Electric double layer for spherical particles in salt-free concentratedsuspensions including ion size effects

Rafael Roa,1 Felix Carrique,1,∗ and Emilio Ruiz-Reina2

1 Fısica Aplicada I, Universidad de Malaga, Spain,2 Fısica Aplicada II, Universidad de Malaga, Spain.

(Dated: February 18, 2011)

The equilibrium electric double layer (EDL) that surrounds the colloidal particles is determinantfor the response of a suspension under a variety of static or alternating external fields. An idealsalt-free suspension is composed by the charged colloidal particles and the ionic counterchargereleased by the charging mechanism. The existing macroscopic theoretical models can be improvedby incorporating different ionic effects usually neglected in previous mean-field approaches, whichare based on the Poisson-Boltzmann equation (PB). The influence of the finite size of the ions seemsto be quite promising because it has been shown to predict phenomena like charge reversal, whichhas been out of the scope of classical PB approximations. In this work we numerically obtain thesurface electric potential and the counterions concentration profiles around a charged particle in aconcentrated salt-free suspension corrected by the finite size of the counterions. The results show thelarge importance of such corrections for moderate to high particle charges at every particle volumefraction, specially, when a region of closest approach of the counterions to the particle surfaceis considered. We conclude that finite ion size considerations are obeyed for the development ofnew theoretical models to study nonequilibrium properties in concentrated colloidal suspensions,particularly the salt-free ones with small and highly charged particles.

I. INTRODUCTION

It is well-known that many transport phenomena insuspensions of charged particles, particularly their sta-bility and electrokinetic response, are closely related tothe properties of the equilibrium electric double layer(EDL) surrounding the particles [1–3]. Most of these the-oretical EDL models are based on the classical Poisson-Boltzmann equation (PB), a mean-field approach whichhas been the cornerstone for the study of the diffuse partof the EDL near charged interfaces in electrolyte solu-tions.

Many of the studies concern suspensions with low par-ticle concentration, but nowadays it is the concentratedregime [4–7] what deserves more attention because ofits practical applications. From the theoretical pointof view, these systems lack of a full understanding dueto the inherent complexity associated with the increas-ing particle-particle electrohydrodynamic interactions asparticle concentration grows.

Very commonly, the electrical interaction between par-ticles is largely screened at typical electrolyte concen-trations, and particle ordering is impeded. However,if the concentration of external salt is sufficiently low,or there is not any salt added to the suspension, theelectrical interactions can give rise to different short orlong-ranged ordered phases at relatively low particle vol-ume fractions. These systems are usually called colloidalcrystals or glasses [8, 9]. Suspensions composed just bythe charged particles and their ionic countercharge (theso-called added counterions) in the liquid medium are

∗E-mail address: carrique@uma.es

named salt-free suspensions, and its study has increas-ingly grown in recent years from experimental and theo-retical points of view.

The theoretical models for such systems are generallyscarce in comparison with those for suspensions in elec-trolyte solutions. In these systems there is a coupling be-tween the added counterions released by each particle andthe opposite charge on the particle surface to preserve theelectroneutrality. If the suspension is concentrated [5],the particle-particle electrohydrodynamic interactions in-troduce severe limitations when dealing with the responseto external fields, not only for the mathematical difficul-ties of dealing with many-body interactions, but also forthe additional numerical problems raised when it comesto solve the equations, no matter the approximation cho-sen. The authors have addressed this problem by using acell model approach to deal with electrokinetic and rhe-ological properties in concentrated suspensions [10–18].

In spite of the achievements of the cell model, it main-tains the drawbacks of the PB treatment, particularly,the neglecting of ionic correlations and the considerationof point-like ions in solution. Fortunately, for many sit-uations including low to moderate particle charges andmonovalent ions in solution, this approach provides anexcellent representation of the equilibrium problem. Onthe contrary, it has been argued that it yields unphys-ical ionic concentration profiles near highly charged in-terfaces, and also that it is not able to predict impor-tant phenomena like, for example, the overcharging, alsocalled charge reversal or charge inversion [19]. In the lit-erature we can find different attempts to overcome thesePB limitations. Some of them concerns microscopic de-scriptions of ion-ion correlations and the finite size ofthe ions [19–23]. Others are based on macroscopic de-scriptions considering average interactions by mean-field

arX

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approximations, that include entropic contributions re-lated with the excluded volume effect when the ions havea finite size [24–40]. While the former ones are physicallymore robust to describe real systems, they are basicallyrestricted to equilibrium conditions. The macroscopicapproaches, however, permit us to make predictions un-der equilibrium and non-equilibrium conditions. Unfor-tunately, it seems that mean-field approaches eliminatesome of the crucial information necessary for explaining,for example, the overcharging phenomena. In the past, ithas been shown with Monte Carlo simulations, and alsowith some theoretical approximations like the HNC-MSA[41], that electrical ion-ion correlations joined to excludeion volume effects could be the key point to understandsuch phenomena [19].

