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Colloids and Surfaces A: Physicochem. Eng. Aspects 436 (2013) 325– 332

Contents lists available at ScienceDirect

Colloids and Surfaces A: Physicochemical andEngineering Aspects

jo ur nal ho me page: www.elsev ier .com/ locate /co lsur fa

population balance equation model to predict regimes of controlledanoparticle aggregation

nand K. Atmuria, Michael A. Hensona, Surita R. Bhatiaa,b,c,∗

Department of Chemical Engineering, University of Massachusetts, Amherst, MA 01003, United StatesDepartment of Chemistry, Stony Brook University, Stony Brook, NY 11794, United StatesCenter for Functional Nanomaterials, Brookhaven National Laboratory, Upton, NY 11793, United States

i g h l i g h t s

Formation of stable finite-sized clus-ters observed in charged colloidaldispersions.A population balance equation (PBE)model was developed to capture thisphenomenon.Model successfully predicts aggrega-tion regimes and final aggregate size.

g r a p h i c a l a b s t r a c t

r t i c l e i n f o

rticle history:eceived 15 February 2013eceived in revised form 27 June 2013ccepted 3 July 2013vailable online 12 July 2013

eywords:lusterggregationggregaterocessingheoryodeling

a b s t r a c t

Forming stable clusters or aggregates of nanoparticles is of interest in a number of emerging applications.While formation of unstable fractal aggregates and flocs has been well-studied with both experimentsand theory, the conditions that lead to stable, finite-sized clusters is not as well understood. Here, wepresent an integrated experimental and modeling study to explore aggregation in concentrated attrac-tive colloidal suspensions. A population balance equation (PBE) model is used to predict the aggregationdynamics of quiescent colloidal suspensions. A DLVO (Derjaguin–Landau–Verwey–Overbeek) type poten-tial is used to describe the interparticle potential, with attractive interactions arising from van der Waalsforces and long-range repulsive interactions caused by electrostatics. The PBE model includes a full cal-culation of stability ratio variations as a function of aggregate size, such that the energy barrier increaseswith increasing size. As the ionic strength is decreased, the model predicts three regimes of behavior:uncontrolled aggregation into large flocs, controlled aggregation into stable clusters, and no aggrega-tion. The model is tested experimentally using latex particles at different salt concentrations and particleconcentrations. When the Hamaker constant and surface potential are fit to aggregate size measure-ments collected at one salt concentration, the model accurately predicts the final mean aggregate sizeand regimes of aggregation at other salt concentrations and the same particle concentration. This resultsuggests that van der Waals and electrostatic forces are the dominant particle interactions in determiningthe final aggregate state. The mean aggregate size and aggregation regimes at different particle concen-trations could be accurately predicted by adjusting the surface potential. This parameter adjustment

is consistent with the expectation that increasing colloid weight fractions cause aggregates to have a more fractal nature and hence

aggregation rates than what areffects or another slow dynamstudy presents the first PBE mregimes of charged colloidal p

∗ Corresponding author at: Department of Chemistry, Stony Brook University, Stony BrE-mail addresses: surita.bhatia@stonybrook.edu, sbhatia@ecs.umass.edu (S.R. Bhatia).

927-7757/$ – see front matter © 2013 Elsevier B.V. All rights reserved.ttp://dx.doi.org/10.1016/j.colsurfa.2013.07.002

have a lower effective repulsion. However, the model predicts much fastere observed experimentally. This discrepancy may be due to hydrodynamicical process which is not accounted for in the model. Nevertheless, thisodel that can successfully predict stable aggregate size and aggregation

articles over a range of salt concentrations and particle concentrations.© 2013 Elsevier B.V. All rights reserved.

ook, NY 111794, United States. Tel.: +1 631 632 7788.

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26 A.K. Atmuri et al. / Colloids and Surfaces A:

. Introduction

Aggregation of colloids has long been studied for processes suchs formation of large flocs for water treatment [1] and formation ofractal colloidal gels. A more recently studied phenomenon is theormation of dense, finite-sized clusters of colloidal particles forpplications such as toner for digital printing [2] microparticles forrug delivery [3] and probes for cellular imaging [4]. We refer torocesses leading to these dense clusters as “controlled aggrega-ion,” as conditions must be such that aggregates grow to a certainesired size and then remain stable in size.

The interaction potential between particles plays a major rolen determining the nature and rate of aggregation in colloidalystems. Clustering and aggregation arising from a number of dif-erent mechanisms have been explored in the literature, includingddition of salt to screen electrostatics [5,6], additional of a non-dsorbing polymer to induce depletion attraction [7,8] polymerridging [9] and “patchy” electrostatic interactions [10]. In colloidalystems with a short-range or moderate-range attraction and long-ange repulsion, there is competition between aggregation from thettractive part of the potential and stabilization from the repulsiveart. This competition can lead to the formation of stable clusterst low volume fractions [11,12], but at higher volume fractions,uspensions undergo structural arrest, either via percolation [13]r a glass transition [14] depending on the range of the repulsion.nterestingly, stable clusters of particles have also been observedn systems with short-range attractions and a screened long-rangeepulsion [7,10] and in systems with steric repulsive interactions15], and some theoretical studies predict stable clusters even inurely repulsive systems that have a soft shoulder in the interpar-icle potential [16].

