J. Phys. Chem. B 2000, 104, 6815-6820
6815
Sedimentation of a Nonconducting Sphere in a Spherical Cavity Eric Lee, Chen-Bin Yen, and Jyh-Ping Hsu* Department of Chemical Engineering, National Taiwan UniVersity, Taipei, Taiwan 10617, R.O.C. ReceiVed: October 18, 1999; In Final Form: March 14, 2000
The sedimentation of a nonconducting spherical particle in a spherical cavity is analyzed theoretically. We show that, for the case where the former is positively charged and the latter uncharged, the sedimentation velocity of the particle decreases with κa, κ and a being respectively the reciprocal Debye length and the radius of a particle. If the surface potential of the particle φr is high, the sedimentation potential has a local maximum as κa varies, which becomes inappreciable if the surface potential is low. If φr is sufficiently high, the sedimentation potential has a local maximum as λ () radius of particle/radius of cavity) varies. It is not observed, however, if κa is large. If an uncharged particle is placed in a positively charged cavity, the sedimentation potential may reverse its sign, and may have a local minimum as κa varies.
Introduction When a charged colloidal dispersion is exposed to a gravitational field, sedimentation occurs to the charged entities in the direction of the field. For spherical colloidal particles, this leads to a distortion in the electrical double layer surrounding a particle, and its spherical symmetry no longer exists.4 An electric field in the opposite direction as that of the gravitational field is induced, which reduces the sedimentation velocity of the particle. This has the effect of reducing the hydrodynamic drag on particle surface, which is absent if the particle is uncharged.4 Gravitational sedimentation is similar to the electrophoresis of charged particles in an applied electrical field, and, in fact, a so-called Onsager relation exists which relates these two phenomena.2,3 A thorough review of the sedimentation of colloidal particles can be found in Deen.1 The theoretical study was pioneered by Smoluchowski.5 It was found that the sedimentation velocity of a charged particle with an infinitely thin double layer is slower than that of an uncharged particle by a term which is proportional to the square of the surface potential of the former. The analysis was also extended to the case of an arbitrary thick double layer.6-8 The result of Booth6 was modified later by Saville9 such that the constraint on the electric current is satisfied. Ohshima et al.3 considered the sedimentation of charged particles in an infinite medium. Under the conditions of low surface potential and large κa, 1/κ and a being respectively the Debye length and the radius of a particle, they were able to derive approximate analytical expressions for both the sedimentation potential and the sedimentation velocity. Numerical results were also provided for these quantities for the case of arbitrary surface potential and double layer thickness. In practice, the effect of the presence of nearby entities on the sedimentaiton of a particle may play a significant role. For example, if the concentration of particles is appreciable, the interactions between adjacent particles may be important. Sedimentaiton in a finite domain is another example in which the presence of rigid boundaries needs to be taken into account. * To whom correspondence should be addressed. Tel: 886-2-23637448. Fax: 886-2-23623040. E-mail:
[email protected] This problem was considered by Pujar and Zydney4 where the sedimentation of a charged particle in a spherical cavity was solved. Assuming small Peclet number and low surface potential, they concluded that the presence of the cavity has the effect of increasing the magnitude of the excess force if 1/κ is small, and the reverse is true if it is large. Although the system consdiered by Pujar and Zydney4 is an idealized one it has the merit that the problem under consideration is essentially onedimensional, and it is capable of simulating the electrokinetic behavior in a porous medium or in a cylindrical geometry.10 Their analysis, however, is limited to low surface potential. In this study, an attempt is made to extend the analysis of Pujar and Zydney4 to the case where the surface potential is not necessarily low. The pseudospectral method based on Chebyshev polynomials proposed by Lee et al.11 was empoloyed, and a numerical scheme for calculating the sedimentation potential is developed. Theory Referring to Figure 1, we consider a nonconducting spherical particle of radius a located at the center of a spherical cavity of radius b, falling steadily under the influence of gravity in a z1: z2 electrolyte solution, z1 and z2 being respectively the valences of cations and anions. If we define R ) -z2/z1, then the electroneutrality in the bulk liquid phase requires that n20 ) (n10/R), n10 and n20 being the bulk concentrations of cations and anions, respectively. Choosing the spherical coordinates (r,θ,φ) with the origin located at the center of the particle is appropriate for the present problem. The conservation of ions leads to
[[
∇‚ Dj ∇nj +
] ]
zjenj ∇φ + njb V ) 0 j ) 1, 2 kBT
(1)
where ∇ is the gradient operator, kB is the Boltzmann constant, e, φ, and b v are respectively the elementary charge, the electrical potential, and the fluid velocity, nj and Dj are the number concentration and the diffusievity of ion species j, respectively, and T is the absolute temperature. Suppose that the time scale for double-layer relaxation is much smaller than that of the
10.1021/jp993706s CCC: $19.00 © 2000 American Chemical Society Published on Web 07/01/2000
6816 J. Phys. Chem. B, Vol. 104, No. 29, 2000
Lee et al. can be approximated by the corresponding linear expressions. The symmetric nature of the problem under consideration suggests that10,13
φ/2 ) Φ2(r)cosθ
(6)
g/1 ) G1(r)cosθ
(7)
g/2 ) G2(r)cosθ
(8)
Ψ* ) Ψ(r)sin2θ
(9)
The original problem reduces to a set of one-dimensional differential equations. The variation of the scaled electrical potential, φ/1, is described by
Lφ/1 ) Figure 1. Schematic representation of the system under consideration. A sphere of radius a is located at the center of a spherical cavity of radius b. U is the sedimentation velocity, g is gravity, and E is the induced electric field.
