Electrokinetics of Concentrated Suspensions of Spherical Colloidal Particles: Effect of a Dynamic Stern Layer on Electrophoresis and DC Conductivity

Journal of Colloid and Interface Science243,351–361 (2001)doi:10.1006/jcis.2001.7903, available online at http://www.idealibrary.com on

Electrokinetics of Concentrated Suspensions of SphericalColloidal Particles: Effect of a Dynamic Stern Layer

on Electrophoresis and DC Conductivity

F. Carrique,∗,1 F. J. Arroyo,† and A. V. Delgado‡∗Departamento de Fısica Aplicada I, Facultad de Ciencias, Universidad de Malaga, 29071 Malaga, Spain;†Departamento de Fısica,

Facultad de Ciencias Experimentales, Universidad de Jaen, 23071 Jaen, Spain; and‡Departamento de Fısica Aplicada,Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain

Received March 26, 2001; accepted August 4, 2001; published online October 5, 2001

In this paper the theory of the electrophoretic mobility and electri-cal conductivity of concentrated suspensions of spherical colloidalparticles, developed by H. Ohshima (J. Colloid Interface Sci. 188,481 (1997); J. Colloid Interface Sci. 212,443 (1999)), has been re-vised and extended to include the effect of a dynamic Stern layeron the surface of the particles. The starting point has been the the-ory developed by C. S. Mangelsdorf and L. R. White (J. Chem.Soc., Faraday Trans. 86, 2859 (1990)) dealing with the calculationof the electrophoretic mobility of a colloidal particle, when lateralmotion of ions in the inner region of the double layer is possible(dynamic Stern layer (DSL)). The effects of Stern layer parameterson the electrophoretic mobility are first discussed and comparedwith the case when a Stern layer is absent. The numerical resultsshow that regardless of the values of the Stern layer and solution pa-rameters chosen, the presence of a DSL causes the electrophoreticmobility to decrease in comparison with the standard case (no Sternlayer present) for every volume fraction. Furthermore, the strongerthe hydrodynamic particle–particle interactions as volume fractionincreases, the lower the mobility for a given zeta potential, bothmechanisms tending to increase the retarding forces that brakethe electrophoretic motion. Concerning direct current conductiv-ity calculations, results show that the presence of a DSL causes theelectrical conductivity to increase in comparison with the standardcase (no Stern layer present) for every volume fraction and zeta po-tential. Obviously, the additional conductivity contribution of everyparticle in the system is related to the presence of an extra mobilelayer, the DSL. The treatment is based on the use of a cell modelto account for hydrodynamic and electrical interactions betweenparticles. We also discuss the use of either Levine–Neale or Shilov–Zharkikh boundary conditions, leading to different results for themobility and direct current conductivity in conditions of both low(where analytical expressions can be reached) and arbitrary zeta po-tentials. The analogies and discrepancies between both approachesare discusesd. C© 2001 Academic Press

Key Words: electrophoretic mobility; electrical conductivity; con-centrated suspensions; electrokinetic equations; electric doublelayer; dynamic Stern layer.

1 To whom correspondence should be addressed. E-mail: carrique@uma

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35

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1. INTRODUCTION

A great deal of effort has recently been devoted to improing the results of the standard electrokinetic theories dealwith different electrokinetic phenomena in dilute colloidal supensions. One of the most remarkable extensions of these etrokinetic models has been the inclusion of a dynamic Stelayer (DSL) onto the surface of the colloidal particles. ThuZukoski and Saville (1) developed a DSL model to reconcthe differences observed between zeta potentials derived felectrophoretic mobility and static conductivity measuremenShortly after, Mangelsdorf and White (2), using the techniqdeveloped by O’Brien and White for the study of the eletrophoretic mobility of a colloidal particle (3), included a generDSL model in the study of electrophoresis. They analyzedrole of different Stern-layer adsorption isotherms on both eletrophoretic mobility and suspension conductivity. More recentKijlstra et al. (4) applied the theory of Stern-layer transportthe study of the low-frequency dielectric response of colloidsuspensions, extending the thin-double-layer theory of Fixm(5, 6). Likewise, Rosenet al.(7) generalized the standard theorof the conductivity and dielectric response of a colloidal supension in alternating electric fields of DeLacey and White (8assuming the model of the Stern layer developed by Zukoand Saville. Very recently, Mangelsdorf and White (9, 10) dveloped a general DSL model to be applied to electrophoreand dielectric response in oscillating electric fields. In generthe DSL models seem to improve the agreement between theand experiments (4, 7, 11, 12) as compared with the standpredictions in dilute suspensions, although there are still imptant discrepancies (in particular, the DSL theory of the primaelectroviscous effect seems to increase the separation betwcalculated and measured data; see Refs. (13–16).

On the other hand, relatively few theoretical studies have dewith the more practical situation of concentrated suspensioFocusing on the problem of electrophoresis, Levine and Ne(17) developed a mobility expression for spherical particles wlow zeta potentials in concentrated suspensions on the bof the Kuwabara cell model (18), in order to account for th

1 0021-9797/01 $35.00Copyright C© 2001 by Academic Press

All rights of reproduction in any form reserved.

