Numerical calculation of the electrophoretic mobility of colloidal particles in weak electrolyte...

ed on able layer

and rateneglected

rtant in thec potentialically ands that are

Journal of Colloid and Interface Science 260 (2003) 107–117www.elsevier.com/locate/jcis

Numerical calculation of the electrophoretic mobility of colloidalparticles in weak electrolyte solutions

Susana Pedrosa,a Vladimir Shilov, and Constantino Grossea,b,∗

a Departamento de Física, Universidad Nacional de Tucumán, Tucumán, Argentinab Consejo Nacional de Investigaciones Científicas y Técnicas, Argentina

Received 15 July 2002; accepted 20 December 2002

Abstract

A numerical calculation of the electrophoretic mobility of colloidal particles in weak electrolyte solution is presented. It is basprevious work (C. Grosse, V.N. Shilov, J. Colloid Interface Sci. 211 (1999) 160–170), where the analytical theory of the thin douconcentration polarization is generalized to the case of weak electrolytes, i.e., when the dissociation–recombination equilibriumconstants both have finite values. The analytical results are first completed by including terms corresponding to co-ions that werein the original presentation. It is shown that these terms that have little bearing in the case of strong electrolytes, become quite impocase when the electrolyte is weak. The problem is then solved using the network method, leading to numerical results for the electriand the concentrations of counterions, co-ions, and neutral ion pairs. Finally, the electrophoretic mobility is calculated both analytnumerically. It is shown that the hypothesis of a weak electrolyte leads to changes of mobility with respect to the classical resulteven stronger than predicted analytically. 2003 Elsevier Science (USA). All rights reserved.

Keywords:Weak electrolytes; Colloidal particles; Electrophoretic mobility; Numerical results; Network method

di-thelyte

eleciont thhatld-ayerand

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1. Introduction

The theoretical treatments of the electrokinetic andelectric properties of colloidal suspensions based onstandard model [1–13], always consider that the electrois strong. This means that no ion pairs are present in thetrolyte solution and that no dissociation or recombinatprocesses take place. The theoretical results show thaelectrophoretic mobility is strongly dependent on, and tthe dielectric increment is mainly determined by, the fieinduced changes of the ion densities outside the double lThis suggests that even a small amount of dissociationrecombination could have an appreciable bearing on tparameters.

A generalization of the theory to the case of weak etrolytes was first presented by Baygents and Saville [These authors developed a numerical theory of electrophsis considering arbitrary values of the equilibrium dissiation–recombination constant but for the limiting case

* Corresponding author.E-mail address:[email protected] (C. Grosse).

0021-9797/03/$ – see front matter 2003 Elsevier Science (USA). All rights rdoi:10.1016/S0021-9797(02)00230-8

-

e

.

-

very fast kinetics, i.e., for infinitely large values of thdissociation–recombination rate constant. An analyticallution considering the general case was presented infor the electrophoretic mobility and in [16] for the condutivity dispersion amplitude, the permittivity increment, athe characteristic time of the low-frequency dielectric dispsion. These last works were developed using the thin doulayer approximation.

In this work we numerically solve the electrodiffusioequations for a rigid nonconducting colloidal particle inweak electrolyte solution. In contrast to Refs. [15,16], whwere developed using the thin double-layer approximatthe calculations are performed for arbitrary values ofdouble-layer thickness to particle radius ratio. We calcuthe electrophoretic mobility and compare the results wthose obtained in [15] and with a corrected analytical expsion obtained by including co-ion related terms that wneglected in the original treatment. It is shown that the vation of the electrophoretic mobility, due to the presenceion pairs and of the corresponding recombination andsociation processes, is even stronger than predicted ancally.

eserved.

http://www.elsevier.com/locate/jcis

108 S. Pedrosa et al. / Journal of Colloid and Interface Science 260 (2003) 107–117

arlytetes:

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2. Mathematical formulation and analytical solution

The system considered and the nomenclature usedidentical to those used in Refs. [15,16]. The weak electrosolution is made of the solvent and three types of solupositive counterions (upper index+), negative co-ions(upper index−), and neutral ion pairs (upper indexn).

