the effect of ion and particle losses in a diffusion charger on reaching a stationary charge...

Aerosol Science 34 (2003) 1647–1664www.elsevier.com/locate/jaerosci

The e�ect of ion and particle losses in a di�usion chargeron reaching a stationary charge distribution

Manuel Alonso∗, Francisco Jos+e Alguacil

National Center for Metallurgical Research (CSIC), Avenida Gregorio del Amo, 8, Madrid 28040, Spain

Received 29 November 2002; accepted 16 June 2003

Abstract

Bipolar charging of nanometer-sized aerosol particles in a tube containing a radioactive source has beeninvestigated theoretically. A model has been developed which accounts for di�usion losses of particles andions to the tube wall, as well as for the spatial dependency of the ion-pair generation rate. The ion generationrate pro3le along the tube axial direction as a function of the source size and of the tube length and radius hasbeen evaluated and, subsequently, used to examine the aerosol charging process. Comparative calculations werealso performed for uniform ion generation and negligible di�usion losses. In a real charger, where di�usionlosses are unavoidable, particles cannot attain a steady charge distribution. On the contrary, provided the ntproduct (ion mean concentration × mean aerosol residence time) is large enough, the number concentration ofcharged particles of a given size reaches a maximum at a certain axial location and thereafter decreases. Theextrinsic charging e9ciency (fraction of originally neutral particles which carry a net charge at the ionizeroutlet) depends in a complex manner on a number of parameters: particle size and polarity, tube length andradius, nt product, and relative aerosol-to-ion concentration.? 2003 Elsevier Ltd. All rights reserved.

Keywords: Aerosol nanoparticles; Bipolar charging; Stationary charge distribution; Non-uniform ion pro3le; Di�usionlosses

1. Introduction

Presently, the most reliable and widely used instrument for the size distribution measurement ofnanometer aerosol particles is the di�erential mobility analyzer (DMA), in which the particles, pre-viously charged, are classi3ed by means of an electric 3eld. The DMA actually gives the electricalmobility distribution of the particles. From this, one can infer the particle size distribution provided

∗ Corresponding author. Tel.: +34-91-553-8900; fax: +34-91-534-7425.E-mail address: [email protected] (M. Alonso).

0021-8502/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved.doi:10.1016/S0021-8502(03)00357-4

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1648 M. Alonso, F.J. Alguacil / Aerosol Science 34 (2003) 1647–1664

that the charge distribution on the particles is known. The device where particles are charged usuallyconsists of a container where bipolar air ions are continuously generated by an ionizing radioactivesource. Before aerosol particles are let into the charger, the ions attain a certain equilibrium con-centration n. In the charging process, aerosol particles are allowed to spend a mean residence timet “long enough” so that the resulting nt product assures attainment of their steady charging state.

The charge distribution on aerosol nanoparticles has been evaluated experimentally by a numberof research groups in the past (see, for instance, Reischl, MGakelGa, Karch, & Necid, 1996; Alonso,Kousaka, Nomura, Hashimoto, & Hashimoto, 1997b, and references therein). For these experiments,one usually employs two geometrically identical devices, one with the ionizing source (charger), theother without any source (reference or dummy unit), and the charge distribution is evaluated fromcomparison of the concentrations measured at the outlet of the two chambers. In this manner, onedoes not need to care about particle di�usion losses, because these are identical for both chambers.The thus-measured fraction of originally neutral particles which become charged upon passing theionizer is customarily referred to as charging probability (or intrinsic charging e2ciency); it canbe regarded as the intrinsic capability of particles of a given size to acquire a net charge in amedium containing an excess of bipolar ions. Particle losses to the wall will result in a lowerrelative concentration of charged particles; this will be termed extrinsic charging e2ciency—thus,extrinsic charging e9ciency is a function of the charging probability and of the particular operatingconditions and geometry of the charger.

In a real case, one must consider particle size distribution modi3cations within the charger due todi�usion losses and Brownian coagulation. While the latter can be minimized by simply diluting theaerosol (although this might result in particle detection problems downstream of the DMA), di�usionlosses of particles and ions cannot be avoided. This has, at least, two implications. First, sincethe ion loss rate is much higher than that of the particles, the relative ion-to-aerosol concentrationdecreases along the charger (though, again, this e�ect might be negligible for dilute aerosols). Second,and this is the most important point, a steady-state charge distribution on the particles can neverbe achieved and, furthermore, the relative concentration of charged particles at the ionizer outlet(extrinsic charging e9ciency) may be much smaller than the charging probability even if the meannt product is large. As a result, the usual DMA inversion procedure assuming a stationary chargingstate for all the particles is not correct and needs to be revised—this will be left for a future work.

Another point which should deserve attention is the fact that the speci3c ionization (number ofion-pairs generated per unit path length) depends on the distance to the source, so that the iongeneration rate is not uniform within the charger; hence, the equilibrium ion concentration in theabsence of aerosol particles is not a constant (n) along the charger, as usually assumed. It would bedesirable to know how the non-uniformity of ion generation a�ects the particle charging process.

