Sedimentation Velocity and Potential in Dilute Suspensions of

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Sedimentation Velocity and Potential in Dilute Suspensions of Charged Porous Shells Yen Z. Yeh and Huan J. Keh*

J. Phys. Chem. B Downloaded from pubs.acs.org by KAOHSIUNG MEDICAL UNIV on 11/10/18. For personal use only.

Department of Chemical Engineering, National Taiwan University, Taipei 10617, Taiwan, ROC

ABSTRACT: The sedimentation of a charged spherical porous shell with arbitrary inner and outer radii, which can model a permeable microcapsule or vesicle, in a general electrolyte solution is analytically examined. The relaxation effect in the electric double layers of arbitrary thickness around the porous shell is considered. The differential equations governing the electric potential profile, ionic electrochemical potential energy (or concentration) distributions, and fluid velocity field are linearized by taking the system to be only slightly distorted from equilibrium. These linearized equations are solved using a perturbation method with the density of the fixed charge of the porous shell as the small perturbation parameter. Closed-form formulas for the sedimentation velocity of a porous shell and sedimentation potential in a suspension of porous shells are obtained from a force balance and a zero current requirement, respectively. Both the charge-induced sedimentation velocity retardation and sedimentation potential are monotonic increasing functions of the relative shell thickness, and these increases are substantial if the shell is thin. The sedimentation velocity and potential are complex functions of the electrokinetic radius and normalized flow penetration length of the porous shell. In the limit of the porous shells with zero inner radius, our formulas for the sedimentation velocity and potential reduce to the results obtained for the intact porous spheres.

1. INTRODUCTION The sedimentation phenomena of charged particles in electrolyte solutions are of fundamental and practical interest in the fields of colloidal science as well as environmental, biomedical, and chemical engineering.1,2 These phenomena are more complicated than those of uncharged particles because the fluid flow around each charged particle distorts its electric double layer. This distortion of the double layer induces an electric field around the charged particle.3,4 The induced electric field in a suspension of particles superimposes to result in a sedimentation potential gradient. Simultaneously, the sedimentation potential alters the fluid velocity distribution via electrostatic interactions and retards the sedimentation of the particle by an electrophoretic effect. Booth5 derived analytical expressions for the sedimentation velocity and potential in a suspension of hard spheres (impermeable to the solvent and ions) for an arbitrary double-layer thickness in power expansions of their small zeta potential using a perturbation method. Later, Stigter6 relieved the assumption of small zeta potential in this analysis to obtain numerical results. Subsequently, Ohshima et al.7 obtained both analytical formulas and numerical results of the sedimentation velocity and potential for the case of a broad © XXXX American Chemical Society

range of zeta potential and double-layer thickness. On the other hand, Booth’s perturbation analysis has also been extended to determine the sedimentation velocity and potential in suspensions of interacting hard spheres,8−12 porous (permeable) spheres,13−16 and soft (composite) spheres17−19 with low fixed charge densities. A porous shell is often employed to model a permeable microcapsule or vesicle,20−22 and the sedimentation and other electrokinetic phenomena of a charged porous shell are of interest.23,24 Recently, analytical formulas of the electrophoretic mobility and diffusiophoretic mobility were derived for a charged spherical porous shell.25,26 These formulas reduce to those obtained for an intact porous sphere13,27−30 when the inner radius of the spherical shell equals zero. However, the sedimentation velocity and potential in solutions of charged porous shells have not been analyzed yet. In this paper, we analyze the sedimentation phenomena in a suspension of charged spherical porous shells. No assumption is made about the double-layer thickness relative to the radii of Received: October 8, 2018 Revised: October 21, 2018 Published: October 22, 2018 A

DOI: 10.1021/acs.jpcb.8b09807 J. Phys. Chem. B XXXX, XXX, XXX−XXX

ÅÄ ÑÉ M zmenm∞ ÅÅÅÅ zmeψ (eq) ÑÑÑÑ 1 ∇ δψ = − ∑ expÅÅ− Ñ(δμ − zmeδψ ) ÅÅ ε m = 1 kT kT ÑÑÑÑ m ÇÅ Ö

Article

The Journal of Physical Chemistry B the porous shells. The governing equations are linearized assuming that the ionic concentrations and electric potential only deviate slightly from equilibrium. The ionic electrochemical potential energy (or concentration), electric potential, and fluid velocity fields are solved using a perturbation method with the fixed charge density of the porous shells as the small perturbation parameter. Closed-form formulas for the sedimentation velocity of the porous shells and sedimentation potential in the suspension are obtained.

