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Sedimentation of Concentrated Charged Spheres at Low Surface Potentials Ming-Hui Chih, Eric Lee, Jhih-Wei Chu, and Jyh-Ping Hsu* Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan 10617, Republic of China Received July 9, 1999. In Final Form: November 1, 1999 The sedimentation of concentrated charged spherical particles in an electrolyte solution is investigated theoretically for an arbitrarily thick double layer taking the effect of double-layer polarization into account. We show that the ratio E*/U* ()scaled sedimentation potential per scaled sedimentation velocity) is a constant if κa is either very small or very large, κ and a being respectively the reciprocal Debye length and the radius of a particle. The variation of E*/U* as a function of κa has a minimum at a medium κa. This minimum increases with the increase in particle concentration. The scaled sedimentation velocity approaches a constant if κa is small, and its variation as a function of κa has a maximum at a medium κa. This maximum decreases with the decrease in particle concentration.
1. Introduction Gravitational sedimentation, a phenomenon similar to electrophoresis, is of both fundamental and practical significance. Booth1 was able to derive both the sedimentation velocity and sedimentation potential of an isolated particle in an infinite fluid under the conditions of low potential and thin double layer. The solution was expressed as a power series in terms of zeta potential. Stigter2 obtained an Onsager relation between electrophoretic velocity and sedimentation potential by adopting a numerical scheme based on Booth’s theory. Saville3 pointed out that the analysis of Booth1 is related to the shape of suspension volume, and the condition that the net current vanishes is not satisfied for an arbitrarily shaped suspension volume. A formula for sedimentation potential was derived which leads to the result of Booth if the shape of the suspension volume becomes an infinite horizontal sheet and the mobilities of all the ions are identical. Ohshima et al.4 extended the analysis of Saville3 by assuming that the net current vanishes. The governing equations derived were solved numerically to obtain both the sedimentation velocity and sedimentation potential for arbitrary levels of potential and double-layer thickness. Approximate analytical expressions were also derived under the conditions of low zeta potential and thin double layer. The analysis on the behavior of a concentrated dispersion was initiated by Levine et al.5 They adopted the cell model proposed by Kuwabara6 and took the effect of double-layer polarization into account. In this approach a concentrated dispersion is modeled by a representative particle surrounded by a liquid shell, and the concentration of particles is measured by the relative sizes of the representative particle and the liquid shell. Analytical expressions for both sedimentation velocity and sedimen* To whom correspondence should be addressed. Fax: 886-223623040. E-mail:
[email protected]. (1) Booth, F. J. Chem. Phys. 1954, 22, 1956. (2) Stigter, D. J. Phys. Chem. 1980, 84, 2758. (3) Saville, D. A. Adv. Colloid Interface Sci. 1982, 16, 267. (4) Ohshima, H.; Healy, T. W.; White, L. R.; O’Brien, R. W. J. Chem. Soc., Faraday Trans. 2 1984, 80, 1299. (5) Levine, S.; Neale, G. H.; Epstein, N. J. Colloid Interface Sci. 1976, 57, 421. (6) Kuwabara, S. J. Phys. Soc. Jpn. 1959, 14, 527.
