PHYSICS AND CHEMISTRY
A COURSE GIVEN AT THE DEPARTMENT OF METEOROLOGY,
UNIVERSITY OF STOCKHOLM, FALL SEMESTER, 1976.
RUDOLF B HUSAR
1. AEROSOL PHYSICAL PROPERTIES
The term aerosol refers to liquid or solid particles suspended in the air. Aerosol particles may be liquid
droplets, aggregates of odd shape, or single crystals of regular, say cubical, or irregular shape. Their
chemical composition may also vary from dilute water solution of acids or salts, organic liquids, to earth's
crust materials (sand) and toxic heavy metals (Pb, Hg, As).
Depending on their origin and visual appearance aerosols have acquired different names in the everyday
language. Dust generally refers to solid airborne material, dispersed into aerosol from grainy powders.
Fumes are produced by various industrial processes such as in a foundary, welding etc.. Combustion
processes produce smoke particles, but the incombustible residue of coal is also called fly ash. Mist is
formed when a vapor condenses to form fine liquid droplets, which may then grow to larger fog droplets.
In the early days, air pollution had the appearance of both smoke and fog, so it was appealing to create a
new word for it: smog. In the open atmosphere, the visibility may often be reduced by haze, originating
from natural or anthropogenic sources.
It is obvious that in the context of physical and chemical treatment of atmospheric aerosol behavior, it is
advantageous to abandon the above set of rather poorly defined terms. Instead, the generic term aerosol is
used to express all types of airborne matter in the 10 to 100 m size range. If further classification is
needed, then it is beneficial to perform the subdivision based on physical or chemical grounds. In case of
atmospheric aerosols it is useful to divide the aerosol population into fine particles, having a diameter less
than about 2 m and course particles with size 2 m. The two aerosol populations have generally
distinctly different physical characteristics (shape, volatility) and chemical composition. Furthermore, the
two populations have different sources and more important - different effects. A scientific rationale for the
separate consideration of fine and coarse particles is given later.
Table 1. Characteristics of particles as a function of their size.
The size of atmospheric aerosols ranges over five orders of magnitude. The smallest observed particles are
of the order of 10 (1 = 10-8 cm) in diameter and they could well be called molecular clusters. In Table
1, the typical size of diatomic molecules is indicated to be between 3 and 6 . On the other extreme, large
dust particles of 100 m (1 m = 10-4 cm) may be kept suspended in the air for extended period of time.
Within the five decade ranges, there are "windows" in which the particles are comparable in size to the
mean free path of atmospheric gases (0.06 m), wavelength of the visible solar radiation (0.4-0.1 m), the
wavelength of the outgoing infrared radiation from the earth (6-40 m) (8-14 m - atmospheric window-
spechrae region over which thermal emission from ground is radiated directly to space. Paetridge - Plalt, 8,
Fig. 7.1) and other important micro-physical length scales.
This is of importance, because the interaction of the particle with the surrounding gaseous medium depends
primarily on the ratio of the mean free path, 1, to the particle diameter, D p, called the Knutsen number Kn.
= 1/Dp. For Dp <<1 the gaseous molecules that are bombarding the particle arriving from far distance and
each collision causes a substantial change in the direction or speed of the particle. Accordingly, such a
particle behaves as if it were of giant molecules and its motion (drag force) and Brownian diffusion may be
described by the rigorous kinetic theory of gases. Such an aerosol is said to be in the free molecular
regime. On the other hand, for a particle which is large compared to 1, i.e. Kn <<1, a change in its
momentum and only the combined effect of many particles yields an effective change. From the point of
view of the particle, it "feels" as if it were suspended in a continuum and for this reason such a particle is
said to be in the continuum regime.
The interaction of a particle with electromagnetic, including visible, radiation depends largely on the ratio
of the particles diameter to the wavelength, , of the incoming radiation. This dimensionless ratio, the
optical size parameter, = Dp/, determines whether the interaction is purely dipole (<<1) or it is
governed by geometrical optics i.e. diffraction, refraction and reflection (>>1).
In addition to the particle size, the behavior of atmospheric aerosols also depends on their shape and
Atmospheric aerosols exhibit a variety of shapes. Most commonly, they are spherical, liquid droplets. The
water solution of sulfuric aced or other salts that constitutes a large fraction of the fine particle mass (e.g.
