Langmuir 2001, 17, 1821-1825
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Conductivity of a Concentrated Cylindrical Dispersion Eric Lee, Ming-Hui Chih, and Jyh-Ping Hsu* Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan 10617, Republic of China Received May 8, 2000. In Final Form: December 27, 2000 The conductivity of a concentrated cylindrical dispersion is evaluated theoretically. We show that the scaled conductivity (K*/K∞) approaches a constant value as κa f 0, which depends on the scaled surface potential φr, where K*, K∞, κ, and a are the apparent conductivity, the conductivity of an infinitely dilute dispersion, the reciprocal Debye length, and the radius of a cylinder. If κa f ∞, K*/K∞ also approaches a constant, but it is independent of φr. The effect of R (-valence of counterions/valence of co-ions) on K*/K∞ is found to be significant if κa is small and becomes negligible as κa f ∞. Also, if R > 1, K*/K∞ decreases monotonically with the increase in κa, and the reverse is true if R < 1. If R > 1, K*/K∞ increases monotonically with φr, and the larger the R the faster the rate of increase. Also, if R < 1, K*/K∞ has a local minimum as φr varies.
1. Introduction The conductivity of a colloidal dispersion is contributed by the movement of the electrolyte ions in the liquid phase and that of the charged dispersed entities. The electrical double layer surrounding a particle may also have a significant effect on the conductivity through its relaxation and polarization. In practice, the surface property of a charged entity can be estimated by measuring the conductivity of its dispersion. A typical application includes the characterization of the charged condition of protein molecules, biological cells, and liquid drops. Theoretical estimation of the conductivity of a colloidal dispersion involves solving a set of electrokinetic equations, which comprise the equations for electric field and for flow field, and the equation for ion conservation. Dukhin and Derjaguin1 analyzed theoretically the conductivity of a planer colloidal dispersion for the case of a thin double layer. That for a spherical dispersion was investigated by several workers. O’Brien,2 for example, discussed the conductivity of a dilute dispersion at low surface potential. Since the interaction between adjacent double layers is neglected, the behavior of the dispersion can be simulated by that of an isolated particle. This model was modified by Saville3,4 to include the effect of nonspecific absorption. Ohshima5 considered a concentrated dispersion by adopting Kuwabara’s cell model.6 The result obtained is limited to a low surface potential and a thin double layer. Stigter7,8 discussed the electrokinetic phenomena of a system containing cylindrical colloids. A survey of the literature reveals that relevant results for cylindrical geometry are mainly for electroosmotic flow,9-11 and there is a lack of corresponding results for conductivity. In the present * To whom correspondence should be addressed. Fax: 886-223623040. E-mail:
[email protected]. (1) Dukhin, S. S.; Derjaguin, B. V. Surface and Colloid Science; Matijevic, E., Ed.; Wiley: New York, 1974. (2) O’Brien, R. W. J. Colloid Interface Sci. 1981, 81, 234. (3) Saville, D. A. J. Colloid Interface Sci. 1979, 71, 447. (4) Saville, D. A. J. Colloid Interface Sci. 1983, 91, 34. (5) Ohshima, H. J. Colloid Interface Sci. 1999, 212, 443. (6) Kuwabara, S. J. Phys. Soc. Jpn. 1959, 14, 527. (7) Stigter, D. J. Phys. Chem. 1979, 83, 1663. (8) Stigter, D. J. Phys. Chem. 1979, 83, 1670. (9) Kozak, M. W.; Davis, E. J. J. Colloid Interface Sci. 1986, 112, 403. (10) Ohshima, H. J. Colloid Interface Sci. 1999, 210, 397. (11) Lee, E.; Lee, Y. S.; Chu, J. W.; Hsu, J. P. J. Colloid Interface Sci. 2000, 223, 223.
