Sedimentation of a Composite Particle in a Spherical Cavity

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Research Article

Sedimentation of a Composite Particle in a Spherical Cavity

Eric Lee, Tzu-Hao Huang, and Jyh-Ping Hsu

Langmuir, 2005, 21 (5), 1729-1737 • DOI: 10.1021/la0476229

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Sedimentation of a Composite Particle in a Spherical

Cavity

Eric Lee, Tzu-Hao Huang, and Jyh-Ping Hsu*

Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan 10617 Received September 24, 2004. In Final Form: November 16, 2004

The boundary effect on the sedimentation of a colloidal particle is investigated theoretically by considering a composite sphere, which comprises a rigid core and an ion-penetrable membrane layer, in a spherical cavity. A pseudo-spectral method is adopted to solve the governing electrokinetic equations, and the influences of the key parameters on the sedimentation behavior of a particle are discussed. We show that both the qualitative and quantitative behaviors of a particle are influenced significantly by the presence of the membrane layer. For example, if the membrane layer is either free of fixed charge or positively charged and the surface potential of the rigid core is sufficiently high, the sedimentation velocity has a local minimum and the sedimentation potential has a local maximum as the thickness of the double layer varies. These local extrema are not observed when the membrane layer is negatively charged. If the double layer is thin, the influence of the fixed charge in the membrane layer on the sedimentation potential is inappreciable.

Introduction

Sedimentation is one of the important phenomena of a colloidal dispersion in a gravitational field. Although it has been studied extensively and relevant results are ample in the literature, the result under general conditions has not been reported. In a study of the sedimentation of an isolated, rigid sphere under the conditions of an infinitely thin double layer, Smoluchowski1_{showed that}

the sedimentation velocity of a charged particle is much smaller than that of the corresponding uncharged particle. Booth2 _{estimated the sedimentation velocity and the}

sedimentation potential of a sphere with an arbitrary double-layer thickness. His analysis was extended by Saville3_{to a more general case. Sedimentation is similar}

to electrophoresis, where an applied electric field provides the driving force for the movement of a charged particle. de Groot et al.4 _{found that the scaled sedimentation}

potential in sedimentation correlates with the electro-phoresis mobility in electroelectro-phoresis, the so-called Onsager relation. Ohshima et al.5_{derived analytical expressions}

for the sedimentation potential and sedimentation velocity for the case of low surface potential and thin double layer. A thorough review for sedimentation was provided by Deen et al.6_{Based on Kuwabara’s cell model, Levine et al.}7_were

able to derive both the sedimentation potential and sedimentation velocity of a colloidal dispersion. An On-sager relation for a concentrated suspension was derived by Ohshima.8_{Pujar and Zydney}9_{evaluated the}

sedimen-tation velocity of a charged particle in a spherical cavity

under the conditions of small Peclet number and low surface potential. Their analysis was extended by Lee et al.10_{to the case of arbitrary electrical potential.}

Nonrigid particles are often encountered in practice. For instance, biological entities such as cells and micro-organisms can be mimicked by a composite particle, which comprises a rigid core and a porous or ion-penetrable membrane layer. The particles in a stable colloidal dispersion are another typical example where the surface of a particle is usually covered by surfactant molecules. In contrast to a rigid particle where the charge it carries is mainly on its surface, the charge of a composite particle comprises that on the surface of its rigid core and that arising from the dissociation of the functional groups in its membrane layer. A human blood cell, for example, has an∼15 nm thick glycoprotein layer near its surface which carries net negative charge under normal conditions.11

The influence of the membrane layer of a composite particle on its sedimentation is twofold. The hydrodynamic drag that arises from the flow of the liquid inside tends to retard its sedimentation. On the other hand, the charge it carries can accelerate its sedimentation. Several at-tempts have been made to analyze the sedimentation of composite particles. Keh and Liu,12_{for instance, derived}

an analytic expression for both sedimentation potential and sedimentation velocity in a dilute suspension under the conditions of low electrical potential. The sedimenta-tion of a concentrated suspension of porous particles was analyzed by Ohshima,13_{and analytic expressions for both}

sedimentation potential and sedimentation velocity were derived for the case of low electrical potential.

