Charging and coagulation influence one another and impact the particle charge and size distributions in the atmosphere. However, few investigations to date have focused on the coagulation kinetics of atmospheric particles accumulating charge. This study presents three approaches to include mutual effects of charging and coagulation on the microphysical evolution of atmospheric particles such as radioactive particles. The first approach employs ion balance, charge balance, and a bivariate population balance model (PBM) to comprehensively calculate both charge accumulation and coagulation rates of particles. The second approach involves a much simpler description of charging, and uses a monovariate PBM and subsequent effects of charge on particle coagulation. The third approach is further simplified assuming that particles instantaneously reach their steady-state charge distributions. It is found that compared to the other two approaches, the first approach can accurately predict time-dependent changes in the size and charge distributions of particles over a wide size range covering from the free molecule to continuum regimes. The other two approaches can reliably predict both charge accumulation and coagulation rates for particles larger than about 0.04 micrometers and atmospherically relevant conditions. These approaches are applied to investigate coagulation kinetics of particles accumulating charge in a radioactive neutralizer, the urban atmosphere, and an atmospheric system containing radioactive particles. Limitations of the approaches are discussed.

Atmospheric particles play an important role in airborne transport of
contaminants, such as radionuclides. Contaminants emitted from anthropogenic
sources (e.g., nuclear plant accidents) can be captured by background
aerosols and are then transported together with pre-existing particles.
Contaminant-laden particles can be deposited onto the ground by dry and wet
deposition (primary contamination) and subsequently resuspended by wind or
heat-driven convection and moved to other areas (secondary contamination).
For instance, due to these atmospheric dispersion patterns, radioactive
particles (e.g.,

The microphysical behavior of atmospheric particles is driven by such properties as charge and size (Fuchs, 1989; Pruppacher and Klett, 1997). Atmospheric particles can acquire charge via self-charging and diffusion charging. Self-charging refers to charge accumulation caused by radioactive decay which typically leads to emission of electrons from particle surfaces. Diffusion charging is attributed to diffusion of ions from the surrounding atmosphere onto the surface of particles. Radioactive particles can be charged through these two charging mechanisms (Greenfield, 1956; Yeh et al., 1976; Clement and Harrison, 1992; Clement et al., 1995; Gensdarmes et al., 2001; Walker et al., 2010; Kweon et al., 2013; Kim et al., 2014, 2015) while natural atmospheric particles are typically charged by diffusion charging (Hoppel, 1985; Yair and Levin, 1989; Renard et al., 2013). Particle charging can modify not only the charge but also the size distribution because charge on the particles modifies their coagulation rate coefficients by generating electrostatic interactions (Fuchs, 1989; Tsouris et al., 1995; Chin et al., 1998; Taboada-Serrano et al., 2005). Coagulation of atmospheric particles can influence their charging because the concentration of atmospheric ions is affected by the particle size distribution (Yair and Levin, 1989), thereby modifying the diffusion charging rates of the particles. Also, particle coagulation can result in charge neutralization or accumulation on atmospheric particles (Alonso et al., 1998). These effects imply that particle coagulation can influence the particle charge distribution. Thus, particle charging and coagulation can mutually affect each other and simultaneously affect both charge and size distributions in the atmosphere.

Theoretical and experimental investigations have been performed to examine the charging of radioactive particles and background aerosols in the atmosphere. However, the effects of coagulation of particles on the charge distribution have been frequently neglected by assuming that the size distribution is constant while they are charged (Greenfield, 1956; Hoppel, 1985; Yair and Levin, 1989). The assumption may be valid if the particle concentration is low or the steady-state charge distribution is instantaneously attained (Hoppel, 1985; Renard et al., 2013; Kim et al., 2015). If the timescale for particle charging is longer than that for particle coagulation, the assumption may no longer be valid (Yair and Levin, 1989). Also, the effects of charging on the particle size distribution are frequently neglected in aerosol transport models involving microphysics of atmospheric particles. A possible reason for neglecting the charging effects may be that the steady-state mean charge of atmospheric particles may rarely affect their coagulation rates (Seinfeld and Pandis, 2006). However, neglecting electrostatic particle–particle interactions may increase uncertainty of prediction results if particles can acquire multiple elementary charges (e.g., radioactive particles). The simplified assumption of omitting electrostatic particle interactions may create uncertainty in transport predictions of radioactive particles. Hence, it may be necessary to take into account the mutual effects of particle charging and coagulation processes in predicting the behavior of atmospheric particles carrying radioactive contaminants.