Figure 1. Schematic representation of the problem considered. The dispersion is simulated by an array of parallel cylinders of radius a. An electric field E is applied perpendicular to the axis of a cylinder.
study, the conductivity of a cylindrical dispersion is discussed. In particular, the effects of double-layer polarization, the concentration of dispersed phase, and the level of surface potential are taken into account. 2. Theory Let us consider a dispersion of positively charged cylindrical particles of radius a in a z1:z2 electrolyte solution, z1 and z2 being respectively the valences of cations and anions. For simplicity, we assume that the length of a particle is much larger than its radius, and the particles are homogeneously distributed to form a regular array. Referring to Figure 1, an electrical field E is applied in the direction shown which leads to a relative velocity between a particle and the surrounding fluid U, which is perpendicular to the axis of the particle. The cell model proposed by Kuwabara6 is adopted to simulate the behavior of the dispersion. Here, a representative cell comprises a particle and a coaxis fluid envelope of thickness (b - a). The concentration of the dispersion can be measured by the parameter H defined by
H ) 1 - (a/b)2
(1)
According to this definition, the smaller the H, the higher the concentration of dispersion. The cylindrical coordinates
10.1021/la0006566 CCC: $20.00 © 2001 American Chemical Society Published on Web 02/15/2001
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Lee et al.
(r,θ,z), with the origin located at the center of the representative particle, are adopted in the discussions below. We assume that the spatial variation of the electrical potential φ can be described by the Poisson equation
∇2φ ) -
F
)
∑i nizie
(2)
where ∇ denotes the gradient operator and and F are the permittivity of the liquid phase and the spatial charge density, respectively, and ni and zi are respectively the number concentration and the valence of ionic species i. The transport of ions in the liquid phase at steady state can be described by
{[
] }
zjenj ∇φ - njv ) 0 kBT
∇‚ Dj ∇nj +
of a particle a is chosen as the characteristic length and UE ()ζa2/µa), the electrophoretic velocity of an isolated particle based upon Smoluchowski’s theory, is chosen as the characteristic velocity. Similar to the treatment of Lee et al.,13 curl is taken on eq 4 to give the governing equation for the stream function Ψ, which can be used to calculate the r and the θ components of v, vr, and vθ, through vr ) (∂Ψ/∂θ)/r and vθ ) ∂Ψ/∂r. The scaled stream function, Ψ*, is defined as Ψ* ) Ψ/aUE. The symmetric nature of the problem under consideration suggests that 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) sin θ. It can be shown that the variation of φ1*, the scaled electrical potential at equilibrium, is governed by
Lφ1* ) -
(3)
where Dj and v are respectively the diffusivity of ionic species j and the velocity of liquid, e is the elementary charge, and kB and T are respectively the Boltzmann constant and the absolute temperature. Suppose that the liquid phase is incompressible and has constant physical properties. Also, the Reynolds number is small so that the flow of liquid phase is in the creeping flow regime. Then the flow field can be described by the Navier-Stokes equation
µ∇2v - ∇p - F∇φ ) 0
(4)
∇‚v ) 0
(5)
(κa)2 [exp(-φrφ1*) - exp(Rφrφ1*)] (9) (1 + R)φr
where the scaled surface potential φr, the linear operator L, and the reciprocal Debye length κ are defined respectively by
L)
(κa)2 )
φr ) ζaz1e/kBT
(9a)
1 1 d d2 + dr*2 r* dr* r*2
(9b)
∑
a2e ni0zi (1 + R)e2n10z12 2 ) a kBT kBT
(9c)
where p and µ denote the pressure and the viscosity, respectively. Following the treatment of O’Brien and White,12 the electrical potential φ is decomposed as
where r* ) r/a. The nature of the present cell model requires that a cell is electrically neutral and there is no net current flow between adjacent cells. These imply that eq 8 should satisfy the boundary conditions below:
φ ) φ 1 + φ2
φ1* ) 1, r* ) 1
(9d)
dφ1* ) 0, r* ) b/a dr*
(9e)
(6)
where φ1 and φ2 represent respectively the electrical potential that would exist in the absence of the applied electrical field (i.e., the equilibrium electrical field) and that due to the applied electrical field. The spatial variation of the concentration of ionic species can be simulated by12
[
nj ) nj0 exp
-zje (φ + φ2 + gj) kBT 1
]
(7)
j ) 1, 2 where gj denotes a perturbation term which accounts for the effect of fluid flow on the concentration of ionic species. If both φ2 and gj are small compared to kBT/e and φ1, eq 7 can be approximated by
(
nj ) nj0 exp -
)[
zjeζa zjeζa φ1* 1 (φ * + gj*) kBT kBT 2
]
(8)
Equation 8 needs to be solved simultaneously with the following equations:
[
L-
]
(κa)2 [exp(-φrφ1*) + R exp(Rφrφ1*)] Φ2 ) 1+a (κa)2 [G exp(-φrφ1*) + RG2 exp(Rφrφ1*)] (10) 1+a 1 LG1 - φr
dφ1* dG1 dφ1* ) Pe1vr* dr* dr* dr*
LG2 + Rφr
j ) 1, 2 ∇4Ψ ) -
(11)
dφ1* dφ1* dG2 ) Pe2vr* dr* dr* dr*
[
(12)
]
dφ1* 1 (κa) (n *G + Rn2*G2) r* 1 + R 1 1 dr*
(13)
where φ1* ) φj/ζa, gj* ) gj/ζa, ζa denotes the zeta potential. It should be pointed out that even if φ2 and gj are small, their effects on nj can be significant.12 For a simpler mathematical treatment, scaled properties are adopted in the following discussions. Here the radius
In these expressions, Pej ) (z1e/kBT)2/µDj, j ) 1, 2, is the electric Peclet number of ion species j, n1* ) exp(-φrφ1*), n2* ) exp(Rφrφ1*), and
(12) O’Brien, R. W.; White, L. R. J. Chem. Soc., Faraday 2 1978, 74, 1607.
