Aerosol Science 35 (2004) 707–730www.elsevier.com/locate/jaerosci
Monte-Carlo simulation of unipolar di#usion chargingfor spherical and non-spherical particles
G. Biskos∗, E. Mastorakos, N. Collings
Hopkinson Laboratory, Department of Engineering, University of Cambridge, Trumpington Street,Cambridge CB2 1PZ, UK
Received 10 August 2003; received in revised form 27 November 2003; accepted 28 November 2003
Abstract
This paper presents a 3D Monte-Carlo model that simulates di#usion charging of aerosol particles inpositive unipolar environments. Calculations are performed for Nit products up to 5 × 1012 ions m−3 s (withNi being the ion concentration and t the charging time), and particles with diameter 5–1000 nm, coveringa wide range of Knudsen numbers at atmospheric pressure. Apart from the average charge, the code allowsfor the calculation of the charge distribution which is shown to be well described by Gaussian statisticsfor monodisperse particles. Standard deviations of the charge distribution calculated with the source-and-sinkapproach show good agreement with the Monte-Carlo results. Comparison of the Monte-Carlo calculations withFuchs’ limiting-sphere theory shows good agreement for the whole size range and highlights the importanceof the image force e#ect for smaller particles. The di#usion-mobility theory of the continuum regime matchesthe simulation results for the larger particle sizes while di#erences with Fuchs’ limiting-sphere theory inthis regime are relatively small. Simulations of non-spherical particles show the power of the code to easilyhandle more complicated situations. Results of rectangular-shaped and elongated chain-aggregate particlesshow di#erent charging behaviour compared to theoretical predictions, and indicate the importance of theassumptions for the surface distribution of charges on the particle. In contrast, calculations of 3D cross-shapedaggregate particles, despite having a very irregular geometry, indicate that the spherical shape assumptionis reasonable.? 2003 Elsevier Ltd. All rights reserved.
1. Introduction
Di#usion charging has become one of the most commonly used methods for charging aerosol par-ticles. Apart from its importance in industrial applications and atmospheric physics, the phenomenon
∗ Corresponding author. Tel.: +44-1223332681; fax: +44-1223765311.E-mail address: [emailprotected] (G. Biskos).
0021-8502/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved.doi:10.1016/j.jaerosci.2003.11.010
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708 G. Biskos et al. / Aerosol Science 35 (2004) 707–730
is of signi@cant interest in aerosol-measurement science when electrical mobility analysis is used.The process can be characterised as unipolar or bipolar depending on the polarity of the ions inthe gas, and both methods are successfully used in such instruments. However, unipolar di#usioncharging proves to be more e#ective when particles are detected by electrometric methods.
In unipolar di#usion charging, aerosols are allowed to interact with an ionised gas for a certaintime, during which randomly moving ions collide with and transfer their charge to the particles.When a particle is charged, the electric @eld formed in the vicinity induces a repulsive force on theincoming ions reducing the probability of further ion–particle collisions. Absence of any externalelectric @eld has to be ensured, although the e#ect is negligible for particles with diameter less than1 �m in weak @elds (Hinds, 1999; Lawless, 1991).
Both unipolar and bipolar di#usion charging have been studied theoretically and various modelsthat describe the phenomena are available in the literature. Naturally, due to the high complexity ofthe problem, there is no unique theory for the whole range of particle sizes and di#erent modelsare often used depending on the relative size of the particles to the ionic mean free path. This, inturn, underlines the importance of the ions in the gas and their associated properties. Several studiesconclude that the most abundant ionic species found in positive unipolar aerosol chargers is in theform of hydrated protons with mean free paths ranging from 10 to 20 nm depending on the numberof water molecules on the ion (Pui, 1976).
The well-established di#usion-mobility equation is used to predict charging levels for particles inthe continuum regime (Arendt & Kallman, 1925; Pauthenier & Moreaut-Hanot, 1932; Fuchs, 1947;Bricard, 1949). For particles in the transition regime, the situation is more complicated and twoapproaches are commonly used. The @rst is the limiting-sphere theory (Fuchs, 1963), which can beconsidered as a correction to the continuum di#usion-mobility equation, while the second is basedon approximate solutions of the Boltzmann equation. Using the Knudsen iteration method, severalresearchers have provided solutions of the BGK Boltzmann approximation equation to estimateion–particle collision rates (Gentry & Brock, 1967; Gentry, 1972; Marlow & Brock, 1975; Huang,Seinfeld, & Marlow, 1990). Transition regime theories are usually extended to describe di#usioncharging in the free molecular regime. However, the controversial model proposed by White (1951),has been extensively used, although its inability to take into account the image force e#ect can resultin signi@cant errors. Models that consider the image force e#ect have been proposed by later authors(Natanson, 1960; Brock, 1969, 1970).
All the above-mentioned theories describe the evolution of the average charge on particles of aspeci@c size. The stochastic nature of the phenomena was pointed out by Boisdron and Brock (1970),who proposed use of the source-and-sink approach to calculate charge distributions on monodisperseparticles. This method, employing combination coeHcients from any di#usion charging theory de-pending on the relative size of the particle to the ionic mean free path, has become common practice.
Several experimental works have been conducted to verify the above-mentioned theories. Liu andPui (1977), and later Kirsch and Zagnit’ko (1981), presented experimental evidence that supportthe validity of the di#usion-mobility equation for the continuum regime. Adachi, Kousaka, andOkuyama (1985) found good agreement between their experimental results and Fuchs’ limiting-spheretheory for particles in the transition regime. Pui, Fruin, and McMurry (1988), comparing theirexperimental results with most of the available theories by that time, showed that the limiting-spheremodel describes well the phenomenon in the transition regime, while Marlow’s approach (Marlow& Brock, 1975) is more appropriate for particles with diameter less than 10 nm at atmospheric
G. Biskos et al. / Aerosol Science 35 (2004) 707–730 709
pressure. Adachi, David, and Pui (1992), using a novel design of a unipolar di#usion charger withminimal particle losses showed that the source-and-sink approach, employing combination coeHcientsdetermined by the limiting-sphere theory, estimates fractions of charged particles within acceptableerror levels.
More complicated phenomena encountered in di#usion charging have been investigated in otherworks. Marlow (1978a), applying the source-and-sink approach managed to incorporate the exis-tence of several ionic species in the gas. In his next paper, Marlow (1978b) considered aerosolpolydispersity and showed that the phenomenon is a#ected by the particle number density of thegas. Moreover, attempts have been made to determine charging levels for particles of arbitrary shapein a theoretical and experimental way (Laframboise & Chang, 1977; Chang, 1981; Wen, Reischl, &Kasper, 1984a, b; Yu, Wang, & Gentry, 1987; Han, Ranade, & Gentry, 1991; Han & Gentry, 1994).
