# Diffusiophoresis of a spherical soft particle in electrolyte gradients.

**ABSTRACT** An analytical study of the diffusiophoresis (consisting of electrophoresis and chemiphoresis) of a charged soft particle (or composite particle) composed of a spherical rigid core and a surrounding porous shell in an electrolyte solution prescribed with a uniform concentration gradient is presented. In the solvent-permeable and ion-penetrable porous surface layer of the particle, idealized frictional segments with fixed charges are assumed to distribute at a constant density. The electrokinetic equations that govern the electric potential profile, ionic concentration distributions, and fluid flow field inside and outside the porous layer of the particle are linearized by assuming that the system is only slightly distorted from equilibrium. Using a regular perturbation method, these linearized equations are solved with the fixed charge densities on the rigid core surface and in the porous shell as the small perturbation parameters. An analytical expression for the diffusiophoretic mobility of the soft sphere in closed form is obtained from a balance between its electrostatic and hydrodynamic forces. This expression, which is correct to the second order of the fixed charge densities, is valid for arbitrary values of κa, λa, and r(0)/a, where κ is the reciprocal of the Debye screening length, λ is the reciprocal of the length characterizing the extent of flow penetration inside the porous layer, a is the radius of the soft sphere, and r(0) is the radius of the rigid core of the particle. It is shown that a soft particle bearing no net charge can undergo diffusiophoresis (electrophoresis and chemiphoresis), and the direction of its diffusiophoretic velocity is decided by the fixed charges in the porous surface layer of the particle. In the limiting cases of large and small values of r(0)/a, the analytical solution describing the diffusiophoretic mobility for a charged soft sphere reduces to that for a charged rigid sphere and for a charged porous sphere, respectively.

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Diffusiophoresis of a Spherical Soft Particle in Electrolyte Gradients

Ping Y. Huang and Huan J. Keh*

Department of Chemical Engineering, National Taiwan University, Taipei 10617, Taiwan, Republic of China

ABSTRACT: An analytical study of the diffusiophoresis

(consisting of electrophoresis and chemiphoresis) of a charged

soft particle (or composite particle) composed of a spherical rigid

core and a surrounding porous shell in an electrolyte solution

prescribed with a uniform concentration gradient is presented. In

the solvent-permeable and ion-penetrable porous surface layer of

the particle, idealized frictional segments with fixed charges are

assumed to distribute at a constant density. The electrokinetic

equations that govern the electric potential profile, ionic

concentration distributions, and fluid flow field inside and

outside the porous layer of the particle are linearized by assuming that the system is only slightly distorted from equilibrium.

Using a regular perturbation method, these linearized equations are solved with the fixed charge densities on the rigid core

surface and in the porous shell as the small perturbation parameters. An analytical expression for the diffusiophoretic mobility of

the soft sphere in closed form is obtained from a balance between its electrostatic and hydrodynamic forces. This expression,

which is correct to the second order of the fixed charge densities, is valid for arbitrary values of κa, λa, and r0/a, where κ is the

reciprocal of the Debye screening length, λ is the reciprocal of the length characterizing the extent of flow penetration inside the

porous layer, a is the radius of the soft sphere, and r0is the radius of the rigid core of the particle. It is shown that a soft particle

bearing no net charge can undergo diffusiophoresis (electrophoresis and chemiphoresis), and the direction of its diffusiophoretic

velocity is decided by the fixed charges in the porous surface layer of the particle. In the limiting cases of large and small values of

r0/a, the analytical solution describing the diffusiophoretic mobility for a charged soft sphere reduces to that for a charged rigid

sphere and for a charged porous sphere, respectively.

1. INTRODUCTION

When a colloidal particle is suspended in a fluid solution

possessing a solute concentration gradient that interacts with its

surface, it will move in the direction of increasing or decreasing

concentration. This motion is known as diffusiophoresis1−5and

has been demonstrated experimentally for both charged6−8and

uncharged9solutes. Being an efficient means to drive particles in

nonuniform solutions, diffusiophoresis is of considerable

importance in numerous practical applications, such as particle

characterization or separation,10latex paint coating processes,11

particle manipulation in microfluidic or lab-on-a chip de-

vices,12,13autonomous motions of micro/nanomotors,14−18and

DNA sequencing.19−22In a solution of nonionic solute, the

solute molecules interact with the particle through the van der

Waals and dipole forces. For charged particles in an electrolyte

solution, the particle-solute interaction is electrostatic in nature,

and its range is the Debye screening length κ−1. Particles with ζ

potentialsoforderkT/e(∼25mV;eisthechargeofaproton,kis

Boltzmann’s constant, and T is the absolute temperature) in

electrolyte gradients of order 0.1 M/mm will move by

diffusiophoresis at speeds of several micrometers per second.

Analytical studies on electrokinetic diffusiophoresis are

limited,incomparison withthose onelectrophoresis, andmostly

restricted to cases of thin electric double layer.1−4When the

double-layer distortion from equilibrium was taken as a small

perturbation, Prieve and Roman23obtained a numerical solution

for the diffusiophoresis of a dielectric sphere of radius a in

concentration gradientsof1:1electrolytes(KClorNaCl), which

was applicable to a broad range of the ζ potential, ζ, and

electrokinetic radius, κa, of the particle. On the other hand,

analytical formulas in closed forms were obtained for the

diffusiophoretic mobilities of a rigid sphere24and a porous

sphere25in symmetric electrolytes at low fixed charge densities

and arbitrary κa, and the results of the former case24are in good

agreement with the numerical calculations23for the entire range

of ζ up to 50 mV.

The surface of a colloidal particle is generally not hard and

smooth as assumed in many theoretical models. For example,

surface layers are purposely formed by adsorbing long-chain

polymers to make the suspended particles stable against

flocculation.26Even the surfaces of model colloids such as silica

and polystyrene latex are “hairy” with a gel-like polymeric layer

extending a substantial distance into the suspending medium

from the bulk material inside the particle.27,28In particular, the

surfaceofabiologicalcellisnotahardsmoothwall,butratherisa

permeable rough surface with various appendages ranging from

protein molecules on the order of nanometers to cilia on the

order of micrometers.29Such particles can be modeled as a soft

particle(orcompositeparticle) havingacentralrigidcoreandan

outer porous shell.30−33When the rigid core vanishes, the

Received:

Revised:

Published: May 31, 2012

March 25, 2012

May 19, 2012

Article

pubs.acs.org/JPCB

© 2012 American Chemical Society

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particle reduces to a permeable one, such as polymer coils or

colloidalflocs.Althoughgeneralexpressionswerederivedforthe

electrophoretic mobility of a spherical soft particle,34−36the

effects of fixed charges on the diffusiophoretic velocity of soft

particles have not yet been analyzed.

In this article, the diffusiophoretic motion of a charged soft

sphere in an unbounded solution of a symmetrically charged

electrolyte with a constant imposed concentration gradient is

analyzed. The densities of the fixed charges and the hydro-

dynamic frictional segments are assumed to be uniform

throughout the porous surface layer of the soft particle, but no

assumption is made for the thickness of the electric double layer

and of the porous layer relative to the dimension of the particle.

The basic electrokinetic equations are linearized assuming that

theelectrolyteionconcentrations,theelectrostaticpotential,and

the fluid pressure have only slight deviations from equilibrium

due to the application of the electrolyte concentration gradient.

Through the use of a regular perturbation method with the fixed

charge densities on the rigid core and inside the porous shell of

the soft particle as the small perturbation parameters, the ion

concentration (or electrochemical potential energy), electro-

static potential, fluid velocity, and dynamic pressure profiles are

determined by solving these linearized electrokinetic equations

subject to the appropriate boundary conditions. A closed-form

expression for the diffusiophoretic velocity of the charged soft

sphere is obtained in eq 23 from a balance between its

electrostatic and hydrodynamic forces.

2. ELECTROKINETIC EQUATIONS

We consider the steady diffusiophoresis of a charged soft sphere

ofradiusainasolutionofasymmetricallychargedelectrolyte.As

illustrated in Figure 1, the soft particle has a porous surface layer

of constant thickness d so that the radius of the rigid core is r0=

a−d. The applied electrolyte concentration gradient ▽n∞is a

constant equal to |▽n∞|ez, and the diffusiophoretic velocity of

theparticle tobedetermined isUez,whereezistheunitvectorin

the z direction. The origin of the spherical coordinate system

(r,θ,ϕ) is taken at the center of the particle, and the polar axis

θ = 0 points toward the positive z direction. Evidently, the

problem is axially symmetric about the z-axis.