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352 CARRIQUE, ARRO

hydrodynamic particle-particle interactions. Kozak and Dav(19, 20) also studied the electrokinetics of concentrated sussions and derived a mobility expression valid for arbitrary zpotential and nonoverlapping double layers. Likewise, Ohsh(21) derived a general mobility expression for spherical pacles in concentrated suspensions tending toward that of Leand Neale for low zeta potentials, and to that of Kozak aDavies for all zeta potentials and nonoverlapping double lers. Ohshima’s result is also based on the Kuwabara cell mas that of Levine and Neale. However, very recently Duket al. (22) have pointed out that the Levine–Neale cell modemployed by many authors to develop theoretical electrokinmodels in multiparticle systems, including sedimentation, etrophoresis, and conductivity in concentrated suspensions21, 23–25), has some deficiencies. According to Dukhinet al.(22) the Levine–Neale cell model is not compatible with tvolume fraction dependence of the exact Smoluchowskiin concentrated suspensions. Instead of the Levine–Nealemodel, Dukhinet al. suggest using the Shilov–Zharkikh cemodel (26), which is based on arguments of nonequilibrithermodynamics, and not only agrees with Smoluchowski’ssult but also correlates with the electrical conductivity of tMaxwell–Wagner theory (27). Thus, it appeared quite intereing to explore in more detail the consequences arising frthe inclusion of the Shilov–Zharkikh cell model into Ohshimatheory of the electrophoretic mobility and direct current (Dconductivity of concentrated suspensions.

In the present paper, we first solve the electrokinetic equat(with Levine–Neale and Shilov–Zharkikh boundary conditionto obtain numerical data of electrophoretic mobility and Dconductivity for arbitrary zeta potential and volume fractiowhen nonoverlapping double layers are assumed.

The DSL correction to the electrokinetic theories is then dewith. A DSL extension of Ohshima’s theory of the sedimentatvelocity and potential in dilute (28) and concentrated (29) spensions has been recently carried out. In this work, we exthe standard Ohshima’s theory of the electrophoretic mob(21) and DC conductivity of spherical particles in a concentrasuspension (25) to include a DSL model. As in previous pap(28, 29), we will use the method that Mangelsdorf and Whdeveloped to allow for the adsorption and lateral motion of ioin the inner region of the double layer (2).

In summary, the aim of this investigation can be describas follows. First, to derive a new mobility formula for low zepotentials according to the Shilov–Zharkikh cell model for tdescription of concentrated suspensions. Second, to obtanumerical solution of the standard Ohshima’s theory of eltrophoresis in concentrated suspensions for the whole rangzeta potential, volume fraction, and nonoverlapping doubleers, and also, when this theory is modified, to allow forconsiderations of the Shilov–Zharkikh cell model. A similanalysis will be carried out using the similar problem of t

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concentrated suspensions (with Shilov–Zharkikh’s conditioto include a DSL on the surface of the particles.

2. GOVERNING EQUATIONS ANDBOUNDARY CONDITIONS

Before proceeding, it will be useful to briefly reviewOhshima’s standard theory of electrophoresis in concentrasuspensions (21), and to show the notation used in thisper. Concerned readers are referred to Ohshima’s paper fcomplete treatment. The standard theory of the electrophsis in a concentrated suspension of spherical colloidal partiwas developed by Ohshima on the basis of the Kuwabaramodel (18) to account for the hydrodynamic particle–particinteractions (see Fig. 1). According to this model, each sphical particle of radiusa is surrounded by a concentric shell oan electrolyte solution, having an outer radiusb such that theparticle/cell volume ratio in the unit cell is equal to the particvolume fraction throughout the entire suspension:

φ = (a/b)3. [1]

The surfacer = a is usually called the “slipping plane.” Thisis the plane outside which the continuum equations of hyddynamics are assumed to hold. Let us consider now a chaspherical particle of radiusa immersed in an electrolyte solution composed ofN ionic species of valencieszi , bulk numberconcentrationsn∞i , and drag coefficientsλi (i = 1, . . . , N). Theaxes of the spherical coordinate system (r, θ, ϕ) are fixed at thecenter of the particle. The latter is assumed to move in an etric field E with a velocityve, the electrophoretic velocity, in theelectrolyte solution of viscosityη. The electrophoretic mobilityue is defined byve = ueE. The polar axis (θ = 0) is set paralleltoE. In the absence of the field the particle has a uniform elecpotential, the zeta potentialζ , atr = a. A complete solution tothe problem would require knowledge of the electric potent9(r ), the number density of each type of ionni (r ), and the driftvelocityvi(r ) of each ionic species (i = 1, . . . , N), the fluid ve-locity v(r ), and the pressurep at every pointr in the system.The fundamental equations governing the problem are (2, 3

∇29(r ) = − ρ(r )

εrsε0, [2]

ρ(r ) =N∑

i=1

zi eni (r ), [3]

