equilibrium sedimentation profiles of charged colloidal suspensions

Eur. Phys. J. E 2, 137–143 (2000) THE EUROPEANPHYSICAL JOURNAL Ec©

EDP SciencesSocieta Italiana di FisicaSpringer-Verlag 2000

Equilibrium sedimentation profiles of charged colloidalsuspensions

G. Telleza and T. Biben

Laboratoire de Physique (Unite Mixte de Recherche du Centre National de la Recherche Scientifique - UMR 5672), EcoleNormale Superieure de Lyon, 69364 Lyon cedex 07, France

Received 5 May 1999 and Received in final form 25 August 1999

Abstract. We investigate the sedimentation equilibrium of a charge-stabilized colloidal suspension in theregime of low ionic strength. We analyze the asymptotic behaviour of the density profiles on the basis ofa simple Poisson-Boltzmann theory and show that the effective mass we can deduce from the barometriclaw corresponds to the actual mass of the colloidal particles, contrary to previous studies.

PACS. 82.70.Dd Colloids – 05.20.Jj Statistical mechanics of classical fluids – 61.20.Gy Theory and modelsof liquid structure

1 Introduction

Under the action of gravity a colloidal suspension sedi-ments to form a stratified fluid. The equilibrium densityprofile of the colloidal particles results from the balancebetween the gravitational force and thermodynamic forcesas derived from the free energy of the system. The densityprofile usually exhibits a dense layer of colloidal particlesat the bottom of the container above which a light cloudof colloidal particles floats. In this last regime, the densityof particles is small enough to treat the fluid as an idealgas. Under the reasonable assumption that density gra-dients can be neglected, the equilibrium colloidal densityobeys the well-known barometric law:

ρcol(z) = ρ0col exp(−z/lg). (1.1)

Here, ρcol(z) denotes the density profile of the colloidalparticles, z is the altitude and lg = (βMg)−1 is the gravi-tational length, where β = (kBT )−1 is the inverse temper-ature, M is the buoyant mass of a colloidal particle andg the intensity of the gravitational field. This exponen-tial law is of practical interest since it gives a prescriptionfor the measurement of the buoyant mass M of the par-ticles. However, a recent experimental study of the sedi-mentation profiles of strongly de-ionized charged colloidalsuspensions [1] leads the authors to challenge the validityof this barometric law. An exponential behaviour was in-deed observed in the asymptotic regime, but the measuredgravitational length l∗g could differ significantly from theexpected one (a factor of two). l∗g was found to systemati-cally overestimate the actual value lg, with the result that

a Present address: Departamento de Fısica, Universidad delos Andes, A.A. 4976, Bogota, Colombia.e-mail: [email protected]

the buoyant mass measured within these experiments issystematically reduced compared to the known buoyantmass of the particles.

Some theoretical efforts have been made to study thisproblem. First Biben and Hansen [2] solved numericallythe problem in a mean-field approach, but unfortunatelydue to numerical difficulties the samples height consideredwhere of the order of the micron while in the experimentsthe samples height are of the order of the centimeter. As aconsequence, the dilute region at high altitude could notbe studied in this approach. Nevertheless, the numericalresults show a positive charge density at the bottom ofthe container and a negative charge at the top while thebulk of the container is neutral. This result shows that anon-zero electric field exists in the bulk of the containerand acts against gravity for the colloids.

More recently one of the authors studied a two-dimensional solvable model for this problem [3]. Thismodel is not very realistic (the valency of the colloids wasZ = 1 and there was no added salt) but has the nice fea-ture of being exactly solvable analytically. It confirmedthe condenser effect noticed for small height containersin reference [2]. For large height containers it showed anew interesting phenomenon: while there is still a positivecharge density at the bottom of the container, the negativecharge density is not any more at the top of the containerbut floating at some altitude. Interestingly, the analyti-cal expression for the density profiles in the asymptoticregime predicts a decay in exp(−z/lg)/z for the colloidaldensity. Besides the 1/z factor that cannot be explainedby a mean-field approach, no mass reduction is predictedby this model. However one should be cautious when com-paring two-dimensional systems to the three-dimensionalcase because the density is not relevant in two-dimensionalCoulomb systems: no matter how small the density in the

138 The European Physical Journal E

system is always coupled, the ideal gas regime is never at-tained. For this reason a decay of the density similar to theone of an ideal gas is in itself surprising in two dimensions.

