Bipolar Diffusion Charging of Soot Aggregates

Abstract

Experimental measurements of bipolar diffusion charging efficiency over the range of 15–400 nm are compared for oil droplets, flame generated soot aggregates, and diesel engine exhaust particulate matter to explore possible effects of particle morphology. Charging efficiency is recorded using a tandem differential mobility analyzer (DMA) approach; the first DMA selects a monodisperse aerosol and the second compares the flux of these particles through a neutralizer versus an identical blank housing. Electrostatic precipitation of mobility selected soot particles onto TEM grids provides comparative data on changes in soot particle morphology with mobility diameter. The measurements yield soot charging efficiencies that slightly, but systematically, differ from those of equal mobility diameter oil droplets. Single positive charging of soot climbs to ∼ 10% higher than oil droplets at 50 nm and then decreases to 15% lower by 400 nm as the soot develops a progressively more fractal-like structure. Negative charging exhibits the same pattern, except the variations are +15% and –10%, respectively. These trends, as well as those for double and triple charging, fall intermediate between Fuchs predictions and the model of Wen et al. (1984a) for charging of fibrous particles.

INTRODUCTION

Potential health effects, global climate change, and energy security have all motivated the recent interest in combustion related particulate matter (PM). One of the principal tools to study the physical properties of these and other aerosol particles is the scanning mobility particle sizer (SMPS) (Knutson and Whitby 1975; Wang and Flagan 1990). Its widespread use often makes this method the standard against which new PM instrumentation is compared (Burtscher 2005). The SMPS measurement consists of two basic steps: The particles are charged and then segregated according to their mobility in an electric field. Particle size is derived from the drift velocity of charged particles in this field. But the measurement of size distributions requires also the number of particles at each size, and for this one needs to know their charging efficiency. Size can be calibrated against standard reference materials, such as polystyrene latex spheres, whereas charging efficiency is widely based on the measurements of Wiedensohler et al. (1986), Wiedensohler (1988), and Wiedensohler and Fissan (1988).

The SMPS method is well established for spherical particles. This is particularly true in the case of size determination, leading Kinney et al. (1991) to examine the feasibility of using mobility classification as the basis for developing a 0.1 μ m reference standard. For non-spherical particles, mobility diameter remains a physically well-defined quantity related to the particle's diffusion constant (Hinds 1999). But the relationships to mass, volume, and surface area become more complex. These have received extensive study via fractal geometry (Forrest and Witten 1979; Jullien and Botet 1987; Cai et al. 1993; Köylü et al. 1995; Megaridis and Dobbins 1990), effective density (Kelly and McMurry 1992; Park et al. 2003; Virtanen et al. 2004; Maricq and Xu 2004), and recently in terms of “idealized aggregates” (Lall et al. 2006a, 2006b).

The study of particle charging has a long history, with the more recent work in this area directed towards ultrafine and nano particles (Hussin et al. 1983; Vijayakumar and Whitby 1984; Adachi et al. 1985; Alonso et al. 1997; Reischl et al. 1996) and the comparison of experimental data to models of diffusion charging (e.g., Fuchs 1963; Hoppel and Frick 1986). But the influence of morphology on particle charging has had relatively little attention. Wen et al. (1984a) reviewed a number of theoretical approaches to describe charging of arbitrary shaped particles. They introduced the notion of a charging-equivalent diameter into the Boltzmann expression and found it to provide a good approximation to the general theory of Laframboise and Chang (1977) for the charging of fibrous particles. Predictions of charging efficiency were generally within about 10% of experimental values for γ -Fe₂O₃ chain aggregates (Wen et al. 1984b). Rogak and Flagan (1992) further examined the question of morphology by comparing the neutral fractions of spherical versus aggregate particles leaving a diffusion charger. They concluded that aggregate and spherical particle charging was very similar, observing only a 5% lower neutral fraction for aggregates between 100 nm ≤ d _m ≤ 800 nm. And indeed, spherical particle charging efficiencies remain the basis to convert raw SMPS data to size distributions.

Very recently, Lall et al. (2006a, 2006b) introduced the idea of “idealized aggregates” to calculate the surface area of particles with low fractal dimension. In a two step approach they make a series of approximations to relate aggregate morphology to mobility, but then they also adopt the method of Wen et al. (1984a) to calculate the aggregate's charging efficiency. The latter step deviates from the conventional use of spherical particle efficiency. Owing to the current interest in combustion aerosols it is, therefore, worthwhile to re-examine the extent to which the bipolar charging of spherical and aggregate particles may differ.

