The effect of ion and particle losses in a diffusion charger on reaching a stationary charge distribution

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

The eect of ion and particle losses in a diusion charger on 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 been investigated theoretically. A model has been developed which accounts for diusion losses of particles and ions to the tube wall, as well as for the spatial dependency of the ion-pair generation rate. The ion generation rate pro3le along the tube axial direction as a function of the source size and of the tube length and radius has been evaluated and, subsequently, used to examine the aerosol charging process. Comparative calculations were also performed for uniform ion generation and negligible diusion losses. In a real charger, where diusion losses are unavoidable, particles cannot attain a steady charge distribution. On the contrary, provided the nt product (ion mean concentration × mean aerosol residence time) is large enough, the number concentration of charged particles of a given size reaches a maximum at a certain axial location and thereafter decreases. The extrinsic charging e9ciency (fraction of originally neutral particles which carry a net charge at the ionizer outlet) depends in a complex manner on a number of parameters: particle size and polarity, tube length and radius, 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; Diusion losses

1. Introduction Presently, the most reliable and widely used instrument for the size distribution measurement of nanometer aerosol particles is the dierential mobility analyzer (DMA), in which the particles, previously charged, are classi3ed by means of an electric 3eld. The DMA actually gives the electrical mobility 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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that the charge distribution on the particles is known. The device where particles are charged usually consists of a container where bipolar air ions are continuously generated by an ionizing radioactive source. Before aerosol particles are let into the charger, the ions attain a certain equilibrium concentration n. In the charging process, aerosol particles are allowed to spend a mean residence time t “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 number of 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), the other without any source (reference or dummy unit), and the charge distribution is evaluated from comparison of the concentrations measured at the outlet of the two chambers. In this manner, one does not need to care about particle diusion losses, because these are identical for both chambers. The thus-measured fraction of originally neutral particles which become charged upon passing the ionizer is customarily referred to as charging probability (or intrinsic charging e2ciency); it can be regarded as the intrinsic capability of particles of a given size to acquire a net charge in a medium containing an excess of bipolar ions. Particle losses to the wall will result in a lower relative 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 operating conditions and geometry of the charger. In a real case, one must consider particle size distribution modi3cations within the charger due to diusion losses and Brownian coagulation. While the latter can be minimized by simply diluting the aerosol (although this might result in particle detection problems downstream of the DMA), diusion losses of particles and ions cannot be avoided. This has, at least, two implications. First, since the ion loss rate is much higher than that of the particles, the relative ion-to-aerosol concentration decreases along the charger (though, again, this eect might be negligible for dilute aerosols). Second, and this is the most important point, a steady-state charge distribution on the particles can never be 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 mean nt product is large. As a result, the usual DMA inversion procedure assuming a stationary charging state 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 of ion-pairs generated per unit path length) depends on the distance to the source, so that the ion generation rate is not uniform within the charger; hence, the equilibrium ion concentration in the absence of aerosol particles is not a constant (n) along the charger, as usually assumed. It would be desirable to know how the non-uniformity of ion generation aects the particle charging process. The qualitative argumentation presented above has hopefully served to illustrate the nature of the problem. 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 241 Am line source. 2. Ion generation rate prole in a tube charger with a line source Past experiments of bipolar charging have been carried out using chargers with dierent geometries and ionizing sources (see the above-cited references). Since the speci3c ionization depends on the

