Particle size analysis of log-normally distributed ultrafine particles using a differential mobility analyser

J. AerosolSci.,Vol.20, No. 5, pp. 547 556,1989. Printed in GreatBritain.

0021-8502/89$3.00+0.00 © 1989MaxwellPergamonMacmillanplc

PARTICLE SIZE ANALYSIS OF LOG-NORMALLY DISTRIBUTED ULTRAFINE PARTICLES USING A DIFFERENTIAL MOBILITY ANALYSER* C. ROTH,'~ U . BERLAUER'~ a n d J. HEYDER~ i'Institut fiir Biophysikalische Strahlenforschung der Gesellschaft fiir Strahlen- und Umweltforschung mbH., Miinchen, Paul-Ehrlich-Str. 20, 6000 Frankfurt am Main, F.R.G. ~Projekt Inhalation d.G. ffir Strahlen- und Umweltforschung mbH., Miinclien, Ingolst/idter LandstraBe 1, 8042 Neuherberg, F.R.G. (Received 6 June 1988; and in final form 2 November 1988)

Almtraet--Polydisperse aerosol size distributions measured with a differential mobility analyser show deviations of the measured size parameters from those parameters of the input distributions. The dependence of the measured modal particle diameters and the geometric standard deviations on the size parameters of log-normal input distributions is derived and the results are proved experimentally by means of fluorescein particles and electron-microscopy. INTRODUCTION

In aerosol literature the term ultrafine or highly dispersed aerosols refers to an aerodisperse system with particles smaller than the wavelength of visible light. Four methods to generate ultrafine aerosols have been accepted. These methods are the exploding wire technique (Phalen, 1972), the burning of metal-organic compounds in a flame (Kasper et al., 1980), the atomization of an aqueous solution (Raabe, 1976), the heterogeneous or homogeneous condensation of a supersaturated vapour (Kanapilly et al., 1982; Scheibel and Porstend6rfer, 1983) and the combination of several methods (Bartz et al., 1987). The characteristics of these four kinds of aerosols have been investigated experimentally and theoretically. The formation process leads in all cases to aerosols with log-normal distributions having a geometric standard deviation in a range between % = 1.3 and a 0 = 2.0. The purpose of this paper is to describe the particle size analysis of these aerosols by measuring the electrical mobility of the particles. DMA

THEORY

For the measurement of the size-dependent electrical mobility of a particle several systems have been proposed. The differential mobility analyser (DMA) has increasingly been used for particle size analysis (Keady et al., 1983). A DMA consists of two concentric cylindrical electrodes. A high voltage supply maintains the central electrode at a precise negative potential. The outer electrode is at ground potential. An inner core of particle-free sheath air and an outer annular ring of aerosol, flow laminarly down between the electrodes with no mixing of the two air streams. Particles with positive charges are attracted through the particle-free sheath air towards the negatively-charged central electrode. The trajectory of such a particle is a function of the flow-rate, analyser geometry, electric field, particle diameter, and the number of charges on the particle. Only those particles within a narrow, predictable mobility range pass through a slit near the bottom of the central electrode. A small air flow carries the particles out of the DMA. Positively-charged particles of other mobilities precipitate onto the central electrode or pass out of the DMA with the excess air. Particles with negative charges precipitate onto the outer electrode. For the measurement of a particle size distribution the DMA has to be connected to a particle counting instrument. It can either be a condensation nucleus counter (CNC) * This work is dedicated in honour of Prof. Dr Wolfgang Pohlit on the occasion of his 60th birthday. 547

548

c. ROTHet al.

(Agarwal and Sem, 1980), or a Faraday cup with an electrometer (Liu et al., 1979). A CNC counts the number of particles grown by detecting their scattered light. An electrometer counts the number of charges and needs a correction for multiply charges. The diameter, d, of the particles passing through the exit slit of the DMA is related to the central electrode voltage, U, the number of charges per particle, n, the sheath air flow-rate, qc, and the DMA geometry, L, rl, r2 (L--length of the cylindrical electrodes, r I and r 2 = diameters of the inner and outer electrodes, respectively): 2neULC(d)

d = 3~/qcIn r2/rl "

