribution of the flow of bulk liquid to conductivity becomes less significant. Also, the higher the φr, the greater the mobility of a particle, and, therefore, the larger the conductivity. Figure 2 also suggests that (K*/K∞) approach a constant as κa f ∞; this constant, however, is independent of φr. This is because if κa is large, the double layer surrounding a particle is thin, and the conductivity is mainly contributed by the flow of bulk liquid which is independent of the surface potential of the particle. Also, if κa f ∞, the double layer is infinitely thin. In this case the system under consideration behaves like an electrolyte solution containing uncharged particles, and, therefore, the conductivity is constant. The simulated variations of the effective conductivity (K*/ K∞) as a function of κa for various levels of surface potential are illustrated in Figure 3. For comparison, the result of Ohshima8 for a low surface potential (φr ) 0.01) is also presented in Figure 3a. This figure shows that the result of Ohshima is satisfactory for κa f ∞. This is expected, since it neglects the interaction between adjacent double layers, which is approximately correct if double layer is thin. The result of Ohshima, however, will underestimate (K*/K∞) as κa f 0, even
Conductivity of a Spherical Colloidal Dispersion
J. Phys. Chem. B, Vol. 105, No. 4, 2001 751
Figure 6. Variation of I1, I2, and (I1 + I2) as a function of φr for the case of Figure 5a.
Figure 7. Variation of effective conductivity (K*/K∞) as a function of κa at various R ()-z2/z1) for the case φr ) 1. Key: same as Figure 5.
Figure 5. Variation of effective conductivity (K*/K∞) as a function of φr at various R ()-z2/z1) for the case κa ) 1. R ) 0.5, (a), R ) 1, (b), R ) 2, (c). Key: λ ) 0.125, Pe1 ) 0.01, and Pe2 ) 0.01.
if φr is low, as revealed by Figure 3a. It is interesting to note that, depending upon the level of φr, the behavior of (K*/K∞) as a function of κa can be different. As can be seen from Figure 3a, if φr is low, (K*/K∞) increases monotonically with κa, and it is less than unity. For a medium φr, (K*/K∞) exhibits a local minimum as κa varies, as shown in Figure 3b. Figure 3c indicates that if φr is high, (K*/K∞) decreases monotonically with κa, and its value may exceed unity. These behaviors can be elaborated as follows. If φr is low, the conductivity is mainly contributed by the motion of particles. Since the thinner the double layer, the greater the mobility of a particle, (K*/K∞) increases with κa, as shown in Figure 3a. However, since the inner part of a particle will not contribute to the conductivity, (K*/K∞) is less than unity. If φr is sufficiently high, the contribution to the conductivity by the bulk flow of liquid-phase becomes more significant than that by the motion of the particle,
and (K*/K∞) may exceed unity, as can be seen in Figure 3c. For a medium φr, the combined effects of the motion of a particle and the bulk flow of liquid phase lead to a local minimum in (K*/K∞) as κa varies, as presented in Figure 3b. Figure 4 shows the variation of the effective conductivity (K*/K∞) as a function of κa at various λ ()a3/b3). Here, λ is a measure for the volume fraction of the dispersed phase; the higher its value the more concentrated the dispersion. Figure 4 suggests that for both κa f 0 and κa f ∞, the smaller the λ the larger the (K*/K∞). This is expected since the smaller the λ, the smaller the volume fraction of the solid phase, which does not contribute to the conductivity, and therefore, the larger the conductivity. It is interesting to note that, however, for a medium κa (K*/K∞) increases with λ. This can be explained as follows. If κa f 0, the double layer is thick, and it fills virtually the whole liquid phase of a cell. As κa increases, the space occupied by the double layer decreases, and the smaller the λ, the less the fraction of the liquid-phase occupied by the double layer. Therefore, the smaller the λ, the earlier the curve (K*/K∞) against κa starts to decrease. The available theoretical results in the literature for dilute dispersions are based on the assumption that the interaction between adjacent double layers is insignificant, and, therefore, the behavior of a dispersion can be simulated by that of an isolated particle. Although it is also applicable to
752 J. Phys. Chem. B, Vol. 105, No. 4, 2001
Lee et al.