In this study we focus on a process-level description of aggre-ation of charged particles that are destabilized with addition of aalt. We use a DLVO-type potential to describe interactions betweenarticles, which includes van der Waals attraction between parti-les and a repulsive component arising from overlapping electricalouble layers [17,18]. Addition of a salt screens the repulsive com-onent of the potential, leading to aggregation. There are severalodel experimental systems that show stability and aggregation

ehavior that are well-described by the classical DLVO theory [18].nder certain conditions, additional interactions that cannot bexplained by DLVO theory may be present, sometimes referreds non-DLVO forces. These include short-range hydration forces,apillary condensation, and specific ion adsorption [17]. However,ue to the difficulty in quantifying these forces and experimen-ally determining the additional parameters that would be neededor their quantification, their utilization in process models is lesseasible [19].

Work by Smoluchowski [20] laid the foundation for the usef particle population balance equation (PBE) models to describeggregation in colloidal suspensions. The two important functionsn the PBE are the collision frequency factor, or collision kernel,nd the collision efficiency factor, or stability ratio. These kernelsake into account aggregation due to Brownian motion and shear9,21–23] with modified kernels available to account for differ-ntial settling [9]. In our present study, we focus on aggregationhat occurs under quiescent conditions [24]. Throughout the paper,e use the term “particles” to include both primary particles and

ggregates formed of primary particles.In the case of particles with a DLVO-type potential, under certain

onditions particles need to cross an energy barrier to aggregateith other particles. So, every collision experienced by a particle

eed not make it to stick to its collision partner, and the collisionutcome is governed in part by the height of the barrier. To take thisarrier into account, we use the stability ratio, which is defined ashe rate of aggregation when the interaction is diffusion-limited

ochem. Eng. Aspects 436 (2013) 325– 332

(i.e., no energy barrier) to the rate when particle interactions arepresent. Thus, a higher value of the stability ratio represents aless efficient collision. We can also consider collision efficiencyin the context of two commonly considered limits of aggrega-tion phenomena, diffusion-limited cluster aggregation (DLCA) andreaction-limited cluster aggregation (RLCA). For DLCA, the collisionefficiency is very high and the stability ratio is the order of unity[25,26]. By contrast, for RLCA the collision efficiency is much lowerand the stability ratio is greater than unity [25,26].

Morbidelli and co-workers have performed extensive exper-iments [5,27], modeling [24,28,29], and simulations [30] tounderstand the kinetics of aggregation and structure of the result-ing aggregates. However, these and other studies [9,26,31,32] havegenerally focused on aggregation leading to gel phases, and thetime scale reported is typically on the order of a few minutes orfor some studies up to a few hours. Thus, formation of finite-sizedclusters is not discussed, and the time scales are such that it isdifficult to determine whether the aggregate size would remainstable over longer times. Additionally, these models generally haveseveral parameters that need to be fit to experimental data. Forexample, in some studies the stability ratio must be fit at differentconditions. Such fitting makes it difficult to extend the model toa larger parameter space and use the model in a predictive man-ner. Additionally, some studies [9,19,26] report PBE-type models ofcolloidal aggregation using the stability ratio only for the primaryparticles. These types of models do not allow the stability ratio tochange as the aggregate size increases, and do not predict a regimeof controlled aggregation.

More detailed molecular dynamics simulations of clusteringphenomena [11,14,33–35] yield some insight into the type ofphysics that must be incorporated into a PBE model to predict theformation of stable clusters. Both Sciortino and co-workers [11]and Groenwold and Kegel [33,34] describe stable cluster formationarising from a balance of short-range attractions and long-rangeelectrostatic repulsion. Short-range attractions initially drive pri-mary particles to aggregate. Further addition of particles to thecluster increases the charge on the cluster. After clusters havegrown to a certain size, they have accumulated so much charge thataddition of additional primary particles is unfavorable [11,33,34].In other words, the clusters become so strongly charged thatthey repel primary particles, and this strong long-range repul-sion dominates overcomes any particle-cluster attraction and setsthe equilibrium cluster size [11]. In the model of Sciortino andcoworkers, the radius of the cluster is explicitly renormalized asthe aggregation process proceeds [11,14] to account for changesin interactions as the aggregation process proceeds. Experimentalstudies have indirectly confirmed these simulations by demon-strating the presence of stable clusters [11,12], although to ourknowledge no direct measurements of the interparticle potentialbetween clusters of varying sizes and primary particles have beenperformed. Nevertheless, we believe molecular simulations of thisphenomenon [11,14,33–35] suggest that any process-level modelmust allow the interaction potential between aggregates and par-ticles, and hence the stability ratio, to change as aggregate sizeincreases.

