cluster size distribution of charged nanopowders in suspensions

(2006) 124–133www.elsevier.com/locate/powtec

Powder Technology 167

Cluster size distribution of charged nanopowders in suspensions

M. Linsenbühler a, J.H. Werth b, S.M. Dammer b, H.A. Knudsen b,H. Hinrichsen c, K.-E. Wirth a,⁎, D.E. Wolf b

a Lehrstuhl für Feststoff-und Grenzflächenverfahrenstechnik, Universität Erlangen-Nürnberg, 91058 Erlangen, Germanyb Fachbereich Physik, Universität Duisburg-Essen, 47048 Duisburg, Germany

c Fakultät für Physik und Astronomie, Universität Würzburg, 97074 Würzburg, Germany

Received 9 November 2004; received in revised form 19 May 2006Available online 7 June 2006

Abstract

In order to control the aggregation phenomena in suspensions of nanoparticles, one often charges the particles electrically in order to delay oreven suppress the aggregation process. The resulting cumulative size distribution of agglomerates of monopolarly charged nanoparticlessuspended in liquid nitrogen is measured experimentally. The distribution is compared to results obtained by mean field theory. The relative widthof the distribution obeys the lower bound predicted by theory. The results are discussed with respect to technical applications, e.g. stabilization ofsuspensions against aggregation.© 2006 Elsevier B.V. All rights reserved.

PACS: 45.70.-n; 82.60.Qr; 83.10.RsKeywords: Granular matter; Charged nanoparticles; Powders; Suspensions

1. Introduction

Aggregation of small granular particles takes place in a largevariety of experimental and industrial situations, e.g. in a shakencontainer [1] or in granular gases [2,3]. The physical properties ofvery fine powders differ significantly from those of coarsegranular matter. While in ordinary granular materials such as drysand contacts between the particles that cannot sustain a tensileload and hence easily break up again, ultra-fine powders aggre-gate in an essentially irreversible way. In fact, most commerciallytraded nanopowders do not consist of primary particles, insteadthey are usually delivered as aggregates of much larger sizes.

The aggregation of such small particles is caused by twotypes of interactions. One of them is hydrogen bonding and theother is the van der Waals force, which is a quantum-mechanicaleffect resulting from the interaction of virtual dipoles. In manycases these forces dominate all other interactions on shortdistances and thus can be considered as a practically irreversiblesticking force. As soon as two small particles or aggregatestouch each other, they stick together in an irreversible manner.

⁎ Corresponding author. Tel.: +49 9131 85 29401; fax: +49 9131 85 29402.E-mail address: [email protected] (K.-E. Wirth).

0032-5910/$ - see front matter © 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.powtec.2006.05.017

As a result the particles aggregate, forming irregular shapedagglomerates, whose structure reminds of fractal flakes. Oncethe contacts between different particles have been established,they may develop sinter necks, enhancing the rigidity of theflakes. Flakes greatly reduce the flowability of such powders,reflecting our everyday experience that fine powders are oftenmore sticky than coarse granular matter. In most applications,aggregation of nanoparticles is an unwanted side effect.

A possibleway to delay or even inhibit aggregation is to chargethe particles electrically. In fact, most materials, in particularinsulators, become charged as soon as the powder is processedmechanically, e.g. by stirring or in a pipe flow. If one manages tocharge all particles with the same polarity, the mutual Coulombrepulsion delays or even suppresses the aggregation process [4].

Another situation, on which we will focus in the present work,occurs when a fine powder is suspended in a liquid. In contrast togranular gases, where particles move and collide ballistically andthus can be characterized by the so-called granular temperature[4], the particles in a sufficiently dilute suspension are highlydamped and perform basically a randomwalk before they collide.Their degrees of freedom are thermodynamically coupled to theheat bath provided by the surrounding liquid. Therefore, eachdegree of freedom carries a kinetic energy kBT/2.

mailto:[email protected]

http://dx.doi.org/10.1016/j.powtec.2006.05.017

Table 1Material properties of Aerosils

Aerosil OX50 R972

Behaviour towards water Hydrophilic HydrophobicBET surface Sm [m2/g] 50±15 110±20dp [nm] 40 16ρs [g/cm

