bipolar diffusion charging characteristics of single-wall carbon nanotube aerosol particles

Aerosol Science 40 (2009) 164 -- 179

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Aerosol Science

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Bipolar diffusion charging characteristics of single-wall carbon nanotubeaerosol particles

Pramod Kulkarni∗, Gregory J. Deye, Paul A. Baron

Centers for Disease Control and Prevention, National Institute for Occupational Safety and Health, 4676 Columbia Parkway, MS: R-3, Cincinnati, OH 45225, USA

A R T I C L E I N F O A B S T R A C T

Article history:Received 1 May 2008Received in revised form5 September 2008Accepted 9 September 2008

Keywords:Single-wall carbon nanotubesDiffusion charging

Bipolar diffusion charging characteristics of airborne single-wall carbon nanotube (SWCNT)agglomerates were investigated in the mobility diameter range of 100–1000nm. Neutral frac-tions of three types of SWCNT aerosols following bipolar charge equilibrium in a radioactivesource were experimentally measured to infer their electrical charging characteristics. Sig-nificant deviation from Boltzmann and Fuchs stationary charge equilibrium was observed,with neutral fractions of SWCNT particles lower by 30–53% compared to that of sphericalparticles of the same mobility. Particles with mobility diameter larger than 400nm showedhigh electrical charging efficiencies compared to that of mobility-equivalent spherical parti-cles. Higher charging efficiencies of SWCNT particles were attributed to their higher electricalcapacitance resulting from complex nonspherical morphologies. Numerical calculations us-ing idealized fiber geometries confirmed the qualitative trend in the experimental data. Theelectrical capacitance of nanotubes particles deduced from experimentally measured neutralfractions were also found to be higher by a factor ranging from 1.6 to 4.6 compared to that ofmobility-equivalent spherical particles, indicating high charge carrying capacity. The charging-equivalent diameters of nanotube particles were computed and were found to be higher thantheir mobility diameter by a factor of 2.85–4.34.

Published by Elsevier Ltd.

1. Introduction

Growth in technological applications of single-wall carbon nanotubes (SWCNT) and their increasing industrial productionhas led to many concerns over the potential health risks from exposure in industrial environments (Lam, James, McCluskey, &Hunter, 2004; Shvedova et al., 2003a,b; Warheit, 2006; Warheit et al., 2004). The potential health hazards of SWCNT remainpoorly understood. Studies have shown that SWCNT are cytotoxic and lead to oxidative stress in vitro (Shvedova et al., 2003a), aswell as indicating pro-inflammatory and fibrogenic responses in rodent lungs (Lam et al., 2004; Shvedova et al., 2005; Warheit,2006; Warheit et al., 2004). Previous studies have shown that as-produced SWCNT material generated using the high pressureCO disproportionation process (HiPCO�) process can release particles into the air subject to controlled agitation (Maynard et al.,2004). Measurement of aerosolized SWCNT showed a bimodal mobility size distribution, withmodal diameters of approximately400 and10nm (Maynard et al., 2004). Although the airborne concentrations of SWCNTobservedduring laboratory and small-scaleproduction conditions were very low, potential health risk from inhalation will depend on the particle toxicity, exposure levels,and the duration of exposures. A recent review of engineered nanomaterial toxicity screening tests emphasized the importanceof physico-chemical characterization of nanomaterial in determining their potential hazard (Oberdorster et al., 2005).

∗ Corresponding author. Tel.: +15138414300; fax: +15138414545.E-mail address: [email protected] (P. Kulkarni).

0021-8502/$ - see front matter Published by Elsevier Ltd.doi:10.1016/j.jaerosci.2008.09.008

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P. Kulkarni et al. / Aerosol Science 40 (2009) 164 -- 179 165

Airborne SWCNTs agglomerates present an entirely new class of aerosols that are distinct from conventional aerosol ag-glomerates of dense, compact, near-spherical primary particles. Transmission electron microscopy (TEM) of SWCNT particlesfrom HiPCO� process shows that these agglomerates consist of four distinct components: iron-rich catalyst particles (about5nm in diameter), discrete SWCNT, bundles of self-aligned carbon nanotubes (CNT) called nanoropes (approximately 5–50nm indiameter), and non-tubular carbon particles. This complex matrix can present unique challenges in terms of their measurementusing conventional aerosol instrumentation. The airborne SWCNT agglomerates have complex morphology with open, fibrousstructure with high internal surface area. SWCNT particles have very high aspect ratios; length of nanotubes and nanoropeswith diameters ranging from 1–50nm can easily exceed a few micrometers in length, resulting in large dynamic shape factors.Furthermore, the presence of catalyst particles and compact, amorphous carbon in the agglomerate matrix can introduce largeuncertainties in their measurement.

While most dense, near-spherical particles and their agglomerates can be described either by their geometrical size or fractalgeometry, it is not known if such descriptions are applicable to SWCNT agglomerates due to their open, porous structures. Inertialimpactor measurements show that overall enveloping physical size of these particles is much larger than their aerodynamic size,by a factor of up to 10 (Baron et al., 2008). This implies that their physical size inferred from microscopy is of little significanceand onemust rely on their property-equivalent diameters to understand their transport in the respiratory system. It is very likelythat there is a large difference in their aerodynamic- and diffusion-equivalent diameters due to their high internal surface area,large dynamic shape factors, and low effective densities. As a result, accurate measurement of diffusion-equivalent (or mobility-equivalent) and aerodynamic diameters is very important.Whilemass-weighted aerodynamic diameters of SWCNT agglomeratescan be reliably measured using inertial impactors, the same cannot be said for electrical mobility-based instrumentation, such asscanning mobility particle sizers (SMPS). The complex shape of the particles can influence their electrical charging and transportwithin the instrument which can significantly bias the measurement of mobility size spectrum. Electrical charging of SWCNTaerosols is also of interest in many other engineering applications.

In this paper we investigate the bipolar diffusion charging characteristics of SWCNT aerosol agglomerates. We examine therelationship between electrical charge, acquired by a SWCNT particle when exposed to a bipolarion atmosphere, and particle'smobility diameter. This information is important in the context of real-time instruments such as the SMPS, the data inversionprocedure of which involves assuming a known charge distribution on the measured aerosol particles.

2. Background

Boltzmann distribution has been widely used to approximate the experimentally measured charge distribution of aerosolparticles (Gunn, 1955; Keefe, Nolan, & Rich, 1959). According to Boltzmann theory, the bipolar charge distribution is the re-sult of equilibrium energy exchange between the particles and the surrounding bipolar ions. The Boltzmann distribution wellrepresents the experimentally measured charge distribution, especially for particles larger than about 100nm in diameter,suggesting that energetics of the system dominate the steady state charge distribution compared to other non-equilibriumeffects.

