H_{2}SO_{4}–H_{2}O–NH_{3} ternary ion-mediated nucleation (TIMN): kinetic-based model and comparison with CLOUD measurements

H_{2}SO_{4}–H_{2}O–NH_{3} ternary ion-mediated nucleation (TIMN): kinetic-based model and comparison with CLOUD measurements

H_{2}SO_{4}–H_{2}O–NH_{3} ternary ion-mediated nucleation (TIMN): kinetic-based model and comparison with CLOUD measurementsH_{2}SO_{4}–H_{2}O–NH_{3} ternary ion-mediated nucleation (TIMN): kinetic-based model and comparison...Fangqun Yu et al.

Fangqun Yu^{1},Alexey B. Nadykto^{1,2,3},Jason Herb^{1},Gan Luo^{1},Kirill M. Nazarenko^{2},and Lyudmila A. Uvarova^{2,3}Fangqun Yu et al. Fangqun Yu^{1},Alexey B. Nadykto^{1,2,3},Jason Herb^{1},Gan Luo^{1},Kirill M. Nazarenko^{2},and Lyudmila A. Uvarova^{2,3}

^{1}Atmospheric Sciences Research Center, University at Albany, Albany,
New York, USA

^{2}Department of Applied Mathematics, Moscow State
Univ. of Technology “STANKIN”, Moscow, Russian Federation

^{3}National Research Nuclear University MEPhI (Moscow Engineering
Physics Institute), Department of General Physics, Moscow, Russian
Federation

^{1}Atmospheric Sciences Research Center, University at Albany, Albany,
New York, USA

^{2}Department of Applied Mathematics, Moscow State
Univ. of Technology “STANKIN”, Moscow, Russian Federation

^{3}National Research Nuclear University MEPhI (Moscow Engineering
Physics Institute), Department of General Physics, Moscow, Russian
Federation

New particle formation (NPF) is known to be an important source
of atmospheric particles that impacts air quality, hydrological cycle, and
climate. Although laboratory measurements indicate that ammonia enhances NPF,
the physicochemical processes underlying the observed effect of ammonia on
NPF are yet to be understood. Here we present a comprehensive
kinetically based H_{2}SO_{4}–H_{2}O–NH_{3}
ternary ion-mediated nucleation (TIMN) model that is based on the
thermodynamic data derived from both quantum-chemical calculations and
laboratory measurements. NH_{3} was found to reduce nucleation
barriers for neutral, positively charged, and negatively charged clusters
differently, due to large differences in the binding strength of
NH_{3}, H_{2}O, and H_{2}SO_{4} to small clusters of
different charging states. The model reveals the general favor of nucleation
of negative ions, followed by nucleation on positive ions and neutral
nucleation, for which higher NH_{3} concentrations are needed, in
excellent agreement with Cosmics Leaving OUtdoor Droplets (CLOUD)
measurements. The TIMN model explicitly resolves dependences of nucleation
rates on all the key controlling parameters and captures the absolute
values of nucleation rates as well as the dependence of TIMN rates on
concentrations of NH_{3} and H_{2}SO_{4}, ionization
rates, temperature, and relative humidity observed in the well-controlled
CLOUD measurements well. The kinetic model offers physicochemical insights into
the ternary nucleation process and provides a physics-based approach to
calculate TIMN rates under a wide range of atmospheric conditions.

New particle formation (NPF), an important source of particles in the
atmosphere, is a dynamic process involving interactions among precursor gas
molecules, small clusters, and preexisting particles (Yu and Turco, 2001;
Zhang et al., 2012). H_{2}SO_{4} and H_{2}O are known to
play an important role in atmospheric particle formation (e.g., Doyle, 1961).
In typical atmospheric conditions, the species dominating the formation and
growth of small clusters is H_{2}SO_{4}. The contribution of
H_{2}O to the nucleation is related to the hydration of
H_{2}SO_{4} clusters (or, in the other words, modification of
the composition of nucleating clusters), which reduces the
H_{2}SO_{4} vapor pressure and hence diminishes the evaporation
of H_{2}SO_{4} from the pre-nucleation clusters. NH_{3}, the
most abundant gas-phase base molecule in the atmosphere and a very efficient
neutralizer of sulfuric acid solutions, has long been proposed to enhance
nucleation in the lower troposphere (Coffman and Hegg, 1995), although it has
been well recognized that earlier versions of the classical ternary nucleation
model (Coffman and Hegg, 1995; Korhonen et al., 1999; Napari et al., 2002)
significantly overpredict the effect of ammonia (Yu, 2006a; Merikanto et
al., 2007; Zhang et al., 2010).

The impacts of NH_{3} on NPF have been investigated in a number of
laboratory studies (Kim et al., 1998; Ball et al., 1999; Hanson and Eisele,
2002; Benson et al., 2009; Kirkby et al., 2011; Zollner et al., 2012; Froyd
and Lovejoy, 2012; Glasoe et al., 2015; Schobesberger et al., 2015; Kürten et
al., 2016)
including those recently conducted at the European Organization
for Nuclear Research (CERN) in the framework of the CLOUD (Cosmics Leaving
OUtdoor Droplets) experiment that has provided a unique dataset for
quantitatively examining the dependences of ternary
H_{2}SO_{4}–H_{2}O–NH_{3} nucleation rates on
concentrations of NH_{3} ([NH_{3}]) and H_{2}SO_{4}
([H_{2}SO_{4}]), ionization rate (Q), temperature (T), and
relative humidity (RH) (Kirkby et al., 2011; Kürten et al., 2016). The
experimental conditions in the CLOUD chamber, a 26.1 m^{3} stainless steel
cylinder, were well controlled, while impacts of potential contaminants were
minimized (Schnitzhofer et al., 2014; Duplissy et al., 2016). Based on CLOUD
measurements in H_{2}SO_{4}–H_{2}O–NH_{3} vapor
mixtures, Kirkby et al. (2011) reported that an increase in [NH_{3}]
from ∼0.03 ppb (parts per billion, by volume) to ∼0.2 ppb can
enhance ion-mediated (or induced) nucleation (IMN) rate by 2–3 orders of magnitude
and that the IMN rate is a factor of 2 to
> 10 higher than that of neutral nucleation under a typical level
of contamination by amines. In the presence of ionization, common highly polar
atmospheric nucleation precursors such as H_{2}SO_{4},
H_{2}O, and NH_{3} molecules tend to cluster around ions, and
charged clusters are generally much more stable than their neutral
counterparts with enhanced growth rates as a result of dipole–charge
interactions (Yu and Turco, 2001).

Despite various laboratory measurements indicating that ammonia enhances
NPF, the physicochemical processes underlying the observed different effects
of ammonia on the formation of neutral, positively charged, and negatively
charged clusters (Schobesberger et al., 2015) are yet to be understood. To
achieve such an understanding, a nucleation model based on the first
principles is needed. Such a model is also necessary to extrapolate data
obtained in a limited number of experimental conditions to a wide range of
atmospheric conditions, in which [NH_{3}], [H_{2}SO_{4}],
ionization rates, T, RH, and surface areas of preexisting particles vary
widely depending on the region, pollution level, and season. The present work
aims to address these issues by developing a kinetically based
H_{2}SO_{4}–H_{2}O–NH_{3} ternary IMN (TIMN) model that is based on the molecular clustering
thermodynamic data. The model predictions are compared with relevant CLOUD
measurements and previous studies.

2 Kinetic-based H_{2}SO_{4}–H_{2}O–NH_{3} ternary ion-mediated nucleation (TIMN) model

Most nucleation models developed in the past for
H_{2}SO_{4}–H_{2}O binary homogeneous nucleation (e.g.,
Vehkamäki et al., 2002), H_{2}SO_{4}–H_{2}O
ion-induced nucleation (IIN; e.g., Hamill et al., 1982; Raes et al., 1986; Laakso
et al., 2003), and H_{2}SO_{4}–H_{2}O–NH_{3}
ternary homogeneous nucleation (Coffman and Hegg, 1995; Korhonen et al.,
1999; Napari et al., 2002) have been based on the classical approach, which
employs capillarity approximation (i.e., assuming that small clusters have
the same properties as bulk) and calculates nucleation rates according to the free
energy change associated with the formation of a “critical embryo”. Yu and
Turco (1997, 2000, 2001) developed a neutral and charged binary
H_{2}SO_{4}–H_{2}O nucleation model using a kinetic
approach that explicitly treats the complex interactions among small air
ions, neutral and charged clusters of various sizes, precursor vapor
molecules, and preexisting aerosols. The formation and evolution of cluster
size distributions for positively and negatively charged cluster ions and
neutral clusters affected by ionization, recombination, neutralization,
condensation, evaporation, coagulation, and scavenging have been named
IMN (Yu and Turco, 2000). The IMN theory
significantly differs from classical IIN theory
(e.g., Hamill et al., 1982; Raes et al., 1986; Laakso et al., 2003), which is
based on a simple modification of the free energy for the formation of a
critical embryo by including the electrostatic potential energy induced
by the embedded charge (i.e., Thomson effect; Thomson, 1888). The classical
approach does not properly account for the kinetic limitation to embryo
development, enhanced stability and growth of charged clusters associated
with dipole–charge interaction (Nadykto and Yu, 2003; Yu, 2005), and the
important contribution of neutral clusters resulting from ion–ion
recombination to nucleation (Yu and Turco, 2011). In contrast, these
important physical processes are explicitly considered in the kinetic-based
IMN model (Yu, 2006b).

Since the beginning of the century, nucleation models based on the kinetic
approach have also been developed in a number of research groups (Lovejoy et
al., 2004; Sorokin et al., 2006; Chen et al., 2012; Dawson et al., 2012;
McGrath et al., 2012). Lovejoy et al. (2004) developed a kinetic ion
nucleation model, which explicitly treats the evaporation of small neutral
and negatively charged H_{2}SO_{4}–H_{2}O clusters. The
thermodynamic data used in their model were obtained from measurements of
small ion clusters, ab initio calculations, the thermodynamic cycle, and some
approximations (adjustment of Gibbs free energy for neutral clusters
calculated based on liquid droplet model, interpolation, etc.). Lovejoy et
al. (2004) did not consider the nucleation on positive ions. Sorokin et
al. (2006) developed an ion cluster–aerosol kinetic (ICAK) model, which uses
the thermodynamic data reported in Froyd and Lovejoy (2003a, b) and empirical
correction terms proposed by Lovejoy et al. (2004). Sorokin et al. (2006)
used the ICAK model to simulate dynamics of neutral and charged
H_{2}SO_{4}–H_{2}O cluster formation and compared the
modeling results with their laboratory measurements. Chen et al. (2012)
developed an approach for modeling NPF based on a sequence
of acid–base reactions, with sulfuric acid evaporation rates (from clusters)
estimated empirically based on measurements of neutral molecular clusters
taken in Mexico City and Atlanta. Dawson et al. (2012) presented a
semiempirical kinetics model for nucleation of methanesulfonic acid (MSA),
amines, and water that explicitly accounted for the sequence of reactions
leading to formation of stable particles. The kinetic models of Chen et
al. (2012) and Dawson et al. (2012) consider only neutral clusters.