In thermodynamic equilibrium (lower index 0), their cocentrationsC0 (number per unit volume) are related by

(1)KCn0 = C+

0 C−0 ,

whereK is the dissociation constant (in the strong eltrolyte limit K → ∞). Without an applied field, the ion anion-pair fluxes vanish,

∓ e

kTD±C±

0 ∇φ0 − D±∇C±0 = 0,

−Dn∇Cn0 = 0,

whereφ0 is the equilibrium electric potential that obeys tPoisson equation,

∇2φ0 = − e

ε

(C+

0 − C−0

),

D± andDn are the diffusion coefficients, and it is assumthat the electrolyte is univalent:z+ = z− = 1.

The solutions of this equation system for the concentions of ions in the neighborhood of a suspended negatcharged particle are

C±0 = C0e

∓φ0e/kT ,

whereC0 is the counterion or co-ion concentration far frothe particle. In contrast, the ion-pair concentration haconstant value, independent of the distance to the partic

Cn0 = C0e

−φ0e/kT C0e+φ0e/kT

K= C2

0

K.

In the presence of a stationary electric fieldE = E0 ×(cosθ er − sinθ eθ ), ion and ion-pair flows appear,

C±v± = ∓ e

kTD±C±∇φ − D±∇C± + C± V ,

Cnvn = −Dn∇Cn + Cn V ,

where v± are the velocities of ions,vn is the velocity ofion pairs, andV is the velocity of the fluid. These flows arelated by the continuity equations

∇(C±v±) = −∇(Cnvn) = S(KCn − C+C−),

whereS is the dissociation–recombination rate constant.first addend in the right-hand side of this equation represthe dissociation rate of ion pairs per unit volume, whthe second corresponds to the recombination rate ofAccording to Onsager’s theory [17], the constantS cannothave a value greater thanSmax = 1/(τC0), whereτ = ε/σ

is the electrolyte solution relaxation time,ε is its absolutepermittivity, andσ is its conductivity. For an aqueous KCelectrolyte solution withC0 = 1024 m−3, the conductivity at

e

.

300 K isσ = 0.025 S/m, so thatSmax = 3.5× 10−17 m3/s.The actual value can be substantially lower due, among ofactors, to the energy barrier associated with the dehydraof the ions required for their recombination.

The electric potential satisfies the Poisson equation,

∇2φ = − e

ε(C+ − C−),

while the velocity of the fluid is determined by the NavieStokes and the incompressibility equations,

η∇2 V − ∇p = e(C+ − C−)∇φ + ρf ( V · ∇) V ,

∇ V = 0,

wherep is the pressure andρf is the fluid density.Writing each field-dependent quantity as an equilibri

value plus a change proportional to the applied field,

C± = C±0 + δC±,

Cn = Cn0 + δCn,

φ = φ0 + δφ,

p = p0 + δp,

and keeping only terms that are linear in the applied fileads to the following equation system:

C±0 v± = −D±∇δC± ∓ e

kTD±C±

0 ∇δφ

(2)∓ e

kTD±δC±∇φ0 + C±

0V ,

(3)∇(C±

0 v±) = S[KδCn − C+

0 δC− − C−0 δC+]

,

(4)Cn0 vn = −Dn∇δCn + Cn

0V ,

(5)Cn0∇vn = −S

[KδCn − C+

0 δC− − C−0 δC+]

,

(6)∇2δφ = − e

ε(δC+ − δC−),

−η∇ × ∇ × V − ∇δp

(7)= e(C+

0 − C−0

)∇δφ + e(δC+ − δC−)∇φ0,

(8)∇ V = 0.

The analytical solution presented in [15] is based onfirst seven equations of this system (Eqs. (2)–(6)). Firstion and the ion-pair velocities are eliminated, takingdivergence of the equations for the flows. The resultingpressions,

−D±∇2δC± ∓ e

kTD±C±

0 ∇2δφ ∓ e

kTD±∇C±

0 · ∇δφ

∓ e

kTD±δC±∇2φ0 ∓ e

kTD±∇δC± · ∇φ0 + ∇C±

0 · V= S

[KδCn − C+

0 δC− − C−0 δC+]

,

Dn∇2δCn = S[KδCn − C+

0 δC− − C−0 δC+]

,

are then considered in the electroneutral electrolyte soluoutside the double layer,φ0 = 0, C± = C0, and δC+ =
0

S. Pedrosa et al. / Journal of Colloid and Interface Science 260 (2003) 107–117 109

of a

ec-

lueinge-

on-in

the

15],h

d too co-

δC− = δC, leading to

−D±∇2δC ∓ e

kTD±C0∇2δφ = S[KδCn − 2C0δC],

Dn∇2δCn = S[KδCn − 2C0δC].This equation system can be transformed into

∇2[KδCn − 2C0δC] = Λ−2[KδCn − 2C0δC],∇2δC = − S

Def[KδCn − 2C0δC],

∇2δφ = −kT

e

S�

C0Def[KδCn − 2C0δC],

where

Λ−2 = SK

Dn

(1+ 2

C0Dn

KDef

),

Def = 2D+D−

D+ + D− , � = D− − D+

D+ + D− .