In a series of papers, Lopez-Garcıa and coworkers [35–38], studied the equilibrium EDL of a particle in a dilutecolloidal suspension in an electrolyte solution account-ing for the finite size of the ions by using a modifiedPoisson-Boltzmann equation (MPB). They also assumedthe existence of a common distance of closest approachto the particle surface for all ionic species. Their first re-sults indicated that the associated exclude volume effectwas very important for equilibrium and non-equilibriumproperties. Shortly after, their model was compared withsome Monte Carlo simulations for general ionic condi-tions [19], and while the latter predicted charge reversalfor the case of multivalent ions, the MPB was unable toshow such behaviours whatever the valence of the ionsmay be. It was concluded that the neglecting of ion-ioncorrelations in the MPB model was its main drawback.

Quite recently, Lopez-Garcıa et al. [42] have extendedtheir previous model to allow for a different distance ofclosest approach to the particle surface for each ionicspecies. The most important fact is that it predictscharge reversal under appropriate circumstances by onlyconsidering such ionic excluded volume effects. To ourknowledge, it is the first time that a phenomenologicaltheory based on macroscopic descriptions for equilibriumand non-equilibrium properties in colloidal suspensions isable to predict charge inversion, mainly with multivalentions. In other words, the charge reversal phenomenon,that has been elusive for such formalisms, now displaysa reasonable explanation from the macroscopic approachbased on mean-field approximations. No consideration ofexplicit ion-ion correlations is then required. Althoughno charge reversal is possible in an ideal salt-free suspen-sion because the whole ionic countercharge equals thatof the particle, and cannot exceed it, these achievementssupport our choice of using a similar macroscopic ap-proach for our salt-free concentrated suspensions, withthe help of a cell model. It has been recently shown that,up to moderately strong electrostatic couplings, the cellmodel proves accurate for the prediction of osmotic pres-sures of deionised suspensions in agreement with MonteCarlo simulations and renormalized-effective interactionapproaches [43].

The macroscopic approach provides reasonable predic-

tions for many non-equilibrium phenomena in concen-trated suspensions, which are in good agreement withexperiments. Other descriptions, like Monte Carlo’s ormicroscopic models, are unable of making feasible pre-dictions out of equilibrium. Summarizing, in this workwe aim at studying the EDL of a particle in a concen-trated salt-free suspension considering that the addedcounterions have finite ion size. Our approach is a mean-field one with an adequate MPB equation inside the cell.The procedure is a generalization of that already used byBorukhov [29] for the special case of a salt-free suspen-sion, valid for the concentrated case. Unlike Borukhov’streatment, our model also incorporates an excluded re-gion in contact with the particle of a hydrated radius size,which has been shown by Aranda-Rascon et al. [37, 38]to yield a more realistic representation of the solid-liquidinterface and also to predict results in better agreementwith experimental electrokinetic data. The equations willbe numerically solved and the equilibrium surface poten-tial and counterion concentration profiles inside a cell willbe analyzed upon changing particle volume fraction, par-ticle surface charge density, and size of the counterions.In order to show the realm of the finite ion size effect, theresults will be compared with MPB predictions that donot take into account a finite distance of closest approach,and also with standard PB predictions for point-like ions.

II. THEORY

A. The cell model

To account for the interactions between particles inconcentrated suspensions, a cell model is used (bareCoulomb interactions among particles are included in anaverage sense, but ions-induced interactions between par-ticles as well as ion-ion correlations, are ignored). Fordetails about the cell model approach see the excellentreview of Zholkovskij et al. [44]. Different electrokineticand rheological phenomena in salt-free concentrated sus-pensions have been developed by the authors accordingto this concept: the DC electrokinetic response to staticelectric fields [10] (electrophoretic mobility and electricalconductivity), their response to oscillating electric fields[11, 12] (dynamic mobility, complex conductivity and di-electric response); the electroviscous effect [13], and somerelated to more realistic descriptions of the phenomenataking into account the possibility of chemical reactionsin the system in agreement with experimental conditions[14–18]. We have learned from those works how impor-tant the description of the EDL is for the non-equilibriumresponses.

Concerning the cell model, Fig. 1, each spherical par-ticle of radius a is surrounded by a concentric shell of theliquid medium, having an outer radius b such that theparticle/cell volume ratio in the cell is equal to the par-ticle volume fraction throughout the entire suspension,that is

3

a

bR

R

FIG. 1: Cell model including the distance of closest approachof the counterions to the particle surface.

φ =(ab

)3

(2.1)

The basic assumption of the cell model is that themacroscopic properties of a suspension can be obtainedfrom appropriate averages of local properties in a uniquecell.

B. Finite size of the counterions

Let us consider a spherical charged particle of radiusa and surface charge density σ immersed in a salt-freemedium with only the presence of the added counterionsof valence zc. The axes of the spherical coordinate system(r, θ, ϕ) are fixed at the center of the particle. In theabsence of any external field, the particle is surroundedby a spherically symmetrical charge distribution.

Within a mean-field approximation, the total free en-ergy of the system, F = U −TS, can be written in termsof the equilibrium electric potential Ψ(r) and the coun-terion concentration nc(r). The configurational internalenergy contribution U is

U =

∫dr

[− ε0εr

2|∇Ψ(r)|2 + zcenc(r)Ψ(r)− µcnc(r)

](2.2)

The first term is the self-energy of the electric field,where ε0 is the vacuum permittivity, and εr is the rela-tive permittivity of the suspending medium. The nextterm is the electrostatic energy of the counterions in theelectrostatic mean field, and the last term couples thesystem to a bulk reservoir, where µc is the chemical po-tential of the counterions.