3] 2.3 2.3SiOH density [nm−2] 2.5 0.6nr [–] 1.45 1.45εr [–] 3.7 3.7

125M. Linsenbühler et al. / Powder Technology 167 (2006) 124–133

While the properties of electrically charged granular matterin vacuum are already fairly well understood, we will focushere on the behaviour of charged powders made of insulatingmaterials suspended in a non-polar fluid such as liquid nitro-gen. As demonstrated in [5], such a suspension can be regardedas stable if the average Coulomb barrier between the particlesexceeds the thermal energy kBT. Contrarily, when the Coulombbarrier is smaller than the thermal energy, the particles (primaryparticles or small agglomerates) begin to aggregate. During theaggregation process the clusters grow, accumulating bothmass and charge. This process continues until the Coulombbarrier between clusters reaches the thermal energy, fromwhere on further aggregation is exponentially suppressed.Obviously, the formed aggregates do not need to have thesame size, rather we expect their sizes to be distributed oversome range with an upper cut-off approximately at the typicalsize where the Coulomb barrier is of the same order as thethermal energy.

The main objective of this paper is to study this size distri-bution both experimentally and theoretically. The basic idea,which triggered the present work, is the concept of self-focussing[6], which works as follows. Since agglomeration is assumed tobe irreversible, the distribution of cluster sizes evolves dyna-mically towards larger clusters until the aforementioned typicalsize is reached, whereas smaller clusters still aggregate. As aresult we expect the size distribution to become sharper, ap-proaching a certain shape.

Regarding experiments the question arises to what extent it ispossible to charge suspended particles monopolarly. Since thereare no countercharges in the suspension, the whole system is notelectrically neutral. Therefore the charged particles are expectedto move to the fluid boundaries, settling on the container walls oreven leaving the container if the Coulomb repulsion is toostrong. In Section 3 we show that in normal experimental con-ditions charged particles do not escape the container. Althoughcharged particles tend to move to the container walls, thusdiluting the suspension, this process is relatively slow comparedto the relevant experimental time scales.

Experimentally, we dispersed nanoparticles (Aerosil R972®and OX50) in liquid nitrogen, charging them triboelectrically. Wemeasured the size distribution of agglomerates by using atransmission electron microscope. Although our quantitative re-sults are based on a small sample ofmeasurements, we do observea distribution with a finite width and a characteristic shape. Inorder to substantiate these findings, in Sections 4 and 5 wecompare the experimental data with the predictions of mean fieldtheory calculations, describing the agglomeration of chargedparticles [6]. This theory allows us to specify a lower bound forthe relative width of the size distribution, which is found to becompatible with the experimental results. In addition, the meanfield theory predicts a universal shape of the distribution in thelong time limit, which is independent ofmicroscopic details of theaggregation process as long as finite size effects are neglected.Our experimental results are not precise enough to fully confirmor disprove the theoretical results. Nevertheless, they indicate thatthe mean field theory is likely to describe the observed aggre-gation phenomenon in a correct way.

The paper is organized as follows. In the following section wefirst describe how the experiments and the data analysis arecarried out. Then, in Section 3, we outline the theoretical frame-work needed to understand the accumulation of charges. Thesetheoretical predictions are then compared with the experimentalresults in Section 4. For a deeper understanding of the experi-mentally observed cumulative size distribution of the agglom-erates we suggest a mean field theory in Section 5. The paperends with concluding remarks, a list of symbols, and two appen-dices in which the maximum charge capacity of a suspension andthe sedimentation of particles at the wall of the container areestimated.

2. Experimental setup and data analysis

2.1. Experimental setup

In each experiment, we dispersed 20 mg of dry nanoparticles,in this case Aerosil® produced by Degussa AG, in approximately400ml of liquid nitrogen in a beaker glass (see Table 1 formaterialproperties). The nanoparticles were dispersed using an Ultra-TurraxDisperser (cf. Fig. 1)with 10,000RPM for 2min. Frequentcontact with the stirrer leads to charge separation, charging theparticles triboelectrically (cf. Fig. 2).