(13) Lee, E.; Chu, J. W.; Hsu, J. P. J. Colloid Interface Sci. 1998, 205, 65.
Conductivity of a Cylindrical Dispersion
∇4 ) (∇2)2 )
Langmuir, Vol. 17, No. 6, 2001 1823
(
)
1 1 d d2 + dr*2 r* dr* r*2
2
(13a)
The boundary conditions associated with eqs 10-13 are assumed as
dΦ2/dr* ) 0, r* ) 1
(13b)
dΦ2/dr* ) -E*, r* ) b/a
(13c)
dGj/dr* ) 0, r* ) 1, j ) 1, 2
(13d)
i)
( )
2φr2 kBT ηa3 zje
3
{
(κa)2 [exp(-φrφ1*) + exp(Rφrφ1*)]v* + (1 + R)
[
R 1 1 exp(-φrφ1*)∇*g1* + exp(Rφrφ1*)∇*g2* φr Pe1 Pe2
(20)
Gj ) -Φ2, r* ) b/a, j ) 1, 2
(13e)
where v* ) v/UE. Substituting eq 20 into eq 14 gives
〈i〉 ) K∞
[
]
1 d 1 d2 + Ψ ) 0 (13g) dr* r* dr* r*2 r* ) b/a
where Ex* ) Exa/ζa and U* ) U/UE, Ex are U being respectively the x component of the electrical field and the terminal velocity of the representative particle. These conditions imply that a particle is nonconducting, the concentration of ions reaches the equilibrium value at the virtual surface (r* ) b/a), which is steady, and satisfies the Kuwabara’s model of zero vorticity. Also, U is constant. Let 〈i〉 and 〈E〉 be the volume average of current density and that of electric field defined respectively by
〈i〉 )
1 V
∫V i dV
()
πLa2Np b 1 2V a R + Pe2 Pe2 φa
2
×
(
)
1 R N G′ + N G′ | δ (21) Pe1 1 1 Pe2 2 2 r*)b/a z
Ψ ) -U*r* and dΨ/dr* ) -U*, r* ) 1 (13f) Ψ ) 0 and
]}
where δz is the unit vector in the Z direction, and the conductivity of an infinitely dilute dispersion, K∞, is defined as 2
∞
K )
∑ i)1
Dizi2e2n1∞ kBT
)
(
2 (κa) UE 1
a 1 + R φ 2 Pe1 r
+
R
)
Pe2
(22)
Performing the integral on the right-hand side of eq 17, we obtain
〈E〉 ) -
(14)
Lπa2Np b φa Φ2|r*)b/a δz 2V a
()
(23)
If we define K*/K∞ as the scaled conductivity, then it can be shown that
and
〈E〉 ) -
1 V
∫V ∇φ2 dV
(15)
Then the apparent conductivity of the present system can be expressed as K* ) 〈i〉/〈E〉. Equation 14 can be written, by applying Gauss’ divergence theorem, as
〈i〉 )
1 V
∫V r‚i(r)n dS
(16)
where r represents the position vector, n is the unit outer normal, and S denotes the virtual surface. Similarly, eq 15 can be written, by employing the gradient theorem, as
〈E〉 ) -
1 V
∫S φ2n dS
(17)
The current density can be evaluated by
i)
∑zjenjvj
(18)
where the velocity of ionic species j, vj, and that of the bulk solution, v, are related by
(
v j ) v - Dj
)
∇nj zje ∇φ ) kBT nj
(19)
Substituting this expression into eq 18 with R ) -z2/zj, n20 ) n10/R, and eq 16 yields
K* )K∞
{(
) (ab)[Pe1
1 R + Pe1 Pe2
-1
dG1 exp(-φrφ1) + 1 dr*
R dG2 exp(Rφrφ1) Pe2 dr*
] }
/(Φ2)r*)b/a (24)
r*)b/a
The governing equations are solved numerically by the pseudospectral method based on Chebyshev polynomials.14 3. Results and Discussion Particles of colloidal sizes such as asbestos fibers, DNA, and protein molecules are frequently treated as infinitely long cylinders in electrokinetic studies.15 For a dilute dispersion, the dispersed particles are usually assumed to have a random orientation. Stigter,15-17 for example, analyzed the electrophoresis and the sedimentation of a dilute dispersion of cylindrical particles based on this assumption. The random orientation assumption becomes unrealistic, however, for the case of a concentrated dispersion. This is because as the concentration of particle becomes high, the interaction between adjacent particles is significant, and particles can no longer assume an arbitrary orientation. In this case, the available space for the free movement of particles in the transverse direction (14) Canuto, C.; Hussaini, M. Y.; Quarteroni, A.; Zang, T. A. Spectral Methods in Fluid Dynamics; Springer-Verlag: New York, 1986. (15) Stigter, D. J. Phys. Chem. 1982, 86, 3553. (16) Stigter, D. J. Phys. Chem. 1978, 82, 1417. (17) Stigter, D. J. Phys. Chem. 1978, 82, 1424.