Here we describe our efforts to develop a PBE model with a min-imum number of fitted parameters capable of mapping out threeregimes: uncontrolled aggregation, controlled aggregation, and noaggregation (a stable suspension of primary particles). By uncon-trolled aggregation, we refer to systems where aggregates continueto grow until either gelation occurs, meaning that the aggregatesare large enough to span the system, or precipitation occurs, mean-

ing that very large aggregates form that are too large to remainsuspended in solution. By controlled aggregation, we refer to sys-tems where aggregates grow to a specified size and remain stableat that size while being suspended in solution. To parameterize and

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A.K. Atmuri et al. / Colloids and Surfaces A:

alidate our model, we have performed experiments on colloidaluspensions with moderate particle concentrations of 5–15 wt%,elevant to some industrial processes that rely on controlled aggre-ation. Recently, we reported experimental evidence of controlledggregation in a shear environment [36], but here we focus onggregation under quiescent conditions where there is no breakageerm in the PBE model. Thus, the particles aggregate irreversibly toorm clusters and the final size of clusters is solely determined byhe interaction potential. To our knowledge, this study representshe first attempt to use PBE models to predict stable aggregate sizend aggregation regimes of charged colloidal particles over a rangef salt and particle concentrations.

. Materials and methods

Aqueous suspensions of electrostatically stabilized polystyreneatex particles with an approximate density of 1.05 g/cm3 and a

ean diameter of 200 nm and a colloid weight fraction of 45 wt%,upplied by Xerox Corporation, were used for the aggregationxperiments. Potassium chloride was used to destabilize the latexuspension. Colloid weight fractions of 5, 10 and 15 wt% at differentonic strengths were prepared by diluting the stock suspension withhe required amount of water. The pH of the suspensions was notltered. After addition of salt, samples were taken out periodicallynd diluted gently to measure the particle mean size and size distri-ution using dynamic light scattering (DLS) with a 200-mW Innovar-ion laser of wavelength 488 nm with a Brookhaven InstrumentsI-9000AT correlator. We report the volume mean diameter as a

unction of time, as opposed to number mean diameter, such thathe formation of large aggregates in emphasized.

. Theory

The population balance equation for pure aggregation of parti-les (e.g., no breakage of aggregates) under quiescent conditions isiven as [20]:

∂n(v)∂t

= 12

∫ v

0

ˇ(v

− v′, v′)n(v − v′, t)n(v′, t)dv′ −∫ ∞

0

ˇ(v, v′)n(v, t)n(v′t)dv′

(1)

here v is the particle volume and n(v, t) is the density of particlesith volume between v and v + dv at time t. In Eq. (1), the term on

he left represents the rate of change of the number density (n),he first term on the right represents the birth rate when parti-les aggregate, the second term on the right represents the deathate when particles undergo aggregation with other particles, and

is the aggregation frequency. For numerical solution, we use aiscretized version of the population balance equation [34]:

dNi(t)dt

=

⎧⎨⎩

j≥k∑xi−1≤(xj=xk)≤xi+1

(1 − 1

2ıj,k

)�ˇj,kNj(t)Nk(t)

⎫⎬⎭

− Ni(t)M∑k=1

ˇi,kNk(t) (2)

here ı is the delta function which avoids double counting of par-icles, � is used to partition particles into their respective size bins,nd ̌ for quiescent aggregation is given by a Brownian-type kernel:

j,k = 2kBT3�Wj,k

(rj + rk)

(1rj

+ 1rk

)(3)

ochem. Eng. Aspects 436 (2013) 325– 332 327

where kB is Boltzmann’s constant, T is absolute temperature and� is the suspending fluid viscosity. The stability ratio Wj,k definesthe efficiency of aggregation when particles of radii rj and rk col-lide, which is governed by the interaction potential between theparticles given by [38]

Wj,k = (rj + rk)

∫ ∞

rj+rk

exp(VT/(kBT))R2

dR (4)

VT represents the total interaction potential, and in the present caseusing DLVO theory is given as the sum of van der Waals attrac-tion (VA) and electrostatic repulsion (VE). The attractive part of thepotential is given by:

VA = −A6

(2r1r2

R2 − (r1 + r2)2+ 2r1r2R2(r1 − r2)2

+ R2 − (r1 + r2)2

R2 − (r1 − r2)2

)(5)

where A represents the Hamaker constant and R is the center-to-center distance between the particles. The electrostatic repulsionis given by

VE = 64�εrε0

(kBT

zce

)2

tanh

(zce 01

4kBT

)tanh

(zce 02

4kBT

)(r1r2r1 + r2

)

exp(−�(R − r1 − r2)) (6)

where e is the elementary charge, zc is valence of the counterion,and ε0 and εr are the dielectric constants of vacuum and the solvent,respectively. The Debye–Huckel parameter � is a function of elec-trolyte concentration, valence of electrolyte ions, and temperature,while the surface potential 0 depends on pH and temperature. Theunknown parameters in Eqs. (5) and (6) are the Hamaker constantand surface potential, which we have used as fitting parameters.

In our model, we consider aggregates as “particles” with a largerradius, and thus use Eqs. (5) and (6) to compute the interactionpotential not only between primary particles, but also betweenaggregates and single particles, between aggregates of differentsizes, and so on. As discussed further below, this simplificationyields an interparticle potential, and hence a stability ratio, thatvaries as aggregation proceeds.

The discretized PBE model (Eq. (2)) was solved numericallyusing the fixed pivot technique [39] with 64 node points for dis-cretizing particle size. This method was chosen due to its relativelylow computational cost and ability to calculate the particle sizedistribution with great precision [37]. The PBE model has beendiscretized at every node point, yielding 64 nonlinear ordinary dif-ferential equations (ODEs) in time which were integrated with theMATLAB code ode15s to calculate the number distribution at eachnode point. The number distribution was converted to volume dis-tribution [40] to calculate the volume mean diameter.