McGrath et al. (2012) developed the Atmospheric Cluster Dynamics Code (ACDC)
to model the cluster kinetics by solving the birth–death equations
explicitly, with evaporation rate coefficients derived from formation free
energies calculated by quantum chemical methods (Almeida et al., 2013;
Olenius et al., 2013). The ACDC model applied to the
H_{2}SO_{4}–dimethylamine (DMA) system considers zero to four base
molecules and zero to four sulfuric acid molecules (Almeida et al., 2013). Olenius et
al. (2013) applied the ACDC model to simulate the steady-state concentrations
and kinetics of neutral and negatively and positively charged clusters
containing up to five H_{2}SO_{4} and five NH_{3} molecules.
In ACDC, the nucleation rate is calculated as the rate of clusters growing
larger than the upper bounds of the simulated system (i.e., clusters
containing four or five H_{2}SO_{4} molecules) (Kürten et al., 2016).

The kinetic IMN model developed by Yu and Turco (1997, 2001) explicitly
simulates the dynamics of neutral, positively charged, and negatively charged
clusters, based on a discrete-sectional bin structure that covers the
clusters containing 0, 1, 2,…, 15,…H_{2}SO_{4}
molecules to particles containing thousands of H_{2}SO_{4} (and
H_{2}O) molecules. In the first version of the kinetic IMN model (Yu
and Turco, 1997, 2001), due to the lack of thermodynamic data for the small
clusters, the compositions of neutral and charged clusters were assumed to be
the same and the evaporation of small clusters was accounted for using a
simple adjustment to the condensation accommodation coefficients. Yu (2006b)
developed a second-generation IMN model that incorporated newer
thermodynamic data (Froyd, 2002; Wilhelm et al., 2004) and physical
algorithms (Froyd, 2002; Wilhelm et al., 2004) and explicitly treated the
evaporation of neutral and charged clusters. Yu (2007) further improved the
IMN model by using two independent measurements (Marti et al., 1997; Hanson
and Eisele, 2000) to constrain monomer hydration in the
H_{2}SO_{4}–H_{2}O system and by incorporating
experimentally determined energetics of small neutral
H_{2}SO_{4}–H_{2}O clusters that became available then
(Hanson and Lovejoy, 2006; Kazil et al., 2007). The first and second
generations of the IMN model were developed for the
H_{2}SO_{4}–H_{2}O binary system, although the possible
effects of ternary species such as the impact of NH_{3} on the
stability of both neutral and charged pre-nucleation clusters have been
pointed out in these previous studies (Yu and Turco, 2001; Yu, 2006b). The
present work extends the previous versions of the IMN model in the binary
H_{2}SO_{4}–H_{2}O system to the ternary
H_{2}SO_{4}–H_{2}O–NH_{3} system, as described
below.

Figure 1Schematic illustration of kinetic processes controlling the
evolution of positively charged (H^{+}S_{s}W_{w}A_{a}), neutral (S_{s}W_{w}A_{a}), and
negatively charged (${\mathrm{S}}^{-}{\mathrm{S}}_{s-\mathrm{1}}{\mathrm{W}}_{\mathrm{w}}{\mathrm{A}}_{\mathrm{a}}$) clusters–droplets that are explicitly simulated in the
ternary ion-mediated nucleation (TIMN) model. Here S, W, and A represent
sulfuric acid (H_{2}SO_{4}), water (H_{2}O), and ammonia
(NH_{3}), respectively, while s, w, and a refer to the number of S, W,
and A molecules in the clusters–droplets, respectively. The TIMN model has
been extended from an earlier version treating binary IMN (BIMN) by adding
NH_{3} into the nucleation system and using a discrete-sectional bin
structure to represent the sizes of clusters–particles starting from a single
molecule up to background particles larger than a few micrometers.

2.2 Model representation of kinetic ternary nucleation processes

Figure 1 schematically illustrates the evolution of charged and neutral
clusters–droplets explicitly simulated in the kinetic
H_{2}SO_{4}–H_{2}O–NH_{3} TIMN model. Here,
H_{2}SO_{4} (S) is the key atmospheric nucleation precursor
driving the TIMN process while ions, H_{2}O (W), and NH_{3} (A)
stabilize the H_{2}SO_{4} clusters and enhance
H_{2}SO_{4} nucleation rates in this way. Ions also enhance cluster
formation rates due to the interaction with polar-nucleating species, leading
to enhanced collision cross sections (Nadykto and Yu, 2003). The airborne
ions are generated by galactic cosmic rays (GCRs) or produced by radioactive
emanations, lightning, corona discharge, combustion, and other ionization
sources. The initial negative ions, which are normally assumed to be
${\mathrm{NO}}_{\mathrm{3}}^{-}$, are converted into
${\mathrm{HSO}}_{\mathrm{4}}^{-}$ core ions (i.e., S^{−})
and then to larger H_{2}SO_{4} clusters in the presence of
gaseous H_{2}SO_{4}. The initial positive ions
H^{+}W_{w} are converted into
${{\mathrm{H}}^{+}{\mathrm{A}}_{\mathrm{1}-\mathrm{2}}\mathrm{W}}_{\mathrm{w}}$ in the presence of NH_{3},
H^{+}S_{s}W_{w} in the presence of H_{2}SO_{4}, or
H^{+}A_{a}S_{s}W_{w} in the case that both NH_{3} and
H_{2}SO_{4} are present in the nucleating vapors. Some of the
binary H_{2}SO_{4}–H_{2}O clusters, both neutral and
charged, transform into ternary ones by taking up NH_{3} vapors. The
molar fraction of ternary clusters in nucleating vapors depends on
[NH_{3}], the binding strength of NH_{3} to binary and ternary
pre-nucleation clusters, cluster composition, and ambient conditions such as
T and RH.

Similar to the kinetic binary IMN (BIMN) model (Yu, 2006b), the kinetic TIMN
model employs a discrete-sectional bin structure to represent
clusters–particles. The bin index i represents the amount of core
component (i.e., H_{2}SO_{4}) and i_{d} is the number of discrete bins. For small clusters (i≤i_{d}= 30 in this study), i is the number of H_{2}SO_{4}
molecules in the cluster (i.e., i=s) and the core volume of the ith bin
${v}_{i}=i\times {v}_{\mathrm{1}}$, where v_{1} is the volume of one
H_{2}SO_{4} molecule. When i > i_{d},
v_{i}= VRAT_{i}×v_{i−1}, where VRAT_{i} is the
volume ratio of the ith bin to the (i−1)th bin. The discrete-sectional bin structure enables the
model to cover a wide range of sizes of nucleating clusters–particles with
the highest possible size resolution for small clusters (Yu, 2006b). For
clusters with a given bin i, the associated amounts of water and
NH_{3} and thus the effective radius of each ternary cluster are
calculated based on the equilibrium of clusters–particles with the water
vapor and/or ammonia, as described in later sections.

The evolution of positive, negative, and neutral clusters due to the
simultaneous condensation, evaporation, recombination, coagulation, and other
loss processes is described by the following differential equations obtained
by the modification of those describing the evolution of the binary
H_{2}SO_{4}–H_{2}O system (Yu, 2006b):

In Eqs. (1)–(6), the superscripts “+”, “−”, and “0” refer to
positive, negative, and neutral clusters, respectively, while subscripts i,
j, and k represent the bin indexes. ${N}_{\mathrm{0}}^{+,-}$ and Q are the
concentration of initial ions not containing H_{2}SO_{4} (i.e.,
H^{+}A_{a}W_{w} and ${\mathrm{NO}}_{\mathrm{3}}^{-}$) and the ionization rate,
respectively. N_{i} is the total number concentration (cm^{−3}) of all
cluster/particles (binary + ternary) in bin i. For small clusters (i≤i_{d}), N_{i} is the number concentration (cm^{−3}) of all clusters
containing iH_{2}SO_{4} molecules. For example, ${N}_{\mathrm{1}}^{\mathrm{0}}$
is the total concentration of binary and ternary neutral clusters containing
one H_{2}SO_{4} molecule. Index i in Eq. (4) refers to the
sum of H_{2}SO_{4} and ${\mathrm{HSO}}_{\mathrm{4}}^{-}$. The second term of
Eq. (2) describes the reaction of ${\mathrm{HSO}}_{\mathrm{4}}^{-}$+HNO_{3}→${\mathrm{NO}}_{\mathrm{3}}^{-}$+H_{2}SO_{4}. Although the rate of this
reaction is generally negligible, we keep the term there for completeness.
${P}_{{\mathrm{H}}_{\mathrm{2}}{\mathrm{SO}}_{\mathrm{4}}}$ is the gas-phase production rate of neutral
H_{2}SO_{4} molecules. ${L}_{i}^{+,-,\mathrm{0}}$ is the loss rate due to
scavenging by preexisting particles and wall and dilution losses in the
laboratory chamber studies (Kirkby et al., 2011; Olenius et al., 2013;
Kürten et al., 2016). ${f}_{j,k,i}$ is the volume fraction of intermediate
particles (volume =v_{j}+v_{k}) partitioned into bin i with
respect to the core component – H_{2}SO_{4}, as defined in
Jacobson et al. (1994). ${g}_{i+\mathrm{1},i}={v}_{\mathrm{1}}/({v}_{i+\mathrm{1}}-{v}_{i})$ is the volume
fraction of intermediate particles of volume (${v}_{i+\mathrm{1}}-{v}_{\mathrm{1}}$) partitioned
into bin i. ${\mathit{\delta}}_{j,\mathrm{2}}=\mathrm{2}$ at j=2 and ${\mathit{\delta}}_{j,\mathrm{2}}=\mathrm{1}$ at j≠2.
${\mathit{\gamma}}_{i}^{+}$, ${\mathit{\gamma}}_{i}^{-}$ , and ${\mathit{\gamma}}_{i}^{\mathrm{0}}$ are the mean (or
effective) cluster evaporation coefficients for positive, negative and
neutral clusters in bin i, respectively. ${\mathit{\beta}}_{i,j}^{+}$,
${\mathit{\beta}}_{i,j}^{-}$, and ${\mathit{\beta}}_{i,j}^{\mathrm{0}}$ are the coagulation kernels for the
neutral clusters–particles in bin j interacting with positive, negative,
and neutral clusters–particles in bin i, respectively, which reduce to the
condensation coefficients for H_{2}SO_{4} monomers at j=1.
${\mathit{\eta}}_{j,k}^{+}$ and ${\mathit{\eta}}_{j,k}^{-}$ are coagulation kernels for
clusters–particles of like sign from bin j and clusters–particles from
bin k. It should be noted that the electrostatic repulsion is too strong
for small clusters to gain more than one charge. However, small charged
clusters can be scavenged by large preexisting particles of the same
polarity. Large preexisting particles serve as the sink for small clusters in
the model and the effect of multiple charge is small and thus is not tracked.
${\mathit{\alpha}}_{i,j}^{+,-}$ is the recombination coefficient for positive
clusters–particles in bin i interacting with negative clusters–particles
in bin j, while ${\mathit{\alpha}}_{i,j}^{-,+}$ is the recombination coefficient of
negative clusters–particles from bin i interacting with positively charged
clusters–particles from bin j.