Lately new results based on an approximate versionof the model introduced in reference [2] lead the authorsof these studies [4,5] to conclude that the mean-field ap-proach was indeed able to predict a mass reduction in theasymptotic regime. We present in the next two sectionssome new results about this problem treated under thePoisson-Boltzmann approximation, and show that it is in-deed not the case.

Obviously, a mean-field treatment of the electrostaticinteractions is questionable in the lower part of the con-tainer where the colloidal particles condense to form asolid layer, and the theories mentioned above are not ableto treat this region of the density profile; rather, thesetheories are meant to treat the asymptotic regime wherecorrelational effects are negligible. The models used in ref-erences [2,4,5] assume that the asymptotic regime is al-ready reached at the bottom of the sample, with the con-sequence that the constraint of global charge neutralityapplies to the asymptotic regime. When a dense sedimen-tation layer is present at the bottom of the container, theconstraint of global charge neutrality is no more restrictedto the asymptotic regime, but to the full density profile.As the asymptotic regime is only constituted by a smallfraction of the particles, the dense sedimented layer canbe considered as a reservoir fixing the chemical potentialof the ionic species in the asymptotic region. As a conse-quence, the theories mentioned above must be solved inthe grand-canonical (rather than canonical) ensemble. Wewill analyze the consequences of a sedimented substrate inthe last section of this article.

2 The model and the Poisson-Boltzmannapproximation

Let us consider some colloidal particles (for example, somelatex spheres) in a solution with some amount of addedsalt. In a polar solvent like water the colloids release somecounterions and therefore acquire a surface electric chargeZe (Z is a entire number usually positive and −e is thecharge of the electron). We consider that the colloidalsample is monodisperse, all colloids have the same va-lency Z, and that the counterions and the salt cationsare both monovalent and therefore we shall not make anydistinction between cations coming from the colloids andsalt cations. We then consider a three-component systemcomposed of colloidal particles with electric charge Zeand mass M , counterions with charge −e and coions withcharge +e. We shall neglect the masses of the counterionsand coions when compared with the mass of the colloids.The solvent shall be considered in a primitive model rep-resentation as a continuous medium of relative dielectricprimitivity ε (for water at room temperature ε ≈ 80). Thesystem is in a container of height h, the bottom of the con-tainer is at z = 0 altitude. We consider that the system isinvariant in the horizontal directions. The density profiles

of each species are denoted by ρcol(z), ρ+(z) and ρ−(z) (zis the vertical coordinate) for the colloids, the cations andthe anions, respectively, at equilibrium. Let us define theelectric charge density (in units of e) ρ = Zρcol− ρ−+ ρ+and the electric potential Φ, solution of the Poisson equa-tion

d2Φdz2

(z) = −4πεeρ(z). (2.1)

It is instructive to recall that the Poisson-Boltzmannequation can be derived from the minimization of the freeenergy density functional

F [ρcol, ρ+, ρ−] =∑i∈{col,+,−}

∫ h

0

kBTρi(z)×[ln(λ3i ρi(z))− 1

]dz

+∫ h

0

Mgzρcol(z) dz +12

∫ h

0

eρ(z)Φ(z), (2.2)

where λi is the de Broglie wavelength of species i. Mini-mization of the grand potential with respect to the densi-ties: δF/δρi(z)−µi = 0, where µi is the chemical potentialof species i, yields

ρcol(z) = ρ0col exp(−βZeΦ(z)− βMgz), (2.3a)

ρ+(z) = ρ0+ exp(−βeΦ(z)), (2.3b)

ρ−(z) = ρ0− exp(βeΦ(z)). (2.3c)

We shall work first in the canonical ensemble, the pref-actors ρ0i which depend on the chemical potentials µi aredetermined by the normalizing conditions∫ h

0

ρi(z) dz = Ni, (2.4)

where Ni is the total number of particles per unitarea of species i. The system is globally neutral so wehave ZNcol −N− +N+ = 0.