Brosset, 1976) is believed to be in droplet form. Photochemically produced aerosol, such as the Los
Angeles smog aerosol consists of droplets formed around a nucleus (e.g. Husar et al. 1976).
Several of the common aerosol salts ((NH4)2SO4, NaCl) exhibit deliquescence (June, 1963; Covert et al.
1974). This behavior is manifested by a rapid transition from crystalline form at low relative humidity to a
droplet at high RH. Dry sodium chloride crystals are almost cubical; ammonium and potassium sulfate
crystals are also generally cubical but with more rounded edges.
Combustion and recondensation of weakly volatile substances lead to the formation of chain aggregates of
solid spherical spheres. The aggregation is generally taking place via coagulation in the vicinity of the
combustion zone and the individual spheres in the chain are held together by molecular forces. When
suspended in the air, a chain aggregate may bend, twist and rotate due to asymmetric molecular
bombardment of fluid shear.
A sample of atmospheric fine particles impacted onto a grid and viewed through electron microscope is
shown in Figure 2. The chain aggregate in the center, probably originated from an automobile exhaust.
Unfortunately, atmospheric liquid particles lose their identity when deposited onto a grid, inserted into the
vacuum of the electron microscope and exposed to the heating due to the absorbed electrons. However,
such particles leave a flat, circular residue as evidence of their original form.
Size segregated samples of coarse atmospheric particles are shown on the scanning electron micrograph
(Figure 3). They are generally non-spherical with major fraction resembling soil dust or fibrous matter.
The characteristic size of spherical particles is their diameter. Both the light scattering and mechanical (e.g.
fluid resistance) properties of nonspherical particles depends strongly on their particular shape. However,
the definition of a characteristic size is difficult. The method of their measurement is often used as the
definition of their size. For example, in case of a cascade impactor, particles are sorted according to their
aerodynamic (or Stockes) diameter which encompasses all the particles having the same trajectory in a
curved flow field. A small but dense particle may have the same aerodynamic size as a large particle with
low density or an aggregate with large aerodynamic resistance. Sorting or sizing of particles with optical
single particle counter/spectrometers, yields an "equivalent optical diameter". Here again, a multiplicity of
physical sizes may correspond to the same equivalent optical diameter if their refractive indexes are
different. The above discussion points to the need to specify the method of measurement whenever particle
size or size distribution are presented.
It is sometimes useful to express the shape of an aerosol by shape factors. For a particle with given
geometrical shape, the shape factors for volume, v, and shape factor for surface, s, are defined as:
V = v Dp3
s = s Dp2
For volume, v and surface, s, respectively. Dp refers to a typical length scale of a particle. The
shape factors of several common shapes are given in Table 1.
Figure 1. Transmission electron micrograph of size classified submicron aerosols. The
white area in the center is where the collodion film is broken and the chain agglomerate
aerosol is directly exposed to electron beam. Most of the submicron aerosol volume is
made up from liquid droplets (which leave circular residues) and of chain agglomerates.
Figure 2. Scanning electron micrograph of size classified (greater than 2 m) atmospheric
aerosol collected by an impactor. The collection substrate is a collodion coated electron
microscope grid. Note the irregular shape of the particles.
Table 1. Shape factors for surface and volume.
Sphere /6 = 0.52
Cube 1 6
Crushed sand 0.28 2
Calcite 0.32 2.5
The aerosol concentration is defined in a similar manner as the density: the number N or mass M of
particles per unit volume, V, of air (N/V). the concentration will vary, depending on the size of the
volume element V. At large V, the concentration, say the total number of particles N, will vary from
one location to another because of macroscopic gradients. With decreasing size of V the number of
particles may decrease such that it may exhibit statistical fluctuations because of the finite number of
Mass concentrations of atmospheric particulate matter may reach several hundred micrograms per cubit
meter. This corresponds to a mass ratio of the order of 10 -7 grams of particulate matter per gram of air.
Thus the contribution of the aerosol to the total mass, even in highly polluted atmospheres, is negligible.
For most applications, the effect of the particles on the fluid motion can be neglected.