4. Results and discussion

4.1. Interaction potential between particles and aggregates

Fig. 1a shows a representative interaction potential betweenequal-sized particles as a function of their distance of separation.The form of the DLVO potential results in a deep primary mini-mum and a shallower secondary minimum [17]. The secondaryminimum is difficult to discern in Fig. 1a due to the large scaleon the y-axis which was used to show the relative contribution ofelectrostatic and van der Waals forces; it is more readily apparentin Fig. 1b. Understanding of aggregation processes typically focuseson the particles entering the primary minimum, where they are in

near-contact and are bound essentially irreversibly. The secondaryminimum can also play a role in aggregation; however, it is thoughtto be related to formation of looser, reversibly bound aggregates[17]. Under appropriate conditions, there is an energy barrier that

328 A.K. Atmuri et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 436 (2013) 325– 332

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Fig. 3. Experimental measurements of particle aggregation for a 5 wt% suspension atdifferent salt concentrations, with aggregate sizes measured via DLS. Uncontrolledaggregation is observed at 0.3 M (red asterisks), controlled aggregation is observed

Similar results were observed for other particle concentrations.Fig. 4 shows aggregation data for a 10 wt% sample at different salt

ig. 1. Interaction energy as a function of distance of separation with param-ters A = 3.08 × 10−20 J and 0 = 59 mV for a particle size of 200 nm (a) ionictrength = 0.2 M and (b) different ionic strengths.

ust be crossed before particles enter the primary minimum andegin to aggregate. The size of this barrier impacts the aggrega-ion efficiency of colliding particles. Addition of salt decreases the

agnitude of electrostatic repulsion and hence the barrier heightecreases and aggregation is enhanced (Fig. 1b).

Fig. 2 shows the total interaction potential energy between aingle particle, fixed in size at 200 nm, and particles of differentizes as a function of their distance of separation. Many groups9,19,26] have performed PBE-type modeling of colloidal aggrega-ion using the stability ratio only for the primary particles. However,s the interparticle potential is dependent on size, the stabilityatio changes as aggregation proceeds. This physics is important innderstanding the mechanism of controlled aggregation of chargedolloids [33,34]. As the aggregates grow in size, although the sur-

ace potential remains the same, the effective repulsion increasesecause of the increased number of particles in the aggregate.rowth increases the barrier against aggregation, so particles find it

ig. 2. Total interaction energy as a function of distance of separation between aarticle of size 200 nm and particles of different sizes. Parameters used are the sames in Fig. 1 at an ionic strength of 0.2 M.

for 0.24–0.29 M (black triangles, stars, diamonds, and circles), and no aggregation isobserved at 0.2 M (blue stars). (For interpretation of the references to color in thisfigure legend, the reader is referred to the web version of this article.)

increasingly difficult to add to an aggregate as its size increases. Asmentioned above, this mechanism of formation of stable clustershas been detailed by Groenwold and Kegel [33,34]. Their resultshighlight the need to include a size-dependent calculation of thestability ratio in our PBE model.

4.2. Aggregation experiments

Aggregation experiments were carried out at colloid weightfractions of 5, 10 and 15 wt% and at different salt concentrations.Addition of salt decreases electrostatic repulsion and thus trigg-ers aggregation. Fig. 3 shows the evolution of the volume meanaggregate diameter as a function of time for salt concentrations of0.2–0.3 M at a colloid weight fraction of 5 wt%, with aggregate sizesmeasured via DLS. At the highest salt concentration used, 0.3 M, thesuspension undergoes uncontrolled aggregation and forms a gelphase. We stopped measuring particle size when the gel phase wasformed. Suspensions with ionic strengths in the range 0.24–0.29 Mshow controlled aggregation that ceases after the clusters reachsome stable size that depends on the salt concentration. At ionicstrengths of 0.2 M and less (data not shown), we do not observeany aggregation, and the suspensions remain as primary particles.

concentrations. This system also displays uncontrolled aggrega-tion at 0.3 M, controlled aggregation with aggregates of a stable

Fig. 4. Experimental measurements of particle aggregation for a 10 wt% suspensionat different salt concentrations, with aggregate sizes measured via DLS. Uncontrolledaggregation is observed at 0.3 M (red asterisks), controlled aggregation is observedfor 0.22–0.28 M (black stars diamonds, circles, and triangles), and no aggregation isobserved at 0.2 M (blue stars). (For interpretation of the references to color in thisfigure legend, the reader is referred to the web version of this article.)

A.K. Atmuri et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 436 (2013) 325– 332 329

Fig. 5. Experimental measurements of particle aggregation for a 15 wt% suspensionat different salt concentrations, with aggregate sizes measured via DLS. Uncontrolledaggregation is observed at 0.26 M (red asterisks), controlled aggregation is observedfor 0.20–0.24 M (black circles, triangles, and stars), and no aggregation is observedat 0.19 M (blue triangles). (For interpretation of the references to color in this figurelegend, the reader is referred to the web version of this article.)

