Langmuir 2008, 24, 11361-11369
11361
Articles Sedimentation Velocity and Potential in a Concentrated Suspension of Charged Liquid Drops Chia-Ping Chiang, Yan-Ying He, and Eric Lee* Department of Chemical Engineering, National Taiwan UniVersity, Taipei, Taiwan 10617 ReceiVed May 20, 2008. ReVised Manuscript ReceiVed July 13, 2008 The sedimentation behavior of a concentrated suspension of charged liquid drops is analyzed theoretically at arbitrary surface potential and arbitrary double-layer thickness; that is, the effects of double-layer polarization and double-layer overlapping are taken into account. Kuwabara’s unit cell model is employed to model the suspension system, and a pseudospectral method based on the Chebyshev polynomial is adopted to solve the governing electrokinetic equations numerically. Several interesting phenomena, which are of significant influence if the internal flow inside a liquid drop is taken into account, are observed. Key factors are examined such as the thickness of the electric double layer, the magnitude of the surface potential, the volume fraction of liquid drops, and the viscosity of the internal fluid. The results presented here add another dimension to the previous studies, which include concentrated suspensions of rigid particles and mercury drops under low ζ potential, with the consideration of the internal flow of liquid drops and double-layer polarization, characterized by its viscosity and the ζ potential respectively. It is found, among other things, that the smaller the viscosity of the internal fluid is, the higher the sedimentation velocity of liquid drops. The higher the ζ potential is, the larger the decrease in sedimentation velocity. In particular, the sedimentation velocity of an inviscid drop (gas bubble) is about three times higher than that of a rigid one. The decrease in sedimentation velocity resulting from the effect of double-layer polarization achieves about 50% if the ζ potential is sufficiently high.
Introduction When a suspension of charged spherical particles falls steadily under gravity, an induced electric field develops as a consequence of the deformation of the double layer surrounding each particle due to fluid motion; this is the so-called polarization (of the double layer) effect. This induced electric field, or the sedimentation potential as generally referred to, tends to reduce the sedimentation velocity of the charged particles in comparison with that of the uncharged ones, since the induced electric field is in the opposite direction of particle motion. This phenomenon was first discovered by Dorn in 1878 when he observed that a vertical electric field developed in a suspension of glass beads in water when they were settling,1 and is often known as the Dorn effect. The first ever theoretical investigation of this problem was given by Smoluchowski2 decades later, who considered the sedimentation potential for a dilute suspension of identical spherical rigid particles settling in an electrolyte solution. In particular, the sedimentation velocity for a single charged spherical particle settling in an infinite medium of electrolyte solution is discussed in detail.1 Under the assumption that the double layer is infinitely thin, he was able to derive analytical formulas for both quantities of interest, namely the sedimentation velocity and the potential, which are found to be influenced significantly by the ζ potential of the charged particle and the specific conductivity of the suspending fluid. However, both effects are small when the double layer is infinitely thin. Smolochowski’s theoretical predictions were found to be in agreement with * To whom correspondence should be addressed. Telephone: 886-223622530. Fax: 886-2-23622530. E-mail:
[email protected]. (1) Booth, F. J. Chem. Phys. 1954, 22, 1956. (2) Von Smoluchowski, M. Graetz Handbuch der Electrizitat und des Magnetismus; Leipzig, 1921; Vol. II, p 385.
experimental observations later as to order of magnitude, as pointed out by Booth.1 Using a regular perturbation analysis, Booth1 obtained formulas for the sedimentation velocity and potential expressed as power series of ζ potential of the charged particle for finite double-layer thickness. His results are limited to low ζ potential though. Yet over another two decades later, the constraint to low ζ potential in Booth’s study in order to treat finite double-layer thickness was relieved by Stigter,3 with the introduction of numerical computations based on Wiersema et al.’s work.4 Moreover, he noted that there is a direct relation between the sedimentation potential and the corresponding electrophoretic mobility of the same particle, the so-called Onsager reciprocal relationship.3 Saville5 contributed to the fundamental theory of sedimentation in general by providing an improved theory in terms of macroscopic field which eliminated the deficiency of Booth’s approach,1 who summed the (microscopic) dipole fields for each particle directly; hence, his macroscopic field strength depends on the shape of the averaging volume, which was obviously incorrect. Saville also noted that the sedimentation potential calculated by simply adding the dipole strength of each particle fails to meet the constraint of zero net electric current except in a special case where all the ions have identical mobilities. As a consequence, Stigter’s results are also confined to this limitation. Following Saville, Ohshima et al.6 derived general expressions for the sedimentation velocity and potential in a dilute dispersion of charged particles with arbitrary ζ potential and double-layer thickness. In addition, they obtained (3) Stigter, D. J. Phys. Chem. 1980, 84, 27580. (4) Wiersema, P. H.; Loeb, A. L.; Overbeek, J. Th. G. J. Colloid Interface Sci. 1966, 22, 78. (5) Saville, D. A. AdV. Colloid Interface Sci. 1982, 16, 267. (6) Ohshima, H.; Healy, T. W.; White, L. R.; O’Brien, R. W. J. Chem. Soc., Faraday Trans. 2 1984, 80, 1299.