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Figure 2. Variation of scaled conductivity (K*/K∞) as a function of κa at various scaled surface potentials of particle φr. Key: H ) 0.75, R ) 1, Pe1 ) 0.01, and Pe2 ) 0.01.
is greatly reduced, and the free space along the longitudinal direction of the cylinders can roughly be simulated by a set of parallel cavities as that assumed in the present study. Figure 2 shows the variation of the scaled conductivity (K*/K∞) as a function of κa at various scaled surface potentials, φr. This figure reveals that if κa f 0, K*/K∞ approaches a constant value, which depends on φr, and if κa f ∞ K*/K∞ approaches a constant value (=0.6), which is independent of φr. The conductivity of the present system comprises that contributed by the bulk liquid phase and that by the union of particles and the surrounding double layers. If κa f 0, the thickness of double layer approaches infinity. In this case, the contribution by the former becomes negligible, and the conductivity is dominated by the latter. Since the mobility is known to approach a constant as κa f 0,18 K*/K∞ also approaches a constant. On the other hand, if κa f ∞, the double layer becomes infinitely thin, and the conductivity is dominated by the transport of ions in the bulk liquid phase. Since the mobility is also known to approach a constant as κa f ∞,18 K*/K∞ approaches a constant too. Figure 2 shows that the limiting effective conductivity as κa f ∞ is less than the volume fraction of the liquid phase (0.75 in this case) is due to the fact that the presence of the cylinders has the effect of increasing the resistance for fluid flow and, therefore, a decrease in the scaled conductivity. The variation of the scaled conductivity (K*/K∞) as a function of κa at various R ()-z2/z1) is illustrated in Figure 3. This figure suggests that the effect of a on K*/K∞ is significant if κa is small, and becomes negligible as κa f ∞. This is mainly due to the shielding effect of counterions; the higher the valence of counterions, the more important the effect. For the case examined, since particles are positively charged, the influence of anions is more significant, and the smaller the R (the smaller the valence the anion) the smaller the scaled conductivity. However, if the double layer is sufficiently thin, the role played by the surface charge of a particle becomes insignificant, and therefore, K*/K∞ is independent of R. Figure 3 also reveals that if R is larger than unity, K*/K∞ decreases monotonically with the increase in κa, and the reverse is true if R is less than unity. The variation of the scaled conductivity (K*/K∞) as a function of φr at various κa is presented in Figure 4. This (18) Hunter, R. J. Foundations of Colloid Science; Clarendon: Oxford, 1989; Vol. I.