(1)

e represents the electrical charge unit, C(d) the slip coefficient, and ~/the air viscosity. For monodisperse aerosols the particle diameter can therefore be taken directly from the calibration curve calculated from the design data of the DMA and the used sheath air flowrate, q~ (Fig. 1). For polydisperse aerosols the particle size distribution is determined by a set of measurements of particle number concentration of the exit aerosol for different high voltages of the central electrode and by an inversion procedure. If no a priori knowledge of the input aerosol size distribution is available a system of linear equations has to be solved and leads to an unstable solution and a great influence of random errors of the measured data. Data inversion techniques have been described by Knutson and Whitby (1975), Hoppel (1978), Kapadia (1980) and Helsper et al. (1982). Because of the difficulties arising with an unknown input size distribution a log-normally distributed incoming aerosol is assumed here. This shortcoming is compensated for by the unique relation between the parameters of the input and output aerosol size distribution. For the DMA set at a fixed voltage, the probability that an aerosol particle of diameter, d,

kv

O

O 'r "e 3.

0

. 1

2

0 5

~ 10

20

50 PARTICLE

100

200 DIAHtETER.

d

Calibrationcurvesof a DMA for differentflow-ratesof sheath and excess air: (A) 461 rain- 1, short version,Vienna; (B) 201 rain-1, TSI model 3071;(C) 101 rain-1 'I'SI Model 3071.

Fig. 1.

Particle size analysis

549

carrying n electrical charges will pass through the DMA is given by:

ca= fZ 2 P(d)F(n, d)H(Z(n, d))dZ,

(2)

1

where Z is the electrical mobility of the particle with Z=

neC(d)

(3)

31t~/d '

P(d) is the particle size distribution, F(n, d) the charge distribution, P(d)'F(n, d) the probability that a particle with the characteristics, n and d, occurs in the aerosol, and

H(Z(n, d)) the transfer function of the instrument. In the present case P(d) is the normalized, log-normal size distribution:

e(d) = x / ~ d l n a ,

lnag

/ j.

(4)

The mean particle diameter, dso, and the modal particle diameter, d, of this distribution are linked by: d=dso exp [ - (In ag)2]. (5) For bipolar charging by gaseous ion diffusion the charge distribution of particles below 100 nm particle diameter has been investigated intensively during the last few years. The size dependence of the fraction of particles carrying one or more negative or positive charges can be described by the Boltzmann function for particles of more than 30 nm in size:

F(n,d)

=

1

x / ~ u exp

[

--l(-n~21 with u = 2~eodkT 2\uJ A e2 '

(6)

where eo is an electrical field constant, k Boltzmann's constant, T the absolute temperature and e again the electrical charge unit. For particles smaller than 30 nm in size this function underestimates the fraction of particles carrying one positive or negative charge. In this size range below 30 nm, where the probability that a particle carrying two or more charges is negligibly small, an approximation proposed by Knutson (1976) is used in a slightly modified form for the calculation of the positively charged particle" fraction (d in nm):

(

1

F(n, d)=l.3 4+(100~_1)1.5444F(n, d ) = 0

)

for n = l

for n > l .

(7a) (7b)

The fraction of particles with a single positive charge calculated from this approximation agrees well with values from the Fuchs theory taking into account the different values of physical parameters for negative and positive ions (Hussin et al., 1983) and with experimental values of the same authors. A more recent correction of the Fuchs theory performed by Hoppel and Frick (1985) also resulting in an asymmetric charge distribution is almost perfectly represented by the modified form of the Knutson formula (Fig. 2). Theoretically, at a fixed voltage of the central electrode only particles with a definite electrical mobility can pass through the instrument. In practice, however, particles in a certain mobility range, defined by Z1 and Z2, are capable of passing through the DMA because of the finite dimensions of the entering and exciting aerosol flow. Knutson (1975) and, later on, Hoppel (19"/8) calculated the electrical mobility trajectories of charged particles through a DMA. The transfer function shown in Fig. 3 is derived from these calculations. For the calculations in this paper the transfer function is replaced by a Gaussian distribution function with maximum value H(Zu) and with a standard deviation of M=0.1 Zu:

H(Z)__~ 2~

F 1/z-z,~V1

550

C. ROTH et at.

0.20 W < I u

-

0 , i 0

0.08 0 &

0

0.05

I

Z 0

0.03

~

0.02

W

/ BOLTZMANN

0,01 4

5

6

7

8

i0

nm

PARTICLE

20

30

DIAMETER.

40 d

Fig. 2. Probabilities of a single, positive charge on particles in bipolar charge equilibrium as a function of the particle diameter according to the Boltzmann theory, to modified Fuchs theories (Hussin and Hoppel), to equation (7) (Knutson) and to experimental values of Hussin (1983).