The qualitative argumentation presented above has hopefully served to illustrate the nature of theproblem. The rest of this paper is devoted to the quanti3cation of its magnitude for a particular case:a tubular charger in which air ions are generated by � radiation emitted by an 241Am line source.

2. Ion generation rate pro�le in a tube charger with a line source

Past experiments of bipolar charging have been carried out using chargers with di�erent geometriesand ionizing sources (see the above-cited references). Since the speci3c ionization depends on the

M. Alonso, F.J. Alguacil / Aerosol Science 34 (2003) 1647–1664 1649

distance to the source, the computation of the ion generation rate pro3le cannot be generalized, butmust be done for each particular type of ionizing source and charger geometry.

2.1. Speci4c ionization

Most of the bipolar chargers employed in research and technical applications use � or � radiation toionize the atoms or molecules of the gas, usually air, where the particles are suspended. The speci3cionization of an � or � particle (number of ions generated per unit path length) at a given locationdepends on its energy and on the concentration of air molecules. The speci3c ionization is relativelylow near the radioactive source where the energy of the emitted particles is a maximum. As the trav-eled distance increases, the kinetic energy E of the particle is progressively reduced and it reachesa point where dE=dz and the speci3c ionization attain their maximum values. At slightly longerdistances, the number of generated ions sharply decreases and vanishes at a certain distance, calledthe stopping distance (Friedlander & Kennedy, 1955). Alpha particles follow straight trajectories andhave a clearly de3ned stopping distance. In contrast, � particles travel along non-straight trajectoriesand do not possess a sharply de3ned stopping distance; as a consequence, � particles present an en-ergy spectrum which complicates the quantitative calculations of their interactions (Coll Buti, 1990).In relation with the main objective of the present work, clearly stated in the title, the e�ect on particlecharging of the non-uniformity of the ion generation rate can be regarded as an additional aspect ofsecondary importance. For these reasons, in spite of the fact that � radiation (e.g. 85Kr) is more com-monly used in commercial chargers, we have preferred to model the charging process with � particles.After all, � ionizers (e.g. 241Am) are also widely used in research laboratories across the world.

The stopping distance depends on the initial energy of the � particles, which is usually in the rangebetween 4 and 6 MeV. The speci3c ionization for 5:5 MeV � particles as a function of the distanceto the 241Am source is shown in Fig. 1, reproduced from Kondrat’ev (1964). Here, I(z) dz is thenumber of ion-pairs generated by an � particle in a length dz around a point situated at a distance zfrom the source, and Imax is the maximum speci3c ionization which in the case of � particles withinitial energy of 5:5 MeV occurs for a distance z ≈ 3:75 cm from the source. The I(z)=Imax ratiowill be denoted as i(z). The curve plotted in Fig. 1 is valid for ionization at atmospheric pressure.

The numerical value of Imax can be estimated from the experimental data of Adachi (1988), whomeasured the ion-pair generation rate in a cylindrical charger of volume 33:8 cm3, containing asource of 241Am with an activity of 100 �Ci (= 3:7 × 106 Bq). The measured ion-pair generationrate was 2:5 × 109 cm−3 s−1; from this, the total ionization results to be 2:3 × 104 (ion-pairs per �particle), which is of the same order of magnitude as that reported by Clement and Harrison (2000)for � decay. Therefore, 2:3×104 = Imax

∫ s0 i(z) dz, where s is the stopping distance of the � particles,

that is, the distance measured from the source beyond which no ion is generated (s = 4 cm in thiscase). The integral, evaluated numerically from the curve plotted in Fig. 1, is equal to 2:26 cm.Thus, we 3nally 3nd Imax = 1:02 × 104 cm−1. This value will be used in the numerical calculationsdiscussed below.

2.2. Evaluation of the ion-pair generation rate pro4le

For the sake of simplicity, we will consider the charger to be a tube of length L and radius Rwith an 241Am source placed on the wall. We are primarily interested in the ion generation rate


0 1 2 3 4 50.0

0.2

0.4

0.6

0.8

1.0

rela

tive

spec

ific

ioni

zatio

n, I(

z)/I m

ax [

- ]

distance from source, z [cm]

Fig. 1. Relative speci3c ionization for 241Am as a function of the distance from the source (reproduced from Kondrat’ev,1964).

dξ

line source element

dx

[x2+(2R)2]1/2

R zdω

ωx

θ

Fig. 2. Tube charger with a radioactive line source element placed at the wall at x = 0, and de3nition of geometricvariables.

pro3le along the tube for which we need to compute the mean generation rates within di�erentialsections perpendicular to the tube axis. For this particular purpose, we are thus allowed to modelthe radioactive source as a line of length � parallel to the tube axis.