2.1.GoverningEquations.Itisassumedthatthemagnitude

of ▽n∞or the particle velocity is not large and hence that the

systemisonlyslightlydistortedfromtheequilibriumstate,where

no bulk electrolyte gradient is imposed and the particle and fluid

are at rest. Therefore, the ionic concentration (number density)

distributions n±(r,θ), the electrostatic potential distribution

ψ(r,θ), and the dynamic pressure distribution p(r,θ) can be

expressed as

δ=+

±±±

nnn

(eq)

(1a)

ψψδψ=+

(eq)

(1b)

δ=+

ppp

(eq)

(1c)

where n±

distributions of ionic concentrations, electrostatic potential,

and dynamic pressure, respectively, and δn±(r,θ), δψ(r,θ), and

δp(r,θ)are the small deviations from the equilibrium state. Here,

thesubscripts +and− referto thecation andanion,respectively.

The equilibrium concentrations n±

brium potential ψ(eq)by the Boltzmann distribution.

It can be shown that the small perturbed quantities δn±, δψ,

andδptogetherwiththefluidvelocity fieldu(r,θ), whichisalsoa

small quantity, satisfy the following set of linearized electro-

kinetic equations:25

∇· =

u

0

(eq)(r), ψ(eq)(r), and p(eq)(r) are the equilibrium

(eq)are related to the equili-

(2)

λ

2

ε

η

ψδψδψ ψ

∇ ∇ ×

= − ∇ × ∇

−∇ ×

(eq)

∇+ ∇∇

h r

( )

uu

()

22

2(eq)

(3)

⎫

⎬

⎭

(4)

δμψδμ

ψ

ω

∇= ±∇·∇−

∇

−

·

−

±±

±

⎪

⎩

⎪

⎪

⎨

⎪

⎧

Ze

kT

kT

(1

h r D

) ( )]

u

[1

2 (eq)

(eq)

δψ

ε

ψ

kT

δμδψ

⎤

⎦

ψ

kT

δμδψ

∇=+

−−−

∞

0

−

+

⎡

⎣

⎛

⎝

⎢

⎢

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

⎜⎜

⎞

⎠

⎟⎟

⎥

⎥

Zen

kT

Ze

Ze

Ze

Ze

exp()

exp()

2

(eq)

(eq)

(5)

Here, δμ±(r,θ) is defined as a linear combination of δn±and δψ

on the basis of the concept of the electrochemical potential

energy,37

δ

δψ

±

±

n

n0∞is the value of the prescribed electrolyte concentration n∞at

the position with z = 0, which can be experimentally taken as the

mean bulk concentration of the electrolyte in the vicinity of the

diffusiophoretic particle; η and ε are the viscosity and

permittivity, respectively, of the fluid; D±and ωD±(with 0 ≤

ω ≤ 1) are the diffusion coefficients of the ionic species outside

and inside the porous surface layer, respectively; Z is the valence

of the symmetric electrolyte, which is positive; λ = (f/η)1/2,

wherefisthehydrodynamicfrictioncoefficientinsidethesurface

layer per unit volume of the fluid (which accounts for the

hindrance to the convective transport of the fluid caused by the

immobile frictional charged segments); h(r) is a unit step

function that equals unity if r0< r < a (inside the surface layer),

andzeroifr>a(outsidethesoftparticle). Thevaluesofη,ε,D±,

ω, and λ are taken to be constant.

δμ=±

±

kT

n

(eq)

Ze

(6)

Figure 1. Geometric sketch for the diffusiophoresis of a charged soft

sphere.

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Equation 3 results from a combination of the Stokes and

Brinkman equations modified with the electrostatic effect.34,35

The reciprocal of the parameter λ is the shielding length

characterizing the extent of flow penetration inside the porous

surface layer. For some model porous particles made of steel

wool (in glycerin−water solution)38and plastic foam slab (in

silicon oil),39experimental values of 1/λ can be as high as

0.4 mm, whereas in the surface regions of human erythrocytes,40

rat lymphocytes,41and grafted polymer microcapsules42in salt

solutions, values of 1/λ were found to be about 3 nm. Note that

1/λ2isthe“permeability”oftheporousmediumintheBrinkman

equation, which is related to its pore size and porosity and

characterizes the dynamic behavior of the viscous fluid in it.

2.2. Boundary Conditions. The boundary conditions at the

surface of the dielectric rigid core of the soft sphere are

=

rr

u

:0

0

δψ

e

0

r

δμ

±

r

where eris the unit vector in the r direction. These conditions

imply that no fluid and ions can penetrate into the rigid core and

the Gauss condition holds at its surface.

The boundary conditions at the surface of the soft particle

(r = a) are

σ ·

ue

andare continuous

r

δψ

andare continuous

δμ

±

where σ is the hydrodynamic stress tensor of the fluid. The

continuity requirements of the fluid velocity and stress in eq 8a,

of the electric potential and electric field in eq 8b, and of the

concentrations and fluxes of the ionic species in eq 8c at the

particle surface are physically realistic and mathematically

consistent for the present problem.43−45The continuity of the

hydrodynamic stress comes from the continuous Maxwell stress,

and the continuity of the electric field results from the assump-

tion that the permittivity of the solution takes the same value

both inside and outside the porous surface layer.

The boundary conditions far from the particle are

→ ∞

rU

ue

:

z

=

(7a)

·∇

·∇

=

=

(7b)

e

0

(7c)

(8a)

δψ∇

(8b)

ωδμ−−∇

±

h r

) ( )]and [1(1are continuous

(8c)

→ −

(9a)

δψβαθ→ −kT

Ze

r

acos

(9b)

δμβ α

)

θ→∓

±

kT

r

a

(1cos

(9c)

whereα=a|∇n∞|/n0∞andβ=(D+−D−)/(D++D−).Equations

7aand9atakeareferenceframethattheparticleisatrestandthe

velocity of the fluid at infinity is the particle velocity in the

opposite direction. Equation 9b for the induced potential field,

which arises spontaneously due to the imposed electrolyte

gradient and the difference in mobilities of the cation and anion

of the electrolyte, is derived from the requirement that the total

fluxes of cations and anions are balanced in order to have no

electric current generated in the bulk solution.1,46

3. SOLUTION FOR THE DIFFUSIOPHORETIC VELOCITY

3.1. Equilibrium Electric Potential. Before solving for the

problem of diffusiophoresis of the charged soft sphere in a

solution of a symmetric electrolyte with a constant bulk

concentration gradient ▽n∞, we need to determine the

equilibrium electrostatic potential first. The equilibrium

potential ψ(eq)satisfies the Poisson−Boltzmann equation and

appropriate boundary conditions (taking the Gauss condition at

the surface of the rigid core of the soft particle, continuous

electricpotentialandelectricfieldattheparticlesurface,andzero

potential far from the particle). It can be shown that, for a

charged soft sphere with a constant surface charge density σ on

the rigid core and a uniform space charge density Q inside the

porous shell,

ψψσψσσσ=

̅+

̅ +

̅̅

̅

̅ ̅

̅

rQQQQ

( ) O(,,,)

(eq)

eq01eq10

3223

(10)

Here, σ̅= Zeσ/εκkT and Q̅ = ZeQ/εκ2kT are the dimensionless

fixed charge densities,

⎛

⎝

Zrr

e 1

0

⎧

⎨

⎩

⎫

⎬

⎭

r

⎧

⎨

⎩

⎫

⎬

⎭

r

and κ = [2Z2e2n0∞/ εkT]1/2is the Debye screening parameter.

The expression in eq 10 for ψ(eq)as a power series in σ̅and Q̅ up

toO(σ̅,Q̅)istheequilibriumsolutionforthelinearizedPoisson−

Boltzmann equation that is valid for small values of the electric

potential (the Debye−Hückel approximation). That is, the fixed

charge densities σ and Q of the soft particle must be sufficiently

small for the potential to remain small. Note that the

contribution from the effects of O(σ̅

only for the case of symmetric electrolytes. The cases of a soft

particle with σ = 0 and Q = 0 represent an uncharged rigid core

with a charged porous surface layer and a charged rigid core with

an uncharged surface layer, respectively.

Experimental data for the surface layers of human

erythrocytes,40rat lymphocytes,41and poly(N-isopropylacryla-

mide) hydrogels47in electrolyte solutions indicate that the

magnitudeofQrangesfromquitelowtoashighas8.7×106C/m3,

depending on the pH value and ionic strength of the electrolyte

solution. As to the surface charge density, an experimental study

on the AgI surface in contact with aqueous solutions reported

that the value of σ changes from 0 to −0.035 C/m2upon

increasing the pAg from 5.6 to 11.48It is widely understood that

the Debye length 1/κ is in the range from less than a nanometer

to about a micrometer, depending on the ionic strength of the

solution. For a soft particle with σ = 2 × 10−3C/m2and Q = 2 ×

106C/m3inanaqueoussolutionofaunivalentelectrolytewith1/

κ = 1 nm, one obtains the dimensionless charge density σ̅≅ 0.1

and Q̅ ≅ 0.1.