H(Z) 1 .OO

o.

0.2

0.1

DISTRIBUTION

0.1

0.2

Z--Zj~

ZH

Fig. 3. Theoretical (triangle) and experimental (Gaussian distribution) transfer functions of the TSI DMA Model 3071, for flow-rates of sheath and excess air of 201 min- 1 and for polyd/spers~ and monodisperse aerosol of 21 min- 1

The replacement does not change the number concentration of the exit aerosol which is half the number concentration of the particles with electrical mobilities between Z M ± 0 . 1 ZM ( K o u s a k a et al., 1985), but it changes its size distribution. The value o f 0.1 for M/ZM is based on our o w n experiments described later on. This value is independent of d and, therefore, does not describe any broadening of the transfer function by diffusion processes of the particles (Reineking and Porstend6rfer, 1986; Karch and Reischl, 1987).

Particle size analysis

551

In case a CNC is used to count the number of particles passing through a DMA at a fixed high voltage of the central electrode, their number concentration is given by:

OCNc(d')=c ~. ~.(z2,P(d)F(n, d)H(Z(n, d))6d n =1

(9)

J d(Zt)

with d' =d(Zu) and c the particle number concentration of the incoming aerosol. In case the CNC is replaced by an electrometer the charge concentration is given by:

Om(d')=c

n n= 1

P(d)F(n,d)H(Z(n, d))6d.

(10)

./d(Zl)

For log-normal size distributions with modal diameters between 10 and 210 nm and geometric standard deviations between 1.20 and 3.00, OEi(d') was calculated. With respect to the resolution of a DMA it proved reasonable to divide the measuring range (0-210 nm for 201 rain- 1 sheath air) into 50 steps; that means a step distance of 4 nm. The program starts with the calculation of the electrical mobility of a singly charged particle, Z u , for the first d' and continues with the determination of the integration limits d(Z1) and d(Z2) being defined by H(Z1)= H(Z2)=0.01 H(ZM). Because the slip coefficient depends on the particle diameter, the diameter corresponding to a certain mobility value is calculated by an iteration process. The first part of the program is finished by the computation of the integral of equation (10). The result describes the number concentration of the singly charged particles extracted out of a DMA set at a fixed high voltage. To calculate the fractions of multiply charged particles represented by the corresponding terms of the sum in equation (10), the previously determined Z Mis divided by the number of charges, then the computations are started again with the thus-found mobility value. Multiple charges are taken into account as long as the fraction of these particles is more than 1% of the fraction of singly charged particles. The computations are continued for the following d's. The results represent a normalized and voltage-dependent function of number concentration of charges measured with the electrometer in case a defined log-normal input aerosol size distribution is applied to a DMA. In case the central rod voltage is related to the diameter of a singly charged particle the value ofd' and therefore Om(d' ) is obtained. It has to be considered that d' will not exceed a maximum value because of the highest available central rod voltage. Figure 4 shows both distributions, P(d) and Om(d' ), plotted vs the particle diameter and the diameter of the singly charged particles for a o= 2.00 and d = 0.050/~m. The measured distribution is broadened and shifted vs larger particle diameters in comparison with the input distribution. The most probable particle diameter, d', of OEl(d') is plotted vs the modal

10 coAd

x l O -a 1,1i,lr1-1 5

0 t

50

0.94

Fig. 4. Input

100

150

200

250

3.'2q

6.'37

9.99

U,'

d ,d',

nm

kV

(P(d)) and calculated output (~b(d')) particle size distribution for the given parameters and for flow-rates of sheath air of 201 rain- 1 and aerosol of 2 1 rain- 1.

552

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_

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--~-/ ~

~0

60

80

100

. i

120

• . ~

lqO

160

18ohm

200

MODAL PARTICLE DIAMETER,d'

Fig. 5. Dependence of themodal diameter ~i', of the calculated or measured aerosol size distribution on the modal diameter, d, of the input aerosol size distribution for different geometric standard deviations, ag.