To start with, consider a line source element of length d� placed at x=0, as shown in Fig. 2. Leta be the source activity per unit length, so that a d� is the activity (measured in Bq, i.e. number ofdisintegrations or decays per second) of the element of length d�. � particles are emitted from thesource element in all directions with equal probability. The fraction f(!) d! of � particles leavingthe source element in a direction speci3ed by an angle between ! and ! + d! is equal to 1=4�times the corresponding solid angle:

f(!) d!= 12sin! d!: (1)


Hence, af(!) d! d� is the number of � particles per second emitted by the line source element inthe direction (!;!+d!). Eq. (1) presents a maximum for !=�=2: the maximum of alpha emissiontakes place in a direction perpendicular to the tube wall.

Consider now a di�erential cylinder of length dx placed at a distance x from the source. Only afraction �=2� of the af(!) d! d� � particles emitted in the (!;! + d!) direction will be able togenerate ions in this di�erential ring, where

�= 2 cos−1( x

2Rtan!

): (2)

The number a�f(!) d! d�=2� of � particles per unit time moving in the direction (!;!+d!) whichreach the di�erential cylinder placed at a distance x from the source have travelled a distance betweenz and dz, where z= x=cos! and dz=dx=cos!. Hence, the number of ion-pairs generated within thering by an � particle moving in the (!;!+ d!) direction can be expressed as I(x=cos!) dx=cos!,where I(x=cos!) is the speci3c ionization evaluated for z = x=cos!. Finally, the number g(x) ofion-pairs generated per unit time and unit volume in the cylindrical section placed at a distance(x; x + dx) from the line source element can be evaluated by considering all the possible directions! of �-ray emission

g(x) =1

�R2 dx

∫ !max

0a d�f(!) d!

�2�I(x=cos!)

dxcos!

= d�aImax

2�2R2

∫ !max

0tan! cos−1

( x2R

tan!)i(x=cos!) d!:

(3)

For a given x, the maximum value of the angle ! is given by

!max = cos−1

(xzmax

); (4)

where

zmax = min(s;√x2 + 4R2): (5)

The meaning of Eq. (5) is as follows:√x2 + 4R2 is the distance between the source and the upper-

most point of the circular section at x; it represents the maximum distance that an � particle hittingthe section at x can travel and, thus, it gives (by Eq. (4)) the maximum angle !max over which theintegral in Eq. (3) must be extended. However, if

√x2 + 4R2 is larger than the stopping distance s,

the integration in Eq. (3) needs not be performed for values of z larger than s, because the speci3cionization vanishes, i(z¿ s) = 0.

From Eq. (3), one can 3nally calculate the ion generation rate pro3le along the tube for a linesource of 3nite length �. If the source is centered at the middle of the tube, so that the midpointof the line source is at x = 0, then the volumetric ion generation rate (number of ion-pairs per unittime and volume) can be evaluated as

G(x) =∫ �=2

−�=2g(x − �) d�; (6)


-0.5 0.0 0.50.1

1

5

tube length, L/s = 1.0source length, λ/s = 0.25

0.10.2

0.30.5

tube radius, R/s =

ion

gene

ratio

n ra

te, G

(X)/

<G

> [

- ]

axial distance from source center, X = x/s [ - ]

Fig. 3. E�ect of tube radius on the ion generation rate pro3le.

where � is the axial coordinate of a speci3c line source element of length d�. Hence,

G(x) =aImax

2�2R2

∫ �=2

−�=2

∫ !max

0tan! cos−1

(x − �2R

tan!)i(x − �cos!

)d! d�: (7)

Similar calculations for this and other types of container geometries can be found in textbooks ofradiation dosimetry and radiological protection (Coll Buti, 1990, for one).

2.3. Examples of ion generation rate pro4le as a function of source size and tube lengthand radius

Ion-pair generation rate pro3les have been calculated for varying tube length L, tube radius R, andline source length �. In order to perform an appropriate comparison between the results obtainedfor di�erent source lengths, the activity per unit length, a, has been varied so as to have a constanttotal activity (= a�) of 100 �Ci for all the cases.

The volumetric ion-pair generation rate G(x) is plotted in Figs. 3–5 for several values of the tuberadius R at 3xed tube length L and source length �, for several values of L at 3xed R and �, and forseveral line source lengths � at 3xed L and R, respectively. Actually, the tube radius and length, andthe line source length are used in dimensionless form by referring them to the � particle stoppingdistance, s; these three parameters, R=s, L=s and �=s (and, of course, the total source activity, notvaried in this work) determine the form of the ion generation rate pro3le. The latter has been alsomade dimensionless by using the mean volumetric ion-pair generation rate, de3ned as

〈G〉 =1L

∫ L=2

−L=2G(x) dx: (8)

As seen in the plots, the ion-pair generation rate is comparatively much larger in the x-sections thatcontain the radioactive source, which is obvious if one recalls that the maximum of alpha emissionoccurs in the direction perpendicular to the source. This is specially seen in Fig. 5, where the G=〈G〉


-1.0 -0.5 0.0 0.5 1.00.1

1

5

tube radius, R/s = 0.3source length, λ/s = 0.25

0.51.0

1.52.0

tube length, L/s =

ion

gene

ratio

n ra

te, G

(X)/

<G

> [

- ]


Fig. 4. E�ect of tube length on ion generation rate pro3le.