Table 1Fit model parameters at different colloid weight fractions.

Parameter Value

Hamaker constant, A 3.08 × 10−20 JSurface potential, 0 (5 wt% suspension) −59.5 mVSurface potential, 0 (10 wt% suspension) −59 mV

dpsdTmo

4

acsratufi

Fcp

Fig. 7. Experimental and predicted stable mean aggregate size as a function of

Surface potential, 0 (15 wt% suspension) −57 mV

iameter at 0.22–0.28 M, and no aggregation at 0.2 M. For a higherarticle concentration of 15 wt% (Fig. 5), uncontrolled aggregationtarts at a lower ionic strength (0.26 M), and the window of con-itions leading to controlled aggregation narrows (0.20–0.24 M).his result is to be expected, since particle collisions will occurore frequently in denser suspensions, increasing the likelihood

f aggregation and formation of large aggregates.

.3. PBE modeling

To utilize the PBE model, we fit experimental data of the finalggregate size at a particle concentration of 5 wt% and a saltoncentration of 0.29 M to obtain the value of the Hamaker con-tant and surface potential, which were 3.08 × 10−20 J and 59.5 mVespectively (Table 1). These parameter values are in reasonable

greement within values cited in literature for similar latex par-icles [18]. Using the parameters fit at 0.29 M, the PBE model wassed to predict the mean aggregate size for 5 wt% suspensions atve different ionic strengths (Fig. 6). At a salt concentration of

ig. 6. PBE model predictions for the mean size of aggregates at different salt con-entrations for a 5 wt% suspension, with values for the Hamaker constant and surfaceotential fit at a single ionic strength, 0.29 M.

ionic strength for dispersions at 5 wt%. Predicted values are based on the Hamakerconstant and surface potential fit at a single ionic strength, 0.29 M.

0.3 M, the mean diameter of aggregates increases very rapidly, andthe aggregate size exceeds the maximum grid size used (unlikethe other ionic strengths). This predicted behavior is consistentwith gel formation, as we observed experimentally (Fig. 3). Atlower ionic strengths of 0.24–0.28 M, aggregates that reach a stable,finite size are predicted, in good agreement with our experimentalobservations. An ionic strength of 0.20 M is predicted to produceno aggregation, consistent with our experiments. However, for allcases the predicted rate of aggregation is much faster than observedexperimentally. We discuss possible reasons for the mismatch inthe predicted aggregation kinetics below. Nevertheless, the goal ofthis work is to develop a PBE model that correctly predicts the con-ditions that lead to stable aggregation and the equilibrium size ofstable aggregates, and in this sense the predictions agree very wellwith experimental results.

It should be noted that the Hamaker constant and surface poten-tial were fit at only one ionic strength, 0.29 M, and these fit valueswere used to predict aggregation behavior at six different ionicstrengths. At a colloid concentration of 5 wt%, the PBE model clearlycaptures the phenomenon of controlled aggregation and can pre-dict the different regimes of aggregation observed experimentally.To quantitatively test the PBE model predictions, we compare thepredicted final mean aggregate size in the controlled aggrega-tion regime with our experimental results at 5 wt%. As shown inFig. 7, the agreement between the final experimental and predictedaggregate sizes is excellent.

Fig. 8 shows the measured and simulated aggregate size distri-butions for a sample at 5 wt% and salt concentration of 0.29 M afterthe stable size is reached. While the mean size values are in agree-ment, the experimental distribution is broader than the distribution

Fig. 8. Comparison of measured and predicted size distributions after stabilizationof a 5 wt% suspension at a salt concentration of 0.29 M. The volume mean sizes are870 nm for the measured distribution and 865 nm for the predicted distribution.

330 A.K. Atmuri et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 436 (2013) 325– 332

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ig. 9. Experimental and predicted stable mean aggregate size as a function of ionictrength for dispersions at 10 wt% and 15 wt%.

redicted by the model. This is often observed in PBE modeling ofggregation-type processes [37]. In our case, one possible cause ofhe mismatch between the experimental and predicted size poly-ispersities is that the PBE model assumes the aggregates formedre compact and spherical. Any fractal structure in the aggregatesould likely broaden the experimentally measured distribution.

The Hamaker constant is an inherent property of the particlend should not change as the colloid weight fraction is increased.owever, there is some evidence that as particle concentration is

ncreased, the aggregates formed become more fractal in nature14] and the effective repulsion experienced by a particle appro-ching the aggregate increases. This effect can be captured by alight decrease in the surface potential. So, to apply the PBE modelo experimental data taken at other weight fractions, we used theamaker constant value from above, but re-fit the surface poten-

ial as before. Table 1 shows that the fitted surface potential doesecrease slightly with particle concentration, as expected. For the0 wt% suspension, data at 0.28 M was used for the fit, resulting

n a surface potential of 59 mV. For the 15 wt% suspension, data at.24 M was used for the fit, resulting in a surface potential of 57 mV.

