Collective Particle Interactions in the Sedimentation of Charged ColloidalSuspensions†

Jan Vesaratchanon, Alex Nikolov, and Darsh T. Wasan*

Department of Chemical and Biological Engineering, Illinois Institute of Technology, Chicago, Illinois 60616

The sedimentation of a similarly charged monodisperse colloidal suspension was studied to investigate therole of collective particle interactions affecting the rate of sedimentation and thereby the stability. The particleconcentration profiles, microstructure formation, and the sedimentation rates were determined using anondestructive back-light scattering technique. Stochastic particle dynamics simulations using the Beresford-Smith and Derjaguin-Landau-Verwey-Overbeek repulsive pair potentials were carried out to predict thesedimentation rates. In addition, Monte Carlo simulations were performed to reveal the particle microstructureformation. The collective particle interactions result in an oscillatory effective potential of interaction betweenthe identically charged particles, even though the pair potential is repulsive, and lead to a more orderedmicrostructure formation and reduction in the sedimentation rate. The theoretical predictions, which were ingood agreement with the experimental measurements, clearly showed the importance of the collective particleinteraction effect in stabilizing the monodisperse suspension.

Introduction

Charged colloidal particle suspensions find widespread usein manufacturing of various industrial products such as latexpaints, inks, cosmetics, pharmaceuticals, and food stuffs. Thestability of colloidal dispersions is the key technologicalparameter that determines the shelf life and quality of theseproducts. The classical Derjaguin-Landau-Verwey-Overbeek(DLVO) theory1,2 has been widely used in colloid chemistry toexplain the stability of a colloidal dispersion. From this theory,the pair interaction between colloidal particles is obtained onthe basis of the van der Waals interaction combined with theelectrostatic double-layer interaction. However, the DLVOtheory is limited to dilute systems and systems consisting ofions and solvent molecules without size in the supporting fluid.3

The limitations of the DLVO theory were directly explainedby various experiments conducted by many research groups.Luck et al.4 and Kose et al.5 observed an ordered microstructure(hexagonal packing) of the concentrated charged latex particlesystem. Similar experiments were carried out by Ise andco-workers6-8 to measure the separation distance betweenparticles using microscopy and the X-ray scattering technique.The coexistence of an ordered and disordered phase wasobserved for highly charged latex particles. Ise and co-workersalso suggested that an ordered structure formation was a resultof a long-range attractive interaction between similarly chargedcolloidal particles. A combination of the condensed (ordered)and the expanded (void) phases was confirmed experimentallyby Ito et al.,9 who investigated the interactions in a highlycharged latex dispersion.

Many efforts have been made to explain the coexistence ofordered and disordered phases. Chu and Wasan10 examined theinteractions of charged particles in a system consisting of amixture of large negatively charged colloidal particles and small

ions, including counterions and some ions from the addition ofan electrolyte. The effective potential of interaction was obtainedby solving the Ornstein-Zernike integral equation of statisticalmechanics with the mean-sphere approximation. At very lowparticle concentrations and low particle charges, the systembehavior followed the repulsive Debye-Huckel theory; uponincreasing the particle concentration and the charge number ofthe particles, the collective particle interactions competed withthe repulsive interactions. A net attractive interaction was foundand the potential of the mean force as a function of interparticleseparation distance becomes oscillatory as the particle concen-tration and charge increase. This oscillation arises as a resultof collective effect because the particles are surrounded by otherneighboring particles.10

Feng and Ruckenstein11 also studied the effective interactionin the dispersion of identically charged colloidal particles usingMonte Carlo simulations and the Debye-Huckel pair potential.When the charge number and the particle volume fraction weresufficiently high (e.g., >15 vol %), the effective potential ofthe interaction becomes oscillatory decay because of thecollective interactions and generates an effective attractiveinteraction between identically charged particles. The transitionof disordered liquid-like to ordered 3D crystal-like structuresoccurred when the particle volume fraction and the charge ofthe particles increased.

In this paper, we investigate the role of collective particleinteractions between similarly charged monodisperse latexparticle dispersions in a sedimentation experiment. The particleconcentration profiles and structure were probed by Kosseldiffraction technique. The stochastic particle dynamics simula-tions using the Beresford-Smith and the DLVO potential wereconducted. The results for the sedimentation of charged particlesare compared with the previously obtained results for thesedimentation of hard-sphere latex particle dispersions.