Let us introduce the following notations: lg =(βMg)−1 is the gravitational length of the particles, l =βe2/ε is the Bjerrum length, φ = βeΦ is the dimensionlesselectric potential and κi = (4πlNi/h)1/2. κ−1

± are the De-bye lengths associated to the counterions and the coionsand (Zκcol)−1 is the Debye length associated to the col-loidal particles. For a quantity q(z) depending on the al-titude, let us define its mean value 〈q〉 = ∫ h

0q(z) dz/h.

With these notations equations (2.1) and (2.3) yield themodified Poisson-Boltzmann equation

d2φdz2

(z) = −Zκ2cole−Zφ(z)−z/lg

〈e−Zφ(z′)−z′/lg〉

+ κ2−eφ(z)

〈eφ(z′)〉 − κ2+

e−φ(z)

〈e−φ(z′)〉 . (2.5)

From equation (2.5) it is clear that the problem has thefollowing scale invariance: if φ(z) is a solution of (2.5) thenφ(αz) is a solution of the problem with the rescaled lengths

G. Tellez and T. Biben: Equilibrium sedimentation profiles of charged colloidal suspensions 139

Fig. 1. From left to right, up to down, starting at the upperleft corner, profiles of: counterions density, colloidal volumefraction in decimal logarithmic scale, coions density, chargedensity, dimensionless electric potential φ, electric field E =−dφ/dz. Full curves correspond to γ = 33.2, dashed curves toγ = 332. The values of the remaining parameters are Z = 100,lg = 0.128mm, h = 30mm and κ = 6.5 (corresponding toCsalt = 0.1mMol/l and ηcol = 0.12 for σ = 180 nm).

αlg, ακ−1i and αh. As a consequence, the shape of the elec-

tric potential is entirely determined by the four dimension-less parameters Z, the charge of the macroions, lg/h andthe two parameters γ = πZ2lNcollg and κ = Csaltlg/Ncol,where Csalt is the salt concentration. γ and κ are the twocontrol parameters introduced in reference [5] and we willuse these parameters for comparison purposes. While γexpresses the competition between Coulomb coupling andgravity, κ is the relative amount of added salt. To clar-ify our scope, it is necessary at this level to recall brieflythe main results obtained in reference [5]. When gravi-tational coupling is strong, γ � 1, the theory presentedin reference [5] predicts no mass reduction, in agreementwith the numerical results presented in reference [2]. Onthe contrary, in strong Coulomb coupling regimes, γ � 1,quite a large mass reduction is predicted in low salinityregimes (κ ≤ 1). Physical values of γ and κ for stronglydeionized suspensions (5 · 10−5Mol/l) are γ � 109 andκ � 1. In the next sections we will focus on the region ofthe phase diagram where a mass reduction is predicted,namely κ � 1 and γ � 1. However, it must be notedthat this region is not favorable to a numerical resolutionof the Poisson-Boltzmann equations (2.5). Indeed, thereare several length scales in this problem: the gravitationallength of the colloids lg, the Debye or screening length, theheight h of the container and eventually the hard-core di-ameter of the particles. In a realistic case h is of the orderof the centimeter, lg of the order of 0.1mm, the screeninglength of the order of 10 nm. We are faced to a practical