We took samples out of the beaker by dip-coating a trans-mission-electron-microscope (TEM) grid. In each experiment wetook three samples: one directly before the end of the dispersionprocess, the next one 20 s after stopping the stirrer, and the thirdone was taken 10min after the end of the dispersion process. In allcases the TEM-grids were prepared by cooling them down inliquid nitrogen, so that the temperature difference between thesuspension and the grid was small. The size distribution of thedeposited agglomerates was studied in a PHILIPS CM 30 T/STEM transmission electron microscope.

Due to impurities such as water, even pure liquid nitrogenwithout Aerosil® particles can be charged electrically. Taking themaximum charge of pure liquid nitrogen into account, the totalcharge of the particles in the suspension is the charge differenceqp. In the case of hydrophobic Aerosil R972® we find qp=−0.171 μC (cf. Fig. 2). The sudden drop in Fig. 2 (at about 120 s)occurs when the stirring device is pulled out of the suspension.

2.2. Data analysis

In order to determine the size and shape of the agglomeratesprojected onto the plane we used the optical picture analysis

Fig. 2. Integral charge as a function of the charging time: Aerosil R972® inliquid nitrogen.

Fig. 1. Experimental set-up: Ultra-Turrax Disperser T50, IKA.

126 M. Linsenbühler et al. / Powder Technology 167 (2006) 124–133

software Optimas 6.2. The program renders various quantities ofinterest, e.g. the number of primary particles n, which can beidentified in the picture, the average primary particle diameter dp,the longest lateral extension of the agglomerate dh (see Fig. 3),and the diameter of a circle of equal area.

The agglomerates found on the TEM-grids show a broad sizedistribution. This size distribution was measured, as well as thenumber of primary particles per agglomerate. As it has beenshown in [7], such agglomerates have fractal properties and canbe characterized by the so-called fractal dimension Df [8]. It isconveniently given by [9]

n ¼ n0dhdp

� �Df

; ð1Þ

which relates the number of primary particles n in an agglom-erate and its size dh in units of dp. The proportionality factor n0is the extrapolation of the power law to dh=dp.

Naively one might expect that it is equal to 1, the number ofprimary particles in an “aggregate” of diameter dp. The maximaldiameter of a dimer consisting of two primary particles is 2dp(corresponding to an exponent Df=1 and n0=1). For growingagglomerates, the fractal dimension crosses over to the properfractal dimension DfN1, thus giving n0b1.

Fig. 3. Illustration of the ‘hydrodynamic diameter’ dh, defined as the largestlateral extension of the agglomerate (Aerosil R972®).

3. Estimation of the involved forces and limits of chargeaccumulation

As outlined in the introduction, the agglomeration process isdominated by thermal Brownian motion, Coulomb interactionsand van der Waals forces. Hydrodynamic interactions are also

present, but it has been shown [5] that to the lowest order theirinfluence leads to a modification of the values of the controlparameters such as the diffusion constant and the collision rate.Therefore, in the theoretical considerations to follow, hydrody-namic interactions are not explicitly taken into account.

3.1. Coulomb energy

During the dispersing process the triboelectric charging of theparticles is monitored by measuring the potential of a Faradaycup [10]. In order to estimate the interaction potential betweenthe charged agglomerates, let us consider the agglomerates asspheres whose radius is given by the mean diameter of theagglomerates. Since the maximal Coulomb force is reached whenboth particles touch each other, we consider that the spherescome into contact due to the Brownian motion (contact distanceaccording to Krupp [11] h0=0.4 nm, cf. Fig. 4).

Fig. 4. Two spheres of equal size subjected to van der Waals' attraction.


The electrostatic repulsion energyφel of two equally chargedagglomerates in contact can be estimated by:

uel ¼q2agg

4ke0erd½J�: ð2Þ

The charge qagg of an agglomerate can be estimated from the totalcharge of the suspension related to the number of agglomerates inthe suspension. The latter one is calculated by relating the mass ofthe solids in the suspension to the average weight of oneagglomerate which has been obtained from the number of primaryparticles per agglomerate using the above mentioned pictureanalysis and the density of the primary particles. Our experimentslead to the estimates qagg≈4.6 e− and for the agglomerate diam-eter d≈400 nm on the average, thus φel≈10−20 J.