3.2. Perturbation Solution of the Electrokinetic

Equations. To solve for the small quantities u, δp, δμ±, and

δψ in terms of the particle velocity U when the parameters σ̅and

ψ

κ

+κ

⎛

⎝

=

κ−−

⎜

⎞

⎠

⎟

kT

rr

e

r r

(

eq01

00

)

0

(11)

ψ

κκ

κκκ

κ κ

r r

=−+

+

−

+‐≤≤

κ−

⎜⎟

⎠

⎞

kT

Zea

e

r

rrr

a

rra

11

1

1

[cosh()

sinh()]

0

if

d

eq10

0

00

0

(12a)

ψ

κκ

κκ

κ

=−+

+

+≥

κ

κ

−

−−

⎜

⎝

⎟

⎠

⎛⎞

kT

Ze a

e

r

rd

d

a

ra

11

1

1

[cosh()

sinh( )]eif

d

r a

(

eq10

0

0

)

(12b)

2,σ̅Q̅,Q̅2) to ψ(eq)disappears

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7577

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Q̅ are small, these variables are written as perturbation

expansions in powers of σ̅and Q̅,

σ

Q

uuuu

011002

σσ=

̅+

̅ +

̅+̅ ̅ +

Q

̅ + ···

Q

uu

2

1120

2

(13a)

δσσσ=

̅+

̅ +

̅+̅ ̅ +

Q

̅ + ···

pp p Q

10

ppp Q

2001 02

2

11

2

(13b)

δμμ

+

μσμμσμσ

μ

=+

̅+

̅ +

̅+ ̅ ̅

̅ + ···

±±±±±±

±

QQ

Q

0001

2

1002

2

11

20

(13c)

δψψψ σ

01

ψψ σ

02

ψ σ

11

ψ=+

̅+

̅ +

̅+̅ ̅ +

Q

̅ + ···

(13d)

QQ

0010

2

20

2

σσσ=

̅+

̅ +

̅+̅ ̅ +

Q

̅ + ···

UUU Q

10

UU U Q

20 0102

2

11

2

(13e)

where the functions uij, pij, μij±, ψij, and Uijare independent of σ̅

and Q̅.The zeroth-order terms of u, δp, and U disappear because

an uncharged particle will not move by applying an electrolyte

concentration gradient if only the electrostatic interaction is

involved.

Substituting the expansions given by eq 13 and ψ(eq)given by

eq 10 into the governing equations given by eqs 2−5 and

boundary conditions in eqs 7−9, and equating like powers of σ̅

and Q̅ on both sides of the respective equations, one can derive a

group of linear differential equations and boundary conditions

foreachsetofthefunctionsuij,pij,μij±,andψij.Aftersolvingthese

perturbation equations,theresults fortherandθcomponents of

u,δp(totheordersofσ̅

of σ̅and Q̅) can be obtained as

2,σ̅Q̅,andQ̅2),δμ±,andδψ(totheorders

η

βασ

⎤

⎦⎥

η

kT

a

βα

η

kT

a

ασ

η

kT

a

ασ

η

α

θ

=−

̅

+−

̅

++

̅

++

̅ ̅

++

̅

⎪

⎩

⎪⎧

⎨

⎡

⎣⎢

⎤

⎦⎥

⎡

⎣⎢

⎡

⎣⎢

⎡

⎣⎢

⎡

⎣⎢

⎡

⎣⎢

⎤

⎦⎥

⎤

⎦⎥

⎤

⎦⎥

uU F

01 00

r

kT

a

Fr

U F

10 00

r

kT

a

F r Q

( )

U F

02 00

rFr

U F

11 00

rFrQ

U F

20 00

rFr Q

( )

( ) ( )

( )

( )( )

( )( )

( )} cos

rrr

rr

rr

rr

rr

2

01

2

10

2

02

2

2

11

2

20

2

(14a)

η

βασ

⎤

⎦⎥

η

kT

a

βα

η

kT

a

ασ

η

kT

a

ασ

η

αθ

=−

̅

+−

̅

++

̅

++

̅ ̅

++

̅

θθθ

θθ

θθ

θθ

θθ

⎪

⎩

⎪⎧

⎨

⎤

⎦⎥

⎡

⎣⎢

⎡

⎣⎢

⎡

⎣⎢

⎡

⎣⎢

⎤

⎦⎥

⎤

⎦⎥

⎤

⎦⎥

u U F

01 00

r

kT

a

Fr

U F

10 00

r

kT

a

Fr Q

( )

U F

02 00

rFr

U F

11 00

rFrQ

U F

20 00

rFr Q

( )

( )( )

( )

( )( )

( )( )

( ) } sin

2

01

2

10

2

02

2

2

11

2

20

2

(14b)

δ

η

a

η

βα

εκ

η

βαψσ

η

βα

εκ

η

βαψ

η

α

εκ

η

αψ

σ

η

α

εκ

η

α ψ

(

ψσ

η

α

εκ

η

αψ

θ

=−−

×

̅+

−

−

̅

+++

×

̅+

+

++

̅ ̅

+++

×

̅

ψ

ψ

ψ

ψψ

ψ

⎪

⎩

⎪⎧

⎨

⎡

⎣⎢

⎤

⎦⎥

⎡

⎣⎢

⎤

⎦⎥

⎡

⎣⎢

⎤

⎦⎥

⎡

⎣⎢

⎤

⎦⎥

⎡

⎣⎢

⎤

⎦⎥

pU F

01

r

kT

a

Fr

akT

Ze

r F

( )

r U F

10

r

kT

a

Fr

akT

Ze

r F

( )

r Q

( )

00

U F

02

r

kT

a

Fr

akT

Ze

r

Fr U F

11

r

kT

a

Fr

akT

Ze

r F

( )

rr F

( )

rQ

U F

20

r

kT

a

Fr

akT

Ze

r

F r Q

( )

10

( )( )

( )( )( )

( ) ( )( )

( )( )( )

( ) ( ))

01

( )( )( )

} cos

pp

pp

pp

pp

pp

00

2

01

2

eq01

00 00

2

10

2

eq10

00

2

02

2

eq01

01

2

00

2

11

2

eq01

10

eq10

00

2

20

2

eq10

2

(14c)

δμ β α

) [

σθ=∓∓

̅∓

̅

μμμ

±

kTFrFrF r Q

( ) ] cos

10

(1( )( )

00 01

(15)

δψαβσθ=−+

̅+

̅

ψψψ

kT

Ze

FrFrFr Q

( ) ] cos

10

[( )( )

0001

(16)

Here, Fijr(r), Fijθ(r), Fpij(r) [with (i,j) equal to (0,1), (1,0),

(0,2), (1,1), and (2,0)], Fμ00(r), Fψ00(r), Fμ01(r),

Fμ10(r), Fψ01(r), and Fψ10(r) are dimensionless functions

of r defined by eqs A1−A6 and A13−A16 in the Appendix.

Note that the solutions for δμ±and δψ to O(σ̅,Q̅) will be

sufficient for the calculation of the particle velocity to

O(σ̅

3.3. Forces Acting on the Particle. The total force acting

on a charged soft sphere undergoing diffusiophoresis

in an electrolyte solution can be expressed as the sum of

the electric force and the hydrodynamic drag force.

The electric force exerted on the soft sphere can be

represented by the integral of the electrostatic force density

over the fluid volume outside the particle. Due to the fact

that the net electric force acting on the particle at the

equilibrium state is zero, the leading order of the electric

force is given by

2,σ̅Q̅,Q̅2).

∫ ∫

θ

d d

πε

2

ψδψδψ ψ

θ

= −∇∇+ ∇ ∇

×

π∞

r

r

F

()

sin

a

e

0

2(eq)2(eq)2

(17)

Substituting eqs 1b, 10, and 16 into eq 17, we obtain

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παβ

εκ

3

ψσ

β

⎡

⎣⎢

⎡

⎣⎢

εκ

3

ψ

εκ

3

ψσ

εκ

3

ψ

⎤

⎦⎥

ψ

σ

εκ

3

ψ

σσσ

=−+∞

̅

−+∞

̅

++∞

̅

++

+∞

̅ ̅ +

Q

+∞

×

̅ +

̅̅

̅

̅ ̅

̅

ψ

ψ

ψ

ψψ

ψ

⎧

⎨

⎩

⎡

⎣⎢

⎤

⎦⎥

⎡

⎣

⎢

⎤

⎦

⎥

⎤

⎦⎥

⎡

⎣⎢

⎤

⎦⎥

⎫

⎬

⎭

kT

a

a

Ze

a F

( )

aJ

a

Ze

a F

( )

aJQ

a

Ze

a F

( )

aJ

a

Ze

a F

( )

a a F

( )

a

J

a

Ze

a F

( )

aJ

QQQQ

F

e

4 ( )()

( )()

( )

01

()

(( )( ))

01

()( )()

O(,,,)

z

e

2 3

eq01

00

01

(3)

2 3

eq10

00

10

(3)

2 3

eq01

02

(3)2

2 3

eq01

10

eq10

11

(3)

2 3

eq10

10

20

(3)

23223

(18)

where the functions Jij(3)(r) are defined by eq A11a.

