Electroviscous Effect of Moderately Concentrated Colloidal Suspensions under OverlappingConditions

Emilio Ruiz-Reina,†,* Pablo Garcıa-Sanchez,‡ and Felix Carrique§

Departamento de Fı´sica Aplicada II, UniVersidad de Ma´ laga, Campus de El Ejido, 29071 Ma´ laga, Spain,Departamento de Electro´nica y Electromagnetismo, UniVersidad de SeVilla, Campus de Reina Mercedes,41012 SeVilla, Spain, and Departamento de Fı´sica Aplicada I, UniVersidad de Ma´ laga, Campus de Teatinos,29071 Malaga, Spain

ReceiVed: NoVember 25, 2004; In Final Form: January 18, 2005

In this work we present a theoretical model for the calculation of the electroviscous coefficient of a colloidalsuspension. The treatment is not limited for dilute suspensions and includes the contribution of the overlappingbetween adjacent ionic layers. The development here used is based on a cell model, which is applicable toNewtonian suspensions under low shear conditions and without crystalline ordering. Also presented are acomplete study of the new numerical results and comparisons with previous results. We find new behaviorsfor the case of moderate volume fractions that do not appear in the dilute limit.

Introduction

The viscosityη of a dilute suspension of spherical, rigid andnonslipping colloidal particles was first calculated by Einstein,1

whereη0 is the viscosity of the suspending medium andφ isthe particle volume fraction. The increase in suspension viscosityis due to the distortion of the applied flow field in theneighborhood of the particles. When the particles are charged,there is an additional increment of the flow distortion and,consequently, of the viscosity, which is caused by the presenceof electric double layers around the particles. This phenomenonis called the primary electroviscous effect and Einstein’sequation is modified to take it into account,

wherep, the primary electroviscous coefficient, is a functionof the charge on the particle and properties of the suspendingelectrolyte. The thickness of the electric double layer is givenby the Debye lengthκ-1,

wheree is the elementary electric charge,kB is Boltzmann’sconstant,T is the absolute temperature,εrs is the relativepermittivity of the suspending medium,ε0 is the vacuumpermittivity, andni

∞ and zi are the bulk number density andvalence ofi-type ions.

The primary electroviscous coefficient is of small magnitudeand its experimental determination is difficult.2,3 Most of the

experimental works4-11 on the primary electroviscous effect findthat it is underestimated by the theoretical models.12-17

Recently, it has been presented various theories that try topredict the experimental results. Rubio-Herna´ndez et al.18-21

and Sherwood et al.22 included in the theoretical models thecontribution of the ionic transport into the Stern layer. Theyfound an improvement of the predictions when the influenceof a dynamic Stern layer was taken into account in both theviscosity calculations and experimental determinations of thestate of charge and properties of the suspension. Their conclu-sion was that it is necessary to consider the presence of a Sternlayer in the complete experimental study of the suspension ina consistent form. Garcı´a-Salinas and de las Nieves3 studiedthe influence of the reduction of ionic diffusion in the neighbor-hood of the colloidal particles and found a better theory-experiment agreement.

If the suspension of charged particles cannot be considereddilute, we distinguish three cases:

(1) The electrical double layer thickness is larger than theinterparticle distance. In this case the electrical interactions (i.e.,the stresses arising from the very high overlapping of electricaldouble layers and the strong alteration of collision trajectories)are considerably greater than the stresses due to hydrodynamicinteractions. The suspension forms a colloidal crystal and itsflow behavior is extremely shear-thinning, being very difficultto find experimentally a low-shear viscosity “plateau”. Ourtheory is, obviously, not applicable in this case.

(2) The electrical double layer thickness is comparable withthe interparticle distance.In this case there is a low overlappingof electrical double layers, in addition to the hydrodynamicinteractions. The suspension does not present crystalline orderingbecause the electrical interactions, although great enough tonoticeably contribute to the viscosity of the suspension, are notsufficiently high.

(3) The electrical double layer thickness is quite smaller thanthe interparticle distance. In this case electrical interactions aremasked by the hydrodynamic interactions (there is not overlap-ping of electrical double layers and the modification of collisiontrajectories is low). The suspension is disordered at rest and

* Corresponding author. E-mail: eruizr@uma.es.† Universidad de Ma´laga, Campus de El Ejido.‡ Universidad de Sevilla.§ Universidad de Ma´laga, Campus de Teatinos.

η ) η0(1 + 52φ) (1)

η ) η0(1 + 52φ(1 + p)) (2)

κ2 )

e2

ε0εrskBT∑i)1

N

ni∞zi

2 (3)

5289J. Phys. Chem. B2005,109,5289-5299

10.1021/jp044619p CCC: $30.25 © 2005 American Chemical SocietyPublished on Web 02/23/2005

shows a wide low-shear Newtonian “plateau”. However, thedistortion of the electrical double layer modifies the hydro-dynamic interactions, because the flow fields around the particlesare different than in the case of uncharged particles.

Ruiz-Reina et al.2 developed a theoretical model of theelectroviscous effect (we eliminate the word “primary” becausethe suspension has not to be necessarily dilute) that includesthe possibility of hydrodynamic interactions and can predict theelectroviscous coefficient for suspensions that fit into case 3described before. The objective of the present work is themodification of that previous model to introduce the overlappingof electrical double layers. In this way, the range of validity ofthe theory is extended for suspensions in case 2 as well.

Theory

Cell Model. To take into account the hydrodynamic particle-particle interactions, Happel’s cell model23 with Simha’shydrodynamic boundary conditions24 at the outer surface of thecell will be used.

The cell model concept has been successfully applied todevelop theoretical models for different electrokinetic phenom-ena in moderately concentrated colloidal suspensions of chargedparticles, such as static electrophoresis and electricalconductivity,28-31 sedimentation velocity and potential,32,33

dynamic electrophoresis,34-36 complex conductivity and dielec-tric response,37 and electroacoustic phenomena,38,39to mentionjust a few.