Experimental Section

Materials. The 8 vol % sulfate latex dispersions producedby Molecular Probes, Inc. were used. The geometrical particlesize was 500 nm in diameter, with a standard deviation of <2%.The particle density was 1.05 g/cm3. The particles had surface

† This paper is dedicated to Professor L. S. Fan on the occasion ofhis 60th birthday. L.S.F. is a generous friend who has made significantresearch contributions to the subject area of particle technology andmultiphase flow. He has been recognized with many honors and awards.The authors wish him many more productive and healthy years.

* To whom correspondence should be addressed. Phone: 312-567-3001. Fax: 312-567-3003. E-mail: wasan@iit.edu.

Ind. Eng. Chem. Res. 2009, 48, 80–8480

10.1021/ie8004176 CCC: $40.75 2009 American Chemical SocietyPublished on Web 08/29/2008

charges of 4.9 µC/cm2. We monitored the sedimentation ratein a dilute suspension (0.001 vol %) and applied Stokes’ lawto estimate the electrostatic Debye length (κ-1) of 120 nm. TheDebye length was calculated for a given surface charge andionic strength as 116 nm according to eq 6 given in the followingsection.

The original suspension was concentrated by settling toprepare a 15 vol % effective concentration, and electrolytesolution of 10-4 M NaCl (1:1) was added to keep the ionicstrength constant. Purified water (cleaned by the Milliporesystem) was used to prepare the suspensions in all of theexperiments.

Sedimentation Measurement. The sedimentation experi-ments were carried out in a cylindrical flat-bottom glass tube(40 mm long and 12 mm in diameter). Sedimentation measure-ments are sensitive to temperature changes and vibration. Theexperimental setup consisted of sedimentation tubes isolated ina partial vacuum to control the temperature and prevent theconvective transfer of heat. The samples were placed on avibration-free table. The temperature was controlled at a constantvalue of 25 ( 1 °C for all experiments. The movement of theseparation boundary was measured as a function of time usinga collimated laser beam produced by Limate Corporation (ModelLS11V2). The calibration of the equipment was performed bythe reproducibility of the measurement and the measurementaccuracy was within ( 0.07 mm.

Monitoring Particle Concentration Profiles Using theBack-light Scattering Technique (Kossel Diffraction). Thelaser beam (wavelength ) 630 nm) produced by LimateCorporation’s laser (Model LS11V2) was scattered by colloidalparticles and rendered a diffraction pattern image on the surfaceof a glass tube. The image was captured by a digital camera(Sony Cyber shot DSC-P41) and then transferred to a computer.The image was analyzed to obtain the intensity profile withimage analyzer software, Image Pro Plus Version 4.0. Thesettling concentration profiles were obtained from the intensityprofile of the diffraction images.

For spherically symmetrical particle systems, the Rayleigh-Gans-Debye theory12 can be applied to obtain the staticstructure factor, S(Q),

I(Q))ANP(Q)S(Q) (1)

where I(Q) is the measured light intensity profile as a functionof the scattering vector, Q. A is the normalized constant, N isthe particle number density, and P(Q) is the form factor, whichis related to the particle shape and size. The form factor wasobtained from the light scattering experiments using dilutecolloidal dispersions. To obtain the particle concentration, thefollowing equation was used in conjunction with eq 1 to obtainS(Q) and N:13

∫0

∞

Q2[S(Q)- 1] dQ)-2π2N (2)

More information about the Kossel diffraction techniqueapplied in this study can be found elsewhere.14,15

Modeling

Many efforts have been made to study the sedimentation ofconcentrated dispersions. Different models were proposed bothempirically16 and theoretically.17 To investigate the collectivehydrodynamic interactions in semidilute and concentratedsuspensions, experimental data from the literature were sum-marized and correlated18-20 in the form of

UU0

) (1-�)-K (3)

where φ is volume fraction of the particles and K consists of alarge negative contribution from the backflow (-5.5), a smallpositive effect from the particle pressure gradient (+0.5), andthe hindrance due to near-field hydrodynamics (-1.55). As aresult of electrostatic double layer, the Stokes velocity (U0) ineq 3 has been corrected by using effective particle diameter,i.e., geometrical particle diameter plus twice the electrostaticdouble layer.

