Diffusiophoresis of a Charged Porous Particle in a Charged Cavity

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Diffusiophoresis of a Charged Porous Particle in a Charged Cavity Ya Ching Chiu, and Huan Jang Keh J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.8b06967 • Publication Date (Web): 03 Oct 2018 Downloaded from http://pubs.acs.org on October 9, 2018

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The Journal of Physical Chemistry

Diffusiophoresis of a Charged Porous Particle in a Charged Cavity Ya C. Chiu and Huan J. Keh* Department of Chemical Engineering, National Taiwan University, Taipei 10617, Taiwan, ROC

ABSTRACT:

The quasisteady diffusiophoresis of a charged porous sphere situated at the

center of a charged spherical cavity filled with a liquid solution of a symmetric electrolyte is analyzed. The porous particle can represent a solvent-permeable and ion-penetrable polyelectrolyte molecule or floc of nanoparticles, in which fixed charges and frictional segments are uniformly distributed, while the spherical cavity can denote a charged pore involved in microfluidic or drug-delivery systems. The linearized electrokinetic differential equations governing the ionic concentration, electric potential, and fluid velocity distributions in the system are solved by using a perturbation method with the fixed charge density of the particle and the zeta potential of the cavity wall as the small perturbation parameters. An expression for the diffusiophoretic (electrophoretic and chemiphoretic) mobility of the confined particle with arbitrary values of a / b , κa , and λa is obtained in closed form, where a and

b are the radii of the particle and cavity, respectively; κ and λ are the reciprocals of the Debye screening length and the length characterizing the extent of flow penetration into the porous particle, respectively. The presence of the charged cavity wall significantly affects the diffusiophoretic motion of the particle in typical cases. The diffusioosmotic (electroosmotic and chemiosmotic) flow occurring at the cavity wall can substantially alter the particle velocity and even reverse the direction of diffusiophoresis. In general, the particle velocity decreases with an increase in a / b , increases with an increase in κa , and decreases with an increase in λa , but exceptions exist.

*Corresponding Author. Telephone: 886-2-33663048.

Fax: 886-2-23623040.

E-mail: [email protected].

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1.

INTRODUCTION Diffusiophoresis regards the motion of colloidal particles under an applied solute

concentration gradient1-4 and provides a mechanism for plenty of practical applications such as particle manipulation and characterization in microfluidic systems,5,6 latex film coating,7 DNA translocation and sequencing,8,9 autonomous motion of micromotors,10-12 and many others.13 For diffusiophoretic motions of charged particles in ionic solutions, the solute-particle interactions are in the electrostatic range characterized by the Debye screening length κ −1 . Most analytical studies for the diffusiophoresis of charged hard particles (impermeable to the ionic solution) are restricted to the case of thin electric double layer ( κa >> 1 , where a is the radius of the particles).14,15 With the assumption of a weak imposed electrolyte concentration gradient, diffusiophoresis was also analytically studied for a charged hard sphere,16,17 a porous sphere (permeable to the ionic solution, such as a charged floc or polyelectrolyte molecule),18,19 and a soft sphere (a charged hard core whose surface adsorbs a charged porous layer)20 with arbitrary values of κa . Colloidal particles are seldom isolated or unbounded in various applications of diffusiophoresis.21,22 In the limit of thin electric double layer ( κa → ∞ ), the normalized velocity field of the fluid caused by a hard particle undergoing diffusiophoresis is identical to that for electrophoresis5 and the boundary effect on electrophoresis, that has been widely investigated,23 can be employed to construe those on diffusiophoresis. Knowing that the boundary effect on diffusiophoresis is different from that on electrophoresis when the double layer polarization is embodied, the diffusiophoretic motions of a hard sphere with a thin but polarized double layer (say, κa ≥ 20 ) parallel24 and perpendicular25 to one or two plates as well as along the axis of a microtube26 were investigated through the use of a semi-analytical boundary collocation method. The system of a colloidal sphere moving in a spherical cavity can be an idealized model for the diffusiophoresis in microfluidic devices or dead-end pores of self-regulated drug delivery systems.27,28 Actually, the diffusiophoretic motions of a hard spherical particle with a thin polarized double layer29 and with an arbitrary value of κa 30 in a concentric charged spherical cavity have been analytically examined. Also, the diffusiophoresis of a soft sphere within a nonconcentric uncharged spherical cavity has been numerically examined.31 As yet, the boundary effects on the diffusiophoresis of a charged porous particle have not been analytical studied. In this article, the diffusiophoresis of a charged porous sphere with an arbitrary value of κa within a concentric charged spherical cavity is analytically studied. The ionic concentration (electrochemical potential energy), electrostatic potential, and fluid velocity 2 ACS Paragon Plus Environment

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distributions are determined in power expansions of the small fixed charge density of the particle and zeta potential of the cavity wall. A closed-form formula for the migration velocity of the porous particle is obtained.

2.

ELECTROKINETIC EQUATIONS We consider the diffusiophoretic motion of a charged porous spherical particle of radius

a , which is solvent-permeable and ion-penetrable, in a concentric charged spherical cavity of radius b filled with a liquid solution of a symmetric electrolyte, as shown in Figure 1, at the quasisteady state. The prescribed electrolyte concentration gradient is a constant ∇n ∞ (viz., the concentration distribution n ∞ is linear) and the migration velocity of the particle to be determined is U , both are in the z direction. The origin of the spherical coordinates ( r,θ , φ ) is set at the center of the cavity/particle, and the problem is axially symmetric about the z axis (independent of φ ). 2.1. Governing Equations. We assume that the relative electrolyte concentration gradient α = a ∇n∞ / n0∞ (where n0∞ is n ∞ at z = 0 ) is small such that the system is only slightly distorted from equilibrium. Thus, the ionic concentration distributions n± (r , θ ) (the subscripts + and − refer to cation and anion, respectively) and the electric potential distribution ψ ( r , θ ) can be decomposed as

n± = n±( eq ) + δn± ,

(1)

ψ =ψ

(2)

( eq )

+ δψ ,

where n±( eq ) (r ) and ψ ( eq ) (r ) are the equilibrium distributions of the ionic concentrations and electric potential, respectively (they are related by the Boltzmann equation), and δn± (r , θ ) and

δψ (r , θ ) are the corresponding small perturbations. The small perturbed quantities δn± , δψ , and the fluid velocity field v ( r ,θ ) are governed by the following linearized equations resulting from the continuity equations of the ionic species, Poisson-Boltzmann equation, and modified Stokes/Brinkman equation, respectively:18 ∇ 2δµ ± = ±