The solutions of these expressions, which have the formHelmholtz and two Poisson equations, are

(9)δC = dc

r2cosθ − Λ2SP

Deff (r,Λ)cosθ,

δφ = −Er cosθ + de

r2 cosθ

(10)− �

C0

kT

e

Λ2SP

Deff (r,Λ)cosθ,

(11)δCn = 2C0

K

dc

r2cosθ + Λ2SP

Dnf (r,Λ)cosθ,

where

f (r,Λ) = e−(r−a)/Λ

r2/a2

1+ r/Λ

1+ a/Λ,

while dc, de, andP are integration constants. The value ofP

is a measure of the deviation from equilibrium of the eltrolyte solution concentrations (the productSP is propor-tional to the rate of dissociation of ion pairs), while the vaof Λ characterizes the thickness of the region surroundthe particle inside which the deviation from equilibrium btween dissociation and recombination is significant.

The boundary conditions required to determine the cstantsP , dc, andde, can be obtained in the case of a thquasi-flat double layer in local equilibrium,

Λ � χ−1,

χa � 1,

whereχ is the reciprocal of the Debye screening length:

χ =√

2e2C0

kT ε.

The results are

(12)Λ2SP = −4DnC0ah

3dc,

Ka

(13)dc = 3(R+ − R−)C0eEa3

kT

C2 + 4C0Dn

KDefK2

,

(14)de = a3EC1 + 4C0D

n

KDefK1

C2 + 4C0Dn

KDefK2

,

where

C1 = (R+ + 2− U+)(R− − 1)

+ (R− + 2− U−)(R+ − 1),

C2 = (R+ + 2− U+)(R− + 2)

+ (R− + 2− U−)(R+ + 2),

K1 = [(1+ �)(hR+ + 1)− hU+]

(R− − 1)

+ [(1−�)(hR− + 1)− hU−]

(R+ − 1),

K2 = [(1+ �)(hR+ + 1)− hU+]

(R− + 2)

+ [(1−�)(hR− + 1)− hU−]

(R+ + 2),

R± = 2

χa

[2(e∓eζ/2kT − 1

)(1+ 3m±)± 3m± eζ

kT

],

U± = 48m±

χaln

(cosh

eζ

4kT

),

m± = 2ε

3ηD±

(kT

e

)2

,

ζ is the equilibrium potential value on the surface, andη isthe electrolyte solution viscosity. In these expressions,parameter

h = −[a∂f (r,Λ)

∂r

∣∣∣∣r=a

]−1

= 1+ a/Λ

2+ 2a/Λ+ a2/Λ2

only depends on the quotientΛ/a and varies betweenh = 0(S → ∞ or K → ∞) andh = 1/2 (S → 0).

These results do not coincide with those presented in [which are simplified forms of the full results from whicthey were obtained, keeping only the main term relateconvection, and assuming that the surface current due tions is negligible as compared to that of counterions:

R+ ≈ 4

χa

(e−eζ/2kT − 1

)(1+ 3m+) = R,

R− ≈ 0,

(15)U± ≈ 0.

These approximations lead again to Eq. (12) and to

(16)dc = C0eEa3

4kT

3R

R + 2

1

1+ 2C0Dn

KDef

[1− R(1+�)(1−2h)

2(R+2)

] ,

(17)de = Ea3

4

R − 4

R + 2

1+ 2C0Dn

KDef

[1+ R (1+�)(1−2h)−3�

R−4

]1+ 2C0D

n

KDef

[1− R(1+�)(1−2h)

2(R+2)

] .


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3. Numerical solution

In order to solve the full equation system (2)–(8) inwhole volume of the electrolyte solution, it is first simplified, expressing the concentration changes in terms ofield-induced changes of the electrochemical potentialsδµ,defined as

δC± = C±0 (δµ± ∓ δφ),

δCn = Cn0δµ

n,

so that

v± = −D±∇δµ± + V ,

vn = −Dn∇δµn + V .