In this study, the boundary effect on sedimentation is investigated by considering the sphere-in-spherical cavity geometry of Pujar and Zydney.9_{We extend their analysis}

to the case of a composite particle at arbitrary electrical potentials. Note that the result for a rigid particle can be recovered from the present study as a limiting case where the thickness of the membrane layer approaches zero. A psuedospectral method based on Chebyshev polynomials

* To whom correspondence should be addressed. Tel: 886-2-23637448. Fax: 886-2-23623040. E-mail: jphsu@ntu.edu.tw.

(1) Smoluchowski, M. Handbuch der Elecktrizitat und des Magne-tismus; Graetz, L., Ed.; Barth: Leipzig, 1921; Vol. II.

(2) Booth, F. J. Chem. Phys. 1954, 22, 1956.

(3) Saville, D. A. Adv. Colloid Interface Sci. 1982, 16, 267. (4) de Groot, S. R.; Mazur, P.; Overbeek, J. Th. G. J. Chem. Phys. 1952, 20, 1825.

(5) Ohshima, H.; Healy, T. W.; White, L. R.; O’Brien, R. W. J. Chem. Soc., Faraday Trans. 2 1984, 80, 1299.

(6) Deen, W. M.; Bohrer, M. P.; Epstein, N. B. Am. Inst. Chem. Eng. J. 1987, 27, 952.

(7) Levine, S.; Neale, G. H.; Epstein, N. J. Colloid Interface Sci. 1976, 57, 424.

(8) Ohshima, H. J. Colloid Interface Sci. 1998, 208, 295. (9) Pujar, N. S.; Zydney, A. L. Am. Inst. Chem. Eng. J. 1996, 42, 2101.

(10) Lee, E.; Yen, C. B.; Hsu, J. P. J. Phys. Chem. B 2000, 104, 6815. (11) Seaman, G. V. The Red Blood Cells; Sergenor, D. M., Ed.; Academic Press: New York, 1975; Vol. 2, pp 1136-1229.

(12) Keh, H. J.; Liu, Y. C. J. Colloid Interface Sci. 1997, 195, 169. (13) Ohshima, H. J. Colloid Interface Sci. 2000, 229, 140.

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is employed to solve the governing equations and the associated boundary conditions. The influences of the key parameters of the present problem, including the surface potential of the rigid core of a particle, the charge density in the membrane layer, the thickness of the double layer, and the relative magnitude of the particle, on its sedi-mentation behavior are examined through numerical simulations.

Theory

Referring to Figure 1, we consider the sedimentation of a spherical particle of radius b at the center of a spherical cavity of radius c. The particle comprises a rigid core of radius a and an ion-penetrable membrane layer of thickness d, b ) a + d. The membrane layer contains uniformly distributed fixed charge. The cavity is filled with an incompressible Newtonian liquid of constant physical properties containing z1:z2electrolyte, z1and z2

being respectively the valences of cations and anions. Let U be the sedimentation velocity of the particle. The deformation of the ionic cloud surrounding the particle yields an induced electric field E. The spherical coordinates (r,θ,φ) are adopted with their origin located at the center of the particle, and the axis θ ) 0 is parallel to both U and E. Let z2) -Rz1and H ) (b/c)3. Suppose that the system

is at a quasi-steady state.

For the present case, the electric field is described by

Here, F ) ∑jnjzje is the space charge density of mobile

ions,∇2_{is the Laplace operator, , F, and F}

fixare respectively

the permittivity of the electrolyte solution, the density of mobile ions, and the fixed charged density in the mem-brane layer. e and φ are respectively the elementary charge and the electrical potential, and njand zjare respectively

the number concentration and the valence of ionic species

j. The conservation of ionic species j leads to

where∇ is the gradient operator, Djis the diffusivity of

ionic species j, k is the Boltzmann constant, T is the absolute temperature, and v is the liquid velocity.

The flow field can be described by

In these expressions, p and η are respectively the pressure and the viscosity, and γ is the frictional coefficient of the membrane layer.