η∇2v(r )−∇ p(r )− ρ∇9(r ) = 0, [4]

∇ · v(r ) = 0, [5]

vi = v− 1

λ i∇µi (i = 1, . . . , N), [6]

µi (r ) = µ∞i + zi e9(r )+ KBT ln ni (r ) (i = 1, . . . , N), [7]

ELECTROKINETICS OF SPHERICAL COLLOIDAL PARTICLES 353

).

tse

t

eyat

FIG. 1. Schematic picture of an ensemble of spherical particles in

wheree is the elementary electric charge,KB is Boltzmann’sconstant, andT is the absolute temperature. Equation [2]Poisson’s equation, whereεrs is the relative permittivity of thesolution,ε0 is the permittivity of a vacuum, andρ(r ) is the elec-tric charge density given by Eq. [3]. Equations [4] and [5] areNavier–Stokes equations appropriate to a steady incompresfluid flow at low Reynolds number in the presence of an eltrical body force. Equation [6] expresses that the ionic flowcaused by the liquid flow and the gradient of the electrocheical potential defined in Eq. [7], and it can be related tobalance of the hydrodynamic drag and electrostatic and therdynamic forces acting on each ionic species. Equation [8] iscontinuity equation expressing the conservation of the numof each ionic species in the system. The drag coefficientλi isrelated to the limiting conductance30

i of the i th ionic speciesby (3)

λi = NAe2|zi |30

i

(i = 1, . . . , N), [9]

whereNA is Avogadro’s number. At equilibrium, that is, in theabsence of the electric field, the distribution of electrolyte io

a concentrated suspension according to the Kuwabara cell model (Ref. (18)

is

heiblec-ism-hemo-theber

obeys the Boltzmann distribution

n(0)i = n∞i exp

(−zi e9(0)

KBT

)(i = 1, . . . , N), [10]

and the equilibrium electric potential9(0) satisfies the Poisson–Boltzmann equation

1

r 2

d

dr

(r 2 d9(0)

dr

)= −ρ

(0)el (r )

εrsε0, [11]

ρ(0)el (r ) =

N∑i=1

zi e n(0)i (r ), [12]

ρ(0)el being the equilibrium electric charge density.The unperturbed or equilibrium electrical potential must ob

the following boundary conditions at the slipping plane andthe outer surface of the cell,

9(0)(a) = ζ, [13]

d9(0)

ns dr(b) = 0. [14]

Y

p

a

px

t

yu

bh

d

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354 CARRIQUE, ARRO

As the coordinate system is set fixed at the center of theticle, the boundary conditions for the liquid velocityv and theionic velocity of each ionic species at the particle surfaceexpressed by the equations

v = 0 at r = a, [15]

vi · r = 0 atr = a (i = 1, . . . , N). [16]

Equation [15] expresses that the fluid layer adjacent to theticle surface is at rest, and Eq. [16] that there are no ion fluthrough the slipping plane (r is the unit normal outward fromthe particle surface). According to the Kuwabara cell model,liquid velocity at the outer surface of the unit cell must satisthe conditions

vr = −ve cosθ = −ueE cosθ at r = b, [17]

ω = ∇ × v = 0 at r = b, [18]

meaning, respectively, that at that surface the liquid velocitparallel to the electrophoretic velocity, and the vorticity is eqto zero.

Following Ohshima, we will assume that the electrical doulayer around the particle is only slightly distorted owing to telectric field (we assume that the external field is low enoughthis condition to be valid; this condition is most often fulfillein practical situations), so that a linear perturbation schemethe above-mentioned quantities can be used,

ni (r ) = n(0)i (r )+ δni (r ) (i = 1, . . . , N), [19]

9(r ) = 9(0)(r )+ δ9(r ), [20]

µi (r ) = µ(0)i + δµi (r ) (i = 1, . . . , N), [21]

(as usual the superscript(0) refers to equilibrium). The perturbations in ionic number density and electric potential are relateeach other through the perturbation in electrochemical poteby

δµi = zi eδ9 + KBTδni/

n(0)i (i = 1, . . . , N). [22]

In terms of the perturbation quantities, the condition thationic species cannot penetrate the particle surface in Eq.(DSL not yet considered) transforms into

∇δµi · r = 0 atr = a (i = 1, . . . , N). [23]

According to Ohshima (21), the boundary condition for tperturbed electric potential at the outer surface of the unitis expressed by

∇δ9 · r = −E · r at r = b. [24]

However, according to the Shilov–Zharkikh cell model (26), th

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latter condition changes to

δ9 = −〈E〉 · r at r = b [25]

and provides, as Dukhinet al. pointed out (22), the connection between the macroscopic, experimentally measured elefield 〈E〉, and local electric properties. The differences betweboth choices of boundary conditions stem from the way in whthe macroscopic (experimentally observable) electric field isfined in connection with local properties. Thus, for Levine aNeale (17) the local electric field atr = b is parallel to the ex-ternally applied electric field. In contrast, Shilov and Zharkidefine a macroscopic field as an average of−∇δ9 performedin such a way that Onsager reciprocity relationships hold,matter the particle concentration (22, 26). For nonoverlappdouble layers, Eq. [22] becomes at the outer region of the(21, 25)

δµi = zi eδ9, [26]

and, correspondingly, Eq. [24] transforms into

∇δµi · r = −zi eE · r at r = b [27]

and Eq. [25] into

δµi = −zi e〈E〉 · r at r = b. [28]

All Ohshima’s equations in the rest of the paper will alsovalid for the Shilov–Zharkikh cell model if the applied electrfield E is substituted by the macroscopic electric field〈E〉.