According to this model (Figure 1), each spherical particleof radiusa is surrounded by a concentric shell of an electrolytesolution, having an outer radiusb such that the particle/cellvolume ratio in the cell is equal to the particle volume fractionthroughout the entire suspension, i.e.,

The surfacer ) a is usually called the “slip-plane”. This is theplane outside which the continuum equations of hydrodynamicsare assumed to hold. In this work we assume that the slip-planecoincides with the true surface of the particle, and that there isnot any effect arising from mobile adsorbed ions in the dynamicStern layer.

The basic assumption of the cell model is that the suspensionproperties can be derived from the study of a unique cell. Thedisturbance due to the presence of each solid sphere isconsidered to be confined inside the cell. According to Simha,each cell enclosing fluid is surrounding by a rigid spherical walland the perturbations of the different magnitudes disappear atthe surface of the cell (r ) b). By its own nature, the cell modelis only applicable when the suspension is homogeneous andisotropic. Therefore, Brownian motion should be dominant over

other potential fields to prevent structural ordering of theparticles. When a flow field is imposed, this is only true in thelow shear region.

Different boundary conditions on the surface of the cell canbe found in the literature. Simha’s boundary condition24 arisesfrom his hypothesis of no disturbance velocity at the cell wall,u′ ) 0. In an alternative treatment, Happel23 introduced adifferent hypothesis about the behavior of the disturbances atr) b. He assumed that only the normal component of theperturbation velocity vanishes at the surface of the cell,supplemented with the condition of no friction on it due to thedisturbance, which corresponds to the vanishing of the tangentialcomponents of the stress perturbation.

Simha’s boundary conditions24 for the cell model leads toEinstein’s result valid for uncharged particles at the limitconditionφ f 0. This is not true when Happel’s conditions areused. Therefore, we decided to use Simha’s instead of the others.

Governing equations. Let us consider now a chargedspherical particle of radiusa immersed in an electrolyte solutioncomposed byN ionic species of valenceszi, bulk numberconcentrationsni

∞, and drag coefficientsλi (i ) 1, ..., N). Theaxes of the spherical coordinate system (r, θ, æ) are fixed atthe center of the particle. In the absence of any flow field, theparticle is surrounded by a radial charge distribution and auniform electric potential, theú-potential, atr ) a.

We will consider the case when a flow field is applied to thesuspension. If a linear shear field is applied to the system, thevelocity field u(r ) can be written as

whereR is a constant second-order tensor andf(r) is a functionthat only depends on the radial coordinate. The second termcorresponds to the perturbation in the flow field due to thepresence of particles in the suspension.25

A complete solution of the problem would require theknowledge of the electric potentialΨ(r ), the number densityni(r ) and the drift velocityvi(r ) of each type of ions, the fluidvelocity u(r ), and the pressureP(r ) at every pointr in thesystem. The governing equations for these fields are16

Equation 6 is Poisson’s equation, whereFel(r ) is the electriccharge density given by eq 7. Equations 8 and 9 are the Navier-Stokes equations appropriate to a steady incompressible fluidflow at low Reynolds number in the presence of an electricalbody force. Equation 10 expresses the balance of hydrodynamicdrag, whereλi are the drag coefficients, electrostatic andthermodynamic forces on ions of typei at positionr . Finally,eq 11 is the continuity equation of the number of type-i ions.

Figure 1. The cell model.

φ ) (ab)3(4)

u(r ) ) R‚r + ∇ × ∇ × [R‚∇f(r)] (5)

∇2Ψ(r ) ) -Fel(r )

εrsε0(6)

Fel(r ) ) ∑i)1

N

zieni(r ) (7)

η0∇2u(r) - ∇P(r ) ) Fel(r )∇Ψ(r ) (8)

∇‚u(r ) ) 0 (9)

-λi[vi(r ) - u(r )] - zie∇Ψ(r ) - kBT∇ ln ni(r ) ) 0i ) 1, ...,N (10)

∇‚[ni(r )vi(r )] ) 0 i ) 1, ...,N (11)

5290 J. Phys. Chem. B, Vol. 109, No. 11, 2005 Ruiz-Reina et al.

For a low shear field, represented by a symmetric (only thesymmetric part of the velocity gradient tensor contributes todissipation of energy and, consequently, to the viscosity of thesuspension) and traceless (because the liquid medium isincompressible) tensorR, the following perturbation schemeapplies,

whereδni(r ), δΨ(r ), andδFel(r ) are the perturbation terms. Theequilibrium quantities (with superscript 0) are related by

The imposed linear shear field is low and the distortion ofthe system from equilibrium is small. It is well-known that thesuspension has a Newtonian behavior in this case (linearrelationship between the volume-averaged stress and shear ratetensors), as can be seen in typical flow curves.40 The concentra-tion of ionsni(r ) around the particles will change slightly, andwe define the perturbation termsΦi(r ) as16

which allows us to write, in combination with eq 12,

Making use of the symmetry of the problem, it is possible toseparate radial and angular dependences,

where ris the unitary vector (r ) r /r, r being the modulus ofthe position vectorr ).