EXPERIMENTAL METHODS

Particle Generation

Three types of aerosols are examined in this work: poly (α -olefin) oligomer (PAO) oil droplets, premixed ethylene flame generated soot particles, and diesel vehicle exhaust particulate matter (PM). Oil droplets are generated by an atomizer followed by a condensation—nucleation particle generator. PAO oil, base stock for synthetic lube oil, is used without the usual additive package present in motor oil. The mean particle size of this polydisperse aerosol is adjusted by atomizer pressure, air diluent added after the atomizer, and condensation—nucleation generator temperature and flow rate.

Flame generated soot particles are produced in a rich premixed ethylene flame. A 6 cm diameter water cooled McKenna burner forms the flame, which a sheath of nitrogen gas (at 40 L/min) protects from the surrounding atmosphere. Tylan flow controllers set the ethylene and air flows to provide equivalence ratios of Φ = 2.1 and 2.4, at cold gas velocities of 31 cm/s and 43 cm/s, respectively. An ejector pump samples soot from the flame, dilutes the sample in nitrogen to ∼ 300C, and introduces it into a residence tube (Maricq 2007). There soot particles are allowed to coagulate at near room temperature over a time of 0 to 20 seconds as a means to adjust their average diameter. Particles are typically sampled from 20 mm above the burner, but to obtain soot down to ∼ 15 nm diameter a lower height of 12 mm is used. Soot particles are then sampled from the residence tube and diluted sufficiently in air to avoid saturating the condensation particle counter (CPC) used to record charging efficiency.

Diesel exhaust PM is obtained from a light duty diesel truck run with low sulfur fuel (< 15 ppm) on a chassis dynamometer. The vehicle has a modern 6.0 L, direct injection, common rail, turbocharged engine. Exhaust aftertreatment includes an oxidation catalyst, but no diesel particulate filter. The vehicle is run at a steady state 50 mph using the vehicle speed control to maintain constant engine operation. Exhaust is sampled at the tailpipe and diluted with air using a two stage ejector pump diluter (Dekati, Ltd.).

Charging Efficiency Measurement

A tandem differential mobility analyzer (t-DMA) method similar to earlier work by Vijayakumar and Whitby (1984) and Alonso et al. (1997) is used to measure bipolar charging efficiency. A schematic diagram of the apparatus is displayed in Figure 1. The test aerosol passes through a ²¹⁰Po neutralizer and DMA1 to select a monodisperse (in electrical mobility), singly charged, aerosol of diameter d (with minor quantities of multiply charged particles at diameter ∼ 2^½ d). A nano DMA (TSI 3085) or long DMA (TSI 3081) and a sheath flow rate of 5 or 10 L/min are chosen depending on the desired particle size in the range of 15–400 nm. The selected particles then pass either through a second ²¹⁰Po neutralizer or an identical “blank” (both use the ADI-100 housing, Aerosol Dynamics Inc.) and are interrogated by sweeping DMA2 through a narrow size range around d, typically a 30 s up-scan from ∼ d/3 to ∼ 1.7d. Particle counts are determined by fitting one or more narrow lognormal distributions to the size distribution (raw counts) recorded by DMA2 and integrating the respective peak areas.

FIG. 1 Schematic diagram of experimental apparatus.

In the “blank” measurement, scanning DMA2 yields a single peak at d. In contrast, the active neutralizer redistributes the predominantly singly charged aerosol from DMA1 back to the equilibrium charge distribution. Here, scanning DMA2 in general reveals a series of peaks dominated by the one at d, but containing subsidiary peaks. Those peaks at smaller diameter correspond to multiply charged particles of diameter d resulting from the second neutralizer. The single, double, and triple charging efficiencies are given by the areas of these peaks relative to the area of the peak recorded with the blank. Because identical aerosol paths are employed, this method avoids systematic errors due to particle losses that occur in the transport lines, DMAs, and neutralizers, as well as due to CPC efficiency. The tradeoff is an increased potential for errors associated with aerosol source fluctuations between “active” and “blank” measurements. This is handled by making each charging efficiency determination from an alternating sequence of blank and active neutralizer measurements that are regressed against time to derive peak areas corresponding to a common measurement time.

