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The general governing equation for aerosol coagulation and condensation can be expressed using the moment formula as follows (Jung and Kim, 2006).

Here, Coag_k and Con_k represent the coagulation and condensation processes, respectively, and k stands for the kth moment.

By substituting for k=0, 3, 6 in Eq. (7) and using these three ordinary differential equations of moment relation, the change in the total number of particles (N), the numerical solutions for geometric standard deviation (σ_g), and the geometric mean diameter (d_g) can be obtained as a function of time (Jung and Kim, 2006; Jung et al., 2004, 2002). The Runge-Kutta 4^th order method can be applied.

Many efforts have been made to derive an analytic expression for the time evolution of aerosol size distribution during coagulation and condensation (Jung and Kim, 2006; Jung et al., 2001; Lee et al., 1984; Lee, 1983). Among them, Lee (1983) obtained the change in particle size distribution due to Brownian coagulation in the continuum regime analytically under the assumption of log-normal aerosol size distribution.

The obtained analytic solution can be expressed as follows:

where K is the collision coefficient (=2kT/3 μ); k, the Boltzmann constant; T, the absolute temperature; and μ, the gas viscosity; the subscript 0 stands for the initial conditions (t=0).

The geometric mean diameter, geometric standard deviation, and total number of particles can be expressed as a function of time according to Lee (1983) under the assumption of the geometric mean diameter does not deviate greatly from the initial geometric mean diameter.

More detailed procedure in obtaining analytic solution for coagulation (Eq. (8)) can be referred in the previous studies (Lee et al., 1984; Lee, 1983).

Likewise, for the condensation process, the moment relations can be usually expressed in terms of the 0th, 1^st, and 2nd moments (Park et al., 2001) as follows:

where, S is the saturation ratio; R, the gas constant; T, the absolute temperature; M, the molecular mass of the condensing vapor; D, the diffusion coefficient; p_s, the saturation vapor pressure; p, the total gas pressure; L, the latent heat; K, the thermal conductivity; and ρ, the particle density.

From Eq. (9), the following analytic solution can be obtained by applying the above moment relation to the condensational equation (Park et al., 2001).

Note that the total number concentration does not change during the condensation process, and the two size parameters of the log-normal distribution-geometric mean diameter (d_g) and geometric standard deviation (σ_g)-are allowed to vary with time (Jung et al., 2002).

Subsequently, from Eqs. (5) and (6), the Ångstrom exponent can be expressed as follows.

The Ångstrom exponent for the aerosol dynamic process can be simulated using the Mie theory, by subtituting in Eq. (11) the size distribution parameters from Eq. (8) for coagulation and from Eq. (10) for condensation. The detailed description of the analytic solution during the two processes can be found in literature (Jung et al., 2006; Park et al., 2001; Lee et al., 1984; Lee, 1983).

During coagulation and condensation, aerosol continuously changes their size as a function of time. Basically, the aerosol extinction coefficient is a function of the size distribution and compositions, which determine the refractive index. During coagulation, the particle number concentration decreases, geometric mean diameter increases, and overall extinction coefficient (b_ext) decreases (Jung and Kim, 2008). A higher extinction coefficient is observed for wide aerosol distributions with a large geometric standard deviation than for narrow aerosol distributions with a small geometric standard deviation. During condensation, the visibility decreases and the overall extinction coefficient (b_ext) increases (Jung and Kim, 2008).

Fig. 1 shows the change in the size distribution and the Ångstrom exponent during the coagulation process. Both, the analytic and numerical driven solutions are shown for comparison.

Fig. 1. 
The changes in the size distribution and the Ångstrom exponent during the coagulation process (Numerical vs. analytic solution, x axis: coagulation-based dimensionless time).

The initial values of the size distribution parameters are as follows: geometric standard deviation, 1.2; geometric mean diameter, 0.2 μm; and total number concentration, 10,000/cm³. KN₀t is the coagulation-based dimensionless time (Lee, 1983).

As Fig. 1 shows, the changes in the Ångstrom exponent with the size distribution as derived from analytical and numerical solutions agree very well; note that the “analytic solution” refers to the coagulation and condensation dynamic solution from previous section. Fig. 1 also shows that the Ångstrom exponent decreases during the coagulation process. The large initial geometric standard deviation with wide size distribution corresponds to a larger Ångstrom exponent than does the small initial geometric standard deviation.

Fig. 2 shows the change in the Ångstrom exponent with the geometric mean diameter during the coagulation process. The initial values are as follows: geometric standard deviation, 1.2; total number concentration, 10,000/cm³; and refractive index, 1.53-0.015i.

Fig. 2. 
The changes in the Ångstrom exponent during the coagulation process for different initial geometric mean diameters (Numerical vs. analytic solution, x axis: coagulationbased dimensionless time, y axis: Ångstrom exponent).

During coagulation, the total number concentration decreases, geometric mean diameter converges to a value around 1.32, and the geometric mean diameter increases (Lee, 1983).

As Fig. 2 shows, the Ångstrom exponent decreases rapidly as the initial geometric mean diameter increases. Fig. 2 also shows that the Ångstrom exponent decreases to a lower value for a large initial geometric mean diameter than for a small initial geometric mean diameter.

Fig. 3 shows the change in the size distribution and the Ångstrom exponent during the condensation process. The condensation rate of 1e-15 cm³/cm³/s was considered for different initial geometric standard deviation values. We used KN₀t for the condensation process as a time variable in order to compare coagulation results for the same time dimension. The Ångstrom exponents derived based on the size parameter from numerical and analytic solutions were compared and showed good agreement. Throughout the condensation process, the number concentration did not change; the geometric mean diameter converged to 1, and the geometric mean diameter increased. As Fig. 3 shows, the Ångstrom exponent decreases as the initial geometric standard deviation increases. The Ångstrom exponent decreases during condensation, and the rate of decrease in the Ångstrom exponent is larger for small initial geometric standard deviation values. Fig. 3 also shows that the difference in the Ångstrom exponent between numerical and analytic solution increase with time. The discrepancy in size parameter estimation between the numerical and analytic results for the condensation process is larger than that for the coagulation process; this is due to the difference between the Ångstrom exponents for the numerical and analytic results.