The hydrodynamic drag force exerted on the soft sphere is

given by the integral of the hydrodynamic stress over the particle

surface,

∫

πδηθ θ

d

=−+∇ + ∇

u

[

·

π

p

Feue

2 a{() ] } sin

rr

h

2

0

T

(19)

Substitution of eq 14 into the above equation results in

⎪⎧

⎨

⎩

⎤

⎦⎥

πηβα

⎡

⎣⎢

ε κ

( )

3

β

αψσηβα

ε κ

( )

3

βαψ

ηα

ε κ

( )

3

αψ

σηα

ε κ

( )

3

α ψ

(

ψσ

ηα

ε κ

( )

3

αψ

σσσ

= −−−

×

̅+

−

−

̅

+++

×

̅+

+

++

̅ ̅

+++

×

̅ +

̅̅

̅

̅ ̅

̅

ψ

ψ

ψ

ψψ

ψ

⎪

⎡

⎣⎢

⎤

⎦⎥

⎡

⎣⎢

⎤

⎦⎥

⎡

⎣⎢

⎤

⎦⎥

⎡

⎣⎢

⎤

⎦⎥

aCU

kT

a

C

a

kT

Ze

a F

( )

aaCU

kT

a

C

a

kT

Ze

a F

( )

a Q

( )

00

aCU

kT

a

C

a

kT

Ze

a

FaaCU

kT

a

C

a

kT

Ze

a F

( )

a a F

( )

a

Q

aCU

kT

a

C

a

kT

Ze

a

F a Q

( )

10

QQQ

F

e

4

( )

( )

( )

01

( )( ))

01

( )

O(,,,)}z

h006 01 016

2

eq01

00006 10106

2

eq10

006 02 026

2

eq01

2

006 11 116

2

eq01

10

eq10

006 20 206

2

eq10

23223

(20)

where the coefficients Cij6are given by eqs A7fand A8f.

3.4. Velocity of the Particle. At the steady state, the total

force exerted on the diffusiophoretic particle is zero. Applying

this constraint to the summation of eqs 18 and 20, we obtain

⎛

⎝

a Ze

εβα

η

κ=

− −

i j

+

i j

⎜⎟

⎠

⎞

U

kT

aH

( )

ijij

(2)

2

2

(21)

where (i,j) equal to (0,1), (1,0), (0,2), (1,1), and (2,0), and Hij

are dimensionless functions of r0/a, κa, and λa defined by

ε

κ

= −

( 1)

−∞

+ −

i j

+

i j

H

Ze

akT

a

C

CJ

()

1

( )

1

[( )]

ij

ij

ij

1

2

2

006

6

(3)

(22)

Fromeqs13eand21,thediffusiophoreticvelocityofthecharged

soft sphere can be expressed as

⎛

⎝

a Ze

a HQa H Q

( ) ( )

11

322

εα

η

+

+

βκ

[

σ β κ

( )

κσ

κσκ

σσσ

=

̅+

̅ +

̅

̅ ̅ +

,

̅

̅̅

̅

̅ ̅

̅

⎜⎟

⎠

⎞

U

kT

aHa H Q a H

( )

QQQ

O(,, )]

2

01

2

10

2

02

2

34

20

2

3

(23)

Note that (κa)σ̅(= aZeσ/εkT) and (κa)2Q̅ (= a2ZeQ/εkT) are

independent of κ or n0∞for constant fixed charge densities σ

and Q.

When there is no permeable layer on the surface of the rigid

core of theparticle, onehas d =0 and r0= a.Then, eq 22reduces

to

1

1{1

⎧

⎨

⎩

a

8( 1)

κ

κκ=

+

−−

κ

H

a

EaEa

e [5 ( )2 ( )]}

5

a

017

(24a)

κ

40e

4

3e [3 ( )

κκ

κκκ

κ

⎫

⎬

⎭

κκ

κ

=

+

2

++

−

+

−

+−

−

κ

κ

κ

H

EaEa

Ea E

( )[ ( )

aEa

EaEaEa

Ea

1

1

1

3e

[10 (2 )

6

7 (2 )]

8

( )]

5

9 ( )

4

7 ( )

5

15 ( )]

6

a

a

a

02

2

2

73

3

(24b)

and H10= H20= H11= 0, where

∫

1

The diffusiophoretic velocity given by eqs 23 and 24 with

H10=H20=H11=0orQ=0isthesameasthatofachargedrigid

sphere derived previously.24

When the particle is a homogeneous porous sphere, one has

r0= 0 and d = a. For the particular case of a charged porous

sphere with λa → ∞ and ω = 1, eq 22 reduces to

2

3( ) e

=

∞

−−

E x

n

tt

( ) ed

nxt

(25)

κα κ

( )

1

=

κ−−

Haa

a

10

3

(26a)

κα κ

( )[ ( )

1

κκκ

α κ

[ ( )]

1

=−+

×

κ−−−

Ha a EaEaa

a

1

9( )

( )]

3

1

12( ) e

a

20

3

5

62

2

(26b)

and H01= H02= H11= 0, where α1(x) is defined by eq A9a in the

Appendix. The diffusiophoretic velocity given by eqs 23 and 26

with H01= H02= H11= 0 or σ = 0 is identical to that of a charged

porous sphere obtained in the literature.25

Atypicalsoftparticleisachargedrigidcoreadsorbingasurface

layerofoppositelyandequivalentlychargedpolyelectrolytes.For

a soft sphere of this type with zero net charge, one has

4

3

where γ = [(a/r0)2−(r0/a)]/3. Analytical studies49,50have

predicted that a nonuniformly charged but “neutral” imperme-

ablesphere(withzeroarea-averagedζpotential)canbedrivento

move by externally applied electric fields. It would be of interest

πσπσγκ+−=

̅= −

̅

rarQ aQ

4

()0 or

0

23

0

3

(27)

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to know whether and how charged but “neutral” soft spheres

undergo electrophoresis or diffusiophoresis. For such spherically

symmetric soft particles with zero net charge, eq 23 with the

substitution of eq 27 becomes

⎛

⎝

aZe

εα

η

β κ

[ ( )

κ=

̅ +̅ +̅

⎜⎟

⎠

⎞

U

kT

a HQ a H Q

( )

Q

O()]

2

2

1

4

2

23

(28)

where

γ=−

HHH

11001

(29a)

γγ=−+

HHHH

22011

2

02

(29b)

In the limit of r0/a = 1 (the porous surface layer of the particle

disappears),γ=0andeq29reducestoH1=H10=0andH2=H20

= 0.

Because the governing equations in the analysis have been

linearized, diffusiophoresis of a charged particle in an electrolyte

solution can be considered as a linear combination of two

effects:2(i) chemiphoresis due to the nonuniform adsorption of

counterions and depletion of co-ions over the surface of the

particle and (ii) electrophoresis due to the macroscopic electric

fieldgeneratedbytheelectrolyteconcentrationgradientgivenby

eq 9b. In eqs 23 and 28 for the diffusiophoretic velocity, the first-

order terms (involving the parameter β) result from the

contribution of electrophoresis, whereas the second-order

terms represent the chemiphoretic contribution. Thus, eqs 23

and 28 with only the first-order terms in the brackets and the

replacement of βαkT/Zea by an applied electric field can also be

used to express the electrophoretic velocity of the charged soft

sphere.

4. RESULTS AND DISCUSSION

Accordingtoeq23,thediffusiophoreticvelocityofachargedsoft

sphere in an electrolyte solution prescribed with a constant

concentration gradient can becalculated to the second ordersσ2,

σQ, and Q2of its fixed charge densities. In this section, we first

discuss the dimensionless coefficients Hijgiven by eq 22 for the

general case, then present the dimensionless coefficients H1and

H2given by eq 29 for a charged but “neutral” soft sphere, and

finally exhibit the diffusiophoretic velocity given by eqs 23 and

28. For conciseness and without loss in physical insight, as

indicated in the analytical result for the diffusiophoresis of a

charged porous particle,25ω = 1 (the same diffusion coefficient

foreachioninsideandoutsidetheporoussurfacelayerofthesoft

particle) will be taken in all the calculations. It is understood that

the particle velocity is hardly dependent on ω at small κa, since

under this condition the particle behaves electrically like a point

charge, and the detail of what happens (such as the diffusion of

ions)insidethesurfacelayeroftheparticleisunimportantforthe

determination of its mobility by a balance between its

electrostatic and hydrodynamic forces.