tation potential were derived by them based on the assumptions of low surface potential, negligible electrical body force, and negligible double-layer interaction. The analysis was modified by Ohshima7 to arrive at a more simplified expression and to obtain an Onsager relation between electrophoresis and sedimentation.8 One of the basic assumptions of Levine et al.5 is that the double layer surrounding a particle is thin, and the interaction between adjacent double layers is insignificant as a result. In this case the equation governing the electrical field can be decoupled from that governing the flow field. Although this makes the subsequent mathematical treatment simpler, the result obtained is limited to high electrolyte concentrations. In addition, if the concentration of dispersed phase becomes high, the probability that the interaction between adjacent double layers increases accordingly, and its effect on the behavior of the system under consideration needs to be taken into account. In this study, the analysis of Levine et al.5 is extended to the case where the electrical body force is considered, and the effects of double-layer interaction and double-layer polarization on the sedimentation behavior of a concentrated colloidal dispersion are investigated. 2. Theory The cell model proposed by Kuwabara6 is adopted to model the behavior of a monodispersed spherical particles in a z1:z2 electrolyte solution, z1 and z2 being the valences of cations and anions, respectively. Let n10 and n20 be the bulk concentrations of cations and anions, respectively. Then n10 ) (n10/R) with R ) -z2/z1. Referring to Figure 1, the nature of the dispersed system is simulated by a representative particle of radius a which is surrounded by a concentric spherical shell of liquid phase of radius b. The concentration of the dispersed phase is measured by the radius ratio H ()a/b). The spherical coordinates (r,θ,φ) with its origin located at the center of the particle are adopted. The sedimentation of the particle in the -z direction induces an electric field E in the z direction (θ ) 0). For simplicity, we assume that the liquid phase is incompressible and has constant physical properties and the motion of particles is sufficiently slow. Under these (7) Ohshima, H. J. Colloid Interface Sci. 1997, 188, 481. (8) Ohshima, H. J. Colloid Interface Sci. 1998, 208, 295.
10.1021/la9908992 CCC: $19.00 © 2000 American Chemical Society Published on Web 01/08/2000
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Langmuir, Vol. 16, No. 4, 2000 1651
concentration of electrolytes as
(
nj ) nj0 exp -
)
zje(φ1 + φ2 + gj) kT
(6)
j ) 1, 2 where gj represents a function which accounts the effects of fluid flow on concentration. Suppose that the surface potential is low and φ2 and gj are small. Then eq 6 can be approximated by the linear expression
[
nj ) nj0 1 -
]
zjeζa / (φ + φ/2 + g/j ) kT 1
(7)
j ) 1, 2 where φ/j ) φj/ζa and g/j ) gj/ζa, ζa being the zeta potential. Following the treatment of Lee et al.11 (Appendix), it can be shown that the variation of the scaled electric potential at equilibrium, φ/1, can be described by Figure 1. Schematic representation of cell model, where a particle of radius a is surrounded by a spherical liquid shell of radius b. The sedimentation of particles induces an electric field E in the z direction.
Lφ/1 ) -(κa)2φ/1
(8)
where φr ()ζaz1e/kT) denotes the scaled surface potential and
conditions, the conservation law yields
[[
]
]
zjenj ∇ B φ + njb ∇ B ‚ Dj ∇ B nj + ν )0 kT
(1)
κ)[
Here, ∇ B represents the gradient operator and Dj, nj, and zj are the diffusivity, the number concentration, and the valence of ion species j, respectively. φ and e are the electrical potential and elementary charge, respectively, b ν is the fluid velocity, and k and T are the Boltzmann constant and the absolute temperature, respectively. We assume that the spatial electrical potential can be described by the Poisson-Boltzmann equation 2
∑ j)1
2 2 d d2 - 2 + 2 r* dr* dr* r*
(8a)
2
j ) 1, 2
∇2φ ) -
L≡
zjenj
(2)
where is the permittivity of the liquid phase. Suppose that the Reynolds number characterizing the liquid flow near the representative particle is sufficiently small. If we let p, η, and F be the pressure, the viscosity, and the space charge density, respectively, then the flow field can be described by
∇ B ‚ν b)0
(3)
ν-∇ B p - F∇ Bφ ) 0 η∇2b
(4)
9
Following the treatment of Lee et al., φ is decomposed as
φ ) φ 1 + φ2
(5)
where φ1 and φ2 are respectively the electrical potential that would exist in the absence of the induced electric field and that outside a particle arising from the induced electric field. According to O’Brien and White,10 doublelayer polarization can be simulated by expressing the (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.