particle diameter, d, of P(d) in Fig. 5. It can be seen in the figure that the modal diameter, d' of Om(d') differs more and more from the modal diameter of P(d) for increasing particle diameter, d, and geometric standard deviation cg. The shift is mainly due to the non-linear dependence of the electrical mobility on particle diameter, therefore the detecting mode (CNC or electrometer) plays no important role for the size of the shift. This size differs by not more than 5% of the particle diameter for the two detecting modes. A log-normal distribution function is defined by its modal particle diameter, d, and its geometric standard deviation, ¢0. The last value can be replaced by the maximum value, P(d), of the normalized function, as equation (4) demonstrates. This replacement is advantageous in the case of the measured distribution Om(d'), because there exists a cut-off of the distribution for all d' greater than a maximum value. The maximum of the measured distribution @El(d') normalized according to the stripped area of Fig. 4 is plotted for different geometric standard deviations cg in Fig. 6. A unique relationship is obtained for nearly the whole size range. For this reason the geometrical standard deviation, ag, of the input distribution can be gained from the maximum value of the measured distribution. The procedure for determining the modal diameter, d, and the geometric standard deviation, ag, of a polydisperse aerosol size distribution is therefore the following: for a set of arbitrarily chosen voltage steps of the central electrode the corresponding number concentrations of the particles or charges are recorded either with a CNC or an electrometer. The measured values are multiplied by the difference of the particle diameters corresponding to the applied voltage and to the next higher voltage step. The so,found values are summed up and the maximum number concentration of particles or charges is divided by this sum. From this value and the matched particle diameter the geometric standard deviation of the measured distribution, a 9, is determined by means of Fig. 6. Knowing cg the modal diameter, d, of the measured distribution can be taken out of Fig. 5. A very similar procedure has to be applied to Ocnc.

Particle size analysis

553

I moQ-

(~El(d"} x l o -3 i

nm -1

Og =

1.50

Og-

1.8o

- - ~ O g

= Og

\

'

1.2o

2.00

3

. O0

\

\ \ \

'\

\ \

\ "~...

0

40

80

___..._ .

.

120 1 6 0 nm 2 0 0 MODAL P A R T I C L E D I A M E T E R . d'

Fig. 6. Maximum value of the normalized distribution of number concentration of particles measuredwith an electrometerfor differentgeometricstandard deviationsof the primaryaerosolsize distribution, ag.

E X P E R I M E N T A L V A L I D A T I O N OF T H E DMA T H E O R Y

Procedure The objective of the following experiments is to check the validity and accuracy of the data inversion method. The experiment performed with the DMA and spherical polydisperse fluorescein particles should show the coincidence of computed and measured size distributions. Within these investigations the standard deviation of the Gaussian transfer function is tested with two DMAs in series.

Generation of the aerosols Ammonium fluorescein solutions were prepared by dissolving an appropriate amount of fluorescein in an aqueous ammonia solution. Aerosols were produced by atomizing the solution and drying the spray with clean air. The jet of a Beckman flame photometer was used as an atomizer. The aerosol size distributions were varied by nebulizing fluorescein solutions of different concentrations. The fluorescein particles were deposited on electronmicroscopic grids by means of an electroprecipitator. The grids were covered with graphite-coated Formvar foils to avoid AS 2 0 : 5 - D

554

c. ROTHet

al.

electrostatic charging of the particles. After sampling of several hours, microphotographs of the deposit were taken with the electron microscope. The a m m o n i u m fluorescein particles proved to be spherical in shape. For a representative size distribution the diameter of 500-800 particles were determined from the film material by means of a light microscope. In parallel with these experiments the size distributions were determined with a D M A and an electrometer. RESULTS For a fixed voltage of the central electrode the 'monodisperse' size distribution of particles leaving a D M A is measured with a second D M A . The first D M A exit aerosol is admitted directly into the second D M A with a higher aerosol-to-sheath-air flow-rate and therefore with a higher resolution. When the central electrode voltage of the first D M A is kept constant the size distribution of the exit aerosol of the first D M A is measured. Such a distribution is shown in Fig. 7 in comparison with a calculated distribution. Both can be L ~

c'&d

~M

nm ' 1. oo-

MEASURED VAI UES

/

0.75"

CALCUt ATE[) VAI UES

!

l o.

5O-

i

n.25-

%

o

i

i

~9 ~ r~n PARTICLE DIAMETER. d

Fig. 7. Measured and calculated distributions of the monodisperse aerosol leaving a DMA for a fixed central electrode voltage (for comparison both distributions are normalized to a maximum value of 1).

Og I .15

1 .oo ~ o

, Io

20

, 30

, 4o

nm

, 50

--

PARTICLE DIAMETER. d

Fig. 8. Geometricalstandard deviation, au, of the monodisperseDMA exit aerosol size distribution (for a fixed voltage of the central electrode) derived theoretically (straight line) and experimentally (crosses connected to a curve).