Substituting eqs 12-19 into the differential eqs 6-11,neglecting products of small perturbation quantities, and ap-plying the symmetry eqs 5, 20, and 21 transform the governingequations into

The functionF(r) is defined by

and the differential operatorsL2 in eqs 23 and 24 andL4 in eq22 are defined by

The N + 2 coupled differential eqs 22-24 must be solvedwith specific boundary conditions. The boundary conditions usedhere are

whereΨp(r ) is the electric potential inside the particle andεrp

is its relative permittivity.The electric state of the particle surface can be specified by

condition (28a), which introduces theú-potential in the usualway, or by condition (28b), whereσ is the surface chargedensity. The condition (29) implies that the cell is electricallyneutral as a whole. Equation 30 reflects the no slip-conditionat the surface of the particle. The condition (31) is that bySimha,24 which considers that the perturbation of the dilatationalflow ∇ × ∇ × (R‚∇f(r)) is zero atr ) b. The condition (32)expresses that the alteration of the equilibrium ion distributiondisappears on the cell boundary. Equation 33 arises from theimpenetrability of the particle surface by the ions. Equation 34comes from the continuity condition for the electric potentialat the particle surface. Expression (35) follows from thediscontinuity of the normal component of the electric displace-ment vector. Finally, eq 36 ensures that the electroneutrality ofthe cell is maintained when the flow field is imposed.

ni(r ) ) ni0(r) + δni(r ) i ) 1, ...,N (12)

Ψ(r ) ) Ψ0(r) + δΨ(r ) (13)

Fel(r ) ) Fel0 (r) + δFel(r ) (14)

ni0(r) ) ni

∞ exp[-zieΨ0(r)

kBT ] i ) 1, ...,N (15)

1

r2

ddr(r2dΨ0(r)

dr ) ) -Fel

0 (r)

εrsε0(16)

Fel0 (r) ) ∑

i)1

N

zieni0(r) (17)

ni(r ) ) ni∞ exp[-

zie

kBT{Ψ(r ) + Φi(r )}] i ) 1, ...,N (18)

δni(r ) ) -( zie

kBT)ni0(r)[δΨ(r ) + Φi(r )] i ) 1, ...,N (19)

δΨ(r ) ) ψ(r)(r ‚R‚r ) (20)

Φi(r ) ) φi(r)(r ‚R‚r ) i ) 1, ...,N (21)

L4F(r) ) -2e2

r2η0kBT(dΨ0

dr )∑i)1

N

ni0(r)zi

2φi(r) (22)

L2φi(r) )ezikBT(dΨ0

dr )[dφi

dr+

λi

zie{r - 3F(r)}] i ) 1, ...,N

(23)

L2ψ(r) -1

εrsε0kBT∑i)1

N

zi2e2ni

0(r)[ψ(r) + φi(r)] ) 0 (24)

F(r) ) ddr(1r df(r)

dr ) (25)

L2ψ ≡ d2ψdr2

+ 2r

dψdr

- 6ψr2

(26)

L4F ≡ d4F

dr4+ 8

rd3F

dr3- 24

r3

dFdr

+ 24

r4F (27)

Ψ0(a) ) ú (a) ordΨ0

dr |r)a

) - σεrsε0

(b) (28)

dΨ0

dr |r)b

) 0 (29)

u(r )|r)a ) 0 (30)

u(r )|r)b ) R‚r (31)

ni(r )|r)b ) ni0(b) i ) 1, ...,N (32)

r ‚vi(r )|r)a ) 0 w r ‚∇Φi(r )|r)a ) 0 (33)

δΨ(r )|r)a ) δΨp(r )|r)a (34)

εrs(∇δΨ(r )‚r )|r)a ) εrp(∇δΨp(r )‚r )|r)a (35)

∇δΨ‚r |r)b ) 0 (36)

Electroviscous Effect under Overlapping Conditions J. Phys. Chem. B, Vol. 109, No. 11, 20055291

According to the symmetry eqs 5, 20, and 21, the boundaryconditions given by eqs 30-36 transform into

The derivation of the last condition (42) is not straightforwardand is detailed in Appendix 1.

Solving the Equations.To solve the coupled eqs 22-24 withthe boundary conditions (37)-(42), the method by DeLaceyand White26 has been used.

The far field asymptotic behaviorr f b of the differentialequations system, using condition (29), is

where we have defined theN + 1 column vectorø(r) and the(N + 1) × (N + 1) matrix A as

with

We introduce a newN + 1 column vectorX(r) ) (Xj), j )1, ..., N + 1, such thatX(r) ) R-1‚ø(r), whereR is a matrixthat diagonalizesA

It can be shown that the eigenvaluesRj are

We finally obtain

and for the components ofX, the latter becomes

with the solution forj ) 1, ...,N

and for j ) N + 1

where

and then

The asymptotic solution for theF(r) function is given by

Using the above-mentioned boundary conditions (39) and (40)at r ) b with the asymptotic functions, the number of constantscan be reduced

The Ak2 constants can be written as a function of theAk1

constants andψ(b) ) -γ, for eachk value:

F(a) ) a3

dFdr |r)a

) 13

(37)

F(b) ) 0dFdr |r)b

) 0 (38)

φi(b) ) -ψ(b) i ) 1, ...,N (39)

dψdr |r)b

) 0 (40)

dφi

dr |r)a

) 0 i ) 1, ...,N (41)

dψdr |r)a

-2εrp

εrsaψ(a) ) 0 (42)

L4F(r) ≈ 0 (43)

L2ø(r) - Aø(r) ≈ 0 (44)

ø(r) ) (φ1

‚‚‚φN

ψ) A ) (0 0 ‚‚‚ 0 0

0 0 ‚‚‚ 0 0‚‚‚ ‚‚‚ ‚‚‚ ‚‚‚ ‚‚‚0 0 ‚‚‚ 0 0

â10 â2

0 ‚‚‚ âN0 ∑

i)1

N

âi0)

âi0 )

zi2e2ni

0(b)

εrsε0kBT(45)

R-1‚A‚R ) diag(R1, R2, ...,RN+1) (46)

Rj ) 0 (j ) 1, ...,N) RN + 1 ) ∑i)1

N

âi0 (47)

L2X - diag(R1, R2, ...,RN+1)X ≈ 0 (48)

L2Xj - RjXj ≈ 0 (j ) 1, ...,N + 1) (49)