The entropic contribution −TS is

− TS = kBTnmaxc

∫dr

[nc(r)

nmaxc

ln

(nc(r)

nmaxc

)+

(1− nc(r)

nmaxc

)ln

(1− nc(r)

nmaxc

)](2.3)

where kB is Boltzmann’s constant, T is the absolute tem-perature, and nmax

c is the maximum possible concentra-tion of counterions due to the excluded volume effect,defined as nmax

c = V −1, where V is the average volumeoccupied by an ion in the solution. The first term insidethe integral is the entropy of the counterions, and thesecond one is the entropy of the solvent molecules. Thislast term accounts for the ion size effect that modifies theclassical Poisson-Boltzmann equation and was proposedearlier by Borukhov et al. [27]

The variation of the free energy F = U − TS withrespect to Ψ(r) provides the Poisson equation

∇2Ψ(r) = − zce

ε0εrnc(r) (2.4)

and the counterions concentration is obtained performingthe variation of the free energy with respect to nc(r)

nc(r) =bc exp

(− zceΨ(r)

kBT

)1 + bc

nmaxc

[exp

(− zceΨ(r)

kBT

)− 1] (2.5)

where bc is an unknown coefficient that represents theionic concentration where the electric potential is zero.

Applying the spherical symmetry of the problem andcombining eqns (2.4) and (2.5), we obtain the modifiedPoisson-Boltzmann equation (MPB) for the equilibriumelectric potential

d2Ψ(r)

dr2+

2

r

dΨ(r)

dr

= − zce

ε0εr

bc exp(− zceΨ(r)

kBT

)1 + bc

nmaxc

[exp

(− zceΨ(r)

kBT

)− 1] (2.6)

As the cell is electrically neutral we have

Q = 4πa2σ = −4π

∫ b

a

ρ(r)r2dr

= −4πzce

∫ b

a

nc(r)r2dr (2.7)

The unknown coefficient bc can be calculated itera-tively substituting the value of nc(r), eqn (2.5), in thelast expression

σ = −zcea2

∫ b

a

bc exp(− zceΨ(r)

kBT

)1 + bc

nmaxc

[exp

(− zceΨ(r)

kBT

)− 1]r2dr

(2.8)We need two boundary conditions to solve the MPB

equation. The first one is

dΨ(r)

dr

∣∣∣∣r=b

= 0 (2.9)

which derives from the electroneutrality condition of thecell and the application of Gauss theorem to the outersurface of the cell. The second one is

Ψ(b) = 0 (2.10)

4

that fixes the origin of the electric potential at r = b.The MPB problem, eqns (2.6), (2.9) and (2.10), can be

solved iteratively using eqn (2.8) to find the unknown co-efficient bc. Fortunately, the authors have recently shown[14] that this kind of problem can be solved avoiding theiterative procedure. For this purpose, we will use dimen-sionless variables, which are defined as

x =r

aΨ(x) =

eΨ(r)

kBTσ =

ea

ε0εrkBTσ

bc =e2a2

ε0εrkBTbc nmax

c =e2a2

ε0εrkBTnmaxc (2.11)

rewritting eqn (2.6) as

g(x) ≡ d2Ψ(x)

dx2+

2

x

dΨ(x)

dx=

−zcbce−zcΨ(x)

1 + bcnmaxc

(e−zcΨ(x) − 1

)(2.12)

where we have defined the function g(x). If we differen-tiate it, after a little algebra, it is possible to eliminatethe unknown coefficient bc and find that

g′(x) + zcg(x)Ψ′(x) +1

nmaxc

g2(x)Ψ′(x) = 0 (2.13)

where the prime stands for differentiation with respectto x. In terms of the electric potential, eqn (2.13) isrewritten as

Ψ′′′(x) +2

xΨ′′(x)− 2

x2Ψ′(x)

+ Ψ′(x)

(Ψ′′(x) +

2

xΨ′(x)

)·[zc +

1

nmaxc

(Ψ′′(x) +

2

xΨ′(x)

)]= 0 (2.14)

Eqn (2.14) is a nonlinear third-order differential equa-tion that needs three boundary conditions to completelyspecify the solution. Two of them are provided by eqns(2.9) and (2.10), which now read

Ψ′(h) = 0 Ψ(h) = 0 (2.15)

where h = (b/a) = φ−1/3 is the dimensionless outer ra-dius of the cell. The third one specifies the electrical stateof the particle, and can be obtained by applying Gausstheorem to the outer side of the particle surface r = a

dΨ(r)

dr

∣∣∣∣r=a

= − σ

ε0εr(2.16)

Its dimensionless form is

Ψ′(1) = −σ (2.17)

Regarding dilute suspensions, it is very common in theliterature to use the surface charge (also regulated sur-face charge) or the surface potential as a boundary condi-tion at the particle surface when solving the equilibrium

Poisson-Boltzmann equation, and all of them are equallyvalid. When it comes to concentrated suspensions, weprefer the use of the particle surface charge as boundarycondition by the following reasoning. The particle chargeis a property that can be often measured experimentally.Instead, the surface potential depends on the choice ofthe potential origin, which is the solution “bulk” for thedilute case (see the discussion in the Appendix). As inthe concentrated case there is not a clear “bulk”, the ori-gin is typically chosen at the outer surface of the cell,r = b, which depends on the particle volume fraction,eqn (2.1). In any case, we always can use the surfacepotential as a boundary condition, representing the po-tential difference between the particle surface and theouter surface of the cell.

