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**Atmospheric Chemistry and Physics**
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ACP | Articles | Volume 20, issue 12

Atmos. Chem. Phys., 20, 7359–7372, 2020

https://doi.org/10.5194/acp-20-7359-2020

© Author(s) 2020. This work is distributed under

the Creative Commons Attribution 4.0 License.

https://doi.org/10.5194/acp-20-7359-2020

© Author(s) 2020. This work is distributed under

the Creative Commons Attribution 4.0 License.

Special issue: The CERN CLOUD experiment (ACP/AMT inter-journal SI)

**Research article**
25 Jun 2020

**Research article** | 25 Jun 2020

Enhanced growth rate of atmospheric particles from sulfuric acid

^{1}Faculty of Physics, University of Vienna, 1090 Vienna, Austria^{2}Institute for Atmospheric and Earth System Research/Physics, University of Helsinki, 00014 Helsinki, Finland^{3}Institute for Atmospheric and Environmental Sciences, Goethe University Frankfurt, 60438 Frankfurt am Main, Germany^{4}School of Earth and Environment, University of Leeds, Leeds LS2 9JT, UK^{5}Finnish Meteorological Institute, 00560 Helsinki, Finland^{6}Atmospheric Chemistry Department, Max Planck Institute for Chemistry, 55128 Mainz, Germany^{7}Department of Chemistry and Cooperative Institute for Research in Environmental Sciences, University of Colorado Boulder, Boulder, CO 80309, USA^{8}Laboratory of Atmospheric Chemistry, Paul Scherrer Institute, 5232 Villigen, Switzerland^{9}Center for Astrophysics and Gravitation, Faculty of Sciences of the University of Lisbon, 1749-016 Lisbon, Portugal^{10}Tofwerk AG, 3600 Thun, Switzerland^{11}Center for Atmospheric Particle Studies, Carnegie Mellon University, Pittsburgh, PA 15217, USA^{12}Institute for Ion Physics and Applied Physics, University of Innsbruck, 6020 Innsbruck, Austria^{13}Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, CA 91125, USA^{14}School of Civil and Environmental Engineering, Pusan National University, Busan 46241, Republic of Korea^{15}Ionicon Analytik GmbH, 6020 Innsbruck, Austria^{16}Department of Applied Physics, University of Eastern Finland, 70211 Kuopio, Finland^{17}P.N. Lebedev Physical Institute of the Russian Academy of Sciences, 119991 Moscow, Russia^{18}CERN, the European Organization for Nuclear Research, 1211 Geneva, Switzerland^{19}Joint International Research Laboratory of Atmospheric and Earth System Sciences, School of Atmospheric Sciences, Nanjing University, 210023 Nanjing, China^{20}Aerosol Physics Laboratory, Tampere University, 33101 Tampere, Finland^{21}Institute Infante Dom Luíz, University of Beira Interior, 6200-001 Covilhã, Portugal

^{1}Faculty of Physics, University of Vienna, 1090 Vienna, Austria^{2}Institute for Atmospheric and Earth System Research/Physics, University of Helsinki, 00014 Helsinki, Finland^{3}Institute for Atmospheric and Environmental Sciences, Goethe University Frankfurt, 60438 Frankfurt am Main, Germany^{4}School of Earth and Environment, University of Leeds, Leeds LS2 9JT, UK^{5}Finnish Meteorological Institute, 00560 Helsinki, Finland^{6}Atmospheric Chemistry Department, Max Planck Institute for Chemistry, 55128 Mainz, Germany^{7}Department of Chemistry and Cooperative Institute for Research in Environmental Sciences, University of Colorado Boulder, Boulder, CO 80309, USA^{8}Laboratory of Atmospheric Chemistry, Paul Scherrer Institute, 5232 Villigen, Switzerland^{9}Center for Astrophysics and Gravitation, Faculty of Sciences of the University of Lisbon, 1749-016 Lisbon, Portugal^{10}Tofwerk AG, 3600 Thun, Switzerland^{11}Center for Atmospheric Particle Studies, Carnegie Mellon University, Pittsburgh, PA 15217, USA^{12}Institute for Ion Physics and Applied Physics, University of Innsbruck, 6020 Innsbruck, Austria^{13}Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, CA 91125, USA^{14}School of Civil and Environmental Engineering, Pusan National University, Busan 46241, Republic of Korea^{15}Ionicon Analytik GmbH, 6020 Innsbruck, Austria^{16}Department of Applied Physics, University of Eastern Finland, 70211 Kuopio, Finland^{17}P.N. Lebedev Physical Institute of the Russian Academy of Sciences, 119991 Moscow, Russia^{18}CERN, the European Organization for Nuclear Research, 1211 Geneva, Switzerland^{19}Joint International Research Laboratory of Atmospheric and Earth System Sciences, School of Atmospheric Sciences, Nanjing University, 210023 Nanjing, China^{20}Aerosol Physics Laboratory, Tampere University, 33101 Tampere, Finland^{21}Institute Infante Dom Luíz, University of Beira Interior, 6200-001 Covilhã, Portugal

**Correspondence**: Paul M. Winkler (paul.winkler@univie.ac.at)

**Correspondence**: Paul M. Winkler (paul.winkler@univie.ac.at)

Abstract

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In the present-day atmosphere, sulfuric acid is the most important vapour for aerosol particle formation and initial growth. However, the growth rates of nanoparticles (<10 nm) from sulfuric acid remain poorly measured. Therefore, the effect of stabilizing bases, the contribution of ions and the impact of attractive forces on molecular collisions are under debate. Here, we present precise growth rate measurements of uncharged sulfuric acid particles from 1.8 to 10 nm, performed under atmospheric conditions in the CERN (European Organization for Nuclear Research) CLOUD chamber. Our results show that the evaporation of sulfuric acid particles above 2 nm is negligible, and growth proceeds kinetically even at low ammonia concentrations. The experimental growth rates exceed the hard-sphere kinetic limit for the condensation of sulfuric acid. We demonstrate that this results from van der Waals forces between the vapour molecules and particles and disentangle it from charge–dipole interactions. The magnitude of the enhancement depends on the assumed particle hydration and collision kinetics but is increasingly important at smaller sizes, resulting in a steep rise in the observed growth rates with decreasing size. Including the experimental results in a global model, we find that the enhanced growth rate of sulfuric acid particles increases the predicted particle number concentrations in the upper free troposphere by more than 50 %.

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How to cite.

Stolzenburg, D., Simon, M., Ranjithkumar, A., Kürten, A., Lehtipalo, K., Gordon, H., Ehrhart, S., Finkenzeller, H., Pichelstorfer, L., Nieminen, T., He, X.-C., Brilke, S., Xiao, M., Amorim, A., Baalbaki, R., Baccarini, A., Beck, L., Bräkling, S., Caudillo Murillo, L., Chen, D., Chu, B., Dada, L., Dias, A., Dommen, J., Duplissy, J., El Haddad, I., Fischer, L., Gonzalez Carracedo, L., Heinritzi, M., Kim, C., Koenig, T. K., Kong, W., Lamkaddam, H., Lee, C. P., Leiminger, M., Li, Z., Makhmutov, V., Manninen, H. E., Marie, G., Marten, R., Müller, T., Nie, W., Partoll, E., Petäjä, T., Pfeifer, J., Philippov, M., Rissanen, M. P., Rörup, B., Schobesberger, S., Schuchmann, S., Shen, J., Sipilä, M., Steiner, G., Stozhkov, Y., Tauber, C., Tham, Y. J., Tomé, A., Vazquez-Pufleau, M., Wagner, A. C., Wang, M., Wang, Y., Weber, S. K., Wimmer, D., Wlasits, P. J., Wu, Y., Ye, Q., Zauner-Wieczorek, M., Baltensperger, U., Carslaw, K. S., Curtius, J., Donahue, N. M., Flagan, R. C., Hansel, A., Kulmala, M., Lelieveld, J., Volkamer, R., Kirkby, J., and Winkler, P. M.: Enhanced growth rate of atmospheric particles from sulfuric acid, Atmos. Chem. Phys., 20, 7359–7372, https://doi.org/10.5194/acp-20-7359-2020, 2020.