where the operator L and the inverse Debye length κ are defined respectively as
sedimentation of the particle. Then the spatial variaiton in the electrical potential is described by the Poisson equation11
∇2φ ) -
F
∑j
)-
2
(κa) 1 [exp(-φrφ/1) - exp(Rφrφ/1)] (10) (1 + R) φr
L≡
d2 2 d 2 + - 2 2 r* dr* dr* r*
(11)
and
zjenj (2)
2
κ)[
nj0(ezj)2/kBT]1/2 ∑ j)1
(12)
where is the permittivity of the liquid phase and F is the space charge density. Suppose that the Reynolds number is small. Then the flow field in the cavity can be described by12
If the particle is charged and the cavity uncharged, then the boundary conditions associated with eq 10 are
∇‚V b)0
(3)
φ/1 ) 1 at r* ) 1
(13a)
0 ) η∇2V - ∇p - F∇φ
(4)
φ/1 ) ζb/ζa at r* ) 1/λ
(13b)
where p and η denote respectively the pressure and the viscosity of the liquid phase. For convenience, φ is decomposed into two parts:13 the electrical potential that would exist in the absence of the induced electrical field, φ1, and the electrical potential outside the particle arising from the induced electric field, φ2. The effect of doublelayer polarization is taken into account by defining12
(
nj ) nj0 exp -
)
zje(φ1 + φ2 + gj) kBT
Similarly, the following equations need to be solved simultaneously:
[
L-
]
(κa)2 [exp(-φrφ/1) + R exp(Rφrφ/1)] Φ2 ) 1+R (κa)2 [G exp(-φrφ/1) + RG2 exp(Rφrφ1)] 1+R 1
(5)
where gj, j ) 1, 2, is a perturbed term. For convenience, the analysis below is based on scaled quantities. To this end, the surface potential of the particle, ζa (or that of the cavity, ζb, if the particle is uncharged), the particle radius a, and the bulk density of cation, n10, are adopted as the scaled factors. We define φ1 ) φ/1ζa (or φ1 ) φ/1ζb), nj ) n/j nj0, and ∇* ) a∇. The electrophoretic velocity of an isolated particle predicted by the Smoluchowski’s theory when an electric field of strength ζa/a is applied, UE ) ζ2a/ηa (or ζ2b/ηa if the particle is uncharged), is chosen as the scaled factor for velocity in the following discussion.11 Also, instead of solving eqs 3 and 4 directly, the equation governing the stream function ψ, which can be obtained by taking curl on eq 4, is solved. If φ2 and gj are small, then the set of nonlinear electrokinetic equations
LG1 - φr
dφ/1 dG1 dφ/1 ) Pe1φ2r V/r dr* dr* dr*
(15)
dφ/1 dG2 dφ/1 2 / LG2 + Rφr ) Pe2φr Vr dr* dr* dr* D4ψ ) -
[
(14)
(16)
]
dφ/1 (κa)2 / (n1G1 + Rn/2G2) 1+R dr*
(17)
where Pej ) (z1e/kBT)2, j ) 1, 2, is the electric Peclet number of ion species j, and the operator D4 is defined as D4 ) D2D2 with
D2 )
d2 2 dr*2 r*2
(18)
Sedimentation of a Nonconducting Sphere
J. Phys. Chem. B, Vol. 104, No. 29, 2000 6817
The associated boundary conditions are