10.1021/la801545j CCC: $40.75 2008 American Chemical Society Published on Web 09/05/2008
11362 Langmuir, Vol. 24, No. 20, 2008
numerical results using a computer program similar to that of O’Brien and White7 for the calculation of the electropohretic mobility of rigid spheres and found that the so-called Onsager reciprocal relation is satisfied between the sedimentation potential6 and the electrophoretic mobility.7 The studies mentioned above focused mainly on the sedimentation of a dilute suspension of charged rigid particles. In other words, the presence of other objects, such as the rigid boundary and nearby entities, is assumed to have a negligible influence on the sedimentation behavior. In practical applications of sedimentation, however, relatively concentrated suspensions of particles are usually encountered, and effects of particle interactions will be important. Using the cell model proposed by Kuwabara8 and enforcing the constraint of zero net electric current, Levine et al.9 were able to derive analytical expressions, which are confined to low surface potential and thin double layer, for sedimentation velocity and potential in concentrated suspensions of identical charged spheres. Their results of sedimentation agreed well with the few experimental data available in the literature, indicating the reliability of applying the cell model to describe the concentrated suspension. Two decades later, Kuwabara’s cell model was also adopted again by Ohshima10 to demonstrate the Onsager reciprocal relation between the sedimentation potential and the electrophoretic mobility under the same conditions of Levine et al.’s study.9 In a recent study, Carrique et al.11 extended Ohshima’s analysis further to the case of arbitrary surface potential and concluded that the Onsager relationship is also valid in concentrated suspensions as well for any surface potential as long as κa is large enough. On the other hand, they also modified Ohshima’s theory to include the presence of a dynamic Stern layer on the particle surface based on the theory developed by Mangelsdorf and White,12 allowing for the lateral motion of ions in the inner region of the double layer. Perhaps the most intriguing observation of the sedimentation problem is its relation with the electrophoresis phenomenon, the so-called Onsager relationship, which relates the sedimentation potential with the electrophoretic mobility directly. Various studies in the literature have been reported to support this relationship by direct analytical or numerical computations, such as the studies mentioned in the above paragraphs.3,6,10,11 As pointed out by Overbeek13 and Booth,1 different electrokinetic processes reflect the same intrinsic phenomena and measurements of one sort of process ought to give essentially the same basic information as any other.5 After all, exactly the same set of electrokinetic equations are solved, whether it is caused by an externally applied electric field (electrophoresis) or by gravity (sedimentation). The (induced) sedimentation potential actually sets up a local electrophoretic motion of the particle. Groot et al.14 first provided a theoretical basis of the Onsager reciprocal relation between sedimentation potential and electrophoretic mobility based on the theory of the thermodynamics of irreversible processes. The assumption that there is no double layer overlapping is essential in all of the theoretical analyses mentioned above, which is generally satisfied in very dilute suspension systems under their consideration. However, it should be noted (7) O’Brien, R. W.; White, L. R. J. Chem. Soc., Faraday Trans. 2 1978, 74, 1607. (8) Kuwabara, S. J. Phys. Soc. Jpn. 1959, 14, 527. (9) Levine, S.; Neale, G.; Epstein, N. J. Colloid Interface Sci. 1976, 57, 424. (10) Ohshima, H. J. Colloid Interface Sci. 1998, 208, 295. (11) Carrique, F.; Arroyo, F. J.; Delgado, A. V. Colloid Surf., A 2001, 195, 157. (12) Mangelsdorf, C. S.; White, L. R. J. Chem. Soc., Faraday Trans. 1990, 86, 2859. (13) Overbeek, J. Th. G. J. Colloid Interface Sci. 1953, 8, 420. (14) DeGroot, S. R.; Mazur, P.; Overbeek, J. Th. G. J. Chem. Phys. 1952, 20, 1825.
Chiang et al.
that the assumption of thin double layer alone in a concentrated suspension can be insufficient to justify it, as pointed out by Keh and Ding,15 who considered the sedimentation problem in a concentrated suspension, taking into account double-layer overlapping. Both the Happel model16 and a modified Kuwabara cell model are adopted to describe the concentrated suspension there. Later, Lee17 extended the analyses of Levine10 and Ohshima11 to the case that the surface potential is not necessarily low and the effect of double-layer polarization may be significant. Several interesting phenomena, which are absent if double-layer polarization is neglected, are observed. All of the studies mentioned above are focused on the suspensions of rigid particles. Corresponding studies on the sedimentation of nonrigid particles have received relatively less research attention. In addition to the obvious academic interest in better understanding the sedimentation phenomenon in general, suspensions of nonrigid charged colloids in fact have a wide variety of potential applications in practice, such as in the electrospraying, biochip apparatus, etc.18-21 Hence, a thorough understanding of their electrokinetic behavior is valuable for various possible future novel applications. Sedimentation is one of the key areas of interest, as the gravity effect is present everywhere on earth to be sure. As for the nonrigid cases, there have been a series of studies on sedimentation of mercury drops in particular, either dilute22,23 or concentrated suspensions.24 This is probably because of the special merit of the mercury drops in classic experimental setups for electrokinetics.22,25 Levich and Frumkin22 first studied the fall of mercury drops in a dilute suspension in a gravitational field. They derived both the sedimentation velocity and potential and verified them by experiments made by Bagotskaya et al.22 It was noted there that, in the limiting case of low electrical conductivity and high surface charge, the falling velocity of such a drop could reduce significantly as to resemble that of a rigid one, the so-called “solidification phenomenon” in their reports. Extensions were carried out by various researches to cover the case of arbitrary ζ potential and arbitrary double-layer thickness in a dilute suspension of mercury drops,23 as well as low ζ potential and thin double layer in a concentrated system.24 General expressions for the sedimentation velocity and potential in a dilute suspension of mercury drops, among other electrokinetic properties of interest, are derived by Ohshima et al.23 Numerical results are displayed as well as some approximate analytic formulas. They concluded that the results of Levich and Frumkin22 are valid only for very small ζ potential and very thin double layer and only when all the ionic species have the same drag coefficient. As for the case of a concentrated suspension of charged mercury drops, Ohshima24 was able to obtain expressions for the sedimentation velocity based on Kuwabara’s cell model. Expression for the sedimentation potential is also derived for the case in which the overlapping of the electrical double layers of adjacent drops is negligible. The Onsager reciprocal relationship was claimed to hold for the above two studies assuming that the double-layer (15) Keh, H. J.; Ding, J. M. J. Colloid Interface Sci. 2000, 227, 540. (16) Happel, J. AIChE J. 1958, 4, 197. (17) Lee, E.; Chu, J. W.; Hsu, J. P. J. Chem. Phys. 1999, 110, 11643. (18) Chen, D. R.; Pui, D. Y. H.; Kaufman, S. L. J. Aerosol Sci. 1995, 26, 963. (19) Liu, H.; Dasgupta, P. K. Microchem. J. 1997, 57, 127. (20) Jayasinghe, S. N.; Townsend-Nicholson, A. Lab Chip 2006, 6, 1086. (21) Shui, L.; Eijkel, J. C. T.; Van der Berg, A. AdV. Colloid Interface Sci. 2007, 133, 35. (22) Levich, V. G. Physicochemical Hydrodynamics; Prentice Hall: New York, 1962. (23) Ohshima, H.; Healy, T. W.; White, L. R. J. Chem. Soc., Faraday Trans. 2 1984, 280, 1643. (24) Ohshima, H. J. Colloid Interface Sci. 1999, 218, 535. (25) Hunter, R. J. Foundations of Colloid Science; Oxford University Press: New York, 1986; Vol. I, p 316.