Similar to the treatment of O’Brien and White14_{for the}

case of electrophoresis, the electrical potential φ is decomposed into the electrical potential arising from the presence of the particle or the equilibrium potential φ1

and that induced by its sedimentation, φ2; that is, φ ) φ1

+ φ2. The deformation of the ionic cloud surrounding the

particle can be described by expressing njas

where gjand nj0are respectively a perturbed potential

and the bulk concentration of ionic species j. Combining eqs 1, 2, and 7 yields

Also, it can be shown that

Substituting eq 7 into eq 3 yields

(10) ∇2 gj -z_je kT∇φ1‚∇gj) 1 Dj v‚∇φ + 1 Dj v‚∇g_j+zje kT∇φ2‚∇gj+ zje kT∇gj‚∇gj (11) E4ψ - γE2ψ ) -sin θ η ∇ × [F∇(φ1+ φ2)] a < r < b (12) E4ψ ) -sin θ η ∇ × [F∇(φ1+ φ2)] b < r < c (13)

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Note that the r- and the θ-component of the liquid velocity,

vrand vθ, can be expressed respectively by

and

Boundary Conditions. We assume that the surface of the rigid core of a particle remains at constant potential

ζa and that of the cavity remains at ζb. Also, both the

electrical potential and the electric field are continuous on the membrane layer-liquid interface. Therefore, the boundary conditions associated with φ1are

The following boundary conditions are assumed for φ2:

Equation 18 arises from the fact that the rigid core of a particle is nonconductive and impermeable to ion species. Equations 19 and 20 imply that both the electrical potential and the electric field are continuous on the membrane layer-liquid interface. The last expression states that the electric field on the cavity surface is that arising from the sedimentation of the particle, where Ez

is the strength of the induced electric field in the

z-direction.

Suppose that the rigid core of a particle is impermeable to ionic species, and both the concentration of an ionic species and its flux are continuous on the membrane layer-liquid interface. Also, the concentration of ionic species reaches its bulk value on the cavity surface. These lead to the following boundary conditions:

In these expressions, fjis the flux of ionic species j, n is the unit normal vector, and δris the unit vector in the

r-direction.

We assume that both the rigid core of a particle and the surface of the cavity are no-slip, and both the velocity of the liquid and its normal and tangential stresses are continuous on the membrane layer-liquid interface. Therefore, the boundary conditions for the flow field are

In these expressions, σH _{and σ}E _{are respectively the}

hydrodynamic stress tensor and the Maxwell stress tensor, and σT_{) σ}H_{+ σ}E_{is the total stress tensor.}

For a simpler treatment, the governing equations and the associated boundary conditions are rewritten in dimensionless forms in subsequent discussions. To this end, the following characteristic variables are adopted: the radius of the rigid core of a particle a, the surface potential on the rigid core of the particle ζa, the bulk

number concentration of ionic species j nj0, and the

characteristic velocity UE) ζa2/ηa. We define the following

dimensionless symbols: the scaled radial distance r* )

r/a, the scaled number concentration of ionic species j,n_j/ ) nj/nj0, the scaled electric fieldEZ

/

) EZ/(ζa/a), the scaled

velocity v* ) v/UE, the scaled sedimentation velocity U*

) U/UE, the scaled equilibrium potentialφ1 /

) φ1/ζa, the

scaled perturbed potential, φ₂/ ) φ2/ζa, the scaled

per-turbed potentialg_j/) gj/ζa, j ) 1,2, and the scaled stream

function ψ* ) ψ/UEa.

Because the induced electric field arising from the sedimentation of a particle is much weaker than that arising from the presence of the particle, φ2, φ1, eq 7 can

be approximated by If both -φr(φ2 /₊ g₁/) and Rφr(φ2 /₊ g₂/) can be treated as perturbed terms, then both eqs 30 and 31 can be further approximated by a linear expression. Therefore, in terms of scaled symbols, eqs 8 and 9 become

In these expressions, κ-1_{) [kT/}_∑_n

j0(ezj)2]1/2is the Debye

length, φr) z1eζa/kT is the scaled surface potential, and Qfix) Ffixa2/ζa is the scaled fixed charge density. The

boundary conditions associated with eqs 32 and 33 are E2) ∂ ∂r2 + sin θ r2 ∂ ∂θ

(

1 sin θ ∂ ∂θ

)