Xj(r) ) Aj1Sj1(r) + Aj2Sj2(r) (50)

XN+1(r) ) AN+1,1SN+1,1(r) + AN+1,2SN+1,2(r) (51)

Sj1(r) ) 1

r3Sj2(r) ) r2 (j ) 1, ...,N) (52)

SN+1,1(r) ) (1

r3+

RN+1

3 r+

xRN+1

r2 ) e-(RN+1)1/2(r-a) (53)

SN+1,2(r) ) (1

r3+

RN+1

3 r-

xRN+1

r2 ) e(RN+1)1/2(r-a) (54)

(φ1(r)‚‚‚φN(r)ψ(r)

) ) (R11 R12 ‚‚‚ R1,N R1,N+1

R21 R22 ‚‚‚ R2,N R2,N+1

‚‚‚ ‚‚‚ ‚‚‚ ‚‚‚ ‚‚‚RN,1 RN,2 ‚‚‚ RN,N RN,N+1

RN+1,1 RN+1,2 ‚‚‚ RN+1,N RN+1,N+1

)‚

(A11S11(r) + A12S12(r)A21S21(r) + A22S22(r)‚‚‚AN,1SN,1(r) + AN,2SN,2(r)AN+1,1SN+1,1(r) + AN+1,2SN+1,2(r)

) (55)

F(r) )B1

r2+

B2

r4+ B3r + B4r

3 (56)

φi(b) ) ∑k)1

N+1

Rik[Ak1Sk1(b) + Ak2Sk2(b)] )

- ∑k)1

N+1

RN+1,k[Ak1Sk1(b) + Ak2Sk2(b)] ) -ψ(b) (57)

dψ

dr(b) ) ∑

k)1

N+1

RN+1,k[Ak1

dSk1

dr(b) + Ak2

dSk2

dr(b)] ) 0 (58)

γ )b

2{-5

b4∑j)1

N

RN+1,jAj1 +

AN+1,1[dSN+1,1

dr(b) -

SN+1,1(b)

SN+1,2(b)

dSN+1,2

dr(b)]} (59)

5292 J. Phys. Chem. B, Vol. 109, No. 11, 2005 Ruiz-Reina et al.

where the matrixR has been chosen to be

Substituting the latter result into the asymptotic forms ofφi

andψ yield

As regards the asymptotic functionF(r), from boundarycondition (38) it becomes

Now, renaming the constants transforms eq 63 into

and

Solutions (65) and (66) are asymptotic, and theN + 3constantsCi will be determined after numerically solving the

full system of differential equations and imposing the generalsolutions to satisfy the boundary conditions given by eqs 37,41, and 42 at the particle surface.

Effective Viscosity. We will follow a formalism similar tothat derived by Batchelor27 for dilute suspensions of unchargedspheres. However, in our case, use will be made of a cell modelto calculate the viscosity of a moderately concentrated suspen-sion. Hydrodynamic interactions between particles will be takeninto account by means of Simha’s hydrodynamic boundaryconditions at the outer surface of the cell.24 Let the velocityflow field and pressure for a pure straining motion (representedby a symmetrical and traceless tensorR) in the liquid mediumbe

wherei, j are Cartesian indexes in three dimensions (i, j ) 1-3)andxi are the Cartesian coordinates.

The quantities with a prime represent deviations from thecorresponding averaged quantities in the suspension (superscript0). The stress tensorσij at any point in the liquid is

whereδij is the Kronecker tensor and the additional perturbationstress tensor,σij ′, is defined as

The rate at which the forces at the outer surface S of the celldo work is

nk (k ) 1-3) being the coordinates of a unit vector normal tothe surface S. In the last expression use has been made of thecondition that, at the outer surface of the cell, the perturbationvelocity field is zero (just Simha’s),

and then

The effective viscosity is defined as that of a homogeneousfluid with the same viscosity of the suspension,η, for whichthe stress tensor is defined as

and the rate of working at the external boundary will be

Operating, we obtain

Ak2 )γ - Ak1Sk1(b)

Sk2(b)k ) 1, ...,N (60)

AN+1,2 ) -SN+1,1(b)

SN+1,2(b)AN+1,1 (61)

R ) (1 0 ‚‚‚ 0 00 1 ‚‚‚ 0 0‚‚‚ ‚‚‚ ‚‚‚ ‚‚‚ ‚‚‚0 0 ‚‚‚ 1 0

-â1

0

∑i)1

N

âi0

-â2

0

∑i)1

N

âi0

‚‚‚-

âN0

∑i)1

N

âi0

1 ) (62)

(φ1(r)‚‚‚φN(r)ψ(r)

) ) (R11 ‚‚‚ R1,N+1

‚‚‚ ‚‚‚ ‚‚‚RN + 1,1 ‚‚‚ RN + 1,N + 1

)‚

(A11[S11(r) -S11(b)

S12(b)S12(r)] + γ

S12(b)S12(r)

A21[S21(r) -S21(b)

S22(b)S22(r)] + γ

S22(b)S22(r)

‚‚‚

AN + 1,1[SN+1,1(r) -SN+1,1(b)

SN+1,2(b)SN+1,2(r)] ) (63)

F(r) ) B1[1

r2- 5r

2b3+ 3r3

2b5] + B2[1

r4- 7r

2b5+ 5r3

2b7] (64)

(φ1(r)‚‚‚φN(r)ψ(r)

) ) (R11 ‚‚‚ R1,N+1

‚‚‚ ‚‚‚ ‚‚‚RN+1,1 ‚‚‚ RN+1,N+1

)‚

(C1[S11(r) -S11(b)

S12(b)S12(r)] + γ

S12(b)S12(r)

C2[S21(r) -S21(b)

S22(b)S22(r)] + γ

S22(b)S22(r)

‚‚‚

CN+1[SN 1,1(r) -SN+1,1(b)