The actual diffusion charging mechanism depends on the diffusion of an ion to the particle surface, and is thus analogous tomass transfer processes, when the electrostatic interactions are negligible. Diffusion charging depends on the sticking probabilityof the ion to particle surface and on the mutual forces between the particle and the diffusing ion. The sticking probability isgenerally taken to be unity as proposed by Fuchs (1963). The rate at which the ions diffuse to the particle surface depends on themobility of the ion, Coulombic force—either attractive or repulsive—between the ion and the particle, and the attractive imageforce and van derWaals force (particularly for smaller particles). Fuchs (1963) proposed amodel to include a Coulomb and imagepotential in the calculations of ion attachment coefficients. The Fuchs model is widely accepted as it is largely consistent withexperimental observations. Hoppel and Frick (1990) showed that the Fuchs theory is not valid for particle diameters close to theionic mean free path (Hoppel & Frick, 1990; Stommel & Riebel, 2007). They used a new calculation method based on the conceptof three-body trapping and showed that for particles smaller than 10nm in diameter, the three-body trapping method predictsmuch lower fractions of charged particles compared to Fuchs theory.

These theories are applicable to spherical particles; however, in case of non-spherical particles their shape and structure mayinfluence the diffusion charging characteristics of the particles. Many studies have attempted to investigate the role of particleshape and structure on charging efficiency of aerosol particles (Chang, 1981; Matsoukas & Friedlander, 1991; Mayya, 1994;Rogak & Flagan, 1992; Wen, Reischl, & Kasper, 1984). Chang studied charging characteristics of prolate spheroids in unipolar ionatmosphere, and found that time history of charging in the continuum regime depends on the capacitance of the particle, whilethat in the free-molecule regime depends on the capacitance as well as the surface area of the aerosol particle (Chang, 1981).Wenet al. (1984) conducted theoretical and experimental investigation of long chain aggregates of spherical particles. They introducedthe concept of charging-equivalent diameter (dQE) which was defined as the diameter of spherical particle which would acquirethe same level of charge as that acquired by the non-spherical particles under similar conditions. For a conducting spheroid witha minor axis of d1 and aspect ratio of �, dQE was given by, dQE = d1�/ln(2�). Wen et al. also compared dQE of prolate spheroidscalculated from the above equation to their surface-equivalent diameter (dSE) and volume-equivalent diameter (dVE) and foundthat for elongated spheroids, dSE was closer to dQE compared to dVE. Their experimental data on fibrous aerosol particles of Fe2O3showed close agreement with the Boltzmann charge distribution.

166 P. Kulkarni et al. / Aerosol Science 40 (2009) 164 -- 179

Rogak and Flagan (1992) also investigated the effect of particle shape on the bipolar diffusion charging. Based on the analogybetween the mass-transfer process and ion diffusion charging they argued that the mass-transfer-equivalent diameter and dQEwould be of the same order of magnitude. They also noted that traditional representation of charging data as a function of volumeormass-equivalent diameter is inadequate and is sensitive to variations in particle bulk properties. Using prolate spheroidmodelcalculations, they showed that dQE could be related to neutral fraction of particles leaving the bipolar charger. Their experimentaldata showed that uncharged fractions of fractal-like TiO2 aggregateswere 5% smaller than that of sphereswith the same electricalmobility diameter (dmob) in the size range of 100–800nm size range, implying that for TiO2 agglomerates dQE∼1.1dmob.

Measurement of neutral fraction (i.e., the fraction of particles with zero electrical charge) of aerosol that is under stationarycharge equilibrium is a reliable technique of ascertaining whether the aerosol charge distribution attains the theoretical chargeequilibrium described by either Boltzmann or Fuchs equilibrium (Matsoukas & Friedlander, 1991; Rogak & Flagan, 1992). Thisapproach is relatively straightforward and robust, however one must take precautions to minimize and account for artifactsarising frommultiple charging of mobility-classified particles in DMA, particularly in case of nonspherical particles. As describedbelow, we employed this technique to experimentally investigate bipolar diffusion charging characteristics of SWCNT particlesand is described below.

3. Experimental

Controlled experimentswere performed tomeasure neutral fractions of CNT aerosol agglomerates over a range of electricmo-bility diameters from 100 to 1000nm. The experimental procedure was validated using spherical aerosol particles of (NH4)2SO4,NaCl, and diethylhexyl sebacate (DEHS). A technique was developed to account for bias introduced by multiple charging of CNTagglomerates. Measured neutral fractions were compared with conventional Boltzmann distribution to determine charging-equivalent diameter (dQE). Morphology of CNT aerosols was studied using TEM.

3.1. Experimental setup

Fig. 1 shows the experimental setup used to measure neutral fractions of CNT. CNT aerosols were generated using twodifferent techniques: (i) aerosolization of aqueous suspension of CNTs using an ultrasonic nebulizer (Model#241CST, SonozapInc., Farmingdale, NY) to generate large aerosol particles; themean droplet diameter from the ultrasonic nebulizer was estimatedto be 1.25�m, and (ii) a custom-built Couette flow apparatus to aerosolize CNT from the dry powder state. The Couette flowapparatus uses high shear forces generated by a Couette flow between two rotating cylinders to aerosolize and break downlarge aerosol agglomerates of CNT. Couette flow system is a modification of a knife-mill generator described by (Baron et al.,2008). Spherical aerosols of (NH4)2SO4 and NaCl were produced by atomization of aqueous salt solutions (model 3076, TSI Inc.,Shoreview, MN). Additionally, a condensation monodisperse aerosol generator (model 3475, TSI Inc.) was also used to generatespherical particles of DEHS.

Aerosol generated using aqueous suspensions was first dried to remove moisture using two diffusion dryers (model 3062, TSIInc.) operating in series. The resulting aerosol was then diluted to achieve desired concentration and flow rate. The conditionedaerosol was then passed through amicro-orifice uniform deposit impactor (MOUDI;model 110;MSP Corp., Shoreview,MN) stageto remove particles larger than 1�m aerodynamic diameter (stages oiled with oleic acid to minimize particle bounce). The flowrate through theMOUDI stagewas adjusted to get a size cut of 1�morbelowas desired. A large reservoir (volume = 65L)was usedimmediately downstreamof theMOUDI stage to facilitate collection of aerosol for a specified amount of time. The concentration ofthe aerosol entering the batch reservoir was kept below 103 cm−3 tominimize coagulation of the aerosol in the reservoir. Aerosolrequired for neutral fractionmeasurementswas then sampled off-line in batches from the reservoir at atmospheric pressure. Thiswas necessary to isolate the aerosol instrumentation (SMPS, CPC) from the low pressures created downstream of theMOUDI. Thereservoir also served to buffer out large fluctuations in concentration that were observed at the outlet of ultrasonic nebulizer andthe Couette flow generator. One batch was sufficient to complete one experiment at one mobility diameter. For neutral fractionmeasurement experiments, the typical residence time of aerosol in the reservoir was about 20–30min. The estimated changein the total number concentration (with Ntot = 103 cm−3) of a 100nm aerosol due to monodisperse coagulation in 30min wasestimated at 0.13%, and was therefore negligible. This ensured practically no agglomeration of the aerosol in the reservoir as itentered DMA-I. The total concentration (Ntot) of all CNT aerosols entering the reservoir was less than 103 cm−3.