4.1. The First-Order Coefficients H01and H10for

Electrophoresis. The first-order coefficients H01and H10in

eq 23 for the diffusiophoretic/electrophoretic velocity of a

charged soft sphere (which also represent the dimensionless

electrophoretic mobilities of a charged rigid core with an

uncharged porous surface layer and an uncharged rigid core with

a charged surface layer, respectively) calculated using eq 22 are

plotted in Figures 2 and 3, respectively, as functions of the

electrokinetic radius κa, the shielding parameter λa, and the

radius ratio r0/a of the particle. As expected, the coefficients H01

and H10forthe electrophoretic contributions are always positive,

and thus the directions of the diffusiophoresis of the particle

caused by these contributions are determined by the signs of the

products of the parameter β (which determines the direction of

theinducedelectricfieldaccordingtoeq9b)andthefixedcharge

densities σ and Q, respectively. The values of H01and H10, which

are in the same order of magnitude for the typical case of r0/a =

0.5, decrease [whereas those of κaH01and (κa)2H10may

increase] monotonically with an increase in κa for specified

values of r0/a and λa, similar to the results of their limiting

cases.24,25

Figures 2 and 3 illustrate that the coefficients H01and H10

decrease monotonically with an increase in λa for given values

Figure 2. Plots of the coefficient H01 given by eq 22 for the

diffusiophoretic/electrophoretic velocity of a charged soft sphere with

variousvaluesofλa,κa,andr0/a:(a)H01versusκaatr0/a=0.5; (b)H01

versus r0/a at κa = 1.

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of r0/a and κa. When λa → ∞, the resistance to the fluid motion

inside the porous surface layer of the soft particle is infinitely

large, and the relative fluid velocity in the surface layer vanishes

(the electrolyte ions can still penetrate the surface layer and the

equilibrium potential distribution ψ(eq)is still given by eq 10).

Therefore, for given values of r0/a and κa with λa → ∞, H01and

H10approach constant minimal values. When λa = 0, the surface

layer does not exert resistance to the fluid motion, and H01and

H10for given values of r0/a and κa approach constant maximal

values.NotethatH01andH10aresensitivefunctionsofλaoverits

range of 1−10.

As expected, Figure 2b shows that the coefficient H01, which

represents the contribution to the electrophoretic velocity of the

soft particle from the surface charge density σ of its impermeable

core, is a monotonic increasing function of r0/a (or the volume

fraction of the impermeable core in the soft particle) for fixed

valuesofκaandλa.Inthespecialcaseofr0/a=0,thesoftparticle

degenerates to a homogeneous porous sphere and H01must

equalzero.Obviously,H01isindependentofλainthespecialcase

of r0/a = 1 in which the soft particle reduces to an impermeable

sphere with no porous surface layer. Conversely, as shown in

Figure 3b, the coefficient H10, which denotes the contribution to

the electrophoretic velocity of the soft particle from the space

charge density Q of its porous surface layer, decreases

monotonically with increasing r0/a and vanishes when r0/a =

1. In the general case of moderate values of r0/a, the fixed charge

densities σand Qcontribute comparably to the diffusiophoretic/

electrophoretic velocity of the soft sphere.

4.2.TheSecond-OrderCoefficientsH02,H11,andH20for

Chemiphoresis. The second-order coefficients H02, H11, and

H20for the diffusiophoretic/chemiphoretic velocity of a charged

soft sphere (where H02and H20also represent the dimensionless

chemiphoretic mobilities of a charged rigid core with an

uncharged porous surface layer and an uncharged rigid core

with a charged surface layer, respectively) can also be calculated

using eq 22, and their results as functions of the parameters κa,

λa, and r0/a are plotted in Figures 4−6. Again, for fixed values of

r0/a and κa, the coefficients H02, H11, and H20generally decrease

monotonically with an increase in λa and are sensitive functions

ofλaoveritsrangeof1−10.NotethatthevaluesofH02,H11,and

H20, which have the same order of magnitude for the typical case

of r0/a = 0.5, in general are also positive (thus, the contributions

ofH02andH20causethechemiphoresisoftheparticletowardthe

side of higher electrolyte concentration, but the contribution of

H11tothedirectionofchemiphoresisisdeterminedbythesignof

theproductofσandQ)andabout2ordersofmagnitudesmaller

than those of H01and H10.

For fixed values of κa and λa, as illustrated in Figures 4b and

6b, the coefficient H02, which represents the contribution to the

chemiphoresisofthesoftparticlefromthesurfacechargedensity

σofitsimpermeablecore,ingeneralincreaseswithanincreasein

thevalueofr0/a(orthevolumefractionoftheimpermeablecore

inthesoftparticle),becomeszeroasr0/a=0,andisindependent

of λa as r0/a = 1, similar to the trend of the coefficient H01,

whereas the coefficient H20, which denotes the contribution to

the chemiphoresis of the soft particle from the space charge

density Q of its porous surface layer, generally decreases with an

increaseinthevalueofr0/aandvanishesasr0/a=1,similartothe

trend of the coefficient H10. On the other hand, the coefficient

H11, which is the contribution to the chemiphoresis of the soft

particle from the interaction between thefixed charge densities σ

and Q, equals zero in both limiting cases of r0/a = 0 and r0/a = 1,

andthusthereexistsamaximalvalueofH11inbetweenthelimits

for specified values of κa and λa, as shown in Figure 5b. The

location of the maximum shifts to greater r0/a as λa increases,

since large volume fraction of the impermeable core (small

volume fraction of the porous surface layer) in the soft particle

favors its movement when the resistance to the fluid motion

inside the surface layer is large. In the general case of moderate

values of r0/a, the three second-order terms of the fixed charge

densities in eq 23 contribute to the diffusiophoretic/chemipho-

retic velocity of the soft sphere comparably.

For specified values of r0/a and λa, as shown in Figures 4a, 5a,

and 6a, the values of the coefficients H02, H11, and H20are

maximal at some values of κa between 0.1 and 1, and fade out

as the value of κa gets small or large. Both the limits κa = 0 and

Figure 3. Plots of the coefficient H10given by eq 22 for the

diffusiophoretic/electrophoretic velocity of a charged soft sphere with

variousvaluesofλa,κa,andr0/a:(a)H10versusκaatr0/a=0.5;(b)H10

versus r0/a at κa = 1.

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κa→∞ resultin H02= H11= H20= 0(but thecontributionfrom

the chemiphoretic effect vanishes only for the case of κa = 0, as

indicated by eq 23). Interestingly, when the value of λa is

sufficiently large (e.g., λa > 10), H02, H11, and H20decrease first,

reach minima (at some values of κa between 1 and 3), which in

general are negative (the direction of the chemiphoresis of the

particle is reversed), and then increase monotonically to zero as

κa increases from its values at the maximal points to infinity. The

locations of the minima (if they exist) and maxima in H02, H11,

and H20shift to smaller κa as λa increases.

The negative behavior of the second-order coefficients H02,

H11, and H20in some cases can be explained as follows. For the

diffusiophoresis/chemiphoresis of a soft particle, the diffusion of

the solute species in the fluid solution affects the particle’s

movement through two mechanisms. Evidently, the concen-

tration gradient of the solute along the external surface of the

particle leads to a diffusioosmotic/chemiosmotic flow, which

drives the soft particle to move in the direction of the

concentration gradient as that for the corresponding motion of

an impermeable particle. On the other hand, the diffusion of the

solute species inside the porous surface layer drags the ambient

fluid, which then drives the soft particle to move in the opposite

direction. For the cases with intermediate κa, large λa, and small

r0/a, the contribution from the diffusion of the solute species

inside the porous surface layer can be dominant over the

contribution from the osmotic flow along the external surface of

the particle so that the values of H02, H11, and H20are negative.

Figure 4. Plots of the coefficient H02given by eq 22 for the

diffusiophoretic/chemiphoretic velocity of a charged soft sphere with

variousvaluesofλa,κa,andr0/a:(a)H02versusκaatr0/a=0.5;(b)H02

versus r0/a at κa = 1.

Figure 5. Plots of the coefficient H11given by eq 22 for the

diffusiophoretic/chemiphoretic velocity of a charged soft sphere with

variousvaluesofλa,κa,andr0/a:(a)H11versusκaatr0/a=0.5; (b)H11

versus r0/a at κa = 1.