v_r) - 1 r2sin θ ∂ψ ∂θ v_θ) 1 r sin θ ∂ψ ∂r φ1) ζa r ) a (14) φ1|r)b-) φ1|r)b+ r ) b (15) ∂φ₁ ∂r|r)b-) ∂φ₁ ∂r|r)b+ r ) b (16) φ1) ξb r ) c (17) ∂φ2 ∂r ) 0 r ) a (18) φ₂|r)b-) φ2|r)b+ r ) b (19) ∂φ2 ∂r|r)b-) ∂φ2 ∂r|r)b+ r ) b (20) ∂φ2 ∂r ) -Ezcos θ r ) c (21) fj‚n ) fj‚δr) 0 r ) a (22) nj|r)b-) nj|r)b+ r ) b (23) f_j|r)b-) fj|r)b+ r ) b (24) nj) nj0 r ) c (25) vr) U cos θ vθ) -U sin θ r ) a (26) vr|r)b-) vr|r)b+ vθ|r)b-) vθ|r)b+ r ) b (27) σ_rθH|r)b-) σrθ H_|r)b + σ_rrT|r)b-) σrr T_|r)b + σrr E_|r)b -) σrr E_|r)b + r ) b (28) v_r) 0 v_θ) 0 r ) c (29) n1 / ) exp(-φrφ1 / )[1 - φr(φ2 / + g1 / )] (30) n₂/) R exp(Rφ_rφ1 / )[1 + Rφ_r(φ₂/+ g₂/)] (31) ∇*2 φ1 /_{) -} (κa)2 (1 + R)φr [exp(-φ_rφ1 / ) - exp(Rφ_rφ1 / )] - Q_fix a < r < b (32) ∇*2 φ₁/) - (κa) 2 (1 + R)φr [exp(-φ_rφ₁/) - exp(Rφ_rφ₁/)] b < r < c (33) φ1 / ) 1 r* ) 1 (34) φ1 / |r*)b-_/a) φ₁ / |r*)b+_/a r* ) b/a (35)

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It can be shown that the corresponding governing equation forφ₂/is

and the associated boundary conditions are

It can be shown thatg₁/andg₂/satisfy

where Pej) UEa/Djis the electric Peclet number for ionic

species j. The boundary conditions associated with eqs 43 and 44 are

In terms of the scaled stream function ψ*, the governing equation for the flow field becomes

The corresponding boundary conditions are

where the dimensionless group (λa)2_{) (γa}2_{/η) is a measure}

for the friction coefficient of the membrane layer. The present problem can be made one-dimensional by applying the method of separation of variables. If all the perturbed variables in eqs 38-56 are linearized, then it can be shown thatφ₂/) Φ2(r) cos θ,g1

/

) G1(r) cos θ,g2 /₎

G2(r) cos θ, and ψ* ) Ψ(r) sin2θ, and Φ2, G1, and G2satisfy

where

The boundary conditions associated with eqs 57-59 are ∂φ1 / ∂r*|r*)b-/a) ∂φ1 / ∂r*|r*)b+/a r* ) b/a (36) ∂φ₁/ ∂r*) 0 r* ) c/a (37) ∇*2 φ2 /_{) -} (κa)2 (1 + R)φ_r{[exp(-φr(φ1 / + φ2 / + g1 / )) -exp(Rφr(φ1 /_{+ φ} 2 /_{+ g} 2 / ))] - [exp(-φrφ1 / ) - exp(Rφrφ1 / )]} (38) ∂φ₂/ ∂r*) 0 r* ) 1 (39) φ2 / |r*)b-/a) φ2 / |r*)b+_/a r* ) b/a (40) ∂φ2 / ∂r*|r*)b-/a) ∂φ2 / ∂r*|r*)b+/a r* ) b/a (41) ∂φ2 / ∂r*) -Ez / cos θ r* ) c/a (42) ∇*2 g₁/- φr∇*φ₁/‚∇g₁/) Pe₁v*‚∇φ₁/+ Pe₁v*‚∇φ₂/+ Pe1v*‚∇g1 /_{+ φ} r∇*φ2 / ‚∇*g1 /_{+ φ} r∇*g1 / ‚∇*g1 / (43) ∇*2 g2 / + Rφr∇*φ1 / ‚∇g2 / ) Pe2v*‚∇*φ1 / + Pe2v*‚∇φ2 /₊ Pe₂v*‚∇g₂/- Rφr∇*φ₂/‚∇*g₂/- Rφr∇*g₂/‚∇*g₂/ (44) ∂gj / ∂r*) 0 r* ) 1 j ) 1,2 (45) g_j/|r*)b-/a) gj / |r*)b+_/a r* ) b/a j ) 1,2 (46) ∂gj / ∂r*r*)b-/a) ∂gj / ∂r*|r*)b+/a r* ) b/a j ) 1,2 (47) gj / ) -φj / r* ) c/a j ) 1,2 (48) E*4ψ* - (λa)2E*2ψ* ) - (κa) 2 (1 + R)×