Sedimentation Velocity and Potential
overlapping is absent. However, no consideration has been given yet so far to the sedimentation of general nonrigid charged colloids in a concentrated suspension, taking into account both the arbitrary ζ potential and the arbitrary double-layer thickness. As a limiting case, under the condition of low ζ potential and thin doublelayer thickness, Ohshima24 was able to derive an analytical expression of the sedimentation velocity of a mercury drop based on Kuwabara’s unit cell model. The velocity is expressed in terms of volume fraction and viscosity of the nonrigid colloids, serving as the limiting case when the nonrigid colloid is either uncharged24 or in the absence of an electrolyte.22 But the viscosity of the mercury drop is a fixed value; hence, the results are not directly applicable to other nonrigid colloidal systems of interest. For the general nonrigid colloid, including the mercury drop, an internal flow will be generated during sedimentation, which has a tendency to increase the sedimentation velocity in general because the shear at the interface is smaller than those of the corresponding rigid colloidal systems, resulting in a smaller hydrodynamic drag force. The key factor determining the strength of the internal flow is its viscosity. An approach taking this fact into consideration is essential in accurately predicting the sedimentation behavior of nonrigid charged colloidal suspensions. In the present study, the sedimentation in a concentrated suspension of charged liquid drops (nonrigid entities) in an electrolyte solution is investigated for the case where both the ζ potential and the double-layer thickness can be arbitrary. The effects of double-layer polarization as well as double-layer overlapping of adjacent entities are taken into account. A pseudospectral method based on Chebyshev polynomials26 is used to solve the resulting general electrokinetic equations. Key parameters such as the ζ potential, the double-layer thickness, the volume fraction of the nonrigid entities, and the viscosity of the inside fluid are examined, and their respective effects on the overall sedimentation behavior of the concentrated suspensions of liquid droplets are analyzed and discussed. An effort to compare the predictions of our theory with the scarce experimental data available in the literature is made with an encouraging outcome, which will be discussed in detail later in the section Results and Discussion. Briefly speaking, our results presented here in the limiting case of solid colloids can be regarded as a generalization of Levine theory extended to arbitrary ζ potential, which has been found to deviate only 5% in experimental trend at a low ζ potential situation. For the general situation of nonrigid liquid drops, our predictions serve as a primary estimation of the sedimentation parameters of interest. To fully determine the deviation range, pertinent experimental data are needed from the specialist colloid chemist in the future. After all, Levine theory9 waited nine years for Marlow et al.27 to support it with solid experimental data. Theory. We consider the sedimentation of a concentrated suspension of charged liquid drops of radius a in a solution that contains z1/z2 electrolytes, with z1 and z2 being respectively the valences of cations and anions. The electroneutrality constraint in the bulk liquid phase requires that n20 ) (n10/R), where n10 and n20 are the bulk concentrations of cations and anions, respectively, and R ) -z2/z1. The cell model proposed by Kuwabara8 is adopted as the basis to describe the concentrated suspension in subsequent analyses. Referring to Figure 1, each liquid drop is surrounded by a concentric spherical shell of electrolyte solution of radius b. The spherical coordinates (r, θ, φ) are used with the origin located at the center of the liquid drop. (26) Canuto, C.; Hussaini, M. Y.; Quarteroni, A.; Zang, T. A. Spectral Methods in Fluid Dynamics; Springer-Verlag: New York, 1988; Chapter 2. (27) Marlow, B. J.; Rowell, R. L. Langmuir 1985, 1, 83.
Langmuir, Vol. 24, No. 20, 2008 11363
Figure 1. Schematic representation of the system under consideration, where a liquid drop (mass density Fd and viscosity ηd) of radius a is surrounded by a spherical shell of an electrolyte solution (mass density Ff and viscosity η) of radius b. U is the sedimentation velocity, g the gravitational acceleration, and E the induced electric field. The spherical coordinates (r, θ, φ) are adopted with the origin located at the center of the liquid drop.