The analytical theories, however, cannot capture easily aerosol polydispersity and/or particles withcomplex shapes. In an e#ort to understand better the di#usion charging process of such particles,a Monte-Carlo code has been developed and is presented in this paper. Monte-Carlo simulationshave been used successfully in modelling a wide range of transport phenomena, for example dif-fusion processes and chemical reactions in the free molecule regime (Bird, 1994), dispersion ofsolid particles in turbulent Lows (Mastorakos, McGuirk, & Taylor, 1990), and particle sintering andcoagulation (Akhtar, Lipscomb, & Pratsinis, 1994). Filippov (1993) used the Monte-Carlo approachto investigate the process of di#usion charging of spherical particles in the size range 5–80 nm. Hisresults showed good agreement with the Fuchs’ limiting-sphere theory for the transition regime andBrock’s model for the free molecular regime.
The power of Monte-Carlo simulations lies in the capability to include the micro-mechanics ofthe phenomena (ion–molecule collisions, ion and particle motion, etc.), which can then be used overa large number of realisations to compile the average behaviour. The code presented in this paperis capable of simulating aerosol di#usion charging for a wide range of Knudsen numbers and Nitproducts. Average number of charges and charge distributions are easily determined by the algorithmfor monodisperse aerosols with diameter 5–1000 nm, and Nit parameters up to 5 × 1012 ions m−3 sat atmospheric pressure. Using the ability of the code to simulate non-spherical aerosols, we presentsome preliminary calculations of arbitrary-shaped particles.
The rest of the paper is organised as follows: in Section 2 we present various theoretical di#usioncharging models that are compared later with the Monte-Carlo results, and provide a discussion onthe properties of the ions. Section 3 describes the structure of the Monte-Carlo code and highlightsthe important aspects of the phenomena, while Section 4 presents simulation results and a comparisonwith the theoretical predictions. Finally, we close with a summary of the most important conclusions.
2. Theoretical background
In this section, we present and discuss various theoretical di#usion charging models that will betested against the Monte-Carlo simulations. Various concepts and de@nitions are also introduced fol-lowed by a description of the ionic properties which both the theoretical models and the Monte-Carlocode use.
The usual approach for solving the problem of di#usion charging in unipolar ionised gases isbased on the source-and-sink theory as proposed by Boisdron and Brock (1970). According to this
710 G. Biskos et al. / Aerosol Science 35 (2004) 707–730
method, the evolution of charge distribution on monodisperse particles is given by the solution ofan in@nite set of di#erential-di#erence equations (DDEs) as follows:
dNp;0
dt= −�0Np;0Ni; (1)
dNp;1
dt= �0Np;0Ni − �1Np;1Ni;
... (2)
dNp; n
dt= �n−1Np; n−1Ni − �nNp; nNi; (3)
where Np; n is the number concentration of particles with n elementary charges, Ni the ion concentra-tion, and �n the combination coeHcient of ions with particles carrying n elementary charges, givenby
�n =JNi: (4)
Here J =dn=dt is the Lux of ions to the particle estimated by various theories discussed in Sections2.1 and 2.2. Simultaneous solution of the above system of DDEs provides the average charge andcharge distribution on particles of a speci@c diameter exposed to given Nit conditions. In practicethe number of equations one has to solve is @nite, although this has to be high compared to theexpected average number of charges.
2.1. Spherical particles
Owing to the high complexity of the problem, there is no unique theory to determine the ionicLux of ions on particles for the whole range of Knudsen numbers, so di#erent models have tobe employed depending on the relative size of the particle with the ionic mean free path, i. Thefollowing paragraphs give a brief review of the main di#usion charging theories for the di#erenttransport regimes. For the rest of the paper a denotes the radius of the particle. However, writtenas a subscript, a indicates the background gas which for all the results presented is considered tobe air at atmospheric pressure.Continuum regime (i�a): The well-established di#usion-mobility equation is used to describe
the process in the continuum regime. In a general formulation, the Lux of ions crossing a sphericalsurfaces of radius r is
J (r) = −4�r2
(Di
dNi
dr− ZiNiE(r)
); (5)
where Di and Zi are the di#usion coeHcient and electrical mobility of the ions, respectively, Ni
the ion concentration, and E the electric @eld strength at distance r from the centre of the particle.The solution of Eq. (5) has been given by many authors independently (Arendt & Kallman, 1925;Bricard, 1949; Fuchs, 1947; Pauthenier & Moreaut-Hanot, 1932) and expressed as the ion Lux to a
G. Biskos et al. / Aerosol Science 35 (2004) 707–730 711
particle of speci@c size it becomes
J =4�DiNi∫∞
a (1=r2) exp(�(r)=kT ) dr: (6)
Here k is Boltzmann’s constant, T the gas temperature and � the potential at distance r from thecentre of the particle. Including the image force the latter is described by
�(r) =∫ ∞
rF dr = KE
[ne2
r− �
a3
2r2(r2 − a2)
]: (7)
In the above equation, F is the ion–particle interaction force, a the particle radius, � = [(� − 1)=(�+ 1)]e2 the image force parameter for particles with dielectric constant � and KE = 1=4��0 with �0the vacuum permittivity. The @rst term in the parenthesis corresponds to the Coulomb force whilethe second term to the image force induced by the ion.