numerical problem, when we will transpose the problemto a lattice, the lattice spacing should be smaller than allthe physical lengths, but since h is much larger than theother lengths, the number of sites in the lattice shouldbe very high (of order 106). A possible approach to dealwith this problem is to study small containers as in refer-ence [2]. In this paper we want to study the case of highcontainers so we will use the physical values for the salin-ity κ � 1 and we will increase the Coulomb coupling asmuch as possible. The values we were able to reach are ofthe order of γ = 102, which is very small compared to thephysical one (109), but is fortunately large enough to bein the region where the parametrization presented in ref-erence [5] predicts a mass reduction. γ = 102 correspondto a value of the screening length of the order of 0.1mm,much larger than in usual physical cases. That way thenecessary number of points in the lattice remains reason-able (a few hundreds). Also, since the screening length isso large, the hard core of the macroions will not changethe results drastically from the case of point particles sowe will concentrate from now on the Poisson-Boltzmannproblem for point particles (Eq. (2.5)).

3 Results

Equation (2.5) is solved numerically by an iterativemethod [7]. Using the Green function of the one-dimensional Laplacian

G(z, z′) =12|z − z′| , (3.1)

the Poisson equation (2.1) can be written as

φ(z) = −4πl∫ h

0

G(z, z′)ρ(z′) dz′. (3.2)

Starting with an arbitrary electric potential, one cancompute the corresponding density profiles using equa-tions (2.3) and derive a new electric potential using equa-tion (3.2), then reiterate the process until a stationarysolution is attained. In practice instead of using the newpotential directly for the next iteration a mixing of theold and new densities is used.

3.1 Generic results

Figure 1 shows the density profiles of each species, thecharge density, the electric potential and the electric-fieldprofile of a typical sample for two values of the couplingγ = 33.2 and γ = 332. The values of the remainingparameters are Z = 100, lg = 0.128mm, h = 30mmand κ = 6.5 which correspond to a salt concentrationCsalt = 0.1mMol/l for a mean colloidal volume fractionηcol = 0.12 (we consider that the particles have a hard-core diameter σ = 180 nm to express the colloidal densityas a volume fraction in order to use units familiar with theexperiments but we do not account for hard-core effectsin the Poisson-Boltzmann equation).


The log plot of colloidal density profiles is similar tothe experimental ones [1]. In the bottom there is a slowdecay whereas at high altitudes there is a faster baromet-ric decay. Since we did not take into account the hardcore of the particles in the theory and we neglected cor-relations we do not find the discontinuity in the densityprofiles near the bottom of the sample observed in the ex-periments [1], due to the phase transition of the colloidsfrom an amorphous solid to a fluid.

The charge density profile confirms the results of ref-erence [3], that there is a strong accumulation of posi-tive charges at the bottom of the container, while thereis a cloud of negative charge density floating at some al-titude z∗. There are clearly two neutral regions in thecontainer: one at low altitude between the positive chargedensity at the bottom and the negative cloud, in whicha non-vanishing electric field exists, and a second neutralregion at high altitude, over the negative cloud. The elec-tric field in the lower region acts against gravity for thecolloids; therefore, as seen in the log plot of the colloidsdensity profile, the decay is much slower than the one foran ideal neutral gas. Numeric results for other series ofsamples suggest that this electric field is proportional toMg/Z. In the upper region the colloidal density dropsexponentially as exp(−z/lg) since the electric potentialis almost constant and the electric field vanishes. Thisfigure also shows the non-trivial behavior of the electricfield when the Coulomb coupling γ is increased. Althoughthe charge density profile is strongly affected (the posi-tive and negative clouds are strongly damped when theCoulomb coupling is increased), the dimensionless electricpotential φ = βeΦ does not change much. This effect il-lustrates very well the complexity of the limit γ → ∞.In this limit, the electric field keeps a finite value in thelower part of the sample and consequently the charge den-sity profile vanishes like 1/γ, and the intuitive conditionof local charge neutrality is then recovered in the strongCoulomb coupling regime. However, the finiteness of theelectric field clearly shows that it is not possible to in-vestigate the γ → ∞ limit by simply forcing the localneutrality condition in the Poisson Boltzmann problem,as done in reference [4]. Doing this would lead to incon-sistencies (the resulting density profiles would not satisfythe a priori constraint of local charge neutrality) and itwould remove the condenser effect which is necessary tocompensate the gravitational force.