The problem under consideration can be decomposed into two sub-problems,11,13 namely, (a) the particle moves at a constant velocity in the absence of the induced electric field, and (b) the particle remains fixed at the center of the cavity. For the present linear system it can be shown that 〈i〉1 ) δU* and 〈i〉2 ) βE*, where 〈i〉1 and 〈i〉2 are the net currents across the plane θ ) π/2 in problems 1 and 2, respectively. The constraint that the net current vanishes implies that E*/U* ) -δ/β. The sedimentation velocity of the particle can be calculated based on the fact the sum of the external forces acting on it, which includes the electric force, FEz, the hydrodynamic forces, FDz, and the gravitational force, Fg, vanishes. These forces can be expressed as11
dΦ2/dr* ) 0 at r* ) 1
(19a)
dΦ2/dr* ) -E/z at r* ) 1/λ
(19b)
dGj/dr* ) 0 at r* ) 1, j ) 1, 2
(20a)
Gj ) -Φ2 at r* ) 1/λ, j ) 1, 2
(20b)
Ψ ) -U*r*2/2 and dΨ/dr* ) -U*r* at r* ) 1 (21a) Ψ ) dΨ/dr* ) 0 at r* ) 1/λ
(21b)
( (
Equation 19a states that the particle is nonconducting, and eq 19b specifies the induced electrical field results from the sedimentation of the particle. Equation 20a states that the particle surface is impenetrable to electrolyte, and eq 20b implies that the concentration of each ionic species reaches the bulk value. Equation 21a states that the particle moves with velocity U,14 and eq 21b implies that the cavity remains fixed and the noslip condition for fluid. The scaled sedimentation potential, E*/U*, where E* ) E/(ζa/a) (or ) E/(ζb/a) if particle is uncharge) and U* ) U/UE, can be evaluated based on the fact that the sedimentation of the particle generates no net current, or the net flow across any horizontal plane vanishes.11 In this study, we apply this constraint to the horizontal plane described by θ ) π/2. We have11
∫abriθdr|θ)(π/2) ) 2π∫abr(∑zjenjVjθ)dr|θ)(π/2)
exp(Rφrφ/1)]Φ2]r*)1 4 ) πζ2a(F/Dhz + F/Dez) 3
F/Ez
(
{
]}
1 1 R exp(-φrφ/1)∇*g/1 + exp(Rφrφ/1)∇*g/2 φr Pe1 Pe2
{
(25) U)-
2φ2r kBT 3 (κa)2 dΨ [exp(-φrφ/1) + exp(Rφrφ/1)] iθ ) 3 z e dr* (1 + R) ηa 1
[
(32)
r*)1
]}
(34)
where f1 is the summation of forces, 2F/Ez, F/Dhz, and F/Dez related to problem 1, and f2 is that related to problem 2. Solving eq 34 for U gives
Also,
( )
( ))
∂ D2Ψ ∂r* r*2
4 4 πζ2a (f1U* + f2E*) + πa3(Fp - Ff)g ) 0 3 3
2φ2r kBT 3 (κa)2 [exp(-φrφ/1) + exp(Rφrφ/1)]V b* + bı ) 3 ηa z1e (1 + R)
[
(31)
r*)1
We have
According to Lee et al.,11 bı can also be expressed as
( )
)
(κa)2 [r*2[exp(-φrφ/1) - exp(Rφrφ/1)]Φ2]r*)1 (1 + R)φr (33)
and
(24)
(
dφ/1 ) r* Φ2 dr*
F/Dhz ) r*4 F/Dez )
)
(30)
In these expressions
(23)
zje ∇n bj ∇φ B+ kBT nj
(29)
4 Fg ) πa3(Fp - Ff)g 3
2
(
r*)1
(κa) 4 + πζ2a [r*2[exp(-φrφ/1) 3 (1 + R)φr
where, iθ and Vjθ are the θ-component of current bı and the θ component of the flow velocity of the jth ionic species, b Vj, defined respectively as11
V - Dj b Vj ) b
(28)
2
(22)
zjenjb Vj ∑ j)1
8 ) πζ2a F/Ez 3
( ))
j)1
bı )
r*)1
∂ D 2Ψ 4 FDz ) πζ2a r*4 3 ∂r* r*2
2
〈i〉 ) 0 ) 2π
)
dφ/1 8 Φ FEz ) πζ2a r* 3 dr* 2
a2(Fp - Ff)g δ -1 f1 - f 2 η β
(
)
(35)
If we let U0 be the sedimentation velocity of an uncharged sphere, then
U0 )
sin θ R 1 1 (26) exp(-φrφ/1)G/1 + exp(Rφrφ/1)G/1 φr Pe1 Pe2 r*
2 2 a (Fp - Ff)g KD0 9 η
(36)
where
Substituting this expression into eq 22 yields
〈i〉 ) 0 ) 2πa Ia 2
∫1
b/a
Iθ(r*)dr*
(27)
KD0 )
1 - (9/4)λ + (5/2)λ3 - (9/4)λ5 + λ6 1 - λ5
(37)
6818 J. Phys. Chem. B, Vol. 104, No. 29, 2000
Lee et al.