Estimating the integral at the denominator of Eq. (6) can be diHcult since the image force at thesurface of the particle becomes in@nite, and indeed an analytic solution of the continuum regimetheory incorporating the image force does not exist. Considering only the Coulomb force one canobtain an analytic solution of the ionic Lux to the particle as follows:
J = KE4�DiNine2
akT[exp(KE
ne2
akT
)− 1] : (8)
It can be easily demonstrated that the e#ect of the image force diminishes with particle size andbecomes negligible for particles greater than a few hundred nm, therefore excluding it from thecontinuum model is a reasonable assumption. For particles in the transition and free molecularregime, however, the image force is important since the associated potential becomes comparable tothe mean thermal energy of the ions.Transition regime (i ≈ a): As mentioned in Section 1, two widely accepted approaches ex-
ist for treating di#usion charging of aerosol particles in the transition regime. The limiting-spheretheory assumes two regions separated by an imaginary sphere concentric to the particle. Betweenthis sphere and the particle surface, motion of the ions is determined by the thermal speed andinteraction potential with the particle, while outside the sphere, this is described by the macroscopicdi#usion-mobility theory. Fuchs (1963), matching the two Luxes at the surface of the limiting-sphere,derived the following expression for the ion Lux to the particle:
J =�� Qci�2Ni exp(−�(�)=kT )
1 + exp(−�(�)=kT ) � Qci�2
4Di
∫ r∞(1=r2) exp(�(r)=kT ) dr
; (9)
where � is the probability of an ion entering the limiting-sphere to collide and transfer its charge tothe particle, Qci the mean thermal speed of the ions, Eq. (20), and � the limiting-sphere radius givenby
�=a3
2i
[(1 + i=a)5
5− (1 + 2
i =a2)(1 + i=a)3
3+
215
(1 +
2i
a2
)5=2]: (10)
In the absence of any electrical forces, the collision probability � reduces to the square of the ratioof the particle radius over the limiting-sphere radius (� = a2=�2). However, for the case of charged
712 G. Biskos et al. / Aerosol Science 35 (2004) 707–730
particles, � is calculated according to the collision parameter b of the minimum apsoidal distance.Natanson (1960) proposed that this be determined by
b2 = r2
[1 +
23kT
[�(�) − �(r)]]: (11)
Setting db2=dr=0, one can calculate the minimum collision parameter bm and the associated collisionprobability as �= b2
m=�2. Hoppel and Frick (1986) provided accurate estimations of collision proba-
bilities for di#erent particle sizes and concluded that, when only attractive encounters are considered,these are not equal to unity as originally stated by Fuchs (1963).
Gentry and Brock (1967) derived an approximate solution of the Boltzmann collision equationby the method of Knudsen iteration. Marlow and Brock (1975), following the same approach andconsidering Coulomb and image force interaction potentials, showed that the Lux of ions to a particlewith radius a is
J =�a2 QciNiE0
1 + E1=2√�E0
; (12)
where E0 and E1 are the zeroth- and @rst-order corrections to the free-molecule Lux, and =a√�=i((ma+mi)=mi). Their calculations demonstrate that the e#ect of taking into account the image
force term results in signi@cantly higher ion transfer rates for Knudsen numbers greater than one.Huang et al. (1990), pointing out a reduction mistake in this theory, and using the same technique,showed that the ionic Lux in the transition and free molecular regime is
J = �a2 QciNi
(E0 − 1
�E1
): (13)
Here, � is the relaxation time of the system given by
�=Di + Dp
a
√mi
kT: (14)
Free molecular regime (i�a): Solution of the di#usion charging problem in the free molecularregime has been based on the kinetic theory of gases, and often the transition regime theories havebeen successfully extended to describe the problem. White (1951), assuming a Boltzmann spatialdistribution of the ions around a charged particle, derived the following expression to predict themean ionic Lux on aerosol particles:
J = �a2 QciNi exp(−KE
ne2
akT
): (15)
White initially derived and compared the above equation for particles in the continuum regime, butlater on it was shown that the model is valid only for the free molecular regime (Liu, Whitby, & Yu,1967; Gentry & Brock 1967). The integrated form of the above equation is widely used and can befound in many basic textbooks (Willeke & Baron, 1993; Hinds, 1999). However, the drawback ofneglecting the image force e#ect in the above theory was pointed out later by Brock (1969, 1970).
G. Biskos et al. / Aerosol Science 35 (2004) 707–730 713
Another early attempt to describe di#usion charging of ultra@ne particles, but this time by takinginto account the image force e#ect, was made by Keefe and Nolan (1962). Considering a Maxwelliandistribution of ion velocities they derived the ion Lux to an uncharged particle as
J = �a2 QciNi
(1 +
√KE
�e2
2akT
): (16)
The term in the parenthesis in the above equation is the enhancement factor of the image force.In a later work, Keefe, Nolan, and Scott (1968) presented numerical calculations of combinationcoeHcients of ions with charged particles.
2.2. Non-spherical particles
Di#usion charging theories assuming spherical shaped particles have been widely and successfullyused to describe the process. However, several experimental studies that examine arbitrary-shapedparticles show signi@cant disagreements when compared with theoretical predictions (Vomela &Whitby, 1967; Kasper & Shaw, 1983), indicating that a more sophisticated analysis is required.
Laframboise and Chang (1977) provided one of the @rst theoretical models of di#usion chargingfor non-spherical particles. Based on the di#usion-mobility equation of the continuum regime andobtaining analytical solutions of the electric @eld around spheroidal charged particles (oblates andprolates), they derived the following equation to calculate the ion Lux to non-spherical particles:
J = 4�NiDiL
ln(2L)� exp(−�)
1 − exp(−�) + Kne�: (17)
Here L is the ratio of the polar to the equatorial diameter of the particle (L¿ 1 for prolate particles),� = KEne2 ln(2L)=akTL is the dimensionless potential on the surface of the particle, and Kne =4�Di=dp Qci ln(2L) is the e#ective Knudsen number.
Chang (1981), in an attempt to provide a simpler and much easier model to use for such particles,derived several approximation equations for the three transport regimes based on Laframboise’stheory. For the particular case of prolate particles in the transition regime the approximation modelis as follows:
� =
2[√
(1+Kn′e)2+(1+Kn′e)(1 + 2Kn′e)K3�−(1+Kn′e)]/
(1+2Kn′e) for �6 1;
K3� for �6 0:1;
(18)
where
K3�=e2
kTNiDit
(1 + Kn′e)�0; Kn′e = Kne
cos−1(1=L)
ln(L+√L2 − 1)
for prolate particles. Wen et al. (1984a), following the above-mentioned theories introduced theterm of the charging equivalent diameter (dqe =dpL=ln(2L)) and in a subsequent work (Wen et al.,1984b), approximating the aspect ratio of the particles with the number of elementary particles onchain aggregates, compared Laframboise’s theory with experimental results for the case of bipolarcharging.