∇ 2δψ =

Ze kT [∇δµ ± ⋅ ∇ψ ( eq ) − v ⋅ ∇ψ ( eq ) ] , kT D±

n0∞ Ze Zeψ ( eq ) Zeψ ( eq ) {(δµ− + Zeδψ ) exp[ ] − (δµ+ − Zeδψ ) exp[− ]} , εkT kT kT

[∇ 2 − λ2 h(r )]∇ × v = −

ε ∇ × [∇ 2δψ∇ψ ( eq ) + ∇ 2ψ ( eq ) ∇δψ ] . η 3 ACS Paragon Plus Environment

(3) (4) (5)


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Here, the ionic electrochemical potential energy perturbations kT δµ ± = ( eq ) δn± ± Zeδψ ; n±

(6)

η and ε are the viscosity and permittivity, respectively, of the fluid; Z is the valence of the symmetric electrolyte; D± are the ionic diffusion coefficients; λ−1 is the Brinkman shielding length characterizing the extent of flow penetration into the porous particle, i.e., λ−2 is its permeability; h( r ) is a step function that equals unity if r ≤ a and zero otherwise. The constants η , ε , and D± are taken to be same inside and outside the porous particle. The boundary conditions for the small perturbed quantities

2.2. Boundary Conditions. at the particle surface are r = a:

δµ ± , ∇δµ ± , δψ , ∇δψ , vr , vθ , τ rr , and τ rθ are continuous,

(7)

where ( vr , vθ ) and ( τ rr , τ rθ ) are the nontrivial components of the fluid velocity and stress, respectively. The boundary conditions at the cavity wall to produce a uniform electrolyte gradient in the absence of the particle are

r =b:

δµ ± = kT (1 m β )α

r cos θ , a

(8a)

kT r βα cos θ , Ze a vr = − U cosθ , vθ = U sin θ ,

δψ = −

where

(8b) (8c) (8d)

β = ( D+ − D− ) /( D+ + D− ) . Expressions 8a and 8b for the perturbed ionic

electrochemical potential energies and induced electric potential result from the linear prescribed ionic concentrations

n± = n0∞ + ∇n ∞ r cos θ

(caused by a salt solubility

distribution, for example) at the wall as well as the equality of the anionic and cationic fluxes in the fluid without the particle. Equations 8c and 8d for the still cavity take a reference frame traveling with the particle.

3.

SOLUTION FOR THE DIFFUSIOPHORETIC VELOCITY 3.1. Equilibrium Electric Potential.

The equilibrium electric potential distribution

about a porous sphere with a uniform space fixed-charge density Q (the known structural charge) in a concentric spherical cavity with a uniform surface (zeta) potential ζ satisfying the continuities in electric potential and current at the particle surface is obtained as 4 ACS Paragon Plus Environment

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3

2

2

3

ψ ( eq ) = ψ eq01 (r )Q + ψ eq10 (r )ζ + O(Q , Q ζ , Qζ , ζ ) , ψ eq01 =

(9)

kT {κ r − csch(κ b)[κ a cosh(κ b − κ a) + sinh(κ b − κ a)]sinh(κ r )} if 0 ≤ r ≤ a , (10a) Zeκ r (κ a) 2

kT csch(κ b)[κ a cosh(κ a) − sinh(κ a)]sinh(κ b − κ r ) Zeκ r (κ a) 2 kTb = csch(κb) sinh(κr ) , Zer

ψ eq01 = ψ eq10

if a ≤ r ≤ b , (10b) (11)

where κ −1 = (εkT / 2Z 2 e 2 n0∞ )1 2 is the Debye screening length, and both Q = Zea 2Q / εkT and 2

2

ζ = Zeζ / kT are dimensionless. The second-order contributions O(Q , Qζ , ζ ) to ψ (eq ) vanish only for the case of symmetric electrolytes. Substituting eqs 9-11 into the Gauss condition at r = b , we obtain the following relation between the zeta potential and the surface charge density σ of the cavity wall confining the porous sphere:

ζ =

[κa cosh(κa) − sinh(κa)]Q + κ 2b sinh(κb)σ . εκ 2[κb cosh(κb) − sinh(κb)]

(12)

Namely, the solution of ψ (eq ) expressed by eq 9 after the substitution of eq 12 is also valid for the case of constant surface charge density at the cavity wall. Note that the surface potential

ψ (eq ) (a) of the porous sphere is equivalent to the zeta potential of a dielectric hard sphere. 3.2. Perturbed Quantities. To solve the perturbed quantities δµ ± , δψ , vr , and vθ in terms of the particle velocity U when the parameters Q and ζ are small, these variables can be expressed as perturbation expansions in powers of Q and ζ , such as 2

2

U = U 01 Q + U10 ζ + U 02 Q + U 11 Qζ + U 20 ζ + ⋅ ⋅ ⋅ ,

(13)

where the coefficients U ij with i and j equal to 0, 1, and 2 to be determined are independent of Q and ζ

but are functions of the particle-to-cavity radius ratio a / b ,

electrokinetic particle radius κa , and hydrodynamic resistance parameter λa . In the expansions of δµ ± and δψ , there should be zeroth-order terms of Q and ζ . Substituting these expansions and eq 9 for ψ (eq) into eqs 3-5, 7, and 8, we obtain the following results for 2

δµ ± , δψ (to the first orders Q and ζ ), vr , and vθ (to the second orders Q , Qζ , and 2

ζ ): δµ± = kT (1 m β )α [ Fµ 00 m Fµ 01 Q m Fµ10 ζ ] cosθ , kT α [− β Fψ 00 + Fψ 01 Q + Fψ 10 ζ ] cos θ , Ze 2 kT kT kT vr = [(U 01F00 − 2 βα F01 )Q + (U10 F00 − 2 βα F10 )ζ + (U 02 F00 + 2 αF02 )Q ηa ηa ηa

δψ =


(14) (15)


+ (U11F00 +

2 kT kT αF11 )Qζ + (U 20 F00 + 2 αF20 )ζ ] cosθ , 2 ηa ηa

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(16a)

tan θ ∂ 2 ( r vr ) , (16b) 2r ∂r where Fµij , Fψij , and Fij are dimensionless functions of r deﬁned by eqs A1, A2, and A13-A15 in the Appendix. vθ = −

3.3. Particle Velocity.

The electrostatic force exerted on the charged porous sphere can

be expressed as

Fe = −2πQ ∫

π

∫

0

a

∇δψ r 2sinθ drdθ .

0

(17)

Substituting eq 15 into the previous equation, we obtain the force up to the second orders as 4πε kT 2 ( ) α Q[ β Fψ 00 − Fψ 01 Q − Fψ 10 ζ ]r =a e z , Fe = (18) 3 Ze where e z is the unit vector in the z direction. The hydrodynamic drag force acting on the porous sphere can be expressed as

Fh = 2πηλ2 ∫

π

0

∫

a

0

v r 2sinθ drdθ .