If we consider point-like counterions, nmaxc = ∞, eqn

(2.14) becomes the expression obtained by Ruiz-Reinaand Carrique [14]. It can be easily demonstrated thatsolving the third-order problem is mathematically equiv-alent to finding the solution of the iterative problem, thatis, that any function satisfying eqns (2.14), (2.15) and(2.17) also satisfies eqns (2.6) and (2.8)−(2.10), and viceversa. We have removed the unknown coefficient bc us-ing this procedure and now the electric potential can beobtained numerically in one single step.

Eqns (2.14), (2.15) and (2.17) form a boundary valueproblem that can be solved numerically using the MAT-LAB routine bvp4c [45], that computes the solution witha finite difference method by the three-stage Lobatto IIIAformula. This is a collocation method that provides a C1

solution that is fourth order uniformly accurate at allthe mesh points. The resulting mesh is non-uniformlyspaced out and has been chosen to fulfill the admittederror tolerance (taken as 10−6 for all the calculations).

Once we have found the electric potential Ψ(x) we can

obtain the coefficient bc by evaluating eqn (2.12) at x = h

bc = − Ψ′′(h)

zc(2.18)

The ionic concentration nc(r) is determined using eqns(2.5) and (2.11).

C. Excluded region in contact with the particle

Following the work of Aranda-Rascon et al. [37], weincorporate a distance of closest approach of the coun-terions to the particle surface, resulting from their finitesize. We assume that counterions cannot come closer tothe surface of the particle than their effective hydrationradius, R, and, therefore, the ionic concentration will bezero in the region between the particle surface, r = a,and the spherical surface, r = a + R, defined by thecounterion effective radius. This reasoning implies thatcounterions are considered as spheres of radius R with apoint charge at its center.

5

The whole electric potential Ψ(r) is now determinedby the stepwise equation

d2Ψ(r)dr2 + 2

rdΨ(r)

dr = 0 a ≤ r ≤ a+R

Eqn (2.6) a+R ≤ r ≤ b(2.19)

To completely specify the problem we now should forcethe potential to be continuous at the surface r = a+ R,and also its first derivative, which is related to the con-tinuity of the normal component of the electric displace-ment at that surface, in addition to boundary conditions,eqns (2.9) and (2.10). Thus, in the region in contact withthe particle [a, a+R], we are solving Laplace’s equation,and, in the region [a+R, b], the MPB equation that weobtained previously in eqn (2.6).

As we have seen before, we can eliminate the coefficientbc changing the system of second order differential equa-tions, eqn (2.19), into one of third order, which hugelysimplify the resolution process

Ψ′′′(x) + 2x Ψ′′(x)− 2

x2 Ψ′(x) = 0 1 ≤ x ≤ 1 + Ra

Eqn (2.14) 1 + Ra ≤ x ≤ h

(2.20)where we have use dimensionless variables. The bound-ary conditions needed to completely close the problemare

Ψ′L(1) = −σ Ψ′L(1 + Ra ) =

−σ(1 +R/a)2

ΨP (h) = 0 Ψ′P (h) = 0 (2.21)

ΨL(1 + Ra ) = ΨP (1 + R

a ) Ψ′L(1 + Ra ) = Ψ′P (1 + R

a )

where subscript L refers to the region in which the poten-tial is calculated using Laplace’s equation, and subscriptP refers to the region in which we evaluate the MPBequation. The electric potential is obtained numericallyin one single step using eqns (2.20) and (2.21). The bc co-efficient and the ionic concentration nc(r) are determinedby eqns (2.18) and (2.5), respectively.

III. RESULTS AND DISCUSSION

For all the calculations, the temperature T has beentaken equal to 298.15 K and the relative electric permit-tivity of the suspending liquid εr = 78.55, which coincideswith that of the deionized water, although no additionalions different to those stemming from the particles havebeen considered in the present model. Also, the valenceof the added counterions zc has been chosen equal to+1 and the particle radius a = 100 nm. Other valuesfor zc could have been chosen. The model for point-likeions is able to work with any value of zc in the Poisson-Boltzmann equation, but we think that the predictions ofthis model will be less accurate in the case of multivalent

counterions, since it is based on a mean-field approachthat does not consider ion-ion correlations. Nevertheless,when we take into account the finite size of the ions, themain objective of this work, we include correlations as-sociated with the ionic excluded volume, solving partlythis problem, because we are still not considering theelectrostatic ion-ion correlations.