The total number concentration may range from 102 particles/cm3 in a very clean atmospheric air to 106
particles/m3 near intensive combustion aerosol sources such as roadways. Even at 10 6 cm-3 the average
distance between individual particles is about 100 m, which is several orders of magnitudes larger than
the characteristic size of combustion aerosols (0.01-0.1 m). Therefore the motion of aerosols is free from
mutual interaction, except when they approach each other and adhere due to molecular forces (coagulate).
The most important physical characteristic of an aerosol population is the particle size distribution. In
principle, the aerosol distribution function should have two independent aerosol variables, viz. particle size
and chemical composition, as well as the usual space and time co-ordinates, x, y, z and t. in practice, it is
convenient and adequate to treat the chemical distribution as discrete set of species, rather tan as a
continuos variable. Thus we may define a distribution function n i; such that
ni (r, x, y, z, t)
represents the number concentration of chemical species I, at position x, y, z and time t, and in the range of
particle radii between r an r + dr. At times it is instructive to consider higher weightings of the number
distribution function such as the distribution of aerosol surface and volume (or mass) with respect to size.
The surface, volume and mass distribution functions, s i, vi and mi respectively, are related to the number
distribution as follows:
si = 4 r2 ni weighting function
vi = 4/3 r3 ni
mi = 4/3 i r3 ni
where i is the mass density of species i. The total aerosol distribution function in each case may be
expressed as the summation over all chemical species. Thus, for example, the total mass distribution
function is defined as
m(r, x, y, z, t) m i (x, y, z, t, r)
Typical total distribution functions for number n, surface s, and volume v, for the Los Angeles smog is
given in Figure 4.
Figure 3. Average size distribution of the Los Angeles smog aerosol (Whitby, Husar & Liu,
Integral Moments of the Distribution Function.
The effects of a polydispersed aerosol population are best described in terms of integral moments of the
size spectrum i.e. integrals of the size spectrum weighted by a function of particle size. If the weighting
function is an integer power law (many people confuse power law (x a) with exponential functions e-x)
function of r, then , the integral moment I, is defined as
I = r n(r)dr
Integral moments have physical significance describing the behavior of a polydisperse system as described
by the following examples.
The zeroth moment is the total number concentration, N, defined as
I0 = n(r)dr
The first moment
I1 = rn(r)dr
gives the total length of a chain if particles were lined up next to each other. A physically more meaningful
parameter is the size (radius) of the average particle, r, defined as
The second moment is proportional to the total aerosol surface area per unit volume, S = 4I2
4 I 2 = 4 r 2 n(r)dr = S
The surface area of the average particle is given by 4I2/Io = S/N.
The third moment is proportional to the total volume, V, of the aerosol suspension per volume of air is,
I 3 = r 3 n(r)dr = V
3 3 0
The total mass, M, is
M = (r)r 3 n(r)dr
where ( r ) is the size dependent particle density.
For = constant, M = V.
The volume of an average particle, V
The fifth moment is proportional to the mass flux, or deposition rate, D, of material sedimenting from air
4 3 8 2 gI 5
D = v s (r)( r )n(r)dr =
where the sedimentation velocity, vs = 2r2g/9, and is the dynamic viscosity of the air.
There are several other important moments of the size spectrum for which the weighting function can not
be conveniently expressed as integer power-law of particle size. In that case the integral moment is defined
I = f(r)n(r)dr
Notable examples for such weighting functions include the light scattering crossection, which yields the
total light scattering of a polydispersed system.
The relative contribution of the different part of the size spectrum to a given moment is determined by the
weighting functional. In a polluted urban atmosphere the number concentration or zeroth moment is
dominated by particles in the 0.01-0.1 m size range; the volume or mass concentration is contributed from
both the 0.1-1.0 m and 0-30 m size range as shown in Fig. 4.
Relationship Between Different Forms of Size Distribution.
The numbersize spectrum of atmospheric particles extends over five orders of magnitude in particle size
and up to ten orders of magnitude in concentration (Junge, 1963; Clark & Whitby, 1967). This extremely
wide range coupled with the different roles of small and large particles lead to the use of several forms of
distribution functions. In this section, attempt is made to facilitate a convenient intercomparison of several
commonly used distribution functions.
The extent of the aerosol size spectrum over several orders of magnitude necessitates the use of a
logarithmic scale for particle size presentation. It is also advisable, that the area under a distribution
function, when displayed graphically, is proportional to the integral (i.e. a moment) of that function.