The liquid drop is nonconducting, and its interior is electrolyte free. Due to the gravity-induced sedimentation, each liquid drop moves with a sedimentation velocity U in the -z direction (θ ) π), and an electric field E is induced in the z direction (θ ) 0). It is assumed that both the liquid phases inside and outside the liquid drop are incompressible Newtonian fluid with constant physical properties. Also, the shape of the liquid drop remains spherical during sedimentation, which is generally true under low-Reynolds-number situations. According to Taylor and Acrivos,28 the deformation of a liquid drop is proportional to Re2, or to the Weber number We ) FaU2/σ1, with Re and σ1 being respectively the Reynolds number and the surface tension of the liquid drop. In our case, because the typical value of Re is on the order of 10-6 and that of We is on the order of 10-13, the deformation of a drop can be neglected in the present analysis. The system is assumed to be at a quasi-steady-state as well due to the low-Reynolds-number situation. Under these conditions, the sedimentation is governed by the well-known set of electrokinetic equations, composed of the Poisson equation, ionflux equation, and Navier-Stokes equation for both the external and the internal region of the droplet. It should be noted that since the liquid drop is electrolyte free, i.e. F ) 0 inside the drop, there is no electric body force in the Navier-Stokes equation governing the fluid flow there. Therefore, there is no need for the other two equations inside the drop. The electric potential, φ, is assumed to be governed by the Poisson equation,
∇2φ ) -
F ε
(1)
where F ()∑2j)1 zjenj) is the space charge density, the permittivity of the electrolyte solution, e the elementary charge, nj the concentration of ionic species j with j ) 1 and 2 indicating respectively the cations and anions, and ∇2 the Laplace operator. At quasi-steady-state, the conservation of ionic species j implies that
∇ · fj ) 0 The flux of ionic species j, fj, can be expressed by (28) Taylor, T. D.; Acrivos, A. J. Fluid Mech. 1964, 18, 466.
(2)
11364 Langmuir, Vol. 24, No. 20, 2008
(
fj ) -Dj ∇nj +
Chiang et al.
)
zje ∇ φ + njv kT
(3)
where k is the Boltzmann constant, T the absolute temperature of the system, v the fluid velocity, Dj the diffusion coefficient of ionic species j, and ∇ the gradient operator. Substituting eq 2 into eq 3, we end up with
∇2nj +
zje 1 (n ∇2φ + ∇ nj · ∇ φ) - ∇ nj · v ) 0, kT j Dj j ) 1, 2
(4)
The flow field is described by the Navier-Stokes equation in the creeping flow region, together with an incompressibility constraint. For the flow field outside the liquid drops, we have
∇·v)0
(5)
η∇ v - ∇ p - F ∇ φ + Ffg ) 0
(6)
2
where p is the pressure, g is the acceleration due to gravity, η and Ff are respectively the viscosity and the mass density of the electrolyte solution, and ∇ · is the divergence operator. For the flow field inside the liquid drops, we have
∇ · vd ) 0
(7)
ηd∇ vd - ∇ pd + Fdg ) 0
(8)
2
where the physical quantities with the subscript d denote the corresponding ones inside the liquid drops. Following the previous treatment by O’Brien and White,7 the electric potential, φ, is decomposed into the equilibrium potential that would exist in the absence of the induced electric field, φe, and the perturbed potential arising from the induced electric field, δφ. Moreover, in virtue of the convection term in eq 2, a function gj is introduced to take the effect of the electric double-layer polarization into account. Then, the distribution of the ionic concentrations in the electrolyte solution can be described in a form similar to the Boltzmann distribution,
[
]
zje nj ) nj0 exp - (φe + δφ + gj) , kT
j ) 1, 2
(9)
where gj can be regarded as a reshaping function of the Boltzmann distribution. Note that eq 9 reduces to the standard Boltzmann distribution as gj ) 0, which implies no account of the polarization effect. If δφ and gj are small compared to φe, such as 1% numerically speaking in practice, then eq 7 can be linearized as
(
)[
]
zje zje nj ) nj0 exp - φe 1 (δφ + gj) , kT kT
j ) 1, 2
For convenience, scaled properties are used in the following analyses. Here, the radius of a liquid drop, a, the surface potential of the liquid drop, ζa, and the electrophoretic velocity of an isolated charged rigid particle under an electric field of strength (ζa/a) based on Smoluchowski’s theory, UE ) (ζa2/ηa), are used as the characteristic physical quantities for the length, the electric potential, and the velocity, respectively. In addition, instead of solving eqs 5-8 directly for velocities and pressures outside and inside the liquid drops, the governing equations for the flow fields can be transformed by taking curl on both eqs 4 and 6 along with the introduction of the stream function, ψ, in the spherical coordinates, which satisfies eqs 5 and 7 automatically. Henceforward, we can define the following dimensionless physical quantities: r* ) r/a, φe* ) φe/ζa, δφ* ) δφ/ζa, gj* ) gj/ζa, j ) 1, 2, and ψ;* ) ψ;/UEa2. The symmetric nature of the spherical coordinates adopted suggests that the present problem under consideration can be reduced to a one-dimensional one,17 i.e., δφ*, gj*, and j ) 1, 2, and ψ* can be expressed respectively as δφ* ) δΦ cos θ, gj* ) Gj cos θ, j ) 1, 2, and ψ* ) ψ sin2 θ. The governing equations, eqs 1-8, can then be simplified accordingly as
L2r φ∗e ) L2δΦ -
L2Gj -
(11)
(κa)2 ∗ (κa)2 ∗ (n1e + Rn∗2e)δΦ ) (n G + Rn∗2eG2) (1 + R) (1 + R) 1e 1 for 1 < r* < b ⁄ a (12)
(
)
zj dφ/e dGj dφ/e 2 φr / / ) Pecjφ2r / - /2 Ψ z1 dr dr dr r for 1, < r* < b ⁄ a,
D4Ψ ) -
dφ∗e ∗ (n G + Rn∗2eG2) ∗ 1e 1
(κa)2 (1 + R) dr
j ) 1, 2 (13)
for 1 < r* < b ⁄ a (14)
and
D4Ψd ) 0 for 0 < r* < 1
(15)
In these expressions, Pecj ) j ) 1, 2, is the electric Peclet number of ionic species j, n1e* ) n1e/n10 ) exp(-φrφe*) and n2e* ) n2e/n20 ) exp(Rφrφe*) are the dimensionless equilibrium concentrations of cations and anions, respectively, φr ) ζa/(kT/z1e) is the dimensionless surface potential, and the reciprocal of the electric double-layer thickness κ (or the so-called Debye-Huckel parameter) and the operators Lr2, L2, and D4 are defined as ε(kT/z1e)2/ηDj,
κ)
(10)
This is done by taking the Taylor series expansion of the exponential term in eq 9 and neglecting all the higher order terms involving the product of δφ and/or gj, as normally treated in the regular perturbation analysis. Note that the exponential term in eq 19, exp(-zjeφe/kT), is the entry point of the arbitrary ζ potential, φe, assumed in this analysis. As pointed out by O’Brien and White,7 small values of δφ and gj do not imply that their effects are insignificant. The ionic number density nj is perturbed by a small quantity δnj as well according to eq 8, and the entire set of governing equations is assumed to be satisfied both before and after the onset of sedimentation, in the sense of regular linear perturbation analysis.