These values of the surface potential, fit at a single ionic strength,ere used to predict the aggregation regime and final stable aggre-

ate sizes for 6–7 other salt concentrations. The predicted stableean aggregate size in the controlled aggregation regime is shown

n Fig. 9, compared with experimental results. Again, the model pre-icts the different regimes of aggregation behavior and the meanize of stable aggregates fairly well. At lower salt concentrationshere the aggregates are relatively small, the measured sizes are

lightly larger than those predicted by the model. This discrepancys likely due to differences between how size is calculated in the

odel versus determined experimentally. DLS gives the effectiveydrodynamic radius, whereas aggregate size in the model is rep-esented by volume mean diameter. If we consider the simple casehere the aggregates are doublets, the model will predict size by

umming of volumes of both particles, whereas the effective hydro-ynamic radius from DLS will be larger. However, at moderate andigh salt concentrations where aggregates are larger, this effectecomes less significant, and there is very good agreement betweenxperiment and model predictions.

The PBE model can be used to generate a “state diagram”howing conditions leading to different aggregation regimes (e.g.,o aggregation, controlled aggregation, and uncontrolled aggre-ation) as a function of colloid concentration and ionic strength.ig. 10 compares the predicted aggregation regimes to experi-ental results. At each colloid concentration, the surface potentialas fit at a single ionic strength, and predictions were performed

t 6–7 additional values of the ionic strength. The agreementetween the predicted state diagram and experimental results

s very good. There is slight disagreement in the predictednd experimental aggregation regimes at 15 wt%. For example,

Fig. 10. Predicted aggregation state diagram as a function of colloid volume fractionand ionic strength, compared to experimental results, showing regions of uncon-trolled aggregation, controlled aggregation, and no aggregation.

experimentally we find that uncontrolled aggregation occurs at0.25 M and above. However, the PBE model predicts the onset ofuncontrolled aggregation at 0.26 M. We suspect this disagreementmay be due to hydrodynamic effects, which are not included in thecurrent PBE model. Stokesian dynamics simulations have shownthat hydrodynamic effects, which are stronger for more con-centrated suspensions, decrease the volume fraction at whichpercolation occurs [41,42]. This effect could potentially widen thewindow of conditions where gelation (e.g., uncontrolled aggrega-tion) experimentally occurs.

As noted above, for all salt and particle concentrations, the pre-dicted rate of aggregation is much faster than what is observedexperimentally. One possible cause is that there is some physicalprocess with slow dynamics not accounted for in our model. Thereare two dynamical processes that could cause the measured clus-ter size to be smaller than the predicted cluster size and slow theaggregation dynamics. One potential process is that particles maybe stuck reversibly, so they may break free from the cluster and dif-fuse away due to thermal motions. This could arise from patchinesson the particle surface, which is most certainly present (e.g., regionson the surface that have more charge than other regions and there-fore experience stronger repulsion). So even if the overall attractionbetween particles is strong enough that we would expect particlesto be irreversibly stuck, there may be local regions that are notexperiencing as strong of an attraction. The second process is thatparticles that have already joined a cluster may re-arrange withinthe cluster. This motion again could be due to patchiness. A particlethat is stuck to a cluster may have some freedom to roll into a posi-tion that makes the cluster denser and could potentially be moreenergetically favorable. If either of these processes occurs on a timescale that is slower than aggregation, the effect would be that thefinal aggregate size could be predicted accurately but the formationrate seen experimentally would be slower than predicted by themodel. We attempted to incorporate the first effect into the model(e.g., reversible aggregation due to particle patchiness), but our pre-liminary work showed that this did not appreciably slow down theaggregation rate and rendered the model unable to predict stableaggregation.

An alternate explanation is that hydrodynamic effects, whichare also not included in the PBE model, may be acting to slowthe aggregation process. Recent Stokesian dynamics simulationsof charge-stabilized particles [42] show that hydrodynamic effectsslow movement of particles into the primary minimum and makeparticle rearrangement difficult at small interparticle separations.These effects would impact the aggregate structure and percolation

volume fraction, but also the overall rate of aggregation. However, itis extremely difficult to incorporate hydrodynamics into a PBE-typemodel. Even state-of-the-art molecular simulations of aggregationthat incorporate hydrodynamics are limited to a relatively small

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A.K. Atmuri et al. / Colloids and Surfaces A:

umber of particles due to the complexity of the interactions, ando accurately model these effects in the concentrated dispersione describe here would be very computationally expensive if not

nfeasible.Although our PBE model does not predict the rate of aggrega-

ion accurately, our goal was to develop a process model capable ofapturing the formation of stable aggregates. The model presenteds able to quantitatively predict the final aggregate mean size at dif-erent ionic strengths and to accurately predict different regimes ofggregation, using only one experimental data set at each particleoncentration to fit the surface potential. Thus, the model has util-ty in designing products based on stable aggregation of chargedanoparticles. Additionally, our results demonstrate the necessaryhysics that must be included in a process model to capture con-rolled aggregation, providing further insight into the mechanismf this phenomenon.