SN+1,2(b)SN+1,2(r)] ) (65)

F(r) ) CN+2[1

r2- 5r

2b3+ 3r3

2b5] + CN+3[1

r4- 7r

2b5+ 5r3

2b7] (66)

ui ) Rijxj + ui′ ) ui0 + ui′

P ) P0 + P′ (67)

σij ) - P0δij + 2η0Rij + σij ′ (68)

σij ′ ) -P′δij + η0(∂uj′∂xi

+∂ui′∂xj

) (69)

dWdt

) ∫Suiσiknk dS) ∫S

Rijxjσiknk dS)

Rij∫S[-P0δik + 2η0Rik - P′δik + η0(∂uix′∂xk

+∂uk′∂xi

)]xjnk dS

(70)

ui′ ) 0 w ui ) ui0 ) Rijxj (71)

2Rij )∂ui

0

∂xj+

∂uj0

∂xi(72)

σij/ ) -P0δij + 2ηRij (73)

dWdt

) ∫SRijxjσik

/nk dS) Rij∫S[-P0δik + 2ηRik]xjnk dS (74)

Electroviscous Effect under Overlapping Conditions J. Phys. Chem. B, Vol. 109, No. 11, 20055293

Equating the two eqs 70 and 74 representing the energydissipation rates, the term involvingP0 can be canceled becauseit is common in both expressions. Taking into account the lasteq 75, the dissipation rate equality becomes

Adding and subtracting the following surface integral

to the above-referred dissipation term, we have

Until this point, the formalism is the same as that shown ina previous paper,2 where overlapping was not considered. Foroverlapping conditions, the perturbation pressure function takesthe form

(see Appendix 2 for its derivation). Substituting this expressionand the velocity fieldu ) R‚r at the outer surface of the cell ineq 78, which is the above-mentioned Simha condition, we obtain

whereD is expressed by

From eq 79 we obtain

When the particles are uncharged, the asymptotic forms forthe far-field behavior of the functions are indeed exact up tothe surface of the particles. For theF(r) function, this meansthat the expression (66) is exact and has to fulfill the boundaryconditions (37) at the surface of the particle. The values of theunknown independent constantsCN+2 and CN+3 can then beobtained. The effective viscosity for the uncharged case turnsout to be

or definingS(φ) (first derived by Simha) as

the effective viscosity becomes

When the suspension is very dilute (φ f 0), the Simhafunction S(φ) tends to 1, and eq 84 transforms into Einsteinresult, as expected.

Electroviscous Coefficient.If the particles are charged, wecan define an electroviscous coefficientp by analogy with thedilute case,

Using eq 81, the electroviscous coefficient is given by

In the limit of a very dilute suspensionφ f 0, it is verifiedthat

and the electroviscous coefficient tends to that calculated byWatterson and White.16

If the charge on the particles is zero (ú ) 0), we have

and the electroviscous coefficient vanishes as expected.In Figures 2-4, the influence of overlapping can be observed

when the volume fraction is varied at constant electrokineticradius, with valuesκa ) 1, κa ) 10 andκa ) 25, respectively.Symbols are plotted in all curves to distinguish between themclearly, although they are not experimental curves. In the firstcase (κa ) 1), the ionic diffuse layer thickness is high and theoverlapping effect is important at any volume fraction in thefigure. Forκa ) 10, a high volume fraction (φ > 0.2) is requiredto appreciate this effect because the diffuse layer thickness isrelatively small, and forκa ) 25, only in theφ ) 0.5 case aredifferences appreciated. The data used in the generation of allfigures can be found in Table 1. The drag coefficientsλi in eq10 are related to the limiting equivalent conductanceΛi

0 ofi-type ions by

whereNA is Avogadro’s constant.The dependence of the electroviscous coefficient against the

variation ofφ in Figures 2-4 presents very interesting features.In the caseκa ) 1 (Figure 2), when the volume fraction isincreased, the electroviscous coefficient diminishes. Forκa )

Rij∫S2ηRikxjnk dS) 2ηRijRik∫S

xjnk dS) 2ηRijRik43πb3δkj )

2ηRijRij43

πb3 (75)

2ηRijRij43πb3 ) Rij∫S[η0(∂ui

0

∂xk+

∂uk0

∂xi) - P′δik +

η0(∂ui′∂xk

+∂uk′∂xi

)]xjnk dS (76)

Rijη0∫S(uinj + ujni) dS (77)

2ηRijRij43πb3 ) ∫S

Rijη0(uinj + ujni) dS+

∫SRij{[-P′δik + η0(∂ui

∂xk+

∂uk

∂xi)]xjnk - η0(uinj + ujni)} dS

(78)

P′(r ) ) -η0(12 d3F

dr3+ 3

rd2F

dr2+ 1

r2

Fel0 (r) ψ(r)

η0)(r ‚R‚r )

2η tr(R2)43πb3 ) 2η0 tr(R2)

43πb3 + 4πCN+12η0tr(R

2) -

2η0 tr(R2)Dφ

543

πb3 (79)

D ) -ψ(b)

φη0

Fel0 (b) ) -

ψ(b)

φη0∑i)1

N

zieni∞ exp[-

zie

kBTΨ0(b)] (80)

η ) η0[1 + φ(3CN+2

a3- D

5)] (81)

η ) η0[1 + 52φ{ 4(1 - φ

7/3)

4(1 + φ10/3) - 25φ(1 + φ

4/3) + 42φ5/3}](82)

S(φ) )4(1 - φ

7/3)

4(1 + φ10/3) - 25φ(1 + φ

4/3) + 42φ5/3(83)

η ) η0[1 + 52φS(φ)] (84)

η ) η0[1 + 52φS(φ)(1 + p)] (85)

p ) 25S(φ)(3CN+2

a3- D

5) - 1 (86)

limφf0

S(φ) ) 1, limφf0

D ) 0 (87)

CN+2ú)0 ) 5a3

6(88)

λi )NAe2|zi|

Λi0

(89)

5294 J. Phys. Chem. B, Vol. 109, No. 11, 2005 Ruiz-Reina et al.

25 (Figure 4), the dependence is the opposite: an increase ofthe volume fraction gives an increase of the electroviscouscoefficient. For intermediateκa values, i.e.,κa ) 10 (Figure3), the electroviscous coefficient increases whenφ changes from0.1 to 0.3 and decreases afterward whenφ increases until 0.5.