The methods for calculating β, γ, η, and α for
binary H_{2}SO_{4}–H_{2}O clusters have been described
in our previous publications (Yu and Turco, 2001; Nadykto and Yu, 2003; Yu,
2006b). Dipole–charge interaction (Nadykto and Yu, 2003), image capture, and
three-body trapping effects (Hoppel and Frick, 1986) are considered in the
calculation of these coefficients. Since β, η, and α depend
on the cluster mass (or size) rather than on the cluster composition, schemes
for calculating these properties in binary and ternary clusters are
identical. In contrast, γ is quite sensitive to cluster composition.
The evaporation rate coefficient of H_{2}SO_{4} molecules from
clusters containing iH_{2}SO_{4} molecules (γ_{i}) is
largely controlled by the stepwise Gibbs free energy change $\mathrm{\Delta}{G}_{i-\mathrm{1},i}$ of formation of an i-mer from an (i−1)-mer (Yu, 2007)

where R is the molar gas constant, N^{o} is the arbitrary number
concentration of a hypothetical gas consisting solely of the species for
which the calculation is performed (generally under the reference vapor
pressure P of 1 atm). ΔH^{o} and ΔS^{o} are enthalpy
and entropy changes under the standard conditions (T=298 K, P=1 atm),
respectively. The temperature dependence of ΔH^{o} and ΔS^{o}, which is generally small and typically negligible over the
temperature range of interest (Nadykto et al., 2009), was not considered.

Figure 2Stepwise Gibbs free energy change under standard conditions for the
addition of a water ($\mathrm{\Delta}{G}_{+\mathrm{W}}^{o}$), ammonia
($\mathrm{\Delta}{G}_{+\mathrm{A}}^{o}$), or sulfuric acid ($\mathrm{\Delta}{G}_{+\mathrm{S}}^{o}$) molecule to form the given positively charged
clusters as a function of the number of water molecules in the clusters
(w). Lines are QC-based values, and symbols are experimental results or
semi-experimental estimates (see notes under Table A2 for the references).

2.3 Thermochemical data of neutral and charged binary and ternary clusters

ΔH, ΔS, and ΔG values needed to calculate cluster
evaporation rates (Eq. 7) for the TIMN model can be derived from laboratory
measurements and computational quantum chemistry (QC) calculation.
Thermochemical properties of neutral and charged binary and ternary clusters
obtained using the computational chemical methods and comparisons of computed
energies with available experimental data and semi-experimental estimates are
given in Tables A1–A4 and discussed in the Appendix. As an example, Fig. 2 shows
ΔG associated with the addition of water
($\mathrm{\Delta}{G}_{+\mathrm{W}}^{o}$), ammonia ($\mathrm{\Delta}{G}_{+\mathrm{A}}^{o}$), and sulfuric acid ($\mathrm{\Delta}{G}_{+\mathrm{S}}^{o}$) to binary and ternary clusters as a function of the
cluster hydration number w. H_{2}O has high proton affinity and
thus H_{2}O is strongly bonded to all positive ions with low w.
$\mathrm{\Delta}{G}_{+\mathrm{W}}^{o}$ expectedly becomes less negative and binding
of H_{2}O to binary and ternary clusters weakens due to the screening
effect as the hydration number w grows (Fig. 2a). The presence of
NH_{3} in the clusters weakens binding of H_{2}O to positive
ions. For example, $\mathrm{\Delta}{G}_{+\mathrm{W}}^{o}$ for
H^{+}A_{1}W_{w}S_{1} is ∼3–4 kcal mol^{−1} less
negative than that for H^{+}W_{w}S_{1} at w=3–6. The addition
of one more NH_{3} to the clusters to form H^{+}A_{2}W_{w}
and H^{+}A_{2}W_{w}S_{1} further weakens H_{2}O binding
by ∼1.5–6 kcal mol^{−1} at w=1–3, while exhibiting a much
smaller impact on hydration free energies at w > 3. Both the
absolute values and trends in $\mathrm{\Delta}{G}_{+\mathrm{W}}^{o}$ derived from
calculations are in agreement with the laboratory measurements within the
uncertainty range of ∼1–2 kcal mol^{−1} for both QC calculations
and measurements. This confirms the efficiency and precision of QC methods in
calculating thermodynamic data needed for the development of nucleation
models.

The proton affinity of NH_{3} is 204.1 kcal mol^{−1}, which is
37.5 kcal mol^{−1} higher than that of H_{2}O
(166.6 kcal mol^{−1}) (Jolly, 1991). The hydrated hydronium ions
(H^{+}W_{w}) are easily converted to H^{+}A_{1}W_{w} in
the presence of NH_{3}. The binding of NH_{3} and H_{2}O
molecules to H^{+}W_{w} exhibits a similar pattern. In particular,
binding of NH_{3} to H^{+}W_{w} decreases as w is growing,
with $\mathrm{\Delta}{G}_{+\mathrm{A}}^{o}$ for H^{+}A_{1}W_{w} ranging
from −52.08 kcal mol^{−1} at w=1 to −8.32 kcal mol^{−1} at
w=9. The binding of NH_{3} to H^{+}W_{w}S_{1} ions is
also quite strong, with $\mathrm{\Delta}{G}_{+\mathrm{A}}^{o}$ for
H^{+}A_{1}W_{w}S_{1} ranging from −33.14 kcal mol^{−1} at
w=1 and to −10.57 kcal mol^{−1} at w=6. The addition of the
NH_{3} molecule to H^{+}A_{1}W_{w} (to form
H^{+}A_{2}W_{w}) is much less favorable thermodynamically than
that to H^{+}W_{w}, with the corresponding
$\mathrm{\Delta}{G}_{+\mathrm{A}}^{o}$ being −22 and −6 kcal mol^{−1} at
w=2 and w=6, respectively. The $\mathrm{\Delta}{G}_{+\mathrm{A}}^{o}$ values for
H^{+}A_{2}W_{w} are 3–5 kcal mol^{−1} more negative than the
experimental values at w=0–1; however, they are pretty close to
experimental data at w=2–3 (Fig. 2b and Table A2). While it is possible
that the QC method overestimates the charge effect on the formation free
energies of smallest clusters, the possible overestimation at w=0–1 will
not affect nucleation calculations because most H^{+}A_{2}W_{w}
clusters in the atmosphere
contain more than two water molecules (i.e., w > 2) due to the
strong hydration (see Table A2 and Fig. 2a).

A comparison of QC and semi-experimental estimates of
$\mathrm{\Delta}{G}_{+\mathrm{S}}^{o}$ values associated with the attachment of
H_{2}SO_{4} to positive ions shown in Fig. 2c indicates that
computed $\mathrm{\Delta}{G}_{+\mathrm{S}}^{o}$ values agree well with observations
for H^{+}W_{w}S_{1} and H^{+}A_{1}W_{w}S_{1} but
differ by ∼2–4 kcal mol^{−1} from semi-experimental values for
H^{+}A_{2}W_{w}S_{1}. As seen from Fig. 2a and c, the attachment
of NH_{3} to H^{+}W_{w}S_{1} weakens the binding of both
H_{2}O and H_{2}SO_{4} to the clusters. This suggests
that the attachment of NH_{3} leads to the evaporation of
H_{2}SO_{4} and H_{2}O molecules from the clusters. In
other words, H_{2}SO_{4} is less stable in
H^{+}A_{1}W_{w}S_{1} than in H^{+}W_{w}S_{1}
(Fig. 2c). While this may be taken for the indication that NH_{3}
inhibits nucleation on positive ions at the first look, further calculations
show that binding of NH_{3} to H^{+}A_{1}W_{w}S_{1} is
quite strong (Fig. 2b) and that H_{2}SO_{4} in the
H^{+}A_{2}W_{w}S_{1} cluster is much more stable than that in
H^{+}A_{1}W_{w}S_{1}, with $\mathrm{\Delta}{G}_{+\mathrm{S}}^{o}$
being more negative by ∼7 kcal mol^{−1} at w > 2. The
H^{+}A_{2}W_{w}S_{1} cluster can also be formed via the
attachment of H_{2}SO_{4} to H^{+}A_{2}W_{w}. In the
presence of sufficient concentrations of NH_{3}, a large fraction of
positively charged H_{2}SO_{4} monomers exist in the form of
H^{+}A_{2}W_{w}S_{1} and hence NH_{3} enhances nucleation
of positive ions. Since positively charged H_{2}SO_{4} dimers
are expected to contain a large number of water molecules, we have not yet
computed and derived quantum chemical data for these clusters. The CLOUD
measurements do indicate that once H^{+}A_{2}W_{w}S_{1}
clusters are formed, they can
continue to grow to larger H^{+}A_{a}W_{w}S_{s} clusters along
$a=s+\mathrm{1}$ pathway (Schobesberger et al., 2015).

Figure 2 clearly shows that the calculated values in most cases agree with
measurements within the uncertainty range that justifies the application of
QC values in the case that no reliable experimental data are available.

2.4 Nucleation barriers for neutral and charged clusters and size-dependent evaporation rates

Nucleation barriers and cluster evaporation rates are critically important
for calculations of nucleation rates. This section describes the methods
employed to calculate the evaporation rates of nucleating clusters of
variable sizes and compositions (i.e., γ in Eqs. 1–6) in the TIMN
model.

2.4.1 Equilibrium distributions of small binary and ternary clusters

In the atmosphere, [H_{2}O] is much higher than
[H_{2}SO_{4}] and thus H_{2}SO_{4}
clusters–particles are always in equilibrium with water vapor (Yu, 2007). In
the lower troposphere, where most of the nucleation events were observed,
[H_{2}SO_{4}] is typically at a sub-parts per trillion to parts per trillion level, while
[NH_{3}] is in the range of sub-parts per billion to parts per billion levels (Butler et al.,
2016; Warner et al., 2016) (note that, in what follows, all references to
vapor mixing ratios – parts per billion and parts per trillion – are by
volume). This means that small ternary clusters can be considered to be in
equilibrium with H_{2}O and NH_{3} vapors. Like the previous
BIMN model derived assuming equilibrium of binary clusters with water vapor,
the present TIMN model treats small clusters containing a given number of
H_{2}SO_{4} molecules as being in equilibrium with both
H_{2}O and NH_{3}. Their relative concentrations are calculated
using the thermodynamic data shown in Tables A1–A4. It should be noted that
the system may deviate from equilibrium and the model scheme is probably not
suitable when [NH_{3}] is less than or close to
[H_{2}SO_{4}]. Under such cases, the equilibrium assumption may
overestimate nucleation rates.

Figure 3Relative abundance (or molar fraction) of small clusters containing
a given number of H_{2}SO_{4} molecules for positive
(a), negative (b), and neutral (c, d) cluster
types at a temperature of 292 K and three different combinations of RHs
(38 % and 90 %) and [NH_{3}] (0.3 and 3 ppb). Some clusters
with different numbers of water molecules were grouped together to make the
plot more clear and neat. For the clusters shown in (d), there are
no hydrate data and thus hydration for these clusters was not calculated.