-0.5 0.0 0.50.1

1

10

tube length, L/s = 1.0tube radius, R/s = 0.3

0.01

0.1

0.25

0.5

0.751.0

source length, λ/s =

ion

gene

ratio

n ra

te, G

(X)/

<G

> [

- ]


Fig. 5. E�ect of line source length on ion generation rate pro3le.

function has been plotted for several values of the line source length �. In the limit of very small �,the system is equivalent to that of a point source. In the other extreme, when the line source lengthequals the tube length, the ion generation rate becomes almost Pat, though not exactly Pat, becauseroughly half of the � particles emitted from source locations close to any of the tube end sectionsis directed toward the region outside the tube and cannot thus generate ions within the charger.

The ion-pair generation rate per unit volume decreases with the charger volume, as can be seenin Fig. 6, where the average ion-pair generation rate has been plotted as a function of the tuberadius for several values of the tube length. Note that the mean ion generation rate is independent


0.1 0.2 0.3 0.4 0.5108

109

1010

2.01.5

1.00.5

tube length, L/s =

mea

n io

n ge

nera

tion

rate

, <G

> [

#/cm

3 s]

tube radius, R/s [ - ]

Fig. 6. Mean ion generation rate within the charger as a function of tube length and radius.

of the source length, as long as the total source activity is maintained constant. Though the meanvolumetric generation rate 〈G〉 decreases with the charger volume, the total number of ion-pairsgenerated per unit time within the charger, �LR2〈G〉, does increase with the charger volume. This isso because, besides the e�ect of L and R2, the angle �, de3ned by Eq. (2), increases with the tuberadius, its maximum value being attained for R→ ∞.

The e�ect of the particular non-uniform ion generation rate pro3le developed above on the aerosolcharging process will be examined in the following two sections, the 3rst one for the special case inwhich di�usion losses are neglected, and the second one for the more realistic case of simultaneouscharging with ion and particle losses.

3. Charging of aerosol nanoparticles in a non-uniform bipolar ion environment withoutdi�usion losses

Let us now discuss the charging of aerosol particles in the tubular charger analyzed in the precedingsection. To simplify the treatment, we shall consider monodisperse nanometer-sized particles which,as is well known, can acquire at most one net charge of either polarity. In this special case, weare left with just four ion attachment rate coe9cients, which will be denoted as �jk , where the 3rstsuperscript refers to the ion polarity and the second one to the particle polarity (a “0” standingfor neutral). As further simpli3cations, we will neglect di�usion losses to the walls and Browniancoagulation. The number concentration of ions and particles will be made dimensionless by referringthem, respectively, to the average equilibrium ion-pair concentration existing in the charger beforethe aerosol particles are let in,

〈n〉 =

√〈G〉�

(9)


and to the total aerosol particle concentration at the charger inlet (it is assumed that all the particlesentering the tubular charger are neutral). In Eq. (9), � is the ion recombination rate constant (=1:6×10−6 cm3 s−1). The mean ion concentration at equilibrium given by Eq. (9) can be straightforwardlyobtained from Eqs. (10) or (11) (see below) applied to the special case where aerosol particles arenot present.

The dimensionless number concentrations of positive and negative ions vary along the tubeaccording to the expressions

dn+

dX=( sL

)(〈n〉t)

[�G〈G〉 − �n

+n− −(N〈n〉

)(�+0n+N 0 + �+−n+N−)

]; (10)

dn−

dX=( sL

)(〈n〉t)

[�G〈G〉 − �n

+n− −(N〈n〉

)(�−0n−N 0 + �−+n−N+)

]: (11)

In the last two equations, X = x=s is the dimensionless axial coordinate, L the tube length, N thetotal aerosol number concentration (constant within the charger, since coagulation and di�usion losseshave been neglected), and Nj is the dimensionless number concentration of particles of polarity j(j = 0 for neutral particles).

Likewise, the fractions of positive and negative particles are given by

dN+

dX=( sL

)(〈n〉t)(�+0n+N 0 − �−+n−N+); (12)

dN−

dX=( sL

)(〈n〉t)(�−0n−N 0 − �+−n+N−): (13)

Finally, the fraction of neutral particles can be determined from the aerosol balance equation

N 0 + N+ + N− = 1: (14)

In the previous section, we chose the origin of the axial coordinate at the midpoint of the radioactiveline source, mainly because the ion generation rate pro3le is symmetric about such point. For thecharging process of an aerosol Powing through the tube, the system is not symmetric any longer,and it is thus preferable to rede3ne the axial coordinate so that now the origin x = 0 is at thetube inlet. To keep the problem as simple as possible, the midpoint of the line source will be stilllocated at the center of the tube, i.e. at x=L=2 (or X =L=2s in dimensionless form). With this newnomenclature in mind, the boundary conditions for system of equations (10)–(14) are

n+ = n− = 0 at X = 0 (15)

and, because only neutral particles enter into the tube charger,

N 0 = 1 and N+ = N− = 0 at X = 0: (16)

Boundary condition (15) for ions is not strictly correct, because at the tube inlet there are actuallyions present, but it is admissible because of the following consideration. The system of di�erentialequations was solved numerically in a forward fashion using a certain QX step for integration(QX = 0:002). Ions start to be generated at exactly X = 0, that is, we assume that G(X ) = 0 for


X ¡ 0; but G(0) “starts” to a�ect the ion concentration in the next interval, i.e. at X = QX , andhence n± di�ers from 0 only for X ¿ 0. At any rate, since the integration step QX is quite small,the inaccuracy introduced by using boundary condition (15) is insigni3cant.