. Conclusion

We present experimental and process modeling results for theggregation of charge-stabilized colloids. Although classical col-oidal aggregation processes are well-studied, here we explore thehenomena of controlled aggregation; e.g., aggregates that growo a fixed, stable size as opposed to uncontrolled aggregation intoractal gels or flocs. Experimentally, we observe three regimes ofggregation behavior as ionic strength is increased: no aggrega-ion, controlled aggregation, and uncontrolled aggregation. Our PBE

odel includes the full calculation of the stability ratio as well asariations in DLVO interactions between particles and aggregatess aggregates grow in size. Incorporation of this physics into theBE model is important for its ability to predict controlled aggrega-ion. As the aggregates reach a certain size, the effective repulsionetween the aggregate and a single particle becomes large enougho prevent any further addition of particles. With minimal param-ter fitting, the model shows good agreement with experiments inredicting the three regimes of aggregation behavior and is alsoble to quantitatively predict the final mean aggregate size. Theodel predicts a much faster aggregation rate than is observed

xperimentally, which we believe could be due to hydrodynamicffects. Nevertheless, to our knowledge this is the first PBE modelble to predict stable aggregate size and aggregation regimes ofharged colloidal particles over a range of salt and particle concen-rations.

cknowledgments

The authors gratefully acknowledge financial support from anSF GOALI award (CBET-0853551), the NSF-funded Center for Hier-rchical Manufacturing (CMMI-1025020) and Xerox Corporation.hese sponsors had no role in study design; in the collection, anal-sis and interpretation of data; in the writing of the report; and inhe decision to submit the article for publication.

eferences

[1] A.D. Karathanasis, D.M.C. Johnson, Stability and transportability of biosolidcolloids through undisturbed soil monoliths, Geoderma 130 (2006) 334–345.

[2] A.J. Turner, S. Nair, Z. Lai, C.-M. Cheng, S.R. Bhatia, Controlled aggregation of col-loidal particles for toner applications, J. Appl. Polym. Sci. 122 (2011) 1358–1363.

[3] Y.H. Kim, S.H. Gihm, C.R. Park, K.Y. Lee, T.W. Kim, I.C. Kwon, H. Chung, S.Y.Jeong, Structural characteristics of size-controlled self-aggregates of deoxy-cholic acid-modified chitosan and their application as a DNA delivery carrier,Bioconjug. Chem. 12 (2001) 932–938.

[4] L.L. Ma, M.D. Feldman, J.M. Tam, A.S. Paranjape, K.K. Cheruku, T.A. Larson, J.O.Tam, D.R. Ingram, V. Paramita, J.W. Villard, J.T. Jenkins, T. Wang, G.D. Clarke, R.Asmis, K. Sokolov, B. Chandrasekar, T.E. Milner, K.P. Johnston, Small multifunc-tional nanoclusters (nanoroses) for targeted cellular imaging and therapy, ACSNano 3 (2009) 2686–2696.

[

[

ochem. Eng. Aspects 436 (2013) 325– 332 331

[5] P. Sandkuhler, J. Sefcik, M. Morbidelli, Kinetics of aggregation and gel formationin concentrated polystyrene colloids, J. Phys. Chem. B 108 (2004) 20105–20121.

[6] P. Sandkuhler, J. Sefcik, M. Morbidelli, Kinetics of gel formation in dilute dis-persions with strong attractive particle interactions, Adv. Colloid Interface Sci.108–109 (2004) 133–143.

[7] P.J. Lu, J.C. Conrad, H.M. Wyss, A.B. Schofield, D.A. Weitz, Fluids of clusters inattractive colloids, Phys. Rev. Lett. 96 (2006) 028306.

[8] A.K. Atmuri, S.R. Bhatia, Polymer-mediated clustering of charged anisotropiccolloids, Langmuir 29 (2013) 3179–3187.

[9] V. Runkana, P. Somasundaran, P.C. Kapur, A population balance model for floc-culation of colloidal suspensions by polymer bridging, Chem. Eng. Sci. 61 (2006)182–191.

10] J. Sabin, G. Prieto, F. Sarmiento, Stable clusters in liposomic systems, Soft Matter8 (2012) 3212–3222.

11] F. Sciortino, S. Mossa, E. Zaccarelli, P. Tartaglia, Equilibrium cluster phases andlow-density arrested disordered states: The role of short-range attraction andlong-range repulsion, Phys. Rev. Lett. 93 (2004) 055701.

12] A.I. Campbell, V.J. Anderson, J.S. van Duijneveldt, P. Bartlett, Dynamical arrest inattractive colloids: the effect of long-range repulsion, Phys. Rev. Lett. 94 (2005)208301.

13] F. Sciortino, P. Tartaglia, E. Zaccarelli, One dimensional cluster growth andbranching gels in colloidal systems with short-range depletion attractive andscreened electrostatic repulsion, J. Phys. Chem. B 109 (2005) 21942–21953.

14] J.C.F. Toledano, F. Sciortino, E. Zaccarelli, Colloidal systems with competinginteractions: from an arrested repulsive cluster phase to a gel, Soft Matter 5(2009) 2390–2398.

15] K. Larson-Smith, D.C. Pozzo, Scalable synthesis of self-assembling nanopar-ticle clusters based on controlled steric interactions, Soft Matter 7 (2011)5339–5347.

16] H. Shin, G.M. Grason, C.D. Santangelo, Mesophases of soft-sphere aggregates,Soft Matter 5 (2009) 3629–3638.

17] W.B. Russel, D.A. Saville, W.R. Schowalter, Colloidal Dispersions, CambridgeUniversity Press, 1989.