Previous attempts to consider charging effects include Oron and Seinfeld (1989a, b), who developed sectional approaches to simultaneously predict the behavior of charged and uncharged atmospheric particles. Laakso et al. (2002) developed a general dynamic equation, including charging and coagulation kinetics of atmospheric particles. Yu and Turco (2001) presented a kinetic approach that involves diffusion charging and particle coagulation. However, the validity of these approaches was not evaluated using analytical solutions. Alonso (1999) and Alonso et al. (1998) developed analytical and numerical approaches to estimate time-dependent changes in the size distributions of singly charged and neutral particles; thus, these approaches cannot be used to investigate the coagulation kinetics of particles acquiring multiple elementary charges. Also, none of these approaches considered self-charging; therefore, the aforementioned approaches may be subject to error when they are used to simulate atmospheric dispersion of radioactive plumes.

Our study presents three approaches to simultaneously predict time-dependent changes of the charge and size distributions of radioactive and nonradioactive particles over a wide size range. Development, validity, application, and limitations of these approaches are discussed.

Many atmospheric processes can generate and remove ions in air. Typical ion
sources in the atmosphere involve natural and artificial radioactivity, as
well as cosmic rays. Ions are generally removed by ion-ion recombination and
ion-particle attachment. Changes in ion concentrations by these processes
can be given by (Kim et al., 2015)

Self-charging generally accumulates positive charge on the surface of
particles, while diffusion charging adds both positive and negative charges,
indicating that the charging mechanisms can compete with one another. For
radioactive particles involved in beta decay, time-dependent changes in
their charge distributions due to competition of the charging mechanisms can
be expressed by (Clement and Harrison, 1992; Kim et al., 2015):

A bivariate population balance model, expressed in terms of particle volume

Three approaches to predict time-dependent changes in the particle size and charge distributions in the atmosphere.

The time evolution of the size distribution of particles can be estimated
using a monovariate population balance model, with only the particle volume
as the variable (Kumar and Ramkrishna, 1996):

Figure 1 shows three approaches which can be used to predict the
time evolution of the charge and size distributions of particles in the
atmosphere. All the approaches can be used to simulate charging and
coagulation kinetics of atmospheric particles carrying contaminants,
including radioactive particles. Approach 1 is a rigorous scheme that
simultaneously computes both charge accumulation and coagulation rates of
particles using the ion balance model (Eqs. 1 and 2), the charge balance
model (Eq. 5), and the bivariate population balance model (Eq. 11).
Approach 2 is a simplified scheme of Approach 1, which can be used to predict
the particle charge distribution using the mean charge of particles (Eq. 7)
and the Gaussian distribution (Eq. 16). In order to easily simulate the
coagulation of charged particles, Approach 2 employs the monovariate
population balance model (Eq. 14) that corrects the collision frequency using
the average collision efficiency (Eq. 15). Approach 2 can be simplified to
Approach 3 by assuming that charge accumulation rates of particles
instantaneously reach a steady state, with a timescale based on 5 times
larger than

Properties of ions used for experimental observations.