Particle size analysis

555

Table 1. Comparison of modal diameters and geometrical standard deviations of fluorescein particle size distributions determined electron. microscopically and with the DMA Electron microscope d(#m) %

aT(#m)

%

1 2 3 4

0.039 0.050 0.060 0.029

2.22 2.24 2.37 2.13

0.039 0.055 0.062 0.031

2.10 2.11 2.25 2.10

5

0.053

2.60

0.050

2.50

Experiment No.

DMA

approximated by a log-normal distribution. The geometrical standard deviations of both distributions are given in Fig. 8 dependent on particle size. The values of the calculated distributions are independent of size as the straight line demonstrates. The deviation of the measured values from the calculated ones is due to diffusion processes of the particles in a DMA. The statistical error is relatively great because of the small number concentration of particles available at the exit of the second DMA. In a size range where the diffusion processes no longer play an important role the curves of the calculated and measured geometrical standard deviations coincide, therefore the assumed value of M = 0.1. Z u is correct. The fluorescein particle size distributions measured with a DMA were normalized and the modal diameters and geometric standard deviations were obtained from Figs 5 and 6. The modal diameters and geometric standard deviations of the fluorescein particles measured with the electron microscope were taken from cumulative plots of the particle sizes since their size distribution is supposed to be log-normal because of the generation process. The modal diameters and geometric standard deviations of the input size distributions measured with a DMA and an electron microscope are compared in Table 1. The agreement is good so that for log-normal distributions the parameters of the input distribution, e.g. the modal particle diameter and the geometric standard deviation, can be deduced from the measured data. CONCLUSION

The computations of the response of a DMA for a set of voltage steps of the central electrode show that a log-normal input size distribution is converted into a log-normal exit or measured distribution. The modal diameter and geometric standard deviation of input and measured distribution are uniquely related. The modal particle diameter of the measured distribution is larger than the input modal diameter. The measured geometric standard deviation is greatly influenced by the cut-off of the particle size distribution of the apparatus.

REFERENCES Agarwal, J. K. and Sere, G. J. (1980) J. Aerosol Sci. 11, 343. Bartz, H., Fissan, H. and Liu, B. Y. H. (1987) Aerosol Sci. Technol. 6, 163. Fuchs, N. (1963) Geofis. pura appl. 56, 185. Helsper, C., Fissan, H., Kapadia, A. and Liu, B. Y. H. (1982) Aerosol Sci. Technol. 1, 135. Hoppel, W. A. (1978) J. Aerosol Sci. 9, 41. Hoppel, W. A. and Frick, G. M. (1986) Aerosol Sci. Technol. 5, 1. Hussin, A., Scheibel, H. G., Becket, K. H. and Porstend6ffer, J. (1983) J. Aerosol Sci. 14, 671. Kanapilly, G. M., Wolff, R. K., DeNee, L. B. and McClellan, R. O. (1982) Ann. occup. Hyo. 26, 77. Kapadia, A. (1980) Data reduction techniques for aerosol size distribution measuring instruments. Ph.D. thesis, University of Minnesota, Minneapolis. Karch, R. and Reischl, G. (1987) The 15th Annual Conference of the Association for Aerosol Research, Hannover. Kasper, G., Shon, S. N. and Shaw, D. T. (1980) Am. ind. HyO. Ass. J. 41, 288. Keady, P. B., Quant, F. R. and Sem, G. J. (1983) TSI Quarterly Voi. IX, p. 3.

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Knutson, E. O. (1976) Fine Particles, pp. 740-762. Academic Press, London. Knutson, E. O. and Whitby, K. T. (1975) J. Aerosol Sci. 6, 443, 453. Kousaka, Y., Okuyama, K. and Adachi, M. (1985) Aerosol Sci. Technol. 4, 209. Liu, B. Y. H., Pui, D. Y. H. and Kapadia, A. (1979) Aerosol Measurement, pp. 341-383. University Press, Florida, Gainesville. Phalen, R. F. (1972) J. Aerosol Sci. 3, 395. Raabe, O. G. (1976) Fine Particles, pp. 57-1 t0. Academic Press, London. Reineking, A. and Porstend6ffer, J. (1986) Aerosol Sci. Technol. 5, 483. Scheibel, H. G. and Porstend6rfer. J. 11983) J. Aerosol Sci. 14, 113.