For the sake of simplicity, we assume that the averagevolume occupied by a counterion is V = (2R)3, being 2Rthe counterion effective diameter. With this considera-tion, the maximum possible concentration of counterionsdue to the excluded volume effect is nmax

c = (2R)−3.This corresponds to a cubic package (52% packing). Inmolar concentrations, the values used in the calculations,nmaxc = 22, 4 and 1.7 M, correspond approximately to

counterion effective diameters of 2R = 0.425, 0.75 and 1nm, respectively. These are typical hydrated ionic radii[46]. We think also that our model could be useful forshort or medium length polyelectrolytes, using an effec-tive value for nmax

c , as long as different ordered phasesdo not coexist.

We will discuss the results obtained from three dif-ferent models, the classical Poisson-Boltzmann equation(PB), the modified Poisson-Boltzmann equation by thefinite ion size effect (MPB), and the MPB equation in-cluding also the distance of closest approach of the coun-terions to the surface of the charged particle (MPBL).

-40

-30

-20

-10

e!(r

)/kBT

PBMPB, ncmax=22 MMPBL, ncmax=22 MMPB, ncmax=4 MMPBL, ncmax=4 MMPB, ncmax=1.7 MMPBL, ncmax=1.7 M

1 1.02 1.04 1.06 1.08 1.1r/a

10-3

10-2

10-1

100

101

n c(M

)

(a)

(b)

"=0.5 #=$40 µC/cm2

FIG. 2: Dimensionless electric potential distribution (a) andcounterions concentration (b) along the cell for different ionsizes, considering (dashed lines) or not (solid lines) the ex-cluded region in contact with the particle. Black lines showthe results of the PB equation.

6

Fig. 2 shows the dimensionless equilibrium electric po-tential distribution, Fig. 2a, and the counterions con-centration profiles, Fig. 2b, along the cell. We displayin solid black lines the predictions of the PB equation,coloured solid lines stand for the results of the MPB equa-tion and dashed lines correspond to the outcomes of theMPBL model. Different colours stand for different coun-terion sizes. The particle surface charge density havebeen chosen equal to −40 µC/cm2 and the particle vol-ume fraction is φ = 0.5 (very concentrated suspension),which implies a normalized cell size of b/a = 1.26.

We can observe in Fig. 2a that the inclusion of thefinite ion size effect (MPB calculations) always rises thesurface electric potential in comparison with the PB pre-dictions, which are recovered in the limit of point-likecounterions (nmax

c = ∞). The reason is that the limi-tation of the counterions concentration in the vicinity ofthe particle seriously diminishes the screening of the par-ticle charge, and consequently, there is an increment ofthe surface potential. An additional significative increaseof the surface potential is obtained when we take into ac-count the distance of closest approach of the counterionsto the particle surface (MPBL model). As expected, theexistence of a Laplace region free of counterions also pe-nalizes the screening of the particle charge.

We can also see in Fig. 2b how the finite ion size(MPB) creates saturated regions which corresponds toplateaus in the counterionic concentration profiles. Thesecondensates extend to larger distances when we considerthe excluded region in contact with the particle (MPBL).

Fig. 3 shows the effective charge divided by the par-ticle charge along the cell for a particle volume fractionφ = 0.5 and for different ion sizes, accounting or not forthe excluded region in contact with the particle. Theeffective charge is the charge seen at a given radial dis-tance from the center of the particle which includes theown particle charge and the ionic volume charge densityin the double layer integrated from the particle surface to

1 1.05 1.1 1.15 1.2 1.25r/a

10-4

10-3

10-2

10-1

100

Qeff/Q

p

PBMPB, ncmax=22 MMPBL, ncmax=22 MMPB, ncmax=4 MMPBL, ncmax=4 MMPB, ncmax=1.7 MMPBL, ncmax=1.7 M

!=0.5 "=#40 µC/cm2

FIG. 3: Effective charge divided by the particle charge alongthe cell for different ion sizes, considering (dashed lines) or not(solid lines) the excluded region in contact with the particle.Black lines show the results of the PB equation.

-40-30-20-100! (µC/cm2)

-60

-50

-40

-30

-20

-10

e"(a

)/kBT

PBMPB, ncmax=22 MMPBL, ncmax=22 MMPB, ncmax=4 MMPBL, ncmax=4 MMPB, ncmax=1.7 MMPBL, ncmax=1.7 M

10-5 10-4 10-3 10-2 10-1

#

-60

-50

-40

-30

-20

-10

(a)

(b)

#=0.5 !=$40 µC/cm2

FIG. 4: Dimensionless surface electric potential for differentvalues of particle surface charge density (a) and particle vol-ume fraction (b) for different ion sizes, considering (dashedlines) or not (solid lines) the excluded region in contact withthe particle. Black lines show the results of the PB equation.

r distance [15]. From this definition, the ratio Qeff/Qp isequal to one at the surface of the particle and zero at theouter surface of the cell. The results of this figure rein-forced those discussed previously in Fig. 2. As expected,the larger the ion size, the softer the screening of the par-ticle charge. Besides, when we consider the distance ofclosest approach of the counterions to the particle sur-face, the screening is even less effective at every givendistance from the latter.