Figure 2. Variation of the scaled sedimentation velocity U/U0 as a function of scaled surface potential φr at various λ for the case κa ) 0.1 and Pe1 ) Pe2 ) 0.01 (Ohshima et al.3).
Figure 3. Variation of scaled sedimentation potential E*/U* as a function of κa at various λ for the case φr ) 4.0 and Pe1 ) Pe2 ) 0.01.
KD0 is the hindrance factor for the sedimentation of an uncharged sphere in a spherical cavity.4,15 Equations 35 and 36 lead to
U0 δ -1 9 f1 - f2 ) U 2KD0 β
(
)
(38)
Discussion Two special cases are considered in the numerical simulation, namely, the particle is positively charged and the cavity is uncharged, and the particle is uncharged and the cavity is positively charged. In the latter, eqs 13a and 13b should be replaced by
φ/1) 0 at r* ) 1
(39a)
φ/1) 1 at r* ) 1/λ
(39b)
Particle Positively Charged and Cavity Uncharged. Let us consider first the case where the particle is positively charged and the cavity is uncharged. Figure 2 shows the variation of the scaled sedimentation velocity of a particle, U/U0, as a function of φr ()ζaz1e/kBT) at various λ () radius of particle/ radius of cavity). For comparison, the result of Ohshima et al.,3 which is valid for the case φr is low and the particle is in an infinite fluid, is also presented. As can be seen in Figure 2, their result can be recovered from the present one when φr f 0, as expected. Figure 2 suggests that U/U0 decrease with the increase in φr. This is because if φr is high, the concentration of ions in the double layer surrounding the particle becomes more concentrated, and the induced electrical field, which is in the opposite direction as the sedimentation of the particle, becomes stronger. Figure 2 also reveals that, for a fixed φr, U/U0 approaches unity as λ increases. According to Zydney,4 the hydrodynamic interaction due to the presence of cavity wall will reduce the relaxation effect of the double layer, which is responsible for the induced electrical field. Therefore, the larger the λ, the less the space between the particle and the cavity, and the less the effect of double-layer relaxation. In this case, the behavior of the particle is close to that of an uncharged particle. The variation of the scaled sedimentation potential, E*/U*, as a function of κa at various λ is illustrated in Figure 3. As can be seen from this figure, E*/U* has a local maximum as
Figure 4. Variation of scaled sedimentation potential E*/U* as a function of λ at various scaled surface potential φr for the case κa ) 0.1 and Pe1 ) Pe2 ) 0.01.
κa varies, which becomes inappreciable if φr is low. As pointed out by Lee et al.,11 the electrical field inside the double layer has the effect of reducing the effect of double-layer relaxation. For constant surface potential, the larger the κa, the smaller the gradient of the electrical potential in a double layer, and, therefore, E*/U* increases with the increase in κa. If φr is too high, however, the induced electrical field will reduce the effect of double-layer relaxation. Figure 3 also suggests that if κa is small, E*/U* may have a local maximum as λ varies. This is justified by Figure 4 where the variation of the scaled sedimentation potential, E*/U*, as a function of λ at various φr is presented. This figure reveals that, if φr is sufficiently high, E*/U* has a local maximum as λ varies. Referring to Figure 3, this is not observed, however, if κa is large. This is because the concentration of ions in the double layer surrounding the particle increases with the increase in λ, and so does E*/U*. On the other hand, the larger the λ, the less the available space between the particle and the cavity. In this case, the hydrodynamic interaction with the cavity wall becomes more significant, which reduces the scaled sedimentation potential. Particle Uncharged and Cavity Positively Charged. Let us consider next the case where the particle is uncharged and the cavity is positively charged. Figure 5 shows the simulated variation in the negative scaled sedimentation potential, -E*/ U*, as a function of κa at various scaled surface potential of
Sedimentation of a Nonconducting Sphere
J. Phys. Chem. B, Vol. 104, No. 29, 2000 6819
Figure 6. Variations of the currents in problem 1 as a function of κa for the case φr ) 6.0, λ ) 0.5, and Pe1 ) Pe2 ) 0.01. Figure 5. Variation of negative scaled sedimentation potential -E*/U* as a function of κa at various scaled surface potential φr for the case λ ) 0.5 and Pe1 ) Pe2 ) 0.01.