From Eqs. (10)–(13), one sees that particle charging is controlled by three parameters: (i) the〈n〉t product (average ion concentration × mean aerosol residence time); (ii) the total aerosol-to-ionconcentration ratio, N=〈n〉; and (iii) the tube length measured as the number of stopping distances,L=s. Though the tube radius R does not appear explicitly in the above system of equations, we mustalso take it as an additional factor inPuencing the charging process because, as we have alreadyseen, it a�ects the ion-pair generation rate pro3le G(X ) and the mean ion generation rate 〈G〉 and,hence, it also a�ects the initial equilibrium ion concentration 〈n〉. We must also consider the linesource length as another inPuencing parameter although, as before, we will keep the total activity,a�, constant for all the calculation runs. The source length � a�ects the ion generation rate pro3leG(X ), but not the mean ion generation rate 〈G〉 because a�= const.

3.1. Examples of particle and ion concentration pro4les without di6usion losses

A few calculations were performed with the above system of equations in order to examinethe e�ect of ion concentration non-uniformity on the particle charging process. The ion-to-aerosolattachment rate coe9cients �jk were calculated using Fuchs’ theory, with ion masses m+ = 150 andm− =80 amu, and ion mobilities Z+ =1:15 and Z− =1:65 cm2 V−1 s−1 (Alonso et al., 1997b). Thecharging probabilities (or intrinsic charging e9ciencies) will be denoted as f± and are given by(Alonso et al., 1997b)

f+ =R+

1 + R+ + R−; (17)

f− =R−

1 + R+ + R−; (18)

where R+ = �+0=�−+ and R− = �−0=�+− are the charging-to-discharging rate ratios for positive andnegative particles, respectively.

The charging probability f± is the quantity usually determined in past experimental works inwhich di�usion losses do not need to be considered because of the use of an additional dummychamber as a reference. Since particle penetration is the same for the charger and for the dummyunit (the particle charging state does not a�ect its di�usion loss rate), the thus de3ned chargingprobability increases with mean aerosol residence time until it reaches a certain constant value whichonly depends on particle size and polarity. This value is usually referred to as equilibrium chargingprobability or, more correctly (since a true equilibrium cannot exist (Fuchs, 1963)), steady-statecharging probability. For brevity, in the rest of this paper this stationary value will be referred tosimply as charging probability.

An example of ion and particle concentration pro3le along the tube charger is shown in Fig. 7.For clarity, only the dimensionless number concentration of negative ions and particles are plotted;the corresponding curves for positive polarity follow a similar trend except for the fact that N+ isalways smaller than N−. In this and other 3gures below, the dotted line representing the steady-state


0.0 0.2 0.4 0.6 0.8 1.010-3

10-2

10-1

100

charging probability (f-)

negative particles (N -)

free negative ions (n -)

non-uniform G: λ/s = 0.25λ/s = 1.0

uniform G:

num

ber

conc

entr

atio

n [

- ]

distance from tube inlet, X = x/s [ - ]

Fig. 7. Typical concentration pro3les of ions and particles for uniform and non-uniform ion generation rates, withoutdi�usion losses. Particle diameter, Dp =5 nm; aerosol-to-ion concentration, N=〈n〉=0:1; tube radius, R=s=0:3; tube length,L=s = 1:0; mean ion concentration × mean aerosol residence time, 〈n〉t = 106 cm−3 s.

charging probability (f− in this case) has been drawn in order to compare it, in an illustrative andeasy manner, with the relative concentration of charged particles; it does not mean, of course, thatthe transient charging probability (i.e., before steady state) is constant all along the tube. For eachspecies, there are three curves; one (full line) describes the situation in which the ion generation rateis uniform along the charger, that is, G = 〈G〉 everywhere within the tube. The discontinuous linesare for non-uniform ion generation rate using two di�erent lengths of the radioactive line source(but, as noted before, having a total activity a� of 100 �Ci in both cases). The case illustrated inFig. 7 has been run for a value of the 〈n〉t product not enough to bring the aerosol to a steady state:note that at the charger outlet, X = 1, the relative particle concentration N− is appreciably lowerthan the (steady-state) charging probability f−. As seen in the 3gure, when the ion generation rateis assumed to be uniform, the aerosol charging rate is larger, which is obvious because 〈G〉¿G(X )for practically the 3rst-half portion of the charger (see the three upper curves). The discrepancybetween the uniform and non-uniform ion generation cases becomes larger as the radioactive linesource length decreases, a result which is quite reasonable if one recalls the pro3les plotted in Fig. 5.