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distance to the source, the computation of the ion generation rate pro3le cannot be generalized, but must 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 to ionize the atoms or molecules of the gas, usually air, where the particles are suspended. The speci3c ionization of an  or  particle (number of ions generated per unit path length) at a given location depends on its energy and on the concentration of air molecules. The speci3c ionization is relatively low near the radioactive source where the energy of the emitted particles is a maximum. As the traveled distance increases, the kinetic energy E of the particle is progressively reduced and it reaches a point where dE=d z and the speci3c ionization attain their maximum values. At slightly longer distances, the number of generated ions sharply decreases and vanishes at a certain distance, called the stopping distance (Friedlander & Kennedy, 1955). Alpha particles follow straight trajectories and have a clearly de3ned stopping distance. In contrast,  particles travel along non-straight trajectories and do not possess a sharply de3ned stopping distance; as a consequence,  particles present an energy 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 eect on particle charging of the non-uniformity of the ion generation rate can be regarded as an additional aspect of secondary importance. For these reasons, in spite of the fact that  radiation (e.g. 85 Kr) is more commonly 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 range between 4 and 6 MeV. The speci3c ionization for 5:5 MeV  particles as a function of the distance to the 241 Am source is shown in Fig. 1, reproduced from Kondrat’ev (1964). Here, I (z) d z is the number of ion-pairs generated by an  particle in a length d z around a point situated at a distance z from the source, and Imax is the maximum speci3c ionization which in the case of  particles with initial energy of 5:5 MeV occurs for a distance z ≈ 3:75 cm from the source. The I (z)=Imax ratio will 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), who measured the ion-pair generation rate in a cylindrical charger of volume 33:8 cm3 , containing a source of 241 Am with an activity of 100 Ci (= 3:7 × 106 Bq). The measured ion-pair generation rate 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) s for  decay. Therefore, 2:3 × 104 = Imax 0 i(z) d z, 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 this case). 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 calculations discussed 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 R with an 241 Am source placed on the wall. We are primarily interested in the ion generation rate

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relative specific ionization, I(z)/Imax [ - ]

1.0

0.8

0.6

0.4

0.2

0.0 0

1

2

3

4

5

distance from source, z [cm]

Fig. 1. Relative speci3c ionization for 1964).

241

Am as a function of the distance from the source (reproduced from Kondrat’ev,

dx [x 2+(2R)2]1/2

R

dξ

dω z

θ

ω x

line source element

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

pro3le along the tube for which we need to compute the mean generation rates within dierential sections perpendicular to the tube axis. For this particular purpose, we are thus allowed to model the 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. Let a be the source activity per unit length, so that a d is the activity (measured in Bq, i.e. number of disintegrations or decays per second) of the element of length d.  particles are emitted from the source element in all directions with equal probability. The fraction f(!) d! of  particles leaving the source element in a direction speci3ed by an angle between ! and ! + d! is equal to 1=4 times the corresponding solid angle: f(!) d! = 12 sin ! d!:

(1)

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Hence, af(!) d! d is the number of  particles per second emitted by the line source element in the direction (!; ! + d!). Eq. (1) presents a maximum for ! = =2: the maximum of alpha emission takes place in a direction perpendicular to the tube wall. Consider now a dierential cylinder of length d x placed at a distance x from the source. Only a fraction =2 of the af(!) d! d  particles emitted in the (!; ! + d!) direction will be able to generate ions in this dierential ring, where   x  = 2 cos−1 tan ! : (2) 2R The number af(!) d! d=2 of  particles per unit time moving in the direction (!; !+d!) which reach the dierential cylinder placed at a distance x from the source have travelled a distance between z and d z, where z = x=cos ! and d z = d x=cos !. Hence, the number of ion-pairs generated within the ring by an  particle moving in the (!; ! + d!) direction can be expressed as I (x=cos !) d x=cos !, where I (x=cos !) is the speci3c ionization evaluated for z = x=cos !. Finally, the number g(x) of ion-pairs generated per unit time and unit volume in the cylindrical section placed at a distance (x; x + d x) from the line source element can be evaluated by considering all the possible directions ! of -ray emission  !max 1 dx  g(x) = I (x=cos !) a df(!) d! 2 R d x 0 2 cos ! (3)  !max   aImax x −1 tan ! i(x=cos !) d!: = d 2 2 tan ! cos 2 R 0 2R For a given x, the maximum value of the angle ! is given by   x !max = cos−1 ; zmax where zmax = min(s;



x2 + 4R2 ):

√

(4)

(5)

The meaning of Eq. (5) is as follows: x2 + 4R2 is the distance between the source and the uppermost point of the circular section at x; it represents the maximum distance that an  particle hitting the section at x can travel and, thus, it gives (by Eq. √ (4)) the maximum angle !max over which the integral 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 speci3c ionization vanishes, i(z ¿ s) = 0. From Eq. (3), one can 3nally calculate the ion generation rate pro3le along the tube for a line source of 3nite length . If the source is centered at the middle of the tube, so that the midpoint of the line source is at x = 0, then the volumetric ion generation rate (number of ion-pairs per unit time and volume) can be evaluated as  =2 G(x) = g(x − ) d; (6) −=2

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ion generation rate, G(X)/ [ - ]

5

tube radius, R/s = 0.5 0.3 0.2 0.1

1

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

0.1 -0.5

0.0

0.5

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

Fig. 3. Eect of tube radius on the ion generation rate pro3le.

where  is the axial coordinate of a speci3c line source element of length d. Hence,      =2  !max x− aImax −1 x −  tan ! i d! d: G(x) = 2 2 tan ! cos 2 R −=2 0 2R cos !