nj0(ezj)2/kT]1/2 ∑ j)1
(8b)
with r* ) r/a. The boundary conditions associated with eq 8 are
φ/1 ) 1,
r* ) 1
dφ/1/dr* ) 0,
(9a)
r* ) b/a
(9b)
It is assumed that the unit cell as a whole is electrically neutral. We have to solve eq 8 simultaneously with10
(κa)2 [G + RG2] 1+R 1
[L - (κa)2]Φ2 )
(10)
LG1 ) Pe1v/r
dφ/1 dr*
(11)
LG2 ) Pe2v/r
dφ/1 dr*
(12)
D4Ψ ) -
[ (
]
dφ/1 (κa)2 (G1 + RG2) 1+R dr*
D4 ) (D2)2 )
)
d2 2 - 2 2 dr* r*
(13)
2
(14)
where Pej ) aUE/Dj, j ) 1, 2, represents the electric Peclet number of ion species j. We assume the following: (a) The representative particle is nonconducting and its surface is impenetrable. (b) The number density of each ionic species equals to the (11) Lee, E.; Chu, J. W.; Hsu, J. P. J. Colloid Interface Sci. 1998, 205, 65.
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Chih et al.
corresponding equilibrium density at the virtual surface, r ) b. (c) The representative particle is moving at velocity U. (d) At the steady state the virtual surface satisfies the Kuwabara’s model of zero vorticity. Under these conditions the boundary conditions associated with eqs 10-14 are
dΦ2/dr* ) 0 at r* ) 1
(15a)
dΦ2/dr* ) -E/z at r* ) b/a
(15b)
dGj/dr* ) 0 at r* ) 1, j ) 1, 2
(15c)
Gj ) -Φ2 at r* ) b/a, j ) 1, 2
(15d)
dΨ 1 ) U*r* at r* ) 1 Ψ ) U*r*2 and 2 dr* Ψ)
[
]
1 d2Ψ 2 Ψ ) 0 at r* ) b/a r* dr*2 r*3
(15e)
) aEz/ζa and U* ) U/UE, Ez and In these expressions U being the z component of the induced electric field and the sedimentation velocity of a particle, respectively, and UE ()ζ2a/ηa) is the electrophoretic velocity of an isolated sphere in an electric field of strength (ζa/a). Equation 15b represents the induced electric field as a result of sedimentation of charged particles, and if double-layer interaction is negligible, then it reduces to φ2 ) 0. The condition of streaming potential requires that the sedimentation of particles generates no net current. Following the treatment of Levine and Neale,12 we let the net current, 〈i〉, across the horizontal plane θ ) π/2 vanish, that is
∫1b/a Iθ(r*) dr* ) 0
〈i〉 ) 2πa2Ia
) δU* + βE* ) 0
)
∇ B nj zje ∇ Bφ + kT nj
(17)
Iθc ) -φr Iθd ) -
[
{
]}
(18)
[
[
with Ia )
2φ3a(κa)2/ηa3,
]
(21b)
∫1a/bIθ,c dr*)problem1
(21c)
∫1a/bIθ,d dr*)problem1
(21d)
δ ) δ c + δd
(21e)
δc ) ( δd ) (
The last expression gives E*/U* ) -δ/β, E*/U* being the induced sedimentation potential per unit sedimentation velocity. The external forces acting on the representative particle in the z direction comprise the electrical force FEz, the hydrodynamic force FDz, and the gravitational force Fg. These forces can be calculated by11
( ) ( ( ))
FEz ) FDz )
dφ/1 8 πζ2a r* Φ 3 dr* 2
4 ∂ D2Ψ πξ2a r*4 3 ∂r* r*2
) r*)1
8 / πζ2a FEz 3
(22)
+
r*)1
4 πξ2a(κa)2[r*2φ/1Φ2]r*)1 3
]}