Peaks in the scan of DMA2 found at diameters larger than d indicate the presence of multiply charged particles transmitted by DMA1 that can interfere with the charging efficiency determination. This is handled in the following way: First, as illustrated for d = 70 and 250 nm in Figure 2, the test aerosol is adjusted (via evaporation/condensation conditions for PAO oil; residence time for soot) such that the particles selected for measurement lie well above the mean diameter of the source aerosol. This ensures that particle concentration falls rapidly above d, minimizing interference. For the PAO oil and flame soot measurements, it was possible to limit multiply charged particles transmitted by DMA1 to less than 10% of the total particle count. Owing to less flexibility in adjusting diesel exhaust PM size, this increased to ∼ 20% at some of the smaller sizes selected in the diesel soot measurements. Second, the area of the peak at size ∼ 2^½ d, if present, is used to estimate the residual concentration of multiply charged particles transmitted by DMA1 and to correct for their presence.

FIG. 2 Typical soot size distributions used in this work: premixed ethylene flame at Φ = 2.1, sampled at 20 mm above the burner and allowed to coagulate for 1 and 20 s. Arrows illustrate the preferred relationship between size distribution and selected size for the examples of ∼ 70 nm and ∼ 250 nm particles.

An important consideration to record accurate charging efficiencies is to ensure that a steady state is achieved between the ion—particle interactions in the neutralizer. The present experiments employ four ²¹⁰Po anti static strips (NRD Staticmaster) in an ADI-100 housing. The following conditions were varied to verify that steady state is established: (a) radiation intensity, (b) initial particle polarity, and (c) aerosol flow rate through neutralizer. ²¹⁰Po strips ranging from 1–12 months in age (half life = 138 days) yield charging efficiencies that are indistinguishable within data scatter for aerosol flow rates of ≤ 1 L/min. Reducing the number of ²¹⁰Po strips from four to two has no discernable effect on charging efficiency (for strips less than ∼ 8 months). The selection of positively versus negatively charged monodisperse particles by DMA1 yields identical charging efficiency data at aerosol flow rates between 0.5–1.2 L/min (see Figure 3).

FIG. 3 Charging efficiency versus aerosol flow rate through ²¹⁰Po neutralizer (ADI housing) for 45 nm PAO oil droplets.

Aerosol flow rate, however, does appear to affect steady state charging levels, as demonstrated in Figure 3 (illustrated for 45 nm oil droplets, but also observed at 200 nm). Between 1.2 L/min and 0.75 L/min, the charging efficiency agrees well with the accepted value (dashed lines indicate values from Wiedensohler et al. 1986), but it deviates significantly at 0.5 L/min. A deviation from steady state is not expected as flow rate decreases. This is confirmed since the same effect is observed irrespective of whether the particles entering the neutralizer are initially charged +1 or −1. Thus, steady state is achieved, but appears to depend on flow rate. It is noteworthy that charge efficiency differences were not observed between 0.5 versus 1.0 L/min flow rates in the original data collection, but only in a second measurement campaign. Also, at times significant background counts were observed through the active neutralizer, but not the blank, indicating the possibility of a contaminant produced by ionizing radiation capable of initiating nucleation in the CPC (this is reduced by increasing sheath flow, but independent of DMA2 voltage). The explanation for the flow rate effect remains unknown, and it may not be universal even to the ADI-100 housing. More insight into how neutralizer design can influence the “equilibrium” charge distribution is provided by Hoppel and Frick (1990). Importantly, the measurements reported in the present article were conducted at an aerosol flow rate of 1.0 L/min, or if taken at 0.5 L/min, were confirmed by repeat measurements at higher flow rates.

Soot TEM Images

Mobility diameter selected soot particles are collected onto transmission electron microscope (TEM) grids using electrostatic precipitation. This is accomplished by mounting a TEM grid onto one of a pair of electrodes, separated by ∼ 5 mm and set with the electric field perpendicular to the monodisperse aerosol stream exiting DMA1. A sufficient voltage of the appropriate polarity is applied to remove a significant fraction, e.g., 50%, of the monodisperse particle stream. Sampling for 5–10 minutes at count rates of ∼ 10⁴/s affords a good number of particles, but not so many that individual particles become difficult to distinguish. A JEOL FasTEM 2010 is used to image the soot particles.

