Real-world aerosols are never composed of particles of a single, uniform size; instead, they exist as populations with a range of sizes. Therefore, describing an aerosol requires statistical distributions. Understanding the nature of these distributions is strategically essential for predicting aerosol transport, quantifying health risks, and modeling environmental impacts such as visibility reduction.

The normalized particle size distribution, denoted f(Dp), describes the relative frequency of particles within a given size range. It is formally defined as:

f(Dp) = (1/N) * (dn/dDp) (Eq. 1)

where N is the total number concentration of particles and dn is the number of particles with diameters between Dp and Dp + dDp. The integral of this function over all possible diameters is equal to one. The cumulative number concentration, F(Dp), gives the fraction of all particles with diameters up to a certain size Dp:

F(Dp) = ∫ f(Dp) dDp (from 0 to Dp) (Eq. 3)

It is critical to distinguish between distributions based on particle number and those based on other quantities like particle mass. A population may be dominated by a high number of very small particles, but the total mass may be dominated by a few very large particles.

The log-normal distribution is a cornerstone model in aerosol science because it accurately represents many real-world distributions while disallowing non-physical negative particle sizes. The log-normal distribution function is given by:

f(Dp) = [1 / (Dp * ln(sg) * √(2π))] * exp[- (ln(Dp) – ln(Dg))^2 / (2 * (ln(sg))^2)] (Eq. 8)

Plotting a log-normal distribution on log-probability coordinates yields a straight line. This graphical method allows for the rapid determination of the distribution’s key parameters:

• Geometric Mean Diameter (Dg) or Number Median Diameter (NMD): This is the central point of the distribution, corresponding to the particle size at the 50% mark on the cumulative frequency axis.
• Geometric Standard Deviation (sg): This parameter describes the spread or width of the distribution. It is calculated from the particle sizes found at the 84.13% and 50% (or 15.87%) cumulative points: sg = Dp(at 84.13%) / Dp(at 50%).

——————————————————————————–

Figure 1: Log-normal size distribution for particles with a geometric mean diameter of 1 μm and a geometric standard deviation of 2.0. The significant difference between the number-based (solid line) and mass-based (dashed line) distributions is clearly visible.

——————————————————————————–

The distinction between number-based and mass-based distributions is significant, as illustrated by the solid and dashed lines in Figure 1. For a log-normal distribution, the Number Median Diameter (NMD) can be converted to the Mass Median Diameter (MMD) using the following relationship:

ln(MMD) = ln(NMD) + 3 * (ln(sg))^2 (Eq. 11)

For any given log-normal distribution defined by its NMD and sg, various average particle diameters can be calculated, each with a different physical significance. These include:

• Number Mean Diameter (D1): The first moment of the distribution, representing the average diameter across the total number of particles.
• Length Mean Diameter (D2): An average weighted by particle length or diameter.
• Surface Mean Diameter (D3): Also known as the Sauter mean diameter, this represents the diameter of a sphere that has the same volume-to-surface-area ratio as the entire particle population.
• Mass Mean Diameter (D4): An average weighted by particle volume or mass.
• Diameter of Average Surface (Ds): The diameter of a particle possessing the mean surface area of the population.
• Diameter of Average Volume (Dv): The diameter of a particle possessing the mean volume (or mass) of the population.
• Harmonic Mean Diameter (Dh): An average that is sensitive to smaller particles in the distribution.
• Number Median Diameter (NMD): The geometric mean diameter; the 50th percentile of the number distribution.
• Mass Median Diameter (MMD): The 50th percentile of the mass distribution, separating the population’s mass into two equal halves.

Atmospheric aerosols often exhibit a bimodal distribution, which is a direct atmospheric manifestation of the two formation mechanisms discussed previously. The surface area distributions shown in Figure 2 reveal a typical minimum around 1 μm. This separates fine particles (submicron), generated by nucleation-based processes like combustion and atmospheric chemistry, from coarse particles (supermicron), which originate from comminution processes like dust storms and sea spray.

Having established how to statistically describe aerosol populations, we can now examine the physical laws that govern their movement and interaction within a fluid.