3.2. Van der Waals energy

The van der Waals potential φvdW in liquids can be derivedfrom the macroscopic Lifschitz theory [12] and, in the limitwhere the distance between the particles h is much smaller thand, is given by

uvdW ¼ Ad24h

½J�; ð3Þ

where A≈10−20 J is the Hamaker constant. This means that onshort distances the van der Waals attraction dominates over theCoulomb repulsion.

Fig. 5. Particle density in a spherical container of radius R as function of thedistance r from the center. For tN0 particles accumulate on the container wall.

3.3. Limits of charge accumulation

Before presenting the experimental result let us return to thequestion to what extent particles in a non-polar liquid canaccumulate charges of the same sign without counterchargesbeing distributed in the fluid. Since the whole system is notelectrically neutral, repulsive forces will drive the particlestowards the boundaries of the system, where either the walls ofthe container or the surface tension γ hinders particles fromleaving the suspension.

As it is shown in Appendix A, particles are not expelled fromthe surface as long as the total charge qtot in the suspension ofvolume V does not exceed an approximate value of

qtot ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi10d 4kere0gV

p: ð4Þ

With the values εr =1.45, γ=83 mN/m, and V=400 ml, onearrives at qtot = 0.23 μC. As it has been shown above(qp≈0.17 μC), the total charge reached in the experiments isbelow this limit.

When the charging process is stopped, the charged particlesin the suspension will eventually settle down on the containerwalls. As with free electrons in a metal, this process will con-tinue until all charges have reached the container walls, i.e. theelectrical field inside the container vanishes. As demonstrated inAppendix B, it turns out that, starting with a given homo-geneous number density of charged particles ρ0, the density ofparticles decreases like

qðtÞ ¼ q02q2q0

3gdpere0t þ 1

� �−1

: ð5Þ

Using this formula it turns out that the time scale, on which thisdelution of the suspension takes place, is much longer than theduration of the experiment because the mobility of particles isrelatively small due to Stokes friction between the particles andthe fluid.

In order to support this argument, we performed additionalmolecular dynamics simulations. In such a simulation, theequation of motion for each particle in the suspension is solved

Fig. 6. Fraction of particles suspended in the bulk of the container as a functionof time.

Fig. 7. First sample: TEM-picture of Aerosil R972® in liquid nitrogen taken at the end of the dispersion process.


explicitly. Since we are only interested in the mean motion ofcharged particles towards the container walls, we can neglectthe Brownian motion. Assuming spherical particles or agglom-erates, the trajectory of each particle can be described by

drY ¼ FYc

3kgdpdt: ð6Þ

Here, drY is the positional shift of a particle of radius dp duringthe time dt, due to the Coulomb force F

Yc.

In principle it is possible to integrate Eq. (6) for all particlesin the suspension in order to get the complete information aboutthe density of particles and agglomerates in the container.However, the number of particles in a real experiment is far too

Fig. 8. Second sample, taken 20 s after

large to be feasible in a simulation. Nonetheless, it is possible toreduce the number of particles needed to be simulated as fol-lows: The Coulomb force acting on an agglomerate in the fluidoriginates from the charges of all other agglomerates in thesuspension. It can be written as a product of the agglomeratecharge q and the electric field E at its position,

drY ¼ EY

3��qddt: ð7Þ

Now we imagine neighboring agglomerates being gathered intovirtual particles of the same total charge, so that the electrical fieldremains approximately unchanged. We attribute to these virtualparticles a radius which is proportional to the charge. Hence the

the end of the dispersion process.

Fig. 9. Third sample, taken 10 min after the end of the dispersion process.

Fig. 10. Best fit of the cumulative size distribution Q0(x) plotted against thecircle with equal area x of the Aerosil R972® agglomerates.


dynamics, Eq. (7), is the same for the virtual particles as for theoriginal agglomerates. In this way we can reduce the number ofparticles needed to be simulated to a sufficiently small value.

Fig. 5 shows the results of such a simulation using typicalexperimental parameters. It is assumed that the suspension isconfined to a spherical vessel, and the graph shows the densityof particles at various positions between the center and thesurface of the suspension at different times. The simulationstarts with a homogeneous distribution of particles. As one cansee, particles (and therefore also charges) accumulate on thesurface during time, but the density of particles inside thesuspension remains homogeneous at all times, as predicted bythe analytical calculation (see Appendix B). Concerning ourexperiments, one can see that on typical timescales (≈10 min) alarge number of particles still remain in the bulk liquid.