(19)

The substitution of eq 16 into eq 19 leads to this force to the second orders, 4π kT kT Fh = ηλ2 a 3 [(U 01 F00 − 2 βα F01 )Q + (U 10 F00 − 2 βα F10 )ζ ηa ηa 3 2 2 kT kT kT + (U 02 F00 + 2 αF02 )Q + (U 11 F00 + 2 αF11 )Qζ + (U 20 F00 + 2 αF20 )ζ ]r = a e z . ηa ηa ηa

(20)

The total force on the particle is zero at the quasisteady state. Applying this constraint to the summation of eqs 18 and 20, we obtain the coefficients for the diffusiophoretic velocity of the porous sphere in eq 13 as ( Ze)2 F01 (a) 1 Fψ 00 (a) − ], U 01 = βU *[ εakT F00 (a) (λa) 2 F00 (a)

(21a)

( Ze) 2 F10 (a) ; εakT F00 (a)

(21b)

( Ze) 2 F02 (a) 1 Fψ 01 (a) − ], εakT F00 (a) (λa) 2 F00 (a)

(22a)

U11 = −U *[

( Ze)2 F11 (a) 1 Fψ 10 (a) − ], εakT F00 (a) (λa)2 F00 (a)

(22b)

U 20 = −U *

( Ze) 2 F20 (a) , εakT F00 (a)

(22c)

U10 = βU *

U 02 = −U *[

where the characteristic particle velocity εα kT 2 U* = ( ) . ηa Ze

(23)

The expansion form of eq 13 with the substitution of eq 12 also applies if the surface charge 6 ACS Paragon Plus Environment

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density (instead of the zeta potential) of the cavity wall is used. In eq 13 for the diffusiophoretic velocity of the charged porous sphere inside the charged cavity, the

O(Q, ζ )

(first-order) and

2

2

O(Q , Qζ , ζ )

(second-order) terms represent

contributions from electrophoresis (due to the induced electric field given by eq 8b) and chemiphoresis, respectively. The charge on the cavity wall modifies the motion of the particle via the wall-corrected electric potential distribution and the diffusioosmotic (electroosmotic and chemiosmotic) recirculation flow produced by the interaction of the electric double layer adjacent to the wall with the imposed electrolyte concentration gradient (significant at large

κb ). Note that U 01Q + U 02 Q

2

and U10 ζ + U 20 ζ

2

can be taken as the diffusiophoretic

(electrophoretic and chemiphoretic) velocity of a charged porous sphere in an uncharged cavity ( ζ = 0 ) and the migration velocity of an uncharged porous sphere ( Q = 0 ) in a charged cavity induced by diffusioosmosis (electroosmosis and chemiosmosis), respectively, and the coupled term U 11 Qζ vanishes in both cases. Also, eq 21 is consistent with the corresponding result for the electrophoretic motion of a charged soft sphere inside a charged spherical cavity.32

4.

RESULTS AND DISCUSSION The diffusiophoretic velocity of a charged porous sphere inside a concentric charged

cavity has been expressed by eqs 13 and 21-23 in the previous section, and its details will be presented in this section. Typical contours of constant ionic electrochemical potential energy and electric potential as well as streamlines around the confined porous sphere can be readily plotted using eqs 14-16, and these contours are similar to those in the context of diffusiophoresis of a soft sphere at an arbitrary position inside an uncharged spherical cavity.31

4.1. The Velocity Coefficients U 01 and U10 for Electrophoresis The normalized velocities U 01 / βU * and U10 / βU * for the electrophoretic motion of a charged porous sphere inside a concentric charged cavity determined from eq 21 are functions of the particle-to-cavity radius ratio a / b , electrokinetic particle radius κa , and hydrodynamic resistance parameter

λa only. For very thin electric double layers ( κa → ∞ ), eq 21 becomes U 01 1 = , * βU (λ a ) 2 U10 1 = A{λa[10ξ 2 (1 − ξ 3 ) − (λa) 2 (3 − 5ξ 2 + 2ξ 5 ) ] cosh λa * βU 3

− [10ξ 2 (1 − ξ 3 ) − (λa) 2 (1 − 5ξ 2 + 4ξ 5 )] sinh λa} , where A = {λa[15ξ 5 − (λa) 2 (1 − ξ 5 )] cosh λa − [15ξ 5 − (λa) 2 (1 − 6ξ 5 )] sinh λa}−1 , 7 ACS Paragon Plus Environment

(24a)

(24b) (25)


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and ξ = a / b . For very thick double layers ( κb → 0 ), eq 21 becomes U 01 A = {λa[270ξ 5 − 6(λa) 2 (1 + 10ξ 3 − 21ξ 5 + 10ξ 6 ) − (λa) 4 (1 − ξ ) 4 (4 + 7ξ + 4ξ 2 )] cosh λa * βU 18(λa) 2

− 3[90ξ 5 − 2(λa ) 2 (1 + 10ξ 3 − 36ξ 5 + 10ξ 6 ) + (λa) 4 ξ (1 − ξ ) 3 (3 + 9ξ + 8ξ 2 )] sinh λa} ,

(26a)

U10 A(κ a)4 (1 − ξ )2 = {−λ a[420(3ξ 3 + 6ξ 2 + 4ξ + 2) + 5(λ a)2 (33ξ 3 + 66ξ 2 + 29ξ − 8 βU * 1260(λ a)2 −24ξ −1 −12ξ −2 ) + (1 − ξ )2 (1 + ξ )(λ a)4 (3 + 9ξ −1 + 11ξ −2 + 9ξ −3 + 3ξ −4 )]cosh λ a

+[420(3ξ 3 + 6ξ 2 + 4ξ + 2) + 15(λ a)2 (39ξ 3 + 78ξ 2 + 47ξ + 16 − 8ξ −1 − 4ξ −2 ) −(1 − ξ )(λ a) 4 (30ξ 2 + 90ξ + 100 + 60ξ −1 + 12ξ −2 − 9ξ −3 − 3ξ −4 )]sinh λ a} .