2

(4) ε [∇2 − h(r )λ 2]∇ × u = − ∇ × [∇2 ψ (eq)∇δψ + ∇2 δψ ∇ψ (eq)] η (5)

where

2. ELECTROKINETIC EQUATIONS We consider the sedimentation of a charged spherical porous shell of inner radius a and outer radius b in an unbounded fluid containing M ionic species at the steady state, as shown in Figure 1. The acceleration of gravity is gez and the velocity of

δμm =

ψ = ψ (eq) + δψ

(2)

δnm + zmeδψ

(6)

r = a , b: ∂δμm ∂δψ , δψ , , ur , uθ , τrr , and τrθ δμm , ∂r ∂r (7)

are continuous

where (ur, uθ) and (τrr, τrθ) are the nonzero components of the fluid velocity and viscous stress, respectively. The conditions far from the porous shell can be written as

the spherical shell equals Uez, where ez is the unit vector in the z direction. The problem is axially symmetric, and the origin of the spherical coordinates (r, θ, ϕ) is set at the particle center. The velocity of the porous shell is assumed to be small so that the neighboring electric double layer is only slightly distorted from equilibrium. Therefore, the distributions of the ionic concentration nm(r,θ) and electric potential ψ(r,θ) can be written as (1)

nm(eq)

In the previous equations, Dm, zm, n∞ m , and δμm(r,θ) are the diffusion coefficient, valence, bulk concentration, and electrochemical potential energy distribution,7 respectively, of the type-m ions; η and ε are the viscosity and permittivity, respectively, of the fluid; h(r) is a step function equal to unity if a ≤ r ≤ b and zero otherwise; λ is the reciprocal of the flow penetration length into the porous shell (i.e., the fluid permeability 1/λ2 = η/f, where f is the friction coefficient per unit volume); e is the elementary charge, k is Boltzmann’s constant, and T is the absolute temperature. The values of Dm, η, ε, and f are taken to be constant. The boundary conditions for solving eqs 3−5 at the porous shell surfaces are

Figure 1. Geometrical sketch for the sedimentation of a charged porous spherical shell.

nm = nm(eq) + δnm

kT

r → ∞:

δμm → 0

(8a)

δψ → 0

(8b)

u → −U ez

(8c)

Equation 8c for the shell velocity in the opposite direction as the fluid velocity at infinity takes a reference frame translating with the shell.

3. SOLUTION OF ELECTROKINETIC EQUATIONS For a charged porous spherical shell with a uniform fixed charge density Q in an unbounded electrolyte solution, the equilibrium potential distribution ψ(eq) satisfying the Poisson− Boltzmann equation and appropriate boundary conditions is

(eq) where n(eq) (r) are the equilibrium profiles of the m (r) and ψ concentration of the species m and electric potential, respectively, and δnm(r,θ) and δψ(r,θ) are the corresponding (eq) small perturbed quantities. In fact, n(eq) are related by m and ψ the Boltzmann equation. The small deviations δnm and δψ as well as the fluid velocity distribution u(r,θ) are governed by the linearized continuity equations of the ions, Poisson−Boltzmann equation, and modified Stokes/Brinkman equations, respectively14 ÄÅ ÉÑ ÑÑ zme ÅÅÅ (eq) kT 2 (eq) ÅÅ∇ψ ·∇δμm − ∇ δμm = ∇ψ ·uÑÑÑÑ, Å ÑÑÖ kT ÅÅÇ Dm m = 1, 2, . . . , M (3)

ψ (eq) = ψeq1(r )Q̅ + O(Q̅ 2)

(9)

where25 ψeq1 =

kT [(1 + κa)e κb − (1 + κb)e κa](e 2κr − 1) e

e−κ(a + b + r) 2κr B

if 0 ≤ r ≤ a

(10a) DOI: 10.1021/acs.jpcb.8b09807 J. Phys. Chem. B XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry B

o kT l o1 + [(1 + κb)(1 − e 2κr ) − (1 + κa)e κ(b − a) ψeq1 = m o e o n + (1 − κa)e κ(a + b)]

o e−κ(b + r) | o } o 2κr o ~

iay K ψ = Fψ (b) − jjj zzz Fψ (a) kb{ 2

π

(10b)