Spherical symmetry considerations led Ohshima to introdthe radial functionsh(r ), φi (r ), andY(r ) then write

v(r ) = (vr , vθ , vϕ) =(−2

rhE cosθ,

1

r

d

dr(r h) E sinθ, 0

),

[29]

δµi (r ) = −zi eφi (r )(E · r ) (i = 1, . . . , N), [30]

δ9 = −Y(r )(E · r ) [31]

to obtain the following set of coupled ordinary differential equtions and boundary conditions at the slipping plane and atouter surface of the cell (21),

L(Lh) = − e

ηr

dy

dr

N∑i=1

n∞i z2i exp(−zi y)φi (r ), with y = e9(0)

KBT,

[32]

Lφi (r ) = dy

dr

(zi

dφi

dr− 2λi

e

h(r )

r

)(i = 1, . . . , N), [33]

1 N∑

εrsεoKBTi=1

zi e ni (r )[Y(r )− φi (r )], [34]

R

d

d

t

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ao

a

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ELECTROKINETICS OF SPHE

with L being a differential operator defined by

L ≡ d2

dr2+ 2

r

d

dr− 2

r 2, [35]

h(a) = dh

dr(a) = 0, Lh(r ) = 0 atr = b, [36]

h(b) = ueb

2, [37]

dφi

dr(a) = 0 (i = 1, . . . , N), [38]

dφi

dr(b) = 1. [39]

However, if we consider the Shilov–Zharkikh boundary contion given by Eq. [25], a different result is found,

φi (b) = b, [40]

where Eq. [30] has been used. In addition to the latter bounconditions, we must impose the constraint that in the stationstate the net force acting on the particle or the unit cell muszero (21).

A numerical method similar to that proposed by DeLaceyWhite in their theory of the dielectric response and conductiof a colloidal suspension in time-dependent fields (8) has bapplied to solve the above-mentioned set of ordinary differenequations of the theory of the electrophoresis in concentrcolloidal suspensions. The numerical computations are shand discussed in Section 5.

3. CALCULATION OF THE CONDUCTIVITYOF THE SUSPENSION

The electrical conductivity,K ∗, of the suspension is defineby

〈i〉 = 1

V

∫V

i(r ) dV = K ∗〈E〉, [41]

with 〈i〉 being the electric current density in the suspension,〈E〉 the macroscopic electric field (i.e., minus the average ofgradient of the electrical potential9(r ) in each position of thesystem), with total volumeV:

〈E〉 = − 1

V

∫V∇9(r ) dV. [42]

According to O’Brien (30) and Ohshima (25), and reca

ICAL COLLOIDAL PARTICLES 355

i-

aryarybe

ndityeentialtedwn

d

ndthe

ll-

(Eqs. [19]–[21]) as

〈i〉 = −N∑

i=1

zi e

λi V

∫V

(KBT∇δni + n∞i zi e∇δ9

)dV

− NP

V

N∑i=1

zi e n∞iλi

∫S{r · ∇δµi (r )− δµi (r )}r dS, [43]

whereS is the outer spherical surface of the cell, andNP thenumber of particles in the (total) volumeV . Note that becausedouble layers are not allowed to overlap, it will be assumedn(0)

i∼= n∞i at the cell surface. Using this approximate equal

and theφi functions defined in Eq. [30], the following expressiofor the average current density can be reached:

〈i〉 =N∑

i=1

zi e2n∞iλi〈E〉 − NP

V

N∑i=1

zi e n∞iλi

×{−zi e

(r

dφi

dr− φi

)r=b

}∫V

(〈E〉 · r )r dS. [44]

Now, using the value of the conductivity of the supportisolutionK∞,

K∞ =N∑

i=1

zi e2n∞iλi

, [45]

and the result∫V

(〈E〉 · r )r dS= (4/3)πb2〈E〉, [46]

Eq. [44] becomes

〈i〉 = K∞

1− 4πNP

V

∑Ni=1

z2i n∞i Ci

λi∑Ni=1

z2i n∞iλi

〈E〉, [47]

where the coefficientsCi were defined by Ohshima as

Ci = −b2

3

(r

dφi

dr− φi

)r=b

. [48]

From Eq. [47], after introducing the volume fraction of soliφ = 4πa3NP/3V , the ratio between the conductivities of thsuspension and the dispersion medium can be writtenfollows:

