## The Size Distribution Function

The atmosphere, whether in urban or remote areas, contains significant concentrations of aerosol particles sometimes as high as 107-108 cm . The diameters of these particles span over four orders of magnitude, from a few nanometers to around 100 pm. To appreciate this wide size range one just needs to consider that the mass of a 10-pm-diameter particle is equivalent to the mass of one billion 10-nm particles. Combustion-generated particles, such as those from automobiles, power generation, and woodburning, can be as small as a few nanometers and as large as 1 pm. Windblown dust, pollens, plant fragments, and seasalt are generally larger than 1 pm. Material produced in the atmosphere by photochemical processes is found mainly in particles smaller than 1 pm. The size of these particles affects both their lifetime in the atmosphere and their physical and chemical properties. It is therefore necessary to develop methods of mathematically characterizing aerosol size distributions. For the purposes of this chapter we neglect the effect of particle shape and consider only spherical particles.

An aerosol particle can be considered to consist of an integer number k of molecules or monomers. The smallest aerosol particle could be defined in principle as that containing two molecules. The aerosol distribution could then be characterized by the number concentration of each cluster, that is, by Nk, the concentration (per cm3 of air) of particles containing k molecules. Although rigorously correct, this discrete method of characterizing the aerosol distribution cannot be used in practice because of the large number of molecules that make up even the smallest aerosol particles. For example, a particle with a diameter of 0.01 pm contains approximately 104 molecules and one with a diameter of 1 pm, around 1010.

A complete description of the aerosol size distribution can also include an accounting of the size of each particle. Even if such information were available, a list of the diameters of thousands of particles, which would vary as a function of time and space, would be cumbersome. A first step in simplifying the necessary accounting is division of the particle size range into discrete intervals and calculation of the number of particles in each size bin. Information for an aerosol size distribution using 12 size intervals is shown in Table 8.1. Such a summary of the aerosol size distribution requires only 25 numbers (the boundaries of the size sections and the corresponding concentrations) instead of the diameters of all the particles. This distribution is presented in the form of a histogram in Figure 8.1. Note that the enormous range of the aerosol particle sizes makes the presentation of the full size

Atmospheric Chemistry and Physics: From Air Pollution to Climate Change, Second Edition, by John H. Seinfeld and Spyros N. Pandis. Copyright © 2006 John Wiley & Sons, Inc.

 Size Range, |xm Concentration, cm-3 Cumulative, cm"3 Concentration, (inT1 cm-3 0.001-0.01 100 100 11,111 0.01-0.02 200 300 20,000 0.02-0.03 30 330 3,000 0.03-0.04 20 350 2,000 0.04-0.08 40 390 1,000 0.08-0.16 60 450 750 0.16-0.32 200 650 1,250 0.32-0.64 180 830 563 0.64-1.25 60 890 98 1.25-2.5 20 910 16 2.5-5.0 5 915 2 5.0-10.0 1 916 0.2

distribution difficult. The details of the size distribution lost by showing the whole range of diameters are illustrated in the inset of Figure 8.1.

The size distribution of a particle population can also be described by using its cumulative distribution. The cumulative distribution value for a size section is defined as the concentration of particles that are smaller than or equal to this size range. For example, for the distribution of Table 8.1, the value of the cumulative distribution for the 0.030.04 pm size range indicates that there are 350 particles cm-3 that are smaller than ±

Diameter, nm

FIGURE 8.1 Histogram of aerosol particle number concentrations versus the size range for the distribution of Table 8.1. The diameter range 0-0.2 pm for the same distribution is shown in the inset.

? 20000

Diameter, nm

FIGURE 8.2 Aerosol number concentration normalized by the width of the size range versus size for the distribution of Table 8.1. The diameter range 0-0.1 |xm for the same distribution is shown in the inset.

0.04 pm. The last value of the cumulative distribution indicates the total particle number concentration.

Use of size bins with different widths makes the interpretation of absolute concentrations difficult. For example, one may want to find out in which size range there are a lot of particles. The number concentrations in Table 8.1 indicate that there are 200 particle cm"3 in the range from 0.01 to 0.02 pm and another 200 particle cm"3 from 0.16 to 0.32 pm. However, this comparison of the concentration of particles covering a size range of 10 nm with that over a 160 nm range favors the latter. To avoid such biases, one often normalizes the distribution by dividing the concentration with the corresponding size range. The result is a concentration expressed in pm"1 cm-3 (Table 8.1) and is illustrated in Figure 8.2. The distribution changes shape, but now the area below the curve is proportional to the number concentration. Figure 8.2 indicates that roughly half of the particles are smaller than 0.1 pm. A plot like Figure 8.1 may be misleading, as it indicates that almost all particles are larger than 0.1 pm. If a logarithmic scale is used for the diameter (Figure 8.3) both the large- and small-particle regions are depicted, but it now erroneously appears that the distribution consists almost exclusively of particles smaller than 0.1 pm.