The aerosol from the batch reservoir was first passed through an electrostatic precipitator (ESP-I) to remove all chargedparticles. The neutral aerosol from ESP-I was then charged using a low-intensity Po-210 bipolar charger (500�Ci, one year old).The aerosol was then classified using a differential mobility analyzer (DMA-I) to obtain near-monodisperse aerosol for neutralfraction measurement experiments. The DMA-I was operated at an aerosol flow rate of 0.6 lmin−1 and a sheath flow rate of3.4 lmin−1. The DMA-classified aerosol was further brought back to stationary charge equilibrium using a second Kr-85 bipolarcharger (model 3022; TSI Inc.; 5mCi, five years-old). The aerosol exiting this bipolar charger represented the test aerosol, theneutral fraction of whichwas to bemeasured. A second electrostatic precipitator (ESP-II) was used to remove all charged particlesfrom the test aerosol. The residence time of this precipitator was estimated to be 2.5 s for the flow rates used in the experiments.The concentration of particles exiting ESP-II was monitored using a CPC (model 3022; TSI Inc.), operating at a constant flow rateof 0.3 lmin−1. Concentration of aerosol was measured with ESP-II on (Con) and off (Coff) to obtain the neutral fraction (f′0) of


TSI 3076Atomizer

DiffusionDryer

ESP- IDMA-I

Kr-85 bipolar charger

CPC

ESP- II

Po-210 bipolarcharger

(500 µCi)

SMPS

MOUDI Reservoir

Vacuumpump

SEM/TEMsampler

Excessair

Dilutionair

MOUDI: Micro-orifice uniform depositimpactor stage with a cut-off diameterof 1µm or below

DMA: Differential Mobility Analyzer

ESP: Electrostatic precipitator

UltrasonicNebulizer

Couette flowgenerator

Neutral fraction measurement setup

Fig. 1. Experimental setup used to measure neutral fractions of SWCNT particles.

the aerosol. Size distribution of the aerosol exiting ESP-II was also measured using an SMPS (model 3936, TSI Inc.) to determinethe fraction of aerosol that acquired multiple charge (i.e., electrical charge of more than unity on one particle) in DMA-I. TheSMPS was operated at an aerosol flow rate of 0.3 lmin−1 and a sheath flow rate of 1.7 lmin−1; at this flow rate the uncertainty inmobility measurement was estimated to be about 17% for particles larger than 100nm. Estimation of multiply charged fractionin DMA-I was necessary to correct the measured neutral fractions (f′0) data for artifacts as discussed later. Electron microscopysamples of aerosol particles exiting ESP-II were also collected to investigate their morphology using a SEM/TEM sampler (InToxProducts, Moriarty, NM), which was also an electrostatic precipitator.

3.2. SWCNT particles

Two types of single-walled CNT powders from twomanufacturers were used to generate carbon nanotube aerosols: (i) CarbonSolutions, Inc., Riverside, CA (CS), and (ii) Carbon Nanotechnologies Inc., Houston, TX (CN). CS particles had a polyethylene glycolas a surface functional group that made the CNT particles hydrophilic, and had a carbonaceous purity of 80–90% and a metalcontent of 6% (Zhao, Hu, Yu, Perea, & Haddon, 2004). As reported by the manufacturer, CS nanotubes had a mean diameterof 1.4nm and lengths of nanotubes that exceeded 5�m. Aqueous dispersions of CS particles (435mg l−1) were prepared indeionized water without the aid of any surfactants. CN nanotubes (CNI, Houston, TX) were produced by the high pressureCO disproportionation process (HiPCO�) (Scott et al., 2003), and were subsequently purified by the manufacturer using acidtreatment to remove metal contaminants (Gorelik, Nikolaev, & Arepalli, 2000). CN was hydrophobic in nature, therefore a traceconcentration of Triton-X surfactant (0.05% v/v) was used to disperse them in water. According to the manufacturer's data, thediameter of nanotubes ranged from 0.7 to 2.4nm and the length exceeded 5�m. Aqueous suspensions of CS particles wereaerosolized using the ultrasonic nebulizer and the resulting aerosol is denoted by CNT-CS. Similarly, aerosols from the aqueous


suspension of CN particles produced using ultrasonic nebulizer were denoted by CNT-CN. Aerosol from dry CN particles producedusing the Couette flow generator is denoted by CNT-CF.

4. Results and discussion

4.1. Validation of experimental protocol

Experimentswere first performedwith near-spherical aerosol particles of (NH4)2SO4, NaCl, and diethylhexyl sebacate (DEHS)to validate the experimental procedure. Fig. 2 shows the concentration history plot obtained from a typical charge fractionmeasurement experiment which consisted of a sequence of operations involving turning the ESP-II on and off for periods of5–7min. Each data point in Fig. 2 represents a running average over 10 s. The inlet concentration to ESP-II fluctuated slowly withtime. To account for the fluctuating concentration, the neutral fractions (f′0) were determined from this time series data usinga procedure similar to that outlined by Rogak and Flagan (1992). In this method, three successive intervals were consideredfrom the time series data: an on state of the precipitator, a preceding off state, and the following off state. Weighted averages ofconcentrations in these three intervals were used to calculate f′0 as the ratio of concentration in on state to that in off state. ThusFig. 2 gives six such values of f′0 for six on states, which were then averaged to obtain an average neutral fraction for the mobilitydiameter under consideration. The method was shown to give reasonably accurate estimates of f′0 by Rogak and Flagan (1992).