The three approaches attained above employ different methods to simulate
charging of particles. These methods were evaluated by comparing their
prediction results with measurements obtained using radioactive charge
neutralizers (Liu and Pui, 1974; Wiedensohler and Fissan, 1991; Alonso et
al., 1997) and radioactive particles (Gensdarmes et al., 2001). Initial
conditions for the simulations were determined from the measurements. The
properties of ions observed during the measurements are shown in Table 1. For
the measurements providing the values of ion mass,

Figure 2 shows the steady-state charge distributions of nonradioactive
particles over a wide size range. Here, the particles were charged by the
diffusion charging mechanism. For particles larger than approximately
0.04

Steady-state charge distributions of particles capturing positive and negative ions. The symbols represent the measurements of the charge distributions of particles.

Analysis of the discrepancies suggests that they originate from the standard
deviation involved in the Gaussian distribution (Eq. 16). At a given
temperature, the width of the particle charge distributions can be
significantly influenced by three parameters: the particle size, ion mass,
and ion mobility (Wiedensohler and Fissan, 1991). In Approaches 2 and 3,
however, the effects of the ion properties are not involved, so particle size
primarily drives the standard deviation, which can differ from what
Approach 1 gives. When Approaches 2 and 3 used the standard deviation values
obtained by Approach 1, their simulation results became closer to the
measurements, although the discrepancies are still seen for negatively and
positively charged particles smaller than about 0.02

In our previous work (Kim et al., 2014, 2015), it has been shown that
Approaches 1 and 3 can reliably simulate charging of radioactive particles.
Thus, in this study, we focused on evaluating the validity of Approach 2 with
the experiments of Gensdarmes et al. (2001) who measured the charge
distributions of

Figure 3 shows the charge accumulation on radioactive particles under two
ionizing conditions:

Charge accumulation on

Charge distributions of

Timescales required for particles to reach steady-state charge.

To evaluate the steady-state assumption of particle charging for atmospheric
conditions, the timescale for reaching steady state (Eq. 8) is evaluated with
Approaches 1 and 2. Figure 5 shows time-dependent changes in the
concentrations of negatively charged particles under two different initial
conditions of Alonso et al. (1997) who measured the charge distributions of
particles of a few nanometers. All particles were initially uncharged or
negatively charged. Because the particle size was very small, Approach 1 was
used to predict the time evolution of the particle concentrations. As time
elapsed, the initially uncharged particles became negatively charged by
capturing negative ions. The diffusion of positive ions led to the
discharging of the initially negatively charged particles. For the initial
conditions used, the charging and discharging rates of the particles reached
a steady state after approximately 0.2 s, respectively. This
charging and/or discharging behavior predicted by Approach 1 is in good agreement
with the measurements of Alonso et al. (1997). However, the timescales
obtained from Eq. (8) are shorter than the prediction results, as well as the
measurements, because Approach 1 and the observations provided exact
timescales, while

Timescale to reach steady-state charge accumulation rates of
0.0071

Equation (8) is based on the assumptions that (i)

Charge accumulation rate of

So far, we have evaluated the validity of the methods used in the three
approaches to predict charge accumulation on atmospheric particles. The
evaluation results suggest that the method employed in Approach 1 can
accurately simulate charging of particles in the free molecule
(

Based on the numerical approach of Alonso et al. (1998), Alonso (1999)
suggested an analytical approach to simultaneously investigate charging and
coagulation kinetics of nonradioactive particles, smaller than
0.02

Validation of the numerical solution for the bivariate population
balance model under a monodispersed initial condition (

Approaches 2 and 3 employ an average collision efficiency and are coupled to
the monovariate, instead of the bivariate, population balance model. These
approaches provided accurate particle charge distributions for various cases
(e.g., Figs. 2 and 4; Kim et al., 2014), suggesting that their validity may
be highly influenced by the accuracy of the average collision efficiency.
Thus, we compared simulation results of Approach 2 with those of Approach 1
to check if the average collision efficiency (Eq. 15) can appropriately
account for the influence of the charge distributions of particles on their
size growth via coagulation. For comparison, simulation results of Approach 2
using the mean charge (Eq. 13), as well as those for uncharged particles,
were included. Similarly to Oron and Seinfeld (1989a, b), we assumed
monodispersed initial size distributions (