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4.3. The Coefficients H1and H2for a Neutral Soft

Particle. The coefficients H1and H2in eq 28, which represent

the dimensionless electrophoretic and chemiphoretic velocities,

respectively, of a charged but “neutral” soft sphere, expressed in

terms of the fixed charge density in its porous surface layer and

calculated using eqs 29 and 22 are plotted as functions of the

parametersκa,λa,andr0/ainFigures7and8,respectively.Itcan

be seen that both H1and H2are always positive and, as expected,

vanish as r0/a = 1 (the porous layer of the particle disappears).

Thus, the “neutral” soft sphere can experience electrophoresis

and chemiphoresis under an electrolyte concentration gradient,

and the directions of the electrophoretic and chemiphoretic

velocities are decided by the fixed charges in the porous layer

(rather than the surface charges of the rigid core, whose effect is

screened entirely by part of the charged porous layer) of the

soft particle. In the limit κa = 0, the effects of the positive and

negative fixed charges on the rigid core and inside the surface

layer on the electrophoresis of the soft particle cancel out with

each other, which leads to H1= 0. Since H02= H11= H20= 0 in

both the limits κa = 0 and κa → ∞; H2= 0 also in these limits

according to eq 29b (but the contribution from the

chemiphoretic effect vanishes only for the case of κa = 0, as

indicated by eq 28). For specified values of r0/a and λa, as

shown in Figure 8a,the value ofH2is maximal at some value of

κa between 1 and 10.

Similar to the coefficients H10and H20presented in Figures 3

and 6, both H1and H2decrease monotonically with an increase

Figure 6. Plots of the coefficient H20given by eq 22 for the

diffusiophoretic/chemiphoretic velocity of a charged soft sphere with

variousvaluesofλa,κa,andr0/a:(a)H20versusκaatr0/a=0.5;(b)H20

versus r0/a at κa = 1.

Figure 7. Plots of the coefficient H1given by eq 29a for the

electrophoretic mobility of a soft sphere with zero net charge with

various values of λa, κa, and r0/a: (a) H1versus κa at r0/a = 0.5; (b) H1

versus r0/a at λa = 1.

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in λa for given values of κa and r0/a, and H2decreases

monotonically with an increase in r0/a for fixed values of κa and

λa.However,H1isnotnecessarilyamonotonicfunctionofeither

κa or r0/a. For given values of κa and λa, there is a maximum of

H1atsomesmallvalueofr0/a,andthismaximumshiftstosmaller

r0/a as κa increases (which favors the electrophoretic

contribution from the porous surface layer rather than that

from the rigid core of the soft particle). The trend of the

dependence of H1on κa and r0/a is quite different from that of

H10presentedinFigure3.Whenκaislarge(greaterthanabout10),

the coefficients H1and H10have the same order of magnitude.

However, when κa is as small as 0.1, the value of H1is about 3

orders of magnitude smaller than that of H10. In general, the

coefficients H2and H20have the same order of magnitude.

4.4. Diffusiophoretic Velocity. The dependence of the

diffusiophoretic velocity of a charged but neutral soft sphere on

its dimensionless fixed charge density (κa)2Q̅ = a2ZeQ/εkT with

various values of κa and r0/a at λa = 1 for the case in which the

cationandaniondiffusivitiesareequal(β=0)is showninFigure9.

Themagnitudeofthediffusiophoreticvelocityisnormalizedbya

characteristic value given by

⎛

⎝

a Ze

εα

η

*=

⎜⎟

⎠

⎞

U

kT

2

(30)

Figure 8. Plots of the coefficient H2 given by eq 29b for the

chemiphoreticmobilityofasoftspherewithzeronetchargewithvarious

values of λa, κa, and r0/a: (a) H2versus κa at r0/a = 0.5; (b) H2versus

r0/a at λa = 1.

Figure 9. Plots of the normalized diffusiophoretic mobility of a charged

soft sphere with zero net charge ina symmetric electrolyte solution with

β = 0 versus the dimensionless fixed charge density (κa)2Q̅: (a) λa = 1

and r0/a = 0.5; (b) κa = 1 and λa = 1.

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Only the results at positive Q̅ are exhibited in Figure 9 since

the particle velocity (due to the chemiphoretic effect entirely

for the case β = 0) is an even function of Q̅ as illustrated by

eq 28. As expected, the reduced diffusiophoretic velocity U/U*

increases monotonically with an increase in (κa)2Q̅ for given

values of κa, λa, and r0/a, with a decrease in λa for specified

values of κa, r0/a, and Q̅ (this result is not shown here for

conciseness), and with a decrease inr0/aforconstant values of

κa,λa, and Q̅.Forfixed values of λa, r0/a, and (κa)2Q̅,the value

of U/U* is maximal at a finite value of κa, and decreases as κa

increases or decreases from this value. There is no

chemiphoretic motion of the particle for the limiting cases

of Q̅ = 0 and κa = 0.

In Figure 10, the normalized diffusiophoretic velocity U/U*

ofa chargedbutneutralsoftspherecalculatedusingeq28with

various values ofκaand r0/aatλa= 1 isplottedasa functionof

(κa)2Q̅ for a case in which the cation and anion have different

diffusion coefficients with β = −0.2. In this case, both the

electrophoretic and chemiphoretic effects contribute to the

motion of the particle, and the net diffusiophoretic velocity is

Figure10.Plotsofthenormalizeddiffusiophoreticmobilityofacharged

softsphere with zero netcharge in asymmetric electrolyte solution with

β = −0.2 versus the dimensionless fixed charge density (κa)2Q̅: (a) λa =

1 and r0/a = 0.5; (b) κa = 1 and λa = 1.

Figure11.Plotsofthenormalizeddiffusiophoreticmobilityofacharged

soft sphere in a symmetric electrolyte solution with σ̅= 1, κa = 1, and

r0/a = 0.5 versus the dimensionless fixed charge density (κa)2Q̅: (a) β =

0; (b) β = −0.2.

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neither an even nor an odd function of Q̅. Again, for given

values ofλa, r0/a, and (κa)2Q̅, the normalized velocity U/U* is

not a monotonic function of κa. For small values of r0/a, the

normalized particle velocity U/U* is not a monotonic

function of (κa)2Q̅. Some of the curves in Figure 10b show

that the particle might reverse direction of movement more

than once as its dimensionless fixed charge density varies from

negative to positive values. The reversals occurring at the

values of (κa)2Q̅ other than zero result from the competition

between the contributions from electrophoresis and chem-

iphoresis. Note that the situations associated with Figures 9

and 10 (β = 0 and −0.2) taking Z = 1 are close to the

diffusiophoresis in the aqueous solutions of KCl and NaCl,

respectively.

To use eq 23 to determine the diffusiophoretic velocity for a

general soft sphere, not only the parameters κa, λa, and r0/a

but also the dimensionless fixed charge densities σ̅and Q̅ of

the particle have to be specified. Results of the reduced

diffusiophoretic velocity U/U* of a charged soft sphere with

the dimensionless surface charge density of its rigid core σ̅= 1

plotted versus the dimensionless space charge density (κa)2Q̅

of its porous layer for various values of λa at κa = 1 and r0/a =

0.5 are given in Figures 11a and 11b for the cases of β = 0 and

β = −0.2, respectively. Again, the magnitude of U/U* decreases

with an increase in the value of λa. For the given case with σ̅=

1, κa = 1, and r0/a = 0.5, eq 27 shows that the soft particle is

“neutral” at Q̅ = −6/7, and the curves in Figure 11 indeed

display minimal magnitudes of the particle velocity in its

vicinity. In the range of |Q̅| < (γκa)−1σ̅, the contribution to the

particle velocity due to σ̅may be important, but the effect of

the fixed charge at the porous layer or Q̅ on the

diffusiophoresis of a soft particle becomes dominant beyond

this range.

5. CONCLUSIONS

In this paper the steady diffusiophoresis (consisting of

electrophoresis and chemiphoresis) of a charged soft sphere

with arbitrary values of the electrokinetic radius κa, the

shielding parameter λa, and the radius ratio r0/a in an

unbounded solution of a symmetric electrolyte with a

uniformly imposed concentration gradient is analyzed. The

porous shell of the soft particle is treated as a solvent-

permeable and ion-penetrable layer in which fixed-charged

groups and frictional segments are distributed at uniform

densities. Solving the linearized Poisson−Boltzmann equa-

tion, continuity equations of ions, and modified Stokes/

Brinkman equations applicable to the system by a regular

perturbation method, we have obtained the electric potential

profile, the ion concentration (or electrochemical potential

energy) distributions, and the fluid velocity field. The

requirement that the total force exerted on the soft

particle is zero leads to eqs 22 and 23 for its diffusiophoretic

velocity as a function of the parameters κa, λa, and

r0/a correct to the second orders of the fixed charge

densities σ and Q. We found that a charged but “neutral”

soft sphere can undergo diffusiophoresis (electrophoresis

and chemiphoresis), and the direction of its diffusiophoretic

velocity is determined by the fixed charges inside its porous

surface layer. Expression 23 for the diffusiophoretic velocity

of a charged soft sphere reduces to the corresponding

formulas for a charged rigid sphere and a charged porous

sphere, respectively, in the limiting cases of r0/a = 1 and

r0/a = 0.