Fig. 8 shows a second example, which is identical to that of Fig. 7 except that now the 〈n〉tproduct has been chosen large enough so that the particles can reach a stationary charging statebefore leaving the charger. It is important to note that, although the calculated transient chargedparticle concentration, N−, di�ers according to whether one considers an uniform or a non-uniformion generation pro3le, the axial coordinate at which N− becomes equal to the charging probabilityf− is practically the same. This trend has been observed in all the calculation runs, and this givesus the possibility of analyzing the e�ect of charger geometry, source length and residence time,using the N±

1 =f± ratio, where N±

1 is the dimensionless number concentration of charged particles atthe tube outlet, i.e. at X = 1. (N±

1 is what one might call the extrinsic charging e2ciency, that is,the relative output of charged particles; when di�usion losses are neglected, the extrinsic charginge9ciency becomes equal to the charging probability if 〈n〉t is su9ciently large.)


0.0 0.2 0.4 0.6 0.8 1.010-3

10-2

10-1

100

charging probability (f -)

free negative ions (n -)

negative particles (N -)

non-uniform G: λ/s = 0.25λ/s = 1.0

uniform G:

num

ber

conc

entr

atio

n [

- ]


Fig. 8. Same as Fig. 7, except that now 〈n〉t = 5 × 106 cm−3 s.

104 105 106 10710-4

10-3

10-2

10-1

100

dp [nm] 3 5 8 12

uniform G

non-uniform G

+ p

artic

le n

o. c

onc.

at o

utle

t, N

1+/f+

[ -

]

<n>t [s/cm3]

Fig. 9. Extrinsic charging e9ciency/charging probability ratio as a function of 〈n〉t and particle size. Comparison betweenuniform and non-uniform ion generation rate. Aerosol-to-ion concentration, N=〈n〉=0:1; tube radius, R=s=0:3; tube length,L=s = 1:0; line source length, �=s = 0:25.

As an illustration of the use of the N±1 =f

± ratio in the examination of the performance of agiven charger, Fig. 9 shows the 〈n〉t product required to attain the stationary state for positive par-ticles (i.e. N+

1 =f+≈1) for di�erent particle diameters. Besides the di�erence between the uniform

and non-uniform G cases already discussed before, the main conclusion that can be drawn fromFig. 9 is that smaller particles require longer values of the 〈n〉t product. Also, when the ionizer isdesigned or operated such that the extrinsic charging e9ciency is smaller than the charging prob-ability (i.e. whenever 〈n〉t is small enough), the calculated charging e9ciency is quite di�erentdepending on whether one considers uniform or non-uniform ion generation rates.


4. Charging of aerosol nanoparticles in a non-uniform bipolar ion environment with di�usion losses

Finally, the more realistic situation in which particles and ions are being lost by di�usion to thecharger wall during the charging process will be examined, assuming laminar Pow. Since chargingequations (10)–(16) must be solved numerically using a certain QX step for integration, we mustconsider particle and ion penetration through a tube of length QX . According to the model ofGormley and Kennedy (1949), penetration is given by

P =

{0:8191 exp(−3:657�) + 0:0975 exp(−22:3�) + 0:0325 exp(−57�) for �¿ 0:0312;

1 − 2:56�2=3 + 1:2� + 0:177�4=3 for �¡ 0:0312;

(19)

� =�DQ

Q x =DstR2L

QX; (20)

where D is the particle or ion di�usion coe9cient, Q the aerosol Pow rate, s the source stoppingdistance, t the mean aerosol residence time in the charger, and R and L the radius and length ofthe tube. Eq. (19) has proved to be also valid for particle diameter as small as that of typicalair ions (Alonso, Kousaka, Hashimoto, & Hashimoto, 1997a). As an explanation of how penetra-tion is included into the model equations, consider for instance Eq. (12) for positive particles; thecorresponding di�erence equation including di�usion losses becomes

N+(X + QX ) = PN+(X ) + QX(

dN+

dX

)no loss

= PN+(X ) + QX( sL

)(〈n〉t)[�+0n+(X )N 0(X ) − �−+n−(X )N+(X )];

(21)

where P is the penetration of positive particles (equal to the penetration of negative and neutralparticles of the same diameter) through the tube section of length QX . The procedure is similarfor ions, except that P must be evaluated for their corresponding di�usion coe9cients. In turn, thelatter are easily computed from the assumed ion mobilities Z+ = 1:15 and Z− = 1:65 cm2 V−1 s−1.