K ∗

K∞=1− 3φ

a3

∑Ni=1

z2i n∞i Ci

λi∑Ni=1

z2i n∞iλi

. [49]

entiesIt is interesting to note that this equation for the conductivity

Y

b

afe

nle

o

aa

r

o

e

ca-

er

L is

o-el

and[38]

–

byct-

etic

356 CARRIQUE, ARRO

differs from that deduced by Ohshima (Eq. [58] in Ref. (25))

K ∗

K∞=1+ 3φ

a3

∑Ni=1

z2i n∞i Ci

λi∑Ni=1

z2i n∞iλi

−1

. [50]

We confirm in Section 5 that the correct limiting cases fufilled by Ohshima’s conductivity formula are also exhibitedour Eq. [49], in particular the low-zeta-potential case. Howevit is worth pointing out that for moderate or high zeta potentithe predictions of both conductivity expressions are very difent, as can be deduced by numerical integration of the theo

4. EXTENSION TO INCLUDE A DYNAMIC STERN LAYER

Let us now consider the possibility of adsorption and iotransport in the inner region of the double layer of the particAs previously mentioned, we will follow the method developby Mangelsdorf and White in their theory of the electrophoreand conductivity in a dilute colloidal suspension (2). This theallows for adsorption and lateral motion of ions in the innregion using the well-known Stern model. Therefore, we wassume a Stern layer that is thin compared to eithera or thedouble-layer thicknessκ−1, whereκ is defined by (31)

κ =[

N∑i=1

n∞i z2i e2

/εrsε0KBT

]1/2

. [51]

Now the condition that ions cannot penetrate the slipping plis no longer valid, and thus, the evaluation of the fluxes of eionic species through the slipping plane gives rise to new sping plane boundary conditions for the functionsφi (r ), replacingEq. [38],

dφi

dr(a)− 2δi

aφi (a) = 0 (i = 1, . . . , N), [52]

δi =[eNi ]

ae10−pKi

(λi

λti

)exp

[zi e

KBTσdC∗2

]NA103+∑N

j=1NA103c∞j10−pK j

exp[− zj e

KBT

(ζ− σd

C∗2

)](i = 1, . . . , N) , [53]

in terms of the so-called surface ionic conductance parameδi of each ionic species, comprising the effect of a mobile sface layer. In fact it is the small thickness of the Stern layecomparison with the other length scales that permits slippplane boundary conditions to be used, including the effectsmobile surface layer (2). These parameters depend on thepotentialζ , the ratio between the drag coefficientλi of eachionic species in the bulk solution and in the Stern layerλt

i , thedensityNi of sites available for adsorption in the Stern lay

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ry.

ices.d

sisryerill

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lip-

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r,

is represented as a dissociation reaction in this theory), thepacityC∗2 of the outer Stern layer, the radiusa of the particles,the electrolyte concentrationc∞j (the equilibrium molar concen-tration of type j ions in solution), and the charge density punit surface area in the double layerσd. It is worth noting thatthe other boundary conditions remain unchanged when a DSassumed.

5. RESULTS AND DISCUSSION

Low-Zeta-Potential Approximations

Following the method described in Ref. (21) for low zeta ptentials but taking into account the Shilov–Zharkikh cell modinstead of the Levine–Neale one, we can solve Eqs. [32][33] subject to the boundary conditions expressed by Eqs.and [40] to obtain

φi (r ) = Y(r ) = 1

1+ φ/2(

r + a3

2r 2

)(i = 1, . . . , N). [54]

Substituting Eq. [54] in Eq. [32], the latter reduces to

L(Lh) = − εrsε0κ2

η(1+ φ/2)

(1+ a3

2r 2

)d9 (0)

dr, [55]

where 9 (0) is now the solution of the linearized PoissonBoltzmann equation, and is given by (21)

9(0)(r ) = ζ a

r

κbcosh [κ(b− r )] − sinh [κ(b− r )]

κbcosh [κ(b− a)] − sinh [κ(b− a)]. [56]

By solving Eq. [55] subject to the boundary conditions givenEq. [36], and considering also the condition of zero net force aing on the particle in the stationary state (21), the electrophormobility can be derived with the help of Eq. [37] to give

ue = 2εrsε0ζ

3η

∫ b

aH (r )

(1+ a3

2r 2

)dr + F, [57]

with

H (r ) = − (κa)2

6(1+ φ/2)

[1− 3r 2

a2+ 2r 3

a3− a3

b3

×(

2

5− r 3

a3+ 3r 5

5a5

)]1

ζ

(d9(0)

dr

)[58]

and

F = 2εrsε0(κa)2

9η9 (0)(b)

[1+ b3

a3− 9b2

5a2− a3

5b3

]. [59]

hesite

We show in Fig. 2 the ratio of the electrophoretic mobilityto the Smoluchowski mobility for zero volume fraction, as a