Fig. 6 shows the fraction of particles suspended in the bulk asa function of time. As one can see, asymptotically the densitydecays as t−1, as predicted by Eq. (5). Gravitation is consideredneither in the simulation nor in our analytical calculation. Com-paring potential and thermal energy under experimentalconditions, we find that this assumption is justified for primaryparticles while for very large agglomerates sedimentation isexpected to play an increasing role. Further corrections areexpected as the cylindrical geometry of the experiment differsfrom the assumed spherical geometry used in the calculation.

4. Experimental results

As described in the previous section, samples were taken atthree different times during and after the dispersing process.

Fig. 7 shows two pictures with agglomerates of the firstsample, which was taken at the end of the dispersion. On the lefthand side one can see a single big agglomerate. We assume thatmost of such big flakes are preexisting agglomerates, whichwere not yet broken up by the stirrer. However, in principle theyalso may have been produced by aggregation of smaller flakes.

On the right hand side one can see small aggregates. Pre-sumably it is not possible to decompose small aggregates all theway into primary particles by the stirring process.

The pictures of the other two samples (Figs. 8 and 9) showthe same characteristics. On the one hand, there are small flakesthat are more or less compact. On the other hand, one observesoccasionally very big agglomerates. As can be seen, in all casesthe dispersion process produces a broad distribution of agglo-merate sizes in the suspension.

Investigating several pictures such as those shown in Figs. 7–9, we obtained the agglomerate size distribution shown in Fig. 10for the three different times.

The first size distribution, showing agglomerate sizes at theend of the dispersion process, is the broadest one. In particular,there is a substantial fraction of small flakes (≈150 nm radius ofan equally sized circle). As one can see, this fraction ofagglomerates disappears later, moving the left tail of the sizedistribution towards larger sizes. As can be seen, the distributionbecomes steeper immediately after dispersing, meaning that

Fig. 12. Best fit of the cumulative size distribution of the Aerosil OX50®agglomerates [13].

Fig. 11. Estimation of the fractal dimension of Aerosil R972® agglomerates inliquid nitrogen by measuring how the number of particles increases with thenormalized hydrodynamic diameter.

Fig. 13. Estimation of the fractal dimension of Aerosil OX50® agglomerates inliquid nitrogen.


small particles have already formed small agglomerates whilelarger ones do not exist.

The agglomerate size distribution in the final sample, i.e.10 min after the end of the dispersion process, shows the smallestwidth. The fraction of flakes bigger than 1μm is similar in all threesamples. The median value of the first and second sample, i.e. endof dispersing and 20 s after dispersing, is x0,50≈415 nm, the value10 min after the dispersing increases slightly to x0,50≈545 nm.

Fig. 11 shows a plot of the number of primary particles in anagglomerate, which characterizes the agglomerate mass, asfunction of the hydrodynamic diameter in units of the primaryparticle diameter. Since both axes are logarithmic, the slope ofthe curve is the fractal dimension of the agglomerates, ac-cording to Eq. (1). Fitting the data of all measurements gives

Df ¼ 2:03F0:08

and

n0 ¼ 0:11F0:03:

We also fitted each data set independently. However, within thestatistical error bars of the fit, all values correspond to Df =2.This result is the maximum for the fractal dimension that can beobtained by evaluating the projection from a three-dimensionalspace onto a two-dimensional plane. Therefore, the true fractaldimension of the agglomerates must be between Df=2 andDf =3. The value of n0 implies that the crossover to the scalingregime happens for aggregates larger than 10 primary particles.

Figs. 12 and 13 show the same kind of plots for AerosilOX50®. Pictures of the corresponding agglomerates were alreadypublished in [13]. Again, we computed the fractal dimension forthree times, i.e. during, directly after and 10 min after the dis-persion process. At early times the fractal dimension is sig-nificantly lower than 2, i.e.Df≈1.4 during the dispersion processandDf≈1.6 shortly after. However, 10min later the agglomeratesare much larger and clearly more compact, i.e. Df≈2.

To summarize, the analysis of the TEM-pictures of the AerosilR972® dispersion in liquid nitrogen shows that the suspensionseems to be stable showing only a slight tendency to agglomeration.