∫0 ∫a

Fh = (10c)

2 ∞ 1/2 In the previous equations, κ = (e2∑M is the m=1zm nm /εkT) 2 Debye screening parameter and Q̅ = eQ/εκ kT is the nondimensional charge density of the porous shell. The small perturbed quantities δμm, δψ, ur, and uθ satisfying the continuity equation ∇·u = 0 as well as eqs 3−8c can be solved using eqs 10a−10c for ψeq1 and the perturbation expansion for the sedimentation velocity of the porous shell

U = U0 + U1Q̅ + U2Q̅ 2 + O(Q̅ 3)

(12)

δψ = U0Fψ (r )cos θQ̅ + O(Q̅ 2)

(13)

3

U1 = 0

ur = {(U0 + U1Q̅ )F0(r ) + [U2F0(r ) + U0F2(r )]Q̅ }cos θ U2 = −

(14a)

4. SEDIMENTATION VELOCITY The net force acting on the sedimenting charged porous shell is the sum of the gravitational force (including buoyant force), electrostatic force, and drag force on the particle. The gravitational force can be expressed as (15)

where ρp and εp are the density and porosity of the shell, respectively, and ρ is the fluid density. The electric force exerted on the porous shell is given by

∫0 ∫a

Fe = −2πQ

∇δψr 2 sin θ dr dθ

4πεkT (κb)2 K ψ U0Q̅ 2 ez + O(Q̅ 3) 3e

U0 ijj εκ 2kT yzz K ψ zz jjjK 2 − K0 k ηλ 2be z{

(22c)

5. SEDIMENTATION POTENTIAL For a dilute suspension of charged porous shells undergoing sedimentation without net electric current, the sedimentation potential field is the mean electric potential gradient in the suspension and can be obtained as14

(16)

Substituting eq 13 into 16, we obtain Fe = −

(22b)

where U0 is the sedimentation velocity of an uncharged porous shell.21,22 Because both the fixed charges of the porous shell and the local induced sedimentation potential gradient are of the order Q, the effect of the fixed charge density on the shell velocity resulting from their interaction starts from the second order Q2. It can be found that U2 depends on the inner-to-outer radius ratio a/b, electrokinetic radius κb (ratio of the outer radius to the Debye length), and shielding parameter λb (ratio of the outer radius to the flow penetration length) of the shell as well as the ionic diffusion coefficients in the electrolyte solution. Because the quantity (κb)2Q̅ (=b2eQ/εkT) does not depend on κ or n∞ m for a fixed charge density Q, the numerical result of the dimensionless coefficient H = −U2/(κb)4U0 calculated from eqs 22a and 22c for the electrokinetic retardation to the sedimentation velocity of the porous shell will be presented in Section 6. For the limiting case of a/b = 0, eqs 22a−22c reduce to the corresponding result obtained for an intact porous sphere.14

(14b)

Here, the functions F0(r), F2(r), Fμm(r), and Fψ(r) are given by eqs A1a−A1c, A2a−A2c, A6, and A7, respectively, in the Appendix. Because Fμm(r) and Fψ(r) are affected by the fluid flow through F0(r), the primary relaxation effect on the mobile ions in the electric double layers is incorporated in the solutions for δμm and δψ up to the first order of Q̅ .

b

(21)

for j = 0, 1, and 2. The net force acting on the sedimenting porous shell vanishes at the steady state. Applying this constraint to the summation of eqs 15, 17, and 20, we obtain the sedimentation velocity of the spherical porous shell as expressed by eq 11 with Ä É 3Ñ (ρp − ρ)g ÅÅÅ ij a yz ÑÑÑÑ Å Å U0 = −(1 − εp) Å1 − jj zz ÑÑ ηλ 2K 0 ÅÅÅÇ k b { ÑÑÖ (22a)

2

4 π (b3 − a3)(1 − εp)(ρp − ρ)g ez 3

(19)