The experimental procedure was validated using near-spherical (NH4)2SO4 and DEHS particles. Fig. 3 shows apparent neutralfractions of (NH4)2SO4 aerosols calculated using the procedure described above. The figure also showsneutral fractions calculatedfrom conventional Boltzmann distribution and Fuchs distribution obtained usingWiedensohler's approximationwhich estimatesthe distributionwith accuracy better than 1% (Wiedensohler, 1988). Themeasured neutral fractions are slightly smaller comparedto the Boltzmann and Fuchs distributions. These calculations do not account for multiple charging of particles in DMA-I, whichtends to create a bias in the measured fraction. For particles with mobility diameter larger than 100nm, the bipolar chargerlocated before DMA-I may impart multiple charges to a single particle. The degree of multiple charging will depend on the natureof particle size distribution of the aerosol entering the bipolar charger, and on the charging probability fq(dp) for charge q atdiameter dp. If the DMA-I is set to diameter dmob1

, the classified aerosol will contain singly charged particles of diameter dmob1,

and possibly doubly charged particles of diameter dmob2and triply charged particles of diameter dmob3

, and so on. The presenceof multiply charged particles in the classified aerosol at the exit of DMA-II results in a lower neutral fraction, as seen in Fig. 3. Theratio of number of multiply charged particles to that of singly charged particles (�) is given by:

� =n(dmob2

)f2(dmob2) + n(dmob3

)f3(dmob3) + · · · + n(dmobN

)fN(dmobN)

n(dmob1)f1(dmob1

)(1)

where n(dmobN) is the number of particles with diameter dmobN

having N unit charges, and fN is the probability that particle withdiameter dmobN

acquires N charges.One way to minimize multiple charging is to sample from that part of the size distribution, just to the right of the peak,

where concentration decays rapidly with diameter. This requires the ability to shift size distribution modes over a range bychanging the operating conditions of the aerosol generator. This was difficult to achieve in this study, particularly for the Couetteflow generator. Instead, two other measures were adopted to minimize the multiply charged particles: (i) using a low activity(500�Ci), one-year-old Po-210 radioactive source before DMA-I (half-life of Po-210 is 138 days). The low intensity source helps

Time (s)

0

Num

ber

conc

entr

atio

n, c

m-3

0

200

400

600

800

1000ESP-II OFF

ESP-II ON

500 1000 1500 2000 2500 3000

Fig. 2. Concentration history plot from a typical neutral fraction measurement experiment.


Diameter (nm)

100

Neu

tral

fra

ctio

n

0.1

1BoltzmannFuchs distribution Sulfate-uncorrected

700

Fig. 3. Experimentally measured neutral fractions of (NH4)2SO4 aerosols. Also shown for comparison are neutral fractions calculated from Boltzmann and Fuchscharge equilibrium theory.

Dp, mm

0

dN, c

m-3

0

20

40

60

80

100

120

140

100 200 300 400 500 600 700 800 900 1000

Fig. 4. Particle size distribution of (NH4)2SO4 aerosols showing multiply charged fraction. Singly, doubly, and triply charged peaks of dmob1 are shown. Multiplycharged fraction � is given by the ratio of two shaded areas.

to reduce the ion density in the charger, therefore minimizing the multiply charged particles. A low activity charger has beenused before to produce singly charged aerosol particles (Gupta & McMurry, 1989), and (ii) truncating part of the size distributionlarger than 1�m using an inertial impactor before the aerosol enters the batch reservoir. Assuming that aerodynamic size issame as mobility size, this truncation significantly eliminates bias due to multiple charging from the super-micron fraction of theaerosol. Effectiveness of truncation, however, will depend on the relationship between mobility and aerodynamic diameter.

Nevertheless, multiple charging from submicron fraction still needed to be accounted for. A procedure was developed toestimate the fraction of multiply charged particles using the size distribution measurements made using the SMPS locatedafter ESP-II. The size distribution of the aerosol entering ESP-II was obtained every time the precipitator was turned off duringthe neutral fraction measurement experiment. Fig. 4 shows one such size distribution spectrum for 400nm mobility-classifiedsulfate particles. As shown in the figure, singly, doubly, and triply charged peaks of 400nm particles are clearly visible. Particles


Mobility diameter (nm)102 103

Neu

tral

fra

ctio

n

0.1

1Boltzmann

Fuchs

Sulfate-uncorrected

Sulfate-corrected

DEHS- corrected

Fig. 5. Corrected neutral fractions of all SWCNT aerosols as a function of mobility diameters. Also shown are neutral fractions of aggregates of ZnO2 (Matsoukas& Friedlander, 1991), and TiO2 (Rogak & Flagan, 1992).

larger than 400nm are particles that acquired more than one charge in DMA-I. The fraction of multiply charged aerosol (�) wasdetermined from this distribution by taking the ratio of total number of particles larger than (dmob1

+ �dmob) to that smallerthan (dmob1

+ �dmob). Here dmob1is the diameter of singly charged particle with mobility Zp corresponding to voltage V in

DMA-I and �dmob is the uncertainty associated with measurement of diameter (corresponding to half-width of DMA transferfunction at diameter dmob1

). This ratio well-approximates the multiply charged fraction present in the aerosol, which in thisstudy ranged from 5% to 25% for sulfate particles and from 3% to 58% for CNT particles. Further, to simplify the calculations itwas also assumed that the entire multiply charged fraction consisted of doubly charged particles. It should be noted that thisassumption merely states that all multiply charged particles have two electrical charges; it does not ignore the presence ofparticles with higher charges. This assumption leads to a rather conservative overestimation of multiply charged fraction. Thisapproximation introduces a minor error in the measurements. For a log-normally distributed aerosol entering DMA-I that hasa geometric mean diameter of 245nm, and geometric standard deviation of 1.9, the above approximation overestimates themultiply charged fraction by about 30% when the DMA-I is set to classify 245nm. A similar assumption was made by Rogak andFlagan (1992) in their analysis.

Other potential sources of biases in the experimental data include orientation of non-spherical SWCNT agglomerates in theelectric field in the DMA. The classified mobility depends on the orientation of the particle. The orientational effects may resultin a wider distribution of mobility than that expected for spherical particles. The contribution of this factor is neglected in thisstudy.

Once the multiply charged fraction � is known, the measured neutral fractions were corrected by using a weighted average(Rogak & Flagan, 1992).

f ′0 = (1 − �)f0(dmob1) + �f0(dmob2

) (2)

where, f′0 is the apparent (uncorrected) neutral fraction obtained from the experiments, f0(dmob1) is the desired neutral fraction

of particles of size dmob1, and f0(dmob2

) is the neutral fraction of particles of size dmob2. It should be noted that particle of size

dmob2with two electrical charges has the same mobility size as that of particle of size dmob1

with one electrical charge. Theneutral fraction for particles of diameter dmob2

(i.e., f0(dmob2)) was obtained by interpolating the experimental data in Fig. 3 as

proposed by Rogak and Flagan (1992).Fig. 5 shows corrected neutral fractions of (NH4)2SO4 and DEHS aerosols, along with the uncorrected data (only for sulfate

aerosols). As shown, the effect of multiple charge correction is minor and results in a net increase of neutral fractions and betteragreement with the theoretical charge distributions. The aforementioned correction procedure could not be applied when theconjugate mobility diameters (i.e., dmob2

,dmob3, . . .) were larger than 1�m (the upper size limit using DMA-II). In this case,


Mobility diameter (nm)

102 103

Neu

tral

fra

ctio

n

0.1

1

Boltzmann

Fuchs

ZnO (Matsoukas & Friedlander, 1991)

TiO2 (Rogak & Flagan, 1992)CNT-CN

CNT-CS

CNT-CF

Fig. 6. Corrected neutral fractions of all SWCNT aerosols as a function of mobility diameters. One-sided error bars on CNT-CS data indicate the magnitude ofmultiple charge correction applied. Multiply charged fraction beyond 1�m diameter was calculated using an extrapolated size distribution entering the firstDMA. Also shown are neutral fractions of aggregates of ZnO2 (Matsoukas & Friedlander, 1991) and TiO2 (Rogak & Flagan, 1992).

multiple charge fraction � was estimated by extrapolating the aerosol size distribution entering DMA-I beyond 1�m using alognormal fit to obtain n(dmobN

) values (Eq. (1)).