1 Introduction

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Sulfuric acid (H_{2}SO_{4}) is the major atmospheric trace compound
responsible for the nucleation of aerosol particles in the present-day
atmosphere (Dunne et al., 2016).
Sulfuric acid participates in new particle formation (NPF) in the upper
troposphere (Brock
et al., 1995; Weber et al., 1999; Weigel et al., 2011), stratosphere
(Deshler, 2008), polar regions
(Jokinen et al., 2018), urban or anthropogenically influenced
environments (Yao et al., 2018), and
when a complex mixture of different condensable vapours is present
(Lehtipalo et al., 2018).
Especially in the initial growth of small atmospheric molecular clusters,
sulfuric acid is likely of crucial importance (Kulmala et
al., 2013). The newly formed particles need to grow rapidly in order to
avoid scavenging by larger, pre-existing aerosols and, thereby, contribute
to the global cloud condensation nuclei (CCN) budget (Pierce
and Adams, 2007). Therefore, the dynamics in this cluster size range of a few nanometres determines the climatic significance of atmospheric NPF, which is
the major source of CCN (Gordon et al., 2017) and can also
affect urban air quality (Guo et al.,
2014).

The main pathway of cluster and particle growth is condensation of low volatility vapours, like sulfuric acid or oxidized organics (Stolzenburg et al., 2018). Nanoparticle growth rates depend on both the evaporation rates of the condensing vapours and the molecular collision frequencies. Uncertainty about the expected behaviour at the collision (“kinetic”) limit influences the interpretation of experimental data. One focus has been on the evaporation rates from small particles and the potential growth rate enhancement from coagulation. In earlier laboratory measurements, it has been shown that bases like ammonia can have a stabilizing effect for growth below 2 nm (Lehtipalo et al., 2016). If amines, which are stronger bases than ammonia, are added, nucleation itself can proceed at the kinetic limit, i.e. evaporation rates from the monomer onwards are zero (Jen et al., 2014; Kürten et al., 2014; Olenius et al., 2013). In this case, cluster coagulation also plays an important role in the growth process due to the strong clustering behaviour of sulfuric acid and amines (Kontkanen et al., 2016; Lehtipalo et al., 2016; Li and McMurry, 2018). However, in the presence of ammonia, the evaporation rates and the magnitude of cluster coagulation remain unmeasured, although ammonia is much more important than amines globally due to its longer atmospheric lifetime. A second focus is on the collisional rate coefficients themselves, which may be enhanced by either charge–dipole interactions (Nadykto and Yu, 2003) or van der Waals forces (Chan and Mozurkewich, 2001). In spite of the importance of these coefficients, there are only few direct measurements of the charge effect on growth (Lehtipalo et al., 2016; Svensmark et al., 2017). Even if the charge–dipole interactions are stronger, an enhancement due to van der Waals forces might be more important at typical atmospheric ionization levels. Several atmospheric studies have demonstrated that sulfuric acid uptake proceeds at close to a collision-limited rate (Bzdek et al., 2013; Kuang et al., 2010), but they could neither provide a measurement of a collision enhancement nor did they consider hydration effects in detail (Verheggen and Mozurkewich, 2002). Both of these factors might be significant in the free molecular regime below 5 nm, where growth measurements are also affected by larger uncertainties (Kangasluoma and Kontkanen, 2017). Here, we address the questions of evaporation and collision enhancement in sulfuric-acid-driven growth with precision measurements (Stolzenburg et al., 2017) at the CERN (European Organization for Nuclear Research) CLOUD experiment (Duplissy et al., 2016).

2 Methods

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The CERN CLOUD chamber is a 26.1 m^{3} stainless steel aerosol chamber that can be kept at a constant temperature within 0.1 K precision. It
offers the possibility to study new particle formation under different
ionization levels. Two high-voltage electrode grids inside the chamber can
efficiently clear ions and charged particles from the chamber within
seconds, ensuring neutral conditions. When there is no electric field in the
chamber, galactic cosmic rays lead to an ion production rate of
∼ 2–4 ion pairs cm^{−3} s^{−1}. Ion concentrations can
also be elevated to upper tropospheric conditions by the illumination of the
chamber with a pion beam from the CERN Proton Synchrotron. The dry air
supply for the chamber is provided by boil-off oxygen and boil-off nitrogen
mixed at the atmospheric ratio of 79 : 21. This ensures extremely low
contaminant levels, especially from organics and sulfuric acid. This was
verified by a PTR3 proton transfer reaction time-of-flight mass spectrometer
(Breitenlechner et al., 2017) and a
nitrate chemical ionization atmospheric pressure interface time-of-flight (nitrate CI-APi-ToF) mass spectrometer (Jokinen et al.,
2012). The absence of any contamination from amines was confirmed by
measurements with a water cluster CI-APi-ToF (Pfeifer et
al., 2020), which did not register dimethylamine mixing ratios above the
detection limit of 0.1 pptv.

We performed measurements of particle growth from sulfuric acid and ammonia
at either +20 or +5 ^{∘}C with the relative
humidity kept constant at either 38 % or 60 %. SO_{2} (5 ppb), O_{3}
(∼120 ppb) and ammonia (varied between 3 and 1000 pptv) were
injected into the chamber. The experiments were initiated by homogeneous
illumination of the chamber at constant O_{3} and SO_{2} levels. The UV
light of four Hamamatsu UV lamps guided into the chamber with fibre optics
induced the photo-dissociation of O_{3} and the production of OH•
radicals. Thus, SO_{2} was oxidized, leading to the formation of
sulfuric acid (varied between 10^{7} and 10^{9} cm^{−3}). A typical
experiment is shown in Fig. S1 in the Supplement. Sulfuric acid monomer
concentrations were measured with the nitrate CI-APi-ToF. Calibration of the
instrument's response to sulfuric acid (Kürten et al.,
2012) was performed before and after the measurement campaign and yielded
comparable results. Compared with previous studies, the measurement of
gas-phase NH_{3} also significantly improved due to the deployment of the
calibrated water cluster CI-APi-ToF. The protonated water cluster reagent
ions selectively ionize ammonia and amines at ambient pressure reaching a
detection limit of approximately 0.5 pptv for ammonia.

Particle growth was monitored using a differential mobility analyser-train (DMA-train; Stolzenburg et al., 2017) for the main size range from 1.8 to 8 nm. We also include measurements from a Caltech nano-radial DMA (Brunelli et al., 2009) with a custom-built diethylene glycol (DEG) counter for sizes between 4 and 8 nm and a TSI Scanning Mobility Particle Sizer (nano-SMPS; model 3936) for sizes larger than 5 nm when investigating the size dependence of the growth. For the growth of the charged fraction, we use a neutral cluster and air ion spectrometer (NAIS; Manninen et al., 2009). All four instruments use electrical mobility classification, and the measured mobility diameters are corrected to mass diameters (Larriba et al., 2011) for the calculation of collision kinetics. Compared with the scanning particle size magnifier (see e.g. Lehtipalo et al., 2014), which was used in Lehtipalo et al. (2016), these instruments that use direct mobility analysis have less systematic uncertainty on the actual size classification. The size ranges of both studies are also not directly comparable. We show the measurements in the lower size interval of the DMA-train (1.8–3.2 nm mobility diameter) as well as the earlier results (size range between 1.5 and 2.5 nm mobility diameter) in Fig. S2.