Since the different densities vary with the altitude wecan define a local screening length which depends on thealtitude by

λ(z) ={4πl

(Z2ρcol(z) + ρ+(z) + ρ−(z)

)}−1/2. (3.3)

The two regions of the sediment are characterized bya very different behavior of this local screening length.In the lower region the colloidal density is so high thatZ2ρcol(z) � ρ+(z) + ρ−(z). In that region the colloidsdominate the screening length. On the other hand, in theupper region the colloidal density is very small and saltnow controls the screening length which is then constantsince at high altitudes the cations and anions densities are

almost constant and equal to the mean salt concentrationas seen in Figure 1. It is interesting to notice that elec-tric charges accumulate in the intermediate region aroundz∗ where there is a change of regime, in agreement withmacroscopic electrostatics principles.

The preceding remark allows us to understand howthe physical parameters (mean volume fraction of colloids,mass of the colloids, amount of added salt) will modifythe altitude z∗ which separates the two regions. For ex-ample if we add more salt, z∗ will diminish since we reachsooner the regime where Z2ρcol(z) < ρ+(z) + ρ−(z). Wehave computed the density profiles in several other caseschanging the values of the parameters in order to find thedependency of z∗ in these parameters. Our numerical re-sults suggest that

z∗ = − c1√lCsalt

+ a2Z

√Ncollg√Csalt

, (3.4)

with c1 = 0.15 ± 0.05 and a2 = 1.0 ± 0.1. The preced-ing equation can be written in a more attracting way,introducing the screening length associated to the saltλsalt = (4πlCsalt)−1/2 and the effective screening lengthassociated to the colloids λeffcol = (4πlZ

2Ncol/lg)−1/2, as

z∗ = λsalt

(−a1 + a2 lg

λeffcol

)(3.5)

with a1 =√4πc1 = 0.5 ± 0.2. We do not consider here

boundary effects: this equation is only valid if z∗ is smallerthan h. The finite height h of the container will have theeffect to “push” the negative cloud downwards if the pa-rameters are such that z∗ approaches the top of the con-tainer. The same holds for the bottom of the container ifz∗ is too small.

Another quantity of interest is the size∆z∗ of the nega-tive cloud, defined as the mid-height width of the negativepeak in the charge density profile (see Fig. 1). Since weknow that at z∗ altitude, Z2ρcol(z∗) is of the same orderof magnitude as ρ+(z∗) + ρ−(z∗) = 2Csalt, the screeninglength at that altitude is proportional to λsalt. From basicelectrostatics we know that the system will only toleratecharges over a length of order of the screening length, wededuce that ∆z∗ is proportional to λsalt. In fact the nu-merical results suggest also a linear dependence of ∆z∗ onlg:

∆z∗ = b1lg + b2λsalt (3.6)

with b1 = 5.0±0.5, b2 = 0.7±0.2, and the same restrictionsconcerning boundary effects as for the equation for z∗.

3.2 The apparent mass

As we mentioned before, at high altitudes (larger thanz∗) the electric potential is almost constant and the elec-tric field vanishes. From this it is clear that the colloidaldensity will decay as exp(−z/lg) and there is no appar-ent reduced mass. Nevertheless, let us notice that in the