Fig. 4 displays the dimensionless equilibrium surfaceelectric potential for a wide range of particle surfacecharge densities, Fig. 4a, and particle volume fractions,Fig. 4b. We repeat this study for different counterionsizes. We find that the surface electric potential increaseswith the particle charge density, Fig. 4a. However, insome cases there is a different behaviour in comparisonwith the point-like case. Initially, a fast and roughly in-crease of the surface potential with the surface chargedensity is observed, which is followed by a much slowergrowth at higher surface charge densities for the PB caseor when the size of the counterions is very small. Thisphenomenon is related to the classical counterion conden-sation effect: for high surface charges a layer of counteri-ons develops very close to the particle surface [5]. Whenthe ion size is taken into account (MPB) we limit theappearance of the classical condensation effect, becausewhen the surface charge is increased, the additional coun-terions join the condensate enlarging it. If the region ofclosest approach of the counterions to the particle surfaceis also considered (MPBL), the mechanism is the same,but now the additional counterions are located in fartherpositions from the particle surface. This explains the fur-ther increase of the surface potential observed for largeion sizes and high surface charges densities.

On the other hand, the surface electric potential de-

7

-40-30-20-100! (µC/cm2)

-30

-25

-20

-15

-10

-5

e"(a

)/kBT

#=10$5

#=10$4

#=10$3

-40-30-20-100! (µC/cm2)

-30

-25

-20

-15

-10

-5

e"(a

)/kBT

#=10$2

#=10$1

#=0.58

ncmax=22 M

FIG. 5: Dimensionless surface potential against the surfacecharge density for different particle volume fraction values.Solid lines stand for the results of the PB equation. Dashedlines stand for the results of the MPBL model.

10-5 10-4 10-3 10-2 10-1

!

-30

-25

-20

-15

-10

-5

e"(a

)/kBT

#=$40 µC/cm2

#=$20 µC/cm2

#=$10 µC/cm2

#=$5 µC/cm2

#=$1 µC/cm2

ncmax=22 M

FIG. 6: Dimensionless surface potential against the particlevolume fraction for different surface charge density values.Solid lines stand for the results of the PB equation. Dashedlines stand for the results of the MPBL model.

creases when the particle volume fraction increases, irre-spectively of the case studied: PB, MPB or MPBL, Fig.4b. When the particle concentration raises, the availablespace for the counterions inside the cell decreases and,consequently, the screening of the particle charge largelyaugments, thus reducing the value of the surface poten-tial.

Figs. 5 and 6 expand the results of Fig. 4. In them,we compare the results of the PB equation (solid lines)with those of the MPBL model (dashed lines) for a givencounterion size (typical of an hydronium ion in solution).Fig. 5 presents the dimensionless surface electric poten-tial at a wide range of particle surface charge densities.The different coloured lines correspond to different par-ticle volume fractions. The most remarkable fact shownin Fig. 5 is the large influence of the finite ion size effect

even for moderately low particle surface charge densities.While for PB predictions the surface potential hardlyincreases with surface charge for each volume fraction,the MPBL results display an outstanding growth for thesame conditions, associated with the lower charge screen-ing ability of the counterions because of their finite size.This fact will surely have important consequences on theelectrokinetic properties of such particles in concentratedsalt-free suspensions, as has already been shown for dilutesuspensions in electrolyte solutions by Aranda-Rascon etal. [37, 38].

Finally, Fig. 6 shows the dimensionless electric po-tential at the particle surface against the particle volumefraction. The different coloured lines correspond to differ-ent negative particle surface charge densities. As previ-ously stated, the particle surface potential monotonouslydecreases with volume fraction at fixed particle chargedensity. For the counterion size chosen, the larger thesurface charge, the larger the relative increase of the po-tential at every volume fraction.

From the results, we think that it is clear that the in-fluence of the finite ion size effect on the EDL descriptioncannot be neglected for many typical particle charges andvolume fractions.

IV. CONCLUSIONS

In this work we have studied the influence of finiteion size corrections on the description of the equilibriumelectric double layer of a spherical particle in a concen-trated salt-free suspension. The resulting model is basedon a mean-field approach, due to its reasonable successin modeling electrokinetic and rheological properties ofconcentrated suspensions, particularly the salt-free oneswhich deserve a special theoretical interest.

We have used a cell model approach to managewith particle-particle interactions, and modified Poisson-Boltzmann equations (MPB and MPBL) to account forsuch ion size effects. The theoretical procedure has fol-lowed that by Borukhov [29] but with the additional in-clusion of a region of closest approach for counterions tothe particle surface [37, 38]. We also have made numer-ical calculations stressing and discussing the results forconcentrated suspensions. The results have clearly shownthat the finite ion size effect (MPB equation) is quite im-portant for moderate to high particle charges at everyparticle volume fraction, and even more if the distanceof closest approach of the counterions to the particle sur-face is taken into account (MPBL model).

This equilibrium model presented in the paper is onthe base of future nonequilibrium models of the responseof a suspension to external electric fields. Experimen-tal results concerning the DC electrophoretic mobility,dynamic electrophoretic mobility, electrical conductivityand dielectric response, should be compared with the pre-dictions of the latter models to test them. To performsuch comparisons highly charged particles are required.

8

Existing highly charged sulfonated polystyrene latexescould be good candidates.

As a final conclusion, we think that it is mandatoryto apply these models to improve the existing theoriesfor predicting non-equilibrium properties in concentratedsuspensions. This task will be addressed by the authorsin the near future.