RESULTS

PAO Oil Droplets

Figure 4 displays PAO oil droplet bipolar charging efficiencies, f _z , as a function of mobility diameter. These measurements were carried out in air. Limited experiments confirm that charging efficiency increases modestly in nitrogen, as reported by Wiedensohler and Fissan (1988). Positive versus negative charging is recorded by reversing the electric field of DMA2. The present data are in excellent agreement with those from Wiedensohler et al. (1986). For singly charged particles, these two data sets agree within their respective scatter. But, the z = +2 and –2 charging efficiencies of Wiedensoholer et al. appear to be ∼ 50% and ∼ 100% too high, respectively. Charging efficiencies for particles in the range of 4–30 nm have also been reported by Adachi et al. (1985) and Hussin et al. (1983). The former agree well with the present work, although with higher data scatter, whereas the latter systematically overestimate present +1 charging by ∼ 15% and –1 charging by ∼ 25%. These data are omitted from Figure 4 for clarity.

FIG. 4 Bipolar charging of PAO oil droplets. Upper panel: z = ± 1. Lower panel: z = ± 2. Solid symbols represent experimental data from this work; open symbols are data from Wiedensohler et al. (1986). Solid lines represent best log-log fits of the data to third order polynomials. Dashed lines depict Fuchs model predictions.

In the case of ±1 charging, best fits of the present and Wiedensohler et al. (1986) data sets to log(f _z ) as polynomial functions of log(d _m ) are virtually indistinguishable, except at the ends of their respective ranges. Therefore these two data sets are combined to produce optimized representations for spherical particle ± 1 diffusion charging efficiency. The resultant coefficients are listed in Table 1. Predictions from the Fuchs model (dashed lines in Figure 4) lie very close to the polynomial fits (solid lines); there are slight deviations at diameters below ∼ 30 nm. The calculations assume ion mobilities of Z ₊ = 1.4 × 10⁻⁴ m²/Vs and Z ₋ = 1.6 × 10^–4 m²/Vs (Hinds 1999), and ion masses of m ₊ = 250 amu and m ₋ = 170 amu.

TABLE 1 Bipolar charging efficiency parameterizations: f _z = 10^{∑limits _k a _z,k log (d _m ) k}

Measurement	z	a z,0	a z,1	a z,2	a z,3	a z,4	a z,5
Soot—present work	+3	−27.6243	19.9539	−3.7776	—	—	—
	+2	−21.2905	22.0034	−7.9597	0.9574	—	—
	+1	−3.2328	2.2417	−0.3731	−0.0553	—	—
	−1	−3.3612	2.9226	−0.8317	0.0380	—	—
	−2	−20.3678	21.1106	−7.5141	0.8845	—	—
	−3	−28.3714	21.3038	−4.1282	—	—	—
Oil—combined fits ^1	+3	−25.4373	17.8044	−3.2907	—	—	—
	+2	−13.1680	9.5095	−1.8950	0.0118	—	—
	+1	−3.2412	2.3299	−0.5102	−0.0106	—	—
	−1	−3.0127	2.1972	−0.4483	−0.0197	—	—
	−2	−15.3056	13.1212	−3.6374	0.2869	—	—
	−3	−26.4814	19.1064	−3.5878	—	—	—
TSI/Wiedensohler 1988 ^2	+1	−2.3484	0.6044	0.4800	0.0013	−0.1553	0.0320
	−1	−2.3197	0.6175	0.6201	−0.1105	−0.1260	0.0297

Due to the limited data reported by Wiedensohler et al. (1986), and systematic differences with the results found here, coefficients for fitting ± 2 charging efficiency are obtained from the present data alone. Experimental data for ± 3 charging of oil droplets are displayed in Figure 5a and the corresponding best fit parameters are listed in Table 1. For both double and triple charging, the best fits are virtually indistinguishable from the Fuchs predictions.

FIG. 5 Triply charged particles. Top panel: PAO oil droplets. Solid lines depict Fuchs model predictions. Dashed lines are polynomial best fits. Bottom panel: flame generated soot. Solid lines are from the Fuchs model. Dashed lines show predictions from the charging theory for fibrous aerosols of Wen et al. (1984a) using d ₀ = 17 nm.