[

∂g1 / ∂r*n1 /₊∂g2 / ∂r*(Rn2 / )

)

∂φ* ∂θ

-(

∂g1 / ∂r*n1 /₊∂g2 / ∂r*(Rn2 / )

)

∂φ* ∂r*

]

[

∂3ψ* ∂r*3 - (λa) 2∂ψ* ∂r*

]

r*)b-/a )

[

∂3ψ* ∂r*3

]

r*)b+_/a r* ) b/a (55) ψ* ) 0

(

∂ 2 ∂r*2 - 2 r*2

)

ψ* ) 0 r* ) c/a (56) L2Φ2 -(κa)2 (1 + R)[exp(- φrφ1 / ) + R exp(Rφrφ1 / )]Φ2) (κa)2 (1 + R)[exp(-φrφ1 / )G1+ R exp(Rφrφ1 / )G2] (57) L2G1- φr 2dφ1 / dr*) Pe1φr 2 vr /dφ1 / dr* (58) L2G2+ Rφr 2dφ1 / dr*) Pe2φr 2 vr /dφ1 / dr* (59) L2≡ d 2 dr*2 + 2 r* d dr*- 2_r*2 dΦ2 dr* ) 0 r* ) 1 (60) Φ2|r*)b-/a) Φ2|r*)b+_/a r* ) b/a (61) dΦ2 dr*|r*)b-/a) dΦ2 dr*|r*)b+/a r* ) b/a (62) dΦ₂ dr* ) -Ez / r* ) c/a (63) dG_j dr*) 0 r* ) 1 j ) 1,2 (64)

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For the flow field, we have

where

The boundary conditions associated with eqs 68 and 69 are

Sedimentation Potential. When a spherical particle bearing net negative charge falls down in the gravitational field, the double layer surrounding it is not spherical; there are excess positive mobile ions beneath the particle and excess negative mobile ions above it. The deformation of the double layer induces an electric field in the z-direction, which is opposite to the direction of the gravitational field. The electrical potential associated with this induced electric field is called the sedimentation potential, E, which can be evaluated based on the fact that the current arising from the sedimentation of a particle is balanced by that arising from the induced electric field.16_{At steady state,}

the sedimentation of the particle generates no net current, and the net flow of current across any horizontal plane vanishes. The current i arising from the sedimentation of a particle can be expressed as

The current across the horizontal plane, θ ) π/2, is

where subscript θ denotes the θ-component. The velocity of ionic species j, vj, and that of liquid v are related by

Substituting this expression into eq 76, we obtain, in terms of scaled symbols,

The θ-component of the current, iθ, can be expressed by

This expression can be rewritten as

where

Substituting eq 81 into eq 77 gives

For a simpler treatment, the present problem is decom-posed into two subproblems.14 _{In the first problem, a}

particle moves with a constant velocity in the absence of the induced electric field, and in the second problem, the particle is held fixed when the induced electric field is present. The current in the first problem can be expressed as

where U* ) U/UE, and U is the z-component of the terminal

velocity. The current in the second problem can be expressed as

where E* ) Eza/ζa, and Ezis the z-component of the induced

electric field. Since the net current across the horizontal plane θ ) π/2 vanishes, that is,〈i〉)〈i〉1+〈i〉2) 0, the (15) Lee, E.; Chu, J. W.; Hsu, J. P. J. Chem. Phys. 1999, 110, 11643.

(16) Hunter, R. J. Foundations of Colloid Science, Vols. 1 & 2; Oxford University Press: London, 1989.