In these expressions the tilde(˜) over a symbol denotesdimensionless quantity:

δφ = e

kTδφ and δµ = 1

kTδµ.

The ion and ion-pair velocities are then eliminated, coming the equations for the flows with the continuity equatioFinally, the change of the pressure is eliminated by takthe curl of Eq. (7). This transforms the equation system

∇2δµ± + SC±0

D± [δµn − δµ− − δµ+]

= ±∇φ0

[∇δµ± − V

D±

],

∇2δµn − SK

Dn[δµn − δµ− − δµ+] = 0,

∇2δφ = e2

kT ε

[(C+

0 + C−0

)δφ +C−

0 δµ− −C+0 δµ+]

,

η∇2(∇ × V ) = e(C+

0 ∇δµ+ − C−0 ∇δµ−) × ∇φ0,

∇ V = 0.

In view of the axial symmetry of the considered systethe dimensionless electric and electrochemical potechanges can be written as

δφ(r) = [δφ(r)− r

]E cosθ,

δµ±(r) = [δµ±(r)∓ r

]E cosθ,

δµn(r) = δµn(r)E cosθ,

wherer = r/a andE = eEa/kT . This transforms the firsfour equations into

L(δµ±) = −SC∓0

D± [δµn − δµ+ − δµ−]

± dφ0

dr

(dδµ±

dr∓ 1

a

)∓ dφ0

dr

Vr

D± ,

L(δµn) = SK

Dn[δµn − δµ+ − δµ−],

L(δφ) = χ2 [(e−φ0 + e−φ0

)δφ − e−φ0δµ+ + eφ0δµ−]

,
2
where

L = d

dr2 + 2

r

d

dr− 2

r2

andVr(r) is the part of the radial component of the fluvelocity that depends on the radial coordinate:

V (r) = Vr(r)E cosθ er − Vθ(r)E sinθ eθ .

The Navier–Stokes equation transforms into

L(b)= e

ηr

dφ0

dr

[C+

0 (δµ+ − r)− C−0 (δµ− + r)

],

whereb(r) is the vorticity, defined as

b(r)≡ ∇ × V =[−dVθ

dr+ (Vr − Vθ)

r

]E sinθ eϕ

= b(r)E sinθ eϕ.

Finally, the incompressibility equation makes it possibleexpress the radial component of the velocity in terms ofvorticity:

d2Vr

dr2= −4

r

dVr

dr− 2

rb.

In order to optimize both the efficiency and the precisof the numerical solution of this equation system, furtvariable changes are necessary. First, the potential chanseparated into two terms: a short-range term mainly relto the field-induced charge in the electrical double laand a long-range one mainly related to the field-induelectrolyte concentration changes outside the double lay

δφ = δψ + 1

2(δµ+ − δµ−).

In this way, the dipolar coefficient is determined exclusivby the difference between the electrochemical potentThe resulting equation forδψ is

L(δψ)= −S

2[δµn − δµ+ − δµ−]

[C+

0

D− − C−0

D+

]

− 1

2

dφ0

dr

d(δµ+ + δµ−)

dr

+ dφ0

dr

Vr

Def+ χ2

2

(e−φ0 + e−φ0

)δψ

+ χ2

4

(e−φ0 − e−φ0

)(δµ+ + δµ−).

The second change consists in replacing the indepenvariable r by x = a/r. The main advantage of this nevariable is that it has a finite range extending from 1 tinstead of the infinite range froma to ∞. Also, a numericasolution requires that the integration interval be divideda finite number of intervals, and constant size intervalthex variable correspond to small intervals in ther variableclose to the particle (where the dependent variables strodepend onr) and large intervals far from the particle. T


di-arti-

e,

ric

i-can

le

of

of

ryicle

and

d tonsor

eeto

berce

resulting final equation system is

d2y±

dx2 = 2

x2y± ± 1

x

dφ0

dxy± − 2

x

dy±

dx+ 1

x3

dφ0

dx

± dφ0

dx

dy±

dx− a2

x4

SC∓0

D±

[δµn

x− (y+ + y−)

]

(18)± 1

x

dφ0

dx

(kT

e

)2ε

D±η

[u− u(x = 1)

x2

],

(19)d2δµn

dx2= 2

x2δµn + a2

x3

SK

Dn

[δµn

x− (y+ + y−)