(26b)

As a / b = 0 (the cavity wall is very far from the porous sphere), eq 21 reduces to

U 01 1 1 1 − e−2κa − 2κa = + + − ( 1 e ) βU * (λa) 2 3(κa) 2 κa

λ 2 κa(1 + e −2κa ) − 1 + e −2κa 1 1 1 + (1 + ) [( ) − 1 + e − 2κa ] , 3 κa (λa) 2 − (κa) 2 κ λa coth λa − 1 U10 = 1. βU *

(27a) (27b)

As a / b = 1 (the particle fills the cavity up entirely), eq 21 leads to U 01 1 = , * βU (λ a ) 2 U10 = 0. βU *

(28a) (28b)

Interestingly, U 01 / βU * in this limit can be finite, in contrast to zero for a confined charged hard sphere.30,32 This singular outcome is understood knowing that the ionic fluid in the permeable and slip porous particle is still driven to flow by the induced electric field shown in eq 8b as long as λa is finite, and thus the particle must migrate according to its force balance. The dimensionless velocities U 01 / βU * and U10 / βU * as calculated from eq 21 are plotted versus the parameters κa , λa , and a / b in Figures 2 and 3, respectively. These dimensionless velocities are always positive (and in the same order of magnitude); thus, the signs of the products β Q and βζ

determine the directions of electrophoretic and

wall-induced electroosmotic effects, respectively, on the particle. For specified values of λa and a / b , the value of U 01 / βU * decreases and the value of U10 / βU * increases with an increase in κa (a decrease in the overlap of electric double layers) with limits as predicted by eqs 24 and 26. For fixed values of κa and a / b , both dimensionless velocities decrease monotonically with an increase in λa as expected; interestingly, U 01 / βU * depends strongly on λa and equals (λa) −2 as λa is small or κa is large, while U10 / βU * is not a sensitive function of λa . For given values of κa and λa , both values of U 01 / βU * and U10 / βU * decrease with an increase in a / b as expected, with limits as predicted by eqs 27 and 28. 8 ACS Paragon Plus Environment

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When the double layers are relatively thick ( κa is small), the boundary effect on these dimensionless velocities is substantial. To determine the first-order migration velocity U 1 = U 01 Q + U 10 ζ of a charged porous sphere undergoing electrophoresis in a concentric charged cavity, not only the parameters κa ,

λa , and a / b , but also the particle charge density Q and cavity zeta potential ζ have to be specified. In Figure 4, the normalized particle velocity U1 / QβU * is plotted versus the ratio

ζ / Q for various values of κa , λa , and a / b (in straight lines with the slope U10 / βU * ). The electroosmotic effect of the cavity wall enhances/reduces the electrophoretic velocity of the porous sphere if their fixed charges are in the same/opposite signs. This effect on the particle migration can be significant if the magnitude of ζ / Q is large. When the value of

ζ / Q is negative with magnitude greater than the order unity, the confined particle may reverse the direction of its velocity from that of electrophoresis in an unbounded fluid. In general, the magnitude of U1 / QβU * decreases with an increase in a / b , increases with an increase in κa , and decreases with an increase in λa .

4.2. The Velocity Coefficients U 02 , U 11 , and U 20 for Chemiphoresis. The * dimensionless velocities U 02 /U * , U11 /U , and U20 /U * for the chemiphoresis of a charged

particle in a charged cavity calculated from eq 22 are also functions of the electrokinetic radius

κa , hydrodynamic resistance parameter λa , and radius ratio a / b only. In the limit κa → ∞ , eq 22 leads to U 02 / U * = U11 / U * = 0 (but finite U 20 /U * ). In the limit κa = 0 , eq 22 becomes U 02 / U * = U11 / U * = U 20 / U * = 0 . On the other hand, as a / b = 0 , eq 22 gives U11 / U * = 0 and U 20 / U * = 1 / 8 . As a / b = 1 , eq 22 becomes U 02 / U * = U11 / U * = U 20 / U * = 0 as expected. The dimensionless velocities U 02 /U * , U11 /U * , and U20 /U * as calculated from eq 22 are plotted versus the parameters κa , λa , and a / b in Figures 5, 6, and 7, respectively. The * values of U11 /U and U20 /U * are always positive, but U02 /U * can be either positive or

negative. Although all of these second-order velocities decrease monotonically with an increase in λa , interestingly, none of them is a monotonic function of either κa or a / b , and local maxima (which are positive) and minima (which are negative and only existent for U02 /U * ) may appear at moderate values of κa and a / b . As expected, all of them vanish in the limits * * * κa = 0 and a / b = 1 , both U 02 /U and U11 /U vanish in the limit κa → ∞ , and U11 /U

vanishes and U 20 / U * = 1 / 8 at a / b = 0 . For fixed values of κa and a / b , both U02 /U * and U11 /U * depend strongly on λa and are proportional to (λa) −2 as λa is small, while

U 20 /U * is not a sensitive function of λa . 2

The second-order migration velocity U 2 = U 02 Q + U11 Qζ + U 20 ζ

2

of a charged porous

sphere undergoing chemiphoresis in a concentric charged cavity depends on the particle charge



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density Q and cavity zeta potential ζ as well as the parameters a / b , κa , and λa . In 2

Figure 8, the normalized particle velocity U 2 / Q U * is plotted versus the ratio ζ / Q for different values of a / b , κa , and λa in parabolic curves which concave upward ( U20 /U * is always positive). When ζ / Q varies, this velocity may reverse twice in its direction due to the combination of the chemiphoretic and wall-induced chemiosmotic effects for specified values of a / b , κa , and λa . Again, the effect of the cavity wall on the migration of the porous 2

particle is significant if the magnitude of ζ / Q is large, and the magnitude of U 2 / Q U * in general decreases with an increase in a / b , increases with an increase in κa , and decreases with an increase in λa .

4.3. Diffusiophoretic Velocity. For the diffusiophoresis of a charged porous sphere in a liquid solution of a symmetric electrolyte solution with equal anion and cation diffusivities (

β = 0 , such as the KCl aqueous solution) at the center of a charged spherical cavity, Figure 8 * has given the results of the dimensionless particle velocity U / U * (identical to U2 /U with

the contributions of chemiphoresis and chemiosmosis only) as a function of the particle charge density Q and cavity zeta potential ζ as well as the parameters a / b , κa , and λa . In Figure 9, a plot of U / U * versus ζ with Q = 1 and various values of a / b , κa , and λa is presented for a symmetric electrolyte whose cation and anion are different in diffusion coefficient ( β = −0.2 , such as the NaCl aqueous solution), and both electrophoretic and chemiphoretic contributions (and wall-induced electroosmotic and chemiosmotic effects) exist (as a combination of Figures 4 and 8). Again, for fixed values of a / b , κa , and λa , the particle velocity may reverse its direction twice when ζ / Q varies, and the effect of the cavity wall is significant as the magnitude of ζ / Q is large.

5.