1 (κa)2 ∗ (n - n∗2e) for 1 < r* < b ⁄ a φr (1 + R) 1e
[
2
j)1
1⁄2
(16)
2 d d2 + dr/2 r/ dr/
(17)
d2 2 d 2 + dr/2 r/ dr/ r/2
(18)
L2r ) L2 )
]
∑ (zje)2nj0/εkT
and
D2 ) D2D2 )
d4 4 d2 8 d 8 - /2 /2 + /3 / - /4 /4 dr r dr r dr r
(19)
respectively. Note that the Poisson equation, eq 1, is replaced by the Laplace equation (∇2φ ) 0) in the conventional infinitely thin double-
Sedimentation Velocity and Potential
Langmuir, Vol. 24, No. 20, 2008 11365
layer approximation in the literature under the assumption that all counterions are confined to the extremely thin double layer; hence, there is no net space charge outside it. Here, however, κa is treated as a finite value via the substitution of eq 14 into relevant equations involving the number density concentration nj and its corresponding perturbation term δnj. In the infinitely thin double-layer approximation, all the terms involving κa in eqs 11, 12, and 14 will vanish. The boundary conditions associated with eqs 11-15 are
φ∗e ) 1 at r* ) 1
(20)
dφ/e (r) ) 0 at r* ) H -1⁄3 dr dδΦ(r) ) 0 at r* ) 1 dr ∗
-1⁄3
(22)
-1⁄3
(
σr
d Ψd(r) dr2
dΨd(r) - 2r dr /
(
)|
In these expressions,
(
(25)
∗ FEz ) 2r/
[
(26)
∗ FDHz ) r/4
(27)
|
dΨd(r) dΨ(r) ) dr r /)1dr r /)1+ /2
4 FG ) πa3(Fd - Ff)g 3
(24)
dΨd(r) Ψd ) ) 0 at r* ) 0 dr Ψ ) Ψd ) 0 at r* ) 1
(28)
d2Ψ(r) dΨ(r) r/2 - 2r/ 2 dr dr
∗ FDEz )
)|
r /)1+
(33)
(34)
)|
dφ/e (r) δΦ(r/) dr
(35)
r /)1
d(r/-2D2Ψ(r)) dr
]|
r /)1
(36)
and
) r /)1-
(32)
and
(23)
Gj ) -δΦ at r* ) H -1⁄3, j ) 1, 2
2
4 ∗ FEz ) πεζ2aFEz 3 4 4 ∗ ∗ ∗ ) πεζ2a(FDHz + FDEz ) FDz ) πεζ2aFDz 3 3
(21)
at r* ) H δΦ ) -E H dGj(r) ) 0 at r* ) 1, j ) 1, 2 dr
|
liquid-liquid interface respectively. Equations 30 and 31 indicate respectively that the fluid is flowing toward the stationary liquid drop with a relative velocity U and satisfies the Kuwabara’s cell model of zero vorticity.8 The sedimentation velocity of a charged liquid drop can be determined from the fact that the sum of the external forces acting on it in the z-direction vanishes at steady state. These external forces include the electric force, FEz, the hydrodynamic forces, FDz, and the gravitational force, FG. These three forces can be expressed respectively as29
|
(κa)2 (r/2F/e (r/) δΦ(r/)) φr(1 + R) r /)1
(37)
The vanishing of the net force acquires FEz + FDz + FG ) 0 or
(29)
1 Ψ ) - U/r/2 at r/ ) H-1⁄3 2
(30)
d2Ψ(r) 2 - /2 Ψ(r) ) 0 at r/ ) H-1⁄3 2 dr r
(31)
In these expressions, E* ) E/(ζa/a), U* ) U/UE, and σ ) ηd/η denoting the ratio of the viscosity of the drop fluid to that of the dispersion medium. Equation 20 means that the surface potential, characterized by ζa, of the liquid drop remains constant. Equation 21 implies that the cell as a whole is electrically neutral. Equation 22 states that the liquid drop is nonconductive, and eq 23 defines the relation between the local perturbed potential and the induced electric field resulting from the sedimentation of the charged liquid drop. Equation 24 suggests that the liquid drop is ionimpenetrable. Equation 25 ensures that the perturbation of the ionic concentration vanishes at the virtual surface. It should be noted that if the overlap between adjacent electric double-layers is negligible, then eqs 23 and 25 shall become respectively δΦ ) 0 and Gj ) 0, j ) 1, 2, and these are the boundary conditions adopted in the analyses of Ohshima10,24 and Carrique et al.11 On the other hand, based on the fact that there are no net flows of electric current and ionic species between adjacent cells, Keh and Ding15 adopted dδΦ/dr ) 0 and dGj/dr ) 0, j ) 1, 2, at the virtual surface. However, these boundaries are only valid when the suspension of the particles is bounded by impermeable, inert, and nonconducting walls. Equation 27 implies that there is no mass change between the liquid drop and the dispersion medium. Equations 28 and 29 represent that the tangential velocity Vθ and the shear stress τrθ of the fluid are continuous across the
4 4 ∗ ∗ ∗ + FDHz + FDEz ) + πa3(Fd - Ff)g ) 0 (38) πεζ2a(FEz 3 3 The problem under consideration can be further decomposed into two subproblems:17 (1) a droplet moves with a velocity, U*, in the absence of an induced electric field, and (2) the droplet is held fixed in an induced electric field, E*. These two problems are denoted as problems 1 and 2 hereafter. Equation 38 can be rewritten as