18] P.C. Hiemenz, R. Rajagopalan, Principles of Colloid and Surface Chemistry, Mar-cel Decker Inc., 1997.

19] L. Ehrl, Z. Jia, H. Wu, M. Lattuada, M. Soos, M. Morbidelli, Role of counterionassociation in colloidal stability, Langmuir 25 (2009) 2696–2702.

20] M.v. Smoluchowski, Versuch einer mathematischen theorie der koagulationskinetik kolloider losungen, Z. Phys. Chem. 92 (1917) 129–168.

21] M. Soos, J. Sefcik, M. Morbidelli, Investigation of aggregation, breakage andrestructuring kinetics of colloidal dispersions in turbulent flows by popula-tion balance modeling and static light scattering, Chem. Eng. Sci. 61 (2006)2349–2363.

22] P.T. Spicer, S.E. Pratsinis, Shear induced flocculation: the evolution of floc struc-ture and the shape of the size distribution at steady state, Water Res. 30 (1996)1049–1056.

23] S.N. Maindarkar, N.B. Raikar, P. Bongers, M.A. Henson, Incorporating emulsiondrop coalescence into population balance equation models of high pressurehomogenization, Colloids Surf. A 396 (2012) 63–73.

24] P. Sandkuhler, J. Sefcik, M. Lattuada, H. Wu, M. Morbidelli, Modeling struc-ture effects on aggregation kinetics in colloidal dispersions, AIChE J. 49 (2003)1542–1555.

25] M.Y. Lin, H.M. Lindsay, R.C. Weitz, D.A. Ball, R. Klein, P. Meakin, Universality incolloid aggregation, Nature 339 (1989) 360–362.

26] V. Runkana, P. Somasundaran, P.C. Kapur, Reaction-limited aggregation in pres-ence of short-range structural forces, AIChE J. 51 (2005) 1233–1245.

27] P. Sandkuhler, M. Lattuada, J. Wu, H. Sefcik, M. Morbidelli, Further insights intothe universality of colloidal aggregation, Adv. Colloid Interface Sci. 113 (2005)65–83.

28] Z. Jia, H. Wu, M. Morbidelli, Application of the generalized stability model topolymer colloids stabilized with both mobile and fixed charges, Ind. Eng. Chem.Res. 46 (2007) 5357–5364.

29] M. Lattuada, P. Sandkuhler, J. Wu, H. Sefcik, M. Morbidelli, Kinetic modelingof aggregation and gel formation in quiescent dispersions of polymer colloids,Macromol. Symp. 206 (2004) 307–320.

30] M. Lattuada, P. Sandkuhler, J. Wu, H. Sefcik, M. Morbidelli, Aggregation kineticsof polymer colloids in reaction limited regime: Experiments and simulations,Adv. Colloid Interface Sci. 103 (2003) 33–56.

31] R. Amal, J.R. Coury, J.A. Raper, W.P. Walsh, T.D. Waite, Structure and kinetics ofaggregating colloidal haematite, Colloids Surf. A 46 (1990) 1–19.

32] E.V. Golikova, Y.M. Chernoberezhskii, V.S. Grigorev, M.P. Semov, Aggregatestability of the sol prepared from crystalline quartz in aqueous solutions ofpotassium chloride, Glass Phys. Chem. 32 (2006) 646–655.

33] J. Groenewold, W.K. Kegel, Anomalously large equilibrium clusters of colloids,J. Phys. Chem. B 105 (2001) 11702–11709.

34] J. Groenewold, W.K. Kegel, Colloidal cluster phases, gelation and nuclear matter,J. Phys.: Condens. Matter 16 (2004) 4877–4886.

35] A.J. Archer, C. Ionescu, D. Pini, L. Reatto, Theory for the phase behaviour of acolloidal fluid with competing interactions, J. Phys.: Condens. Matter 20 (2008)415106.

36] A.K. Atmuri, Ph.D. Thesis, University of Massachusetts Amherst, 2012.

37] M. Tourbin, C. Frances, Experimental characterization and population balance

modelling of the dense silica suspensions aggregation process, Chem. Eng. Sci.63 (2008) 5239–5251.

38] G.H. Bogush, C.F. Zukoski, Uniform silica particle precipitation: an aggregativegrowth model, J. Colloid Interface Sci. 142 (1991) 19–34.

3 Physic

[

[

32 A.K. Atmuri et al. / Colloids and Surfaces A:

39] S. Kumar, D. Ramkrishna, On the solution of population balance equations bydiscretization – I. A fixed pivot technique, Chem. Eng. Sci. 51 (1996) 1311–1332.

40] N.B. Raikar, S.R. Bhatia, M.F. Malone, M.A. Henson, Experimental studies andpopulation balance equation models for breakage prediction of emulsion dropsize distributions, Chem. Eng. Sci. 64 (2009) 2337–2433.

[

[

ochem. Eng. Aspects 436 (2013) 325– 332

41] A. Furukawa, H. Tanaka, Key role of hydrodynamic interactions in colloidalgelation, Phys. Rev. Lett. 104 (2010) 245702.

42] X.J. Cao, H.Z. Cummins, J.F. Morris, Hydrodynamic and interparticle potentialeffects on aggregation of colloidal particles, J. Colloid Interface Sci. 368 (2012)86–96.