Figure 9. Variation of I1, I2, and (I1 + I2) as a function of κa for the case φr ) 1 and R ) 0.5. Key: same as Figure 5.
Figure 10. Variation of I1, I2, and (I1 + I2) as a function of κa for the case φr ) 3 and R ) 0.5. Key: same as Figure 5.
I1 + I2 K* )∞ (Φ K 2)r*)b/a Figure 8. Variation of effective conductivity (K*/K∞) as a function of κa at various R ()-z2/z1) for the case φr ) 3. R ) 0.5, (a), R ) 1, (b), R ) 2, (c). Key: same as Figure 5.
dilute dispersions, our analysis focused mainly on the case where the volume fraction of dispersed phase becomes appreciable so that the interaction between adjacent double layers needs to be considered. The limitation in the volume fraction of a dilutedispersion model can be estimated based on the fact that it should be sufficiently low so that adjacent double layers do not overlap with each other. For example, if a ) 1 µm, I ) 10-3 M, F ) 2.5 g/cm3 (density of particle), kB ) 1.38 × 10-23 J/K, ) 8.679 × 10-9 coulombs/(V‚m), e ) 1.6022 × 10-19 coulombs, n10 ) 6.02 × 1020 no./m3, then the volume fraction of a dilute-dispersion model should not exceed about 0.06. The simulated variations of the effective conductivity (K*/K∞) as a function of φr at various R ()-z2/z1) are illustrated in Figure 5. This figure reveals that if R is small, that is, the valence of anions (counterions) is lower than that of cations (co-ions), (K*/K∞) has a local minimum as φr varies. This can be explained by rewriting eq 23 as
where
I1 ) I2 )
( (
1 R + Pe1 Pe2 R 1 + Pe1 Pe2
) ( )[ ) ( )[ -1
-1
(24)
] ]
b 1 dG1 exp(-φrφ1) a Pe1 dr*
r*)b/a
b R dG2 exp(Rφrφ1) a Pe1 dr*
r*)b/a
(24a) (24b)
This implies that the current associated with the conductivity can be decomposed into two parts: the current contributed by cations I1 and that by anions I2. Figure 6 shows the variations of I1, I2, and (I1 + I2) as a function of scaled surface potential φr for the case of Figure 5a. This Figure 6 reveals that the higher the surface potential the smaller the contribution to current by cations, but the larger the contribution to current by anions. For the case R ) 0.5 if φr is low, I1 is larger than I2, but if φr is high, I1 becomes smaller than I2. The net effect yields a local minimum in (I1 + I2) as φr varies, which leads to a local minimum in conductivity. However, as R becomes large, it can be shown that I1 is smaller than I2 for all φr, and (I1 + I2) does not have a local minimum.