Gj|r*)b-/a) Gj|r*)b+_/a r* ) b/a j ) 1,2 (65) dGj dr*|r*)b-/a) dGj dr*|r*)b+/a r* ) b/a j ) 1,2 (66) G1) -Φ2and G2) -Φ2 r* ) c/a (67) D4Ψ - (λa)2D2Ψ ) - (κa) 2 1 + R

[

(n1 / G₁+ n₂/G₂)dφ1 / dr*

]

a < r < b (68) D4Ψ ) - (κa) 2 1 + R

[

(n1 / G1+ n2 / G2) dφ₁/ dr*

]

b < r < c (69) D4≡ D2D2)

(

d 2 dr*2 - 2 r*2

1 Pe₁exp(-φrφ1 / )∇*g1 /₊ R Pe₂exp(Rφrφ1 / )∇*g2 /

]

}

(79) i_θ) 3 φr 2 ηa3

(

kT z₁e

)

3 (κa)2 (1 + R){[exp(-φrφ1 / ) -exp(Rφrφ1 / )]}dΨ dr*- 1φ_r

[

1 Pe₁exp(-φrφ1 / )G1 / R Pe₂× exp(Rφrφ1 / )G2 /

]

sin θ r* (80) i_θ) IaIθ(r*) sin θ r* (81) Ia) 3φr 2 ηa3

(

kT z₁e

)

3 (κa)2 (1 + R) (82) I_θ)

{

[exp(-φrφ1 / ) - exp(Rφrφ1 / )]dΨ dr* -1 φr

[

1 Pe1 exp(-φ_rφ1 / )G₁/ R Pe2 exp(Rφ_rφ1 / )G₂/

]

}

(83) 〈i〉) 0 ) 2πa2I_a

∫

₁c/aI_θ(r*) dr* (84) 〈i〉1) δU* (85) 〈i〉₂) βE_z/ (86)

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scaled sedimentation potentialE_z//U* is

The terminal velocity of a particle can be determined from the fact that the sum of the external forces acting on it in the z-direction vanishes at the steady state. For the present case, these include the electric force, FEz, the

hydrodynamic forces, FDz, and the gravitational force, Fg,

and they can be expressed respectively as

with

In these expressions, Ff, Fp, and Fs are respectively the

densities of the fluid, the rigid core, and the membrane layer. We have FDz+ FEz+ Fg) 0, or

where Vsis the effective volume of the membrane layer.

The term in the first parenthesis of this expression can be rewritten as f1′U* + f2′E*, where f1′ and f2′ are

respectively the sum of the forces in problems 1 and 2. The sedimentation velocity can be expressed as

where φp) (a/c)3and φs) Vs/(4πc3/3) are respectively the

volume fraction of the rigid core and the effective volume of the membrane layer. For convenience, the terminal

velocity of a sphere with uncharged membrane layer U0

is used as a reference velocity in subsequent discussions. Results and Discussion

The sedimentation behavior of a particle is examined through numerical simulation. The governing equations and the associated boundary conditions are solved nu-merically by a pseudo-spectral method based on Cheby-shev polynomials, which has been justified to be an efficient approach for electrokinetic phenomena of the present type.17_{For illustration, an aqueous KCl solution is chosen}

as a model system, and the following values are used: T ) 298.15 K, r) 6.954 × 10-10F/m, η0) 8.904 × 10-3P, F_f_{) 0.99704 g/cm}3_{, Pe} 1) Pe2) 0.01, Fs) Fp) 1.05 g/cm3, d/a ) 0.5, Z1) ZK+) 1, Z2) ZK-) -1, D1) DK+) 1.962297 × 10-5_cm2_{/s, and D} 2) DCl-) 2.037051 × 10-5cm2/s.

Influence of Double-Layer Thickness. Figure 2 shows the variation of the scaled sedimentation velocity

U/U0as a function of double-layer thickness κa at various

fixed charge densities Qfix. Let us consider first the results

shown in Figure 2a where the membrane layer of a particle is uncharged (Qfix) 0). This figure indicates that U/U0 (17) Canuto, C.; Hussaini, M. Y.; Quarteroni, A.; Zang T. A. Spectral Method in Fluid Dynamics; Springer-Verlag: Berlin, 1988. Ez / U*) - δβ (87) F_Ez) 4 3πσa 2

(

r*2

(

dφ1 / dr*

)

f1′- δ_βf2′

)

-1 (95)

Figure 2. Variation of scaled sedimentation velocity U/U0as a function of κa at various φrand Qfixfor the case when λa ) 10 and H ) 0.421875: (a) Qfix) 0, (b) Qfix) 20, (c) Qfix) -20.