714 G. Biskos et al. / Aerosol Science 35 (2004) 707–730
Table 1Di#usion charging theories compared with the Monte-Carlo simulations
Theory Reference Equation regime Particle shape Imageforce
Di#usion mobility theory Fuchs (1947)a Eq. (8) Continuum Spherical NoLimiting-sphere theory Fuchs (1963) Eq. (9) Transition Spherical YesApprox. of Boltzmann’s equation Marlow and Brock (1975) Eq. (12) Transition Spherical YesApprox. of Boltzmann’s equation Huang et al. (1990) Eq. (13) Transition Spherical YesFree molecular regime theory White (1951)b Eq. (15) Free molecular Spherical NoNon-spherical particle theory Laframboise and Chang (1977) Eq. (17) Continuum Non-sphericalc NoNon-spherical particle theory Chang (1981) Eq. (18) Transitiond Non-spherical No
aThe theory has been proposed independently by other authors (Arendt & Kallman, 1925; Pauthenier & Moreaut-Hanot1932; Bricard, 1949).
bAlthough White, 1951 originally derived the equation for the continuum regime, other researchers showed that this isonly applicable to the free molecular regime (Liu et al., 1967; Gentry & Brock, 1967).
cFor prolate and oblate spheroids.dApproximations for the other regimes (continuum and free molecular) are also given by Chang, 1981.
Table 1 gives a descriptive summary of the theories reviewed and compared with the Monte-Carloin this paper.
2.3. Ion properties
Use of any of the above models to describe di#usion charging requires information on the prop-erties of the ions. There is a substantial amount of literature on mobility measurements for ionsproduced in atmospheric or laboratory environments. Works ranging from early investigations ofionised gases by Thomson and his group in the Cavendish Laboratory (Thomson, 1896, 1898), tomost recent publications (Sakata & Okada, 1994), indicate that ion mobility depends on the chemicalcomposition of the background gas and aging of the ions. It is generally agreed that high mobilityions, once formed, undergo a clustering process resulting in heavier stable ions of lower mobility.
For the particular case of positive ions, the relative humidity of the gas is of high importancebecause water molecules lead to the formation of hydrated proton clusters (Mohnen, 1977; Kebarle,Searles, Zolla, Scarbrough, & Arshadi, 1967). Many authors have used Langevin’s theory (Langevin,1905) to determine the ionic mass based on mobility measurements. However, the theory is onlyvalid for monoatomic ions and fails for molecular or cluster ions since the e#ective ionic size andits associated collision cross-section with neutral gas molecules are diHcult to predict (BRohringeret al., 1987).
It has been common practice to use experimental data for both of these quantities. Mass andelectrical mobility of ions are usually determined by combination of mass spectrometer and drifttube measurements. Kilpatrick (1971) provided one of the most popular experimental data sets onmobilities of ions with masses ranging from 35.5 to 2122 amu. Huertas, Marty, and Fontan (1974)reported similar measurements but speci@cally for hydrated proton ions, while Meyerott and Reagan(1980), making a review of works on ionic properties, concluded that Kilpatrick’s @tted model agreeswith most ion-cluster species of mass up to a few thousands amu.
G. Biskos et al. / Aerosol Science 35 (2004) 707–730 715
Using experimental indications of the electrical mobility, we can calculate the di#usion coeHcientof the ions by the Stokes–Einstein equation
Di =kTZi
e; (19)
while the mean velocity of the speci@c ions is given by
Qci =
√8kT�mi
(20)
with mi being the mass of the ions.Once the di#usion coeHcient and mean speed of the ions is determined, the mean free path can
be calculated according to the theory of binary di#usivity. Huang and Seinfeld (1988) made a briefreview on the available models that connect binary di#usivity with the mean free path to concludethat depending on the relative magnitude of the mass of the ions mi, to that of the background gasmolecules ma, one of the following expressions should be used:
Di =
13i Qci for mi�ma;
0:5985i Qci for mi ≈ ma;3�mi32ma
i Qci for mi�ma:
(21)
However, most works on aerosol di#usion charging use the Maxwell–Chapman–Enskog theory todetermine the mean free path of the ions. According to Pui et al. (1988), this is expressed as
Di =38
(1 + �i; a)√�i
(mi + ma
mamikT)1=2
; (22)
where �i; a is a correction factor that depends on the relative mass of the two species (�i; a rangesfrom 0.016 to 0.132 for ions and molecules of equal and unequal masses, respectively). Assumingthe hydrated proton H+(H2O)6, to be the most abundant ionic species with an average mobilityZi = 1:4× 10−4 m2 V−1 s−1 (Pui, 1976), mean free paths calculated by Eqs. (21) and (22) are 13.5and 14:6 nm respectively for T = 300 K .
Another way to determine the ionic mean free path is based on information of the physical sizeof the ions. Wei and Salahub (1994) determined the geometric structure of the hydrated proton ionsby calculating the physical lengths of the bonds. According to their results, H–O bonds in the ionbecome stronger while the hydrogen bonds become weaker as the number of water molecules onthe ion increases. For the particular case of hydrated protons with six water molecules two isomersexist and their physical size, despite the non-spherical shape, is of the order of 1 nm.
In the light of this information, the mean free path of the randomly moving ions can be calcu-lated by simple considerations of the kinetic theory of binary gases (Chapman & Cowling, 1990).According to this theory, the ionic mean free path within the background gas a is
i = Qci�i; a; (23)
where �i; a is the mean time between successive collisions of the ions determined by the followingequation:
�i; a =Ni
"i; i + "i; a: (24)
716 G. Biskos et al. / Aerosol Science 35 (2004) 707–730
Here "i; i and "i; a are the ion–ion and ion–molecule collision frequencies, respectively:
"i; i =
√2
2�N 2
i #2i; i Qci; "i; a =
√2�NiNa#2
i; a Qci; a: (25)
In the above equations #i; i is the ion collision diameter, #i; a= 12(#i+#a) the binary collision diameter,
Na the background gas concentration and Qci; a is the relative speed of the ions with the gas molecules:
Qci; a =
√4kT�mi; a
; (26)
where mi; a = mima=(mi + ma) is the reduced mass of the system. Noting that ion concentrationsare signi@cantly lower than that of the background gas molecules, the binary collision frequency ismuch higher compared to the ion–ion collision frequency ("i; a�"i; i). Eliminating "i; i in Eq. (24),and substituting back to Eq. (23) we can write
i =Qci√
2�Na#2i; a Qci; a
: (27)
Substituting Eqs. (20) and (26) for the mean speeds we can rearrange to obtain
i =1
�Na#2i; a
√1 + ma=mi
: (28)
It is obvious that the mean free path of the ions is independent of their concentration and thermalspeed. At atmospheric pressure the air molecule concentration is Na ≈ 2:5 × 1025 m−3, while thediameter of air molecules is #i =0:25 nm. Using Eq. (28), calculations based on the hydrated protonsize (#i ≈ 1 nm) as estimated by Wei and Salahub (1994), corroborate that the ionic mean free pathis of the order of 15 nm.