4.2. Neutral fractions of SWCNT particles

Fig. 6 shows corrected neutral fractions of all SWCNT particles measured in this study, along with those of agglomeratesof near-spherical primary particles of ZnO (Matsoukas & Friedlander, 1991) and TiO2 (Rogak & Flagan, 1992) available fromprevious studies. The one-sided error bars on CNT-CS data in Fig. 6 indicate the magnitude of multiple charge correction applied.The neutral fractions of all CNT particles above 400nm in Fig. 6 are significantly lower than the Boltzmann values.

Before one examines the experimental data in Fig. 6 in greater detail, it is worth asking if the magnitude of experimentalbiases could cause the deviations of magnitude shown in Fig. 6. As discussed earlier, the multiple charging can introduce a biastowards lower neutral fractions; however, themagnitude of such a biaswill depend on the accurate knowledge of size distributionbeyond 1�m. Such a bias could be particularly important for particles � 700nm where all conjugate mobility diameters fall inthe supermicron size range. Fortunately, the CNT-CS aerosol size distribution entering the DMA-I had a broad distribution with ageometric countmean diameter of 246nm and a geometric standard variation of 1.9 (size distribution of CNT-CN aerosol was alsosimilar). The size distribution significantly tailed off at 1�m; the total particle concentration at 1�mdropped to less than 3% of thepeak concentration. The droplet size distribution from the ultrasonic nebulizer was unimodal with a mean diameter of 1.25�m,so it is safe to assume that there is no additional peak in the supermicron size range. Additionally, truncation using MOUDIalso helped to substantially reduce the fraction above 1�m. Multiple charge fractions, based on the extrapolated lognormalsize distribution beyond 1�m, for the CNT-CS particles were 4.73%, 3.8%, and 3% for 800, 900, and 1000nm mobility diametersrespectively. The corresponding corrections to neutral fractions were negligible, as evident from the one-sided error-bars inFig. 6. Similar corrections for the other two CNT particles were also negligible in this size range.

While the above observations suggest low interference from multiply charged particles in the supermicron size range, itmay still be instructive to probe the quantitative extent of the bias in case the supermicron fraction is underestimated by theextrapolated size distribution. Consider for instance, a 500nm particle whose uncharged fraction is to be determined using theexperimental setup shown in Fig. 1. In this case, dmob1

=500nm, dmob2=889nm, and dmob3

=1270nm. During a typical chargingexperiment, when the DMA-I is set to classify 500nm particles, the bias in neutral fraction will come from 889 and 1270nmparticles with two and three charges respectively that escape DMA-I (higher charged particles are neglected). If it is assumedthat the concentration of all three mobility conjugate particles is same in the mobility classified aerosol exiting DMA-I, then themeasured neutral fractions f′0 using Eq. (2) would be 0.15, which is about 20% lower than the true neutral fraction of 0.19 for the500nm particle. This bias of 20% is still much lower compared to 35% deviation from Boltzmann distribution that we observed


Table 1Mean measured neutral fractions for all CNT particles at various mobility diameters

Mobility diameter (nm) CNT-CS CNT-CN CNT-CF

f′0 �expa f′0 �exp f′0 �exp

100 0.458 4.1 0.394 3.3 – –200 0.270 0.8 0.274 1.0 – –300 – – 0.222 3.9 – –400 0.164 3.4 0.207 7.0 – –500 0.132 0.9 0.171 4.0 0.091 6.3600 0.130 3.4 0.136 9.4 0.108 16.0700 0.097 3.3 0.113 6.2 0.088 10.5800 0.079 4.9 – – 0.081 0.7900 0.071 6.9 – – 0.076 1.41000 0.063 9.0 – – 0.079 6.2

aPercent relative standard deviation associated with measurement of f′0.

Fig. 7. Morphology of CNT-CS particles from ultrasonic nebulizer at several mobility diameters: (a) 200nm; (b) 500nm and (c) 700nm.

for the 500nm CNT-CS particle in our experiments (Fig. 6). Since there is no evidence to indicate that the number concentrationrises with increasing diameter beyond 1�m, this calculation estimates a realistic upper bound for the bias introduced bymultiplycharged supermicron particles. Therefore it is safe to say that the interference from the multiply charged particles, if any, willonly partially explain the deviations observed in Fig. 6.

The observed deviations also need to be assessed in the context of measurement precision of the experimental technique usedin this work. Of particular concern is the fluctuation of aerosol concentration during the experiment which may result in largeuncertainties. Fortunately, the uncertainties associated withmeasured neutral fractions were low. Table 1 shows percent relativestandard error associated with each neutral fraction measured for all three particle types. These uncertainties were calculatedby propagating errors associated with each quantity involved in computing the neutral fractions. The uncertainties in Table 1range from 0.8% to 16%, and mainly represent the uncertainties originating from fluctuations in aerosol concentration during theexperiment. These values are comparable to those reported by Rogak and Flagan (1992) for their experimental system.

Lastly it is worth investigating if the nonequilibrium conditions, if any, in Kr-85 charger lead to deviations noted in Fig. 6.Nonequilibrium conditions created by ion depletion due to losses to particle surface and walls (Hoppel & Frick, 1990) can leadto deviation from Boltzmann or Fuchs stationary charge equilibrium. However, the effect of this non-stationary equilibrium is toincrease the neutral fractions to values higher than Boltzmann/Fuchs equilibrium values—a trend opposite to that seen in Fig. 6.

Measured neutral fractions of all SWCNT particles in this study are lower than those from Boltzmann and Fuch's stationarycharge equilibrium for mobility diameters larger than about 400nm (Fig. 6). Lower neutral fractions for SWCNT agglomeratesindicate their higher electrical charging efficiency. For CNT-CS particles, which were produced by ultrasonic nebulization, theneutral fractions were lower by approximately 30% at 500nm and by 53% at 1�m compared to that from Boltzmann distribution.Data for CNT-CN and CNT-CF also follow a similar qualitative trend. However, there was a slight difference in charging behaviorof the two types of SWCNT materials studied. CNT-CN aerosols showed lesser deviation from theoretical charge distributions,compared to CNT-CS aerosols. Neutral fractions of CNT-CN agglomerates were found to be lower by about 5% at 500nm and by32% at 1000nm. Neutral fractions of CNT-CS agglomerates were found to be lower by 22% at 400nm, and by 53% at 1000nm,indicating increased charging probabilities. It is worth noting that TiO2 agglomerates don't exhibit similar behavior in spite oftheir non-spherical, dendritic shape. The neutral fractions of TiO2 agglomerates are 5% lower compared to Boltzmann values.Rogak and Flagan considered this as a minor deviation and assumed that these agglomerates follow Boltzmann distribution(Rogak & Flagan, 1992).