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approaches a constant as κa f 0. This is because if κa f 0, the double layer surrounding a particle is infinitely thick, and the mobile ions are uniformly distributed in the cavity. In this case, the sedimentation of a particle is influenced by the hydrodynamic force and the gravitational force only. If κa is sufficiently large, the effect of double-layer polarization comes into play. In this case, because the deformed double layer yields an induced electric field, which is in the z-direction, the sedimentation velocity declines. Note that if the double layer becomes thinner than the cavity radius, the absolute value of the potential gradient increases with the increase in κa, and the distribution of ionic species in the cavity becomes more nonuniform. The latter provides a driving force for liquid flow, which decelerates the sedimentation of a particle. If

κa increases further, the internal electric field arising from

the potential gradient near the rigid core of a particle becomes stronger, which makes it more difficult for the double layer to deform. The competition between the effect of double-layer polarization and that of the internal electric field leads to a local minimum in U/U0as κa varies, as can

be seen in Figure 2a. This local minimum may disappear if φris sufficiently low. This is because the effect of

double-layer polarization is significant only if φris sufficiently

high and, furthermore, because the higher the φr, the

stronger the induced electric field and, therefore, the smaller the U/U0. After passing the local minimum, U/U0

increases with the increase in κa, and the higher the φr,

the faster the rate of increase of U/U0. This is because if

κa is sufficiently large, the effect of double-layer

polariza-tion is insignificant, and the internal electric field dominates the behavior of a particle. In Figure 2b the membrane layer of a particle is positively charged. The behavior of U/U0in this figure is similar to that observed

in Figure 2a for the case when the membrane layer is free of fixed charge. Since the surface of the core of the particle is also positively charged, the presence of the membrane layer is equivalent to raising φrto a higher level. As can

be seen in Figure 2b, this is reflected by the fact that the influence of double-layer polarization occurs at a smaller

κa (=10-2_{) compared with that when the membrane layer}

if free of fixed charge. The variation of U/U0when the

membrane layer is negatively charged is illustrated in Figure 2c. In this case, although U/U0also approaches a

constant as κa f 0, it does not have a local minimum as

κa varies. This is because the induced electric field arising

from double-layer polarization is in the same direction as that of sedimentation. That is, double-layer polarization has the effect of accelerating the sedimentation of a particle. In this case, the higher the φr, the larger the U/U0. For a fixed φr, U/U0increases with the increase in

κa, as expected.

The simulated variation of the scaled sedimentation potential E*/U* for the case of Figure 2 is presented in Figure 3. This figure reveals that, regardless of the charged conditions of the membrane layer of a particle, E*/U* approaches a constant for both κa f 0 and κa f∞. The former is because if the double layer is infinitely thick, the ionic species are distributed uniformly in the cavity, and the equilibrium potential becomes independent of κa. The latter arises from the fact that if κa is large, double-layer polarization is compressed by the internal electric field, and therefore, the sedimentation potential is no longer sensitive to the effect of double-layer polarization. For the case when the membrane layer of a particle is either free of fixed charge or positively charged, if φris

sufficiently high, E*/U* has a local maximum as κa varies, which is consistent with the result of Lee et al.15_{for the}

case of rigid particles and can be explained by the

competition between the effects of double-layer polariza-tion and the internal electric field. If the membrane layer is negatively charged, because the induced electric field has the same direction as that of the gravitational field, the local maximum does not appear, and E*/U* increases monotonically with κa.

Influence of Surface Potential. The influence of the surface potential of the rigid core of a particle on its sedimentation velocity at various κa is illustrated in Figure 4 for different Qfix. Figure 4a reveals that for the case

when the membrane layer of a particle is positively charged, the scaled sedimentation velocity U/U0declines

with the increase in φr. This is expected because the higher

the φr, the more significant the effect of double-layer

polarization, which tends to retard the sedimentation of the particle. On the other hand, if the membrane layer is negatively charged, U/U0increases with the increase in

φr, as is shown in Figure 4b. This is because the electric

field induced by double-layer polarization is now in the same direction as that of sedimentation.

Influence of Fixed Charge Density. The simulated variations of the scaled sedimentation potential E*/U* and the scaled sedimentation velocity U/U0as a function

of κa at various scaled fixed charge densities in the membrane layer of a particle Qfixare presented in Figure

5. Figure 5a reveals that for a fixed κa, E*/U* increases with Qfix. However, as κa f ∞, the influence of Qfixon

Figure 3. Variation of scaled sedimentation potential E*/U* as a function of κa for the case of Figure 2.