To summarise, ionic species of unipolar ions found in aerosol chargers are dynamically changingspecies growing by a series of clustering reactions that depend on the composition of the gas. Owingto this dynamic change, such gases consist of several kinds of ions that vary in electrical mobility.In order to keep analysis of such systems simple it is a common practice to assume a single specieswith representative properties. For the particular case of the present Monte-Carlo simulations, thisassumption, apart from simplifying the algorithm, it signi@cantly reduces the required computationalpower. All calculations presented in the following sections assume a mean free path of the ions of14:5 nm.
3. The Monte-Carlo simulation
The basic concept of the Monte-Carlo algorithm is presented in Fig. 1. The simulation volumeconsists of a system of uniformly distributed particles and ions that move randomly according to theirthermal velocities. Particles, being signi@cantly more massive compared to ions, have much smallerthermal speeds; thus, considering @xed particle positions in the simulation volume is a reasonableassumption that saves computational power. The error of this assumption is inversely proportional toparticle size. However, even for the smallest particles investigated in this work (i.e. dp = 5 nm), themean speed is one order of magnitude lower compared to that of the ions if we assume a typicalparticle density of 2 g cm−3. All the results presented in this work regard particles @xed within the
G. Biskos et al. / Aerosol Science 35 (2004) 707–730 717
x
y
z
x
y
z
�
�
(a) (b)
Fig. 1. Schematic representation of the Monte-Carlo simulation code. (a) Monte-Carlo simulation volume, and(b) Ion particle system.
simulation space, although the e#ect of their random thermal motion can be investigated at the costof computational time.
Having the positions of the particles @xed in the simulation volume, great amount of computationalpower is spent to describe the random motion of the ions in the gas. This motion is fully describedby the thermal speed and mean free path of the ions. As mentioned in the previous section, ionconcentrations in unipolar environments is suHciently low that ion–ion collisions can be neglected.For this reason, we can assume that the velocity distribution of the ions is completely determinedby ion–neutral molecule interactions. Therefore, far away from charged particles (i.e. in the absenceof any external force), the ions move in straight paths between collisions. The algorithm uses asimulation time step St equal to the mean time between successive ion–neutral molecule collisionscalculated by Eq. (24). At the beginning of every time step, an ion–molecule collision occurs resultingin a new random velocity of the ion. A pseudo-random number generator produces Maxwellianthermal velocities for the ions at every time step. Ascribing new velocities at the beginning, andcalculating the new positions at the end of every time step, random-walk paths of the ions arefollowed throughout the simulation.
When an ion is close to a charged particle, its velocity is signi@cantly altered by the ion–particleinteraction force, and the associated speed distribution departs from Maxwellian. The path of the ionin this case is calculated by solving the equation of motion for the duration of St. In the vicinityof a charged particle this becomes
midvi
dt= qE; (29)
where E=−∇�(r) is the electric @eld induced by the charged particle. Solving the above equationnumerically along the ionic free Light results in the calculation of the ion trajectories close to acharged particle. Despite the curved trajectories of the ions now, their mean free path change is as-sumed to be insigni@cant, allowing for a collision with a neutral molecules at the end of every timestep. A new random velocity is ascribed to the ion and its new path calculated according to Eq. (29).
718 G. Biskos et al. / Aerosol Science 35 (2004) 707–730
L =
l l
dp
dp
dpdp
(a) (b)
(c)
1
L = dp
1
Fig. 2. Schematic layout of non-spherical particles investigated with the Monte-Carlo simulation code. (a) Rectangular-shape particle, (b) Chain aggregate, and (c) 3D Cross-shape aggregate.
Although solution of the equation of motion fully describes the ionic trajectories close to a chargedparticle, one important point regarding calculations of the repulsive forces (in other words, the electric@eld E) was raised by some preliminary simulations of non-spherical particles. All charging theoriesconsidering spherical particles assume that the total charge is located at the centre of the particle,and the induced forces are calculated accordingly. However, ionic Lux variations can be expectedwhen the location of the charge is acentric, or in more complicated cases, when the elementarycharges are distributed on the surface of the particle. Information of the charge distribution on theparticle can be diHcult to obtain, and two extreme cases can be distinguished: (a) the total chargeis located at the centre of the particle and (b) every elementary charge is located at the point onthe surface of the particle where an ion has collided.
It is obvious that the latter case results in much more complicated analysis, and in case one wantsto simulate the phenomena using Monte-Carlo techniques, the computational power required willbe large. In order to overcome this problem, we considered that every particle consists of smallerelementary spherules. This in fact is a very realistic situation since aggregates are very commonparticles. Ion trajectories are then determined based on the total induced force on the ion, calculatedfrom the centres of every elementary particle. At the end of the simulation, the total number ofcharges on the particle is the sum of the charges on every elementary particle. Two di#erent kindsof aggregates where the elementary particles were arranged in a simple line-chain and 3D cross-shapecon@gurations were investigated. Simulations of rectangular-shaped particles were also performed tocompare their charging behaviour with spherical and aggregate particles. Fig. 2 shows a diagram ofthe three types of non-spherical particles investigated in this paper.
G. Biskos et al. / Aerosol Science 35 (2004) 707–730 719
-80 -60 -40 -20 0 20 40 60 80-60
-40
-20
20
40
60
x-Distance (nm)
y-D
ista
nce
(nm
)
b
-80 -60 -40 -20 0 20 40 60 80-60
-40
-20
20
40
60
y-D
ista
nce
(nm
)
x-Distance (nm)
1
2
3
4 5
6
7
89b
(a) (b)
Fig. 3. Ion trajectories in the vicinity of singly charged spherical particles and aggregates. (a) Spherical particle, and(b) Cross-shape aggregate.
Assuming no background gas, Fig. 3 shows 2D projections of calculated ion trajectories close to50 nm spherical and aggregate particles with one elementary charge. Comparing these trajectories,we can see that ion-particle collision probabilities, apart from being a function of the collisionparameter b, also depend on the location of the charge on the particle, or to be more precise, onwhich elementary particles carry charges.
The induced repulsive force on ions close to a charged spherical particle lies along the linethat passes through the centres of the two species (ions and particles) as demonstrated in Fig. 1b.Considering that this has to be calculated and converted into rectangular coordinates for all thetime steps of integration of Eq. (29), it is obvious that these sub-calculations can take much of thecomputational power. For computational eHciency, a sphere is de@ned around the particle equal tofew times its size. When an ion is within this sphere the sub-routine that solves Eq. (29) is used.Outside this region, motion of particles is described by the random thermal velocity only.