Figs. 7–9 show TEM images ofmobility classified particles of all three types of CNTs studied, and exhibit a variety ofmorpholo-gies. SWCNT particle structures ranged CNT-CN aerosols to more compact and densely packed agglomerates of nanotubes for


Fig. 8. Morphology of CNT-CN particles from ultrasonic nebulizer at several mobility diameters. (a) 100nm; (b) 200nm; (c) 400nm; (d) 500nm and (e) 700nm.

Fig. 9. Morphology of CNT-CF particles from Couette flow apparatus at three mobility diameters. (a) 500nm; (b) 800nm and (c) 1000nm.

CNT-CS aerosols. Clearly, the aerosolization technique had the maximum influence on the morphological features. The nebulizednanotubes consisted of mainly nanoropes (i.e., bundles of nanotubes) packed closely in a relatively smaller volume. Nanotubesin an aerosolized droplet can reorganize and align with each other under the influence of interparticle colloidal and van derWaals forces, resulting in abundance of nanoropes. Also evaporation dynamics of a drying droplet can promote more compactstructures of nanotubes agglomerates. TEM images showed a variety of structures ranging from folded nanoropes, simple U-loops, to loops that resembled the number eight. The SWCNT particles from Couette flow system represented the other endof morphological spectrum with more open, sparse, and intricate agglomerates of nanoropes and nanotubes. Closer exami-nation of micrographs also show presence of catalyst particles (approximately 5nm in diameter) attached to nanoropes andnanotubes.

Caution must be exercised to interpret the observed deviation seen in Fig. 6 to isolate the deviation originating from en-hanced charging of SWCNT particles. We have chosen to represent neutral fractions as a function of mobility diameter, whichstrongly depends on the dynamic shape factor of SWCNT agglomerates. A SWCNT agglomerate can acquire a range of dy-namic shape factors (and hence electrical mobilities) depending on the structure of the fibrous network and its orientation in


Table 2Electrical capacitance of single-wall carbon nanotubes agglomerates calculated frommeasured neutral fractions of SWCNTparticles using themodified Boltzmannapproach

Mobility diameter (nm) Capacitance,×10−18 F

Spherical CNT-CS CNT-CF

400 22.2 36.4 –500 27.8 56.3 119.0600 33.4 58.1 83.5700 38.9 103.6 125.0800 44.5 159.0 151.0900 50.0 197.0 173.51000 55.6 256.0 –

the electrical field in the DMA. Thus a SWCNT particle, which exhibits similar electrical charging characteristics as that of aspherical particle of diameter dp, may have its mobility diameter very different from dp. Plotting neutral fractions as a func-tion of volume equivalent diameters may decouple the effect of dynamic shape factor, and could facilitate understanding therole of particle morphology on its diffusion charging characteristics. Volume-equivalent diameters were not measured in thisstudy.

To further qualitatively investigate the effect ofmorphology on charging, we performed numerical calculations using idealizednonspherical particle geometries of linear straight chains for spherical primary particles and cylindrical straight fibers. Since theconventional Boltzmann stationary charge equilibrium theory implicitly assumes spherical particle shape, it was necessary tomodify this framework to account for nonspherical particle shape. The details of themodified Boltzmann approach are presentedin Appendix A. The approach allows one to calculate charge distribution on nonspherical particles by expressing the electrostaticenergy term in the conventional Boltzmann expression as a function of particle's electrical capacitance. Capacitance of a particleindicates its ability to store electrostatic energy and is directly a function of particle shape and morphology. Calculations werefurther performed to compute neutral fractions of nonspherical particles using this modified approach for idealized geometriesof linear chains of spherical particles, and straight cylindrical fibers, and are shown in Figs. A2–A3. It should be noted that thepurpose of these calculations was only to qualitatively explore the influence of particle morphology on its diffusion chargingcharacteristics. The particle shapes used in the calculations, though fibrous, were far too idealized compared to those of SWCNTparticles studied in this work.

The neutral fractions of chains and fibers plotted as a function of their mobility diameter presented in Appendix A showdeviation from Boltzmann charge equilibrium, and confirm the qualitative experimental observation. These calculations clearlyshow that particle morphology has significant influence on its diffusion charging characteristics. Primary particle size has apronounced effect on the electrical charge distribution of linear aggregates—chains with smaller primary particle size (or fiberswith smaller fiber diameter) has lower neutral fractions. A key assumption for applicability of the proposed modified Boltzmannapproach is particle's high electrically conductivity. Single-wall CNTs have been shown to possess high electrical conductivities,exhibiting metallic behavior (Tans et al., 1997), allowing the application of modified approach.

The calculations presented in Appendix A do help to interpret the observed deviation in Fig. 6 in the context of influence ofparticle morphology on its capacitance, and in turn on its electrical charging properties. While some deviation is attributable tothe different electrical mobilities that particles of same volume can acquire depending on its dynamic shape factor, it is clearfrom the calculations that the shape of the particle also plays an important role in determining its electrical capacitance andtherefore must influence its equilibrium charge distribution. Table 2 lists electrical capacitance of SWCNT particles at differentmobility diameters calculated from the modified Boltzmann equation using the experimentally measured neutral fraction data.Also shown in the table are electrical capacitances of spherical particles. Capacitance of SWCNT particles is higher by a factor upto 4.6, indicating their high energy storing capacity.

It is also worth noting that SWCNT particles smaller than 300nm in this study did not show significant deviation fromBoltzmann and Fuch's charge equilibrium (Fig. 6). No additional experiments were performed to further probe this observa-tion. It could be surmised that the smaller particles ( < 400nm) produced by nebulization in this study could have a morecompact, non-fibrous structure with a relatively high fraction of residue material in the particle matrix. The residue par-ticles, created by complete evaporation of soluble contaminants in aqueous droplets may result in more compact particlesthat exhibit similar charging behavior as the compact spherical particles. Filtrate of aqueous suspension of CNT-CS, after dry-ing and evaporation, produced a size distribution with a mode of 250nm using the ultrasonic nebulizer. Such large residuesize can easily modify morphology of SWCNT particle matrix to a more compact structure, and may explain why no devia-tion was observed below 400nm in Fig. 6. Unfortunately the Couette flow generator, which does not suffer from interferencedue to residue particles, was not capable of producing SWCNT particles smaller than 400nm in sufficient concentration toperform reliable measurements. Therefore the hypothesized role of residue particles could not be confirmed, and it is notclear whether SWCNT particles smaller than 400nm with more open and porous structures will exhibit enhanced chargingprobabilities.