(7)

Similar calculations for this and other types of container geometries can be found in textbooks of radiation 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 length and radius Ion-pair generation rate pro3les have been calculated for varying tube length L, tube radius R, and line source length . In order to perform an appropriate comparison between the results obtained for dierent source lengths, the activity per unit length, a, has been varied so as to have a constant total 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 tube radius R at 3xed tube length L and source length , for several values of L at 3xed R and , and for several line source lengths  at 3xed L and R, respectively. Actually, the tube radius and length, and the line source length are used in dimensionless form by referring them to the  particle stopping distance, s; these three parameters, R=s, L=s and =s (and, of course, the total source activity, not varied in this work) determine the form of the ion generation rate pro3le. The latter has been also made dimensionless by using the mean volumetric ion-pair generation rate, de3ned as  1 L=2 G = G(x) d x: (8) L −L=2 As seen in the plots, the ion-pair generation rate is comparatively much larger in the x-sections that contain the radioactive source, which is obvious if one recalls that the maximum of alpha emission occurs in the direction perpendicular to the source. This is specially seen in Fig. 5, where the G=G

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ion generation rate, G(X)/ [ - ]

5

tube length, L/s = 2.0 1.5 1.0 0.5

1

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

0.1 -1.0

-0.5

0.0

0.5

1.0

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

Fig. 4. Eect of tube length on ion generation rate pro3le.

ion generation rate, G(X)/ [ - ]

10 source length, λ/s = 0.01 0.1 0.25 0.5 0.75 1.0

1

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

0.1 -0.5

0.0

0.5

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

Fig. 5. Eect 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 length equals the tube length, the ion generation rate becomes almost Pat, though not exactly Pat, because roughly half of the  particles emitted from source locations close to any of the tube end sections is 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 seen in Fig. 6, where the average ion-pair generation rate has been plotted as a function of the tube radius for several values of the tube length. Note that the mean ion generation rate is independent

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mean ion generation rate, [#/cm3 s]

10

10

tube length, L/s = 0.5 1.0

10

10

1.5

9

2.0

8

0.1

0.2

0.3

0.4

0.5

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 mean volumetric generation rate G decreases with the charger volume, the total number of ion-pairs generated per unit time within the charger, LR2 G, does increase with the charger volume. This is so because, besides the eect of L and R2 , the angle , de3ned by Eq. (2), increases with the tube radius, its maximum value being attained for R → ∞. The eect of the particular non-uniform ion generation rate pro3le developed above on the aerosol charging process will be examined in the following two sections, the 3rst one for the special case in which diusion losses are neglected, and the second one for the more realistic case of simultaneous charging with ion and particle losses.

3. Charging of aerosol nanoparticles in a non-uniform bipolar ion environment without diusion losses Let us now discuss the charging of aerosol particles in the tubular charger analyzed in the preceding section. 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, we are left with just four ion attachment rate coe9cients, which will be denoted as jk , where the 3rst superscript refers to the ion polarity and the second one to the particle polarity (a “0” standing for neutral). As further simpli3cations, we will neglect diusion losses to the walls and Brownian coagulation. The number concentration of ions and particles will be made dimensionless by referring them, respectively, to the average equilibrium ion-pair concentration existing in the charger before the aerosol particles are let in,  G n = (9) 

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and to the total aerosol particle concentration at the charger inlet (it is assumed that all the particles entering 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 straightforwardly obtained from Eqs. (10) or (11) (see below) applied to the special case where aerosol particles are not present. The dimensionless number concentrations of positive and negative ions vary along the tube according to the expressions  