(Junge, 1963; Berry, 1967; Whitby et al., 1972)/
If that criterion is satisfied, then simple visual impection of a graph reveals the value of that integral (total
number, mass) as well as the contributions of different size ranges to the integral. Such presentation is
facilitated by the semi-log plot of a properly defined spectrum function (linear) vs. log of particle size
(Berry, 1967; Whitby, 1972). If the distributed variable is. For example the total number such distribution
function, n(log r), is defined as,
dN = n(log r) d(log r)
where dN is the number of particles in the radius range between log r and log r + d (log r). Conventionally
decimal logarithms are used. Unfortunately, practical problems impose another problem of presentation.
The measurements of aerosol size or size distribution is most frequently performed in terms of particle
diameter, Dp, rather than radius. The corresponding number distribution function is n(log D p). In this case
the numerical values of the two distribution functions are identical as shown below. Following the
definition of distribution functions, that the number of particles, dN, in an infinitesimal range of the
independent variable, dr, and d(log r) must be identical. So we may write
dN = n(r) dr = n(log r) d(log r)
Thus the transformation from one independent variable, x, to another, z, follows the equation
y(z) = y(x) J
where the Jacobian transformation function J = dx/dy. The Jacobian of the transformation from r to log r is
J = dr/d(log r) = r/log e. Thus n(r)dr is related to the distribution function n(log r) as follows:
n(log r) = n(r)dr
Following similar transformation, the functional relationship between the most commonly used distribution
functions are given in Table 2. The functions in the table are the multipliers of f 2 to yield f1. For example a
transformation of the number radius distribution n(r) to the volume distribution with (log Dp) is:
For practical reasons, in the following text, we shall almost exclusively be using (log D p) as the size
4 r 4
v(log Dp) = n(r)
3 log e
Commonly Used Distribution Functions.
There is no universal distribution function that adequately describes the majority of aerosol populations,
including their various weightings. It is often desirable, however, to approximate the form of an aerosol
spectrum with an analytical function. In the past , the log-normal distribution, the power-law (Junge)
distribution, the gamma function and the "bimodal" distribution have been used most frequently.
A. Log-normal distribution.
Crushing and atomization processes often give skewed distributions over broad size range as shown
schematically in Figure 4.
Figure 4. Features of the log-normal distribution.
When such distribution are plotted on semi-log paper their shape resembles that of the normal Gaussian
distribution. Functionally, the only difference from the Gaussian distribution is that the independent
variable, r, is replaced by (lnr) or (log r). Thus the log-normal distribution is defined as
1 - (log r - log rg ) 2
n(log r) d(log r) = exp d(log r)
2 log g 2(log g )
where rg is the geometrical mean size and g is the geometrical mean deviation.
In analogy to the calculation of the mean radius (p. 12) the geometrical mean radius r g is obtained as
log rg = log r n(log r) d (log r)
Similarly, the logarithmic standard deviation is defined as
(log g ) 2
= (log r
- log rg ) 2 n(log r) d(log r)
The mode of a distribution is at size r, where the maximum of the distribution occurs. The median of the
distribution is defined as the size which divides the total population into two equal halves (see Figure 5 a).
by definition, the median radius, r m is such that
f (log r) d(log r) f (log r) d(log r)
where f(log r) may be any number surface or other distribution function. The mass median radius has often
been used to indicate the characteristic size at which the bulk of the aerosol mass is contained.
The median is best determined from normalized cumulative plots of the quantity (number, surface, volume
or mass) vs. log r. Such plots are also referred to as log-probability graphs.
Figure 5. Normalized cumulative distributions. Log-probability plot.
A convenient property of this function is that of n(r) is log-normal, then all the other weighting, rn( r ) are
also log-normal with the same g. In fact, it can be shown that the geometric mean radius log rg of a
higher (-th) weighting, can be calculated from the geometric mean size of the zeroth weighting log r g0
log rg = (log g)2 + log rg0
As an example, if geometric number mean radius is r g = 0.1m and g = 2.0 then the volume mean diameter
log rg,3 = 3(log 2)2 + log(0.1) = -0.728
rg,3 = 0.187 m
It is worth pointing out that the integral of the log-normal distributions unity. Such a distribution is defined
by three parameters, the mean size, rg, standard deviation and a scaling factor for the concentration.