Mobility diameter (nm)

0

Cha

rgin

g-eq

uiva

lent

dia

met

er (

nm)

0

500

1000

1500

2000

2500

3000

3500CNT-CSSoot, flame top, Karasev et al. (2004)

CNT-CN

= 3.0dMOB

dQE

= 4.34dMOB

dQE

= 2.85dMOB

dQE

Spheres

200 400 600 800 1000 1200

Fig. 10. Charging-equivalent diameters of SWCNT particles calculated from experimentally measured neutral fraction data. Also shown for comparison areexperimental charging-equivalent diameters measured by Karasev et al. (2004) for soot particles.

4.3. Charging-equivalent diameters of SWCNT particles

Charging-equivalent diameter—diameter of a spherical particle that results in the same neutral fraction as the SWCNTparticle—was computed using data in Fig. 6, and are shown in Fig. 10. Karasev et al. (2004) measured charging-equivalentdiameters for soot aggregates; these measurement are also shown in the figure for comparison. They report that the charging-equivalent diameter of soot aggregates formed by propane combustion in a diffusion flame is larger than the aggregate meanmobility diameter by a factor of 3.0 for soot sampled from the top of the flame. Charging-equivalent diameters for the CNT-CSparticles in this study were found to be larger by a factor of 4.34 in the mobility size range of 400–1000nm. While those forCNT-CN were larger by a factor of 2.85 between mobility diameters of 400–700nm. Charging-equivalent diameters for CNT-CFparticles (not shown in the figure) were identical to that of CNT-CS particles in the size range 600–900nm.

5. Summary and conclusions

Bipolar charge distribution of SWCNT aerosol agglomerates were investigated experimentally by measuring the neutralfraction of the aerosol exiting a bipolar radioactive charger over a range of 100–1000nm electrical mobility diameters. Measuredneutral fractions of nanotubes agglomerates (larger than 400nm mobility diameter) were found to be lower compared to thosefrom Boltzmann stationary charge equilibrium, indicating their higher charging efficiencies for particles larger than 400nm. Theterm that accounts for the electrostatic energy in Boltzmann stationary charge equilibrium equation was modified to includeparticle's electrical capacitance,which allowed computation of neutral fractions for nonspherical particles. Numerical calculationsof idealized nonspherical geometries, such as linear chains and straight fibers, using themodified Boltzmann equation confirmedthe qualitative trend in the experimental data. The electrical capacitance of nanotubes particles deduced frommeasured neutralfractions were higher by a factor ranging from 1.6 to 4.6 compared to spherical particles of the samemobility diameter, indicatinghigh charge carrying capacity of nanotube particles. TEM images of SWCNT aerosols used in this study revealed their complexmorphologies ranging from compact to more open fibrous structures. The overall physical size of these agglomerates was foundto be larger than their mobility size.

The findings have implications in measurement of nanotube aerosols using electrical mobility techniques and other instru-ments that rely on electrical charging of particles, such as electrical low pressure impactors and diffusion charging based surfacearea monitors. The multiple charging could significantly bias the exposure levels measured using these instruments. The resultssuggest exercising caution in interpreting results from these instruments. The DMA inversion procedures developed for compactspheres may need to be modified to account for observed deviation from Boltzmann stationary charge equilibrium, especiallyfor larger particles. The charging-equivalent diameters of nanotubes aerosols in this study were found to be higher than theirmobility diameter by a factor of 2.85 to 4.34. The charging-equivalent diameters could be used to obtain a more accurate size


distribution from the mobility distribution. Whether single charging-equivalent diameter, deduced from neutral charge state,can represent all other charge states is not know.

Acknowledgments

This work was supported by funding from NIOSH (CAN 927Z2MK) and NORA program (CAN 927ZBCQ). PK thanks Dr. LeeTurkevich for the insightful discussion on the experimental results. Disclaimer: The findings and conclusions in this abstract havenot been formally disseminated by the National Institute for Occupational Safety and Health and should not be construed torepresent any agency determination or policy.

Appendix A.

We extend the conventional Boltzmann stationary charge equilibrium theory to nonspherical particles by introducing elec-trical capacitance in the regular Boltzmann expression. Consider a spherical particle of diameter dp, much larger than the ionmean free path, suspended in a bipolar ion atmosphere at temperature T. Then according to the Boltzmann charge equilibriumtheory, the fraction of particles with charge i is given by (Friedlander, 2000)

NiN0

= exp(

− �kT

)(3)

where Ni is the number concentration of particles with charge i and N0 is the number concentration of neutral particles. � isthe electrostatic potential energy of particle of size dp carrying elementary charge q ( = i×e). From elementary electrostatics,the potential energy � of an isolated single conducting sphere is given by, � = 1

2qV , where V is the electrostatic potentialat the particle surface corresponding to charge q and is given by, V = q/2��0dp. Then the electrostatic energy � is given by�= q2/4��0dp. Substituting this value of � in the Eq. (3) above, yields the widely used Boltzmann formula to calculate stationarycharge equilibrium of spherical aerosol particles in the continuum regime. The assumption of particle sphericity is implicit inthe above analysis through the use of spherical diameter dp to describe potential distribution around the sphere. A more generalapproach, one that does not require particle shape to be spherical, can be readily developed by using electrical capacitance Cp inthe above analysis. For a conducting aerosol particle, electrical capacitance indicates its ability to store electrical charge at a givenpotential. The capacitance of a spherical particle is given by Cp = 2��0dp. Capacitance is directly related to electrostatic energy �of the particle through the following expression, � = 1

2qV = q2/2Cp. Substituting this in Eq. (3) above, the modified Boltzmannexpression can be written as,

NiN0

= exp

⎛⎜⎜⎜⎝−

q2

2CpkT

⎞⎟⎟⎟⎠ (4)

The above equation gives the stationary charge equilibriumof aerosol particleswithoutmaking any specific assumption regardingtheir shape. As long as a particle's capacitance, which is directly dependent on its shape and morphology, is known the chargedistribution can be calculated from the above equation. We used the above modified Boltzmann expression to investigatetheoretically the effect of nonspherical particle shape on its diffusion charging properties using idealized geometries of straightchains (of spherical particles) and straight cylindrical fibers. It should be noted that the purpose of these calculations is toexplore the effect of morphology of aerosol particles on their stationary charge distribution, particularly to decouple the dynamicshape factor (which influences particle's mobility diameter) and the electrical capacitance (which affects electrical charging).The idealized geometries of particles used in the calculations do not represent the SWCNT particle morphologies studied in thiswork; however they can provide an insight into qualitative understanding of charging of nonspherical particles.