Another difference between the instruments is the treatment of the sample
relative humidity. In the DMA-train, the aerosol sheath flow is dried using
silica gel, achieving a relative humidity measured at the sheath inlet of the
DMA below 5 % for all experiments in this study. The nano-SMPS uses a
water trap to keep the relative humidity of the DMA sheath flow below 20 % during the reported experiments. The Caltech nano-radial DMA, the NAIS
and the particle size magnifier used in Lehtipalo et al. (2016) do not deploy any humidity conditioning for the sheath or sample
flow, except for the possible decrease in relative humidity as a result of a
temperature increase between the measurement device and chamber. This effect
occurred for all instruments to some extent, even if the sampling lines were
insulated. The effect of aerosol dehydration during the measurement is
usually described by the hygroscopic growth factor gf, relating the
measured diameter *d*_{p,m} to the actual diameter *d*_{p} as follows: ${d}_{\mathrm{p}}=\mathrm{gf}\phantom{\rule{0.125em}{0ex}}\cdot {d}_{\mathrm{p},\mathrm{m}}$.

From the measured aerosol size distributions, we inferred particle growth
rates using two complementary methods in order to limit systematic biases in
the analysis. In the first method, particle growth rates were measured with
the appearance time method, which requires a growing particle population that
can be clearly identified (Dada et
al., 2020; Lehtipalo et al., 2014; Stolzenburg et al., 2018). Figure S1d demonstrates how the signal in each size channel is fitted by
an empirical sigmoidal shape curve estimating the time at which 50 % of the
maximum signal intensity is reached. These appearance times are fitted with
a linear function over the size intervals from 1.8 to 3.2 and from 3.2 to 8 nm, with the
slope yielding an average growth rate over the interval (shown in Fig. S1b).
In the second method, we applied the size- and time-resolving growth rate
analysis method “INSIDE” (INterpreting the change rate of the Size-Integrated general Dynamic Equation; Pichelstorfer et al., 2018) to
cross-check our results. The INSIDE method uses the measured particle size
distribution at a time *t*_{1} and simulates the expected aerosol dynamics
(coagulation, wall losses and dilution) until time *t*_{2}. By comparing this to the measured data at *t*_{2} and evaluating the general dynamics
equation, it infers the condensational growth rate at specified diameters
for this time step. The time- and size-resolved growth rates for each
experiment were time-averaged for all sizes to yield a statistically more
robust result. Compared to the appearance time method, INSIDE requires
accurate absolute number size distributions, whereas the appearance time
method only depends on the relative signal increase. However, INSIDE can
confirm the absence of systematic biases like changing precursor vapour
concentrations or coagulation and wall loss effects. Therefore, a combined assessment
with both methods should yield a solid estimate of the observed
growth rates.

If the evaporation rates of the growing particles are effectively zero due to the extremely low vapour pressure of the condensing vapour, particle growth rates are limited by the collision frequencies of vapour molecules with the growing particles. Our description of particle growth follows the approach of Nieminen et al. (2010), which, in comparison to the equations of mass transfer that can be found in e.g. Seinfeld and Pandis (2016), include the non-negligible effect of vapour molecular size using a collision frequency between vapour and particle in analogy to coagulation theory (Lehtinen and Kulmala, 2003):

$$\begin{array}{}\text{(1)}& \begin{array}{rl}\mathrm{GR}& ={\displaystyle \frac{\mathrm{d}{d}_{\mathrm{p}}}{\mathrm{d}t}}={\displaystyle \frac{\frac{\mathrm{d}{V}_{\mathrm{p}}}{\mathrm{d}t}}{\frac{\mathrm{d}{V}_{\mathrm{p}}}{\mathrm{d}{d}_{\mathrm{p}}}}}={\displaystyle \frac{{k}_{\mathrm{coll}}\left({d}_{\mathrm{v}},{d}_{\mathrm{p}}\right)\phantom{\rule{0.125em}{0ex}}\cdot \phantom{\rule{0.125em}{0ex}}{V}_{\mathrm{v}}\cdot \phantom{\rule{0.125em}{0ex}}{C}_{\mathrm{v}}}{\frac{\mathrm{d}}{\mathrm{d}{d}_{\mathrm{p}}}\left[\frac{\mathit{\pi}}{\mathrm{6}}{d}_{\mathrm{p}}^{\mathrm{3}}\right]}}\\ & ={\displaystyle \frac{{k}_{\mathrm{coll}}\left({d}_{\mathrm{v}},\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}{d}_{\mathrm{p}}\right)\phantom{\rule{0.125em}{0ex}}\cdot \phantom{\rule{0.125em}{0ex}}{V}_{\mathrm{v}}\cdot \phantom{\rule{0.125em}{0ex}}{C}_{\mathrm{v}}}{\mathit{\pi}/\mathrm{2}\cdot {d}_{\mathrm{p}}^{\mathrm{2}}}},\end{array}\end{array}$$

where *d*_{p} is the growing particle mass diameter, *V*_{p} and *V*_{v} are
the volume of the particle and vapour molecule respectively, *C*_{v} is the vapour monomer
concentration and *k*_{coll}(*d*_{v},*d*_{p}) is the kinetic
collision frequency between particle and vapour. Following
Fuchs and Sutugin (1971), the collision frequency for
the transition regime is defined by

$$\begin{array}{}\text{(2)}& \begin{array}{rl}& {k}_{\mathrm{coll}}\left({d}_{\mathrm{v}},{d}_{\mathrm{p}}\right)=\mathrm{2}\mathit{\pi}\cdot \left({d}_{\mathrm{v}}+{d}_{\mathrm{p}}\right)\cdot \left({D}_{\mathrm{v}}+{D}_{\mathrm{p}}\right)\\ & \phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\cdot {\displaystyle \frac{\mathrm{1}+\mathit{Kn}}{\mathrm{1}+\left(\mathrm{0.377}+\frac{\mathrm{4}}{\mathrm{3}\mathit{\alpha}}\right)\mathit{Kn}+\frac{\mathrm{4}}{\mathrm{3}\mathit{\alpha}}{\mathit{Kn}}^{\mathrm{2}}}},\end{array}\end{array}$$

where, according to Lehtinen and Kulmala (2003), the Knudsen
number (*Kn*) and mean free path (*λ*) need to be specified as
$\mathit{Kn}=\mathrm{2}\mathit{\lambda}\cdot {\left({d}_{\mathrm{v}}+{d}_{\mathrm{p}}\right)}^{-\mathrm{1}}$ and $\mathit{\lambda}=\mathrm{3}\left({D}_{\mathrm{v}}+{D}_{\mathrm{p}}\right)\cdot {\left({\stackrel{\mathrm{\u203e}}{c}}_{\mathrm{v}}^{\mathrm{2}}+{\stackrel{\mathrm{\u203e}}{c}}_{\mathrm{p}}^{\mathrm{2}}\right)}^{-\mathrm{1}/\mathrm{2}}$, which depend on the diameters *d*_{v∕p}, the
masses *m*_{v∕p} (within the calculation of the mean thermal velocities
${\stackrel{\mathrm{\u203e}}{c}}_{\mathrm{v}/\mathrm{p}})$ and the diffusion coefficients *D*_{v∕p} of the colliding
vapour molecules or particles respectively. Assuming that the accommodation
coefficient *α* is unity and relating the volume *V*_{v} of the
condensing monomer to its molecular mass and (bulk) density ${V}_{\mathrm{v}}={m}_{\mathrm{v}}/{\mathit{\rho}}_{\mathrm{v}}$, Eqs. (1) and (2) determine the
hard-sphere kinetic limit for particle growth.