CONCLUDING REMARKS The diffusiophoretic motion of a charged porous sphere within a concentric charged

spherical cavity under an imposed symmetric electrolyte concentration gradient at the quasisteady state is analytically investigated for arbitrary values of the electrokinetic particle radius κa , hydrodynamic resistance parameter λa , and particle-to-cavity radius ratio a / b . Solving the relevant linearized governing equations by a regular perturbation method, we obtain the ionic electrochemical potential energy, electric potential, and velocity fields in the fluid in power expansions of the dimensionless particle charge density Q and cavity zeta potential ζ . The balance of the hydrodynamic and electrostatic forces exerted on the porous particle leads to a closed-form formula, eq 13 (together with eqs 21-23), for the particle velocity in terms of a / b , κa , and λa to the second orders of Q and ζ . The 10 ACS Paragon Plus Environment

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dimensionless diffusiophoretic velocity components U 01 / βU * , U 02 /U * (which can be either positive or negative), and U11 /U * depend strongly on λa and are proportional to (λa) −2 as

λa is small, while the dimensionless diffusioosmosis-induced velocity components U10 / βU * and U 20 /U * are not sensitive functions of λa . The diffusioosmotic flow occurring at the cavity wall can substantially alter the particle velocity and even reverse the direction of diffusiophoresis. In general, the particle velocity decreases with an increase in a / b , increases with an increase in κa , and decreases with an increase in λa . In this work, we only consider the exceptional case where the particle is at the center of the cavity. It would be needed in practice to know what happens when the particle is not at the center. Recently, a combined analytical and numerical solution has been obtained for the diffusiophoresis of a hard spherical particle surrounded by a thin but polarized particle-solute interaction layer located at an arbitrary position within a spherical cavity filled with a nonelectrolyte solution.33 The normalized diffusiophoretic velocity of the particle was found to decrease monotonically with an increase in the normalized distance between the particle and cavity centers (this decrease is more substantial when the particle is closer to the cavity wall) and vanish in the touching limit.

APPENDIX For conciseness some functions in eqs 14-16 are listed here. In eq 16, a F00 (r ) = C001 + C002γ (λr )( )3 if 0 ≤ r < a , r a a r F00 (r ) = C003 ( )3 + C004 + C005 + C006 ( ) 2 if a ≤ r ≤ b ; r r a a 2 a 3 ( 3) ( 0) Fij (r ) = Cij1 + Cij 2γ (λr )( )3 + [ I ( r ) − I ij (r )] ij r 3(λa ) 2 r3 2 + 5 2 3 [γ (λr ) I ijδ (r ) − δ (λr ) I ijγ (r )] if 0 ≤ r < a , λar a a r a 3 (5) a Fij (r ) = Cij 3 ( )3 + Cij 4 + Cij 5 + Cij 6 ( ) 2 − J (r ) + J ij(3) (r ) 3 ij r r a 15r 3r 1 ( 2) r 2 (0 ) − J ij (r ) + J ij (r ) if a ≤ r ≤ b , 3 15a 2

for ( i , j ) equal to (0,1), (1,0), (0,2), (1,1), and (2,0), where 6 C001 = {[ 45ξ 3 + (λa ) 2 (6ξ 3 − 5ξ − ξ −2 )]γ (λa ) − 15(λa ) 2 sinh( λa )ξ 3 } , H 6 C002 = − (λ a ) 2 (3ξ 3 − 5ξ + 2ξ −2 ) , H


(A1a) (A1b)

(A2a)

(A2b)

(A3a) (A3b)


Page 12 of 31

1 C003 = − ξ −2 (C 004 − 2ξ −3C006 ) , 3 1 C004 = − (λ a ) 2 [C 001 + γ (λ a )C002 ] , 3 1 C005 = C 001 + (λa ) 3 cosh( λa )C002 − 5C 006 , 3 3 C006 = (λa ) 2 ξ 3{2(λa ) 2 sinh( λa ) − [6 + (λa ) 2 (1 − ξ −2 )]γ (λ a )} ; H − 2 ( 0) C C Cij1 = I (a) + 001 J ij( 2 ) (a) − [ 0032 + 2γ (λa)Ω1 ]J ij(0 ) (a) − Ω 2 I ijγ (a) 2 ij 3(λa) 3 (λ a ) 1 − Ω 3 J ij(3) (a) − Ω 9 J ij(5) (a) + {λa cosh(λa)[180ξ 3 − L3 ] − 180ξ 4γ (λa) 3H + 3L4 sinh( λa )}I ij(3) ( a ) ,

−2 δ C I (a) + 002 J ij( 2) (a) + 2Ω1 J ij( 0) (a) − Ω 2 I ij(3) (a) − Ω 4 J ij(3) (a) − Ω5 J ij(5) (a) 5 ij (λ a ) 3 2 + [(λa) L1 sinh(λa) − 3L2 cosh(λa)]I ijγ (a) , (λ a ) 5 H C C = 003 J ij( 2 ) (a ) − [ 0032 + 2γ (λa)Ω1 ]I ij(3) (a ) + ξΩ 8 J ij(3) (a ) + 2Ω1 I ijγ (a ) 3 (λ a )

(A3c) (A3d) (A3e) (A3f)

(A4a)

Cij 2 =

Cij 3

ξ −2

(λ a ) 2 − 2 ξ {[6(5ξ 3 + 1) 5 15H + (λa ) 2 (5ξ 3 − 9ξ + 4)]γ (λa ) − 2(λa ) 2 sinh( λa )(5ξ 3 − 2)}J ij(5) ( a ) , −

(A4b)

{2Ω8 + C003 }J ij( 0) (a) +

1 Cij 4 = [C 001 I ij(3) (a) + C002 I ijγ (a) + C005 J ij(3) (a) + C004 J ij( 2 ) (a) 3 + C 003 J ij(0 ) ( a ) + C 006 J ij(5) ( a )] , Cij 5 = −Ω I

( 3) 3 ij

γ

(a ) − Ω 4 I ij ( a ) + Ω 6 J

( 5) ij

( a ) − ξΩ 7 J

( 3) ij

(A4c)

(A4d)

( a ) + ξΩ J

(0) 8 ij

(a)

( λa ) [Ω3 + γ (λa)Ω 4 ]J ij( 2 ) (a) , 3 1 2(λa) 2 = {C005 + [{−3(10ξ 4 − ξ −2 ) − 2(λa) 2 (ξ 4 − ξ −2 )}γ (λa) 15 H C + 2(λa) 2 (5ξ 4 + ξ −2 ) sinh(λa)]}J ij( 0) (a) + 006 J ij( 2) (a) − Ω5 I ijγ (a) 3 2 3 ( a ) λ ξ + Ω6 J ij(3) (a) − Ω9 I ij(3) (a) − {[−3 + 2(λa)2 (ξ − 1)]γ (λa) 5H − 2(λa ) 2 sinh( λa )}J ij(5) ( a ) . 2