G dn+  s  N + − +0 + 0 +− + − = (nt)  − n n − ( n N +  n N ) ; (10) dX L G n  

G N dn−  s  = (nt)  − n+ n− − (−0 n− N 0 + −+ n− N + ) : (11) dX L G n In the last two equations, X = x=s is the dimensionless axial coordinate, L the tube length, N the total aerosol number concentration (constant within the charger, since coagulation and diusion losses have been neglected), and N j 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 +  s  (nt)(+0 n+ N 0 − −+ n− N + ); (12) = dX L dN −  s  = (nt)(−0 n− N 0 − +− n+ N − ): (13) dX L 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 radioactive line source, mainly because the ion generation rate pro3le is symmetric about such point. For the charging 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 the tube inlet. To keep the problem as simple as possible, the midpoint of the line source will be still located at the center of the tube, i.e. at x = L=2 (or X = L=2s in dimensionless form). With this new nomenclature 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, N0 = 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 actually ions present, but it is admissible because of the following consideration. The system of dierential equations 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

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X ¡ 0; but G(0) “starts” to aect the ion concentration in the next interval, i.e. at X = QX , and hence n± diers 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 nt product (average ion concentration × mean aerosol residence time); (ii) the total aerosol-to-ion concentration 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 must also take it as an additional factor inPuencing the charging process because, as we have already seen, it aects the ion-pair generation rate pro3le G(X ) and the mean ion generation rate G and, hence, it also aects the initial equilibrium ion concentration n. We must also consider the line source length as another inPuencing parameter although, as before, we will keep the total activity, a, constant for all the calculation runs. The source length  aects the ion generation rate pro3le G(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 examine the eect of ion concentration non-uniformity on the particle charging process. The ion-to-aerosol attachment rate coe9cients jk were calculated using Fuchs’ theory, with ion masses m+ = 150 and m− = 80 amu, and ion mobilities Z + = 1:15 and Z − = 1:65 cm2 V−1 s−1 (Alonso et al., 1997b). The charging 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 and negative particles, respectively. The charging probability f± is the quantity usually determined in past experimental works in which diusion losses do not need to be considered because of the use of an additional dummy chamber as a reference. Since particle penetration is the same for the charger and for the dummy unit (the particle charging state does not aect its diusion loss rate), the thus de3ned charging probability increases with mean aerosol residence time until it reaches a certain constant value which only depends on particle size and polarity. This value is usually referred to as equilibrium charging probability or, more correctly (since a true equilibrium cannot exist (Fuchs, 1963)), steady-state charging probability. For brevity, in the rest of this paper this stationary value will be referred to simply 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 + is always smaller than N − . In this and other 3gures below, the dotted line representing the steady-state

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λ/s = 0.25 λ/s = 1.0

non-uniform G:

number concentration [ - ]

uniform G:

10

0

free negative ions (n -)

10

-1

charging probability (f -)

10

-2

negative particles (N -)

10

-3

0.0

0.2

0.4

0.6

0.8

1.0

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

Fig. 7. Typical concentration pro3les of ions and particles for uniform and non-uniform ion generation rates, without diusion 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, nt = 106 cm−3 s.

charging probability (f− in this case) has been drawn in order to compare it, in an illustrative and easy manner, with the relative concentration of charged particles; it does not mean, of course, that the transient charging probability (i.e., before steady state) is constant all along the tube. For each species, there are three curves; one (full line) describes the situation in which the ion generation rate is uniform along the charger, that is, G = G everywhere within the tube. The discontinuous lines are for non-uniform ion generation rate using two dierent lengths of the radioactive line source (but, as noted before, having a total activity a of 100 Ci in both cases). The case illustrated in Fig. 7 has been run for a value of the nt 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 lower than the (steady-state) charging probability f− . As seen in the 3gure, when the ion generation rate is 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 discrepancy between the uniform and non-uniform ion generation cases becomes larger as the radioactive line source 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 nt product has been chosen large enough so that the particles can reach a stationary charging state before leaving the charger. It is important to note that, although the calculated transient charged particle concentration, N − , diers according to whether one considers an uniform or a non-uniform ion generation pro3le, the axial coordinate at which N − becomes equal to the charging probability f− is practically the same. This trend has been observed in all the calculation runs, and this gives us the possibility of analyzing the eect of charger geometry, source length and residence time, using the N1± =f± ratio, where N1± is the dimensionless number concentration of charged particles at the tube outlet, i.e. at X = 1. (N1± is what one might call the extrinsic charging e2ciency, that is, the relative output of charged particles; when diusion losses are neglected, the extrinsic charging e9ciency becomes equal to the charging probability if nt is su9ciently large.)

