Charge distribution among non-spherical particles in a bipolar ion environment

Pergamon

J Aerosol Sc~, Vol 25, No 4, pp. 611-615, 1994 Elf,nor $c~nc~ Ltd Pnntod m Great Britain 0021-8502/94 $700+000

0021-8502(94)E0031-R

CHARGE

DISTRIBUTION AMONG NON-SPHERICAL IN A BIPOLAR ION ENVIRONMENT

PARTICLES

A. V. FILIPPOV* Process and Aerosol Measurement Technology, University of Duisburg, D-47048 Dmsburg, Germany (First received 7 October 1993; and in final form 6 January 1994) Al~tract--The charge distribution on conducting aerosol particles of arbitrary shape in bipolar ion environment is determined, assuming that the mean free path of ions is small compared to the particle size It is shown that the charge balance equattons yield the modified Boltzmann distribution as the stationary soluUon, rather than the standard Boltzmann law. Application of the latter is proved to contradict the irreversibility of the process, as well as the charge balance equations. In the obtained analytical solutton the particle capacitance acts as a shape factor, which deternunes the effective particle radius. INTRODUCTION

The fact that the Boltzmann distribution frequently approximates the experimentally measured charge distributions of aerosol particles (Lissowski, 1940), motivated Keefe et al. (1959) to explain it as the result of an equilibrium energy exchange between particles and bipolar ions. This straightforward thermodynamic approach has been criticised by Fuchs (1963) who pointed out at the irreversibility of the charging process. Meanwhile, further experiments have shown closeness of the measured charge distribution to the Boltzmann law in the cases of fibrous particles (Wen et al., 1984) and of fractal-like agglomerates (Rogak and Flagan, 1992). Matsoukas (1994) finds, that the Boltzmann distribution approximates the solution of the continuum variant of the charge balance equations, provided the particles are spherical and sufficiently large. He also tries to explain this closeness by energy arguments, which are critiqued in detail by Mayya (1994). This begs the larger question: why does the Boltzmann law in some situations describe real charge distributions, despite the obvious inappropriateness of thermodynamic equilibrium considerations. Here, the general problem of determining the charge distribution on conducting aerosol particles of arbitrary shape in bipolar ion environment in considered. The ion mean free path is assumed to be small compared to the particle size. The stationary analytical solution of the charge balance equations is found and appears to yield a modified Boltzmann distribution, rather than the standard Boltzmann law. For spherical particles, the same distribution was found by Clement and Harrison (1991, 1992) and independently by Poluektov et al. (1991). However, in the general case of nonspherical particles and aggregates it is the capacitance that plays the role of effective particle radius. Apparent universality of the Boltzmann distribution is then discussed along with the general properties of the modified Boltzmann distribution obtained as the general solution. POSING

THE

PROBLEM

AND ITS SOLUTION

Consider bipolar ionic charging of monodisperse conducting aerosol particles. Suppose that ion mean free paths are small in comparison with particle diameter d. The image force can be neglected, if d is sufficiently large. We assume also, that the particle surface is absolutely absorbing for ions. Then the fluxes J~ of positive and negative ions to an * On leave from the Institute of Mechanics, Moscow Umverstty, 119899 Moscow, Russia. *Also Clement and Harrison (1992) and independently by Poluetkov et al. (1991) 611

612

A V. FiLlPPOV

individual particle with charge Q = me are described by the expression (Laframboise and Cheng, 1977; Mayya, 1990) am

J+ = +_4nCD+ n+

- exp( + 0tm)- 1

-

(1)

e2

(2)

= CkT'

where n+_ and D_+ are the number concentration and diffusion coefficient of positive and negative ions and e is the elementary charge. The material properties of the particles enter into equation (1) only through the electrical capacitance C, which for spherical particles simply equals the particle radius R. The aerosol charge distribution can be found by solving the basic equations of bipolar charging (e.g. Marlow and Brock, 1975) m=0, +1, ___2, +_3. . . .

dN'=-N,,,(J+,,,+J~)+N,,,_IJ~_I+N,+IJT,+I

dt

(3)

In these equations, N, denotes the number concentration of aerosol particles with charge me and t is the time. The stationary solution of the balance equations (3) can be obtained, if one assumes N,n

N,,-I

= J m+J;