The reason for this behavior has to be found in the relativesizes of the electrical double layer and the cell. For lowκavalues, the diffuse layer is comparable or greater than the fluidshell inside the cell for allφ values presented (Figure 2), andan increase of the volume fraction sets the boundary conditionsat the cell surface nearer to the particle surface, confining thedouble layer in a smaller region than it would have in the infinitediluted case for the sameκa value. As a consequence, the

growing electrical body force acting inside the electrical doublelayer reduces its distortion, resulting in a diminution of energydissipation and, consequently, of the electroviscous coefficientp.

On the contrary, for highκa values the double layer thicknessis lower than the particle-cell surface distance for allφ valuesin Figure 4. The electroviscous coefficient rises withφ becausethe velocity gradients inside the electrical double layer augmentwhen the cell radius decreases, without contracting the diffuselayer. The enhancement of the velocity gradients involves anincrease of the distortion of the double layer.

Finally, at intermediateκa values, there is a competitionbetween both mechanisms described above, which explains theobservedp-maximum againstφ in Figure 3.

In Figure 5 it is shown the electroviscous coefficient againstthe volume fraction at different values of the electrokineticradiusκa and constant dimensionlessú-potentialeú/kBT ) 6.In this figure we see more clearly theφ dependence explainedbefore: for allκa values there is a maximum of the electro-viscous coefficient against the volume fraction. The maximumappears at a lowerφ value when the electrokinetic radius isdiminished.

The horizontal dashed lines indicate the results of Wattersonand White16 (W-W). These results are reproduced by our modelin the limit of very dilute suspensions, as stated above.

In Figure 6 the dependence ofp with κa and eú/kBT ispresented at a constant volume fractionφ ) 0.01. The volumefraction is low and at highκa values the results match those ofWatterson and White.16 However, at lowκa values and highúpotential the electroviscous coefficient is larger than those ofWatterson and White due to the overlapping of adjacent ioniclayers. This is better shown in Figure 7, wherep is calculatedboth taking into account the overlapping and neglecting it(D ) 0 in eq 86). Only forκa ) 1 do differences appear dueto overlapping.

In Figure 8 the volume fraction is far from the dilute limit.The effect of particle concentration is now noticeable and wecannot compare with earlier results. It can be seen that theelectroviscous coefficient has a remarkable maximum for all

Figure 2. Electroviscous coefficient against dimensionlessú-potentialat κa ) 1 for variousφ values.

Figure 3. Electroviscous coefficient against dimensionlessú-potentialat κa ) 10 for variousφ values.

Figure 4. Electroviscous coefficient against dimensionlessú-potentialat κa ) 25 for variousφ values.

Figure 5. Electroviscous coefficient dependence withφ at constantú-potential (eú/kBT ) 6) and variousκa values. W-W is theWatterson-White’s16 limit for very dilute suspensions.

TABLE 1: Data Used in the Generation of the Figures

particle radius a ) 100 nmtemperature T ) 298.16 Krel permittivity of suspending liquid εrs ) 78.55rel permittivity of solid particles εrp ) 2electrolyte KClco-ion K+ limiting equiv

conductanceΛ-

0 ) 76.31 cm2/(ohm‚g.equiv)

counterion Cl- limiting equivconductance

Λ-0 ) 73.48 cm2/(ohm‚g.equiv)

Electroviscous Effect under Overlapping Conditions J. Phys. Chem. B, Vol. 109, No. 11, 20055295

κa and dimensionlessú-potential values. It is usual to find amaximum of the primary electroviscous coefficient (i.e., in thevery dilute limit of Watterson and White16) as a function ofú-potential. Our interpretation of the maximum againstú-po-

tential is as follows: the dissipation of energy (and, conse-quently, the viscosity) increases withú when the potential islow because there are more ions in the double layer and thedistortion of the flow field around the particles is higher. But ifthe potential rises sufficiently, the electrical body forces actingon the fluid inside the double layer reduces the distortion ofthe latter and tends to recover the equilibrium distribution. Thismeans that the region of flow distortion is smaller and the systemapproaches the Einstein problem with an effective particle radiuslarger than the true radius of the solid particles.

The maximum ofp against the electrokinetic radiusκa isdue to the overlapping of adjacent ionic layers and, therefore,has not any analogue in the model of Watterson and White.16

The physical interpretation is that, as the electrokinetic radiusdecreases, the Debye length is greater and the region of doublelayer distortion increases, leading to more energy dissipation.On the other hand, the number density of ions inside the diffuselayer is also decreasing and this contributes to a lower viscosity.The competition between them explains the maximum.

Finally, in Figure 9, we show the calculations of theelectroviscous coefficient for a volume fraction close to the limitof what, we believe, is the limit of validity of the model (φ )0.5). In this case the cell is very small; therefore,κa must bevery high for the effect of overlapping to disappear. Themaximum inp occurs at lowerú-potential asκa increases. Thismeans that, as the ionic diffuse layer is smaller, the effect ofthe electrical body force in it, which makes it less deformable,happens at lower potentials (according to the interpretationsgiven above).