■APPENDIX

The definitions of some functions in Section 3 are listed here. In

eq 14,

α λ

( )

003 1

β λ

( )]

004 1

=+++

<<

⎛

⎝

⎜

⎝

⎟

⎠

⎛⎞

FrCCCrCr

a

r

rra

( )[

if

r

00001 002

3

0

(A1a)

=+−>

⎜⎟

⎠

⎞

FrC

a

r

C

a

r

r

a

( )1 if

r

00005

3

006

(A1b)

α λ β λ

( )]

004 2

= −+−−

<<

θ

⎛

⎝

⎜⎟

⎠

⎞

FrCCCrCr

a

r

rra

( )

1

2[

( )

if

00001002003 2

3

0

(A2a)

=−+>

θ

⎜

⎝

⎡

⎣

⎟

⎠

⎛⎞

Fr

C

a

r

C

a

r

r

a

( )

22

1 if

00

005

3

006

(A2b)

λ=−<<

⎜

⎝

⎟

⎠

⎢

⎛⎞

⎤

⎦

⎥

Fra

C

a

r

C

r

a

rr

a

( )()

2

if

p00

2

002

2

0010

(A3a)

=>

⎜

⎝

⎟

⎠

⎛⎞

FrC

a

r

r

a

( )if

p00006

2

(A3b)

α λ

( )

3 1

β λ

( )]

4 1

λλ

α λ

[ ( ) ( )

1

β λ

⎤

⎦

λ

=+++

+−

+−<<

βα

ij

⎜

⎝

⎟

⎠

⎜

⎝

⎟

⎠

⎛⎞

⎡

⎣

⎢

⎛⎞

⎥

F r

ijr

CCCrCr

a

r

a

2

r

r J rr Jr

a

Jr

a

r

Jrrr

a

( )[

6

2

() ( )

( ) ( )]

1

()

( )( ) if

ij ijij ij

ij

ijij

12

3

3

2

(0)

3

(3)

0

(A4a)

⎞

⎠

=

⎛

⎝

5

+−∞ +

)

∞

+−+−

>

⎜

⎝

⎟

⎠

⎜

⎝

2

⎟

⎜⎟

⎠

⎜

⎝

⎟

⎠

⎛⎞⎛

⎞⎛⎞

F r

ijr

C

a

r

C

a

r

JJ

r

a

a

r

Jr

a

rJ

rJr

r

a

Jr

ra

( )(

1

5

()

1

( )( )( )

1

5

( )

if

ij ij

ijij

ijij ij ij

5

3

6

(2)(0)

2

3

(5)(3)(2)(0)

(A4b)

⎞

⎠

r

α λ

3 2

β λ

( )]

4 2

λλ

α λ

[ ( ) ( )

2

β λ

⎤

⎦

λ

= −+−−

−−

−+<<

θ

βα

ij

⎜

⎝

⎟

⎜

⎝

⎟

⎠

⎛

⎡

⎣

⎢

⎛⎞

⎥

FrCCCrCr

a

a

2

r

r Jrr Jr

a

Jr

a

r

Jrrra

( )

1

2[

( )

3

2

( ) ( )

( ) ( )]

()

( )( ) if

ij ijijij ij

ij

ijij

12

3

3

2

2

(0)

3

(3)

0

(A5a)

⎛

⎝

a

=−+ ∞ −

)

∞

++−

+>

θ

⎜

⎝

⎟

⎠

⎜⎟

⎠

⎜

⎝

r

a

⎟

⎠

2

⎜

⎝

⎟

⎠

⎛⎞⎞

⎛⎞

⎛⎞

Fr

C

a

r

C

a

r

JJ

r

a

r

Jr

a

2

rJ

rJr

Jrra

( )

22

(

2

5

()

1

10

2

5

( ) ( ) ( )

( )if

ij

ijij

ijij

ijij ij

ij

5

3

6

(2)(0)

2

3

(5) (3)(2)

(0)

(A5b)

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7586

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λ=−−

−<<

⎜

⎝

⎟

⎠

⎜

⎝

⎟

⎠

⎡

⎣

⎢

⎛⎞

⎤

⎦

⎥

⎛⎞

Fra

C

a

r

C

r

a

r

aJ

r

a

r

Jrrra

( )()

2

2 ( )

( )if

pij

ij

ij

ij

ij

2

2

2

1

(0)

2

(3)

0

(A6a)

=+∞−

−>

⎜

⎝

⎛

⎝

⎟

⎠

⎞

⎠

⎜⎟

⎛⎞

FrC

a

r

a

r

J

r

a

r

aJ

r

Jrra

( )2()2 ( )

( )if

pijij

ijij

ij

6

2

(0)(0)

2

(3)

(A6b)

for (i,j) = (0, 1), (1, 0), (0, 2), (1, 1), and (2, 0), where

λλλλ=

Ω

−+

Craaa

3(

cosh sinh)

0010

(A7a)

λ

=

Γ

)

ΩΔ

+

C

a

C

(

2

002

3

001

(A7b)

λ

λ β λ

[ (

0

1

β λ

(

1

=

−

a

Ω

+−

CArar

3

)(

{3) )]}

0003

3

(A7c)

λ

λλ

λλ α λ

[ (

0

α λ

(

1

=

Ω

+

×+−

C

a

sinh

ar

rrar

3

)(

{[2()( ) ]

0

3) )]}

0

004

3

33

01

(A7d)

α λ

(

β λ

(

=+

C

++

C

+−

CCCCa

a

1)

)

005 001002 003 1

004 1 006

(A7e)

λ

=

Γ

ΩΔ

a

C

2

006

(A7f)

λ

λλ

λ

λ

λλ

= −+−

+−

α

ij

β

C

a

r

0

CrCr

a

Jr

a r

JrrJrr

3

(sinhcosh)

2

()

( )

0

2

[ ( ) cosh

0

( ) sinh

0

]

ijijij

ij

ij

1

2 3

3040

2

(0)

3 2

0

00

(A8a)

λ

=++∞

CC

a

CJ

2

2

()

[2()]

ij ij ij

ij

21

2

6

(0)

(A8b)

λ

λ β λ

[ (

0

1

β λ

(

1

λλλλβ λ

(

1

λλ

λ

+

λ

λλλ β λ

3 (

2

)

sinh

β λ

3 (

1

λ

λ

λλλλ

λ

λ

λλλλ

=

Ω

−

−+−∞

+++

+ ∞ +

)

×+−

×+Δ−

×−

+−

+

α

ij

β

C

a

(

ArarJ

Araaar

ar J

)

0

r

a

a

arrar

Jra Jr

a

Brra r J

] ( )

0

r

a

r J

] ( )}

ij

0

Brrar

r

3

)(

{ (3) )])

0

()

3[sinhcosh

2

0

2

)]

0

6cosh([2()

sinh() cosh

0

) )]

0

( )

0

2 ( )

0

(

[ cosh

2

(

3

() sinh

0

)

[ sinh() sinh

0

cosh

ij

ij

ij

ij ij

3

3

(2)

0

(0)3

2

0

1

(0)(3)

2

0

2

0

2

0

2

0

0

(A8c)

λ

λλ

λλλλ

λ

r

λ

λλ

=

Δ

−∞ +

)

∞

−−+

+−

βα

ij

C

a

r J

0

r J

0

arar C

)

0

r

a

Jr

a

JrrJr

1

)(

{ 3(3()

() (coshsinh

6

( )

0

6

()

[ ( ) sinh

ij

0

( ) cosh]}

ij

ijij

ij

ij

4

3

(2)(0)

3

03

0

2

(0)

2

00

(A8d)

α λ

(

1

5

β λ

(

1

=+++−

+ ∞ −

)

∞

CCCa C

)

a C

)

C

JJ

(()

ijijijijijij

ijij

5121346

(2)(0)

(A8e)

=∞ +

)

∞ −

)

−−−

α

ij

β

ij

CCJCJCJr

CJrCJrCJr

((( )

0

( )

0

( )

0

( )

0

ij

ijijij

ij

6005

(0)

006

(2)

001

(3)

002

(0)

003004

(A8f)

α=−

xxxx

( )

1

coshsinh

(A9a)

α=+−

xxxxx

( )

2

(1) sinhcosh

2

(A9b)

β=−

xxxx

( )

1

sinhcosh

(A9c)

β=+−

xxxxx

( )(1) cosh sinh

2

2

(A9d)