4 4 πεζ2a(c1U∗ + c2E∗) + πa3(Fd - Ff)g ) 0 3 3
(39)
where c1 is the sum of FEz*, FDHz*, and FDEz*, which are calculated in problem 1, and c2 is that obtained in problem 2. The dimensionless sedimentation velocity, U/U0, can be expressed as17
(
c2 E∗ U 9 -1 ) c1 1 + U0 2 c1 U∗
)
-1
(40)
where -c2/c1 is the electrophoretic mobility of the corresponding electrophoresis problem,17 U0 [)2/9(a2(Fd - Ff)g/η] is the sedimentation velocity of an isolated uncharged rigid particle, and (E*/U*) is a ratio, which is the sedimentation potential gradient per unit sedimentation velocity, evaluated based on the fact that the sedimentation of the charged liquid drops produces zero net current, or the net flow of current across any horizontal plane vanishes. We apply this condition at the horizontal plane θ ) π/2.9 Then, the net current, 〈i〉, can be defined as17 (29) Lee, E.; Chu, J. W.; Hsu, J. P. J. Colloid Interface Sci. 1999, 209, 240.
11366 Langmuir, Vol. 24, No. 20, 2008
〈i〉 ) 2π
Chiang et al.
∫ab riθ dr|θ)π⁄2 ) 2πa2Ia∫1b⁄a Iθ(r/) dr/
(41)
where
( )
ε2φ2r kT 3 (κa)2 ηa3 z1e (1 + R) dΨ 1 R n G n G Iθ(r/) ) Fe/ / φrPec1 1e/ 1 φrPec2 2e/ 2 dr Ia )
with Fe* ) n1e* - n2e*, and iθ is the θ-component of current i defined by
i)
2
2
j)1
j)1
[
(
)]
ze
∑ zjenjvj ) ∑ zje njv - Dj ∇nj + kTj nj∇φ
(42)
The net current across the horizontal plane θ ) π/2 in problems 1 and 2 can be respectively expressed as 〈i〉1 ) c3U* and 〈i〉2 ) c4E*, where c3 and c4 are proportional constants. The constraint of zero net current yields the ratio (E*/U*) ) -c3/c4.17 It should be noted that a similar methodology was proposed by O’Brien and White7 to solve the electrophoretic mobility of a spherical particle where the constraint of zero net force is employed. Equations 11-31 are also applicable to the case of the electrophoresis of a concentrated suspension of colloidal particles29 and those of nonrigid particles and mercury drops.30,31 The methodology proposed by O’Brien and White7 is used to determine the dimensionless electrophoretic mobility on the basis of a force balance in the previous research,29-31 while it is adopted to obtain the ratio E*/U* for the sedimentation of a concentrated suspension of charged liquid drops based on the constraint of zero net current in the present study. Then, the dimensionless sedimentation potential, ESED*, which is a measure for the induced electrical field per unit gravitational force can be defined as
Figure 2. Variations of the dimensionless sedimentation velocity, (U/ U0), for a concentrated suspension of charged rigid particles (or charged liquid drops with infinite viscosity) as a function of κa at two values of φr. The solid curves express the numerical results based on the boundary conditions adopted in the present study, while the dashed curves are the results given by Keh and Ding.15 Key: H ) 0.125.
K∞ESED 1 η 2 1 1 (κa)2 ESED* ) ) × (Fd - Ff)g H ε(kT/z1e) 9 H φr (1 + R)
(
)
1 R E∗ U + (43) Pec1 Pec2 U ∗ U0
2 where K∞ (≡∑j)1 z2j e2nj0Dj/kT) is the specific conductivity of an infinitely dilute dispersion.
Results and Discussion The governing equations, eqs 11-15, and the associated boundary conditions, eqs 20-31, are solved numerically by a pseudospectral method based on Chebyshev polynomials, which is found to be both accurate and efficient for solving electrokinetic problems.17,29 The dimensionless sedimentation velocity, (U/U0), and the dimensionless sedimentation potential
(
ESED* )
η K ∞ESED 1 (Fd - Ff) H ε(kT/z1e)
)
for a concentrated suspension of charged liquid drops are calculated. A typical aqueous solution of KCl at 25 °C is used here, which yields Pec1 ) Pec2 ) 0.26. A. Effect of Double-Layer Thickness. We first compare our results with those of Keh and Ding,15 who considered sedimentation velocity and potential based on a cell model which is similar to the traditional Kuwabara’s cell model but with different boundary conditions for perturbed electric potential and ionic concentration at the virtual surface adopted in previous (30) Lee, E.; Fu, C. H.; Hsu, J. P. Langmuir 2003, 19, 3035. (31) Lee, E.; Hu, J. K.; Hsu, J. P. J. Colloid Interface Sci. 2003, 257, 250.