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number concentration [ - ]

non-uniform G: uniform G:

10

0

-

free negative ions (n )

10

-1 -

charging probability (f )

10

-2 -

negative particles (N )

10

-3

0.0

0.2

0.4

0.6

0.8

1.0

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

Fig. 8. Same as Fig. 7, except that now nt = 5 × 106 cm−3 s.

+ particle no. conc. at outlet, N1+/f+ [ - ]

10

0

dp [nm]

uniform G

10

3 5 8 12

-1

non-uniform G

10

10

10

-2

-3

-4

10

4

10

5

10

6

10

7

3

t [s/cm ]

Fig. 9. Extrinsic charging e9ciency/charging probability ratio as a function of nt and particle size. Comparison between uniform 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 N1± =f± ratio in the examination of the performance of a given charger, Fig. 9 shows the nt product required to attain the stationary state for positive particles (i.e. N1+ =f+ ≈1) for dierent particle diameters. Besides the dierence between the uniform and non-uniform G cases already discussed before, the main conclusion that can be drawn from Fig. 9 is that smaller particles require longer values of the nt product. Also, when the ionizer is designed or operated such that the extrinsic charging e9ciency is smaller than the charging probability (i.e. whenever nt is small enough), the calculated charging e9ciency is quite dierent depending on whether one considers uniform or non-uniform ion generation rates.

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4. Charging of aerosol nanoparticles in a non-uniform bipolar ion environment with diusion losses Finally, the more realistic situation in which particles and ions are being lost by diusion to the charger wall during the charging process will be examined, assuming laminar Pow. Since charging equations (10)–(16) must be solved numerically using a certain QX step for integration, we must consider particle and ion penetration through a tube of length QX . According to the model of Gormley and Kennedy (1949), penetration is given by  0:8191 exp(−3:657) + 0:0975 exp(−22:3) + 0:0325 exp(−57) for  ¿ 0:0312; P= 1 − 2:562=3 + 1:2 + 0:1774=3 for  ¡ 0:0312; (19) =

D Dst Q x = 2 QX; Q RL

(20)

where D is the particle or ion diusion coe9cient, Q the aerosol Pow rate, s the source stopping distance, t the mean aerosol residence time in the charger, and R and L the radius and length of the tube. Eq. (19) has proved to be also valid for particle diameter as small as that of typical air ions (Alonso, Kousaka, Hashimoto, & Hashimoto, 1997a). As an explanation of how penetration is included into the model equations, consider for instance Eq. (12) for positive particles; the corresponding dierence equation including diusion losses becomes  + dN + + N (X + QX ) = PN (X ) + QX dX no loss (21) s + +0 + 0 −+ − + (nt)[ n (X )N (X ) −  n (X )N (X )]; = PN (X ) + QX L where P is the penetration of positive particles (equal to the penetration of negative and neutral particles of the same diameter) through the tube section of length QX . The procedure is similar for ions, except that P must be evaluated for their corresponding diusion coe9cients. In turn, the latter 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 ) = P X=Q X ;

(22)

where P is again the particle penetration. It must be stressed that the number concentration of particles in whatever state of charge is made dimensionless by referring it to the total aerosol number concentration. When diusion losses were neglected, as in the previous section, the total aerosol number concentration, N , was constant all along the tube charger (Eq. (14)). When diusion losses 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 diusion losses are considered, is shown in Fig. 10. The case plotted here is the same as that of Fig. 8, except that in the latter

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M. Alonso, F.J. Alguacil / Aerosol Science 34 (2003) 1647 – 1664 10

1

number concentration, [ - ]

N

10

0

n

+

0

n

10

-1

f

10

-

-2

+

N

f

-

-

N

+

with diffusion losses

10

-3

0.0

0.2

0.4

0.6

0.8

1.0

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

Fig. 10. Typical concentration pro3les of ions and particles for non-uniform ion generation rate, when diusion losses are 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, nt = 5 × 106 cm−3 s; line source length, =s = 0:25.