R

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ta’ohli

tc.toefie

e

8h

5

ethe

la

le-

ver,woer-are

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thend,. 3

ELECTROKINETICS OF SPHE

FIG. 2. Ratio of the electrophoretic mobility to the Smoluchowski mobilfor zero volume fraction, as a function ofκa and volume fraction of the suspensions. Solid lines show Ohshima’s mobility formula, Eqs. [33] and [34Ref. (17). Dotted lines show mobility expressed by Eqs. [57]–[59] in this pa

function ofκa and the volume fraction of the suspensions. Infigure, solid lines display the mobility according to Ohshimmobility formula (Eqs. [33]–[34] in Ref. (21), Levine–Nealecell model), and dotted lines show the mobility calculated frour Eqs. [57]–[59] (Shilov–Zharkikh’s cell model). It is wortpointing out that the volume fraction dependence of the mobiκa curves is quite different when both approximations are copared, especially in the region of higher values ofκa and volumefraction. The nature of such discrepancies is based on theistence of two different definitions of electrophoretic mobilibeing identical when dilute suspensions are concerned butsiderably different in the case of concentrated suspensionsreader is referred to Ref. (22), where the details of this conversy are given. Briefly, the difference refers again to the chof the electric field to be used to obtain the mobility from the eltrophoretic velocity. Thus, Ohshima (21) assumes that thisis precisely the externally applied one,E; on the other hand, thShilov–Zharkikh cell model uses an average〈E〉 of the electricfield, that for concentrated suspensions differs from the exnally applied fieldE that, following Ohshima, would exist in thsolution at large distance from the particle. The Shilov–Zharkchoice is again compatible with nonequilibrium thermodynaics (22).

In the case of DC conductivity, it can be seen from Eqs. [4[50] that the calculation of the conductivity ratio requires tprevious knowledge of the functionsφi and their first derivativeson the cell surfacer = b. For low zeta potentials, Ohshima (2proved that rather simple expressions can be found forCi , sothat Eqs. [49] and [50] become, respectively,

K ∗ 1− φ (

eζ) ∑N

i=1z3

i n∞iλi

1+ φ/21−3φKBT

L(κa, φ)∑Ni=1

z2i n∞iλi

[60]

ICAL COLLOIDAL PARTICLES 357

y

iner.

he’ssm

ty-m-

ex-y,on-Thero-icec-eld

ter-

ikhm-

]–e

)

and

K ∗

K∞= 1− φ

1+ φ/2

1+ 3φ

(eζ

KBT

)L(κa, φ)

∑Ni=1

z3i n∞iλi∑N

i=1z2

i n∞iλi

−1

,

[61]

with L(κa, φ) defined by (25)

L(κa, φ) = − 1

3a3ζ (1− φ)(1+ φ/2)

×∫ b

a

(a3

2+ r 3

)(1− a3

r 3

)d9(0)

drdr, [62]

where9 (0) is the potential distribution in the equilibrium doubllayer, Eq. [56]. Note that our Eq. [60] is in agreement wiO’Brien’s conductivity formula for low zeta potential and dilutsolutions (30). Furthermore, when particles are uncharged (ζ =0), Eqs. [60] and [61] lead to the well-known Maxwell formufor uncharged spheres (32):

K ∗

K∞= 1− φ

1+ φ/2. [63]

Let us also mention that in the limit of infinitesimally thin doublayers (κa→∞), the predictions of Eqs. [60] and [61] are identical and independent of zeta potential, becauseL(κa, φ)→ 0,just as when dilute suspensions are considered (30). Howewhen the zeta potential is no longer low, the results of the tconductivity expressions, Eqs. [60] and [61], are quite diffent, as shown in the next section where numerical datapresented.

Numerical Calculations

In Fig. 3, the electrophoretic mobility is represented as a fution of zeta potential for different volume fractions. In this figurnumerical calculations of the electrophoretic mobility followinOhshima’s and Shilov–Zharkikh’s models are shown for coparison. We can observe some remarkable features in theseFirst, for a given zeta potential in the region of not very high zvalues, Ohshima’s mobility is higher than the Shilov–Zharkiprediction at every volume fraction. Likewise, Ohshima’s mbility maximum shifts to lower zeta potentials as volume fractiincreases, whereas the Shilov–Zharkikh one not only shiftthe opposite zeta region, it also broadens and almost disappBoth mobilities also diminish when volume fraction increasesa fixed zeta potential, owing to the increasing importance ofhydrodynamic particle–particle interactions. On the other hathe numerical data in the region of low zeta potentials of Figconfirm, according to Dukhinet al. (33), the zero-frequency

Y

io,e

rsz.

tid

or-to

e,con--in

e ap-our

thes atis

forgesc-

ionrly,

ame. 4.

ng

ins ofop-met isuteherowap-thential

358 CARRIQUE, ARRO

FIG. 3. Scaled electrophoretic mobility of a spherical particle as a functof zeta potential for different volume fractions of the suspensions in a KCl stion at 25◦C. Particle radius, 100 nm;κa = 100. For nonzero volume fractionssolid lines show results calculated by numerical integration of Ohshima’s thof electrophoresis in concentrated suspensions; dotted lines show the samwith Shilov–Zharkikh’s boundary conditions.