The ion concentration is kept constant throughout the simulations by assuming periodic boundaryconditions (i.e. once an ion exits the simulation volume from one boundary, a new one enters fromthe opposite boundary in the following time step). Moreover, when an ion is attached to a particle,a new ion enters the simulation volume in a random manner from the boundaries. Summarising allthe individual steps described in this section, Fig. 4 shows the Low chart of the Monte-Carlo code.
The simulations involve statistical error. This error is inversely proportional to the square rootof the number of independent observations, or in terms of the present model, from the number ofsimulated particles. Hence, the error in the mean number of elementary charges collected by theparticles is
Error =#√Np
: (30)
Here # is the standard deviation of the charge distribution. Depending on the required accuracyand the available computational power, the simulation box length can vary from few tens to few
720 G. Biskos et al. / Aerosol Science 35 (2004) 707–730
Determine random initial positions of ions & particles
START
END
NO
NO
NOYES
YES YES
t = t + ∆t
t = 0
Apply randomvelocities to the ions
Is any ion in the vicinity of
a particle?
Has anion-particle collision
occured?
Has the simulation finished?
Add charge to the particle& insert new ion in the sim.
Calculate new positions for the ions (∆x=u × ∆t )
Calculate ion trajectory andestimate new position Eq.(29)
�t given by Eq.(24)
Fig. 4. Flow chart of the Monte-Carlo simulation code.
hundreds of �m. Results presented in this paper required simulation times of the order of weeks ona modern 2 GHz IBM-PC compatible computer for typical systems of 1000 particles.
4. Results and discussion
This section presents the results of the Monte-Carlo simulations. Calculations for spherical aerosolsand comparison with theoretical models are given @rst, followed by simulations of non-sphericalparticles that include rectangular elongated and aggregate shapes. All the theoretical models used tocompare with the Monte-Carlo results were produced by numerical solution of the source-and-sinkmethod using the di#erent ion-Lux equations as described in Section 2.
4.1. Spherical particles
The output of the simulations is given in a particle-charge versus time format. Fig. 5 shows theevolution of average charge for di#erent monodisperse aerosols when the image force e#ect is takeninto account. It is evident that calculations for larger particles agree better with the continuum model,while those of smaller particles with the limiting-sphere theory. The increasing pattern of the averagecharge is similar for all particle sizes, and in most cases agreement with the theoretical models isachieved to within 1%. This good agreement with the widely accepted theories of the continuumand transition regime builds con@dence for the present Monte-Carlo method.
Fig. 6 presents in more detail the evolution of di#erent particle species (characterised accordingto the number of charges they carry). Taking all the particles to be uncharged at the beginningof the simulation, concentrations of particles carrying 1; 2; : : : ; n elementary charges are calculated.
G. Biskos et al. / Aerosol Science 35 (2004) 707–730 721
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
2
4
6
8
10
12
14
16
Time (s)
Ave
rage
Cha
rge
Ni = 1012 ions m−3
dp = 500 nm
dp = 100 nm
dp = 1 �m
Monte-Carlo SimulationsContinuum, (Fuchs, 1947)Limiting-Sphere, (Fuchs, 1963)
Fig. 5. Evolution of the average number of charges on monodisperse spherical particles.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time (s)
Frac
tion
of P
artic
les
Source and Sink Method
Ni = 1012 ions m−3
Np,0 Monte-Carlo
Np,1 Monte-Carlo
Np,2 Monte-Carlo
Np,3 Monte-Carlo
Fig. 6. Evolution of charged particle species on monodisperse particles with diameter dp = 100 nm.
The calculations shown in this @gure are for 100 nm spherical particles and an interaction potentialthat takes into account the image force e#ect. It is clear that the Monte-Carlo simulations agreewell with the source-and-sink approach, if the limiting-sphere theory is used to estimate combinationcoeHcients.
Figs. 7 and 8 show the average charge as a function of particle size with and without taking intoaccount the image force e#ect, respectively. It is evident that White’s theory under-predicts charginglevels for particles in the free molecular and transition regimes where the image force parameteris signi@cant. For particles with diameter greater than 200 nm, White’s theory over-predicts averagecharge by as much as a factor of two. The crossing point of the Monte-Carlo data and White’stheory, however, can be di#erent depending on the Nit product.
722 G. Biskos et al. / Aerosol Science 35 (2004) 707–730
101 102 10310-3
10-2
10-1
100
101
102
Particle Diameter (nm)
Ave
rage
Cha
rge
Monte-Carlo SimulationsContinuum, (Fuchs, 1947)Limiting-Sphere, (Fuchs, 1963)BKG Approx., (Huang et al., 1990)Free Molecular, (White, 1951)Power Law Fitting
Nit = 1012 ions m−3s
Fig. 7. Average charge of spherical particles as a function of size (with image force consideration).
101 102 10310-3
10-2
10-1
100
101
102
Particle Diameter (nm)
Ave
rage
Cha
rge
Monte-Carlo SimulationsContinuum, (Fuchs, 1947)Limiting-Sphere, (Fuchs, 1963)Free Molecular, (White, 1951)Power Law Fitting
Nit = 1012 ions m−3s
Fig. 8. Average charge of spherical particles as a function of size (without image force consideration).
When the image force e#ect is ignored, the limiting-sphere theory matches White’s equation for thesmaller particle sizes. For the particular simulations shown in Figs. 7 and 8, the e#ect of the imageforce appears to be signi@cant for particles less than 100 nm, while for particles in the continuumregime it is negligible. Simulation results agree with the limiting-sphere theory for the whole rangeof particles investigated in this work. Agreement of the results with the continuum theory is betterfor particles with diameter greater than 500 nm, although due to the logarithmic scale of the graphsthis is not so obvious.