Appendix: Invariance of MPB equation underchanges in the origin of the electric potential

In this Appendix we demonstrate that the new MPBequation is invariant under changes in the origin of theelectric potential, as it must be according to basic phys-ical grounds. This is not a trivial question, because theclassical Poisson-Boltzmann equation for the low particleconcentration case in electrolyte solutions is not invariantunder changes in the origin of the electric potential: it isonly valid when we take the potential zero at an infinitedistance from the particle surface (bulk). When we studysalt-free and/or concentrated suspensions, we do not havea bulk, because the electroneutrality is achieved at theouter surface of the cell, and we have to take care aboutthe correct invariance of the equation. In this case we donot have an “infinity” to set the origin of the potentialand we have also an unknown coefficient bc that mustbe correctly defined to account for the latter statements.Our PB description for the salt-free case [14] satisfies theabove-mentioned invariance requirement.

As we said before, the unknown coefficient bc, that ap-pears in the MPB equation, eqn (2.6), represents the ionicconcentration where the equilibrium electric potential iszero. Our choice along this work has been Ψ(b) = 0. Ac-cording to this definition and using eqn (2.5), we can seethat nc(b) = bc.

The potential difference between two points r, r1 doesnot depend on the origin of potentials, Fig. 7,

Ψ(r)−Ψ(r1) = Ψ∗(r)−Ψ∗(r1) (A-1)

where the magnitudes without asterisks refer to the originΨ(b) = 0 and those with asterisks, to an arbitrary origin,Ψ∗(r1) = 0, for the electric potential. The relationshipbetween the potentials referred to both choices of electricpotential origins is

Ψ(r) = Ψ∗(r) + Ψ(r1) (A-2)

Substituting the last expression in the MPB equation,eqn (2.6), we obtain

d2Ψ∗(r)

dr2+

2

r

dΨ∗(r)

dr

= − zce

ε0εr

bc exp(− zceΨ∗(r)

kBT

)exp

(− zceΨ(r1)

kBT

)1 + bc

nmaxc

[exp

(− zceΨ∗(r)

kBT

)exp

(− zceΨ(r1)

kBT

)− 1]

(A-3)

r r1r2 = ba

FIG. 7: Changes in the origin of the electric potential.

After a little algebra, and defining

b∗c =bc(

1− bcbmaxc

)exp

(− zceΨ(r1)

kBT

)+ bc

bmaxc

(A-4)

it yields

d2Ψ∗(r)

dr2+

2

r

dΨ∗(r)

dr

= − zce

ε0εr

b∗c exp(− zceΨ∗(r)

kBT

)1 +

b∗cnmaxc

[exp

(− zceΨ∗(r)

kBT

)− 1]

= − zce

ε0εrn∗c(r) (A-5)

where we have defined

n∗c(r) =b∗c exp

(− zceΨ∗(r)

kBT

)1 +

b∗cnmaxc

[exp

(− zceΨ∗(r)

kBT

)− 1] (A-6)

Eqn (A-5) shows that Ψ∗(r) satisfies the MPB equa-tion, eqn (2.6), with a different coefficient b∗c . Let’s finallycheck that the concentration of counterions, n∗c(r), is thesame irrespective of the origin of the electric potential:n∗c(r) = nc(r), in spite of the different values of the un-known coefficients bc and b∗c .

By inserting eqn (A-4) in eqn (A-6), we obtain

n∗c(r) =bc(

1− bcbmaxc

)exp

(− zceΨ(r1)

kBT

)+ bc

bmaxc

·exp

(− zceΨ∗(r)

kBT

)1 + bc

nmaxc

[exp

(− zceΨ∗(r)

kBT

)− 1]

(1− bc

bmaxc

)exp

(− zceΨ(r1)

kBT

)+ bc

bmaxc

=bc exp

(− zceΨ(r)

kBT

)1 + bc

nmaxc

[exp

(− zceΨ(r)

kBT

)− 1] = nc(r) (A-7)

as we wanted to demonstrate.

9

Acknowledgements

Junta de Andalucıa, Spain (Project P08-FQM-3779),MEC, Spain (Project FIS2007-62737) and MICINN,

Spain (Project FIS2010-18972), co-financed with FEDERfunds by the EU. Helpful discussions with Dr. Juan J.Alonso are gratefully acknowledged.

[1] J. Lyklema, Fundamentals of interface and colloid sci-ence: vol. II, Solid-liquid interfaces (Academic Press,London, 1995).

[2] R. J. Hunter, Foundations of colloid science (Oxford Uni-versity Press, London, 1995).

[3] J. H. Masliyah and S. Bhattacharjee, Electrokinetic andcolloid transport phenomena (Wiley-Interscience, NewJersey, 2006).

[4] R. W. OBrien, J. Fluid Mech. 212, 81 (1990).[5] H. Ohshima, Theory of colloid and interfacial electric

phenomena (Academic Press, The Netherlands, 2006).[6] A. S. Dukhin, H. Ohshima, V. N. Shilov, and P. J. Goetz,

Langmuir 15, 3445 (1999).[7] F. Carrique, F. J. Arroyo, M. L. Jimenez, and A. V.