G/1 2φ3a RG/1 sin θ 1 2 / dΨ (κa) -φ + iθ ) 1 dr* 1 + R Pe1 Pe2 r* ηa3 sin θ ) IaIθ(r*) r*
(21a)
RG2 G1 1 + 1 + R Pe1 Pe2
and
{
dΨ dr*
For convenience, we define
In our study, two ionic species are present with R ) -z2/z1 and n20 ) n10/R, and therefore
B *g/2 2φ2a B *g/1 R∇ 1 ∇ / 2 (κa) -φ b v * + + bı ) 1 1 + R Pe1 Pe2 ηa3
(21)
Equation 19a suggests that Iθ(r*) comprises the current due to the transport of ions through convection, Iθ,c, and that through diffusion, Iθ,d, where
j ) 1, 2
(
(20)
〈i〉 ) 〈i〉1 + 〈i〉2
(16)
v - Dj b vj ) b
(19a)
The linear problem under consideration can be separated into two subproblems:10 (a) a particle moves at a velocity U in the absence of the induced electric field and (b) the particle is held fixed in the induced electric field, referred as problems 1 and 2 hereafter. The net currents across the plane θ ) π/2 in problems 1 and 2, 〈i〉1 and 〈i〉2, can be expressed respectively as 〈i〉1 ) δU* and 〈i〉2 ) βE*. Since 〈i〉 must vanish at steady state, we have
∫abriθ dr|θ)π/2 ) 2π∫abr(∑zjenjvjθ) dr|θ)π/2
ı which can be where iθ is the θ component of current b, evaluated by b) ı ∑zjenjb vj. vjθ is the θ component of the flow velocity of the jth ionic species, b vj, which can be calculated by
]}
Equations 16 and 19 lead to
(15f)
E/z
〈i〉 ) 0 ) 2π
[
{
RG/2 G/1 dΨ 1 + dr* 1 + R Pe1 Pe2
Iθ(r*) ) -φ/1
(19) and
(12) Levine, S.; Neale, G. H. J. Colloid Interface Sci. 1974, 47, 520.
)
4 / / πξ2a(FDhz + FDez ) 3 Fg )
4 3 πa (Fp - Ff)/g 3
(23) (24)
where g and Ff are respectively the gravitational ac/ ) [r*(dφ/1/dr*)Φ2]r*)1, celeration and fluid density, FEz / / FDhz ) [r*4∂(D2Ψ/r*2)/∂r*]r*)1, and FDez ) [(κa)2/(1 + R)]/ [r*2φ1Φ2]r*)1. At steady state the net force acting on the
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Langmuir, Vol. 16, No. 4, 2000 1653
Figure 2. Variation of E*/U* as a function of κa at various radius ratios H ()a/b) for the case φr f 0. Key: R ) 1, Pe1 ) Pe2 ) 0.01.
Figure 3. Variation of δ, δc, and δd as a function of κa for the case of Figure 2 except that H ) 0.5.
representative particle vanishes
FDz + FEz + Fg )
4 2 / / / πζ (2FEz + FDhz + FDez )+ 3 a 4 3 πa (Fp - Ff)g ) 0 (25) 3
This expression can be rewritten as
4 4 2 πζ (f U* + f2E*) + πa3(Fp - Ff)g ) 0 3 a 1 3 / 2FEz ,
/ FDhz ,
(26)
/ FDez
and in problem 1 where f1 is the sum of and f2 is that in problem 2. The sedimentation velocity U can be expressed as
a2(Fp - Ff)g δ U) f1 1 + M E η β
((
))
-1
(27)
where ME ) -f2/f1 is the mobility of the corresponding electrophoresis problem. Let v0 be the sedimentation velocity of an isolated particle in an infinite liquid. Then it can be shown that5 2 2 a (Fp - Ff)g v0 ) 9 η
The last two expressions give
(28)
Figure 4. Variation of v0/U as a function of κa: (a) H ) 1/6; (b) H ) 1/4; (c) H ) 1/2. Key: same as that in given Figure 2.