4 πηλ 2b3[K 0(U0 + U1Q̅ ) + (K 0U2 + K 2U0)Q̅ 2]ez + O(Q̅ 3) 3 (20)

iay Kj = Fj(b) − jjj zzz Fj(a) kb{

(11)

δμm = U0Fμm(r )cos θQ̅ + O(Q̅ 2)

tan θ ∂ 2 (r ur ) uθ = − 2r ∂r

ur 2 sin θ dr dθ

where

and the results are

+ O(Q̅ 3)

b

Fh = 2πf

The substitution of eqs 14a and 14b into the previous equation results in

if r ≥ b

π

(18)

The drag force exerted on the porous shell is given by if a ≤ r ≤ b

kT ψeq1 = [(1 − κa)e κ(2a + b) + (1 + κb)e κa e e−κ(a + b + r) − (1 − κb)e κ(a + 2b) − (1 + κa)e κb] 2κr

Fg =

Article

ESED =

(17)

where C

ϕeU0 3

∞

b kT Λ

i dFμm

∑ zmnm∞Dmjjjjjr 3 M

m=1

k

yz − r 2Fμmzzzz Q̅ ez + O(Q̅ 2) dr {r →∞ (23) DOI: 10.1021/acs.jpcb.8b09807 J. Phys. Chem. B XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry B

Figure 2. Coefficient H for a sedimenting charged porous spherical shell in the aqueous KCl solution vs its inner-to-outer radius ratio a/b: (a) λb = 1 and (b) κb = 1.

Figure 3. Coefficient H for a sedimenting charged porous spherical shell in the aqueous KCl solution vs its shielding parameter λb: (a) κb = 1 and (b) a/b = 0.999 (solid curves) and a/b = 0 (dashed curves).

Figure 4. Coefficient H for a sedimenting charged porous spherical shell in the aqueous KCl solution vs its electrokinetic radius κb: (a) λb = 1 and (b) a/b = 0.999 (solid curves) and a/b = 0 (dashed curves).

where ϕ is the apparent volume fraction of the porous shells in 2 ∞ the suspension and Λ∞ = e2∑M m=1zm nm Dm/kT is the electric conductivity of the shell-free electrolyte solution. Substituting

eq 22a for U0 and eq A6 for Fμm(r) into eq 23, we obtain this field as D

DOI: 10.1021/acs.jpcb.8b09807 J. Phys. Chem. B XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry B

Figure 5. Dimensionless sedimentation potential ηλ2μE/Q in a dilute suspension of charged porous spherical shells vs the radius ratio a/b: (a) λb = 10 and (b) κb = 1.

Figure 6. Dimensionless sedimentation potential ηλ2μE/Q in a dilute suspension of charged porous spherical shells vs the shielding parameter λb: (a) κb = 1 and (b) a/b = 0.999 (solid curves) and a/b = 0 (dashed curves).

ESED = −(1 − εp)(ρp − ρ)g

ϕ μ ez Λ∞ E

which is the same as the formula for μE obtained for an intact porous sphere.13,14

(24)

É ÅÄ 3Ñ| 2 o K1 Qb2 l 1 ÅÅÅÅ ij a yz ÑÑÑÑo o o e − + O(Q̅ 2) μE = 1 − jj zz ÑÑ} m 2 2Å o o Å o o Å Ñ ηK 0 n εbkT (κb) (λb) ÅÇ k b { ÑÖ~

where

6. RESULTS AND DISCUSSION The sedimentation velocity of a charged porous spherical shell can be evaluated to the second order of the fixed charge density Q according to eqs 11 and 22a−22c. Figures 2−4 are plotted for the numerical values of the dimensionless coefficient H = −U2/(κb)4U0 (which is positive) for the charge-induced retardation to this velocity in an aqueous solution of KCl at room temperature (with M = 2, z1 = −z2 = 1, D1 = D2 = D, and εk2T2/ηDe2 = 0.25932) as a function of the inner-to-outer radius ratio a/b, electrokinetic radius κb, and shielding parameter (reciprocal of normalized flow penetration length) λb of the porous shell. For fixed values of λb and κb, the coefficient H decreases monotonically with an increase in the radius ratio a/b of the porous shell (or a decrease in the relative thickness 1 − a/b) because of a decrease in its total fixed charge 4π(b3 − a3)Q/3. Especially, the decrease of H with a/b is quite substantial as the porous shell turns thin. For specified values of a/b and λb, the coefficient H is not a monotonic function of κb, has a maximum at a finite value of κb, and equals zero in the limiting cases of κb → 0 and κb →