cavity, φr () ζbz1e/kBT). As can be seen from this figure, -E*/ U* may change its sign and may have a local minimum as κa varies. The rationale behind this can be elaborated as following. Let us decompose the current Iθ(r*) into two parts,11 namely, the current due to the convective motion of liquid phase, Iθ,c, and that due to the diffusion of ionic species, Iθ,d, defined respectively as
∂Ψ Iθ,c ) [exp(-φrφ/1) - exp(Rφrφ/1)] ∂r*
(40)
and
Iθ,d ) -
[
]
1 1 R exp(-φrφ/1)G1 + exp(Rφrφ/1)G2 (41) φr Pe1 Pe2
For convenience, we define11
∫1a/bIθ,cdr*)problem 1
-δc ) (
∫1a/bIθ,cdr*)problem 2
βc ) (
-δd ) (
∫1a/bIθ,ddr*)problem1
(42) (43) (44)
∫1a/bIθ,ddr*)problem2
(45)
δ ) δc + δd
(46)
β ) βc + βd
(47)
βd ) (
The variations of -δ, -δc, and -δd as a function of κa are illustrated in Figure 6. This figure reveals that the current due to convection, which arises from the motion of the particle in the -Z direction, -δc, is positive and decreases with the increase in κa. This is because the concentration of electrolyte decreases with κa. Figure 6 reveals that -δd have a local minimum as κa varies. This is because the particle is moving downward, and the effect of double-layer relaxation leads to a higher concentration of electrolyte near the bottom of the cavity. This effect is pronounced as κa increases since the change in electrolyte concentration becomes more significant. However, the potential gradient ∇φ1 increases with κa also, and the net effect is that
Figure 7. Variations of the currents β, βc, and βd in problem 2 as a function of κa for the case of Figure 6.
-δd has a local minimum, or |δd| has a local maximum as κa varies. Figure 6 also suggests that if κa is small, the current due to convection, -δc, is dominant and, for a medium κa, the current due to diffusion, -δd, becomes more significant. Therefore -δ may change its sign and may have a local minimum as κa varies. Figure 7 illustrates the variations of β, βc, and βd as a function of κa. As can be seen from this figure, βc is negligible compared with βd, that is β = βd. Note that in problem 2, the induced electrical field is in the Z direction. Therefore, the concentration of anion at the bottom of cavity is higher than that at its top, and βd is negative. Similar to the behavior of -δd, βd has a local minimum as κa varies. On the basis of the above discussions, E*/U* () -δd/β) may reverse its sign and may have a local minimum as κa varies. Acknowledgment. This work is partially financially supported by the National Science Council of the Republic of China. References and Notes (1) Deen, W. M. AIChE J. 1987, 27, 952. (2) de Groot, S. R.; Mazur, P.; Overbeek, J. Th. G. J. Chem. Phys. 1952, 20, 1825. (3) Ohshima, H.; Healy, T. W.; White, L. R.; O’Brien, R. W. J. Chem. Soc., Faraday Trans. 2 1984, 80, 1299. (4) Pujar, N. S.; Zydney, A. L. AIChE J. 1996, 42, 2101. (5) Smoluchowski, M. In Handbuch der Elecktrizitat und des Magnetismus, Vol. II; Graetz, L., Ed., Barth: Leipzig, 1921.
6820 J. Phys. Chem. B, Vol. 104, No. 29, 2000 (6) (7) (8) (9) (10) 338. (11)
Booth, F. J. Chem. Phys. 1954, 22, 1956. Overbeek, J. Th. G. Kolloid-Beih. 1943, 54, 287. Booth, F. Proc. R. Soc. London, Ser. A 1950, 203, 514. Saville, D. A. AdV. Colloid Interface Sci. 1982, 16, 267. Pujar, N. S.; and Zydney, A. L. J. Colloid Interface Sci. 1997, 192, Lee, E.; Chu, J. W.; Hsu, J. P. J. Chem. Phys. 1999, 110, 11643.
Lee et al. (12) Lee, E.; Chu, J. W.; Hsu, J. P. J. Colloid Interface Sci. 1998, 205, 65. (13) O’Brien, R. W.; White, L. R. J. Chem. Soc., Faraday Trans. 2 1978, 74, 1607. (14) Happel, J.; Brenner, H. Low-Reynolds Number Hydrodynamics; Martinus: Nijhoff, 1983. (15) Zydney, A. L. J. Colloid Interface Sci. 1995, 169, 476.