But the most remarkable feature on this graph is the increaseof p asκa increases up toκa ) 10 and the following decreaseat κa ) 25. The reason for this curious behavior is that we areat very high particle concentration, leading to a very small cellsize (the conditions for no perturbation are imposed close tothe particle surface). When the Debye length increases (κa )25 f κa ) 10) the ionic diffuse layer grows up inside the celland, then, the distorted region is bigger. Nevertheless, whenthis layer is larger than the cell, what one gets whenκa decreasesis that the number of ions inside the cell is decreasing and,consequently, the dissipated energy is smaller.

It can be observed in Figures 2-4, 7, and 9 presented abovethat when the contribution of the overlapping of adjacent ionicdouble layers is included in the theoretical model (overlapcurves), the electroviscous coefficient is higher than in the caseit is not taken into account (no-overlap curves).

It also follows from the precedent discussions that generallythe electroviscous coefficient diminishes when the electrokinetic

Figure 6. Electroviscous coefficient againstκa and dimensionlessú-potential for constant volume fractionφ ) 0.01.

Figure 7. Electroviscous coefficient against dimensionlessú-potentialfor constant volume fractionφ ) 0.01.

Figure 8. Electroviscous coefficient againstκa and dimensionlessú-potential for constant volume fractionφ ) 0.1.

Figure 9. Electroviscous coefficient against dimensionlessú-potentialat φ ) 0.5 for variousκa values.

5296 J. Phys. Chem. B, Vol. 109, No. 11, 2005 Ruiz-Reina et al.

radius is increased and rises with particle volume fraction andú potential. This assertion is always true provided that thephenomenon of “rigidization” of the electrical double layer doesnot appear. The appearance of “rigidization” occurs if theparticle volume fraction and/or theú-potential are augmentedat constant electrokinetic radius. In Figures 5 and 8 it can beobserved that the bigger the electrical double layer (i.e., thesmaller the electrokinetic radius), the lower theφ value or thehigher theú-potential value where the phenomenon takes place.

We think that the model presented in this work can improvethe agreement between theory and experiment in the primaryelectroviscous effect. As mentioned in the Introduction, thereis an underestimation of the effect by the theoretical models,especially under conditions of very low ionic concentrations.In its experimental determination, it is usual to reachφ valuesof the order ofφ ) 0.01. This means that, for colloidal particlesof 100 nm radius and a 1:1 electrolyte concentration 10-5 M,the particle-cell surface distance (b - a) is about only 3.7 timesthe double layer thicknessκ-1 in that case, and for a 2:1electrolyte concentration 10-6 M, we find that (b - a) ≈2.05κ-1. As we can infer from these examples, there must besome influence of the hydrodynamic interactions and overlap-ping of adjacent diffuse layers that is not being taken intoaccount.

In addition, with a new theoretical model able to predict theelectroviscous coefficient of moderately concentrated suspen-sions, it is possible to extend the range ofφ values in theexperimental studies of the electroviscous effect, making thecomparison theory-experiment more reliable and easier to getthan in the dilute limit. We stress the necessity of newexperimental works to check the validity of the present theory.A Fortran program for the calculation of the electroviscous effectis available from the authors.

Appendix 1: Boundary Condition for the Function ψ(r)at the Particle Surface

By substituting the expressions

in the boundary condition (35), we obtain

Operating in this equality, we have

We need an expression for the derivative in the secondmember of eq 93.

The Laplace equation for the electric potential inside theparticle must be satisfied, so

By substituting expression (91) in this equation, we obtainthe equation for the functionψp(r),

The general solution of this equation is

The constantC2 must be zero becauseδΨp(r ) has to be finitefor r ea, and thenψp(r) ) C1r2.

According to the boundary condition (34),

and substituting the latter result, we obtain

and then

The radial perturbation in the interior electrical potential is,finally,

Operating with the latter potential function in eq 93, theboundary condition (42) for the perturbation potential at theparticle surface is obtained

Appendix 2. Perturbation Pressure Function

We need a full solution for the perturbation pressure functionof the Navier-Stokes equation

Substituting the following perturbed quantities

in the Navier-Stokes equation (102) yields

In equilibrium conditions the equation

is satisfied and the corresponding terms in eq 107 are canceled.In addition, the last nonlinear term in eq 107 can be eliminated

δΨ(r ) ) ψ(r)(r ‚R‚r ) (90)

δΨp(r ) ) ψp(r)(r ‚R‚r ) (91)

εrs[ r ‚∇(ψ(r)(r ‚R‚r ))]|r)a ) εrp[ r ‚∇(ψp(r)(r ‚R‚r ))]|r)a (92)

εrsdψdr

(a) ) εrp

dψp

dr(a) (93)

∇2δΨp(r ) ) 0 (94)

L2ψp(r) )d2ψp(r)

dr2+ 2

r

dψp(r)

dr- 6

r2ψp(r) ) 0 (95)

ψp(r) ) C1r2 + C2

1

r3(96)

δΨ(r)|r)a ) δΨp(r)|r)a (97)

C1r2(r ‚R‚r )|r)a ) ψ(r)(r ‚R‚r )|r)a (98)

C1 )ψ(a)

a2(99)

ψp(r) )ψ(a)

a2r2 (100)

εrsdψdr

(a) ) εrp

2ψ(a)a

wdψdr

(a) -2εrp

εrs

ψ(a)a

) 0 (101)

η0∇2u(r ) - ∇P(r ) ) Fel(r )∇Ψ(r ) (102)

ni(r ) ) ni0(r) + δni(r ) i)1, ...,N (103)

P(r ) ) P0(r) + P′(r ) Ψ(r ) ) Ψ0(r) + δΨ(r ) (104)

Fel(r ) ) Fel0 (r) + δFel(r ) δFel(r ) ) ∑

i)1

N

zieδni(r ) (105)

δni(r ) ) -( zie

kBT)ni0(r)[δΨ(r ) + Φi(r )] )