v0 2 δ ) f 1 + ME U 9 1 β
(
)
(29)
3. Results and Discussion Figure 2 shows the simulated variation of the scaled sedimentation potential per scaled sedimentation velocity, E*/U*, as a function of κa at various H ()a/b). This figure reveals that E*/U* approaches a constant value if κa is either very small or very large. The variation of E*/U* as a function of κa has a minimum at a medium value of κa. This minimum increases with the increase in H. These behaviors can be elaborated as follows. Figure 3 illustrates the variation of net current across the plane θ ) π/2 in problem 1, δ, as a function of κa. According to eq 21e, δ consists of two parts, δc and δd. The former, which is the current arises from the convective motion of ions, is related to -φr(dΨ/dr*). The latter, which is the current arises from the diffusion of ions, is related to Iθd defined in eq
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Langmuir, Vol. 16, No. 4, 2000
Figure 5. Variation of v0/U as a function of H for the case κa ) 10: solid line, result of present study; dashed line, result based on Ohshima.8 Key: same as that given in Figure 2.
21b. The variations of δc and δd as a function of κa are also presented in Figure 3. If κa , 1, the double layer is very thick, and φ1 becomes insensitive to r*, (dΨ/dr*) is a weak function of κa in problem 1, and, therefore, δc is almost constant. Also, if κa is small, the rate of diffusion of ions is low, and δd is inappreciable. Therefore, δ is almost constant if κa is very small. On the other hand, as κa is getting large, the double layer surrounding the representative particle becomes thin, the gradient of electrical potential near the particle is large, and the diffusion of ions is more important than their convection motion. That is, δd is large and δc becomes small, and the net effect is that δ has a minimum, as can be seen from Figure 3. Since E*/U* ) -δ/β, E*/U* also exhibits a minimum, as shown in Figure 2. Figure 4 shows the variations of the scaled sedimentation velocity v0/U as a function of κa at various H ()a/b). This figure reveals that v0/U approaches a constant if κa
Chih et al.
, 1. This is because if κa , 1, f1 is a constant and Me vanishes,9 and eq 29 yields v0/U ) 2f1/9. As κa increases, Me increases accordingly, but δ/β has a minimum at a medium κa. Also, f1 decreases with the increase in κa. The net effect is that v0/U has a maximum at a medium κa, as can be seen in Figure 4. This maximum decreases with the decrease in H. If κa f ∞, the double layer surrounding the representative particle is infinitely thin, and the problem under consideration reduces to that for an isolated particle.13 In this case, ME, f, and -δ/β all approach to constant and, therefore, v0/U also approaches a constant. Figure 5 shows the simulated variation of v0/U as a function of H. The corresponding result based on Ohshima’s model8 is also shown for comparison. This figure reveals that Ohshima’s model will overestimate v0/U. However, if H is small (dilute suspension), its performance becomes satisfactory. It can be shown that v0/U approaches unity as H f 0. Acknowledgment. This work is financially supported by the National Science Council of the Republic of China. Appendix The Navier-Stokes equation, eq 4, is transformed by taking curl under the constraint expressed by eq 3 to yield the governing equation for the stream function, Ψ. The v, vθ, can be r component of b v, vr, and the θ component of b evaluated by vr ) -(1/r2 sin θ)(∂Ψ/∂θ) and vθ ) (1/r sin θ)(∂Ψ/∂r). For convenience, we define the scaled stream potential, Ψ*, as Ψ* ) Ψ/a2UE. The symmetric nature of the problem under consideration suggests that φ/2, g/1, g/2, and Ψ* can be expressed respectively as φ/2 ) Φ2(r) cos θ, g/1 ) G1(r) cos θ, g/2 ) G2(r) cos θ, and Ψ* ) Ψ(r) sin2 θ. Equations 1-4 can be simplified accordingly. LA9908992 (13) Smoluchowski, M., Z. Phys. Chem. 1918, 92, 129.