The total particle balance equation (14) must also be modi3ed thus

N 0(X ) + N+(X ) + N−(X ) = PX=QX ; (22)

where P is again the particle penetration. It must be stressed that the number concentration ofparticles in whatever state of charge is made dimensionless by referring it to the total aerosolnumber concentration. When di�usion losses were neglected, as in the previous section, the totalaerosol number concentration, N , was constant all along the tube charger (Eq. (14)). When di�usionlosses are taken into account, the total aerosol number concentration decreases with X (Eq. (22));for this reason we now de3ne N as the total particle number concentration at the charger inlet.

4.1. Examples of particle and ion concentration pro4les with di6usion losses

An example of ion and particle concentration pro3les when di�usion losses are considered, isshown in Fig. 10. The case plotted here is the same as that of Fig. 8, except that in the latter


0.0 0.2 0.4 0.6 0.8 1.010-3

10-2

10-1

100

101

with diffusion losses

N+N-

n-

n+N0

f+ f-

num

ber

conc

entr

atio

n, [

- ]


Fig. 10. Typical concentration pro3les of ions and particles for non-uniform ion generation rate, when di�usion lossesare taken into account. Particle diameter, Dp = 5 nm; aerosol-to-ion concentration, N=〈n〉 = 0:1; tube radius, R=s = 0:3;tube length, L=s= 1:0; mean ion concentration × mean aerosol residence time, 〈n〉t = 5× 106 cm−3 s; line source length,�=s = 0:25.

di�usion loss was neglected. In addition, we have also included the pro3les for positive ions andparticles. Concentration of negative ions at the charger outlet is smaller than that of positive ions,because of the di�erence in electrical mobility and, hence, di�usion coe9cient. When di�usion losseswere neglected (Fig. 8), the dimensionless number concentration of negative particles at the chargeroutlet (extrinsic charging e9ciency, N−

1 ) was equal to the charging probability (f−). However, whenlosses are considered, the population of negative particles is unable to attain the “equilibrium” state:N−

1 ¡f−. In contrast, the charging e9ciency for positive particles becomes equal to the chargingprobability (N+

1 ≈f+). The reason for this di�erence resides in the unequal concentrations of positiveand negative ions due to their di�erent deposition loss rates.

Fig. 11 shows the same example as that of Fig. 10 except that now the mean aerosol residencetime in the charger is 10 times higher. In this case, the dimensionless number concentration ofpositive and negative particles at the charger outlet are both smaller than the corresponding chargingprobabilities. This is a typical example of an overdimensioned (or overdesigned) charger.

In summary, too short values of 〈n〉t prevent the particles from achieving their stationary chargedistribution. Too large 〈n〉t values result in so high di�usion losses that the extrinsic charginge9ciency is much lower than the charging probability. Therefore, for a given tube geometry, thereexists an optimum value of 〈n〉t for which the extrinsic charging e9ciency is a maximum but, ingeneral, the maximum e9ciency will be smaller than the charging probability. This will be seenmore clearly in the next subsection.

4.2. Charged particle concentration at the charger outlet (extrinsic charging e2ciency)

In this last part of the paper, we will analyze the e�ect of tube geometry, source length, aerosol-to-ion concentration, and 〈n〉t product on the relative number concentration of charged particles at


0.0 0.2 0.4 0.6 0.8 1.010-3

10-2

10-1

100

101


N-N+

f-f+

n-n+

N0

num

ber

conc

entr

atio

n [

- ]


Fig. 11. Same as Fig. 10, except that now 〈n〉t = 5 × 107 cm−3 s.

104 105 106 107 10810-3

10-2

10-1

100


particle diameter Dp [nm]

3 5 8 25 40

nega

tive

part

icle

con

c. a

t out

let,

N1- /f- [

- ]

<n>t [s/cm3]

Fig. 12. Extrinsic charging e9ciency/charging probability ratio as a function of 〈n〉t and particle size. Aerosol-to-ionconcentration, N=〈n〉 = 0:1; tube radius, R=s = 0:3; tube length, L=s = 1:0; line source length, �=s = 0:25.

the charger outlet. In order to present the plots as clear as possible, only negatively charged particleswill be considered, keeping in mind that the corresponding results for positive particles follow thesame tendency although the numerical values are di�erent. The 3gures that follow are plots of theextrinsic charging e9ciency/probability ratio for negative particles, N−

1 =f−, against the 〈n〉t product,

calculated for non-uniform ion generation rate.Fig. 12 shows the e�ect of particle size on the N−

1 =f− ratio for the conditions speci3ed in the

legend. For these speci3c conditions, the charging e9ciency is maximized, regardless the particle size,for a value of 〈n〉t of about 3×106 cm−3 s. However, the maximum value of the N−

1 =f− ratio does


104 105 106 107 10810-3

10-2

10-1

100


tube radius R/s [-]

0.1 0.2 0.3 0.5

nega

tive

part

icle

con

c. a

t out

let,

N1- /f- [

- ]

<n>t [s/cm 3]

Fig. 13. Extrinsic charging e9ciency/charging probability ratio as a function of 〈n〉t and tube radius. Particle diameter,Dp = 5 nm; aerosol-to-ion concentration, N=〈n〉 = 0:1; tube length, L=s = 1:0; line source length, �=s = 0:25.