Flame Generated Soot Aggregates

Monodisperse soot particles are selected from size distributions such as exhibited in Figure 2. Previous analysis of the shapes of these distributions and how they evolve with time in the residence tube indicate that the Φ = 2.1 and 2.4 flames produce soot particles with fractal dimensions of D _f = 2.0 and 1.9, respectively, as defined by their mobility diameter (Maricq 2007). The charging efficiency of these soot particles is measured with the same procedure as for oil droplets. The soot diffusion charging results are plotted in Figure 6 (single and double charging) and Figure 5b (triple charging). Best fits of these data to polynomial expressions are listed in Table 1. These are matched to the spherical particle trends at small size without sacrificing the quality of fit to the remaining data, and are valid over the 10-500 nm range.

FIG. 6 Bipolar charging of flame generated soot aggregates. Upper panel: z = ± 1. Lower panel: z = ± 2. Symbols represent experimental data. Solid lines depict the best fits to the spherical particle data from Figure 4. Dashed lines show predictions from the charging theory for fibrous aerosols of Wen et al. (1984a) using d ₀ = 17 nm.

There is no noticeable difference in charging efficiency between soot formed at Φ = 2.1 and Φ = 2.4 equivalence ratios. But there is a distinct, if small, difference relative to spherical particles. This is evident in Figure 6 from the comparison of the soot data (symbols) to the PAO oil best fits taken from Figure 4 (solid lines). At mobility diameters below ∼ 20 nm, soot and PAO oil charging is essentially equal. As particle size increases to ∼ 50 nm, soot charging efficiency increases by about 10% for +1 and 15 % for –1 charge relative to that of oil droplets. Near 150 nm the charging efficiency curves cross, and singly charged soot populations decrease more rapidly with increasing d _m than their spherical counterparts. By 400 nm, +1 charging is lower by 15%, and –1 charging lower by 10%, than for oil droplets.

± 2 charging (Figure 6b) exhibits the same trends, except that the enhancement at 50 nm is ∼ 80% and the crossover occurs at 300 nm. Because oil droplet and soot measurements were conducted on separate occasions, it is conceivable that subtle changes in experimental procedure might be responsible for the observed differences. This was checked by carrying out back-to-back oil and soot charging measurements, particularly at 50 nm and 200 nm. These reproduced the original data and support the differences observed in the charging of soot versus oil droplets.

Figure 7 illustrates the morphological changes in soot particles that accompany increasing mobility diameter. The projected area equivalent diameters of these particles are 201 ± 18, 104 ± 7, 53 ± 3, and 25 ± 2 nm (±1 st. dev.), in excellent agreement with the mobility diameters selected. The relative increase in soot charging efficiency coincides with the change from spheroidal shape at d _m = 25 nm (Figure 7d) to aggregates of roughly 10 primary particles at d _m = 50 nm (Figure 7c). The subsequent decrease in charging efficiency of soot relative to oil droplets parallels the more filamentary appearance of the aggregates as they increase in size (Figure 7a,Figure 7b).

FIG. 7 TEM images of mobility size selected soot particles. (a) d _m = 200 nm. (b) d _m = 100 nm. (c) d _m = 50 nm. (d) d _m = 25 nm.

This filamentary structure invites comparison to the charging model develop by Wen et al. (1984a) for fibrous particles. Dashed lines in Figure 6 (and 5b) illustrate predictions from this model, based on the d ₀ = 17 ± 2 nm primary particle diameter observed in the TEM images of Figure 7 and on the same ion mobilities as employed in the Fuchs model above. Here, the probability of finding an aggregate with z charges is given by the Gaussian expression

where D _qe = d ₀ N _p /ln (2N _p ) is the charging equivalent diameter, = D _qe kT/2e ² ln (N ₊ Z ₊/N ₋ Z ₋) is the mean charge number, N _p is the number of primary particles in the aggregate, d ₀ is the primary particle radius, e is the electron charge, N _± are the positive and negative ion number densities, and Z _± are their mobilities. Following Lall and Friedlander (2006a), d ₀ and N _p are related to mobility diameter via d _m = C(d _m ) c* N _p d ₀ ²/12π λ, where C(d _m ) is the Cunningham slip correction, λ is the mean free path, and c* is the dimensionless drag force, set here to a value of 8 appropriate for randomly oriented aggregates. In each case, ± 1 and ± 2 charging in Figure 6 and ± 3 charging in Figure 5b, the observed populations of charged soot aggregates lie intermediate between the spherical particle (Fuchs) values and the fibrous particle model predictions.