Charge distributions of a population of two types of particle shapes were computed: (1) linear straight chains of Np sphericalprimary particles of diameter dprim and (2) linear fibers of diameter dcyl, and length lcyl. The following steps were involved incomputing charge distribution on chains and fibers using the modified Boltzmann equation: (i) geometry of straight chain (or afiber) was first defined using suitable values for Np and dprim, (ii) capacitance of this straight chain or fiber was then computedaccording to procedures described below, (iii) electrical mobility of this straight chain or fiber was then computed by computingthe frictional drag on this model particle, (iv) fraction of particles having different charge levels were then computed usingregular Boltzmann equation, and the modified Boltzmann equation (Eq. (4)), and finally (v) neutral (or uncharged) fractions werecomputed and compared over a range of electric mobility diameter to study the effect of particle shape.

Electrical mobility diameter (dmob) of straight linear chains aggregates of known geometry was computed using the approachof Chan and Dahneke (1981) to calculate free molecular drag on the straight chain aggregate. Calculation of electrical mobilitydiameter of cylindrical fibers of known geometry was calculated using the slip correction and adjusted diameter approach ofDahneke (1973). Capacitance of linear chains was computed according to the approach outlined by Brown and Hemingway


Charge

-5

Prob

abili

ty

0.0

0.1

0.2

0.3

0.4

0.5

dmob = 785.6 nm

dmob = 785.6 nm

dmob = 69.7 nm

dmob = 69.7 nm

Np = 200, dprim = 30 mn

Np = 200, dprim = 5 nm

-4 -3 -2 -1 0 1 2 3 4 5

Fig. A1. Charge distributions of two linear chain aggregates with two primary particle diameters of 5 and 30nm with the same number of primary particles(Np = 200). The mobility diameters of shorter and the longer chain are 69.7 and 785.6nm, respectively. Also shown for comparison are charge distributions ofspherical particle of corresponding mobility diameter. Primary particle size has significant influence on charge distribution of chains (annotations of sphere andchains are not to the scale).

Mobility diameter of chain (nm)

1000100

Neu

tral

fra

ctio

n

0.1

1

Cap

acita

nce

of s

trai

ght c

hain

par

ticle

, x 1

0-18 F

1

10

100

1000

Boltzmanndprim = 10 nm

dprim = 25 nm

Np=100

Np=15

…Np

dprim

1 2…

Fig. A2. Neutral fractions and electrical capacitance of straight chains as a function of chain mobility diameter at two different primary particle sizes calculatedusing modified Boltzmann equation. At a given primary particle size, length of the chain was varied to span the mobility diameter range shown.

(1995), and that of the straight fibers was computed using an expression given by Falco, Panariello, Schettino, and Verolino(2003).

Calculations were performed for a range of geometries with varying primary particle number and size in a straight chainaggregate. Fig. 1 shows charge distribution of two linear chains with different primary particle size (dprim = 5 and 30nm) butwith the same number of primary particles (Np = 200), and that of their corresponding equivalent spherical mobility diameters.Charge distribution of chains was calculated using modified Boltzmann equation (Eq. (4)). Charge distributions of chains tend tobe wider compared to that of spherical particles with same mobility diameter. Comparing charge distributions of the two chainswith dprim = 5 and 30nm shows that as the primary particle size in a chain increases, the distribution becomes broader. It isworth noting in Fig. A1 that at large primary particle size the charge distribution of a chain is closer to that of spherical particlewith the same electrical mobility. Primary particle size has a pronounced effect on the electrical charge distribution of linearaggregates. The capacitance of a linear chain with smaller primary particle size is higher compared to that of a chain with samemobility diameter but with larger primary particle size. For instance a longer linear chainwith dprim = 5nm andNp = 100 has the


Property-equivalent diameter (nm)

102 103

Neu

tral

fra

ctio

n

0.1

1

Boltzmann

dcyl

dmobdseqdveq

Lcyl

Fig. A3. Neutral fraction of cylindrical particles with dcyl = 15nm plotted as a function of mobility-, volume-, and surface area-equivalent diameters, calculatesusing modified Boltzmann approach for random orientation of fibers. Length of fiber (Lcyl) was varied to span the mobility range shown. The deviation of neutralfraction of cylindrical particles fromBoltzmann charge equilibrium ismagnified if the neutral fractions are expressed as a function of volume-equivalent diameter,followed by surface- and mobility-equivalent diameters respectively.

same mobility as that of a much shorter chain with dprim = 29nm and Np = 3. The longer chain (dprim = 5nm) has a capacitanceof 5.6×10−18 F, while the shorter chain (dprim = 29nm) has a less than half the capacitance, equal to 2.6×10−18 F. This differencein capacitance explains the effect of primary particle size observed in Fig. A1.

Fig. A2 shows neutral fraction (on left y-axis), calculated using modified Boltzmann equation, of linear chain aerosol as afunction of mobility diameter (calculated using unit charge on the particle). Two curves corresponding to two different primaryparticle sizes (dprim = 10, 25nm), along with a Boltzmann distribution for spherical particles are shown. Also, shown in the samefigure is electrical capacitance (on right y-axis) of chain aggregates as a function of their mobility diameter. As expected, in allcases the neutral fractions decreasewith increasingmobility diameter. It is worth noting that neutral fractions of linear chains aresmaller compared to that of their mobility equivalent spherical diameter. The deviation from Boltzmann distribution increaseswith increasing mobility diameter of the straight chain. The figure also shows that the neutral fraction is a function of primaryparticle size (dprim) of a straight chain. As discussed above, dprim influences the electrical capacitance of the chain, plotted on righty-axis in Fig. A2, and therefore its charging behavior. In general, for a given primary particle size, the deviation from sphericalcapacitance increases with increasing mobility diameter. At a given mobility size, primary particle size has a pronounced effecton the capacitance of the particle, with smaller primary particle size resulting in larger capacitance. These calculations clearlydemonstrate the coupling between particle morphology, its capacitance, and diffusion charging characteristics.

It may also be worth considering the apparent deviation introduced by plotting the charging data as a function of mobilitydiameter, instead of other property-equivalent diameters, such as volume- and surface area-equivalent diameters.Mobility diam-eter depends on dynamic shape factor of the particles; two particles with different morphologies may have similar capacitances,but different mobility diameters. For example, a longer linear chain with dprim = 5nm and Np = 100 have the same capacitance(5.6×10−18 F) as that of a shorter chain with dprim = 14nm and Np = 30; however their mobility diameters are different: 48nmfor a longer chain compared to 76.2nm for a shorter chain. This shows that two nonspherical particles can have same capacitancebut different mobility diameters. Fig. A3 shows neutral fractions of cylindrical fibers as a function of mobility-, volume-, andsurface area-equivalent diameters. The figure serves to illustrate the apparent deviation caused by expressing neutral fractionsas a function of various property-equivalent diameters. It is worth noting that there is least deviation from Boltzmann chargeequilibrium when charging data are plotted as a function of mobility-equivalent diameters.

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