Fig. 2. Colloidal density profile in decimal logarithmic scalefor different salt concentrations and restricted to volume frac-tions higher than 10−5. Common parameters to all curvesare: lg = 0.128mm, Z = 100, σ = 180 nm, ηcol = 0.12 andh = 30mm. The salt concentration in mMol/l from left toright is 0.4, 0.3, 0.2, 0.1, 0.05, 0.04, 0.03, 0.02, 0.01, 0.005. Theapparent gravitational length l∗g obtained from the slope of thelow density wing is, from left to right in mm: 0.131 ± 0.003,0.131 ± 0.003, 0.134 ± 0.003, 0.140 ± 0.004, 0.151 ± 0.012,0.155 ± 0.12, 0.162 ± 0.013, 0.172 ± 0.014, 0.187 ± 0.009,0.217± 0.011.

regime where the electric potential is almost constant inour calculations the corresponding colloidal volume frac-tion is smaller than 10−9. Such volume fractions cannot bemeasured experimentally. In practice the optical methodsused in [1] allow to measure only volume fractions largerthan 10−5. To investigate the consequences of such a trun-cation, we made a log plot of several colloidal volume frac-tion profiles restricting the plot to volume fractions higherthan 10−5 (Fig. 2). We computed the slope of the wingof the colloidal density to find an effective gravitationallength l∗g which is larger than the actual gravitationallength lg as observed in the experiments. Furthermore,when we plot the colloidal volume fraction profile and thecorresponding electric field profile together (Fig. 3) we no-tice that for volume fractions higher than 10−5 the electricfield is not zero.

The different plots in Figure 2 where obtained us-ing different salt concentrations, so the sediment height(which is proportional to z∗) varies. In this case we foundthat the apparent mass is a decreasing function of theheight of the sediment, in agreement with the experiments.However, the sediment height can be changed by changingother parameters like the mean colloidal density or theirvalency Z. Computing the apparent gravitational lengthl∗g as defined before for other series of samples, we foundthat the apparent gravitational length l∗g does not dependon Z or the mean colloidal density. In our model the ratio

Fig. 3. Colloidal volume fraction decimal logarithmic pro-file and the corresponding electric field profile in the caseCsalt = 0.005mMol/l, the other parameters being those of Fig-ure 2. Notice that in the low density wing used to compute theapparent gravitational length l∗g the electric field is not zero.

Fig. 4. The ratio of apparent gravitational length by the actualgravitational length l∗g/lg versus the salt screening length λsalt,for two different values of the gravitational length.

l∗g/lg is only a function of the salt density. Figure 4 showsthe ratio l∗g/lg as a function of salt screening length λsalt.

It must be noted however that these results have beenobtained for a value of the coupling γ = 33.2 that is quitesmall compared to the experimental value γ � 109. In-creasing γ will reduce this effect since the range of theresidual electric field is related to the Debye length whichis of the order of 10 nm in physical situations. An expla-nation of the experimental data in terms of a slow conver-gence due to a residual electric field does not seem to bevery realistic, on the basis of the model presented here.


Table 1. Comparison between the reduced mass predicted by Lowen’s theory and our numerical results from the free mini-mization. In all situations lg = 0.128mm and h = 30mm.

γ κ Csalt Z M∗/M : theory [5] M∗/M : free minimization

7.2 · 10−3 6 · 104 3.27 · 10−4Mol/l 100 0.9915 0.9994± 0.000133.2 0.33 5 · 10−6Mol/l 100 0.0204 0.986± 0.01533.2 6.5 10−4Mol/l 100 0.0590 0.999± 0.00133.2 325 5 · 10−3Mol/l 100 0.3103 1.00001± 0.00001133 6.5 10−4Mol/l 200 0.0296 0.9998± 0.0003

4 Comparison with previous approaches

As mentioned in the introduction the model presentedabove has motivated several studies both numerically [2]and analytically [4,5]. The purpose of this section is tocompare our numerical results with the most achieved ver-sion of the theory presented in reference [5]. This theoret-ical approach is based on a constrained minimization ofthe free energy functional (2.2) assuming an exponentialansatz for the density profiles:

ρcol(z) =Ncol a

lgexp(−az/lg), (4.1a)

ρ+(z) = Csalt, (4.1b)

ρ−(z) = Csalt +ZNcol b

lgexp(−bz/lg). (4.1c)

With this parametrization a = M∗/M is the ratio of thereduced mass M∗ by the buoyant mass M of a colloidalparticle, Csalt denotes the fixed salt concentration, andNcol is the fixed overall colloidal density per unit area,i.e.