We then additionally consider a collision enhancement of neutral vapour monomers and particles due to attractive van der Waals forces, where the collision frequency can be described according to Sceats (1989):

$$\begin{array}{}\text{(3)}& {k}_{\mathrm{coll}}\left({d}_{\mathrm{v}},{d}_{\mathrm{p}}\right)={k}_{\mathrm{K}}\cdot \left(\sqrt{\mathrm{1}+{\left({\displaystyle \frac{{k}_{\mathrm{K}}}{\mathrm{2}{k}_{\mathrm{D}}}}\right)}^{\mathrm{2}}}-\left({\displaystyle \frac{{k}_{\mathrm{K}}}{\mathrm{2}{k}_{\mathrm{D}}}}\right)\right),\end{array}$$

with the enhanced collision frequency for the continuum regime described by

$$\begin{array}{}\text{(4)}& {k}_{\mathrm{D}}=\mathrm{2}\mathit{\pi}\cdot \left({d}_{\mathrm{v}}+{d}_{\mathrm{p}}\right)\cdot \left({D}_{\mathrm{v}}+{D}_{\mathrm{p}}\right)\cdot E\left(\mathrm{0}\right)\end{array}$$

and the enhanced collision frequency for the kinetic regime described by

$$\begin{array}{}\text{(5)}& {k}_{K}={\displaystyle \frac{\mathit{\pi}}{\mathrm{4}}}\cdot {\left({d}_{\mathrm{v}}+{d}_{\mathrm{p}}\right)}^{\mathrm{2}}\cdot {\left({\displaystyle \frac{\mathrm{8}kT}{\mathit{\pi}}}\right)}^{\mathrm{1}/\mathrm{2}}\cdot {\left({\displaystyle \frac{\mathrm{1}}{{m}_{\mathrm{v}}}}+{\displaystyle \frac{\mathrm{1}}{{m}_{\mathrm{p}}}}\right)}^{\mathrm{1}/\mathrm{2}}\cdot E\left(\mathrm{\infty}\right)\end{array}$$

Equation (3) is designed such that it reaches the correct limits of the free
molecular and diffusion regime which is comparable to the approach of
Fuchs and Sutugin (1971), i.e. Eq. (2). However, it
includes the collision enhancement factors *E*(∞) and *E*(0). These
factors can be linked to the attractive potential of van der Waals forces.
For the continuum regime, this is done by solving the following integral:

$$\begin{array}{}\text{(6)}& E\left(\mathrm{0}\right)={\left[\underset{\left({r}_{\mathrm{v}}+{r}_{\mathrm{p}}\right)}{\overset{\mathrm{\infty}}{\int}}\left({\displaystyle \frac{{r}_{\mathrm{v}}+{r}_{\mathrm{p}}}{{x}^{\mathrm{2}}}}\right)\mathrm{exp}\left({\displaystyle \frac{\mathit{\varphi}\left(x\right)}{kT}}\right)\mathrm{d}x\right]}^{-\mathrm{1}},\end{array}$$

Here, *x* is the relative distance between the centres of the two colliding
entities, and *ϕ*(*x*) is the van der Waals potential
(Hamaker, 1937), which is expressed in terms of
the vapour and particle radii *r*_{v∕p}:

$$\begin{array}{}\text{(7)}& \begin{array}{rl}{\displaystyle \frac{\mathit{\varphi}\left(x\right)}{kT}}& =-{\displaystyle \frac{\mathrm{1}}{\mathrm{6}}}{\displaystyle \frac{A}{kT}}\left({\displaystyle \frac{\mathrm{2}{r}_{\mathrm{v}}{r}_{\mathrm{p}}}{{x}^{\mathrm{2}}-{\left({r}_{\mathrm{v}}+{r}_{\mathrm{p}}\right)}^{\mathrm{2}}}}+{\displaystyle \frac{\mathrm{2}{r}_{\mathrm{v}}{r}_{\mathrm{p}}}{{x}^{\mathrm{2}}-{\left({r}_{\mathrm{v}}-{r}_{\mathrm{p}}\right)}^{\mathrm{2}}}}\right.\\ & \left.+\mathrm{ln}\left({\displaystyle \frac{{x}^{\mathrm{2}}-{\left({r}_{\mathrm{v}}+{r}_{\mathrm{p}}\right)}^{\mathrm{2}}}{{x}^{\mathrm{2}}-{\left({r}_{\mathrm{v}}-{r}_{\mathrm{p}}\right)}^{\mathrm{2}}}}\right)\right)\end{array}\end{array}$$

Chan and Mozurkewich (2001) provide a fit to the solution of the numerically evaluated integral from Sceats (1989):

$$\begin{array}{}\text{(8)}& E\left(\mathrm{0}\right)=\mathrm{1}+{a}_{\mathrm{1}}\cdot \mathrm{ln}\left(\mathrm{1}+{A}^{\prime}\right)+{a}_{\mathrm{2}}\cdot {\mathrm{ln}}^{\mathrm{3}}\left(\mathrm{1}+{A}^{\prime}\right),\end{array}$$

where *a*_{n} are the fit parameters, and *A*^{′} is the reduced
Hamaker constant, which relates to the Hamaker constant *A* by ${A}^{\prime}=\mathrm{4}A\cdot {k}^{-\mathrm{1}}{T}^{-\mathrm{1}}\cdot {d}_{\mathrm{v}}{d}_{\mathrm{p}}\cdot {\left({d}_{\mathrm{v}}+{d}_{\mathrm{p}}\right)}^{-\mathrm{2}}$ (Chan and Mozurkewich,
2001; Hamaker, 1937). However, the measurements of this study are conducted
completely in the free molecular regime; thus, the derivation of the
continuum case will not significantly affect our results. For the free
molecular regime enhancement factor *E*(∞), an overview of
its relation to the Hamaker constant is given in Ouyang
et al. (2012). Chan and Mozurkewich (2001) also used
a fit to the solution from Sceats (1989) with the fit
parameters *b*_{n}:

$$\begin{array}{}\text{(9)}& \begin{array}{rl}E\left(\mathrm{\infty}\right)& =\mathrm{1}+{\displaystyle \frac{\sqrt{{A}^{\prime}/\mathrm{3}}}{\mathrm{1}+{b}_{\mathrm{0}}\sqrt{{A}^{\prime}}}}+{b}_{\mathrm{1}}\cdot \mathrm{ln}\left(\mathrm{1}+{A}^{\prime}\right)\\ & +{b}_{\mathrm{2}}\cdot {\mathrm{ln}}^{\mathrm{3}}\left(\mathrm{1}+{A}^{\prime}\right)\end{array}\end{array}$$

In this study,
we compare the results of Sceats (1989), who used
Brownian coagulation to describe the collisions, to the simple ballistics
approach of Fuchs and Sutugin (1965). In that approach, the minimum distance *x*_{min} along the trajectory of two colliding particles
with impact parameter *b* is calculated from the conservation of angular
momentum and energy:

$$\begin{array}{}\text{(10)}& b={x}_{\mathrm{min}}\sqrt{\mathrm{1}+\left({\displaystyle \frac{\mathrm{2}\left|\mathit{\varphi}\left({x}_{min}\right)\right|}{\mathit{\mu}{v}^{\mathrm{2}}}}\right)},\end{array}$$

where *ϕ* is the interaction potential, *μ* is the reduced mass of the
colliding entities and *v* is their relative speed. The critical impact
parameter *b*_{crit} is obtained as the minimum value of *b* for
which the minimum distance still takes a real value larger than (*r*_{v}+*r*_{p}). The enhancement factor is than related to the critical
impact parameter *b*_{crit} :

$$\begin{array}{}\text{(11)}& E\left(\mathrm{\infty}\right)={\displaystyle \frac{\mathrm{4}\phantom{\rule{0.125em}{0ex}}{b}_{\mathrm{crit}}^{\mathrm{2}}}{{\left({d}_{\mathrm{v}}+{d}_{\mathrm{p}}\right)}^{\mathrm{2}}}}\phantom{\rule{0.125em}{0ex}}\sqrt{{\displaystyle \frac{\mathrm{3}}{\mathrm{2}}}}\end{array}$$

Note, that this approach is oversimplified, as the initial velocity of the colliding entities is assumed to be fixed but should actually follow a (Maxwell–Boltzmann) distribution. However, Ouyang et al. (2012) concluded that the difference in the derived Hamaker constant is almost negligible.