(25)

which is the electrophoretic mobility of the charged porous spherical shell.25,26 Equation 24 is an Onsager reciprocal relation connecting the electrophoretic mobility with the sedimentation potential.31 Note that μE is a function of the parameters κb, λb, and a/b but independent of the ionic diffusion coefficients in the electrolyte solution. For the limiting case of a/b = 0, eq 20 reduces to μE = ij jj1 k

l 2 o Q o 1 i λ y ji 1 − e−2κb zyz 1 jij λ 2 zyz zz + jj 2 1 + jjj zzz jjjj1 + e−2κb − zz 2m o z 3kκ { k 3 jk λ − κ 2 z{ κb ηλ o o { n ÉÑ| ÄÅ 2 ÑÑo 1 yzÅÅÅÅij λ yz κb(1 + e−2κb) − 1 + e−2κb Ño zzÅÅjj zz + − 1 + e−2κbÑÑÑ} + O(Q̅ 2) ÑÑo κb {ÅÅÅÇk κ { λb coth(λb) − 1 o ÑÖo ~

(26) E

DOI: 10.1021/acs.jpcb.8b09807 J. Phys. Chem. B XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry B

Figure 7. Dimensionless sedimentation potential ηλ2μE/Q in a dilute suspension of charged porous spherical shells vs the electrokinetic radius κb: (a) λb = 10 and (b) a/b = 0.999 (solid curves) and a/b = 0 (dashed curves).

sedimentation potential is obtained to the first order Q, which indicates that its dimensionless value (ηλ2μE/Q) is an increasing function of λb and a decreasing function of κb. Both the charge-induced sedimentation velocity retardation and sedimentation potential are monotonic increasing functions of the relative shell thickness 1 − a/b, and these increases are substantial as a/b is close to unity.

∞. As shown in Figure 4, the location of this maximum shifts to smaller κb as λb increases. For given values of κb and a/b, the coefficient H increases monotonically with decreasing λb, is proportional to (λb)−2 in wide ranges (especially as λb is small, κb is large, or a/b is close to unity), and approaches a finite value if λb is large (unless κb is large or a/b is close to unity, where H → 0 and the sedimentation velocity of the porous shell limits to U0 for an uncharged one as λb → ∞). The normalized sedimentation potential ηλ2μE/Q (which is positive) for a dilute suspension of porous spherical shells calculated using eq 25 is plotted versus the parameters κb, λb, and a/b in Figures 5−7. For specified values of λb and κb, the value of ηλ2μE/Q is a decreasing function of the radius ratio a/ b of the porous shell (or an increasing function of the relative thickness 1 − a/b), also manifesting the effect of its overall fixed charge 4π(b3 − a3)Q/3. Again, the decrease of ηλ2μE/Q with a/b is more conspicuous as the shell is thin. When κb is smaller or λb is larger (viz., κ/λ is smaller), the influence of a/b on ηλ2μE/Q is stronger. For given values of λb and a/b, ηλ2μE/Q decreases monotonically with an increase in κb, is proportional to (κb)−2 in wide ranges, and approaches unity in the limit κb → ∞. For given values of a/b and κb, ηλ2μE/Q increases with an increase in λb, approaches unity in the limit λb → 0, and is proportional to (λb)2 in wide ranges.

■

APPENDIX For conciseness, some functions in Section 3 are defined here. In eqs 14a and 14b, ir y F0(r ) = C01 + C02jjj zzz ka{

2

if r < a

(A1a)

γ (λ r ) δ(λr ) iay F0(r ) = C03jjj zzz + C04 + C05 3 3 + C06 3 3 λr λr kr { 3

(A1b)

if a < r < b

iby iby F0(r ) = C07jjj zzz + C08jjj zzz − 1 kr { kr { 3

if r > b

(A1c)

r 2 (0) 1 ir y Fi(r ) = Ci1 + Ci2jjj zzz − Ii (0, r ) + Ii(2)(0, r ) a 15 3 k { 1 (3) 1 (5) Ii (0, r ) if r < a − Ii (0, r ) + (A2a) 3r 15r 3 2