Using a number of size bins to describe an aerosol size distribution generally results in loss of information about the distribution structure inside each bin. While this may be acceptable for some applications, our goal in this chapter is to develop a rigorous mathematical framework for the description of the aerosol size distribution. The issues discussed in the preceding example provide valuable insights into how we should express and present ambient aerosol size distributions. Diameter, nm

FIGURE 8.2 Aerosol number concentration normalized by the width of the size range versus size for the distribution of Table 8.1. The diameter range 0-0.1 |xm for the same distribution is shown in the inset.

Diameter, nm

Diameter, nm

? 20000

15000

5000

0.01

1.00

0.10

Diameter, |xm

FIGURE 8.3 Same as Figure 8.2 but plotted against the logarithm of the diameter.

8.1.1 The Number Distribution nN(Dp)

In the previous section, the value of the aerosol distribution n, for a size interval i was expressed as the ratio of the absolute aerosol concentration Ni of this interval and the size range ADp. The aerosol concentration can then be calculated by

The use of arbitrary intervals ADp can be confusing and makes the intercomparison of size distributions difficult. To avoid these complications and to maintain all the information regarding the aerosol distribution, one can use smaller and smaller size bins, effectively taking the limit ADp —> 0. At this limit, ADp becomes infinitesimally small and equal to dDp. Then one can define the size distribution function n^(Dp), as follows:

nN(Dp) dDp = number of particles per cm3 of air having diameters in the range Dp to (Dp + dDp)

The units of nN(Dp) are pm_1 cm-3, and the total number of particles per cm3, N, is then just Diameter, |xm

FIGURE 8.4 Atmospheric aerosol number, surface, and volume continuous distributions versus particle size. The diameter range 0-0.4 pm for the number distribution is shown as an inset.

Diameter, |xm

FIGURE 8.4 Atmospheric aerosol number, surface, and volume continuous distributions versus particle size. The diameter range 0-0.4 pm for the number distribution is shown as an inset.

By using the function nN(Dp) we implicitly assume that the number distribution is no longer a discrete function of the number of molecules but a continuous function of the diameter Dp. This assumption of a continuous size distribution is valid beyond a certain number of molecules, say, around 100. In the atmosphere most of the particles have diameters smaller than 0.1 pm and the number distribution function nN(Dp) usually exhibits a narrow spike near the origin (Figure 8.4).

The cumulative size distribution function N(DP) is defined as

N(DP) = number of particles per cm3 having diameter smaller than Dp

The function N{DP), in contrast to n^(Dp), represents the actual particle concentration in the size range 0-Dp and has units of cm"3. By definition it is related to nN(Dp) by

D* is used as the integrating dummy variable in (8.2) to avoid confusion with the upper limit of the integration, Dp. Differentiating (8.2), the size distribution function can be written as nN{Dp) = dN jdDp (8.3)

and /¡,v (Dp) can be also viewed as the derivative of the cumulative aerosol size distribution function N(Dp). Both sides of (8.3) represent the same aerosol size distribution, and the notation dN/dDp is often used instead of n^(Dp). To conform to the common notation, we will also express the distributions in this manner.

Example 8.1 For the distribution of Figure 8.4, how many particles of diameter 0.1 pm exist?

According to the inset of Figure 8.4, nw(0.1 ¡im) = 13,000 pm_1 cm-3. However, this is not the number of particles of diameter 0.1 ¡im (it even has the wrong units). To calculate the number of particles we need to multiply nN by the width of the size range ADp. But if we are interested only in particles with Dp = 0.1 pm this size range is zero and therefore there are zero particles of diameter exactly equal to 0.1 pm. Let us try to rephrase the question posed above.

Example 8.2. For the distribution of Figure 8.4, how many particles with diameter in the range 0.1-0.11 pm exist?

The size distribution is practically constant over this narrow range with n^(O.lpm) = 13,000 pm 1 cm^.The width of the region isO.l 1 - 0.1 = 0.01 pm and there are 0.01 x 13,000= 130 particles cm 3 with diameters between 0.1 and 0.11 pm for this size distribution.

The above examples indicate that while nN is a unique description of the aerosol size distribution (it does not depend on definitions of size bins, etc.), one should be careful with its physical interpretation.

8.1.2 The Surface Area, Volume, and Mass Distributions

Several aerosol properties depend on the particle surface area and volume distributions with respect to particle size. Let us define the aerosol surface area distribution n\$(Dp) as n\$(Dp)dDp = surface area of particles per cm3 of air having diameters in range Dp to (Dp + dDp)

and let us consider all particles as spheres. All the particles in this infinites!mally narrow size range have effectively the same diameter Dp, and each of them has surface area it D2. There are ««(Op) dDp particles in this size range and therefore their surface area is KD^nN(Dp)dDp. But then by definition rts(DP) = K&pnuipJ) (pmcm"3) (8.4) 