Using the description of an enhanced collision kernel, the particle growth
rates measured with the DMA-train can be fitted with the Hamaker constant as
the single free parameter of the fit. As the theoretical growth rates are
compared to the appearance time growth rates, which are measured as a time
difference in signal appearance Δ*t* over a certain size interval
Δ*d*_{p} (ranging from *d*_{init} to *d*_{final}), a
comparison with experimental values requires the integration of Eq. (1):

$$\begin{array}{}\text{(12)}& \begin{array}{rl}& \mathrm{GR}\left({d}_{\mathrm{init}},{d}_{\mathrm{final}}\right)={\displaystyle \frac{\mathrm{\Delta}{d}_{\mathrm{p}}}{\mathrm{\Delta}t}}\\ & \phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}=\left({d}_{\mathrm{final}}-{d}_{\mathrm{init}}\right)/\underset{{d}_{\mathrm{init}}}{\overset{{d}_{\mathrm{final}}}{\int}}{\displaystyle \frac{\mathit{\pi}/\mathrm{2}\phantom{\rule{0.125em}{0ex}}\cdot \phantom{\rule{0.125em}{0ex}}{d}_{\mathrm{p}}^{\mathrm{2}}}{{k}_{\mathrm{coll}}\left({d}_{\mathrm{v}},{d}_{\mathrm{p}}\right)\cdot {V}_{\mathrm{v}}\cdot {C}_{\mathrm{v}}}}\mathrm{d}{d}_{\mathrm{p}}\end{array}\end{array}$$

Equation (12) includes several properties of the condensing vapour and the growing particles. Sulfuric acid molecules are usually hydrated at typical ambient relative humidity. While the thermodynamic model E-AIM (Extended Aerosol Inorganics Model; Wexler et al., 2002) predicts an average of two water molecules attached to a sulfuric acid monomer at 298 K and 40 %–60 % relative humidity, quantum chemical studies predict an average hydration of one to two water molecules for these conditions (Henschel et al., 2014; Kurtén et al., 2007; Temelso et al., 2012). Moreover, the hydration state of the particles in the chamber is also not directly measured and might be altered during the sampling process, which requires information on the hygroscopic growth factor (see Sect. 2.1).

We examine the effect of hydration using three different approaches. In the
first naïve approach, we assume that no dehydration occurs during
measurement and that the particle sulfuric acid mass fraction is equal to the
vapour mass fraction, i.e. $w={M}_{{\mathrm{H}}_{\mathrm{2}}{\mathrm{SO}}_{\mathrm{4}}}/{m}_{\mathrm{v}}$, with ${m}_{\mathrm{v}}={M}_{{\mathrm{H}}_{\mathrm{2}}{\mathrm{SO}}_{\mathrm{4}}}+\mathrm{2}{M}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ (assuming two water molecules attached to the
sulfuric acid monomer), where ${M}_{{\mathrm{H}}_{\mathrm{2}}{\mathrm{SO}}_{\mathrm{4}}}$ and ${M}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ are the
molecular mass of sulfuric acid and water respectively. In the second
approach, we assume a dry measurement, and in this case the growth of the
measured dry particles is described by uptake of sulfuric monomers only,
i.e. ${m}_{\mathrm{v}}={M}_{{\mathrm{H}}_{\mathrm{2}}{\mathrm{SO}}_{\mathrm{4}}}$. However, for the actual vapour and particle size
used in the collision kernel *k*_{coll}(*d*_{v},*d*_{p}), the
hydrated sizes are used. We again assume an average hydration for the
monomer with two water molecules as above and an average hygroscopic growth
factor of 1.25 for all particle sizes and RH values in our experiments. The
latter is an average value of the results of Biskos et al. (2009) for highly
acidic sulfuric acid sub-10 nm particles at 40 %–60 % relative humidity.
In the third approach, we consider that the extent of hydration
might vary with size and relative humidity. We use modelled composition data
from MABNAG (Model for Acid–Base chemistry in Nanoparticle Growth; Yli-Juuti et al., 2013) in order to
predict the sulfuric acid mass fraction *w*(RH,*T*) (see Fig. S4a) and calculate the hygroscopic growth factor:

$$\begin{array}{}\text{(13)}& \mathrm{gf}={\left({\displaystyle \frac{w\left({\mathrm{RH}}_{\mathrm{m}},{T}_{\mathrm{m}}\right)\cdot {\mathit{\rho}}_{\mathrm{sol}}\left(w\right({\mathrm{RH}}_{\mathrm{m}},{T}_{\mathrm{m}}),{T}_{\mathrm{m}})}{w(\mathrm{RH},T)\cdot {\mathit{\rho}}_{\mathrm{sol}}\left(w\right(\mathrm{RH},T),T)}}\right)}^{\mathrm{1}/\mathrm{3}},\end{array}$$

where *ρ*_{sol} is a parametrization of the density of the sulfuric acid–water solution (Myhre et al., 1998), and
*w*(RH,*T*) and *w*(RH_{m}*T*_{m}) are the mass fractions of sulfuric acid in
the growing and measured particles respectively. We follow the
considerations of Verheggen and Mozurkewich (2002) in order to separate the growth by sulfuric acid addition and water
uptake by differentiating the hydrated particle volume ${V}_{\mathrm{p}}={m}_{{\mathrm{H}}_{\mathrm{2}}{\mathrm{SO}}_{\mathrm{4}}}/\left(w{\mathit{\rho}}_{\mathrm{sol}}\right)$. Both the numerator (particle
sulfuric acid mass ${m}_{{\mathrm{H}}_{\mathrm{2}}{\mathrm{SO}}_{\mathrm{4}}}$) and the denominator (sulfuric acid mass
fraction and solution density) depend on time. The addition of sulfuric
acid is again described in analogy to coagulation theory, resulting in

$$\begin{array}{}\text{(14)}& \begin{array}{rl}& {\displaystyle \frac{\mathit{\pi}}{\mathrm{2}}}{d}_{\mathrm{p}}^{\mathrm{2}}{\displaystyle \frac{\mathrm{d}{d}_{\mathrm{p}}}{\mathrm{d}t}}={\displaystyle \frac{{k}_{\mathrm{coll}}\left({d}_{\mathrm{v}},{d}_{\mathrm{p}}\right)\cdot {m}_{\mathrm{v}}\phantom{\rule{0.125em}{0ex}}\cdot \phantom{\rule{0.125em}{0ex}}{C}_{\mathrm{v}}}{w\phantom{\rule{0.125em}{0ex}}\cdot \mathit{\rho}}}-{\displaystyle \frac{\mathit{\pi}{d}_{\mathrm{p}}^{\mathrm{3}}}{\mathrm{6}}}{\displaystyle \frac{\mathrm{d}\mathrm{ln}\left(w\mathit{\rho}\right)}{\mathrm{d}t}}\\ & \phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}={\displaystyle \frac{{k}_{\mathrm{coll}}\left({d}_{\mathrm{v}},{d}_{\mathrm{p}}\right)\phantom{\rule{0.125em}{0ex}}\cdot \phantom{\rule{0.125em}{0ex}}{m}_{\mathrm{v}}\phantom{\rule{0.125em}{0ex}}\cdot \phantom{\rule{0.125em}{0ex}}{C}_{\mathrm{v}}}{w\cdot \mathit{\rho}}}-{\displaystyle \frac{\mathit{\pi}{d}_{\mathrm{p}}^{\mathrm{3}}}{\mathrm{6}}}{\displaystyle \frac{\mathrm{d}\mathrm{ln}\left(w\mathit{\rho}\right)}{\mathrm{d}{d}_{\mathrm{p}}}}{\displaystyle \frac{\mathrm{d}{d}_{\mathrm{p}}}{\mathrm{d}t}}\end{array}\end{array}$$