7. CONCLUSIONS In this work, the steady-state sedimentation phenomena of charged spherical porous shells in electrolyte solutions with arbitrary values of the inner-to-outer radius ratio a/b, electrokinetic radius κb, and shielding parameter λb are analyzed. Solving the linearized electrokinetic equations applicable to the system by a perturbation method, we obtain the ionic electrochemical potential energy (or concentration), electrostatic potential, and fluid flow distributions for the case of low fixed charge density Q. The requirement that the net force acting on the porous shell vanishes leads to a closed-form formula for its sedimentation velocity, which shows that the retardation effect of the fixed charge begins at the second order Q2. For specified values of λb and a/b, this effect is maximal at a finite value of κb and vanishes when κb becomes infinity and zero. For given values of κb and a/b, this effect is a monotonic decreasing function of λb. An analytical expression for the

iay iay iay Fi(r ) = Ci3jjj zzz + Ci 4 + Ci5jjj zzz γ(λr ) + Ci6jjj zzz δ(λr ) kr { kr { kr { 2 (0) 2 (3) 2 γ + 2 Ii (a , r ) − 2 3 Ii (a , r ) − 5 3 Ii (r )δ(λr ) 3λ 3λ r λr 2 δ if a < r < b + 5 3 Ii (r )γ(λr ) (A2b) λr 3

3

3

iby iby 1 r 2 (0) Fi(r ) = Ci7jjj zzz + Ci8jjj zzz − Ji (r ) + Ji(2)(0, r ) 15 3 kr { kr { 1 1 (5) − Ji(3)(r ) + if r < b J (r ) (A2c) 3r 15r 3 i 3

where i = 1 and 2 F

DOI: 10.1021/acs.jpcb.8b09807 J. Phys. Chem. B XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry B Ji(n)(r ) =

r

∫∞ r nGi(r) dr

Ii(n)(r1 , r2) =

∫r

r2

r nGi(r ) dr

(A3b)