However, analyzing the agglomerates with the TEM, one hasto take into account that the three-dimensional structure of theclusters is projected into a two-dimensional plane. For thisreason the number of particles n will be underestimated sincesome primary particles shadow the other ones, meaning that thefull structure of the flakes can be characterized only approxi-mately. Nevertheless, this method may be useful for comparingdifferent experiments with similar particles, where these sys-tematic errors are expected to be of the same order of magnitude.

5. Mean field theory

In order to understand the experimentally measured cumu-lative size distribution in Fig. 10, let us consider a mean fieldtheory for monopolarly charged particles in a suspension whichhas been introduced recently in [6]. As will be shown, this theorypredicts a lower bound for the width of the distribution, which isfound to be compatible with the experimental observations.

To keep the equations as simple as possible we assume thatthe agglomeration process starts with primary particles which allhave the same mass m⁎, the same radius dp⁎, and carry the samecharge q⁎. In this case mass and charge are proportional to eachother and can be expressed in a single index n, denoting thenumber of primary particles in a given cluster. Moreover, we

Fig. 14. Scaling function of the cumulative size distribution, cf. Eq. (13).


assume the clusters themselves to be essentially spherical,capturing their fractal properties in a non-trivial effective radius.Under these assumptions Smoluchowski's coagulation equationreads

dcnðtÞdt

¼ 12

Xkþl¼n

RklckðtÞclðtÞ−cnðtÞXlk¼1

RnkckðtÞ: ð8Þ

Here, cn(t) denotes the concentration of cluster with n primaryparticles, having the mass m=nm⁎, the charge q=nq⁎, and theradius dh=n

αdp⁎, where the exponent α=1 /Df accounts for thefractal structure of the agglomerates. For example, for spheresα=1 /3 while for sparse structures we expect some valuebetween 1 /3 and 1. For simplicity the initial condition assumesprimary particles, i.e. cn(0)=δn,1.

The key quantity in the mean field equation, which inheritsmost of the physical properties, is the so-called coagulationkernel Rnk. As explained in Ref. [6], for equally chargedparticles a natural choice is

Rnk ¼ ðna þ kaÞðn−a þ k−aÞ jnkexpðjnkÞ−1 ; ð9Þ

jnk ¼ a2nkðna þ kaÞ ; ð10Þ

where

a2 ¼ ðq*Þ22ke0erd*p kBT

:

Investigating the mean field equation, one arrives at the fol-lowing results. Initially the suspension behaves as if it wasuncharged, forming aggregates whose average size n̄(t) in termsof the number of primary particles per aggregate grows linearlywith time. The agglomeration process continues until a typicalcrossover time

tc ~ a−2

2−a ð11Þ

is reached at which the average cluster size is of the ordern̄c∝ tc. Due to the repulsive Coulomb forces the system thencrosses over to a regime of very slow (exponentially sup-pressed) growth. Remarkably, in this regime the number densitydistribution and similarly the cumulated size distribution Qi(t)obey a simple scaling law

QiðtÞ ¼Xi

n¼1

cnðtÞ=Xln¼1

cnðtÞiUn

n̄ðtÞ� �

; ð12Þ

where Φ converges towards a universal function in theasymptotic limit t→∞. The asymptotic shape of this functioncan be computed exactly and is given by

UðnÞ ¼

0 if nb1

2lnð2Þ2−

1nlnð2Þ if

12lnð2ÞVnb

1lnð2Þ

1 if nz1

lnð2Þ

8>>>>>><>>>>>>:

ð13Þ

as sketched in Fig. 14. Here ξ=n / n̄ denotes the argument of thescaling function in Eq. (12).The experimentally observedevolution of the size distribution (cf. Fig. 14) is in accordancewith the assumption that asymptotically the distribution is givenby Eq. (13).

While details of the calculation are given in [6], let usmotivate it here from a phenomenological point of view. Assoon as the Coulomb barrier exceeds the thermal energy, furtheragglomeration is strongly suppressed, i.e., the rate for mergersdecreases exponentially with their size. The time evolution ofthe cluster size distribution is dominated by coagulation eventsamong two (equally sized) clusters from the left edge of thedistribution. Once merged, they form a cluster which is twice aslarge, being located on the right edge of the size distribution.Hence the dominating process transfers clusters from the leftedge to the right edge of the size distribution, explaining whythe positions of the two edges just differ by a factor of 2.