Equation (14) contains a first term for the addition of pure sulfuric acid and a second term for water uptake. It can be solved for the particle growth rate $\frac{\mathrm{d}{d}_{\mathrm{p}}}{\mathrm{d}t}$:

$$\begin{array}{}\text{(15)}& \mathrm{GR}={\displaystyle \frac{\mathrm{2}\phantom{\rule{0.125em}{0ex}}\cdot \phantom{\rule{0.125em}{0ex}}{k}_{\mathrm{coll}}\left({d}_{\mathrm{v}},{d}_{\mathrm{p}}\right)\phantom{\rule{0.125em}{0ex}}\cdot \phantom{\rule{0.125em}{0ex}}{m}_{\mathrm{v}}\phantom{\rule{0.125em}{0ex}}\cdot \phantom{\rule{0.125em}{0ex}}{C}_{\mathrm{v}}}{w\left(\mathrm{RH},T\right)\phantom{\rule{0.125em}{0ex}}\cdot \mathit{\rho}\left(\mathrm{RH},T\right)\cdot \mathit{\pi}\cdot {d}_{\mathrm{p}}^{\mathrm{2}}\phantom{\rule{0.125em}{0ex}}\cdot \phantom{\rule{0.125em}{0ex}}\left(\mathrm{1}+\phantom{\rule{0.125em}{0ex}}\frac{{d}_{\mathrm{p}}}{\mathrm{3}}\phantom{\rule{0.125em}{0ex}}\cdot \phantom{\rule{0.125em}{0ex}}\frac{\mathrm{d}\mathrm{ln}\left(w\mathit{\rho}\right)}{\mathrm{d}{d}_{\mathrm{p}}}\right)}}\end{array}$$

In this case, we assume
${m}_{\mathrm{v}}={M}_{{\mathrm{H}}_{\mathrm{2}}{\mathrm{SO}}_{\mathrm{4}}}$, but we use the hydrated monomer diameter *d*_{v} in the
collision kernel. For the particles, we now use the hydrated size, i.e. ${d}_{\mathrm{p}}=\mathrm{gf}\cdot {d}_{\mathrm{p},\mathrm{m}}$ with gf and *w*(RH,*T*)
taken from the model. We compare the MABNAG predictions in Fig. S4b with SAWNUC (Sulfuric Acid Water Nucleation model; Ehrhart et al.,
2016), which only takes sulfuric acid and water into account, whereas MABNAG
also includes ammonia. MABNAG predicts a significantly lower water content
at larger sizes (>2.5 nm), even at 3 pptv ammonia. In addition,
previous experiments in the CLOUD chamber have suggested that even background
level ammonia has an influence on the hygroscopic growth factor
(Kim et al., 2016); this is similar to Biskos et al. (2009), who also indicated
some extent of neutralization for sub-10 nm particles at low ammonia. Due to
the presumably better prediction of the particle hydration by MABNAG for
sizes larger than 2.5 nm, we choose the results from Fig. S4a, even if they might overestimate the hydration at small sizes. We
have neglected the effect of ammonia addition on collisions in all three
approaches so far, but we test the assumption ${m}_{\mathrm{v}}={M}_{{\mathrm{H}}_{\mathrm{2}}{\mathrm{SO}}_{\mathrm{4}}}+\mathrm{2}{M}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}+\mathrm{1}{M}_{{\mathrm{NH}}_{\mathrm{3}}}$ and different vapour hydrations in
our systematic uncertainties estimate in Fig. S5. All of the
parameters for vapour and particles used for all approaches are summarized in
Table S1 in the Supplement.

We implement the results of our growth rate measurements for sulfuric-acid-driven growth in a global model (Mann et al., 2010; Mulcahy et al., 2018), which includes sulfuric acid–water binary nucleation. However, the model does not include ternary nucleation schemes (Dunne et al., 2016) and pure biogenic nucleation (Gordon et al., 2016) and will, therefore, underestimate the impact of nucleation on the global aerosol and CCN budget. In the model, growth between the nucleation size and 3 nm is treated with the equation from Kerminen and Kulmala (2002), which gives the fraction of particles surviving to 3 nm at a given growth and loss rate. Here, as a baseline case, we use the geometric hard-sphere kinetic growth rate based on bulk density (Eqs. 1–2) and compare this to the collision-enhanced growth (Eqs. 3–9). For larger sizes, aerosol growth in the model is calculated by solving the condensation equations. Therefore, no direct growth parametrization can be altered, but as condensational growth scales linearly with the diffusion coefficient of the condensing vapour, we increased sulfuric acid diffusion for condensation in the nucleation mode (2–10 nm) and in the Aitken mode (10–100 nm). The enhancement factors are derived for the median diameters of the modes (7.6 and 57 nm respectively) at the cloud-base level (1 km). However, this constant factor of increase in the diffusion coefficient (and, hence, flux onto particles) for all particles of the entire mode might underestimate the impact of the collision enhancement. Rapid growth is increasingly important for the smallest particles, which actually have a higher collision enhancement than particles with the size of the mode median diameters.

3 Results

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Figure 1 shows the particle growth rates for two size intervals (Fig. 1a, 1.8–3.2 nm mobility diameter; Fig. 1b, 3.2–8.0 nm mobility diameter) versus the sulfuric acid monomer concentration, which correlate linearly. No significant dependence on temperature, ionization levels in the chamber or the concentration of ammonia is evident. While the effect of temperature expected from theory is small and cannot be discerned within the statistical uncertainties of our measurements (Nieminen et al., 2010), the insignificant influence of ammonia and the ionization level on the growth rate differs from previous findings (Lehtipalo et al., 2016).

We compare the measured growth rates from this study with the results from Lehtipalo et al. (2016) in Fig. S2. In contrast to our results, elevated ammonia (∼1000 pptv) led to increased growth rates in that study. The major difference is the narrower size range for the growth rate measurements (1.5–2.5 nm mobility diameter) due to different instrumentation. For smaller sizes and at low ammonia, sulfuric acid evaporation likely plays a role due to an increased Kelvin term. The stabilizing effect of ammonia is certainly relevant at the sizes of the nucleating clusters (Kirkby et al., 2011). For our results, we confirm the absence of significant evaporation rates above 2 nm using an independent experiment presented in Fig. 2. This demonstrates that, in the absence of gas-phase sulfuric acid, the coagulation- and dilution-corrected loss rates of particles (${k}_{\mathrm{tot}}^{\mathrm{meas}}-{k}_{\mathrm{dil}}-{k}_{\mathrm{coag}}^{\mathrm{avg}}$) over all sizes follow the expected size dependence of wall losses which is inferred from the sulfuric acid monomer decay. Evaporation would cause another term and would distort the balance equation (also depending on the relative abundances of the particles during the decay), causing a deviation from the expected wall loss rate.