Figure 3. The corresponding dimensionless sedimentation potential, ESED*, for the same conditions as those in Figure 2.
analyses of electrophoresis of concentrated suspensions of particles. Adopting their boundary conditions, we can reproduce exactly their predictions for sedimentation results, as shown in Figure 2, indicating the correction and reliability of our calculations. The derivation at intermediate range of κa in Figure 2 shows the error bound of their approach, which is limited to very thick and very thin double layers; that is, κa is very small and very large. Figure 2 also provides the numerical comparison needs to determine the range of applicability of Keh and Ding’s predictions, as mentioned in their concluding remarks. Corresponding profiles of sedimentation potential as a function of κa under the same conditions are presented in Figure 3. Again the agreement is excellent when either κa is very large or very small, under the same conditions as explained in Figure 2. B. Effect of Viscosity. We now turn to the cases of charged liquid drops. In Figure 4, the sedimentation velocity for various viscosity ratios σ ()ηd/η) is depicted as a function of κa. In general, the smaller the viscosity ratio is, the higher the sedimentation velocity, regardless of the value of κa. This is because when the viscosity of the internal fluid is smaller, the shear stress and hence the viscous drag at the interface between the liquid drop and the suspending medium gets smaller accordingly, as can be seen clearly for the corresponding case of an uncharged liquid drop during sedimentation, where an analytical formula for the terminal velocity is available.24 Indeed,
Sedimentation Velocity and Potential
Figure 4. Variations of (U/U0) as a function of κa at various viscosity ratios, σ, for the case where (a) φr ) 1 or (b) φr ) 3: dashed curves, rigid particle case; dotted curves, gas bubble case. Key: Same as Figure 2.
the charged drops only sediment slower due to the induced electric field, but the key hydrodynamic behavior of the drops remains essentially the same. It is found that when σ ) 100, the sedimentation behavior of a liquid drop reduces to that of a rigid particle, and when σ ) 0.01, behavior corresponds to that of a gas bubble. In particular, the sedimentation velocity for a bubble is about three-times larger than that of a rigid particle. For typical emulsions, on the other hand, σ would be around unity, and the sedimentation velocity is about two times larger than that of the corresponding rigid particles. Both are very useful rule-of-thumb values in practical applications. Despite the quantitative results, the profiles are qualitatively similar for various viscosity ratios. The decrease around κa ) 1.0 is a result of the polarization effect, the deformation of an ionic cloud around the liquid drop in motion, which tends to retard the motion significantly as the surface potential gets high, as φr ) 1 here (Figure 4a). Once the double layer gets very thin, indicated by a large κa, around, say, 10, the double-layer deformation is no longer that significant, and the polarization effect weakens accordingly, resulting in a general recovery of sedimentation velocity. Similar behavior can be found in Figure 4b except for the sharp drop around κa ) 3 due to the polarization effect explained earlier. Corresponding dimensionless sedimentation potentials are shown in Figure 5 for various values of viscosity ratio, with rigid particle and gas bubble serving as the limiting case for σ ) ∞ and σ ) 0, respectively, where the suspension under consideration consists of liquid drops with a volume fraction of H ) 0.125. Monotonic behaviors with increasing κa are observed. However,
Langmuir, Vol. 24, No. 20, 2008 11367
Figure 5. The corresponding ESED* for the same condition as those in Figure 4.
our result is consistent with the findings of Ohshima25 in his study of sedimentation of concentrated mercury drops. He noticed that, under the constraint of low surface potential and neglecting a double-layer overlapping effect, the sedimentation potential of a mercury drop is roughly κa-times that of a rigid particle. In other words, the deviation gets more and more severe as κa gets small, as depicted in this figure. However, no such simple rule can be derived in general, as shown in our results, where both the surface potential is higher and the double-layer overlapping, as well as the polarization effect, is considered. Our results add another dimension to Ohshima’s by considering various σ as well. C. Effect of ζ Potential. The effect of surface potential on the sedimentation velocity as a function of κa is presented in Figure 6 for a representative liquid drop suspension with σ ) 1 and H ) 0.125. The decrease of velocity as κa increases to around unity is owing to the deformation of the double layer: the so-called double-layer polarization effect. The higher the surface potential φr is, the more significant this drop in velocity, as has been observed in numerous related electrokinetic phenomena.22,23,29-31 Local minima are observed around unity for various φr. The deviation of sedimentation velocity between charged and uncharged liquid drops is shown in Figure 7 as a function of κa at various surface potentials φr. The deviation is defined as
D% ) |(U - Ud0) ⁄ Ud0| × 100%
(44)
where Ud0 represents the sedimentation velocity of an uncharged liquid drop in the cell and it can be expressed as24
11368 Langmuir, Vol. 24, No. 20, 2008
Chiang et al.
Figure 7. Variations of the deviation of sedimentation velocity between a concentrated suspension of charged and uncharged liquid drops, D%, as a function of κa at various φr for the case where σ ) 1. Key: Same as Figure 2.
[ 3(σ3σ ++ 21) - 59 H
Ud0 ) U0
1⁄3
+
Figure 8. Variations of (U/U0) as a function of σ at two values of φr for the case where κa ) 1. Key: Same as Figure 2.