-( zie

kBT)ni0(r)[ψ(r) + φi(r)]( r ‚R‚r ) (106)

∇P0(r) + ∇P′(r ) ) -η0∇ × ∇ × u(r ) - Fel(r )∇Ψ(r ) )

-η0∇ × ∇ × u(r ) - Fel0 (r)∇Ψ0(r) - Fel

0 (r)∇δΨ(r ) -

δFel(r )∇Ψ0(r) - δFel(r )∇δΨ(r ) (107)

∇P0(r) ) -Fel0 (r) ∇Ψ0(r) (108)

Electroviscous Effect under Overlapping Conditions J. Phys. Chem. B, Vol. 109, No. 11, 20055297

because we are looking for the linear response inR, i.e.,

the functiont(r) being defined in eq 109.Simplifying and rearranging terms, we obtain

As the velocity field from eqs 5 and 25 can be expressed by

operating, we obtain

where the functiong(r) is given by

Substituting the latter results in the linearized Navier-Stokesequation (112) for the perturbation pressure yields, afterrearranging terms, yields

On the other hand, symmetry conditions permit us to expressthe perturbation pressure function as

and then, its gradient turns out to be

Equating the terms with the same vector dependences in thetwo expressions (116) and (118), we obtain the following twoequations

The h(r) function, and then, the pressure function, can bederived from the second one

Replacing now the functiong(r) and its first derivative inthe latter equation, we finally obtain

We now have to check that eq 119 is satisfied for theh(r)function we have just calculated. Replacingh(r) in the firstmember of eq 119 and substituting thet(r) function and thefirst derivative of theg(r) function in the second member weobtain, after rearranging terms,

which is the differential eq 22 satisfied by theF(r) function.

Acknowledgment. Financial support for this work byMinisterio de Ciencia y Tecnologı´a, Spain (Project MAT2003-04688) is gratefully acknowledged by E. Ruiz-Reina and F.Carrique.

References and Notes

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Merino, A. I.; Garcı´a-Sanchez, P.J. Phys. Chem B2003, 107, 9528.(3) Garcı´a-Salinas, M. J.; de las Nieves, F. J.Colloids Surf. A2003,

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Colloids Surf. A1999, 159, 373.

δFel(r ) ) ∑i)1

N

zieδni(r ) ) -∑i)1

N (zi2e2

kBT)ni0(r)[ψ(r) +

φi(r)]( r ‚R‚r ) ) t(r)(r ‚R‚r ) (109)

∇δΨ(r ) ) ∇[ψ(r)(r ‚R‚r )] ) ∇[ψ(r)

r2(r ‚R‚r )] )

∇[ψ(r)

r2 ](r ‚R‚r ) +ψ(r)

r2∇(r ‚R‚r ) ) r2 d

dr[ψ(r)

r2 ]r (r ‚R‚r ) +

ψ(r)r

2(R‚r ) (110)

δFel(r )∇δΨ(r ) ) [t(r)(r ‚R‚r )]{r2 ddr[ψ(r)

r2 ]r (r ‚R‚r ) +

ψ(r)r

2(R‚r )} ≈ O(R2) (111)

∇P′(r ) ) -η0∇ × ∇ × u(r ) - Fel0 (r)∇δΨ(r ) -

δFel(r )∇Ψ0(r) ) -η0∇ × ∇ × u(r ) -

Fel0 (r)r2 d

dr[ψ(r)

r2 ]r (r ‚R‚r ) - Fel0 (r)

2ψ(r)r

(R‚r ) -

t(r)(dΨ0(r)dr )r (r ‚R‚r ) (112)

u(r ) ) (R‚r )[r - 3F(r) - r2 ddr(F(r)

r )] + r2 ddr(F(r)

r )r (r ‚R‚r )(113)

∇ × ∇ × u(r ) ) r2dg(r)dr

r (r ‚R‚r ) - [r2dg(r)dr

+ 3rg(r)](R‚r )(114)

g(r) ) - 1r

d2F(r)

dr2- 4

r2

dF(r)dr

+ 4F(r)

r3(115)

∇P′(r ) ) {-η0r2dg(r)

dr- Fel

0 (r)r2 ddr[ψ(r)

r2 ] -

t(r)(dΨ0(r)dr )}r (r ‚R‚r ) + {η0[r2dg(r)

dr+ 3rg(r)] -

Fel0 (r)

2ψ(r)r }(R‚r ) (116)

P′(r ) ) h(r)(r ‚R‚r ) (117)

∇P′(r ) ) [r2 ddr(h(r)

r2 )]r (r ‚R‚r ) + [2h(r)r ](R‚r ) (118)

[r2 ddr(h(r)

r2 )] ) -η0r2dg(r)

dr- Fel

0 (r)r2 ddr[ψ(r)

r2 ] -

t(r)(dΨ0(r)dr ) (119)

[2h(r)r ] ) η0[r2dg(r)

dr+ 3rg(r)] - Fel

0 (r)2ψ(r)

r(120)

P′(r ) ) {η0[r3

2dg(r)

dr+ 3r2

2g(r)] - Fel

0 (r) ψ(r)}(r ‚R‚r )(121)

P′(r ) ) -η0{r2

2d3F

dr3+ 3r

d2F

dr2+

Fel0 (r) ψ(r)

η0}(r ‚R‚r )

(122)

d4F(r)

dr4+

8

r

d3F(r)

dr3-

24

r3

dF(r)

dr+

24F(r)

r4)

-2e2

r2η0kBT(dΨ0(r)

dr)∑

i)1

N

zi2ni

0(r) φi(r) (123)

5298 J. Phys. Chem. B, Vol. 109, No. 11, 2005 Ruiz-Reina et al.

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Electroviscous Effect under Overlapping Conditions J. Phys. Chem. B, Vol. 109, No. 11, 20055299