Fig. 3. 
The changes in the size distribution and the Ångstrom exponent during the condensation process (Numerical vs. analytic solution, x axis: coagulation-based dimensionless time).

Fig. 4 shows the change in the Ångstrom exponent during condensation under different condensation conditions. As Fig. 4 shows, the exponent decreases to a minimum value and then increases as condensation proceeds. Fig. 4 shows that increasing the condensation rate accelerates these tendencies because the size distribution parameters change more rapidly as the condensation rate (=C(S-1)) increases.

Fig. 4. 
The changes in the Ångstrom exponent during condensation under different condensation conditions (Numerical vs. analytic solution, x axis: coagulation-based dimensionless time).

As Figs. 1-4 show, the change in the Ångstrom exponent is closely related to the refractive index of particles as well as size distribution. Fig. 5 shows the sensitivity of the Ångstrom exponent as a function of geometric standard deviation (σ_g) and geometric mean diameter (d_g) for different refractive indices. In Fig. 5, the Ångstrom exponent has been calculated for the same size distribution with different refractive indices. The Ångstrom exponent approaches 0 for large particles and 1 for small particles with diameters less than 0.01 mm. For nuclei-mode particles with geometric mean diameters less than 0.1 mm, the Ångstrom exponent approaches 4 in the case of a real refractive index for nonabsorbing particles. However, for absorbing particles with an imaginary refractive index, the Ångstrom exponent approaches 1 (Jung and Kim, 2013); this explains the behavior observed at the right bottom edge of Fig. 5(b). The geometric mean diameter at which the Ångstrom exponent has a peak value decreases as the initial geometric standard deviation increases and the imaginary part of the refractive index decreases. The peak value decreases as the imaginary part of the refractive index increases (Jung and Kim, 2013).

Fig. 5. 
Sensitivity of the Ångstrom exponent as a function of the geometric standard deviation and geometric mean diameter for different refractive indices (x axis: geometric standard deviation, y axis: geometric mean diameter).

In Fig. 5(a), (b) and (c), the results for refractive indices 1.53, 1.53-0.01i, and 1.53-0.1i are compared to determine the effect of absorbing particles (the imaginary part) on the Ångstrom exponent. In Fig. 5(a) and (d), the results for refractive indices 1.53 and 2 are compared to determine the effects of the real part of the refractive index. As Fig. 5 shows, the Ångstrom exponent is sensitive to the refractive index; the figure shows that for nonabsorbing particles, where only real part of refractive index considered, the Ångstrom exponent decreases as the geometric standard deviation and geometric mean diameter increases. For absorbing particles (i.e., taking into account the imaginary part), however, the Ångstrom exponent may increase or decrease depending on the initial geometric standard deviation and geometric mean diameter. Generally, the Ångstrom exponent has a maximum value for a geometric mean diameter of ~0.1 μm and small geometric standard deviation. As the geometric standard deviation increases, the geometric mean diameter at which the Ångstrom exponent becomes maximum decreases.

The size distribution and refractive index are very crucial in estimating the Ångstrom exponent for aerosol dynamic processes.

In order to quantify the effect of refractive index, Fig. 6 also describes the sensitivity of the Ångstrom exponent under different size distributions and refractivity conditions. Fig. 6 shows the sensitivity of the Ångstrom exponent as a function of the real and imaginary part of the refractive indices with different size parameters.

Fig. 6. 
Sensitivity of the Ångstrom exponent as a function of the real and imaginary parts of the refractive indices for different size parameters (x axis: the real part of the refractive index, y axis: the imaginary part of the refractive index).

Initial geometric mean diameters of 0.3 μm and 0.6 μm with geometric standard deviations of 1.5 and 2.0 were considered. As Fig. 6 shows, the Ångstrom exponent decreases as the geometric standard deviation increases and as the geometric mean diameter increases. Fig. 6 also shows that there exist several pairs of real and imaginary refractive index corresponding to a given Ångstrom exponent. For example, for a geometric mean diameter of 0.3 μm and geometric standard deviation of 1.5, the refractive indices 1.1-0.1i and 1.57 have the same Ångstrom exponent of 1.0. This means that the Ångstrom exponent and refractive index do not have a one-to-one relation, which makes it difficult to retrieve particle size distribution or particle characteristics from only the measurement of optical properties.

The Ångstrom exponent is a function of aerosol size distribution and aerosol refractive index. The refractive index is related to the chemical composition of the aerosol, and the real and imaginary parts of the refractive index represent the scattering and absorption of aerosol, respectively. During the coagulation and condensation processes, aerosol size distribution changes continuously as a function of time, and this change affects the change in the Ångstrom exponent.

For example, the total number concentration decreases and the geometric mean diameter converges to a value of around 1.32 during coagulation. One the while, the geometric standard deviation approaches 1, indicating that the particle size distribution converges to a monodisperse distribution during the condensation process.

The degree of change in the Ångstrom exponent is closely related to the refractive index and initial aerosol size distribution, and it is difficult to pinpoint the parameter to which the Ångstrom exponent is more sensitive. However, Figs. 5 and 6 show the degree of change in the Ångstrom exponent as a function of the size parameters such as the geometric mean diameter and geometric standard deviation, and show that the change is more sensitive to the above mentioned size parameters than to refractive index under the simulation conditions.