],

d2δψ

dx2 = 2

x2δψ − dφ0

dx

x

2

d(y+ + y−)

dx− dφ0

dx

(y+ + y−)

2

+ (χa)2

2x4

(eφ0 + e−φ0

)δψ

+ (χa)2

4x3

(eφ0 − e−φ0

)(y+ + y−)

− SC0a2

2x3

[e−φ0

D− − eφ0

D+

][δµn

x− (y+ + y−)

]

(20)− dφ0

dx

(kT

e

)2ε

ηDef

[u− u(x = 1)

x2

],

d2b

dx2 = 2b

x2 − (χa)2

2

dφ0

dx

(21)×[(

y+e−φ0 − y−eφ0) − 1

x2

(e−φ0 + eφ0

)],

(22)d2u

dx2= − 2

x5b + 2

x2u− 2

x

du

dx,

where

y± = δµ± r

a,

b = e2ηa2

(kT )2εb,

u = e2η

(kT )2ε

r2

a

(Vr − V ∞

r

).

In order to solve this equation system, boundary contions for each dependent variable on the surface of the pcle and at infinity must be specified.

3.1. Conditions on the surface of the particle

• Continuity of the field-induced electric potential chang

δφ(a) = δφi(a),

whereφi is the electric potential inside the particle.• Continuity of the normal component of the elect

displacement,

ε∇δφ|r=a · er = εi∇δφi |r=a · er ,

whereεi is the permittivity of the particle. This condtion specifies that no field-induced surface chargesbe created.

• Zero value of both components of the fluid velocity,

Vr(a)= Vθ(a) = 0,

which follows from the impenetrability of the particand the adherence conditions.

• Zero value of the derivative of the radial componentthe fluid velocity,

dVr

dr

∣∣∣∣a

= 0,

which follows from the previous conditions and thatfluid incompressibility.

• Value of the vorticity and its derivative,

e

∞∫a

(C+

0 δµ+ − C−0 δµ−)dφ0

drr2dr

+ a2[ηa

db

dr

∣∣∣∣a

− ηb(a)

]= 0,

which follows from the condition that, in the stationasituation considered, the total force acting on the partmust be equal to zero [18],

Fm + Fe = 0,

where the addends denote forces of mechanicalelectric origin, respectively.

The mechanical force, due to pressure gradients anviscosity, is obtained by integrating the viscous stress teover the surface of the particle,

Fm =∫S

σd S =∫S

(σrr cosθ − σθr sinθ) dS k

= −E

∫S

[p(a)E cos2 θ + η

∂Vθ

∂r

∣∣∣∣a

E sin2 θ

]dS k

= −4

3πa2E

[p(a)+ 2ηb(a)

]k,

where k is a unit vector pointing in the direction of thapplied field. The pressure can be determined from thθ

component of the Navier–Stokes equation, which leads

p(a) = −e(C+

0 − C−0

)δφ(a)− η

[adb

dr

∣∣∣∣a

+ b(a)

],

so that the mechanical force becomes

Fm = −4

3πa2E

×[−e

(C+

0 − C−0

)δφ(a)− ηa

db

dr

∣∣∣∣a

+ ηb(a)

]k.

As for the electric force acting on the particle, it cancalculated as minus the volume integral of the body fo


es-

ual

theliedical,[18,hats theucesanctricwas

icalsus-Theentnter-ele-orkntaryingce ofuitscor-

acting on the fluid:

Fe = −e

∫V

[(δC+ − δC−)∇φ0 + (

C+0 − C−

0

)∇δφ]dV

= −4

3πeE

(C+

0 − C−0

)δφ(a)a2.

3.2. Boundary conditions at infinity

• All the field-induced perturbations related to the prence of the particle tend to zero:

δµ±(r → ∞) = δµn(r → ∞) = δφ(r → ∞) = 0.

• The value of the fluid velocity is constant (that is, eqto minus the electrophoretic velocity):

dVr

dr

∣∣∣∣r→∞

= dVθ

dr

∣∣∣∣r→∞

= 0.

• The value of the vorticity is zero,

b(r → ∞) = 0,

which follows from the incompressibility condition.