The relative width of the asymptotic distribution can becomputed easily using the moments Mk=∑i ikIni of theagglomerates size distribution and is given by

rr ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiM2

M 21

−1

s; ð14Þ

hence rrYtYl rl with

rl ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=½2ðln2Þ2�−1

qc0:2017: ð15Þ

Although this asymptotic width is reached only after anenormous time and depends on various idealizations on whichthe mean field equation is based, it can be interpreted as a lowerbound to any experimentally measured width. In fact, the


cumulative distribution reported in the previous section (seeFig. 10) gives a relative width 10 min after the dispersionprocess for Aerosil R972®

rexc0:72 ð16Þwhich is indeed larger than σ∞. From Fig. 12 it is obvious thatfor the hydrophilic Aerosil OX50® the width of the agglomeratesize distribution after the dispersion process is larger in com-parison to the hydrophobic Aerosil R972®. Therefore bothnanopowders show a larger width as calculated.

The measured widths, which deviates significantly from0.2017 may be explained as follows. Initially the particlesaggregate as if they were uncharged. For uncharged particles thetheory predicts an increase of the relative width towards theasymptotic value of approximately σr=1 (see Ref. [6]). At alater stage, when the Coulomb interaction becomes relevant, therelative width decreases again, self-focussing towards theasymptotic value σr≈0.2017. Therefore, the experimentallyobserved width should be between σr =1 and the valueσr≈0.2017. Moreover, one has to keep in mind that the initialconditions in the experiment differ from those in the theory. Inthe experiment already in the beginning the particles havedifferent sizes and may carry different charges including evenuncharged particles, broadening the distribution.

6. Conclusions

In this paper we have investigated the agglomeration of cha-rged nanoparticles suspended in a non-polar liquid. To this endwehave carried out experimentswithAerosil® particles suspended inliquid nitrogen, analyzing TEM-grids taken at different stages ofthe agglomeration process after dispersion. Although theavailable data set is small we have been able to compute thecumulative size distribution and to estimate its width.

The agglomeration process can be modelled theoreticallyby a mean field approach based on Smoluchowski's rateequation. This calculation predicts that the influence of elec-trostatic charges leads to the phenomenon of self-focussing,rendering a universal size distribution in the limit t→∞ with awell-defined relative width. We have argued that this widthcan be interpreted as a lower bound to any experimentallyobservable width of the size distribution in agglomeratingcharged granular matter subjected to Brownian motion andCoulomb forces.

The phenomenon of self-focussing may be used to controlthe agglomeration behaviour of very fine powders and toproduce agglomerate sizes in a narrow window.

List of symbolsa Ratio of electrostatic and thermal energy [–]A Hamaker constant [J]
cn Concentration of agglomerates of size n [m−3]d Diameter [m]Df Fractal dimensiondh Hydrodynamic diameter [m]dp Primary particle diameter [m]
E Electrical fieldFc Coulomb force [N]h Distance [m]h0 Contact distance [h0=0.4 m]j Flux of particles [m−2 s−1]kB Boltzmann constant [J/K]m Mass [kg]Mr Moment of distribution rn Number of primary particlesn̄ Average cluster sizen0 Dimensionless constantnr Refractive index [–]p Pressure [Pa]pcap Capillarity pressure [Pa]q Charge [C]qagg Electrostatic charge of an agglomerate [C]qp Total electrostatic charge of particles [C]qtot Maximal obtainable charge [C]Q0 Cumulative size distributionr Position [m]R Radius of the container [m]Rij Coagulation kernelSm BET surface [m2/g]T Temperature [K]t Time [s]V Volume [m3]x0,50 Mean particle diameter of number distributionα =1/Df, inverse fractal dimensionγ Surface tension [N/m]δ(r) Dirac functionε0 Dielectric constant [F/m]εr Relative dielectric constant [–]η Viscosity [N s/m2]φel Electrostatic energy [J]φvdW van der Waals energy [J]Φ(ξ) Scaling functionρ(t) Number density of particles in suspension [m− 3]ρ0(t) Initial number density [m− 3]ρs Mass density [kg m− 3]σ Relative width of distribution

Acknowledgements

We gratefully acknowledge the financial support by theDeutsche Forschungsgemeinschaft (DFG), grants WI 972/14-1and HI 744/2-1.