Figure 3 shows the relative abundance (or molar fractions) of small positive,
negative, and neutral clusters (${f}_{s,a,w}^{+,-,\mathrm{0}}$) containing a given
number of H_{2}SO_{4} molecules at the ambient temperature of
292 K and three different combinations of RH and [NH_{3}] values. As
a result of relative instability of H_{2}SO_{4} in
H^{+}A_{1}W_{w}S_{1} compared to H^{+}W_{w}S_{1} or
H^{+}A_{2}W_{w}S_{1} (Fig. 2c), most of positive ions with one
H_{2}SO_{4} molecule exist in the form of either
H^{+}W_{w}S_{1} or H^{+}A_{2}W_{w}S_{1} (i.e,
containing either zero or two NH_{3} molecules; Fig. 3a). When
[NH_{3}] = 0.3 ppb (with T=292 K), most of the positive ions
containing one H_{2}SO_{4} molecule do not contain
NH_{3} and their composition is dominated by
H^{+}W_{w}S_{1} ($\stackrel{\mathrm{\u203e}}{w}=\sim \mathrm{7}$). At the given T and
[NH_{3}] = 0.3 ppb, around 17 % of positive ions with one
H_{2}SO_{4} molecule contain two NH_{3} molecules at
RH = 38 %. The fraction of positive ions containing one
H_{2}SO_{4} and two NH_{3} molecules decreases to
0.9 %, when RH=90 %. At T=292 K and RH=38 %, the
increase in [NH_{3}] by a factor of 10 to 3 ppb leads to the
domination of H^{+}A_{2}W_{w}S_{1} (∼95 %) in the
composition of positively charged H_{2}SO_{4} monomers. As
expected, the composition of positive ions and their contribution to
nucleation depends on T, RH, and [NH_{3}]. The incorporation of the
quantum chemical and experimental clustering thermodynamics in the framework
of the kinetic nucleation model enables us to study all these dependencies.

As a result of very weak binding of H_{2}O and NH_{3} to small
negative ions (Table A4), nearly all negatively charged clusters with
s= 0–1 do not contain water and ammonia (not shown). In the case that s grows to 2, all S^{−}S_{2}A_{a}W_{w} clusters still do
not contain NH_{3} (i.e., a=0), while only 20 %–40 % of them
contain one water molecule (w=1) (Fig. 3b). As s further increases to 3,
NH_{3} begins to enter some of the negatively charged ions. The
fraction of S^{−}S_{3}A_{a}W_{w} clusters containing one
NH_{3} molecule is 9 % at RH = 38 % and
[NH_{3}] = 0.3 ppb, 3 % at RH = 90 % and
[NH_{3}] = 0.3 ppb, and 50 % at RH = 38 % and
[NH_{3}] = 3 ppb. Most S^{−}S_{3}W_{w} clusters are
hydrated while the fraction of S^{−}S_{3}A_{a}W_{w} clusters
containing two NH_{3} molecules at these ambient conditions is
negligible. The fraction of negative cluster ions containing two
NH_{3} molecules becomes significant at s=4 (Fig. 3b) and increases
from 28 % at [NH_{3}] = 0.3 ppb to 80 % at
[NH_{3}] = 3 ppb at RH = 38 %. At
[NH_{3}] = 0.3 ppb, the increase in RH from 38 % to 90 %
reduces the fraction of NH_{3}-containing S^{−}S_{3}A_{a}W_{w}
clusters (i.e, a > =1) from to 95 % to 70 %,
demonstrating a significant impact of RH on cluster compositions and
emphasizing the importance of accounting for the RH in calculations of
ternary nucleation rates.

The equilibrium distributions of neutral clusters are presented in Fig. 3c
(H_{2}SO_{4} monomers and dimers) and Fig. 3d
(H_{2}SO_{4} trimers and tetramers). Hydration is accounted for
in the case of monomers and dimers and not included, due to lack of
thermodynamic data, in calculations for trimers and tetramers. Based on the
thermodynamic data shown in Table A3, the dominant fraction of neutral
monomers is hydrated (79 % at RH = 38 % and 94 % at
RH = 90 %) while the fraction of monomers containing NH_{3} is
negligible (0.02 % at [NH_{3}] = 0.3 ppb and 0.2 % at
[NH_{3}] = 3 ppb, RH = 38 %). As a result of the growing
binding strength of NH_{3} with the cluster size (Table A3), the
fraction of neutral sulfuric acid dimers containing one NH_{3}
molecule reaches 18 % at [NH_{3}] = 0.3 ppb and 69 % at
[NH_{3}] = 3 ppb when T= 292 K and RH = 38 %. In
the case of H_{2}SO_{4} trimers and tetramers, data shown in
Fig. 3d are limited to the relative abundance of unhydrated clusters only.
Under the given conditions, most trimers contain two NH_{3}
molecules while most tetramers contain three NH_{3} molecules. At
[NH_{3}] = 3 ppb, ∼2 % of trimers contain three
NH_{3} molecules (i.e., $s=a=\mathrm{3}$) and 55 % of tetramers contain
four NH_{3} molecules (i.e., $s=a=\mathrm{4}$). As a result of a significant
drop of $\mathrm{\Delta}{G}_{+\mathrm{A}}^{o}$ in the case that the a∕s ratio exceeds
one (Table A3), the fraction of neutral clusters with $a=s+\mathrm{1}$ is negligible.
The cluster distributions clearly indicate that small sulfuric acid clusters
are still not fully neutralized by NH_{3} even if [NH_{3}] is
at a parts per billion level and that the degree of neutralization (i.e., a:s ratio)
increases with the cluster size.

2.4.2 Mean stepwise and accumulative Gibbs free energy change and impact of ammonia

In the TIMN model, the equilibrium distributions are used to calculate number-concentration-weighted stepwise Gibbs free energy change for adding one
H_{2}SO_{4} molecule to form a neutral, positively charged, and
negatively charged cluster containing sH_{2}SO_{4} molecules
(${\stackrel{\mathrm{\u203e}}{\mathrm{\Delta}G}}_{s-\mathrm{1},s}$):

where ${f}_{s,a,w}^{+,-,\mathrm{0}}$ is the equilibrium fraction of a particular cluster
within a cluster type as shown in Fig. 3.

In the atmosphere, where substantial nucleation is observed, the sizes of
critical clusters are generally small (s < ∼5–10) (e.g.,
Sipilä et al., 2010) and nucleation rates are largely controlled by the
stability (or γ) of small clusters with s < ∼5–10.
QC calculations and experimental data on clustering thermodynamics available
for clusters of small sizes (Tables A2–A4) are critically important as the
formation of these small clusters is generally the limiting step for
nucleation. Nevertheless, thermodynamics data for larger clusters are also
needed to develop a robust nucleation model that can calculate nucleation
rates under various conditions. Both measurements and QC calculations (Tables
A2–A4) show significant effects of charge and charge signs (i.e., positive
or negative) on the stability and composition of small clusters. These charge
effects decrease quickly as the clusters grow due to the short-ranged nature
of dipole–charge interaction and the quick decrease in electrical field
strength around charged clusters as cluster sizes increase (Yu, 2005). Based
on experimental data (Kebarle et al., 1967; Davidson et al., 1977; Wlodek et
al., 1980; Holland and Castleman, 1982; Froyd and Lovejoy, 2003b), the
stepwise ΔG values for clusters decrease exponentially as the cluster
sizes increase and approach to the bulk values when clusters contain more
than ∼8–10 molecules (Yu, 2005). Cluster compositions measured with an
atmospheric pressure interface time-of-flight (APi-TOF) mass spectrometer
during CLOUD experiments also show that the difference in the composition of
positively and negatively charged clusters quickly decreases as the number of
H_{2}SO_{4} molecules increases from 1 to ∼10 and exhibits
little further changes (Schobesberger et al., 2015).

In the present TIMN model, we assume that both neutral and charged clusters
have the same composition when s≥10 and the following extrapolation
scheme is used to calculate $\mathrm{\Delta}{G}_{s-\mathrm{1},s}$ for clusters up to s=10:

where $\mathrm{\Delta}{G}_{{s}_{\mathrm{1}}-\mathrm{1},{s}_{\mathrm{1}}}$ is the stepwise mean Gibbs free energy
change for H_{2}SO_{4} addition for a specific type (neutral,
positive, or negative) of clusters at s=s_{1} that can be derived from QC
calculation and/or experimental measurements, and $\mathrm{\Delta}{G}_{{s}_{\mathrm{2}}-\mathrm{1},{s}_{\mathrm{2}}}$ is the corresponding value for clusters at s=s_{2} (= 10 in
the present study) that is calculated in the capillarity approximation
accounting for the Kelvin effect. c in Eq. (10) is the exponential
coefficient that determines how fast $\mathrm{\Delta}{G}_{s-\mathrm{1},s}$ approaches to bulk
values as s increases. In the present study, c is estimated by fitting
$\mathrm{\Delta}{G}_{s-\mathrm{1},s}$ at s=2 and s=3 based on Eq. (10) to
the corresponding $\mathrm{\Delta}{G}_{s-\mathrm{1},s}$ from experimental (Hanson
and Lovejoy, 2006; Kazil et al., 2007) or QC data (Table A3). Apparently the
interpolation approximation Eq. (10) is subject to uncertainty. Nevertheless,
it is a reasonable approach to connect thermochemical properties of QC data
for small binary and ternary clusters that cannot be adequately described by
the capillarity approximation with those for large clusters that can be
adequately described by the very same capillarity approximation, and it is
the best approach we can come up with at this point in order to develop a
model that can be applied to all conditions. Further QC and experimental
studies of the thermodynamics of relatively larger clusters can help to
reduce the uncertainty.

For clusters with s≥s_{2}, the capillarity approximation is used to
calculate $\mathrm{\Delta}{G}_{s-\mathrm{1},s}$ as

where P is the H_{2}SO_{4} vapor pressure and P_{s}
is the H_{2}SO_{4} saturation vapor pressure over a flat surface with
the same composition as the cluster. σ is the surface tension and
v_{1} is the volume of one H_{2}SO_{4} molecule. r_{s} is the
radius of the cluster and N_{A} is the Avogadro's number.

The scheme to calculate bulk $\mathrm{\Delta}{G}_{s-\mathrm{1},s}$ (s≥10) for
H_{2}SO_{4}–H_{2}O binary clusters has been described
in Yu (2007). For ternary nucleation, both experiments (Schobesberger et al.,
2015) and QC calculations (Table A4) indicate that the growth of relatively
large clusters follows the s=a line (i.e, in the composition of ammonia
bisulfate). In the present TIMN model, the bulk $\mathrm{\Delta}{G}_{s-\mathrm{1},s}$ values for
ternary clusters are calculated based on parameterized
H_{2}SO_{4} saturation vapor pressure over ammonia bisulfate as
a function of temperature, derived by Marti et al. (1997) from vapor
pressures measured at a temperature between 27 and 60 ^{∘}C and surface
tension measured at 298 K from Hyvärinen et al. (2005). The uncertainty in saturation vapor
pressures and surface tension used in the calculation of the bulk $\mathrm{\Delta}{G}_{s-\mathrm{1},s}$ values is another source of uncertainty in the TIMN model,
although it is likely to be small compared to other uncertainties as the
nucleation is generally limited by the formation of small clusters.