Figure 8 shows the time evolution of the particle size distributions induced
by particle charging and coagulation. The simulation conditions led to the
accumulation of more negative than positive charges on the particles. At

Time evolution of the particle size distributions predicted by the
monovariate population balance model with the average collision efficiency
(Eq. 15) under a monodispersed initial condition (

While most particles were negatively charged, some particles captured
positive ions. Owing to electrostatic attractive forces, the positively
charged particles can more frequently coagulate with the negatively charged
particles and grow. Therefore, the coagulation rates predicted by Approach 2
with the average collision efficiency were slightly higher than those for the
case assuming that all particles were negatively charged (Approach 2 (Eq. 13)
vs. Approach 2 (Eq. 15)). These coagulation patterns predicted by Approach 2
using the average collision efficiency were in good agreement with those
given by Approach 1 (Approach 1 vs. Approach 2 (Eq. 15)), as well as the
particle charge distributions in various size ranges (Fig. S1 in the
Supplement). Similar results were obtained for different initial particle
size distributions (

Radioactive neutralizers are typically used to control the charge of
atmospheric particles in many laboratory-scale experiments. The applicability
of the three approaches to studies using radioactive neutralizers was
evaluated using the experiments of Alonso et al. (1998) who measured the size
distribution of nanometer-size particles passing by a

Figure 9 shows the size distribution of negatively charged particles when the
residence time,

Evolution of the size distribution of negatively charged particles
in a

Time evolution of the charge

As shown in Figs. 2–5, Approach 1 can accurately predict the charge accumulation rate of radioactive and nonradioactive particles in the free-molecule, transition, and continuum regimes. Approach 1 employs the interpolation formula of Fuchs that can be used to compute the collision frequency of the particles in these regimes, revealing that this approach can also precisely predict the charge distribution of larger particles undergoing coagulation. These results suggest that Approach 1 can be a reasonable option to simultaneously simulate charging and coagulation of particles of any size in laboratory-scale experiments.

Hoppel (1985) simulated charging of 0.06

Figure 10 presents changes in the particle charge and size distributions vs. time. The simulation results performed by Hoppel (1985) showed that the particle charge distribution approached its steady-state value after approximately 90 min. However, as time elapsed, the particles grew in size due to coagulation. The size growth led to the generation of large particles capturing many ions, thereby modifying the particle charge distribution.

The simulation results of Hoppel (1985) also indicated that the ion concentrations became unchanged after reaching a steady state. However, coagulation reduced the particle number concentrations which can affect the loss rate of ions by diffusion charging (see Eqs. 1 and 2). The reduction in the particle concentrations increased the ion concentrations, thereby enhancing the electrical conductivity of the postulated atmosphere (Fig. 11). The ion concentrations and electrical conductivity are expected to increase until ion-ion recombination becomes the major ion removal mechanism. These results suggest that coagulation can affect the electrical properties in the atmosphere, as well as the particle charge distribution.

Time evolution of the mean ion concentration,

Nuclear events can release particles carrying radionuclides.
Greenfield (1956) simulated time evolution of the charge distribution of
0.1

Figure 12 shows changes in the particle charge distributions vs. time. Both self-charging and diffusion charging influenced the charge accumulation on radioactive particles. Due to many ion pairs produced by beta radiation (Fig. S2), positive charge accumulated on the particles by self-charging was rapidly neutralized by capturing negative ions. Thus, the particle charge distribution given by Approach 1 was slightly shifted to the right of the zero elementary charge in Fig. 12a, although large particles with a high level of radioactivity were generated by coagulation. As time elapsed, the particle charge distribution was slightly moved to the left. Because the decay rates of the highly radioactive particles were reduced over time, their self-charging rates also decreased, and this led to the slight movement of the charge distribution to the left in Fig. 12a.