Ohshima,ue−OHS (34), and Shilov,ue−SHI (27):

ue−SHI = ue−OHS1− φ

1+ φ/2. [64]

Concerning DC conductivities, Fig. 4 shows numericalsults of the ratio between the conductivities of the suspenand the electrolyte solution, as a function of dimensionlesspotential for different volume fractions of the suspensions

FIG. 4. Ratio of the suspension conductivityK ∗ to the conductivity of thesupporting electrolyte solutionK∞, as a function of the dimensionless zepotential for different volume fractions of the suspensions in a KCl solutat 25◦C. Particle radius, 100 nm;κa = 100. Solid lines show results obtaine

O, AND DELGADO

onlu-

orye but

e-ionetaIn

aon

the figure, solid lines correspond to Ohshima’s conductivity fmula (Levine–Neale cell model), Eq. [50], and dashed linesour Eq. [49] (Shilov–Zharkikh cell model). As above, we usrespectively, the subscripts OHS and SHI to denote theseductivity ratios. It is worth pointing out that for low zeta potential the conductivities obtained with both formulae arevery good numerical agreement, as expected (see also thproximate analytical expressions, Ohshima’s Eq. [61] andEq. [60], both valid for lowζ ), but they differ considerablyfor moderate-to-high zeta values. Furthermore, the highervolume fraction, the higher the deviation between the ratioevery zeta potential. Likewise, Ohshima’s conductivity ratioa monotonically decreasing function of the volume fractiona fixed zeta potential. On the other hand, our prediction chanfrom a monotonically decreasing behavior with volume fration at low zeta values to a monotonically increasing functof volume fraction for the highest zeta values studied. SimilaFig. 5 displays the usual “conductivity increment” (2, 30) asfunction of the dimensionless zeta potential for different volufractions, with the same considerations as those given in FigAs observed, the conductivity increment tends to the limitivalue of (−3/2) when volume fraction andζ tend to zero, thusverifying the standard result for the conductivity incrementdilute suspensions (30). On the other hand, the predictionthe conductivity increment according to both models showposite behavior in the region of high zeta potentials as volufraction increases. While Ohshima’s conductivity incremenalways lower than the corresponding prediction for the dilcase in that zeta region, our conductivity ratio is always higthan that of the dilute case. This different behavior reveals hsensitive the theoretical predictions have turned out to be toparently small changes in boundary conditions, in particularone expressing the behavior of the perturbed electric poteat the outer surface of the unit cell (see Eqs. [24] and [25]).

theFIG. 5. Same as Fig. 4, but for the conductivity increment, (K ∗ − K∞)/K∞φ.

R

fe.

ea

ccl

is

n

t

l

a

rs

nifi-in-canpor-entialed

ela-ite

SLon-

ions

ELECTROKINETICS OF SPHE

FIG. 6. Ratio of the DSL electrophoretic mobility to the standard eletrophoretic mobility of a spherical particle, as a function of zeta potentialtwo volume fractions and different values of the DSL parameterN− (expressedaseN−, inµC/cm2). Dispersion medium, KCl solution at 25◦C. Particle radius,100 nm;κa = 100. Other DSL parameters are shown in the figure.

Effect of a Dynamic Stern Layer

Regarding the DSL corrections, let us first consider their efon the electrophoretic mobility in concentrated suspensionspresent in Fig. 6 the ratio of the DSL corrected to the standelectrophoretic mobility, both calculated according the ShiloZharkikh cell model, as a function of the dimensionless zpotential, for two extreme volume fractions, and different vues of the Stern layer parameterN−, the density of counterionadsorption sites. Recall that the higher this parameter, the msignificant the role of Stern-layer conductance in the electronetics of the system. Figure 6 demonstrates that the presena DSL reduces the mobility, the effect being more pronounwhen the role of the DSL is increased on increasing the vaof the parameterN−.

A qualitative explanation for these facts can be given, takinto account the rather complex and interrelated mechanismsponsible for the electrophoretic mobility dependence onζ . Asthe latter rises, the electrokinetic charge increases as well andoes the electrophoretic velocity. On the other hand, the streof the dipole moment induced on the particles by the electric fialso increases with zeta (35), tending to decrease the mobThe presence of a conducting Stern layer will favor the formtion of the dipole by ionic migration. As a consequence,mobility will be further reduced as compared to the standarduation. As the zeta potential is increased, the charge in the Slayer also rises, and the ratio (u∗e)DSL/u∗e decreases, as observein Fig. 6. However, at sufficiently high zeta potentials, the Stelayer charge must tend to saturate, and hence, the diffuseshould play the essential role. As a consequence, the trend omobility ratio changes, and the latter increases until eventu

ICAL COLLOIDAL PARTICLES 359

c-for

ctWeardv–tal-

oreki-e ofed

ues

ngre-

d sogth

eldility.a-

hesit-terndrn-ayerf thelly

FIG. 7. Same as Fig. 6, but for differentκa values, and the DSL parameteindicated.

mobility from the standard prediction appears to be less sigcant the higher the volume fraction, or equivalently, the moretense the hydrodynamic particle–particle interactions. Thisbe explained by considering that such interactions are so imtant that the presence or not of a DSL ceases to be an essfactor in interpreting electrokinetic behavior. This is confirmby the weaker variations of the mobility ratio withζ when thesuspensions are concentrated.