Figure 4(a) Average stepwise Gibbs free energy change for the
addition of one H_{2}SO_{4} molecule to form a neutral (black),
positively charged (red), or negatively charged (blue) binary
H_{2}SO_{4}–H_{2}O (dashed lines or empty circles) or
ternary H_{2}SO_{4}–H_{2}O–NH_{3} (solid lines or
filled circles) cluster containing sH_{2}SO_{4} molecules
(${\stackrel{\mathrm{\u203e}}{\mathrm{\Delta}G}}_{s-\mathrm{1},s}$). (b) Same as (a) but for
the cumulative (total) Gibbs free energy change in each case. Filled and
empty circles in (a) refer to ${\stackrel{\mathrm{\u203e}}{\mathrm{\Delta}G}}_{s-\mathrm{1},s}$ obtained
using measurements and/or quantum-chemical calculations. ${\stackrel{\mathrm{\u203e}}{\mathrm{\Delta}G}}_{s-\mathrm{1},s}$ for larger clusters with s≥ 10, which approach the
properties of the equivalent bulk liquid (20), is calculated using the
capillarity approximation. Interpolation is used to calculate
${\stackrel{\mathrm{\u203e}}{\mathrm{\Delta}G}}_{s-\mathrm{1},s}$ for clusters up to s=10 (Eq. 11).
Calculations were carried out at T= 292 K, RH = 38 %,
[H_{2}SO_{4}] = 3 × 10^{8} cm^{−3}, and
[NH_{3}] = 0.3 ppb. The inset diagrams represent equilibrium
geometries for the most stable isomers of selected binary clusters
((H_{3}O^{+})(H_{2}SO_{4})(H_{2}O)_{6},
(H_{2}SO_{4})_{2}(H_{2}O)_{4}, and
(${\mathrm{HSO}}_{\mathrm{4}}^{-}$)(H_{2}SO_{4})_{4}(H_{2}O)_{2}) and
ternary clusters
((${\mathrm{NH}}_{\mathrm{4}}^{+}$)(H_{2}SO_{4})(NH_{3})(H_{2}O)_{4},
(${\mathrm{HSO}}_{\mathrm{4}}^{-}$)(H_{2}SO_{4})_{4}(H_{2}O)(NH_{3}),
(H_{2}SO_{4})_{4}(NH_{3})_{4}).

Figure 4 presents stepwise (${\stackrel{\mathrm{\u203e}}{\mathrm{\Delta}G}}_{s-\mathrm{1},s}$) and cumulative
(total) ${\stackrel{\mathrm{\u203e}}{\mathrm{\Delta}G}}_{\mathrm{s}}$ Gibbs free energy changes associated with
the formation of neutral, positively charged, and negatively charged binary
and ternary clusters containing sH_{2}SO_{4} molecules under
the conditions specified in the figure caption. The clusters are assumed to
be in equilibrium with water (Yu, 2007) and ammonia (Fig. 3). As seen from
Fig. 4, the presence of NH_{3} reduces the mean ${\stackrel{\mathrm{\u203e}}{\mathrm{\Delta}G}}_{s-\mathrm{1},s}$ for larger clusters, which can be treated as the bulk binary
H_{2}SO_{4}–H_{2}O solution (Schobesberger et al.,
2015), by ∼3 kcal mol^{−1}, indicating a substantial reduction in
the H_{2}SO_{4} vapor pressure over ternary solutions (Marti et
al., 1997). The comparison also shows that the influence of NH_{3} on
${\stackrel{\mathrm{\u203e}}{\mathrm{\Delta}G}}_{s-\mathrm{1},s}$ of small clusters ($s\le \sim \mathrm{4}$) is much lower
than that on larger ones and bulk solutions. For example, at
[NH_{3}] = 0.3 ppb, the differences in ${\stackrel{\mathrm{\u203e}}{\mathrm{\Delta}G}}_{s-\mathrm{1},s}$ between binary and ternary positive ions with s=1 and neutral
clusters with s=2 are only 0.45 and ∼1 kcal mol^{−1},
respectively. In the case of negative ions, zero and
0.27–0.45 kcal mol^{−1} differences at s≤2 and s=3–4,
respectively, were observed. The reduced effect of ammonia on smaller
clusters is explained (Tables A2–A4) by ammonia's weaker bonding to smaller
clusters than to larger ones, which in turn yields lower average
NH_{3}-to-H_{2}SO_{4} ratios (Fig. 3). It should be
noted that QC data for positively charged clusters are very limited and the
interpolation approximation is subject to large uncertainty. In order for the
nucleation on positive ions to occur, the first step is for
H_{2}SO_{4} to attach to a positive ion that does not contain
H_{2}SO_{4}. Unlike negative ions, the effect of charge on the
bonding of H_{2}SO_{4} with positive ions is much weaker and
thus the stepwise Gibbs free energy change for the addition of one
H_{2}SO_{4} molecule to form a positively charged cluster is
likely to be similar to that of neutral clusters, i.e., decreasing with
cluster size. Therefore, the QC data for positively charged clusters
containing one H_{2}SO_{4} molecule provide a critical
constraint. The success of the model in predicting the [NH_{3}] needed
for nucleation on positive ions to occur (see Sect. 3) shows the usefulness of
the first-step data and approximation.

As seen from Fig. 4, bonding of H_{2}SO_{4} to small negatively
charged clusters (s < 3) is much stronger than that to neutrals
and positive ions. As a result, at s < 3 the formation of
negatively charged clusters is barrierless (${\stackrel{\mathrm{\u203e}}{\mathrm{\Delta}G}}_{s-\mathrm{1},s}$ < 0). These small clusters cannot be considered
nucleated particles because ${\stackrel{\mathrm{\u203e}}{\mathrm{\Delta}G}}_{s-\mathrm{1},s}$ (Fig. 4a) first
increases and then decreases with growing s, reaching the maximum barrier
values at $s=\sim \mathrm{3}$–6. ${\stackrel{\mathrm{\u203e}}{\mathrm{\Delta}G}}_{s-\mathrm{1},s}$ can become positive for
larger clusters due to the charge effect decreasing quickly as the clusters
are growing. The effect of NH_{3} on negative ions becomes important
at $s\ge \sim \mathrm{4}$, when bonding between the clusters and NH_{3} becomes
strong enough to contaminate a large fraction of binary clusters with ammonia
(Fig. 3). In contrast, the impact of NH_{3} on neutral dimers and
positively charged monomers of H_{2}SO_{4}, as well as on
${\stackrel{\mathrm{\u203e}}{\mathrm{\Delta}G}}_{s-\mathrm{1},s}$ for both positively charged and neutral
clusters, monotonically decreases for all s values, including s≤5.

${\stackrel{\mathrm{\u203e}}{\mathrm{\Delta}G}}_{s-\mathrm{1},s}$ for charged and neutral clusters converges into
the bulk values at $s=\sim \mathrm{10}$, when impact of the chemical identity of the
core ion on the cluster composition becomes diffuse (Schobesberger et al.,
2015) and when the contribution of the electrostatic effect to
${\stackrel{\mathrm{\u203e}}{\mathrm{\Delta}G}}_{s-\mathrm{1},s}$ becomes less than
∼0.5 kcal mol^{−1}. The comparison of cumulative (total)
${\stackrel{\mathrm{\u203e}}{\mathrm{\Delta}G}}_{\mathrm{s}}$ (Fig. 4b) indicates the lowest nucleation barrier
for the case of negative ions, followed by positive ions and neutrals. The
barrierless formation of clusters with s ranging from 1 to 3 substantially
reduces the nucleation barrier for negatively charged ions and facilitates
their nucleation. The presence of 0.3 ppb of NH_{3} lowers the
nucleation barrier for negative, positive, and neutral clusters from
∼17, 24, and 38 kcal mol^{−1} to 2, 7, and 16 kcal mol^{−1},
respectively. A relatively low nucleation barrier for charged ternary
clusters is explained by the simultaneous effect of ionization and
NH_{3}, which also reduces the size of the critical cluster (s^{*}).

It is important to note that the size of the critical cluster, commonly used
to “measure” the activity of nucleation agents in the classical nucleation
theory (Coffman and Hegg, 1995; Korhonen et al., 1999; Vehkamäki et al.,
2002; Napari et al., 2002; Hamill et al., 1982) is no longer a valid
indicator, when charged molecular clusters and small nanoparticles are
considered. As seen from Fig. 4, positively charged ternary critical clusters
(${s}^{*}=\mathrm{3}$–4) are smaller than the corresponding negatively charged ones
(${s}^{*}=\mathrm{4}$–5); however, the nucleation barrier for ternary positive clusters
under the condition specified in the figure caption is more than 3 times
higher than that for ternary negatives ones.

2.4.3 Size- and composition-dependent H_{2}SO_{4} evaporation rates

As we mentioned earlier, H_{2}SO_{4} is the key atmospheric
nucleation precursor driving the formation and growth of clusters in the
ternary H_{2}SO_{4}–H_{2}O–NH_{3} system while
ions, H_{2}O, and NH_{3} act to stabilize the
H_{2}SO_{4} clusters. The clustering thermodynamic data derived
from QC calculations and measurements (Sect. 2.3) are used to constrain size-
and composition- dependent Gibbs free energy changes and evaporation rates of
H_{2}SO_{4}, which are critically important. Average or
effective rates of H_{2}SO_{4} molecule evaporation from
positively charged, negatively charged, and neutral clusters containing sH_{2}SO_{4} molecules (${\stackrel{\mathrm{\u203e}}{\mathit{\gamma}}}_{\mathrm{s}}^{+,-,\mathrm{0}}$) are
calculated from ${\stackrel{\mathrm{\u203e}}{\mathrm{\Delta}G}}_{s-\mathrm{1},s}$ as

where N^{o} is as defined in Eq. (6). The present model assumes
only a single H_{2}SO_{4} molecule evaporates, i.e., no water
ligands, for instance, are attached to it. This is likely the dominant
evaporation pathway as hydrated H_{2}SO_{4} molecules are
generally more stable.

Figure 5The number-concentration-weighted mean evaporation rates
($\stackrel{\mathrm{\u203e}}{\mathit{\gamma}}$) of H_{2}SO_{4} molecules from neutral
clusters (black), positively charged clusters (red), and negatively charged
clusters (blue) for binary (H_{2}SO_{4}–H_{2}O, dashed
lines) and ternary (H_{2}SO_{4}–H_{2}O–NH_{3},
solid lines) nucleating systems containing sH_{2}SO_{4}
molecules (${\stackrel{\mathrm{\u203e}}{\mathrm{\Delta}G}}_{s-\mathrm{1},s}$). T= 292 K,
RH = 38 %, and [NH_{3}] = 0.3 ppb for the ternary system.

Figure 5 gives the mean evaporation rate ($\stackrel{\mathrm{\u203e}}{\mathit{\gamma}}$) of a
H_{2}SO_{4} molecule from these clusters under the conditions
corresponding to Fig. 4. The shapes of $\stackrel{\mathrm{\u203e}}{\mathit{\gamma}}$ curves are similar
to those of ${\stackrel{\mathrm{\u203e}}{\mathrm{\Delta}G}}_{s-\mathrm{1},s}$ (Fig. 4a) as $\stackrel{\mathrm{\u203e}}{\mathit{\gamma}}$
values are largely controlled by ${\stackrel{\mathrm{\u203e}}{\mathrm{\Delta}G}}_{s-\mathrm{1},s}$ (Eq. 12). The
presence of ammonia, as expected, significantly reduces the vapor pressure of
H_{2}SO_{4} over bulk aerosol (Marti et al., 1997) and hence
the H_{2}SO_{4} evaporation rate. The evaporation rates of both
neutral and positive clusters decrease as s increases, and the positive
clusters are uniformly more stable than corresponding neutral clusters.
$\stackrel{\mathrm{\u203e}}{\mathit{\gamma}}$ for negative ions first increases and then decreases as
s increases, peaking at around $s=\sim \mathrm{3}$–6. The presence of
NH_{3} reduces the evaporation rates of larger clusters by more than
2 orders of magnitude and the effect decreases for smaller clusters, as the
binding of NH_{3} to small neutral and charged clusters is weaker
compared to that for larger clusters (Fig. 4). [NH_{3}] influences the
average NH_{3}:H_{2}SO_{4} ratio (Fig. 3) and the
evaporation rates of these small clusters. The nucleation rates, limited by
formation of small clusters (s < ∼5), depend strongly on
the stability or evaporation rate of these small clusters. While the binding
of NH_{3} to small neutral and charged clusters is weaker compared to
that to larger clusters, small clusters containing NH_{3} are much
more stable than those without (Fig. 4) and thus ammonia is important for
nucleation.