+

Cij 6

In the above equations, ξ = a / b , γ ( x) = x cosh x − sinh x , δ ( x) = x sinh x − cosh x ; H = λa cosh(λa) L1 − 3 sinh(λa) L2 ; −2

L2 = 2(λa) (10ξ − 36ξ + 10ξ + ξ ) + (λa) L4 − 90ξ , 4

3

(A4f) (A5a) (A5b) (A6)

L1 = 6(λa) 2 (10ξ 4 − 21ξ 3 + 10ξ + ξ −2 ) + (λa) 2 L3 − 270ξ 3 , 2

(A4e)

2

3


(A7a) (A7b)

Page 13 of 31 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


L3 = (λa) 2 (4ξ 4 − 9ξ 3 + 10ξ − 9ξ −1 + 4ξ −2 ) ,

(A8a)

L4 = (λa) 2 (8ξ 4 − 15ξ 3 + 10ξ − 3ξ −1 ) ;

(A8b)

Ω1 = Ω2 = Ω3 = Ω4 Ω5 Ω6 Ω7

Ω8

ξ

−2

H

[6 + (λa) 2 (ξ 3 − 3ξ + 2)] ,

(A9a)

2 [90ξ 3 − (λa) 2 (20ξ 4 − 27ξ 3 + 5ξ + 2ξ −2 )] , 2 (λ a ) H

ξ

{[60ξ 3 + (λa) 2 (8ξ 3 − 5 − 3ξ −2 )]γ (λa) − 10(λa) 2 sinh(λa)(2ξ 3 + 1)} ,

H 2 = ξ [15 + (λa) 2 (−2ξ 3 + 5 − 3ξ −2 )] , H 2 = ξ 3 [(λa) 2 (2ξ − 3 + ξ −2 ) − 9] , H (λ a ) 2 =− ξ {[3 − 12ξ 3 − 2(λa) 2 (ξ 3 − 1)]γ (λa) + 2(λa) 2 sinh(λa)(2ξ 3 + 1)} , 3H 1 = {[60ξ 3 + (λa) 2 (28ξ 3 − 3ξ − 2 ) + 2(λa) 4 (ξ 3 − ξ −2 )]γ (λa) H − 2[10ξ 3 + (λa) 2 (4ξ 3 + ξ −2 )](λa)2 sinh(λa)} , 1 = − {[30 + 2(λa) 2 (7 − 3ξ −2 ) + (λa) 4 (1 − ξ −2 )]γ (λa) H − [10 + 2(λa) 2 (2 − ξ −2 )](λa) 2 sinh(λa)} ,

Ω9 =

ξ H

(λa) 2 [6ξ 2 sinh(λa) − (4ξ 3 − 3ξ 2 − 1)γ (λa)] ;

r r I ij( n ) (r ) = ∫ ( ) n Gij (r )dr , 0 a

(A9b) (A9c) (A9d) (A9e) (A9f)

(A9g)

(A9h) (A9i) (A10a)

r

I ijγ (r ) = ∫ γ (λr )Gij (r )dr ,

(A10b)

I ijδ (r ) = ∫ δ (λr )Gij (r )dr ,

(A10c)

0 r 0

b r J ij( n ) (r ) = ∫ ( ) n Gij (r )dr ; r a dψ εκ 2 a 4 G01 (r ) = Fµ 00 (r ) eq01 , Zer dr 2 4 dψ εκ a G10 (r ) = Fµ 00 (r ) eq10 , Zer dr 2 4 dψ εκ a W01 (r ) eq01 , G02 (r ) = − Zer dr 2 4 dψ dψ εκ a [W01 (r ) eq10 + W10 (r ) eq 01 ] , G11 (r ) = − Zer dr dr 2 4 d ψ εκ a eq10 G20 (r ) = − W10 (r ) , Zer dr Ze Wij (r ) = [ Fµij (r ) + ψ eqij (r ) Fµ 00 (r )] . kT

In eqs 14, 15, and A12, 13 ACS Paragon Plus Environment

(A10d) (A11a) (A11b) (A11c) (A11d) (A11e) (A12)


Fµ 00 (r ) = Fψ 00 (r ) =

r , a

Page 14 of 31

(A13)

dψ eqij b dψ eqij b r Zer a 3 r r 3 dψ eqij [( ) ∫ ( ) dr + ∫ dr − ∫ ( ) 3 dr ] , 0 r 0 3kTa r a dr dr b dr r b 1 Fψij (r ) = − 2 {(κr + 1)e −κr ∫ γ (κr )Wij (r ) dr + γ (κr ) ∫ (κr + 1)e −κrWij (r )dr 0 r κr − κb b γ (κr ) (κb + 1)e − γ (κr )Wij (r ) dr} , ∫ 0 γ (κb)

Fµij (r ) =

(A14)

(A15)

for ( i , j ) equal to (0,1) and (1,0). ACKNOWLEDGMENT 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.

REFERENCES

(1) Dukhin, S. S.; Derjaguin, B. V. In Surface and Colloid Science; Matijevic, E., Ed.; Wiley: New York, 1974; Vol. 7. (2) Prieve, D. C.; Anderson, J. L.; Ebel, J. P.; Lowell, M. E. Motion of a Particle Generated by Chemical Gradients. Part 2. Electrolytes. J. Fluid Mech. 1984, 148, 247-269. (3) Khair, A. S. Diffusiophoresis of Colloidal Particles in Neutral Solute Gradients at Finite Péclet Number. J. Fluid Mech. 2013, 731, 64-94. (4) Keh, H. J. Diffusiophoresis of Charged Particles and Diffusioosmosis of Electrolyte Solutions. Curr. Opin. Colloid Interface Sci. 2016, 24, 13-22. (5) Anderson, J. L. Colloid Transport by Interfacial Forces. Annu. Rev. Fluid Mech. 1989, 21, 61-99. (6) Abecassis, B.; Cottin-Bizonne, C.; Ybert, C.; Ajdari, A.; Bocquet, L. Osmotic Manipulation of Particles for Microfluidic Applications. New J. Phys. 2009, 11, 075022. (7) Smith, R. E.; Prieve, D. C. Accelerated Deposition of Latex Particles onto a Rapidly Dissolving Steel Surface. Chem. Eng. Sci. 1982, 37, 1213-1223. (8) Wanunu, M.; Morrison, W.; Rabin, Y.; Grosberg, A. Y.; Meller, A. Electrostatic Focusing of Unlabeled DNA into Nanoscale Pores Using a Salt Gradient. Nature Nanotechnol. 2010, 5, 160-165. (9) Hatlo, M. M.; Panja, D.; van Roij, R. Translocation of DNA Molecules through Nanopores with Salt Gradients: The Role of Osmotic Flow. Phys. Rev. Lett. 2011, 107, 68101. (10) Sen, A.; Ibele, M.; Hong, Y.; Velegol, D. Chemo and Phototactic Nano/Microbots. Faraday Discuss. 2009, 143, 15-27. (11) Brown, A.; Poon, W. Ionic Effects in Self-Propelled Pt-Coated Janus Swimmers. Soft Matter 2014, 10, 4016-4027. 14 ACS Paragon Plus Environment