Delgado, J. Chem. Phys. 118, 1945 (2003).[8] A. K. Sood, Solid State Physics (Academic Press, 1991),

vol. 45, p. 1.[9] M. Medebach and T. Palberg, J. Chem. Phys. 119, 3360

(2003).[10] F. Carrique, E. Ruiz-Reina, F. J. Arroyo, and A. V. Del-

gado, J. Phys. Chem. B 110, 18313 (2006).[11] F. Carrique, E. Ruiz-Reina, F. J. Arroyo, M. L. Jimenez,

and A. V. Delgado, Langmuir 24, 2395 (2008).[12] F. Carrique, E. Ruiz-Reina, F. J. Arroyo, M. L. Jimenez,

and A. V. Delgado, Langmuir 24, 11544 (2008).[13] E. Ruiz-Reina and F. Carrique, J. Phys. Chem. C 111,

141 (2007).[14] E. Ruiz-Reina and F. Carrique, J. Phys. Chem. B 112,

11960 (2008).[15] F. Carrique and E. Ruiz-Reina, J. Phys. Chem. B 113,

8613 (2009).[16] F. Carrique and E. Ruiz-Reina, J. Phys. Chem. B 113,

10261 (2009).[17] E. Ruiz-Reina and F. Carrique, J. Colloid Interface Sci.

345, 538 (2010).[18] F. Carrique, E. Ruiz-Reina, F. J. Arroyo, and A. V. Del-

gado, J. Phys. Chem. B 114, 6134 (2010).[19] J. Ibarra-Armenta, A. Martın-Molina, and M. Quesada-

Perez, Phys. Chem. Chem. Phys. 11, 309 (2009).[20] P. Chodanowski and S. Stoll, J. Chem. Phys. 115, 4951

(2001).

[21] F. Jimenez-Angeles and M. Lozada-Cassou, J. Phys.Chem. B 108, 7286 (2004).

[22] A. Diehl and Y. Levin, J. Chem. Phys. 129, 124506(2008).

[23] A. Martın-Molina, C. Rodrıguez-Beas, R. Hidalgo-

Alvarez, and M. Quesada-Perez, J. Phys. Chem. B 113,

6834 (2009).[24] J. J. Bikerman, Philos. Mag. 33, 384 (1942).[25] C. W. Outhwaite and L. B. Bhuiyan, J. Chem. Soc., Fara-

day Trans. 2 79, 707 (1983).[26] Z. Adamczyk and P. Warszynsky, Adv. Colloid Interface

Sci. 63, 41 (1996).[27] I. Borukhov, D. Andelman, and H. Orland, Phys. Rev.

Lett. 79, 435 (1997).[28] I. Borukhov, D. Andelman, and H. Orland, Electrochim.

Acta 46, 221 (2000).[29] I. Borukhov, J. Polym. Sci. B Polym. Phys. 42, 3598

(2004).[30] L. Lue, N. Zoeller, and D. Blankschtein, Langmuir 15,

3726 (1999).[31] V. Kralj-Iglic and A. Iglic, J. Phys. II 6, 477 (1996).[32] K. Bohinc, V. Kralj-Iglic, and I. A., Electrochim. Acta

46, 3033 (2001).[33] P. M. Biesheuvel and M. van Soestbergen, J. Colloid In-

terface Sci. 316, 490 (2007).[34] M. Z. Bazant, M. S. Kilic, B. D. Storey, and A. Ajdari,

Adv. Colloid Interface Sci. 152, 48 (2009).[35] J. J. Lopez-Garcıa, M. J. Aranda-Rascon, and J. Horno,

J. Colloid Interface Sci. 316, 196 (2007).[36] J. J. Lopez-Garcıa, M. J. Aranda-Rascon, and J. Horno,

J. Colloid Interface Sci. 323, 146 (2008).[37] M. J. Aranda-Rascon, C. Grosse, J. J. Lopez-Garcıa, and

J. Horno, J. Colloid Interface Sci. 335, 250 (2009).[38] M. J. Aranda-Rascon, C. Grosse, J. J. Lopez-Garcıa, and

J. Horno, J. Colloid Interface Sci. 336, 857 (2009).[39] L. B. Bhuiyan and C. W. Outhwaite, Phys. Chem. Chem.

Phys. 6, 3467 (2004).[40] L. B. Bhuiyan and C. W. Outhwaite, J. Colloid Interface

Sci. 331, 543 (2009).[41] E. Gonzalez-Tovar and M. Lozada-Cassou, J. Phys.

Chem. 93, 3761 (1989).[42] J. J. Lopez-Garcıa, M. J. Aranda-Rascon, C. Grosse, and

J. Horno, J. Phys. Chem. B 114, 7548 (2010).[43] A. R. Denton, J. Phys.: Condens. Matter 22, 364108

(2010).[44] E. K. Zholkovskij, J. H. Masliyah, V. N. Shilov, and S.

Bhattacharjee, Adv. Colloid Interface Sci. 134-135, 279(2007).

[45] J. Kierzenka and L. F. Shampine, ACM Trans. Math.Softw. 27, 299 (2001).

[46] J. N. Israelachvili, Intermolecular and surface forces(Academic Press, London, 1992).