Time evolution of the charge

The discrepancies between the predictions of Approach 1 and Greenfield (1956) result mainly from the ion-particle attachment coefficient used in the simulation. The values assumed by Greenfield (1956) were beyond the ion-particle attachment coefficient found by other researchers (e.g., Hoppel and Frick, 1986), leading to the discrepancies observed (Kim et al., 2015).

Beta radiation caused by radioactive decay rapidly increased the ion concentrations, thereby enhancing the electrical conductivity in the atmosphere (Fig. S2). In contrast to the case shown in Fig. 11, the ion concentrations and air conductivity significantly decreased with time because the ionization rate of air molecules decreased considerably, and ion-ion recombination was responsible for the change in the concentrations. Nevertheless, the air conductivity enhancement by beta radiation was much higher than that by cosmic rays and natural radioactivity.

After the Chernobyl and Fukushima accidents, short- and long-range transport
of particles carrying radionuclides, such as

The steady-state assumption of particle charging can be useful to simulate coagulation of radioactive particles in model studies of radioactivity transport. Charging and coagulation kinetics of radioactive particles were investigated using Approaches 2 and 3 to evaluate the validity of the steady-state assumption of radioactive particle charging. For comparison, the size growth of particles by coagulation was simulated by assuming the Boltzmann charge distribution.

Computational costs of the approaches used.

We used the simulation condition employed to validate the average collision
efficiency, but additionally presumed that radioactive decay of

Figure 13 shows the charge and size distributions of the

The charge

The computational costs to predict transport of particles containing
contaminants depends on the number of ordinary differential equations (ODEs)
solved during simulation. Thus, the number of ODEs involved in the three
approaches was evaluated by assuming 30 size bins, which corresponds to those
used in the two-moment aerosol sectional microphysics model, covering
particle diameters from 0.01 to 10

Table 3 shows an example of the computational costs of the three approaches. For Approach 1, we assumed that atmospheric particles can acquire up to fifteen elementary charges regardless of their sign, thereby resulting in 932 ODEs. Because Approaches 2 and 3 employed the monovariate population balance model, a fewer number of ODEs were involved in these Approaches than in Approach 1, suggesting that they are computationally more efficient. For instance, compared to Approach 1, Approaches 2 and 3 more quickly computed the charge accumulation and coagulation rates of urban aerosols.

A simple way to reduce the number of ODEs included in Approach 1 is to assume
that atmospheric particles acquire only a few electrical charges. For
example, Laakso et al. (2002) assumed that submicron particles can acquire
elementary charges from

Approach 3 includes all the physics of charging and coagulation to predict the particle size and/or charge distribution, but, compared to Approaches 1 and 2, it is computationally more suitable for use in a 3-D global transport model to predict the transport of radioactivity in the environment after a radiological event such as a nuclear plant accident.

Understanding the behavior of atmospheric particles is important to accurately predict short- and long-range transport of contaminants. Particle charging and coagulation processes can strongly affect the behavior of atmospheric particles because these processes can change their important physical and electrical properties, such as size and charge. This study has shown three approaches with a wide range of complexity and applications to involve the mutual effects of charging and coagulation processes in the simulation of particle charge and size distributions vs. time. Depending on the initial conditions, these approaches can be employed to accurately predict the behavior of atmospheric particles carrying radioactive contaminants. We have shown the approaches to be applicable to a wide variety of atmospheric (laboratory and field) applications. The accuracy of the approaches depends on the assumptions made to reduce computational cost. The developed approaches can be readily incorporated into microphysical and transport models of any scale to account for charging phenomena of atmospheric particles.

This work was supported by the Defense Threat Reduction Agency under grant number DTRA1-08-10-BRCWMD-BAA. The manuscript has been co-authored by UT-Battelle, LLC, under Contract No. DEAC05-00OR22725 with the US Department of Energy. Edited by: M. Kanakidou