The insignificant effect of ammonia on growth (Fig. 1) and the same high ratio (>100, Fig. S3a) between sulfuric acid monomer and dimer concentrations for all experiments point towards a negligible influence of clustering on our measured growth rates (Li and McMurry, 2018). Moreover, in Fig. S3b, using a model including sulfuric acid/ammonia clustering and evaporation, we show that no cluster contribution is indeed expected, even at elevated ammonia concentrations (Kürten, 2019).

In the absence of evaporation and strong clustering, our growth rate data
provide a direct measurement of the condensational growth at the kinetic
limit caused by sulfuric acid monomers only. We find the measured growth
rates both with and without the addition of ammonia to be significantly above
the geometric hard-sphere limit (Eqs. 1–2) of kinetic condensation
(Nieminen et al., 2010). For this comparison, we followed a
naïve approach, assuming an average hydration of the monomer by two water
molecules and applied the resulting mass fraction to find the bulk density
(Myhre et al., 1998). The observed enhancement
is similar to Lehtipalo et al. (2016) in the case where
evaporation was suppressed by ammonia (see Fig. S2). We also measure a
growth rate enhancement for the larger size range (Fig. 1b), which should be
less sensitive to evaporation. The faster growth rates might be due to an
enhanced collision frequency, which can be attributed to van der Waals
forces, either permanent dipole–(induced) dipole interactions between polar
sulfuric acid molecules and particles or London dispersion forces
(London, 1937). The magnitude of the enhancement is described
by the Hamaker constant *A* (Hamaker, 1937),
which we use as the single free parameter to fit a collision-enhanced
kinetic limit. For the Brownian coagulation model linking the Hamaker
constant to the collision kernel, i.e. Eqs. (3)–(9) (Sceats,
1989), we find $A=\left(\mathrm{4.6}\phantom{\rule{0.125em}{0ex}}\pm \mathrm{1.5}\phantom{\rule{0.125em}{0ex}}\left(\mathrm{stat}.\right)\right)\times {\mathrm{10}}^{-\mathrm{20}}$ J. If
we apply a ballistics approach in the free molecular regime (Fuchs and
Sutugin, 1965; Ouyang et al., 2012), we derive a slightly higher value of
$A=\mathrm{8.7}\times {\mathrm{10}}^{-\mathrm{20}}$ J, but both yield comparable values to previous results (Chan and
Mozurkewich, 2001; McMurry, 1980).

An enhancement due to charge–dipole interactions between the polar sulfuric acid monomers and charged particles is not significant in our total (neutral plus charged particle) growth rate measurements (as shown in Fig. 1), where we observe no difference between growth rates under neutral and galactic cosmic ray ionization levels. From average dipole orientation theory (Su and Bowers, 1973), a small enhancement is expected in the collision frequency for charged particles above 2 nm (Nadykto and Yu, 2003), which should affect the growth rate (Laakso et al., 2003; Lehtipalo et al., 2016). We find an enhancement factor of 1.45 by comparing the total to the ion growth rate (as shown in Fig. 3), which is in good agreement with theory. However, the total growth rate is influenced on a minor level by faster ion growth because, at the representative galactic cosmic ray ionization levels and sulfuric acid concentrations in our experiments, most (more than 75 %) of the growing particles are neutral (see Fig. 3). However, the effects of ion condensation and charge–dipole enhancement might be stronger at lower sulfuric acid concentrations (Svensmark et al., 2017).

Condensational growth at the geometric kinetic limit predicts increasing growth rates with decreasing particle sizes due to the non-negligible effect of vapour molecule size on the collision cross section (Nieminen et al., 2010), which has not yet been shown experimentally. Furthermore, the collision enhancement due to van der Waals forces and the collision enhancement due to charge–dipole interactions also depend on the comparative size of the condensing vapour and the growing particle. Figure 4a illustrates the theoretical predictions of the size dependence of the collision rate of sulfuric acid monomers with larger particles, including van der Waals forces and charge–dipole interactions. The enhancement factor compared to the hard-sphere kinetic limit is shown for both the Brownian coagulation model (Sceats, 1989) and the ballistics approach (Fuchs and Sutugin, 1965), which is 2.1 and 2.3 for the free molecular regime respectively, and is comparable to previous experimental results (Kürten et al., 2014; Lehtipalo et al., 2016) and quantum chemical calculations (Halonen et al., 2019).

In addition to the approach for calculating the kinetic enhancement factor,
the description of particle hydration might also play a crucial role. Until now,
we have used the naïve assumption that vapour and particle hydration are the
same and that particles are measured at their hydrated size. However, during
sampling the measured particles are potentially dried. To investigate the
effect of particle hydration, we use the DMA-train data from Fig. 1 to fit the
collision enhancement for two alternative approaches: one approach where we assume
that particles are measured dry and one approach where we separate the uptake of
water and sulfuric acid condensation
(Verheggen and Mozurkewich, 2002) using
modelled particle composition data from SAWNUC
(Ehrhart et al., 2016) or MABNAG
(Yli-Juuti et al., 2013). We compare the predictions
for the size dependence of all approaches with the measured growth rates of
all instruments normalized to a sulfuric acid concentration of 10^{7} cm^{−3} in Fig. 4b. In addition, we
show the growth rates using the time- and size-resolving growth rate
analysis method INSIDE (Pichelstorfer et al., 2018), which
agrees with the appearance time method, demonstrating a minor systematic bias
in our growth rate determination. All approaches reproduce the
size dependence at an acceptable level (*R*^{2} larger than 0.87). The
separation approach yields higher growth rates at the smallest sizes due to
the overestimation of hydration by MABNAG below 2.5 nm. However, for SAWNUC
composition data, which presumably describe the cluster hydration better,
the *R*^{2} is only 0.66 and does not reproduce the observed
size dependence. This is possibly caused by the overly high hydration assumed
for larger sizes. Thus, the simple dry measurement approach might be a good
approximation for the predictions of both MABNAG and SAWNUC for the size
range of interest (see Fig. S4b). We estimate the
systematic uncertainty of the results in Fig. S5, including the effects of different vapour hydration, ammonia addition and
sulfuric acid measurement uncertainty. All approaches overlap largely
within their systematic uncertainties with $A=\left({\mathrm{5.2}}_{-\mathrm{3.4}}^{+\mathrm{9.7}}\left(\mathrm{syst}.\right)\right)\times {\mathrm{10}}^{-\mathrm{20}}$ J
as the best estimate of a combined assessment (assuming the Brownian
coagulation model). We also give a first-order approximation for our measured
growth rates and their size dependence for the conditions in our
experiments:

$$\begin{array}{}\text{(16)}& \begin{array}{rl}\mathrm{GR}\left(\mathrm{nm}\phantom{\rule{0.125em}{0ex}}{\mathrm{h}}^{-\mathrm{1}}\right)& =\left[\mathrm{2.68}\cdot {d}_{\mathrm{p}}{\left(\mathrm{nm}\right)}^{-\mathrm{1.27}}+\mathrm{0.81}\right]\\ & \cdot \left[{\mathrm{H}}_{\mathrm{2}}{\mathrm{SO}}_{\mathrm{4}}\left({\mathrm{cm}}^{-\mathrm{3}}\right)\times {\mathrm{10}}^{-\mathrm{7}}\right]\end{array}\end{array}$$