1

Iiγ(r ) =

∫a

(3) Gopmandal, P. P.; Bhattacharyya, S.; Barman, B. Effect of Induced Electric Field on Migration of a Charged Porous Particle. Eur. Phys. J. E 2014, 37, 104. (4) Khair, A. S. Strong Deformation of the Thick Electric Double Layer around a Charged Particle during Sedimentation or Electrophoresis. Langmuir 2018, 34, 876−885. (5) Booth, F. Sedimentation Potential and Velocity of Solid Spherical Particles. J. Chem. Phys. 1954, 22, 1956−1968. (6) Stigter, D. Sedimentation of Highly Charged Colloidal Spheres. J. Phys. Chem. 1980, 84, 2758−2762. (7) Ohshima, H.; Healy, T. W.; White, L. R.; O’Brien, R. W. Sedimentation Velocity and Potential in a Dilute Suspension of Charged Spherical Colloidal Particles. J. Chem. Soc., Faraday Trans. 2 1984, 80, 1299−1317. (8) Levine, S.; Neale, G.; Epstein, N. The Prediction of Electrokinetic Phenomena within Multiparticle Systems II. Sedimentation Potential. J. Colloid Interface Sci. 1976, 57, 424−437. (9) Ohshima, H. Sedimentation Potential in a Concentrated Suspension of Spherical Colloidal Particles. J. Colloid Interface Sci. 1998, 208, 295−301. (10) Keh, H. J.; Ding, J. M. Sedimentation Velocity and Potential in Concentrated Suspensions of Charged Spheres with Arbitrary Double-Layer Thickness. J. Colloid Interface Sci. 2000, 227, 540−552. (11) Carrique, F.; Arroyo, F. J.; Delgado, A. V. Sedimentation Velocity and Potential in a Concentrated Colloidal Suspension: Effect of a Dynamic Stern Layer. Colloids Surf., A 2001, 195, 157−169. (12) Keh, H. J.; Cheng, T. F. Sedimentation of a Charged Particle in a Charged Cavity. J. Chem. Phys. 2011, 135, 214706. (13) Hermans, J. J. Sedimentation and Electrophoresis of Porous Spheres. J. Polym. Sci. 1955, 18, 527−534. (14) Liu, Y. C.; Keh, H. J. Sedimentation Velocity and Potential in a Dilute Suspension of Charged Porous Spheres. Colloids Surf., A 1998, 140, 245−259. (15) Keh, H. J.; Chen, W. C. Sedimentation Velocity and Potential in Concentrated Suspensions of Charged Porous Spheres. J. Colloid Interface Sci. 2006, 296, 710−720. (16) Chang, Y. J.; Keh, H. J. Sedimentation of a Charged Porous Particle in a Charged Cavity. J. Phys. Chem. B 2013, 117, 12319− 12327. (17) Keh, H. J.; Liu, Y. C. Sedimentation Velocity and Potential in a Dilute Suspension of Charged Composite Spheres. J. Colloid Interface Sci. 1997, 195, 169−191. (18) Ohshima, H. Sedimentation Potential and Velocity in a Concentrated Suspension of Soft Particles. J. Colloid Interface Sci. 2000, 229, 140−147. (19) Chiu, Y. S.; Keh, H. J. Sedimentation Velocity and Potential in a Concentrated Suspension of Charged Soft Spheres. Colloids Surf., A 2014, 440, 185−196. (20) Jones, I. P. Low Reynolds Number Flow Past a Porous Spherical Shell. Proc. Cambridge Philos. Soc. 1973, 73, 231−238. (21) Bhatt, B. S.; Sacheti, N. C. Flow Past a Porous Spherical Shell Using the Brinkman Model. J. Phys. D: Appl. Phys. 1994, 27, 37−41. (22) Keh, M. P.; Keh, H. J. Slow Motion of an Assemblage of Porous Spherical Shells Relative to a Fluid. Transp. Porous Media 2010, 81, 261−275. (23) Fransaer, J.; Celis, J.-P.; Roos, J. R. Electro-osmophoresis of a Charged Permeable Microcapsule with Thin Double Layer. J. Colloid Interface Sci. 1992, 151, 26−40. (24) Suárez, I. R.; Leidy, C.; Téllez, G.; Gay, G.; Gonzalez-Mancera, A. Slow Sedimentation and Deformability of Charged Lipid Vesicles. PloS One 2013, 8, No. e68309. (25) Li, W. C.; Keh, H. J. Electrophoretic Mobility of Charged Porous Shells or Microcapsules and Electric Conductivity of Their Dilute Suspensions. Colloids Surf., A 2016, 497, 154−166. (26) Yeh, Y. Z.; Keh, H. J. Diffusiophoresis of a Charged Porous Shell in Electrolyte Gradients. Colloid Polym. Sci. 2018, 296, 451−459. (27) Ohshima, H. Electrophoretic Mobility of Soft Particles. J. Colloid Interface Sci. 1994, 163, 474−483.

(A3a)

r

γ(λr )Gi(r )dr

(A3c)

δ(λr )Gi(r ) dr

(A3d)

r

Ii δ(r ) =

∫a

G1(r ) =

εκ 2b dψeq1 dr e

(A4a)

G2(r ) =

M e dψeq1 ∑ zmnm∞Fμm(r) ηkT r dr m = 1

(A4b)

γ(x) = x cosh x − sinh x

(A5a)

δ(x) = x sinh x − cosh x

(A5b)

and the dimensionless constants Cjn for n = 1, 2, ..., 8 and j = 0, 1, 2 are given in ref 26. In eqs 12 and 13, ÄÅ dψeq1 zme ÅÅÅÅ 1 r 3 dr Fμm(r ) = r F0(r ) ÅÅ 2 dr 3Dm ÅÅÅÇ r 0 É dψeq1 ÑÑÑÑ ∞ +r dr ÑÑÑ F0(r ) ÑÑ r dr (A6) ÑÖ

∫

∫

M

Fψ (r ) =

{

e ∑ zmnm∞ (κr + 1) εkTκ 3r 2 m = 1 e‐κr

∫0

r

γ(κr )Fμm(r )dr + γ(κr )

∫r

∞

(κr + 1)

}

e‐κr Fμm(r )dr

■

(A7)

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: 886-2-33663048. Fax: 8862-23623040. ORCID

Huan J. Keh: 0000-0002-8156-2122 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This research was supported by the Ministry of Science and Technology of the Republic of China (Taiwan) under the grant MOST106-2221-E-002-167-MY3.

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