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E*/U* becomes unimportant, and E*/U* approaches a

constant, which is independent of Qfix. The behavior of

U/U0as a function of κa depends on the nature of Qfix. For

Qfixg0, the larger the Qfix, the smaller the U/U0. Also,

U/U0 has a local minimum as κa varies, which can be

explained by the effect of double-layer polarization. On the other hand, if Qfix< 0, U/U0increases monotonically

with the increase in κa. Furthermore, the relative magnitudes of U/U0at different values of Qfixdepend on

the value of κa. This is because the influence of double-layer polarization is insignificant when|Qfix| is small.

Boundary Effect. The boundary effect on the sedi-mentation velocity of a particle is illustrated in Figure 6, and that on its sedimentation potential is shown in Figure 7. For illustration, we assume that the membrane layer of a particle is free of fixed charge. In Figure 6, if φris low,

that is, the effect of double-layer polarization is unim-portant, the scaled sedimentation velocity U/U0is almost

independent of both φrand H. On the other hand, if φris

sufficiently high, the effect of double-layer polarization becomes significant. In this case, the larger the H, that is, the more important the boundary effect, the larger the

U/U0. Since double-layer polarization has the effect of

retarding the sedimentation of a particle for the present case, this implies that the presence of the cavity has the effect of reducing the influence of double-layer polarization. This is because the liquid phase is confined by the cavity, and the larger the H, the smaller the space between the particle and the cavity available for the ionic cloud surrounding the former to deform. A similar result is also observed by Pujar and Zydney9 _{for the case of rigid}

Figure 4. Variation of scaled sedimentation velocity U/U0as a function of φrat various κa for the case when λa ) 10 and H ) 0.421875: (a) Qfix) 20, (b) Qfix) -20.

Figure 5. Variation of scaled sedimentation potential E*/U* (a) and scaled sedimentation velocity U/U0(b) as a function of

κa at various Qfixfor the case when φr) 1.0, λa ) 10, and H ) 0.421875.

Figure 6. Variation of scaled sedimentation velocity U/U0as a function of scaled surface potential φrat various H for the case when Qfix) 0, κa ) 1.0, and λa ) 10.

Figure 7. Variation of scaled sedimentation potential E*/U* as a function of κa at various H for the case when Qfix) 0 and

λa ) 10: (a) φr) 1.0, (b) φr) 3.0.

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particles. Figure 7 reveals that if φris low, the scaled

sedimentation potential E*/U* increases with the increase in κa or the decrease in H. This is expected because the effect of double-layer polarization is insignificant if φris

low. On the other hand, if φris sufficiently high, E*/U*

exhibits a local maximum as κa varies. Furthermore, the relative magnitude of E*/U* at different values of H depends on the value of κa. These behaviors can be explained by the interaction between the effect of double-layer polarization and that of the boundary.

Conclusions

We show that the influence of the presence of a boundary on the sedimentation of a composite particle depends largely on the nature of the membrane layer of the particle, in particular, its charged conditions. This arises mainly from the effect of double-layer polarization. If the surface of the rigid core of a particle is positively charged, the main results can be summarized as follows: (a) If the membrane layer is either free of fixed charge or positively charged and the surface potential of the rigid core of the

particle is sufficiently high, the sedimentation velocity has a local minimum and the sedimentation potential has a local maximum as the thickness of the double layer varies. These local extrema do not appear when the membrane layer is negatively charged. (b) If the membrane layer of a particle is positively charged, its sedimentation velocity decreases with the increase in the surface potential of its rigid core. The reverse is true if the membrane layer of the particle is negatively charged. (c) For a fixed double-layer thickness, the sedimentation potential increases with the density of fixed charge in the membrane layer. This trend becomes inappreciable as the double layer ap-proaches infinitely thin. (d) The presence of a boundary has the effect of reducing the influence of double-layer polarization. The behavior of a particle when the surface of its rigid core is negatively charged can be inferred from the above observations.

Acknowledgment. This work is supported by the National Science Council of the Republic of China. LA0476229

Sedimentation of a Composite Particle in a Spherical Cavity

Research Article

Sedimentation of a Composite Particle in a Spherical Cavity

Eric Lee, Tzu-Hao Huang, and Jyh-Ping Hsu

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Sedimentation of a Composite Particle in a Spherical

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Eric Lee, Tzu-Hao Huang, and Jyh-Ping Hsu*

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