1

(4)

Consequent application of the recurrence relations (4) with numbers m, m - 1. . . . . 1 yields (Hoppel and Frick, 1986) m-1

H J¢

N,._ k=0

(5)

m--1

No

I-I ,//-+1 k=0

If the ion fluxes to the particle are given by equations (1)-(2), the solution (5) takes the form N,,,

2 y " . . /am'X

[

~m2Xx

N o - -~m slab k-~-) exp~--~--) n + D+ Y-n_D_

(6) (7)

'

where 7 is the conductivity ratio. The form of equation (6) coincides with that for the modified Boltzmann distribution, obtained for spherical particles by Clement and Harrison (1991). In the present case of arbitrarily shaped particles the particle radius is substituted by capacitance C Equation (6) yields the following useful formula 2 smh . (-2-)exp ~m N" Ns = a-ram ( °t(m2 s)2)

(8)

s=llny,

(9)

0t

where N, is a normalizing parameter independent on m and s is the equilibrium number of elementary charges, at which fluxes (1) of positive and negative ions to the particle compensate each other

J+ =s~-.

(lO)

Charge distribution among non-spherical particles

613

Actual number concentrations of the charged particles can be found from (8) by simple recalculation

where Nt is the total number concentration of aerosol particles. For large particles with C >>k T / e 2 and not too high m, the distribution (8) approaches the Boltzraann law

~=exp(-°~(m~s)2-).

(12)

Note however, that equation (12) itself does not satisfy the system of charge balance equations (3). DISCUSSION

Form of the distribution

The form of equation (8) explains why in many experiments the Charge distribution appears to be close to the Boltzmann law. As it could he expected from the formula (8), this is true for conducting particles of various shapes, provided that they are large enough in comparison with the ion mean free path and • ¢ 1. For example, distributions close to the Boltzmann law have been observed in fibrous aerosols (Wen et al., 1984) and in the case of fractal-like agglomerates (Rogak and Flagan, 1992). Under the conditions of these experiments, the results could also agree with the modified Boltzmann distribution (8). However, if the condition 0 ~ 1 is not fulfilled, formula (8), which is an exact solution of the balance equations (3), and the Boltzmann distribution (12) provide differing results. For instance, Table I shows the difference of values N , / N o , predicted for particles with a capacitance 0.1/~m (or for spherical particles with such radius) at the room temperature. This difference is in no case negligible, and if one considers fractions with high number of charges, it can be substantial for higher particle sizes as well. On the other hand, the Boltzmann distribution is also not an analytical solution of the continuum approximations of the charge balance equations (Matsoukas, 1994). It can therefore be considered only as a convenient, and sometimes incorrect, approximation of the exact solutions and experimental results. The inconsistency of the Boltzmann distribution with the balance equations is a direct confirmation of the conclusion of Fuchs (1963), who stressed inapplicability of the equilibrium thermodynamic approach of Keefe et al. (1959) to the problem, due to irreversibility of the charging process. For instance, Keefe et al. (1959) based their arguments on the classical definition of electrostatic energy E, which in our terms can be written as E=

,

(13)

with C = R for spherical particles. This definition corresponds to a process of reversible charge transfer from infinity to the particle surface, in the case of infinitely small unipolar charges. However, in the case of bipolar charging by finite charges (like those of ions) the Table 1. Ratios N,,/No predicted by distributions (8) mid (12) for particles with capacitance Cffi0.1/ml. Positive and negative ion conductivities are equal

AS ~'5:4-B

N.INo

~INo

~21No

~31No

N, INo

Exact-solution (8) Boitzmann formula (11) Discrepancy %

0.7621 0.7519 1.34

0.3373 0.3196 5.22

0.08655 0.07683 11.23

0.01286 0.01044 18.79

614

A.V. FILIPPOV

energy level of the particle can not be uniquely defined in terms of the net charge alone. To prove it, suppose the contrary and assign the zero value to the energy of the neutral particle. If for example, some positive ion is absorbed, no work is done, so the energy of the particle must remain the same. The following absorbtion of an ion with an opposite sign results in work - e 2 / C and zero charge of the particle. This ideal process can be repeated, each time producing the same work - e2/C on ions although resulting in the same neutral state of the particle. Therefore, the energy of the considered system can not be a function of the particle charge alone. To calculate the work of the electric field, one has to take into account not only the initial and final charge of the particle, but a whole sequence of particle charge changes. Use of the definition (9) in such a situation is clearly erroneous. This fact manifests the irreversibility of the process (Sedov, 1971) and makes impossible thermodynamic approach of Keefe et al. (1959), based on assumption of reversible exchange of energy between the particles and the ions. On the other hand, the basic charge balance equations (3), and their approximation for large particle sizes (Matsoukas, 1994) are easy to apply and provide rigorous physical and mathematical grounds for a study.