We used the Haile method21 of discrete particle dynamics tomodel the separation processes in a gravity field. The Langevinequation governs the movement of a particle in the sedimenta-tion process.22 The total force includes terms from the inter-particle interactions, Brownian motion, the gravity effect, andthe friction force.

mi

dUi

dt)-∑

j

∂Wij

∂rij+ fi +

π6

Di,eff3∆Fg+∑

j

�ijUjδij (4)

where mi is the mass of particle i, Uj is the velocity of eachparticle, Wij is the particle-particle interaction energy, and rij

is the distance between particles i and j. fi is the Brownian forcedue to Brownian motion by diffusion, di,eff is the effectivediameter of the particle, �ij is the friction tensor, ∆F is the densitydifference between the dispersed particles and continuousmedium, and g is the gravitational acceleration. The first termon the right side of the force balance equation gives themovement of particle i from the interparticle interactionaccording to the interparticle distance, rij. The second term arisesfrom the stochastic Brownian force contributed from the particlediffusion. The third term represents the downward gravitationalforce due to the density difference between the particle andsolvent. The last term is the hydrodynamic interaction, whichis the combination of the Stokes viscous drag, �ij ) -3πηDi,eff

(where η represents the viscosity of the solvent), and the velocityof the particle, according to the hydrodynamic backflow givenby eq 3; K accounts for the hydrodynamic effect and is -6.55,according to Batchelor.23 Therefore, the collective multiparticlehydrodynamic interactions were accounted for by the term(1 - φ)6.55 × (π/6)∆FgDi

3.To calculate the effective (collective) particle interactions, a

dynamics simulation method was applied.21 A pairwise elec-trostatic potential was used in the computational program; themulti particle-particle interaction among the neighboringparticles was considered by accounting for other particles withinthe interaction distance within three particle diameters (fromcenter-to-center).

The Brownian force of particles can be explained by thediffusion coefficient of the settling particles.24 However, thecalculation based on the diffusion coefficient (D0) shows thatthe Brownian effect is negligible (D0 = 10-8 cm2/s for 500 nmparticles in water estimated by Stokes-Einstein equation), asthe particle size we used is quite large and the gravitationalterm is much stronger.

The total forces acting on the particles are calculated at eachtime step (∆t) and each particle configuration is replaced withthe new position (done in accordance with the Langevinequation). We were able to obtain the results for the sedimenta-tion velocity from eq 4 by considering the total forces actingon the particles.

Pair Potential of Interaction between Charged Par-ticles. The pair potential of interaction (W) between a pair ofcharged colloidal particles was proposed by Beresford-Smith

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et al.25 In the presence of an electrolyte in the form of thescreened Coulombic potential,

W(r)) { ∞ (for r < 2R)(Ze)2

εrexp[-κ(r- 2R)]

(1+ κR)2(1+�)2 (for rg 2R)

(5)

where r is the center-to-center distance between colloidalparticles, R is the geometrical radius of the particles, Z is thenumber of charges per particle, e is an electron’s charge, ε isthe dielectric constant, φ is the volume fraction of particles,and κ is a reciprocal Debye length as calculated by theexpression,

κ2 ) 4πe2

εkT (NZzc +∑i

nizi2) (6)

where N indicates the particle concentration by the numberdensity, ni is the concentration of ions of species i and valencez, and zc is the valence of the counterions of the particles. Inthis case, zi and zc are considered to be unity.

Other than the potential of interaction proposed by Beresford-Smith, the DLVO theory has been used in colloidal systems.The electrostatic part of the DLVO potential between a pair ofcharged particles is described as1,2

W(r)) (Ze)2

4πεrexp[-κ(r- 2R)]

(1+ κR)2(for r > 2R) (7)

where κ is a reciprocal Debye length.Effective Potential of Interaction between Charged

Particles. We calculated the effective interaction energy betweenparticles using Monte Carlo simulations with the pair potentialof interaction given by the Beresford-Smith and the DLVOtheories. The Monte Carlo simulations were performed in a fixedcubic box of 15 × 15 × 15 (scaled by the particle diameter) indimension. The simulation results were independent of the boxsize as the box size was normally scaled by the particle diameterin the simulations. The number of particles in the simulationbox was varied from 50 to 1000 particles, depending upon theparticle concentration. The first million runs were discarded toeliminate the initial particle configuration. The simulations werecarried out until equilibrium was reachedswhen the radialdistribution function did not change further.