3 TIMN rates and comparisons with CLOUD measurements

The evolution of cluster/particle size distributions can be obtained by
solving the dynamic Eqs. (1)–(6). Since the concentrations of clusters of
all sizes are predicted, the nucleation rates in the kinetic model can be
calculated for any cluster size larger than the critical size of neutral
clusters (i > i^{*}) (Yu, 2006b),

where ${J}_{i}^{+}$, ${J}_{i}^{-}$, and ${J}_{i}^{\mathrm{0}}$ are nucleation rates
associated with positive, negative, and neutral clusters containing iH_{2}SO_{4} molecules. As a result of scavenging by
preexisting particles or wall loss, the steady-state J_{i} decreases as
i increases. To compare with CLOUD measurements, we calculate nucleation at
a
cluster mobility diameter of 1.7 nm (J_{1.7}).

Many practical applications require information on the steady state
nucleation rates. For each nucleation case presented in this paper, constant
values of [H_{2}SO_{4}] (i.e., ${N}_{\mathrm{1}}^{\mathrm{0}}$), [NH_{3}],
T, RH, Q, and ${L}_{i}^{+,-,\mathrm{0}}$ are assumed. The preexisting particles with
fixed surface area or wall loss serve as a sink for all clusters. Under a
given condition, cluster distribution and nucleation rate reach steady state
after a certain amount of time. We calculate size-dependent coefficients for
a given case and then solve Eqs. (1)–(6) to obtain the steady-state cluster
distribution and nucleation rate, with the approach described in Yu (2006b).

Figure 6Effect of ammonia concentrations ([NH_{3}]) on effective
nucleation rates calculated at a cluster mobility diameter of 1.7 nm
(J_{1.7}, lines) under the stated conditions with two ionization rates (Q)
– background ionization, bg (blue), and ionization enhanced by a pion beam,
pi (red). Also shown are predictions from the TIMN model, the Atmospheric
Cluster Dynamics Code (ACDC) with thermochemistry obtained using the
RI-CC2//B3LYP method (McGrath et al., 2012; Kürten et al., 2016), and an
empirical parameterization of CLOUD measurements (CLOUDpara) (Dunne et al.,
2016) indicated by solid, dashed, and dot-dashed lines, respectively. The
symbols refer to CLOUD experimental data (Kirkby et al., 2011; Dunne et al.,
2016), with the uncertainties in measured [NH_{3}] and J_{1.7} shown
by horizontal and vertical bars, respectively. To be comparable, the CLOUD
data points given in Dunne et al. (2016) under the conditions of
T= 292 K and RH = 38 % with [H_{2}SO_{4}]
close to 1.5 × 10^{8} cm^{−3} have been interpolated to the same
[H_{2}SO_{4}] value (= 1.5 × 10^{8} cm^{−3}).

Figure 6 shows a comparison of the model TIMN rates J_{1.7} with CLOUD
measurements, as a function of [NH_{3}] under two ionization rates. It
should be noted that Dunne et al. (2016) developed a simple empirical
parameterization (denoted hereafter as “CLOUDpara”) of binary, ternary, and
IIN rates in CLOUD measurements as a function of
[NH_{3}], [H_{2}SO_{4}], T, and negative ion
concentration. The predictions of CLOUDpara (Dunne et al., 2016) and ACDC
based on nucleation thermochemistry obtained using the RI-CC2//B3LYP method
(McGrath et al., 2012; Kürten et al., 2016) are also presented in Fig. 6 for
comparisons.

Like the CLOUD measurements, the TIMN predictions reveal a complex dependence
of J_{1.7} on [NH_{3}], and an analysis of the TIMN results shows
this behavior can be explained by the differing responses of negative,
positive, and neutral clusters to the presence of ammonia (Fig. 4). Under the
conditions specified in Fig. 6, nucleation is dominated by negative ions for
[NH_{3}] < ∼0.5 ppb, by both negative and positive
ions for [NH_{3}] from ∼0.5 ppb to ∼10 ppb (with
background ionization) or ∼20 ppb (with pion-enhanced ionization),
and by neutrals at higher [NH_{3}]. According to TIMN, [NH_{3}]
values
of at least 0.6–1 ppb are needed before positive ions contribute
significantly to nucleation rates – in good agreement with the threshold
found in the CLOUD experiments (Kirkby et al., 2011; Schobesberger et al.,
2015). TIMN simulations also extend CLOUD data at [NH_{3}] of
∼1 ppb to include a “zero-sensitivity zone” in the region of
1–10 ppb, followed by a region of strong sensitivity of J_{1.7} to
[NH_{3}] commencing at
[NH_{3}] > ∼10–20 ppb. The latter zone may have
important implications for NPF in heavily polluted regions, including much of
India and China, where [NH_{3}] may exceed 10–20 ppb (Behera and
Sharma, 2010; Meng et al., 2018). It is noteworthy in Fig. 6 that the
dependence of J_{1.7} on [NH_{3}] and Q predicted by the ACDC
model (McGrath et al., 2012) and the CLOUD data parameterization (Dunne et
al., 2016) deviate substantially from the experimental data as well as the
TIMN simulations. CLOUDpara does not consider impacts of positive ions
and key controlling parameters such as RH and surface area of preexisting
particles. Dunne et al. (2016) reported that CLOUDpara is also very sensitive
to the approach to parameterize T dependence, showing that the contribution
of ternary IIN to NPF below 15 km in altitude has grown from
9.6 % to 37.5 %, after the initial empirical temperature function was
replaced with a simpler one.

Figure 7Comparison of TIMN simulations (solid lines), CLOUDpara predictions
(Dunne et al., 2016) (dot-dashed lines), and CLOUD measurements (symbols and data
from Dunne et al., 2016) of the dependences of nucleation rates on
(a) [H_{2}SO_{4}] at five different temperatures (T= 292,
278, 248, 223, and 208 K) and (b) RH at two sets of conditions as specified.
[NH_{3}] is in parts per trillion, by volume. Error bars for the
uncertainties in measured [H_{2}SO_{4}] (−50 %,
+100 %), [NH_{3}] (−50 %, +100 %), and J_{1.7}
(overall a factor of 2) are not shown. To be comparable, the CLOUD data
points given in Dunne et al. (2016) under the conditions (T, RH, ionization
rate) with [NH_{3}] or [H_{2}SO_{4}] close to the
corresponding values specified in the figure legends have been interpolated
to the same [NH_{3}] (Fig. 7a) or [H_{2}SO_{4}]
(Fig. 7b) values.

Figure 7 presents a more detailed comparison of TIMN simulations with CLOUD
measurements of J_{1.7} as a function of [H_{2}SO_{4}], T,
and RH. The TIMN model reproduces both the absolute values of J_{1.7} and
its dependencies on [H_{2}SO_{4}], T, and RH in a wide range
of temperatures (T= 208–292 K) and [H_{2}SO_{4}]
(5 × 10^{5}–5 × 10^{8} cm^{−3}). As expected,
nucleation rates are very sensitive to [H_{2}SO_{4}] and T.
For example, J_{1.7} increases by 3 to 5 orders of magnitude with an
increase in [H_{2}SO_{4}] of a factor of 10 and by roughly 1
order of magnitude for a temperature decrease of
10 ^{∘}C, except in cases in which the nucleation
rate is limited by Q (for example,
[H_{2}SO_{4}] =∼10^{8}–10^{9} cm^{−3} at
T= 278 and 292 K, shown in Fig. 7a). The key difference between
CLOUDpara and TIMN predictions is that the
dlnJ_{1.7}∕dln[H_{2}SO_{4}] ratio predicted by
CLOUDpara is nearly constant while TIMN shows that this ratio depends on both
[H_{2}SO_{4}] and T. The CLOUD measurements taken at
T= 278 K clearly show (in agreement with the TIMN) that
dlnJ_{1.7}∕dln[H_{2}SO_{4}] is not constant.
CLOUDpara overestimates J_{1.7} compared to both measurements and TIMN
simulations, except for the case in which T= 278 K and
[H_{2}SO_{4}] range from ∼7× 10^{6} to
5 × 10^{7} cm^{−3}, with a deviation of CLOUDpara from
experimental data and TIMN growing with the lower temperature.

Both CLOUD measurements and TIMN simulations (Fig. 7b) show an important
influence of RH on nucleation rates. In particular, CLOUD measurements
indicate a 1–5 order of magnitude rise in J_{1.7} after RH increases from
10 % to 70 %–80 % and a stronger effect of RH on nucleation rates at
higher temperatures under the conditions shown in Fig. 7b. The RH dependence
of J_{1.7} predicted by the TIMN model is consistent with measurements,
being slightly weaker than that measured at high RH.

Figure 8Model predicted (J_{model}) versus observed (J_{obs})
nucleation rates under various conditions of all 377 data points of CLOUD
measurements reported in Table S1 of Dunne et al. (2016), with (a) and
without (b) the presence of ionization. The data points are grouped according
to temperatures as specified in the legend. Vertical error bars show the
range of J_{model} values calculated at 50 % and 200 % of measured
[H_{2}SO_{4}], corresponding to the uncertainties in measured
[H_{2}SO_{4}] (−50 %, +100 %). Error bars associated
with the uncertainties in measured [NH_{3}] (−50 %, +100 %)
and J_{obs} (overall a factor of 2) are not shown.

Figure 8 compares TIMN model predictions with all 377 data points of CLOUD
measurements reported in data Table S1 of Dunne et al. (2016). The vertical
error bars show the range of J_{model} associated with the
uncertainty in the [H_{2}SO_{4}] measured (−50 %,
+100 %). The effect of uncertainty in measured [NH_{3}]
(−50 %, +100 %) is not included. In the presence of ionization
(Fig. 8a), J_{model} agrees with CLOUD measurements within the
uncertainties under mainly all conditions, although J_{model} tends
to be slightly lower than J_{obs} when T= 292–300 K and
J_{obs} is relatively small
(< ∼1 cm^{−3} s^{−1}). For the neutral nucleation
(Fig. 8b), the model agrees well with observations at low T
(T= 205–223 K) but deviates from observations as T increases. The
underprediction of the model for neutral nucleation at T= 278–300 K
cannot be explained by the uncertainties in measured
[H_{2}SO_{4}] and [NH_{3}]. Apparently for neutral
nucleation the model predicts much stronger temperature dependence than the
CLOUD measurements. The possible reasons for the difference include the
uncertainties in both the model (especially the thermodynamics data and
approximation) and measurements. It should be noted that under the
conditions of high T and absence of ions, the role of cluster evaporation
(i.e., thermodynamics) becomes more important (i.e., higher evaporation and/or
generally less tightly bound clusters) and the effect of the possible biases
of the used thermochemistry can be more clearly revealed. The contamination
(by amines) in the CLOUD measurements (Kirkby et al., 2011) can be another
possible reason. The level of contamination in the cloud chamber appears to
increase with temperature (Kürten et al., 2016), which may explain the good
agreement at low T and increased deviation at higher T. Further research
is needed to identify the source of the difference for neutral ternary
nucleation at high T.