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Figure 7 shows the variation of the effective conductivity (K*/K∞) as a function of κa at various R ()-z2/z1) for the case φr ) 1. That for a higher surface potential (φr ) 3) is presented in Figure 8. Figure 7 reveals that for the case φr is low, (K*/K∞) increases monotonically with κa if the valence of anions (counterions) is lower than that of cations (co-ions). On the other hand, if the valence of anions (counterions) is higher than that of cations (co-ions), (K*/K∞) decreases monotonically with κa. If the valence of anions is comparable to that of cations, (K*/K∞) has a local minimum as κa varies. As can be seen from Figure 8, if φr is sufficiently high, (K*/K∞) has a local minimum as κa varies even if the valence of anions (counterions) is lower than that of cations (co-ions). Again, the qualitative behaviors of (K*/K∞) as κa varies presented in Figures 7 and 8 can be explained by the variations of I1, I2, and (I1 + I2) as a function of κa illustrated in Figures 9 and 10. Figure 9 shows that for the case R ) 0.5, I1 increases with κa, but I2 decreases with κa. Also, if κa is small, I1 is smaller than I2 but the reverse is true if κa is large. However, the difference between I1 and I2 for a small κa is not large, (I1 + I2) does not have a local minimum as κa varies. On the other hand, at a higher φr, the difference between I1 and I2 for a small κa becomes appreciable, and (I1 + I2) has a local minimum as illustrated in Figure 10. Acknowledgment. This work is supported by the National Science Council of the Republic of China. Appendix The pressure term in eq 4 in the text can be eliminated by adopting the stream function representation
1 B × [F∇ B (φ1 + φ2)] B δ φE4ψ ) - sin θ∇ η
(A-1)
where B δφ denotes the unit vector in the φ-direction, and the operator E4 ) E2E2 with
E2 )
∂2 sin θ ∂ 1 ∂ + 2 ∂r2 r ∂θ sin θ ∂θ
(
)
(A-2)
In the discussion below a symbol with an asterisk denotes a scaled quantity. In the case the applied electric field is weak so that the expressions for the distortion of double layer, the electric potential, and the flow field near a particle can be linearized. The scaled equilibrium potential, φ1*, is described by the scaled
form of eq 2 in the text with the scaled number concentrations of ions n1* ) exp(-φrφ1*) and n2* ) exp(Rφrφ1*). We have 2
∇*2φ1* ) -
(κa) 1 [exp(-φrφ1) - exp(Rφrφ1)] (1 + R) φr (A-3)
The Laplacian of the perturbed potential φ2, ∇2φ2, can be expressed as ∇2φ2 ) ∇2φ - ∇2φ1. Applying eqs 2 and 7 in the text and eq A-3 gives
(κa)2 [exp(-φrφ1*) + R exp(Rφrφ1*)]φ2* ) (1 + R) (κa)2 [exp(-φrφ1*)g1* + exp(Rφrφ1*)Rg2*] (A-4) (1 + R)
∇*2φ2* -
Substituting eq 7 in the text into eq 1 in the text and neglecting the terms which involve the product of two perturbed terms, we obtain
B *φ1*‚∇ B *g1* ) φr2Pe1b ν *‚∇ B *φ1* (A-5) ∇*2g1* - φr∇ Substituting eq 7 in the text into eq A-1 yields the scaled
B *φ1*‚∇ B *g2* ) φr2Pe2b ν *‚∇ B *φ1* (A-6) ∇*2g2* + Rφr∇ equation for the stream function
E*4ψ* )
[(
) ]
∂g2* ∂φ1* (κa)2 ∂g1* sin θ (A-7) n *+ Rn2* ∂θ ∂r* (1 + R) ∂θ 1
Applying the relations φ2* ) Φ2(r) cos θ, g1* ) G1(r) cos θ, g2* ) G2(r) cos θ, ψ* ) Ψ(r) sin2 θ to eqs A-4 through A-7, eqs 9 through 12 in the text can be recovered. References and Notes (1) Levine, S.; Neale, G. H. J. Colloid Interface Sci. 1974, 47, 520. (2) Kuwabara, S. J. Phys. Soc. Jpn. 1959, 14, 527. (3) Dukhin, S. S.; Derjaguin, B. V. In Surface and Colloid Science; Matijevic, E. Ed.; Wiley: New York, 1974. (4) O’Brien, R. W. J. Colloid Interface Sci. 1981, 81, 234. (5) Watillon, A.; Stone-Masui, J. J. Electroanal. 1972, 37, 143. (6) Saville, D. A. J. Colloid Interface Sci. 1979, 71, 447. (7) Saville, D. A. J. Colloid Interface Sci. 1983, 91, 34. (8) Ohshima, H. J. Colloid Interface Sci. 1999, 212, 443. (9) Lee, E.; Chu, J. W.; Hsu, J. P. J. Colloid Interface Sci. 1999, 209, 240. (10) O’Brien, R. W.; White, L. R. J. Chem. Soc., Faraday Trans. 2 1978, 74, 1607.