The numerical calculations were performed usingNetwork method [19], which has been successfully appover the past years to solve a variety of physicochemelectrochemical, dielectric, and electrokinetic problems20–24]. It consists in designing an electric network thas the same balance and constitutive equations amathematical model of the considered system. This redthe original problem to the solution of a circuit, which cbe accomplished using any commercial program for elecircuit analysis. We used the PSPICE package, whichfound most useful for this purpose.

In order to generate the electric network, the physspace representing the electrolyte solution around thepended particle is divided into a series of compartments.system of Eqs. (18)–(22) written for any given compartmis then discretized, and the resulting equation terms are ipreted as lumped circuit elements. This leads to a set ofmentary circuits, one for each equation. The whole netwis made connecting in series each one of these elemecircuits with the corresponding circuits of the neighborcompartments. At both ends, corresponding to the surfathe particle and to infinity, the series of elementary circare terminated with lumped elements that represent the

Fig. 1. Schematic representation of the electrical network used to solve the system of Eqs. (18)–(22).


ep-ingr ofset

citybed on

erwois

en-l toheirface

(9)

nal

re-

ndnceand

ng

sus-is-liedned

ubleap-

lec-,

lytek

responding boundary conditions. The whole network is rresented in Fig. 1, while the definitions of all the intervenelements are presented in Appendix A. The total numbecompartments in the network represented in Fig. 1 wasto N = 400.

4. Electrophoretic mobility

The analytical expression for the electrophoretic veloVeph of a spherical particle with a thin double layer canobtained using the method presented in [15] that is basethe well-known expression [25]

(23)Veph= −2

3

Vt(a, θ)

sinθ,

whereVt is the tangential fluid velocity outside the outboundary of the double layer. This velocity is made of tcontributions [26–28]: the electroosmotic term, whichproportional to the tangential gradient of the electric pottial, and the capillary osmotic term, which is proportionathe tangential gradient of the electrolyte concentration. Tvalues can be calculated considering a charged flat interwhich leads to

Vt = εζ

η∇t φ − 4εkT

ηeC0

kT

eln

(cosh

eζ

4kT

)∇t δC.

Combining this expression with Eq. (23) and using Eqs.and (10) gives

(24)

Veph= 2

3E

{εζ

η

[1− de

Ea3 + kT

eE

Λ2SP�

C0aDef

]

− 4εkT

ηeln

(cosh

eζ

4kT

)

×[−kT

eE

dc

C0a3 + kT

eE

Λ2SP

C0aDef

]}.

This result, together with Eqs. (12)–(14), leads to the fiexpression:

ηVeph

Eε= ζ +

{4(R+ − R−)

[2kT

eln

(cosh

eζ

4kT

)

×(

1+ 4C0Dnh

KDef

)− 2ζ

C0Dnh�

KDef

]

− ζ

{2C1 + C2

3+ 4C0D

n

KDef

2K1 + K2

3

}}

(25)

/{C2 + 4C0D

n

K2

}.

KDef

,

This expression does not coincide with the result psented in [15],

ηVeph

Eε= ζ +

{4R

[2kT

eln

(cosh

eζ

4kT

)

×(

1+ 4C0Dnh

KDef

)− 2ζ

C0Dnh�

KDef

]

− ζR

{2+ 4C0D

n

KDef(1− �)

}}

(26)

/{4(R + 2)+ 4C0D

n

KDef

× [2(1+ �)(hR + 1)+ (1− �)(R + 2)

]},

which was obtained combining Eq. (24) with Eq. (12) athe approximate equations (16) and (17). The differebetween these two expressions is usually less than 5%it does not decrease with theζ potential, at least for thecases andζ potential range considered in the forthcomidiscussion.

5. Results and discussion

The calculations were performed assuming that thepended particle is negatively charged. In the following dcussion we shall assume that the electric field is appfrom left to right. Figure 2 represents the results obtai

Fig. 2. Field-induced ion concentration changes outside the thin dolayer on the right-hand side of a negatively charged particle with fieldplied from left to right (cosθ = 1). Strong electrolyte andD− = D+: (2)full theory, (dashed line) simplified theory, (thin line) numerical. Weak etrolyte with D− = D+: (1) full theory, (dashed line) simplified theory(thick line) numerical. Weak electrolyte withD− = D+/5: (E) full the-ory, (dashed line) simplified theory, (thick line) numerical. Weak electrowith D− = 5D+: (!) full theory, (dashed line) simplified theory, (thicline) numerical. Values used:a = 0.2 µm,ζ = −200 mV,C0 = 1024 m−3,K = 1024 m−3, S = 10−18 m3/s, D+ = Dn = 2 × 10−9 m2/s,T = 298.16 K, ε = 78.54ε0, η = 8.904× 10−4 poise.


theheyory,ntedrre-po-l andandn-

tionex

ef-ical

eensidewerthe

). Inthe

e

yte.ong

. 2)arealsionsTheticlesity,nearhes

thethe

andhenr).ieshree

of

Fig. 3. Same as Fig. 2 but for the ion-pair concentration changes.