In the early stages of atmospheric aerosol science, (Junge, 1963) atmospheric aerosol spectra have been
fitted with a power-law function
n(r) = ar- a = constant
where the exponent ranges from 3 < < 5. Such fits were arising as straight lines in plots of n(log r) on
log-log paper. The power-law fit brought to the researcher's attention some of the regularities exhibited by
the size spectrum of atmospheric aerosols (Friedlander, 1951; Clark & Whitby, 1961). More detailed
examination of aerosol size spectrum (Whitby et al., 1972), in particular of its higher weighting (surface ad
volume), revealed that simple functions such as the power-law, are inadequate to describe the complex and
dynamic behaviors of atmospheric aerosols. For this reason, the use of power-law distributions is no
recommended. A direct comparison of different distribution functions is given later.
The modified gamma function.
The calculation of radiation transfer through have layers, on a prior assumption needs to be made about the
aerosol size spectrum. For such radiation calculations, a modified gamma function has been proposed by
Diermendian (1969) and used extensively (e.g. Dave, 1971). The form of the function is
n(r) = a1ra2 exp(-b1rb2)
where a1, a2, b1,b2 are positive constants.
a1 a2 b1 b2
Haze M 5.33104 1 8.94 1/2
Haze L 4.97106 2 15.11 1/2
Cumulus C 2.38 6 3/2 1
Corona cloud C 1.0810-2 8 1/24 3
B. Bimodal distribution.
Measurements of atmospheric aerosol size distributions in urban areas indicate a frequent occurrence of a
bimodal distribution for the aerosol volume (mass) concentration. The two modes of the volume
distribution are illustrated in Figure 4. There is accumulating evidence that the aerosol populations in the
two mass modes are formed by different mechanisms, come from different sources and have drastically
different chemical compositions. The two modes are separated by approximately one decade of particle
diameter or more, while the width of the two distributions generally does not exceed g = 2.5. This allows
a separation such that the overlap of the two modes is less than 10-20%. Since the two modes of aerosol
mass are contributed by different sources, they are practically decoupled from each other in their spatial
and temporal pattern.
The existence of bimodal aerosol volume distributions was first pointed out by Whitby et al, (1972), while
some of the physical mechanisms that shaped the bimodal distribution were discussed by Husar et al,
(1972). They have compiled a number of their own size spectrum measurements obtained in Los Angeles
by electrical mobility analyzer and an optical counter and compared whit data of other investigators
obtained in variety of locations (Figure 9).
The data shown in Figure 9 indicate the existence of two aerosol populations, the fine and the coarse
particles. It should be noted, however, that the relative magnitudes of the two modes (the integral under the
two modes) may vary drastically from location to location.
At present there is no generally accepted expression (formula) which would adequately describe the
bimodal distribution. However, visual inspection of volume or mass distributions such as shown in Figure
4 and 9 suggests that the two modes of the volume distribution can be reasonably well approximated by
two log-normal distributions.
We have pointed out previously that the size and the size distribution of a polydispersed aerosol population
depends on the method of measurement. It is, therefore, important to test whether the bimodal mass
distribution emerges when the size distribution is measured by other methods.
Recently, Kadowaki, (1976), reported mass distribution data obtained by a cascade impactor. His data are
shown in Table 10 for comparison. The normalized cumulative form of the mass distribution function is
shown in Table 11. The mass measurements of the individual stages of the cascade impactor, the size
ranges for each stage, and the calculated mass distribution function are given in Table 3. The above data
clearly indicate the existence of the bimodal distribution when the aerosol is characterized in terms of its
Figure 6. Comparison of volume distributions measured by several investigators in
different locations: Additional data are given in Table VIII. Note the universal bimodal
nature of all of these data and that the data obtained by Clark, Peterson, and from the
more recent Los Angeles and Colorado studies, were obtained under pollution-free
conditions such that it may be assumed that a background aerosol was being measured,
is rising sharply at 10 m. (Whitby, Husar, & Liu, 1972).
Figure 7. Histogram and size distribution curve of total aerosols in Nagoya (16-22 May
1974. Conc 131 gm3).