∫ +∞0

dzρcol(z) ≡ Ncol. The system considered in [5]is semi-infinite, z = 0 corresponds to the bottom of thesample and h = +∞. a and b are the two variational di-mensionless parameters of the theory, and the equilibriumdensity profiles ρcol(z) and ρ−(z) correspond to the valuesof a and b that minimize the free energy functional (2.2).Following reference [5], the minimization conditions are

b(a) = a

(2√

1 + (1− a)/γ − 1), (4.2a)

Zb(a)− κI(Zb(a)κ

)− γ + 4γb2(a)

(a+ b(a))2= 0, (4.2b)

where γ = πZ2lNcollg is the coupling parameter (l isthe Bjerrum length introduced previously), and κ =Csaltlg/Ncol is the relative amount of added salt. Func-tion I is defined by I(x) =

∫ x

0dy(ln(1 + y))/y.

Although equations (4.2) require a numerical treat-ment, it is possible to extract asymptotic expressions whenthe coupling parameter γ is vanishingly small (stronggravitational coupling regime) or large compared to unity(strong Coulomb coupling regime). Such an analysis ispresented in reference [5], and we only reproduce herethe main features. When gravitational coupling is strong,

γ � 1, the reduced mass is given by a � 1 − 3γ (for allvalues of the salinity κ) and therefore no mass reductionis observed in this regime (in agreement with the numeri-cal results presented in reference [2]). On the contrary, instrong Coulomb coupling regimes, γ � 1, quite a largemass reduction is predicted, even in low salinity regimesκ� 1 (in such a situation the mass reduction is given bya � [1+ κ

2 ln2(κ(1+ 1

Z ))]/(Z+1)+O(1/γ)). Our numericalresults based on a free minimization of the functional (2.2)show that it is indeed not the case, even though we observenice exponential asymptotic behaviours at high altitudes.To emphasize this point we present in Table 1 data ob-tained mostly in the strong Coulomb coupling regime. Ex-cepted for the first value γ = 7.2 ·10−3, which correspondsto the opposite strong gravitational coupling regime, theagreement between the theory presented in reference [5]and the present free minimization is very poor. The ef-fective mass predicted by the free minimization procedurecorresponds to the actual mass within less than 0.1% inmost situations, whereas in certain cases the theory pre-dicts effective masses that are only 2% of the actual mass!

To our opinion, the failure of the parametrization pre-sented in reference [5] is not due to the exponential ansatzitself, but to the constraint of global charge neutrality ap-plied to the asymptotic regime. The theory presented inreference [5] assumes that the profiles are exponential fromthe bottom of the sample to the top, as a result the freeenergy functional (2.2) has to be minimized with the con-straint of global charge neutrality. However, the actualsituation is quite different. If we refer to the experimentalwork done by Piazza et al. [1], the exponential regime isreached only above a macroscopic layer of strongly inter-acting colloidal particles. Data presented in the previoussection (see, e.g., Fig. 1) resulting from a free minimiza-tion of the functional also exhibit a dense macroscopiclayer of colloidal particles in the bottom of the cell, andthese profiles cannot be simply represented by a single ex-ponential. This feature can be incorporated to the modelsuggested by Lowen, by splitting the cell into two parts.The upper part of the cell (above a given altitude “z0”)corresponds to the asymptotic region where the profilescan be accurately represented by an exponential, whereasbelow z0 the profiles are more complicated. As we cansee, z0 is defined by the condition that the profiles areexponential above it. There is then no upper bound onthe value of z0 and the asymptotic profiles should not