The observed steep increase of the growth rates with decreasing size shows that the collision enhancement due to van der Waals forces is especially important for the smallest particles. As these are the most vulnerable for losses to pre-existing aerosols, their survival probability in the atmosphere is directly affected, altering the CCN budget (Pierce and Adams, 2007) or promoting NPF in urban environments (Kulmala et al., 2017). In order to test the effects of collision enhancement on sulfuric acid growth on a global scale, we use the atmosphere-only configuration of the United Kingdom Earth System Model (UKESM1; Mulcahy et al., 2018; Walters et al., 2019) which includes the GLOMAP (Global Model of Aerosol Processes) aerosol microphysics module describing nucleation and growth (Mann et al., 2010). Figure 5 illustrates the global model results comparing the baseline case (no collision enhancement) with a collision enhancement simulation (with enhancement factors of 2.2, 1.8 and 1.3 for the cluster, nucleation and Aitken modes respectively) for the present-day atmosphere. The absolute particle number concentrations averaged over all longitudes are shown in Fig. 5a, indicating changes of more than 50 %, especially at high altitudes (> 10 km; Fig. 5b) where most aerosol particles originate from pure sulfuric-acid-driven NPF. The importance of the nucleation process and, therefore, the growth rate enhancement is lower at lower altitudes and in the Northern Hemisphere, which is mainly due to the higher condensation sink and the restriction of the model to sulfuric acid–water binary nucleation. However, the significant enhancement of sulfuric-acid-driven nanoparticle growth in the upper troposphere may be important in quantifying sources of stratospheric aerosols and cirrus CCN (Brock et al., 1995; Deshler, 2008) and needs to be accounted for in future model development.

4 Discussion

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Understanding nanoparticle growth driven by sulfuric acid is extremely
important for modelling the present-day atmosphere. Our measured growth
rates cover a wide range of representative atmospheric conditions below 20 ^{∘}C and reveal that sulfuric acid growth proceeds faster than the
geometric hard-sphere kinetic limit. These faster growth rates in the cluster
size range could be partially responsible for the occurrence of NPF in
polluted environments (Kulmala et al.,
2017). Our results suggest that this collision
enhancement due to van der Waals forces can be more important than
charge–dipole interactions or base stabilization by ammonia for sizes larger than 2 nm. However, a
better knowledge of the chemical composition of the condensing vapour and
growing sub-10 nm particles could further improve our understanding of
molecular collision rates. For smaller sizes, the evaporation of sulfuric acid
and charge effects need to be considered, but the size range covered by our
measurements is sufficient for the global model used, which nucleates
particles at 1.7 nm. We find significantly increased upper tropospheric
aerosol concentrations, but the global impact of van der Waals forces in
nanoparticle growth might be even higher due to the model limitations to
binary sulfuric acid–water nucleation. Therefore, our results should be
considered in future model development, especially when discussing the
importance of changing sulfuric acid levels due to the reduced anthropogenic
emissions of SO_{2}. Moreover, our parametrization of pure sulfuric acid
growth rates will help to identify the contribution of other
co-condensing vapours in ambient and laboratory experiments to growth, as they set a
new baseline for the kinetic condensation of sulfuric acid. Several
simplifications have often been applied to kinetic particle growth,
including hard-sphere collision based on bulk density and the omission of vapour
size in the collision cross section; our results provide clear experimental
verification that these simplifications are no longer fit for increasingly
accurate measurements at these tiny yet critical sizes.

Data availability

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Data availability.

All of the datasets presented in this paper are available from the corresponding author upon reasonable request.

Supplement

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Supplement.

The supplement related to this article is available online at: https://doi.org/10.5194/acp-20-7359-2020-supplement.

Author contributions

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Author contributions.

DS, MSim, AK, KL, HF, XH, SBri, MX, RB, AB, SBrä, LCM, DC, BC, AD, JDo, JDu, IEH, LF, LGC, MH, CK, WK, HL, CPL, ML, ZL, VM, HEM, TM, EP, JP, MP, MPR, SScho, SSchu, JS, MSip, GS, YS, YJT, AT, ACW, MW, YW, SKW, DW, PJW, YW, QY, MZW, UB, JC, RCF, RV, JK and PMW prepared the CLOUD facility or measuring instruments. DS, MSim, AR, KL, XH, SBri, MX, AA, RB, AB, LB, SBrä, LCM, DC, LD, AD, JDu, IEH, HF, LF, LGC, MH, CK, TKK, WK, HL, CPL, ML, ZL, HEM, RM, TM, WN, EP, JP, MPR, BR, SSchu, GS, CT, YJT, AT, MVP, ACW, MW, SKW, DW, PJW, YW, QY and MZW collected the data. DS, MSim, AR, HG, TN, LP, LD, HF, SE, MH, CK, ACW and SKW analysed the data. DS, MS, AR, AK, KL, TN, XH, MX, JDo, JDu, IEH, TKK, TP, MPR, MSip, UB, KSC, JC, NMD, RCF, AH, MK, JL, RV, JK and PMW were involved in the scientific discussion and the interpretation of the data. DS, AK, KL, HG, NMD, JK and PMW wrote the paper.

Competing interests

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Competing interests.

The authors declare that they have no conflict of interest.

Special issue statement

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Special issue statement.

This article is part of the special issue “The CERN CLOUD experiment (ACP/AMT inter-journal SI)”. It is not associated with a conference.

Acknowledgements

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Acknowledgements.

We thank CERN for supporting CLOUD with technical and financial resources and for providing a particle beam from the CERN Proton Synchrotron. We are also grateful to Patrick Carrie, Louis-Philippe De Menezes, Jonathan Dumollard, Katja Ivanova, Francisco Josa, Timo Keber, Ilia Krasin, Robert Kristic, Abdelmajid Laassiri, Osman Maksumov, Benjamin Marichy, Herve Martinati, Robert Sitals, Albin Wasem, Sergey Vitaljevich Mizin and Mats Wilhelmsson for their contributions to the experiment.

Financial support

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Financial support.

This research has received funding from the European Commission Seventh Framework Programme and the European Union's Horizon 2020 programme (Marie Skłodowska-Curie action no. 764991 “CLOUD-MOTION”; MC-COFUND grant no. 665779 and ERC projects nos. 616075 “NANODYNAMITE” and 714621 “GASPARCON”), the German Federal Ministry of Education and Research (grant no. 01LK1601A “CLOUD-16”), the Swiss National Science Foundation (project nos. 200020_152907, 20FI20_159851, 200021_169090, 200020_172602 and 20FI20_172622), the Academy of Finland (project nos. 296628, 299574, 307331 and 310682), the Austrian Science Fund (FWF; project nos. J-3951, P27295-N20 and J-4241), the Portuguese Foundation for Science and Technology (FCT; project no. CERN/FIS-COM/0014/2017), the U.S. National Science Foundation (grant nos. AGS-1649147, AGS-1801280, AGS-1602086 and AGS-1801329). Open access funding was provided by University of Vienna.

Review statement

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Review statement.

This paper was edited by Jonathan Abbatt and reviewed by David R. Hanson and two anonymous referees.

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Special issue

Short summary

Sulfuric acid is a major atmospheric vapour for aerosol formation. If new particles grow fast enough, they can act as cloud droplet seeds or affect air quality. In a controlled laboratory set-up, we demonstrate that van der Waals forces enhance growth from sulfuric acid. We disentangle the effects of ammonia, ions and particle hydration, presenting a complete picture of sulfuric acid growth from molecular clusters onwards. In a climate model, we show its influence on the global aerosol budget.

Sulfuric acid is a major atmospheric vapour for aerosol formation. If new particles grow fast...

Atmospheric Chemistry and Physics

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