Using di#usion charging models based on Boltzmann approximation techniques, speci@callyEq. (13), seems to over-predict average charges for particles in the transition regime. The dif-ference between simulations and this model lies in the Brownian motion of the particles which is
G. Biskos et al. / Aerosol Science 35 (2004) 707–730 723
0 1 2 3 4 5 6 7 8 9 100
0.15
0.3
0.45
0.6
0.75
Number of Charges
Frac
tion
of C
harg
ed P
artic
les
dp = 50 nm dp = 100 nm
dp = 500 nm dp = 1 �m
Continuum, (Fuchs, 1947)Limiting-Sphere, (Fuchs, 1963)
0 1 2 3 4 5 6 7 8 9 100
0.15
0.3
0.45
0.6
0.75
Number of Charges
Frac
tion
of C
harg
ed P
artic
les
0 2 4 6 8 10 12 14 16 18 200
0.05
0.1
0.15
0.2
0.25
Number of Charges
Frac
tion
of C
harg
ed P
artic
les
5 7 9 11 13 15 17 19 21 23 250
0.05
0.1
0.15
0.2
0.25
Number of Charges
Frac
tion
of C
harg
ed P
artic
les
(a) (b)
(c) (d)
Fig. 9. Charge distributions on monodisperse spherical particles at Nit = 1012 ions m−3 s.
not considered by the code at present. Keeping in mind that all the simulations refer to non-movingparticles while Eq. (13) takes into consideration both ion and particle di#usivities, di#erences, asexpected, are higher for the smaller particles whose Brownian motion is more intense.
Fitting a power-law model to the Monte-Carlo data, we observe that the slope of the curveis di#erent depending on whether the image force e#ect is included in the calculations or not.Considering only Coulomb interaction potentials, a power law of the order of 1.1 @ts the data, whilefor interaction potentials determined by Eq. (7), the estimated slope is equal to 1.3 when � = e2.Slightly di#erent slopes are predicted for other Nit products.
Apart from the average charge on monodisperse particles, information on the charge distributionis equally important when di#usion charging is used in particle spectrometers prior to [emailprotected] Fig. 9 shows, the Monte-Carlo code and the source-and-sink theory indicate that the number ofcharges on monodisperse particles follow a Gaussian distribution. Combination coeHcients used forthe theoretical calculations in this @gure are estimated by the continuum and limiting-sphere models.
724 G. Biskos et al. / Aerosol Science 35 (2004) 707–730
101 102 1030
0.5
1
1.5
2
2.5
3
Particle Diameter (nm)
Stan
dard
Dev
iatio
n
Monte-Carlo SimulationsContinuum, (Fuchs, 1947)Limiting-Sphere, (Fuchs, 1963)
Ni = 1012 ions m−3
Fig. 10. Standard deviation of charge distribution on monodisperse particles as a function of their size.
It is obvious that simulations of large particles agree better with the former and results of the smallparticles with the latter. Fig. 10 shows that the standard deviations of charge distributions increaseswith particle size indicating wider charge and mobility distributions. This is captured by both thesource-and-sink theory and the Monte-Carlo calculations.
In summary, average number of charges on spherical particles predicted by the Monte-Carlo sim-ulations show good agreement with predictions of the di#usion-mobility and limiting-sphere theoriesfor the continuum and transition regime, respectively. The importance of the image force e#ect forthe smaller particles is demonstrated by comparing the results with White’s equation. The resultsindicate a change in the slope of the average-charge versus particle-diameter curve depending onthe image force consideration. Finally, the simulations show that charge on monodisperse particlesfollows Gaussian distributions, as also predicted by the source-and-sink approach.
4.2. Non-spherical particles
In an attempt to investigate the e#ect of particle shape, some initial simulations were performed forrectangular-shaped and aggregate particles. Fig. 11 shows simulations of three di#erent rectangular-shaped particles with aspect ratios 1 (square particle), 5 and 10. In order to make these particlescomparable, their dimensions were such that the total surface of all three particles was approximatelythe same. Image forces are not included in those calculations due to the shape and size of the particles(note that image force potential is comparable to the mean ion thermal energy at distances similarto their mean free path).
Despite the fact that all three particles have equal surface area, charging behaviour appears to bequite di#erent. In fact, the mean number of charges the aggregates acquire appears to be a strongerfunction of the aspect ratio rather than the physical size of the particles. The results, however,raise the question of whether assuming a uniform potential according to Coulomb’s law calculatedfrom the metacentre of the charged particle is realistic. Indeed, motion of ions near the cornersof the rectangular particles is dominated by their thermal speed, while ions near the metacentre
G. Biskos et al. / Aerosol Science 35 (2004) 707–730 725
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
2
4
6
8
10
12
14
16
18
20
Time (s)
Ave
rage
Cha
rge
dp = 113 nm, L
= 10
dp = 157 nm, L = 5
dp = 157 nm, L = 5
dp = 113 nm, L = 10
dp = 300 nm, L = 1
Monte-Carlo SimulationsLimiting-Sphere, (Fuchs, 1963)Non-Spherical, (Laframboise & Chang, 1977)Non-Spherical, (Chang, 1981)
Ni = 5 × 1012 ions m−3
Fig. 11. Average charge of rectangular-shape elongated particles in comparison with theoretical predictions of sphericaland non-spherical particles.
of the particle (where the particle charge is located) are retarded by the repulsive electric @eld.Therefore, dependence of ion Lux on the charging state of the particle decreases as the aspect ratioincreases.
Comparing results of spherical with square particles of the same size one can see the di#erentcharging pattern. For the 300 nm particle, the charging slope (in other words the ionic Lux), isslightly higher compared to the 340 nm spherical particle. For elongated particles the discrepancyis higher when compared to non-spherical theories and can be attributed to the di#erence in cal-culating the interaction potential as shown in Fig. 11. Mean numbers of charges calculated by thenon-spherical particle theories are actually reduced compared to the charged particle, indicating anopposite behaviour compared to the Monte-Carlo simulations. Predictions made with Laframboise’stheory (Laframboise & Chang, 1977) indicated that charging behaviour of the two elongated particlesare very similar while Chang’s theory (Chang, 1981) shows mean charge to be inversely proportionalwith aspect ratio.
To investigate a more realistic situation, chain aggregate particles were simulated with the Monte-Carlo code. Ten elementary spherical particles of 50 nm diameter were placed along a straight lineto form a 500 nm long aggregate (Fig. 2b). Assuming that the particles are electrically isolated fromeach other, interaction potentials of the ion-aggregate system were determined based on the totalforce on the ion (sum of forces induced by every elementary particle). Simulations performed forthis type of particles included consideration of the image force.