Appendix A. Maximum capacity of a suspension

We want to compute the maximum capacity of a suspensioncontaining equally charged particles. In such a suspension, theelectrostatic repulsion between particles leads to pressure on thesurface of the suspension, which must be compensated e.g. bythe surface tension.

The particles arrange in a way that the electrostatic potentialenergy in the system is lowered. Thus, in reality the potential


energy is clearly smaller than it would be in a suspension withhomogeneously distributed charges (since this situation doesnot correspond to the minimum energy state). A homogeneouscharge distribution can therefore be used to calculate an upperbound of the potential energy of the system. Assuming that thesuspension with total charge q on the particles is confined by aspherical vessel with radius R and volume V, this upper bound is

uelVq

V

� �2 1

4kere01

2

Zvd3r

Zvd3r V

1

jrY−rYVj ¼3

5

q2

4kere0R: ð17Þ

The pressure on the suspension surface is given by the totalCoulomb force acting on the suspensions surface, i.e.

p ¼ −1

4kR2

Auel

AR¼ 1

3uelðV ÞV

: ð18Þ

This pressure must be compensated by the capillarity pressureof the suspension, given by

pcap ¼ 2gR: ð19Þ

Stability requires that pbpcap, and therefore qbqtot with

qtot ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi10d4pere0gV

p: ð20Þ

Appendix B. Sedimentation of particles at the walls of thecontainer

In order to estimate the settling of particles onto the containerwalls, we consider a simplified setup: all particles are dis-tributed in the suspension homogeneously with an initialdensity ρ0. The evolution of the number density of particleswith time is given by

AqAt

¼ −j!d j

Y;

where j is the flux of particles inside the suspension, driven byCoulomb interaction between the particles. We assume that thesuspension is contained in a spherical vessel of radius R. Due tosymmetry, j will always point away from the center of thecontainer. The electrical field at a distance r from the middle ofthe container (0≤ r≤R) is given by

EðrÞ ¼ 43kq0qr

3 14kere0r2

¼ qq0r3ere0

:

The flux of particles is given in the overdamped limit by

jðrÞ ¼ qqEðrÞ3kgdp

¼ q2q2r18gaere0

:

Thus, we arrive at

AqAt

¼ −q2q2

3gdpere0:

Note that the term on the right hand side does not depend on r,so that an initial homogeneous distribution of particles willremain homogeneous for all times. The solution of thedifferential equation is

qðtÞ ¼ q02q2q0

3gdpere0t þ 1

� �−1

:

References

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[4] T. Scheffler, Kollisionskühlung in elektrisch geladener granularer Materie.PhD thesis, University of Duisburg, 2000.

[5] J.H. Werth, M. Linsenbühler, S.M. Dammer, Z. Farkas, H. Hinrichsen,K.-E. Wirth, D.E. Wolf, Powder Technology 133 (2003) 106–111.

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[7] S. Kütz, In-situ Methoden zur Bestimmung von Struktureigenschaftengasgetragener Agglomerate. PhD thesis, University of Duisburg, 1994.

[8] B.B. Mandelbrot, The Fractal Geometry of Nature, W.H. Freeman andCompany, New York, 1983.

[9] R. Lange, Entwicklung und Überprüfung einer Methode zur Bestimmungdes interzeptionsäquivalenten Durchmessers agglomerierter Partikel, PhDthesis, University of Duisburg, 2000.

[10] G. Huber, Elektrostatisch unterstütztes Mischen und Beschichten hoch-disperser Pulver in flüssigem Stickstoff, PhD thesis, University ofErlangen-Nürnberg, 2001.

[11] H. Krupp, Particle adhesion — theory and experiment, Advances inColloid and Interface Science 1 (1967) 111–239.

[12] E.M. Lifschitz, The theory of molecular attractive forces between solids,Journal of Experimental and Theoretical Physics 28 (1956) 73–83.

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cluster size distribution of charged nanopowders in suspensions

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