The effective potential of interaction between the particles(i.e., the potential of the mean force), Weff in the units of kT,was obtained from the radial distribution function, g(r):

Weff )-kT ln g(r) (8)

Results and Discussion

The concentration profiles in the charged particle settlingexperiments were determined using the Kossel diffractiontechnique. As shown in Figure 1, the vertical movement (h/H)of different concentration planes (φ, φeff) with time wasmonitored to determine the sedimentation rate. Figure 1 showsthe experimentally measured concentration profiles after 40 and70 days as a vertical distance scaled by the total height of thetube versus the particle volume fraction. Initial effective particlevolume fraction in the suspension was 0.15, with an ionicstrength of 10-4 mol/L NaCl. The effective volume fractionbased on the effective diameter with the Debye length (i.e., thereal diameter plus twice the Debye length) is also shown onthe x-axis.

Figure 2 shows the sedimentation rate (normalized by thebackflow) of charged latex particles (500 nm in diameter)

determined from the local concentration profiles. The theoreticalmodel prediction using the discrete (stochastic) particle dynamicmethod and the Beresford-Smith (B-S) and the DLVO potentialsare also shown on this figure. The parameters used in the theoreticalcalculations are given in Table 1. The effective density, whichaccounts for the double layer around the particle, was used in thetheoretical calculations. The model predictions based on the B-Spotential are in better agreement with the experimental data,especially at high particle concentrations, compared to those basedon the DLVO theory; the latter is usually limited to dilute particlesystems.

The equilibrium particles configurations calculated using theMonte Carlo method are also displayed as insets in Figure 2. Thecharged particles were randomly structured, up to 30 vol %(effective concentration). However, at an effective particle con-centration of 45 vol %, the particles started to form 3-D orderedstructure due to the many-body (collective) interactions. Thereduction in the settling rate was observed as a result of the orderedparticle microstructuring.

Figure 1. Particle concentration profile after 40 and 70 days. Latex particlesize (geometrical): 500 nm.

Figure 2. Normalized sedimentation rate as a function of particle volumefraction when the number of charges per particles Z ) 300.

82 Ind. Eng. Chem. Res., Vol. 48, No. 1, 2009

We also performed a settling experiment using low-charge(∼0.005 µm/cm2), hard-sphere hydroxylate latex particles as areference model.26 These data are taken from ref26 and are shownin Figure 2. An acceleration of the settling rate was observed forhard-sphere particles. This is due to the attractive self-depletioncaused by the excluded volume effect. The data for hard spheresis also in good agreement with the model predictions based on eq4 and the hard-sphere potential.26 Also, Okubo’s27 experimentaldata are plotted in Figure 2; he used 937 nm charged polystyrenespheres dispersed in various electrolyte solutions. Once again, ourcalculations based on the discrete particle dynamic method usingthe B-S potential agree with Okubo’s experimental data. In ourmodel calculations, we assumed that the surface charge isindependent of the particle concentration.

The effective interaction energy (i.e., the potential of the meanforce) between particles was obtained using the Boltzmannexpression and the radial distribution function calculated by theMonte Carlo method (Figure 3). The effective interaction energyusing the B-S potential is an oscillatory curve with relatively small

amplitude at a low particle concentration. These results werecompared with the simulation results obtained by Feng andRuckenstein, who used the Debye-Huckel screened potential.

An inset in Figure 3 shows the equilibrium particle configurationfor the low value of the particle concentration. The particles movedrandomly at low particle concentrations. Upon increasing theparticle concentration to a higher volume fraction of φ ) 0.06 andthe number of charges per particle of Z ) 1000, we found that theeffective potential of interaction is strongly oscillatory with aminimum energy of about 1 kT, as depicted by the center of theparticle (Figure 4). The nature of this attractive minimum is dueto the collective interactions between identically charged particles.The particle configuration (inset) shows a more ordered structureas the particle concentration increased from φ ) 0.015 to φ )0.06. Once again, our simulation results, which are based on theBeresford-Smith potential, are in fair agreement with the simulationdata of Feng and Ruckenstein,11 who based it on the Debye-Huckelscreened potential.

In contrast to the effective potential of interaction (which isoscillatory, including both attractive and repulsive contributions),the pair potential of interaction calculated between similarly chargedparticles from eq 5 is purely repulsive at all separation distances(Figure 5). As the charge on the particle increases, the pair potentialexhibits a stronger repulsive interaction between the particles.