diusion loss was neglected. In addition, we have also included the pro3les for positive ions and particles. Concentration of negative ions at the charger outlet is smaller than that of positive ions, because of the dierence in electrical mobility and, hence, diusion coe9cient. When diusion losses were neglected (Fig. 8), the dimensionless number concentration of negative particles at the charger outlet (extrinsic charging e9ciency, N1− ) was equal to the charging probability (f− ). However, when losses are considered, the population of negative particles is unable to attain the “equilibrium” state: N1− ¡ f− . In contrast, the charging e9ciency for positive particles becomes equal to the charging probability (N1+ ≈f+ ). The reason for this dierence resides in the unequal concentrations of positive and negative ions due to their dierent deposition loss rates. Fig. 11 shows the same example as that of Fig. 10 except that now the mean aerosol residence time in the charger is 10 times higher. In this case, the dimensionless number concentration of positive and negative particles at the charger outlet are both smaller than the corresponding charging probabilities. This is a typical example of an overdimensioned (or overdesigned) charger. In summary, too short values of nt prevent the particles from achieving their stationary charge distribution. Too large nt values result in so high diusion losses that the extrinsic charging e9ciency is much lower than the charging probability. Therefore, for a given tube geometry, there exists an optimum value of nt for which the extrinsic charging e9ciency is a maximum but, in general, the maximum e9ciency will be smaller than the charging probability. This will be seen more 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 eect of tube geometry, source length, aerosolto-ion concentration, and nt product on the relative number concentration of charged particles at

M. Alonso, F.J. Alguacil / Aerosol Science 34 (2003) 1647 – 1664 10

1

number concentration [ - ]

N

10

10

n

0

+

n

-

0

-1

f

10

1661

+

f

-2

N

-

+

N

-

with diffusion losses

10

-3

0.0

0.2

0.4

0.6

0.8

1.0

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

negative particle conc. at outlet, N1-/f- [ - ]

Fig. 11. Same as Fig. 10, except that now nt = 5 × 107 cm−3 s.

10

10

10

0

-1

particle diameter D p [nm] 3 5 8 25 40

-2

with diffusion losses

10

-3

10

4

10

5

10

6

10

7

10

8

3

t [s/cm ]

Fig. 12. Extrinsic charging e9ciency/charging probability ratio as a function of nt and particle size. 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.

the charger outlet. In order to present the plots as clear as possible, only negatively charged particles will be considered, keeping in mind that the corresponding results for positive particles follow the same tendency although the numerical values are dierent. The 3gures that follow are plots of the extrinsic charging e9ciency/probability ratio for negative particles, N1− =f− , against the nt product, calculated for non-uniform ion generation rate. Fig. 12 shows the eect of particle size on the N1− =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 nt of about 3 × 106 cm−3 s. However, the maximum value of the N1− =f− ratio does

M. Alonso, F.J. Alguacil / Aerosol Science 34 (2003) 1647 – 1664 negative particle conc. at outlet, N1-/f- [ - ]

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10

10

10

0

-1

tube radius R/s [-] 0.1 0.2 0.3 0.5

-2

with diffusion losses

10

-3

10

4

10

5

10

6

10

7

10

8

3

t [s/cm ]

Fig. 13. Extrinsic charging e9ciency/charging probability ratio as a function of nt 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 eect of diusion losses on charging becomes negligible. (For particles larger than about 40 nm, multiple charging cannot be neglected (Hoppel & Frick, 1988) and, hence, the above-presented calculation model, valid for singly charged particles, cannot be used.) For 40 nm particles, the maximum attainable extrinsic charging e9ciency is quite close to the corresponding charging probability, but too large residence times in the charger also result in a progressive reduction of the extrinsic e9ciency. Therefore, a truly stationary charge distribution is unattainable even for a particle diameter as “large” as 40 nm, although for this particle size the eect of diusion losses is much less drastic. Figs. 13 and 14 show the eect of tube geometry. The value of nt at which the outlet concentration of charged particles is maximized slightly depends on the tube radius, but not on the tube length. Again, the maximum attainable concentration of charged particles may be much smaller than that inferred from the charging probability, specially if the tube charger is too narrow (small radius) or too long. Also note that, since diusion losses cannot be avoided in practice, overdimensioning the charger operation (e.g. too long residence time) leads to a considerable reduction of the charged particle output. As regards to the line source length, , even though it exerts a large inPuence on the ion generation rate pro3le (as was shown in Fig. 5), it has practically no eect on the relationship between N1− =f− and nt. For this reason, no plot is provided. Finally, we have also examined the eect of the relative aerosol-to-ion concentration, given by the ratio N=n (= total aerosol number concentration at the charger inlet/mean equilibrium ion number concentration before particles are fed to the charger). This is shown in Fig. 15. The N1− =f− vs. nt curves for relative aerosol-to-ion concentrations down to 10−4 coincide with that shown for N=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