depend on particle diameter. As an exception, curves are also presented for larger particle diameters(25 and 40 nm) as an attempt to estimate the particle size above which the e�ect of di�usion losseson charging becomes negligible. (For particles larger than about 40 nm, multiple charging cannotbe neglected (Hoppel & Frick, 1988) and, hence, the above-presented calculation model, valid forsingly charged particles, cannot be used.) For 40 nm particles, the maximum attainable extrinsiccharging e9ciency is quite close to the corresponding charging probability, but too large residencetimes in the charger also result in a progressive reduction of the extrinsic e9ciency. Therefore, atruly stationary charge distribution is unattainable even for a particle diameter as “large” as 40 nm,although for this particle size the e�ect of di�usion losses is much less drastic.

Figs. 13 and 14 show the e�ect of tube geometry. The value of 〈n〉t at which the outlet concen-tration of charged particles is maximized slightly depends on the tube radius, but not on the tubelength. Again, the maximum attainable concentration of charged particles may be much smaller thanthat inferred from the charging probability, specially if the tube charger is too narrow (small radius)or too long. Also note that, since di�usion losses cannot be avoided in practice, overdimensioningthe charger operation (e.g. too long residence time) leads to a considerable reduction of the chargedparticle output.

As regards to the line source length, �, even though it exerts a large inPuence on the ion generationrate pro3le (as was shown in Fig. 5), it has practically no e�ect on the relationship between N−

1 =f−

and 〈n〉t. For this reason, no plot is provided.Finally, we have also examined the e�ect of the relative aerosol-to-ion concentration, given by the

ratio N=〈n〉 (= total aerosol number concentration at the charger inlet/mean equilibrium ion numberconcentration before particles are fed to the charger). This is shown in Fig. 15. The N−

1 =f− vs.

〈n〉t curves for relative aerosol-to-ion concentrations down to 10−4 coincide with that shown forN=〈n〉=0:1. Departure from this curve starts to be observed for values of N=〈n〉 as high as about 50.Since no one would carry out particle charging under these unusual conditions, from a practical point


104 105 106 107 10810-3

10-2

10-1

100


tube length L/s [-]

0.5 1.0 1.5 2.0

nega

tive

part

icle

con

c. a

t out

let,

N1- /f- [

- ]

<n>t [s/cm3]

Fig. 14. Extrinsic charging e9ciency/charging probability ratio as a function of 〈n〉t and tube length. Particle diameter,Dp = 5 nm; aerosol-to-ion concentration, N=〈n〉 = 0:1; tube radius, R=s = 0:3; line source length, �=s = 0:25.

104 105 106 107 10810-3

10-2

10-1

100


ion-to-aerosolconc. N/<n> [-]

0.1 10 100

nega

tive

part

icle

con

c. a

t out

let,

N1- /f- [

- ]

<n>t [s/cm3]

Fig. 15. Extrinsic charging e9ciency/charging probability ratio as a function of 〈n〉t and aerosol-to-ion concentration.Particle diameter, Dp = 5 nm; tube radius, R=s = 0:3; tube length, L=s = 1:0; line source length, �=s = 0:25.

of view we can conclude that, in “normal” circumstances, the aerosol-to-ion relative concentrationhas no e�ect on the charged particle concentration at the charger outlet.

5. Conclusions

As a brief summary of the results discussed above, we can state that (1) di�usion losses preventachievement of a steady-state distribution of charged particles in a real charger; (2) the relative


concentration of charged particles at the charger outlet (extrinsic charging e9ciency), which is thevariable of primary interest in practice, may be much smaller than that inferred from the experimen-tally measured charging probability; (3) the magnitude of the e�ect of di�usion losses decreases asparticle size increases, but it is still noticeable for 40 nm singly charged particles; (4) the di�erencebetween extrinsic charging e9ciency and charging probability depends in a complex manner oncharger geometry, particle size, and 〈n〉t product, and this complexity makes hard to 3nd a suitablecorrelation to use in practical situations.

These results have large implications in the size distribution measurement of nanometer aerosolsby electrical mobility analyzers. For a given charger geometry, ionizing source type, and aerosolPow rate, one should calculate the theoretical extrinsic charging e9ciency as a function of particlesize in order to transform the measured electrical mobility distribution into the desired particle sizedistribution. The model calculations presented in this paper are valid for a speci3c type of ionizer,in fact the simplest one; similar models need to be developed for alternative designs.

Finally, it should be also interesting to determine the particle size range in which di�usion lossprevents the attainment of a stationary charge distribution. In order to accomplish this goal wemust reformulate the model to include multiple charging because, as we have seen, di�usion lossesdo exert a measurable e�ect on particle charging in the whole particle size range where multiplecharging is negligible (diameter less than about 40 nm).

Acknowledgements

This work was supported by Spanish Ministerio de Ciencia y Tecnolog+Ua, under Grant No.MAT2001-1659.

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the effect of ion and particle losses in a diffusion charger on reaching a stationary charge...

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