Lee et al.
Figure 3. Variation of scaled conductivity (K*/K∞) as a function of κa at various R ()-z2/z1) for the case φr ) 1. Key: same as that for Figure 2.
Figure 4. Variation of scaled conductivity (K*/K∞) as a function of φr at various κa. Key: same as that for Figure 2.
figure reveals that if double layer is thin (κa is large), the effect of surface charge on the scaled conductivity is insignificant, as expected. On the other hand, if the thickness of double layer becomes appreciable, the effect of surface charge should be taken into account. Figure 5 illustrates the variation of the scaled conductivity (K*/K∞) as a function of φr at various R. This figure suggests that if R is greater than unity, that is, the valence of counterions is greater than that of co-ions, K*/K∞ increases monotonically with φr, and the larger the R the faster the rate of increase. Again, this is due to the shielding effect of counterions. Figure 5 also shows that if R is less than unity, K*/K∞ has a local minimum as φr varies. This can be explained by rewriting eq 24 as
I1 + I2 K* )∞ (Φ2)r*)b/a K
(25)
where
I1 + I2 +
( (
) ( )[ ) ( )[
R 1 + Pe1 Pe2
-1
R 1 + Pe1 Pe2
-1
] ]
b 1 dG1 exp(-φrφ1) a Pe1 dr*
r*)b/a
b 1 dG2 exp(Rφrφ1) a Pe2 dr*
r*)b/a
(25a) (25b)
Conductivity of a Cylindrical Dispersion
Figure 5. Variation of scaled conductivity (K*/K∞) as a function of φr at various R ()-z2/z1) for the case κa ) 1. Key: same as that for Figure 2.
Figure 6. Variations of -I1, -I2, defined in eqs 25a and 25b, and -(I1 + I2) as a function of φr for the case given in Figure 5.
Here, I1 is a measure for the contribution to the scaled conductivity by cations, and I2 is that for the contribution to the scaled conductivity by anions. Figure 6 shows the variations of -I1, -I2, and -(I1 + I2) as a function of the scaled surface potential -(I1 + I2). This figure reveals that -I1 decreases monotonically with φr, but I2 increases monotonically with φr. The net effect leads to a local minimum in -(I1 + I2) as -(I1 + I2) varies. As R gets large, the contribution to the scaled conductivity by anions becomes more significant than that by cations, I1 becomes less significant than I2, and the local minimum disappears. Figure 7 shows the variation of the scaled conductivity (K*/K∞) as a function of κa at various H values for the case of a higher surface potential, and that for a lower surface potential is illustrated in Figure 8. As defined in eq 1, H is a measure for the concentration of the dispersed phase. Figures 7 and 8 suggest that the limiting value of K*/K∞ as κa f ∞ is a function of H, and this value is slightly less than H. The rationale behind this is stated in the discussion of Figure 2. Figures 7 and 8 also reveal that
Langmuir, Vol. 17, No. 6, 2001 1825
Figure 7. Variation of scaled conductivity (K*/K∞) as a function of κa at various H for the case φr ) 3.0. Key: R ) 1.0, Pe1 ) 0.01, and Pe2 ) 0.01.
Figure 8. Variation of scaled conductivity (K*/K∞) as a function of κa at various H for the case φr ) 1.0. Key: same as that given in Figure 7.
K*/K∞ approaches a constant as κa f 0, and the limiting value depends on both H and φr. For a fixed H, this limiting value increases with φr as expected. Figure 7 shows that for a fixed H, K*/K∞ remained at about a constant value for small κa and starts to decrease if κa exceeds a certain value; the larger the H, the earlier the K*/K∞ starts to decrease, and also the faster the rate of decrease. This is mainly due to the relative magnitude of the thickness of double layer l/κ and the radius of liquid envelope b. If 1/κ g b, K*/K∞ will remain at constant. For a fixed particle radius a, a larger H implies a larger b, and the earlier K*/K∞ starts to decrease. As suggested by Figure 8, this phenomenon may become inappreciable, however, if the surface potential is low. Acknowledgment. This work is supported by the National Science Council of the Republic of China. LA0006566