Fig. 12 shows the charging behaviour of such chain aggregates in comparison with a rectangular-shape elongated particles of similar dimensions (50 nm width and 500 nm length). The total numberof elementary charges on the chain aggregates appears to be approximately 10% lower compared tothe rectangular-shaped particles. Such behaviour is expected since the elementary charges are nowbetter distributed along the particle, and the e#ect of the repulsive electrostatic force can be equallyimportant at any point. However, in an elementary particle basis, the mean number of chargesis signi@cantly reduced compared to the case where 50 nm monodisperse particles are uniformly
726 G. Biskos et al. / Aerosol Science 35 (2004) 707–730
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
2
4
6
8
10
12
Time (s)
Ave
rage
Cha
rge
Chain Aggregate, Monte-CarloRectangular, Monte-CarloSpherical, Monte-CarloLimiting-Sphere, (Fuchs, 1963)Non-Spherical, (Laframboise & Chang, 1977)Non-Spherical, (Chang, 1981)
Ni = 5 × 1012 ions m−3
dp = 50 nm, L = 10
Fig. 12. Average charge of chain aggregates in comparison with rectangular-shape elongated and spherical particles.
distributed in space. Monte-Carlo simulations and theoretical predictions of 50 nm spherical particlesare also included in the @gure.
The Monte-Carlo results show signi@cant di#erences from the charging theories of non-sphericalparticles. Both theories predict mean charges on the aggregate that are 65–75% lower comparedwith the Monte-Carlo simulations. It should be mentioned, however, that Laframboise’s theory(Laframboise & Chang, 1977) is applicable to the continuum regime, and only Chang’s approx-imation (Chang, 1981) for the transition regime can be thought as comparable with the resultspresented in Fig. 12. This comparison highlights the di#erence in the distribution of the electric @eldaround a non-spherical charged particle. Theoretical models assume a uniform potential around theparticle by taking into account their aspect ratio and calculating the forces from the centre of thespheroids. On the other hand, the Monte-Carlo algorithm determines the electric @eld around theaggregates as the superposition of @elds induced by the individual elementary particles, and there-fore its characteristic is a function of the charge distribution on the aggregate. This di#erence canbe partly attributed to the limitation of the non-spherical particle theories to take into account theimage force e#ect. However, the image force e#ect compared to the surface distribution di#erenceis believed to be less important.
Simulations of elongated rectangular-shaped and chain aggregate particles highlight the e#ect ofsurface charge distribution on their charging behaviour. Taking that into account and consideringthat geometry of real-life agglomerate particles have a more arbitrary con@guration, we performedsome preliminary calculations of slightly more complicated aggregates. To form these agglomerates,elementary spherical particles of 10 nm diameter were placed in a 3D cross-shape [emailprotected] 13 elementary particles an agglomerate of 50 nm e#ective length was formed (Fig. 2c).
Fig. 13 presents simulations of the average charge evolution on these 3D cross-shape aggregateparticles. The results are compared with theoretical predictions for spherical and non-spherical par-ticles as well as with Monte-Carlo simulations of spherical particles of the size of the aggregate(50 nm). Charging behaviour of 3D cross-shaped aggregates seem very similar to those of sphericalparticles of the same size, while the limiting-sphere theory assuming spherical shapes appears to
G. Biskos et al. / Aerosol Science 35 (2004) 707–730 727
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Time (s)
Ave
rage
Cha
rge
Cross-Shape Aggregate, Monte-CarloSpherical, Monte-CarloLimiting-Sphere, (Fuchs, 1963)Non-Spherical, (Chang, 1981)
Ni = 5 × 1012 ions m−3
dp = 50 nm, L � 1
dp = 10 nm, L = 15
Fig. 13. Average charge of 3D cross-shape aggregates in comparison with Monte-Carlo simulations of spherical particlesand theoretical predictions of spherical and non-spherical particles.
interpret both cases well. Agreement with the non-spherical particle theory on the other hand is verypoor, and calculations seem to under-predict charging levels of such aggregates even at extreme as-sumptions of their aspect ratio. Using the same theory but for a primary particle diameter of 50 nmwith an aspect ration close to unity gives better agreement with the Monte-Carlo calculations.
These results indicate that depending on the con@guration of the aggregate particles, the sphericalshape assumption can be reasonable to use with existing di#usion charging theories. However, oneshould keep in mind that the most probable number of elementary charges on 50 nm particles wouldbe one as shown in Fig. 9, making the e#ect of the electrostatic repulsion less signi@cant. Also,it should be pointed out that due to the high complication of geometry de@nition of the aggregateparticles, simulations were only feasible for a system of 10 particles within the simulation volume.The associated statistical error of such a system is of the order of 20% for aerosols of this size.
To summarise, simulations of rectangular-shape and elongated chain-agglomerate particles appearedto have di#erent charging behaviour compared to theoretical predictions. This increasing di#erencewith aspect ratio of the particles shows the importance of the assumption of the surface chargedistribution, and whether considering all the charge to be located at the metacentre of the aggregatesis meaningful. Simulations of 3D cross-shape aggregate particles on the other hand, indicate that thespherical shape assumption is reasonable.
5. Conclusions
In this paper, we used the Monte-Carlo approach to determine charging levels of spherical andnon-spherical particles exposed to unipolar ions. The algorithm simulates random motion of ionsin aerosol gases, and allows for the calculation of the average charge and charge distribution onparticles for a wide range of Knudsen numbers.
Results were presented for monodisperse aerosols in the size range 5–1000 nm and Nit products ofup to 5×1012 ions m−3 s. For spherical particles comparison of the results with the di#usion-mobility
728 G. Biskos et al. / Aerosol Science 35 (2004) 707–730
equation of the continuum regime shows agreement only for the larger particles, while the limiting-sphere theory matches the simulations for the whole size range. The calculations highlight the impor-tance of the image force e#ect, especially for the smaller particle sizes, and predict di#erent slopesof the average-charge versus particle-diameter curves depending on the consideration of the imageforce e#ect.
Simulations for non-spherical particles show the power of the method to provide a better under-standing of di#usion charging phenomena. The simulations indicate that the behaviour of rectangular-shaped particles is strongly dependent on their aspect ratio. Charging behaviour of such particlesshowed signi@cant di#erences when compared with theoretical predictions. Chain aggregates alsoshowed a di#erent behaviour when compared with analytical theories of elongated spheroids, whereascharging of 3D cross-shape aggregates showed reasonably good agreement with theories assumingspherical particle shapes.
The results of non-spherical particles indicate that shape is an important parameter for di#usioncharging, and although some theories are available to describe the phenomena, the Monte-Carloapproach can provide a useful tool for studying such processes further. The Monte-Carlo algorithmpresented in this paper can be easily extended to investigate additional phenomena encountered inaerosol di#usion chargers (i.e. the e#ect of external electric @eld, initial charge on the particles,di#erent ion–particle interaction potentials, Brownian motion of particles, etc.). The method can alsohandle bipolar di#usion charging, and its power can be used to consider situations like particlepolydispersity or ions with a range of mobilities that are diHcult to treat theoretically.
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