When the double-layer thickness is decreased (κa increased),while keeping all other parameters unaltered (Fig. 7), the rtive mobility reduction brought about by the Stern-layer fin

FIG. 8. Ratio of the standard suspension conductivity (no DSL) to the Dsuspension conductivity (both calculated with Shilov–Zharkikh’s boundary cditions) as a function of dimensionless zeta potential, for two volume fractand different values of the DSL parameterN− (expressed aseN−, in µC/cm2).

ticDispersion medium, KCl solution at 25◦C. Particle radius, 100 nm;κa = 100.Other DSL parameters are shown in the figure.

Y

-

t

omima

o

u

t

h

i

h

terisstr toce

theetale

ionsinrbi-ex-peduc-nsto

u-ns

rs)ndhensessncebil-ithtiv-

onnd,

ytets

yerthe

ndd.

360 CARRIQUE, ARRO

FIG. 9. Same as Fig. 8, but for differentκa values, and for the DSL parameters indicated.

conductance also decreases. Since in Fig. 7 we assume thaa isconstant for all cases, increasingκ is equivalent to raising theionic concentration in the medium. It is hence to be expectedthe change in the diffuse layer (for a given zeta potential) walso rise: As a consequence, the effects of DSL will be increingly hidden by those of the diffuse atmosphere. This bringsmobility ratio (u∗e)DSL/u∗e closer to one, as numerically showin Fig. 7.

Regarding the DSL correction to the conductivity of concetrated suspensions, we represent in Fig. 8 the ratio of the sdard suspension conductivity (no DSL) to the DSL value, bcalculated according to the Shilov–Zharkikh cell model (siilar effects are predicted for Ohshima’s theory), as a functof the dimensionless zeta potential, for low and high volufractions and the same values of Stern layer parametersFig. 6. As observed, in the presence of a conducting Stern lathe conductivity of the suspensions is higher than that dedufrom the standard predictions (note that the conductivity rais always less than unity). Also, the higher the volume fractithe lower the conductivity ratio, clearly indicating the importarole of the mobile Stern layer in the explanation of the condtivity of the suspension. In fact, note how reducingeN− leads tocloser proximity between standard and DSL calculations ofconductivity. This behavior is easy to explain because inlatter case a new ionic transport process develops in theturbed inner region of the double layer, giving rise to a higconductivity at every zeta potential and volume fraction.

On the other hand, the relative deviation of the DSL suspsion conductivity from the standard prediction seems to be mimportant the lower the zeta potential, for every volume fract(Fig. 8). Furthermore, the conductivity ratio tends to unitythe limit of high zeta potentials for every volume fraction: T

O, AND DELGADO

t

hatillas-then

n-tan-th-

one

s inyer,cedtion,

ntc-

theheper-er

en-oreonine

Finally, Fig. 9 allows us to analyze the effect of the parameκa on the conductivity differences. As the ionic concentrationreduced (κ decreases), the conductivity in the diffuse layer mualso decrease, so the relative contribution of the Stern layethe overall conductance is likely to be more significant: henthe larger diferences betweenK ∗ and (K ∗)DSL observed in thefigure.

6. CONCLUSIONS

In this work we have first derived a general expression forconductivity of a concentrated suspension valid for arbitrary zpotential and volume fraction when nonoverlapping of doublayers is assumed. Likewise, approximate analytical expressfor the electrophoretic mobility and electrical conductivityconcentrated suspensions, valid for low zeta potential and atraryκa and volume fraction, have also been obtained. Thesepressions have been calculated following the method develoby Ohshima in his theory of the electrophoresis and condtivity of concentrated suspensions, substituting the conditioimposed by the Levine–Neale cell model by those accordingthe Shilov–Zharkikh cell model. Furthermore, numerical calclations are presented for both quantities in arbitrary conditio(with the only restriction being nonoverlapping double layeof volume fraction and zeta potential, for both Ohshima aShilov–Zharkikh’s models. In addition, we have extended ttheory corresponding to Shilov–Zharkikh’s boundary conditioto include a DSL into the model. The results show that regardlof the particle volume fraction and zeta potential, the preseof a DSL gives rise to a decrease in the electrophoretic moity, and an increase in the DC conductivity, in comparison wthe standard predictions. These DSL effects on the conducity are relatively more important the higher the volume fractiand the lower the electrolyte concentration. On the other hathe higher the volume fraction and the higher the electrolconcentration, the lower the relative influence of DSL effecon the electrophoretic mobility. Furthermore these Stern-laeffects tend to decrease on increasing the zeta potential inhigh-zeta-potential region.

ACKNOWLEDGMENTS

Financial support for this work by MEC, Spain (Projects MAT98-0940 aBFM 2000-1099), and INTAS (Project 99-00510) is gratefully acknowledge

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