A comprehensive kinetically based
H_{2}SO_{4}–H_{2}O–NH_{3} TIMN model, constrained with thermodynamic data from
QC calculations and laboratory measurements, has been developed
and used to shed new light on physicochemical processes underlying the
effect of ammonia on NPF. We show that the stabilizing effect of
NH_{3} grows with the cluster size and that the reduced effect of
ammonia on smaller clusters is caused by weaker bonding that in turn yields
lower average NH_{3}-to-H_{2}SO_{4} ratios.
NH_{3} was found to impact nucleation barriers for neutral, positively
charged, and negatively charged clusters differently due to the large
difference in the binding energies of NH_{3}, H_{2}O, and
H_{2}SO_{4} to small clusters of different charging states. The
lowest and highest nucleation barriers are observed in the case of negative
ions and neutrals, respectively. Therefore, nucleation of negative ions is
favorable, followed by nucleation of positive ions and neutrals. Different
responses of negative, positive, and neutral clusters to ammonia result in a
complex dependence of ternary nucleation rates on [NH_{3}]. The TIMN
model reproduces both the absolute values of nucleation rates and their
dependencies on the key controlling parameters and agrees with the CLOUD
measurements for all the cases at the presence of ionization. For the neutral
ternary nucleation, the model agrees well with observations at low
temperature but deviates from observations as temperature increases.

The TIMN model developed in the present study may be subject to uncertainties
associated with the uncertainties in thermodynamic data and interpolation
approximation for pre-nucleation clusters. Further measurements and quantum
calculations, especially for relatively larger clusters, are needed to reduce
the uncertainties. While the TIMN model predicts nucleation rates in a good
overall agreement with the CLOUD measurements, its ability to explain the NPF
events observed in the real atmosphere is yet to be quantified and will be
investigated in further studies.

Thermochemical data for small neutral and charged binary
H_{2}SO_{4}–H_{2}O and ternary
H_{2}SO_{4}–H_{2}O–NH_{3} clusters have been
reported in a number of earlier publications (Bandy and Ianni, 1998; Ianni
and Bandy, 1999; Torpo et al., 2007; Nadykto et al., 2008; Herb et al., 2011,
2013; Temelso et al., 2012a, b; DePalma et al., 2012; Ortega et al., 2012;
Chon et al., 2014; Husar et al., 2014; Henschel et al., 2014, 2016; Kürten
et al., 2015). The PW91PW91/6-311$++$G(3df,3pd) method, which is a
combination of the Perdue–Wang PW91PW91 density functional with the largest
Pople 6-311$++$G(3df,3pd) basis set, has thoroughly been validated and agrees
well with existing experimental data. In earlier studies, this method has
been applied to a large variety of atmospherically relevant clusters (Nadykto
et al., 2006, 2007, 2008, 2014, 2015; Nadykto and Yu, 2007; Torpo et al.,
2007; Zhang et al., 2009; Elm et al., 2012, 2013; Leverentz et al., 2013; Xu
and Zhang, 2012, 2013; Zhu et al., 2014; Bork et al., 2014; Elm and
Mikkelsen, 2014; Peng et al., 2015; Miao et al 2015; Ma et al., 2016) and has
been shown to be well suited to study the
H_{2}SO_{4}–H_{2}O and
H_{2}SO_{4}–H_{2}O–NH_{3} clusters, as
evidenced by a very good agreement of the computed values with measured
cluster geometries, vibrational fundamentals, dipole properties, formation
Gibbs free energies (Nadykto et al., 2007, 2008, 2014, 2015; Nadykto and Yu,
2007; Herb et al., 2013; Elm et al., 2012, 2013; Leverentz et al., 2013; Bork
et al., 2014), and high-level ab initio results (Temelso et al., 2012a, b;
Husar et al., 2012; Bustos et al., 2014).

Table A1Number of isomers successfully converged at the 6-311 level for selected
clusters, along with the enthalpy, entropy, and Gibbs free energy of the most
stable isomers (1 hartree = 627.5 kcal mol^{−1}).

Table A2QC-based stepwise Gibbs free energy change (kcal mol^{−1}) for the
addition of one water ($\mathrm{\Delta}{G}_{+\mathrm{W}}^{o}$), ammonia
($\mathrm{\Delta}{G}_{+\mathrm{A}}^{o}$), or sulfuric acid ($\mathrm{\Delta}{G}_{+\mathrm{S}}^{o}$) molecule to form the given positively charged
clusters under standard conditions and the corresponding experimental data
or semi-experimental estimates.

^{a} Froyd and Lovejoy (2003a). ^{b} Meot-Ner
(Mautner) et al. (1984). ^{c} Payzant et al. (1973).
^{d} Froyd (2002). ^{e} Froyd and Lovejoy (2012).
^{f} The $\mathrm{\Delta}{G}_{+\mathrm{S}}^{o}$ values given
here were calculated based on experimental $\mathrm{\Delta}{G}_{+\mathrm{S}}^{o}$
values at T= 270 K from Froyd and Lovejoy (2003a) and ΔS
values from quantum calculation.

We have extended the earlier QC studies of binary and ternary clusters to
larger sizes. The computations have been carried out using a Gaussian 09 suite
of programs (Frish et al., 2009). In order to ensure the quality of the
conformational search we have carried out a thorough sampling of conformers.
We have used both a basin-hopping algorithm, as implemented in Biovia Materials
Studio 8.0, and locally developed sampling code. The sampling code is based
on the following principle: mesh, with molecules to be added to the cluster
placed in the mesh nodes, is created around the cluster, and a blind search
algorithm is used to generate the guess geometries. The mesh density and
orientation of molecules, as well as the minimum distance
between molecules and cluster, are variable. Typically, for each cluster of a given
chemical composition a thousand to several thousands of isomers have been
sampled. We used a three-step optimization procedure, which includes

i.

pre-optimization of initial and guess geometries using the semiempirical PM6
method, separation of the most stable isomers located within
15 kcal mol^{−1} of the intermediate global minimum and duplicate
removal, followed by

ii.

optimization of the selected isomers meeting the aforementioned
stability criterion by the PW91PW91/CBSB7 method and

iii.

the final
optimization of the most stable isomers at the PW91PW91/CBSB7 level within
5 kcal mol^{−1} of the current global minimum using the
PW91PW91/6-311$++$G(3df,3pd) method.

Typically, only ∼4 %–30 % of initially sampled isomers reach
the second (PW91PW91/CBSB7) level, at which ∼10 %–40 % of
isomers optimized with PW91PW91/CBSB7 are selected for the final run.
Typically, the number of equilibrium isomers of hydrated clusters is larger
than that of unhydrated ones of similar chemical composition. Table A1 shows
the numbers of isomers converged at the final PW91PW91/6-311$++$G(3df,3pd)
optimization step for selected clusters and HSG (enthalpy, entropy, and Gibbs
free energy) values of the most stable
isomers used in the present study. The number of isomers optimized at the
PW91PW91/6-311$++$G(3df,3pd) level of theory varies from case to case,
typically being in the range of ∼10–200.

The computed stepwise enthalpy, entropy, and Gibbs free energies of cluster
formation have been thoroughly evaluated and used to calculate the
evaporation rates of H_{2}SO_{4} from neutral, positively, and
negatively
charged clusters.

A1 Positively charged clusters

Table A2 presents the computed stepwise Gibbs free energy changes under
standard conditions (ΔG^{o}) for positive binary and ternary
clusters, along with the corresponding experimental data or semi-experimental
estimates. Figure 2 in the main text shows ΔG associated with the
addition of water ($\mathrm{\Delta}{G}_{+\mathrm{W}}^{o}$), ammonia
($\mathrm{\Delta}{G}_{+\mathrm{A}}^{o}$), and sulfuric acid ($\mathrm{\Delta}{G}_{+\mathrm{S}}^{o}$) to binary and ternary clusters as a function of the
cluster hydration number w. Both the absolute values and trends in
$\mathrm{\Delta}{G}_{+\mathrm{W}}^{o}$ derived from calculations are in agreement
with the laboratory measurements within the uncertainty range of
∼1–2 kcal mol^{−1} for both QC calculations and measurements.
This confirms the efficiency and precision of QC methods in calculating
thermodynamic data needed for the development of nucleation models.
Nevertheless, it should be noted that the uncertainties in computed free
energies of 1–2 kcal mol^{−1} may lead to large uncertainty in predicted
particle formation rates. By increasing or decreasing all Gibbs free energies
by 1 kcal mol^{−1}, Kürten at al. (2016) showed that, depending on
the conditions, the modeled particle formation rate can change from less than
an order of magnitude to several orders of magnitude. Uncertainties estimated
by Kürten at al. (2016) represent the upper limit because computed free
energies may be overestimated for some clusters and underpredicted for others, which leads to partial or, in some case, full error cancelation.

A2 Neutral clusters

Table A3 presents the computed stepwise Gibbs free energy changes for the
formation of ternary S_{s}A_{a}W_{w} clusters under standard
conditions. The corresponding binary electrically neutral clusters can be
found in previous publications (e.g., Nadykto et al., 2008; Herb et al.,
2011). The thermodynamic properties of the S_{1}A_{1} have been reported
in a number of computational studies (e.g., Herb et al., 2011; Kurtén et al.,
2007; Nadykto and Yu, 2007). However, most of these studies, except for
Nadykto and Yu (2007) and Henschel et al. (2014, 2016), did not consider the
impact of H_{2}O on cluster thermodynamics. We have extended the
earlier studies of Nadykto and Yu (2007) and Herb et al. (2011) to larger
clusters up to S_{4}A_{5} (no hydration) and up to S_{2}A_{2}
(hydration included). The free energy of binding of NH_{3} to
H_{2}SO_{4} (or H_{2}SO_{4} to NH_{3}) obtained
using our method is −7.77 kcal mol^{−1}, which is slightly more negative
than values reported by other groups (−6.6–7.61 kcal mol^{−1}) and
within less than 0.5 kcal mol^{−1} of the experimental value of
−8.2 kcal mol^{−1} derived from CLOUD measurements (Kürten et al.,
2015).