Diesel Soot

Diesel PM diffusion charging measurements are made from the exhaust of a light duty diesel truck, with no diesel particulate filter, operated at a steady state 50 mph cruise and low sulfur fuel. Separate measurements on quartz filter collected PM made using the Horiba 1370 thermal analysis system, not for this vehicle, but similar light duty diesel vehicles, indicate the PM typically to contain ∼ 80% elemental carbon, ∼ 20% organic carbon, and negligible sulfate. The single and double charging efficiencies for this light duty diesel soot are illustrated in Figure 8. The results resemble very closely the ethylene flame soot data in Figure 6. In particular these data too fall intermediate between the PAO oil values (solid lines) and the fibrous particle model predictions (dashed lines) of Wen et al. (1984a). The latter continue to assume d ₀ = 17 nm, which is in the midrange of diesel primary particle diameters. Diesel PM with high semivolatile fraction, such as from ship engines, high sulfur fuel, or engines with high oil consumption, would be expected to exhibit charging efficiency closer to those of spherical particles.

FIG. 8 Bipolar charging of diesel engine exhaust particulate matter. Upper panel: z = ± 1. Lower panel: z = ± 2. Symbols represent experimental data. Solid lines depict the best fits to the spherical particle data from Figure 4. Dashed lines show predictions from the charging theory for fibrous aerosols of Wen et al. (1984a) using d ₀ = 17 nm.

DISCUSSION AND CONCLUSION

The spherical particle (PAO oil droplet) diffusion charging data presented here are virtually indistinguishable from the results of Wiedensohler et al. (1986). Thus, they support the current widespread use of these values for SMPS data inversion. Agreement is good, as well, with the work of Adachi et al. (1985) and Hussin et al. (1983), and Fuchs model predictions. All this provides a good foundation against which to compare soot aggregate charging.

The present measurements yield consistent charging efficiencies between flame generated and light duty diesel soot, at least in cases that sulfate and semivolatile hydrocarbon components of the latter are small. In contrast, these aggregate particles exhibit small, but distinct, differences compared to the bipolar charging of spherical oil droplets measured under identical circumstances. The initial increase in soot charging efficiency observed relative to that of spherical particles, and the subsequent decrease, track the transition of soot from spherical to aggregate morphology and mimic the fibrous model predictions of Wen et al. (1984a). The peaks in the experimental plots of soot ± 1 charging efficiency versus mobility diameter shift to smaller size in qualitative agreement with the model, but not to the extent predicted. Double and triple soot charging levels follow the same pattern, namely they fall intermediate between fibrous model predictions and the charging efficiency for spherical particles. This is perhaps not surprising since the TEM images in Figure 7 show that soot morphology, while clearly aggregate, does not reach that of the “idealized” aggregates envisioned by Lall et al. (2006a, 2006b).

For particle size measurement, the principal interest is in the +1 (or –1) charging efficiencies needed to convert raw SMPS data into particle size distributions. Here, application of the standard charging efficiency to soot measurements potentially leads to distortions in the shape of the size distributions amounting to 10–15%. In practical terms, these are comparable, if not smaller, than other factors that can affect the accuracy of an aerosol size distribution measurement; for example, the impact of using nitrogen instead of air as the suspending gas is of the same magnitude. As pointed out by Liu and Deshler (2003), there are many issues that must be addressed to make quantitatively accurate SMPS measurements, including calibration of sheath and aerosol flow rates, especially at pressures different from factory calibration, ensuring that a steady state is achieved during neutralization, and accounting for particle losses during transport, within the SMPS, and the CPC. The influence of morphology on charging, however, remains of fundamental interest and may impact precision measurements on aggregates.

Acknowledgments

I would like to thank Mike Loos and Adolfo Mauti for their generous help with the diesel vehicle tests, Thanasis Mamakos (Aristotle University) for help with the Fuchs calculations, and Yi Liu (Wayne State University) for his expert help creating the TEM images.

Notes

¹For ± 1, fits are to combined present PAO oil and Wiedensoholer et al. (1986) data. For ± 2 and ± 3, fits are to present data only.

²The TSI and Wiedensohler (1988) values for a _{+ 1,4} differ: It is −0.1553 in the former, and −0.1544 in the latter case.