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

negative particle conc. at outlet, N1-/f- [ - ]

10

10

10

1663

0

-1

tube length L/s [-] 0.5 1.0 1.5 2.0

-2

with diffusion losses

10

-3

10

4

10

5

10

6

10

7

10

8

t [s/cm3]

negative particle conc. at outlet, N1-/f- [ - ]

Fig. 14. Extrinsic charging e9ciency/charging probability ratio as a function of nt 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. 10

10

10

0

-1

ion-to-aerosol conc. N/ [-] 0.1 10 100

-2

with diffusion losses

10

-3

10

4

10

5

10

6

10

7

10

8

3

t [s/cm ]

Fig. 15. Extrinsic charging e9ciency/charging probability ratio as a function of nt 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 concentration has no eect 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) diusion losses prevent achievement of a steady-state distribution of charged particles in a real charger; (2) the relative

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concentration of charged particles at the charger outlet (extrinsic charging e9ciency), which is the variable of primary interest in practice, may be much smaller than that inferred from the experimentally measured charging probability; (3) the magnitude of the eect of diusion losses decreases as particle size increases, but it is still noticeable for 40 nm singly charged particles; (4) the dierence between extrinsic charging e9ciency and charging probability depends in a complex manner on charger geometry, particle size, and nt product, and this complexity makes hard to 3nd a suitable correlation to use in practical situations. These results have large implications in the size distribution measurement of nanometer aerosols by electrical mobility analyzers. For a given charger geometry, ionizing source type, and aerosol Pow rate, one should calculate the theoretical extrinsic charging e9ciency as a function of particle size in order to transform the measured electrical mobility distribution into the desired particle size distribution. 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 diusion loss prevents the attainment of a stationary charge distribution. In order to accomplish this goal we must reformulate the model to include multiple charging because, as we have seen, diusion losses do exert a measurable eect on particle charging in the whole particle size range where multiple charging 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. References Adachi, M. (1988). Electrical behavior of aerosol particles. Ph.D. thesis, Osaka Prefecture University. Alonso, M., Kousaka, Y., Hashimoto, T., & Hashimoto, N. (1997a). Penetration of nanometer-sized aerosol particles through wire screen and laminar Pow tube. Aerosol Science and Technology, 27, 471–480. Alonso, M., Kousaka, Y., Nomura, T., Hashimoto, N., & Hashimoto, T. (1997b). Bipolar charging and neutralization of nanometer-sized aerosol particles. Journal of Aerosol Science, 28, 1479–1490. Clement, C. F., & Harrison, R. G. (2000). Enhanced localised charging of radioactive aerosols. Journal of Aerosol Science, 31, 363–378. Coll Buti, P. (1990). Fundamentos de Dosimetr9:a Te9orica y Protecci9on Radiol9ogica (pp. 51– 67). Barcelona: UPC. Friedlander, G., & Kennedy, J. W. (1955). Nuclear and radiochemistry (pp. 192–195). New York: Wiley. Fuchs, N. A. (1963). On the stationary charge distribution on aerosol particles in a bipolar ionic atmosphere. Geo4sica Purae Applicata, 56, 185–193. Gormley, P. G., & Kennedy, M. (1949). Diusion from a stream Powing through a cylindrical tube. Proceedings of the Royal Irish Academy, 52A, 163–169. Hoppel, W. A., & Frick, G. M. (1988). Aerosol charge distributions produced by radioactive ionizers. NRL report 9108, Naval Research Laboratory. Kondrat’ev, V. N. (1964). Chemical kinetics of gas reactions (pp. 543). Oxford: Pergamon. Reischl, G. P., MGakelGa, J. M., Karch, R., & Necid, J. (1996). Bipolar charging of ultra3ne particles in the size range below 10 nm. Journal of Aerosol Science, 27, 931–949.