As it may be seen from Table A3, the NH_{3} binding to S_{1−2}W_{w}
weakens as w increases. The average $\mathrm{\Delta}{G}_{+\mathrm{W}}^{o}$ for
S_{1}W_{w} formation derived from a combination of laboratory measurements
and quantum chemical studies are −3.02, −2.37, and
−1.40 kcal mol^{−1} for the first, second, and third hydrations,
respectively (Yu, 2007). This indicates that a large fraction of
H_{2}SO_{4} monomers in the Earth's atmosphere are likely
hydrated. Therefore, the decreasing NH_{3} binding strength to
hydrated H_{2}SO_{4} monomers implies that RH (and T) will
affect the relative abundance of H_{2}SO_{4} monomers
containing NH_{3}. Currently, no experimental data or observations are
available to evaluate the impact of hydration (or RH) on $\mathrm{\Delta}{G}_{+\mathrm{A}}^{o}$. Table A3 shows that the presence of NH_{3}
in H_{2}SO_{4} clusters suppresses hydration and that
$\mathrm{\Delta}{G}_{+\mathrm{W}}^{o}$ for S_{2}A_{2} falls below
−2.0 kcal mol^{−1}. This is consistent with earlier studies by our
group (Herb et al., 2011) and others (Henschel et al., 2014, 2016) showing
that large S_{n}A_{n} clusters (n > 2) are not hydrated
under typical atmospheric conditions. In the present study, the hydration of
neutral S_{n}A_{n} clusters at n > 2 is neglected, due
to the lack of thermodynamic data.

The number of NH_{3} molecules in the cluster (or
H_{2}SO_{4} to NH_{3} ratio) significantly affects
$\mathrm{\Delta}{G}_{+\mathrm{S}}^{o}$ and $\mathrm{\Delta}{G}_{+\mathrm{A}}^{o}$ values.
For example, $\mathrm{\Delta}{G}_{+\mathrm{S}}^{o}$ for S_{3}A_{a} clusters
increases from −7.08 to −16.92 kcal mol^{−1} and
$\mathrm{\Delta}{G}_{+\mathrm{A}}^{o}$ decreases from −16.14 to
−8.93 kcal mol^{−1} as a grows from 1 to 3. For S_{4}A_{a}
clusters, $\mathrm{\Delta}{G}_{+\mathrm{S}}^{o}$ increases from −7.48 to
−16.26 kcal mol^{−1} and $\mathrm{\Delta}{G}_{+\mathrm{A}}^{o}$ decreases
from −17.16 to −11.34 kcal mol^{−1} as a increases from 2 to 4.
$\mathrm{\Delta}{G}_{+\mathrm{A}}^{o}$ for the S_{4}A_{1} cluster is less negative by
1.38 kcal mol^{−1} than that for S_{4}A_{2}. $\mathrm{\Delta}{G}_{+\mathrm{S}}^{o}$ for the S_{4}A_{1} cluster is also quite low
(−4.16 kcal mol^{−1}), which might indicate the possible existence of a
more stable S_{4}A_{1} isomer, which is yet to be identified. In the
presence of NH_{3}, the uncertainty in the thermochemistry data for
S_{4}A_{1} will not significantly affect ternary nucleation rates because
most S_{4} clusters contain three or four NH_{3} molecules.

For the S_{s}A_{a} clusters with s=a, $\mathrm{\Delta}{G}_{+\mathrm{A}}^{o}$
increases as the cluster grows while $\mathrm{\Delta}{G}_{+\mathrm{S}}^{o}$ first
increases significantly as S_{1}A_{1} is converted into S_{2}A_{2}
and then levels off as S_{2}A_{2} is converted into S_{4}A_{4}. We
also observe a significant drop in $\mathrm{\Delta}{G}_{+\mathrm{A}}^{o}$ when the NH_{3}∕H_{2}SO_{4} ratio exceeds 1. This
finding is consistent with the ACDC model calculation showing that growth of
neutral S_{s}A_{a} clusters follows the s=a pathway (Schobesberger et
al., 2015).

A3 Negative ionic clusters

Table A4 shows ΔG_{+W}, ΔG_{+A}, and ΔG_{+S} needed to form negatively charged clusters under standard
conditions, along with available semi-experimental values (Froyd and Lovejoy,
2003b). H_{2}O binding to negatively charged S^{−}S_{s} clusters
significantly strengths with increasing s, from $\mathrm{\Delta}{G}_{+\mathrm{W}}^{o}$=−0.61 – −1.83 kcal mol^{−1} at
s=1–2 to $\mathrm{\Delta}{G}_{+\mathrm{W}}^{o}$=−3.5 kcal mol^{−1}
at w=1 and −2.25 kcal mol^{−1} at w=4 at s=4. $\mathrm{\Delta}{G}_{+\mathrm{W}}^{o}$ values at s=3 and 4 are slightly more negative (by
∼0.1–0.9 kcal mol^{−1}) than those reported by Froyd and
Lovejoy (2003b). Just like H_{2}O binding, NH_{3} binding to
S^{−}S_{s} at s < 3 is very weak, with $\mathrm{\Delta}{G}_{+\mathrm{A}}^{o}$ ranging from +2.81 kcal mol^{−1} at s=0 to
−4.85 kcal mol^{−1} at s=2. However, it significantly increases as
s grows. In particular, at $s\ge \mathrm{3}\mathrm{\Delta}{G}_{+\mathrm{A}}^{o}$ ranges
from −11.89 kcal mol^{−1} for S^{−}S_{3}A_{1} to
−15.37 kcal mol^{−1} for S^{−} S_{4}A_{1}. NH_{3} clearly
cannot enter small negative ions. However, it can easily attach to larger
negative ions with s≥3, which is consistent with CLOUD measurements
(Schobesberger et al., 2015). Since hydration weakens NH_{3} binding
in S^{−}S_{3}A_{1}W_{w} and S^{−}S_{4}A_{1}W_{w} clusters,
its impacts on the cluster formation and nucleation rates may potentially be
important.

In contrast to H_{2}O and NH_{3}, binding of
H_{2}SO_{4} to small negative ions (s < 3) is very
strong. These ions are very stable even when they contain no NH_{3} or
H_{2}O molecules. High electron affinity of H_{2}SO_{4}
molecules results in the high stability of S^{−}S_{s} at s=1–2.
However, the charge effect reduces as s grows. In particular, $\mathrm{\Delta}{G}_{+\mathrm{S}}^{o}$ of S^{−}S_{s} drops from
−32.74 kcal mol^{−1} at s=1 to −10.58 kcal mol^{−1} and
−8.28 kcal mol^{−1} at s=3 and 4, respectively. At the same time,
$\mathrm{\Delta}{G}_{+\mathrm{A}}^{o}$ increases from 0.08 kcal mol^{−1} (s=1)
to −11.89 kcal mol^{−1} (s=3) and −15.37 kcal mol^{−1} (s=4).
The hydration of S^{−}S_{s} at s=3 and 4 enhances the strength of
H_{2}SO_{4} binding, especially at s=4. $\mathrm{\Delta}{G}_{+\mathrm{S}}^{o}$ values for S^{−}S_{3−4}W_{w} are consistently
∼1.5–3 kcal mol^{−1} less negative than the corresponding
semi-experimental estimates (Table A4). The possible reasons behind the
observed systematic difference are yet to be identified and include the use
of the low-level ab initio Hartree–Fock method to compute reaction
enthalpies and uncertainties in experimental enthalpies in studies by Froyd
and Lovejoy (2003b).

NH_{3} binding to S^{−}S_{3} significantly enhances the stability
of H_{2}SO_{4} in the cluster by ∼7 kcal mol^{−1}
compared to $\mathrm{\Delta}{G}_{+\mathrm{S}}^{o}$ for the corresponding binary
counterpart. The binding of the second NH_{3} to S^{−}S_{3}A to
form S^{−}S_{3}A_{2} is much weaker ($\mathrm{\Delta}{\mathrm{G}}_{+\mathrm{A}}^{o}$=−7.27 kcal mol^{−1}) than that of
the first NH_{3} molecule ($\mathrm{\Delta}{\mathrm{G}}_{+\mathrm{A}}^{o}=-\mathrm{11.89}$ kcal mol^{−1}). This indicates that most of S^{−}S_{3}A_{a}
can only contain one NH_{3} molecule, in a perfect agreement with the
laboratory study of Schobesberger et al. (2015). In the case of
S^{−}S_{4}, binding of the first ($\mathrm{\Delta}{\mathrm{G}}_{+\mathrm{A}}^{\mathrm{o}}$=−15.37 kcal mol^{−1}) and
second (and −12.23 kcal mol^{−1}) NH_{3} molecules to the
cluster is quite strong, while the attachment of NH_{3} leads to
substantial stabilization of H_{2}SO_{4} in the cluster, as
evidenced by $\mathrm{\Delta}{G}_{+\mathrm{S}}^{o}$ growing from
−8.28 kcal mol^{−1} at a=0 to −11.76 kcal mol^{−1} and
−16.71 kcal mol^{−1} at a=1 and a=2, respectively. The
NH_{3} binding free energy to S^{−}S_{4}A_{2} (to form
S^{−}S_{4}A_{3}) drops to −7.59 kcal mol^{−1}, indicating, in
agreement with the CLOUD measurements (Schobesberger et al., 2015), that most
S^{−}S_{4} clusters contain one or two NH_{3} molecules.

FY designed the study, led the development
of the TIMN model, analyzed results, and wrote a major part of the
paper. ABN, JH, KMN, and LAU designed, carried out, and evaluated
quantum calculations. ABN also contributed to the writing of the paper.
GL contributed to the result analysis and discussion. All contributed to the
reviewing and editing of the paper.

The authors thank Richard Turco (Distinguished Professor Emeritus, UCLA) for
comments that helped to improve the paper. This study was supported by the
NSF under grant 1550816, NASA under grant NNX13AK20G, and NYSERDA under
contract 100416. ABN, KMN and LAU would like to thank the Russian Science Foundation
(under grant 18-11-00247) and
the Ministry of Science and Education of Russia (under grants 1.6198.2017/6.7
and 1.7706.2017/8.9) for support and the Center of Collective Use of MSTU
“STANKIN” for providing resources. Edited by: Veli-Matti
Kerminen Reviewed by: three anonymous referees

Almeida, J., Schobesberger, S., Kürten, A., Ortega, I. K., Kupiainen-Määttä,
O., Praplan, A. P., Adamov, A., Amorim, A., Bianchi, F., Breitenlechner, M.,
David, A., Dommen, J., Donahue, N. M., Downard, A., Dunne, E., Duplissy, J.,
Ehrhart, S., Flagan, R. C., Franchin, A., Guida, R., Hakala, J., Hansel, A.,
Heinritzi, M., Henschel, H., Jokinen, T., Junninen, H., Kajos, M.,
Kangasluoma, J., Keskinen, H., Kupc, A., Kurtén, T., Kvashin, A. N.,
Laaksonen, A., Lehtipalo, K., Leiminger, M., Leppä, J., Loukonen, V.,
Makhmutov, V., Mathot, S., McGrath, M. J., Nieminen, T., Olenius, T., Onnela,
A., Petäjä, T., Riccobono, F., Riipinen, I., Rissanen, M., Rondo, L.,
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Aerosol nucleation exerts important influences on the climate, hydrological cycle, and air quality. We have developed an advanced physical–chemical model that describes ion-induced and neutral nucleation involving ammonia, sulfuric acid, and water vapors. The model is shown to reproduce laboratory measurements taken under a wide range of conditions, offers physiochemical insights into the ternary nucleation process, and provides an accurate approach to calculate ternary rate in the atmosphere.

Aerosol nucleation exerts important influences on the climate, hydrological cycle, and air...