Page 15 of 31 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(12) Oshanin, G.; Popescu, M. N.; Dietrich, S. Active Colloids in the Context of Chemical Kinetics. J. Phys. A: Math. Theor. 2017, 50, 134001. (13) Velegol, D.; Garg, A.; Guha, R.; Kar, A.; Kumar, M. Origins of Concentration Gradients for Diffusiophoresis. Soft Matter 2016, 12, 4686-4703. (14) Pawar, Y.; Solomentsev, Y. E.; Anderson, J. L. Polarization Effects on Diffusiophoresis in Electrolyte Gradients. J. Colloid Interface Sci. 1993, 155, 488-498. (15) Tu, H. J.; Keh, H. J. Particle Interactions in Diffusiophoresis and Electrophoresis of Colloidal Spheres with Thin but Polarized Double Layers. J. Colloid Interface Sci. 2000, 231, 265-282. (16) Prieve, D. C.; Roman, R. Diffusiophoresis of a Rigid Sphere through a Viscous Electrolyte Solution. J. Chem. Soc., Faraday Trans. 2 1987, 83, 1287-1306. (17) Keh, H. J.; Wei, Y. K. Diffusiophoretic Mobility of Spherical Particles at Low Potential and Arbitrary Double-Layer Thickness. Langmuir 2000, 16, 5289-5294. (18) Wei, Y. K.; Keh, H. J. Diffusiophoretic Mobility of Charged Porous Spheres in Electrolyte Gradients. J. Colloid Interface Sci. 2004, 269, 240-250. (19) Huang, H. Y.; Keh, H. J. Diffusiophoresis in Suspensions of Charged Porous Particles. J. Phys. Chem. B 2015, 119, 2040-2050. (20) Huang, P. Y.; Keh, H. J. Diffusiophoresis of a Spherical Soft Particle in Electrolyte Gradients. J. Phys. Chem. B 2012, 116, 7575-7589. (21) Joo, S. W.; Lee, S. Y.; Liu, J.; Qian, S. Diffusiophoresis of an Elongated Cylindrical Nanoparticle Along the Axis of a Nanopore. ChemPhysChem 2010, 11, 3281-3290. (22) Lee, S. Y.; Yalcin, S. E.; Joo, S. W.; Sharma, A.; Baysal, O.; Qian, S. The Effect of Axial Concentration Gradient on Electrophoretic Motion of a Charged Spherical Particle in a Nanopore. Microgravity Sci. Technol. 2010, 22, 329-338. (23) Lee, T. C.; Keh, H. J. Electrophoresis of a Spherical Particle in a Spherical Cavity. Microfluid. Nanofluid. 2014, 16, 1107-1115 (24) Chen, P. Y.; Keh, H. J. Diffusiophoresis and Electrophoresis of a Charged Sphere Parallel to One or Two Plane Walls. J. Colloid Interface Sci. 2005, 286, 774-791. (25) Chang, Y. C.; Keh, H. J. Diffusiophoresis and Electrophoresis of a Charged Sphere Perpendicular to One or Two Plane Walls. J. Colloid Interface Sci. 2008, 322, 634-653. (26) Chiu, H. C.; Keh, H. J. Diffusiophoresis of a Charged Particle in a Microtube. Electrophoresis 2017, 38, 2468-2478. (27) Kar, A.; Chiang, T.-S.; Rivera, I. O.; Sen, A.; Velegol, D. Enhanced Transport into and out of Dead-End Pores. ACS Nano 2015; 9, 746-753. (28) Shin, S.; Um, E.; Sabass. B.; Ault, J. T.; Rahimi, M.; Warren, P. B.; Stone, H. A. Size-Dependent Control of Colloid Transport via Solute Gradients in Dead-End Channels. PNAS 2016, 113, 257-261. (29) Chiu, H. C.; Keh, H. J. Electrophoresis and Diffusiophoresis of a Colloidal Sphere with 15 ACS Paragon Plus Environment


Double-Layer Polarization in a Concentric Charged Cavity. Microfluid. Nanofluid. 2017, 21, 45. (30) Chiu, Y. C.; Keh, H. J. Diffusiophoresis of a Charged Particle in a Charged Cavity with Arbitrary Electric-Double-Layer Thickness. Microfluid. Nanofluid. 2018, 22, 84. (31) Zhang, X.; Hsu, W.-L.; Hsu, J.-P.; Tseng, S. Diffusiophoresis of a Soft Spherical Particle in a Spherical Cavity. J. Phys. Chem. B 2009, 113, 8646-8656. (32) Chen, W. J.; Keh, H. J. Electrophoresis of a Charged Soft Particle in a Charged Cavity with Arbitrary Double-Layer Thickness. J. Phys. Chem. B 2013, 117, 9757-9767. (33) Lee, T. C.; Keh, H. J. Diffusiophoresis of a Colloidal Sphere in Nonelectrolyte Gradients within a Spherical Cavity. Am. J. Heat Mass Transfer 2014, 1, 130-148. Figure Captions Figure 1. Geometric sketch for the diffusiophoresis of a charged porous sphere at the center of

a charged spherical cavity. Figure 2. The dimensionless velocity U 01 / βU * for the electrophoresis of a charged porous

sphere in a spherical cavity calculated from Eq. (21a): (a) κa = 1 ; (b) λa = 10 ; (c) a / b = 0.5 . Figure 3. The dimensionless velocity U10 / βU * for a porous sphere in a charged spherical

cavity with electroosmosis calculated from Eq. (21b): (a) κa = 1 ; (b) λa = 10 ; (c) a / b = 0.5 . Figure 4. The normalized velocity U1 / QβU * for the electrophoresis of a charged porous

sphere in a charged spherical cavity versus the ratio ζ / Q : (a) κa = 1 and λa = 10 ; (b)