Fig. 4. Same as Fig. 2 but for the electric potential changes.

for the field-induced ion concentration changes alongsymmetry axis on the right-hand side of the particle. Tare compared to the full analytical results of the theEqs. (9), (12), (13), and to the approximate results presein [15], Eqs. (6), (9), (12). Figure 3 represents the cosponding results for ion pairs and Fig. 4 for the electrictential changes. Figures 2 and 4 also include numericaanalytical results corresponding to strong electrolytesD− = D+. It should be noted that in the approximate aalytical expressions for strong electrolytes, the value ofD−has no bearing either on the field-induced ion concentrachanges or on the electric potential changes, since thepression ofR (Eqs. (15)) does not include the diffusion coficients of co-ions. In contrast, since both the full theoretexpressions and the numerical results do depend onD−, thecurves corresponding to strong electrolytes andD− �= D+were omitted in order to simplify Figs. 2 and 4.

As can be seen, there is fairly good agreement betwthe numerical results and the theoretical predictions outthe double layer (this agreement further improves for lo|ζ | values) and even up to the surface of the particle incase of the concentration changes of ion pairs (Fig. 3contrast, close to the particle, the numerical results for

-

Fig. 5. Electrophoretic mobility versusζ potential of a spherical particlin a weak and in a strong electrolyte solution withD− = D+. Numericalresults: (solid symbols) weak electrolyte, (thick line) strong electrolFull theory: (empty symbols) weak electrolyte, (thick dashed line) strelectrolyte. Values used: (2,1) K = 1022 m−3, (F,E) K = 1024 m−3,(",!) K = 1026 m−3. Remaining values as in Fig. 2.

Fig. 6. Same as Fig. 5, but forD− = D+/5.

concentration changes of counterions and co-ions (Figand for the changes of the electric potential (Fig. 4)totally different from those obtained from the analyticexpressions (as expected, since the analytical expreswere deduced for the region outside the double layer).strong decrement of the electric potential close to the par(Fig. 4) corresponds to the field-induced charge denwhich is positive near the particle, becomes negativethe outer boundary of the double layer, and finally vanisfar from the particle (Fig. 2).

The figures also show that the differences betweennumerical and the analytical results are comparable todifferences between the results obtained using the fullthe approximate analytical expressions (especially so wthe diffusion coefficients of counterions and co-ions diffe

The results obtained for the electrophoretic mobilitare represented in Figs. 5–7, which correspond to tdifferent relationships between the diffusion coefficients


-ym-

aredtheutof

en-ionthe

dednd-icprecan

lity,hatct is

m ofsus-n ofthepo-the

ichultsoryfromthe

ults.bil-icalhatlyteofent

.

Fig. 7. Same as Fig. 5, but forD− = 5D+.

Fig. 8. Electrophoretic mobility versusζ potential of a spherical particlein a weak electrolyte solution withD− = D+, relative to the corresponding value in a strong electrolyte with the same ion content. Solid sbols: numerical results, empty symbols: full theory. Values used: (2,1)K = 1022 m−3, (F,E) K = 1024 m−3, (",!) K = 1026 m−3. Remain-ing values as in Fig. 2.

counterions and co-ions. The numerical results are compwith the full analytical results, Eq. (25). As can be seen,theory works fairly well for moderately charged particles boverestimates the electrophoretic mobility at low valuestheζ potential and underestimates it for highζ .

Nevertheless, it correctly predicts the qualitative depdence of the electrophoretic mobility on the dissociatconstantK. This can be appreciated in Figs. 8–10, wherevalue of the electrophoretic mobility of particles suspenin a weak electrolyte solution divided by the correspoing value in a strong electrolyte solution with equal ionstrength is presented. These figures confirm the findingssented in [15]: the dissociation–recombination processesmarkedly increase the value of the electrophoretic mobiespecially when the mobility of co-ions is greater than tof counterions. The numerical results show that this effeeven greater than the theoretical prediction.