Figure 6 shows the effective particle interaction energy calculatedbased on Monte Carlo simulations at three different particleconcentrations (3, 12, and 20 vol %) and at a fixed particle chargeof Z ) 300. This charge value corresponds to the particle chargepresent in our sedimentation experiments. Again, it is observedthat upon increasing the particle concentration from 3 to 20 vol%, the potential of the mean force becomes oscillatory and a moreordered structure results. Our experimental results show a reductionin the settling rate at higher particle concentrations. This is attributed

Table 1. Parameters Used in the Model (eq 4)

symbol parameter

D 500 nmDeff 740 nmκ-1 120 nmη at 25 °C 0.95 cPF 1.05 g/cm3

Feff 1.02 g/cm3

W(r) eqs 5, 7Z 300T at 25 °C 298 Ke 1.60 × 10-19 Cε ) εrε0

εr for water at 25 °C 78.54ε0 8.85 × 10-12 C2 J-1 m-1

Figure 3. Effective interaction energy (potential of mean force) as a functionof center-to-center distance. Particle size is 500 nm in diameter. Particlevolume fraction, φ ) 0.015; charge per particles, Z ) 300; electrolyteconcentration, 10-4 M.

Figure 4. Effective interaction energy (potential of mean force) as a functionof center-to-center distance. Particle size is 500 nm in diameter. Particlevolume fraction, φ ) 0.06; charge per particles, Z ) 1200; electrolyteconcentration, 10-4 M.

Figure 5. Pair potential of interaction as a function of center-to-centerdistance using Beresdord-Smith model (eq 5) for various number of chargesper particles. Particle size is 500 nm in diameter. Electrolyte concentrationis 10-4 M. (1) Z ) 50; (2) Z ) 100; (3) Z ) 200.

Figure 6. Effective interaction energy (potential of mean force) as a functionof center-to-center distance for different effective particle volume fraction.Particle size, 500 nm in diameter; number of charges per particles, Z )300. (1) φ ) 0.03; (2) φ ) 0.12; (3) φ ) 0.20. Electrolyte concentration of10-4 M.

Ind. Eng. Chem. Res., Vol. 48, No. 1, 2009 83

to the particle structuring, i.e., from disordered (random) liquid-like to ordered crystal-like structures.

Figure 7 shows the effective interaction energy between particlesat two different electrolyte strengths: 10-4 M and 10-2 M for aparticle size of 500 nm and at a fixed charge of Z ) 300. Theinteraction between the particles changed significantly when weincreased the electrolyte concentration from 10-4 M to 10-2 M.The minimum in energy is much smaller at 10-2 M electrolyteconcentrations. At high electrolyte concentrations, the chargedparticle system starts to behave more like a hard-sphere system,as seen from the particle configuration in the inset. Feng andRuckenstein’s results show the same trend of a less orderedstructure formation when the electrolyte concentration increasesfrom 10-5 M to 10-2 M.

Conclusions

The effect of collective interactions between similarly chargedmonodispersed particles on the sedimentation rate was investigatedexperimentally by using the nondestructive Kossel diffractiontechnique and theoretically by the stochastic particle dynamicssimulations using the Beresford-Smith and DLVO repulsive pairpotentials. Monte Carlo simulations were also carried out to revealthe particle microstructure formation. A comparison with a hard-sphere system was made.

The experimental results and theoretical calculations revealedthat a more ordered structure exists, especially at high concentra-tions (e.g., >15 vol %) and charges (e.g., Z > 300) of the parti-cles. This is because the effective interaction potential in a like-charged multiparticle system is oscillatory, including both theattractive and repulsive contributions, even though the pair potentialis repulsive. Consequently, this many-body effect results instabilization of the colloidal dispersion. The experimental resultsat high particle concentrations were in better agreement with themodel predictions based on the Beresford-Smith potential than withthe DLVO theory, which is limited to very dilute systems.

The effect of the increased electrolyte concentration was thatthe charged particle suspension behaved more like a hard-spheresystem. The acceleration in the sedimentation rate was observedexperimentally. This was attributed to an attractive self-depletioneffect. The model predictions based on the hard-sphere potentialagreed well with our experimental data.

Acknowledgment

This work was supported with funds from the National ScienceFoundation Grant CTS-0553738.

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ReceiVed for reView March 13, 2008ReVised manuscript receiVed July 9, 2008

Accepted July 10, 2008

IE8004176

Figure 7. Effective interaction energy (potential of mean force) as a functionof center-to-center distance. Particle size is 500 nm in diameter. Particlevolume fraction, φ ) 0.20; Z ) 300. With electrolyte concentration of 10-4

M and 10-2 M (1:1).

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