3σ 3(σ - 1) 2 HH 3σ + 2 5(3σ + 2) (45)
]
The deviation reflects the extent of the effect of double-layer polarization which decreases the sedimentation velocity of the charged liquid drops. It should be noted that the local maximum in each curve is close to κa ) 1, which corresponds to the occurrence of electric double-layer overlapping. Note that the corresponding deviation in a concentrated suspension of charged rigid particles (or charged liquid drops with infinite viscosity) is roughly half of the magnitude observed here. The deviation can go as high as 50% for φr ) 5 (about 128 mV on the droplet surface at room temperature) at κa around unity. As can be seen in Figure 8, we now show the detailed dependence of sedimentation velocity on the viscosity ratio at a fixed value of κa ) 1 for a liquid drop suspension of volume fraction H ) 0.125. Two different surface potentials (φr ) 1 and 3) are shown simultaneously, representing typical situations of low and high surface potential, respectively. It is observed that the dependence on viscosity ratio is most significant around σ ) 1, indicated by the steep profile there. As a result, the consideration of internal flow for liquid drops is important, since in practice σ is expected to be around unity in typical electrolyte suspension. D. Effect of Volume Fraction. In Figure 9, the sedimentation velocity as a function of κa at various volume fractions, H, is
Figure 6. Variations of (U/U0) as a function of κa at various φr for the case where σ ) 1. Key: Same as Figure 2.
Figure 9. Variations of (U/U0) as a function of κa at various volume fractions H for the case where φr ) 3 and σ ) 1.
depicted for a liquid drop suspension with φr ) 3 and σ ) 1. In general, the higher the volume fraction is, the lower the sedimentation velocity. This is mainly due to the hydrodynamic hindrance effect with the presence of crowded neighboring drops. A corresponding dependence of dimensionless sedimentation potential on the volume fraction is shown in Figure 10 for both φr ) 3 and φr ) 1, as typical cases for high and low surface potential situations. The effect of volume fraction is essentially the same for both cases. E. Comparison with Experimental Data in the Literature. After an extensive literature survey on the possible sources of experimental data available for direct comparison purposes, it was found that no such data were reported for the sedimentation of concentrated liquid drops of colloid size. However, we did find a valuable paper by Marlow et al.27 which considered the sedimentation of a concentrated suspension of solid particles in aqueous electrolytes by taking experimental data of 97 µm glass microspheres with a volume fraction up to nearly 4% and compared it with the theoretical predictions by Levine et al.9 He found, among other things, that the Levine theory deviates only 5% from the experimental trend, and he noted the following: “Thus, good agreement is obtained using the cell-model theory as with electrophoresis.” Levine theory is actually our special case valid for rigid particles at low ζ potential (φr e 1) only. Direct calculation from our general theory gives essentially the
Sedimentation Velocity and Potential
Langmuir, Vol. 24, No. 20, 2008 11369
To fully determine the deviation range, pertinent experimental data are needed from the specialist colloid chemist in the future. After all, Levine theory9 waited nine years for Marlow et al.27 to support it with solid experimental data.
Conclusion
Figure 10. Variations of ESED* as a function of H at two values of φr for the case where κa ) 1 and σ ) 1. The solid curves express the numerical results based on the boundary conditions adopted in the present study, while the dashed curve is the result given by Ohshima.24 Note that the dashed curve coincides with the abscissa for H g 0.31.
same prediction as Levine’s, indicating the 5% deviation in reliability is also the situation in our results. Levich22 indeed referred to some experimental sedimentation data of an extremely dilute dispersion (volume fraction H ) 1.4 × 10-6) of mercury drops of diameter 0.442 mm in glycerin. As the viscosities are so different (0.01 poise for mercury vs 7.2 poise for glycerin), the system actually resembles a bubble suspension as far as the viscosity ratio is concerned. The reciprocal of the double-layer thickness, κa, involved in the experiment is so high that it indicates an extremely thin double layer, which is beyond the scope of our theoretical approach. No meaningful comparison can be carried out. In summary, our results presented here in the limiting case of solid colloids can be regarded as a generalization of Levine theory extended to arbitrary ζ potential, which has been to deviate only 5% in experimental trend at a low ζ potential situation. For the general situations of liquid drops, our predictions serve as a primary estimation of the sedimentation parameters of interest.
The sedimentation behavior of a concentrated suspension of charged liquid drops with arbitrary surface potential and arbitrary double-layer thickness is investigated theoretically here. The effect of the volume fraction of liquid drops, the ζ potential, and the viscosity ratio is examined in particular. In summary, we conclude the following: (i) The smaller the viscosity of the droplet fluid is, the larger the magnitude of the sedimentation velocity. An inviscid liquid drop (gas bubble) has a magnitude about three times higher than the corresponding rigid ones. (ii) If the surface potential of charged liquid drops is high and the double-layer thickness is finite, the effect of double-layer polarization is significant. The velocity of the liquid drops exhibits a local minimum as a result. (iii) The higher the ζ potential is, the more significant the decrease of the sedimentation velocity. The decrease in sedimentation velocity resulting from the effect of double-layer polarization achieves 50% if the ζ potential is sufficiently high. (iv) The magnitude of the sedimentation velocity decreases with increasing volume fraction of liquid drops in general due to the hydrodynamic hindrance effect from neighboring drops. (v) The effect of double-layer overlapping gets even more significant in a very concentrated suspension as the double-layer polarization is taken into account as well. (vi) Comparison with the scarce experimental data available in the literature shows a 5% deviation in experimental trend for the limiting case of solid colloid at low ζ potential with volume fraction less than 4%, as Levine theory did as well, which is a special case valid for rigid particles at low ζ potential only, of our general theory presented here. Acknowledgment. This work is partially financially supported by the National Science Council of the Republic of China. LA801545J