(a− 1){1 + z∗0 + z

∗02}+ a

{µcolkBT

− ln(λ3col

Ncola

lg

)}z∗0

+ γ

[e−z∗

0(1 + 2z∗0

)+ 4a2e−z∗

0 b/a

(a2

(a+ b)2− 12− z∗0

(a2 + b2

)b(a+ b)

)]= 0,

(4.5)

depend on its precise value. As a result z0 can be chosenarbitrarily large. As a consequence, the part of the fluidlocated below z0 can be considered as a reservoir fixing thechemical potential of the ionic species µcol and µ− (µ+ isirrelevant since the local density ρ+(z) is held fixed, and isthus not a variational parameter). Although the full sys-tem must be charge neutral, the asymptotic part above z0has no reason to be neutral. We are then lead to minimizethe free energy of the upper part of the cell in the grand-canonical ensemble. Assuming that parametrization (4.1)is valid above z0 the minimization equation associated tothe colloidal particles reads:

∂F [ρcol, ρ+, ρ−]∂a

= µcol∂Ncol

∂a, (4.3)

where F [ρcol, ρ+, ρ−] is now the free energy functionalabove z0:

F [ρcol, ρ+, ρ−] =∑i∈{col,+,−}

∫ +∞

z0

kBTρi(z)[ln

(λ3i ρi(z)

) − 1]dz+

∫ +∞

z0

Mgzρcol(z) dz +12

∫ +∞

z0

eρ(z)Φ(z) (4.4)

and Ncol =∫ +∞

z0ρcol(z)dz is the number of colloidal parti-

cles above z0 per unit area. After some algebra, this min-imization equation can be written in the form

see equation (4.5) above

where z∗0 ≡ az0/lg. We can easily check that when z∗0 = 0we recover the first equation of condition (4.2). As z∗0 canbe chosen arbitrarily large in our model, we easily see thatthis equation implies a = 1 (no mass reduction) and

µcolkBT

= ln(λcol

Ncol

lg

). (4.6)

This new version of the theory is consistent with our nu-merical results predicting no mass reduction.

5 Conclusion

A free minimization of the Poisson-Boltzmann theory usedin references [2,5] has been performed in this article whichlead us to conclude that this simple mean-field theory doesnot predict any mass reduction contrarily to previous ap-proximate minimization of the same functional. These newresults are fully consistent with the analytical results ob-tained in a two-dimensional case by Tellez [3]. In par-ticular, we observe the same condenser effect between thebottom of the container and the top of the dense region, re-sulting from a competition between electroneutrality andentropy of the microions. Data plotted in Figure 2 ex-hibit a behavior similar to experimental results obtainedby Piazza et al. [1]. Although in the asymptotic regime weobserve no mass reduction, this regime is attained for verylow values of the colloidal packing fractions, below the ex-perimental resolution (10−5). Above volume fractions of10−5 our computations show an apparent mass reductiondue to a residual electric field. However, the mechanismresponsible for this effect in our computations should nottransfer to physical situations since the results presentedin Figure 2 correspond to a value of the Coulomb couplingmuch smaller than the actual one.

References

1. R. Piazza, T. Bellini, V. Degiorgio, Phys. Rev. Lett. 71,4267 (1993).

2. T. Biben, J.P. Hansen, J. Phys. C: Condens. Matter 6, A345(1994).

3. G. Tellez, J. Chem. Phys. 106, 8572 (1997); G. Tellez, J.Phys. A: Math. Gen. 31, 5277 (1998).

4. J.P. Simonin, J. Phys. Chem. 99, 1577 (1995).5. H. Lowen, J. Phys.: Condens. Matter 10, L479 (1998).6. N.F. Carnahan, K.E. Starling, J. Chem. Phys. 51, 635(1969).

7. J.P. Badiali, M.L. Rosinberg, D. Levesque, J.J. Weis, J.Phys. C: Solid State Phys. 16, 2183 (1983).

equilibrium sedimentation profiles of charged colloidal suspensions

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