λa = 10 and a / b = 0.5 ; (c) κa = 1 and a / b = 0.5 . Figure 5. The dimensionless velocity U 02 /U * for the chemiphoresis of a charged porous

sphere in a spherical cavity calculated from Eq. (22a): (a) κa = 1 ; (b) λa = 10 ; (c) a / b = 0.5 . Figure 6. The dimensionless velocity U11 /U * for a charged porous sphere in a charged

spherical cavity calculated from Eq. (22b): (a) κa = 1 ; (b) λa = 10 ; (c) a / b = 0.5 . Figure 7. The dimensionless velocity U 20 /U * for a porous sphere in a charged spherical

cavity with chemiosmosis calculated from Eq. (22c): (a) κa = 1 ; (b) λa = 10 ; (c) a / b = 0.5 . 2

Figure 8. The normalized velocity U 2 / Q U * for the chemiphoresis of a charged porous

sphere in a charged spherical cavity versus the ratio ζ / Q : (a) κa = 1 and λa = 10 ; (b)

λa = 10 and a / b = 0.5 ; (c) κa = 1 and a / b = 0.5 . Figure 9. The normalized velocity U /U * for the diffusiophoresis of a charged porous sphere


Page 16 of 31

Page 17 of 31 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


in a charged spherical cavity with β = −0.2 and Q = 1 versus the zeta potential ζ : (a)

κa = 1 and λa = 10 ; (b) λa = 10 and a / b = 0.5 ; (c) κa = 1 and a / b = 0.5 .



Page 18 of 31

Figure 1

z

θ

Ue z

r

a

b


∇n∞

Page 19 of 31 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 2a

0.40

0.35

λa = 2 0.30

U 01 βU *

0.25

3

0.20

0.15

5 0.10

10

0.05

∞

0.00 0.0

0.2

0.4

0.6

0.8

1.0

a/b Figure 2b

0.24

a/b = 0 0.20

0.1 0.16

U 01 βU * 0.12

0.3

0.08

0.5 0.04

1 0.00 0.01

0.1

1

10

κa


100


Page 20 of 31

Figure 2c

100

10

1

U 01 βU * 0.1

κa = 0

2

0.01

5

∞

10

1E-3 0.1

1

10

100

λa

Figure 3a

1

0.1

λa = 0

0.01

U 10 βU *

10

1E-3

∞ 1E-4

1E-5 0.0

0.2

0.4

0.6

0.8

a/b


1.0

Page 21 of 31 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 3b

10

1

a/b = 0 0.1

U 10 βU *

1

0.01

0.1

0.3

0.6

0.9

1E-3

1E-4 0.01

0.1

1

10

100

κa

Figure 3c

10

∞

1

10 0.1

U 10 βU *

0.01

1 1E-3

0.5 1E-4

1E-5

κa = 0.1 1E-6 0.1

1

10

100

λa


1000


Page 22 of 31

Figure 4a

0.2

a/b = 0 0.3 0.4

0.1

0.5

U1 Qβ U *

1

1 0.0

-0.1 -4

-3

-2

-1

0

1

2

0

1

2

ζ /Q

Figure 4b

1.0

0.5

κa = 0 0.0

U1 QβU *

3 -0.5

5 10

-1.0

∞ -1.5 -4

-3

-2

-1

ζ /Q


Page 23 of 31 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 4c

1.5

λa = 1

1.0

U1 Qβ U *

1

0.5

2

5 0.0

∞

-0.5 -4

-3

-2

-1

0

1

2

ζ /Q

Figure 5a

0.004

λa = 2 0.003

U 02 U* 0.002

3

0.001

5 10

∞

0.000 0.0

0.2

0.4

0.6

0.8

a/b


1.0


Page 24 of 31

Figure 5b

0.0004

0.3 0.0003

a/b = 0

0.0002

U 02 U*

0.5 0.0001

0.7 0.0000

1 -0.0001 0.01

0.1

1

10

100

κa

Figure 5c

1

0.1

κa = 2

0.01

U 02 U*

0.1 10

1E-3

100 1E-4

0.02 5

1

1E-5 0.1

1

10

λa


100

Page 25 of 31 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 6a

0.12

λa = 2 0.10

0.08

U11 U*

3 0.06

5

0.04

10 0.02

∞

0.00 0.0

0.2

0.4

0.6

0.8

1.0

a/b

Figure 6b

0.025

a/b = 0.3

0.020

0.015

U11 U* 0.010

0.5 0.005

0.7 0.000

0 0.01

0.1

1

1

10

κa


100


Page 26 of 31

Figure 6c

10

1

U11 U*

κa = 2

0.1

5 1

10 100 0.01

0.1 0.02 1E-3 0.1

1

10

100

λa

Figure 7a

1

0.1

λa = 0

0.01

U 20 U* 10

1E-3

∞ 1E-4

1E-5 0.0

0.2

0.4

0.6

0.8

a/b


1.0

Page 27 of 31 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 7b

1

0.1

a/b = 0

0.01

U 20 U*

1

1E-3

0.3

0.1

0.6

0.9

1E-4

1E-5 0.01

0.1

1

10

100

κa

Figure 7c

1

κa = 1000

0.1

U 20 U*

0.01

1

1E-3

0.5

1E-4

0.2 1E-5

0.1 1E-6 0.1

1

10

100

λa


1000


Page 28 of 31

Figure 8a

0.8 0.7

0.3

0.6

a/b = 0

0.5

U2 2

QU

0.4

1

*

0.3

0.4 0.2 0.1

0.5

0.0

1 -4

-3

-2

-1

0

1

2

ζ /Q

Figure 8b

1.5

10 1.2

5 1000

0.9

U2

3

2

Q U*

0.6

0.3

1 0.0

κa = 0 -4

-3

-2

-1

0

1

ζ /Q


2

Page 29 of 31 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 8c

0.2

∞

0.1

5 3

U2

1

0.0

2

Q U* 2 -0.1

λa = 1 -0.2 -4

-3

-2

-1

0

1

2

ζ /Q Figure 9a

0.20

0.15

0

0.3

0.10

U

0.4

0.05

1

U*

0.5 0.00

a/b = 1 -0.05

-0.10

-2

-1

0

1

ζ


2


Page 30 of 31

Figure 9b

0.6

κa = 1000 0.5

0.4

5 10

0.3

U

3

U*

0.2

0.1

1 0.0

0 -0.1 -2

-1

0

1

2

ζ Figure 9c

0.10

λa = 1 0.05

∞ 0.00

U

5

1

U* -0.05

3

-0.10

2 -0.15 -2

-1

0

1

ζ


2

Page 31 of 31 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


TOC Graphic

ψ (eq ) = ψ eq01 (r )Q + ψ eq10 (r )ζ 3

2

2

3

+ O(Q , Q ζ , Qζ , ζ ) U = U 01 Q + U 10 ζ 2

+ U 02 Q + U 11 Qζ + U 20 ζ


2

Diffusiophoresis of a Charged Porous Particle in a Charged Cavity

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