λλλΔ =−

rar

sinhcosh

00

(A10a)

λ

[3(

λλ

λ

λλλ

λ

Ω = −+

r

+++

+−

rarard

d

6[2(

2

)()33] cosh

)3] sinh

0

3

0

3

0

0

(A10b)

λλ

3

β λ

λ

r

(

β λ

λ

r

3

λ

λλ

β λ

+

3

β λ

] (

1

λλ

λλβ λ

(

1

β λ

(

1

λ

Γ =−++−

+

+

+

++

−

r A

0

raraA

raa

arrd

Arrar

d

39() [ (

0

)()]

0

3{

[2(

0

2

)))

3()} cosh

0

3{[(

0

)))]} sinh

0

2

11

0

3

0

0 1

0

2

(A10c)

λλλ=+

Aarr

[2()() ] cosh

0

33

0

(A10d)

λλλλλλ=+++−

Bararaa

[2()()33] sinh3 cosh

3

0

3

0

(A10e)

∫

=

⎜

⎝

⎟

⎠

⎛⎞

Jr

r

a

G r

ij

r

( )( ) d

ij

n

a

r

n

( )

(A11a)

∫ α λ

a

∫ β λ

a

=

α

ij

Jrr G r

( )

1

r

( )( ) d

r

ij

(A11b)

=

β

Jrr G r

( )

1

r

( )( ) d

ij

r

ij

(A11c)

εκ

3

ψ

d

ψ

d

=

μ

Gr

a

ZerF

r

r

( )( )

d

01

2 4

00

eq01

(A12a)

εκ

3

=

μ

Gr

a

ZerF

r

r

ψ

d

( )( )

d

10

2 4

00

eq10

(A12b)

εκ

3

= −

Gr

a

ZerW rr

( )( )

d

02

2 4

01

eq01

(A12c)

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Page 14

εκ

3

ψ

d

ψ

d

= −+

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

Gr

a

Zer

W r

01

r

W r

10

r

( ) ( )

d

( )

d

11

2 4

eq10eq01

(A12d)

εκ

3

ψ

d

= −

Gr

a

ZerW rr

( ) ( )

d

20

2 4

10

eq10

(A12e)

ψ=+

μμ

⎡

⎣⎢

⎤

⎦⎥

W r

ij

Fr

Ze

kT

r F

( )

ij

eq

r

( ) ( )

ij

( )

00

(A12f)

In eqs 15, 16, and A12,

ωω=−+++

>

μ

⎜

⎝

⎟

⎠

⎜

⎝

⎟

⎠

⎜

⎝

⎟

⎠

⎡

⎣

⎢

⎛⎞

⎛⎞

⎤

⎦

⎥⎛

⎞

Frs

r

a

a

r

r

a

ra

( )1

1

2

if

002

0

32

(A13a)

=+<<

μ

⎜

⎝

⎟ ⎜

⎠

⎟

⎠

⎡

⎣

⎢

⎛⎞⎛

⎝

⎞

⎤

⎦

⎥

Frs

r

a

a

r

r

a

rr

a

( )3

1

2

if

002

0

32

0

(A13b)

ω

κκ

κ

κ

κ

κ

2

κ

=−−−

+−−−−

×

+

>

ψμ

κ−−

⎜

⎝

⎟

⎠

⎜

⎝

⎟

⎠

⎪

⎩

⎪

⎨

⎡

⎣

⎢

⎛⎞

⎤

⎦

⎥

⎧

⎡

⎣⎢

⎤

⎦⎥

⎡

⎣⎢

⎛⎞

⎤

⎦⎥

FrFrs s

1 2

r

a

r

r

a

d

a

r

r

a

d

r

r

ra

( )( )3(1)1

2()2cosh

2

2

sinh}1

( )

e if

r r

(

0000

0

3

0

2

0

0

0

)

0

(A14a)

ω

κ

κ

κκκ

κκ

=−−−

+

3

+

r

0

−

)

0

−

−+−−<<

ψμ

κ−

⎜

⎝

⎟

⎠

⎡

⎣

⎢

⎛⎞

⎤

⎦

⎥

FrFrs s

1 2

r

a

a

2

ar

r r

0

rrr

rrrrrr

(A14b)

a

( ) ( ) 3(1)e1

1

{ [(2)2 ] cosh (

0

)

[ (22] sinh ( )}

0

if

d

0000

0

3

22

0

2

0

ωω

=−−+

×+−

⎤

⎦

−+∞

⎫

⎬

⎭

×++

<<

μ

⎜

⎝

⎟

⎠

⎜

⎝

⎛

⎝

⎟

⎠

⎞

⎠

⎜

⎝

⎟

⎠

⎜⎟⎜

⎝

⎟

⎠

⎜

⎝

⎟

⎠

⎜

⎝

⎟

⎠

⎪

⎩

⎪

⎪

⎨

⎪

⎧

⎡

⎣

⎢⎛

⎞

⎤

⎦

⎥

⎡

⎣

⎡

⎣

⎢⎛

⎞

⎛⎞

⎤

⎦

⎥

⎢⎛

⎞

⎥⎛

⎞⎛⎞

Frs

r

aK

rs

r

a

KrKr

a

r

r

a

L

r

a

r

a

a

r

a

r

Kr

rra

( )

ij

( )( )

0

2( )

0

1

1

2

(1)()

1

2

( )

if

ijij ij

ij

ij

2

(0)

2

0

3

(0)(3)

2

(0)

3

0

322

(3)

0

(A15a)

ω

ω

= −+

+−++

×+−−−∞

(

>

μ

⎜

⎝

⎟

⎠

⎜

⎝

⎟

⎠

⎜

⎝

⎟

⎠

⎜

⎝

⎟

⎠

⎤

⎦

⎜

⎝

⎟

⎠

⎜

⎝

⎟

⎠

⎜

⎝

⎟

⎠

⎪

⎩

⎪

⎭

⎪

⎨

⎪

⎬

⎡

⎣

⎢⎛

⎞

⎤

⎦

⎥⎛

⎞

⎛⎞

⎧

⎡

⎣

⎢

⎛⎞

⎤

⎦

⎥

⎡

⎣

⎢⎛

⎞

⎤

⎦

⎥

⎡

⎣

⎢

⎛⎞

⎤

⎦

⎥

⎡

⎣

⎢⎛

⎞

⎥

⎫

Frs

r

a

KrKr

a

r

a

r

Lr

r

aL

r

s

6

r

a

a

r

r

a

r

a

a

r

r

a

L

ra

( )

ij

3

2

( )

0

2( )

0

1

3

( )

1

3

( )

2

221)

if

ijij

ijij

ij

2

2

0

3

(0)(3)

2

2

(3) (0)

2

0

3

2

0

32

(0)

(A15b)

∫

κ

M r

ij

α κ

1

α κ

[ (

1

α κ

(

1

κκ

α κ

( )

1

= −−−

×+++

×

ψ

κκ−−−

Fr

r

r M rsrr

rr

r W rr

( )

ij

1{ ( )

2

( )))]

0

( )(1

r

)e(1)e

( )d }

ij

ij

r r

(

r

r

20

0

)

0

0

(A16)

where

∫

ψ

d

=−

⎜

⎝

⎟

⎠

⎜

⎝

⎟

⎠

⎛⎞

⎡

⎣

⎢

⎛⎞

⎤

⎦

⎡

⎣

⎥

⎥

Kr

Ze

kT

r

a

r

rr

r

( )1

d

d

ij

n

a

r

n

ij

( )

0

3

eq

(A17a)

∫

ψ

d

=−++

×

−

⎜

⎝

⎟

⎠

⎜

⎝

⎟

⎠

⎜

⎝

⎟

⎠

⎪

⎭

⎪

⎬

⎪

⎩

⎪

⎨

⎛⎞

⎧

⎢

⎛⎞

⎤

⎦

⎥

⎛⎞

⎫

Lr

Ze

kT

r

a

s

r

a

r

a

r

r

( )232

d

d

ij

n

a

r

n

ij

( )

3

2

0

33

eq

(A17b)

∫

κ=+

κ

∞

−

M r

ij

r W r

ij

r

( )(1)e( )d

r

r

(A17c)

κκ=++

−

srr

[2

⎡

⎣

(2

0

)]

01

1

(A18a)

ωω=++−

−

⎜

⎝

⎟

⎠

⎢

⎛⎞

⎤

⎦

⎥

s

r

a

2(1)

2

0

3

1

(A18b)

■AUTHOR INFORMATION

Corresponding Author

*Telephone: 886-2-33663048. Fax: +886-2-2362-3040. E-mail:

huan@ntu.edu.tw.

Notes

The authors declare no competing financial interest.

■ACKNOWLEDGMENTS

This research was supported by the National Science Council of

the Republic of China.

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