Specifics of the charoino of non-spherical particles The particle shape enters into the ion flux equations (1) and resulting distributions (6) and (8) via the particle capacitance. For instance, in the particular case of uncharged particles, the ion flux expressions (1) yield

J~ =4~D± n± C.

(14)

Since the model of Laframboise and Cheng does not take into account the image force acting on ions, equations (14) hold for fluxes of any diffusing species to the particle, provided that the condition of the absolute absorption on the surface is fnlfil!ed. These equations are also valid for diffusion fluxes of ions to a large uncharged dielectric particle of the same shape as conducting particle with capacitance C. This means that the capacitance has also a meaning of a diffusion equivalent radius, which scales with the radius of gyration R s of the agglomerate (Schmidt-Ott et al., 1990). These scaling relations yield C = ARg,

(15)

where A is a prefactor depending on aggregate shape and fractal dimension. Mayya (1990) came to the same scaling law for the capacitance, comparing the formulae (14) with the results of Ball and Witten (1984), who analyzed the absorption rate of the diffusing particles by a large cluster. So far the experimental and theoretical results on diffusion to random aggregates agree with the scaling relation (15). The distribution law (8) provides other opportunities to test its validity. For example, Rogak and Flagon (1992) have found that the charge distribution on bipolar charged aggregates is the same as that of spherical particles with diameter equal to the mobility diameter of the aggregate. Like the diffusion diameter, the mobility diameter scales with the gyration radius R s. On the other hand, formula (8) implies that equivalent particle diameter, used in the interpretation of these experimental data, must equal twice the particle capacitance C. This confirms the scaling law (15) between the capacitance and the gyration radius. Acknowledgement--The author appreciates kind support of this work by the Alexander yon Humboldt Foundation, Bonn, Germany.

REFERENCES Bali, R. C. and Witten, T. A. (1994) Phys. Rev. A ~ , 2966. Clement, C. F. aad Harmon, IL (3. 0991) Inst. Phys. Conj:.Ser. No. 118, Sec. 5, 275-280. Clement, C. F. and Haxtimn, IL O, (1992) J. Aerosol 8ci. 7,3, 481. Coniglio, A. and Stanley, H. E. (1983) Phys. Rev. Lett. 52, 1068.

Charge distribution among non-spherical particles Hoppcl, W. A. and Frick, G. M. (1986) Aerosol Sd. TachnoL & 1. Keefe, D, Nohm, P. J. and Rich, T. A. (1959) Proc. R. Irish Acad. f ~ A, 27. Laframboise, J. G. and Cheng, J. S. (1977) J. Aerosol $cf. L 331. Limowski, P. (1940) Acta Phy~.ochfmica URS$13, 157. Marlow, W. H. and Brock, J. It. (1975) .I. Colloid Interface $cl. 51, 23. Matsoukas, T. (1994) J. Aerosol $cf. 2S, 599. Mayya, Y. S. (1990) J. Colloid Interface $cL 140, 185. Mayya, Y. S. (1994) J. Aerosol Sd. 25, 617. Poluektov, P. P~ Emets, E. P. and Kascheev, V. A. (1991) J. Aerosol $~ 22, $237. Ro~k~ S. N. and Flagan, R. C. (1992) J. Aerosol $cl. 23, 693. Sedov, L. L (1971) A Cour~ in Mechanfcs of Continuum, VoL 2. Wolter~Noordho~, Groninge~ Sdmfidt-Ott, A., Baltansperger, U, G/lggkr, H. W. and Jost, D. T. (1990) J. Aerosol $cL 21, 711. Wen, H. Y, Reischl, G. P. and Kasper, G. (1983) J. Aerosol $cL 15, 103,

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