-

Fig. 9. Same as Fig. 8, but forD− = D+/5.

Fig. 10. Same as Fig. 8, but forD− = 5D+.

6. Conclusion

We present a numerical solution of the standard systeelectrokinetic equations for a charged spherical particlepended in a weak electrolyte solution and under the actioa weak stationary electric field. The results obtained forion and ion-pair concentrations, as well as the electrictential distribution around the particle, generally confirmanalytical solution of these equations presented in [15].

However, important quantitative differences exist, whare partly due to the simplifications of the analytical resappearing in [15]. We present the full results of the theand show that the differences between values obtainedthe simplified and the full expressions are comparable todifferences between the analytical and the numerical res

The numerical calculations of the electrophoretic moity are also in good qualitative agreement with the analytpredictions. They confirm the finding presented in [15] tdissociation–recombination processes in a weak electrosolution can greatly increase the electrophoretic mobilitysuspended particles. Moreover, they show that this incremis even stronger than predicted by the analytical solution


hel de

s it

e

e

ons,

yer.

77

. 88

. 93

.

lero,

Acknowledgment

This work was partially supported by a grant of tConsejo de Investigaciones de la Universidad NacionaTucumán.

Appendix A

Equation (18) can be written in the form

dhy±

dx= − 2

x2y± ∓ 1

x

dφ0

dxy± + 2

x

dy±

dx− 1

x3

dφ0

dx

+ a2

x4

SC∓0

D±

[δµn

x− (y+ + y−)

]

(A.1)∓ 1

x

dφ0

dx

(kT

e

)2ε

D±η

[u− u(x = 1)

x2

],

where

hy±(x)= −dy±(x)

dx.

This last expression can be discretized, which transforminto

hy±(xj ± �xj

2

)= ±y±(xj ) − y±(

xj ± �xj2

)�xj

2

.

This equation has the same form as Ohm’s Law, wherehy±is the current,y± the voltage, and�xj/2 the resistance. Thchange in the current is generated by a current sourceG

j

y±and a resistanceRj

y± . Their values are obtained from thdiscretized form of Eq. (A.1):

Gj

y± = ∓(y±

x± 1

x3

)�φ0j + 2

x�y±

j

+ a2�x

x4

SC∓0

D±

[δµn

x− (y+ + y−)

]

∓ 1

x�φ0j

(kT

e

)2 ε

D±η

[u − u(x = 1)

x2

],

(A.2)Rj

y± = x2

2�x.

In Eqs. (A.2), and in the following equations,

�fj = f

(xj + �xj

2

)− f

(xj − �xj

2

),

wheref stands for a function ofx: φ0, y+, y−, or u. Thecorresponding circuit is shown schematically in Fig. 1.

Using this same procedure with the remaining equati(19)–(22), leads to

hδµn(x)= −dδµn(x)

dx,

Gδµn = −a2

3

SK�x

n

[δµn

− (y+ + y−)

],

x D x

Rδµn = x2

2�x,

hδψ (x) = −dδψ(x)

dx,

Gδψ = x

2�x�φ0j

(�y+

j + �y−j

) + 1

2�φ0j (y

+ + y−)

− (χa)2�x

2x4

(eφ0 + e−φ0

)δψ

− (χa)2�x

4x3

(eφ0 − e−φ0

)(y+ + y−)

+ SC0a2�x

2x3

[e−φ0

D− − eφ0

D+

][δµn

x− (y+ + y−)

]

+ �φ0j

(kT

e

)2ε

ηDef

[u− u(x = 1)

x2

],

Rδψ = x2

2�x,

hb(x) = −db(x)

dx,

Gb = (χa)2

2�φ0j

[(y+e−φ0 − y−eφ0

) − 1

x2

(e−φ0 + eφ0

)],

Rb = x2

2�x,

hu(x) = −du(x)

dx,

Gu = 2�x

x5 b + 2

x�uj ,

Ru = x2

2�xu.

References

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[19] B. Speiser, in: A.J. Bard, I. Rubinstein (Eds.), Electroanalytical Chistry, Vol. 19, Dekker, New York, 1996, p. 64.

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Numerical calculation of the electrophoretic mobility of colloidal particles in weak electrolyte...

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