Collisions of molecules and clusters play a key role in determining the rate of atmospheric new particle formation and growth. Traditionally the statistics of these collisions are taken from kinetic gas theory assuming spherical noninteracting particles, which may significantly underestimate the collision coefficients for most atmospherically relevant molecules. Such systematic errors in predicted new particle formation rates will also affect large-scale climate models.
We studied the statistics of collisions of sulfuric acid molecules in a vacuum using atomistic molecular dynamics simulations. We found that the effective collision cross section of the

New particle formation from condensable vapours provides an important contribution to the composition of aerosols in the atmosphere which affects air quality as well as the Earth's climate. The positive and negative contributions of atmospheric aerosols to the planet's radiative balance are still not fully understood, and currently constitute one of the largest uncertainties in climate modelling. The earliest stage of new particle formation involves the collisions of individual molecules leading to the appearance of a new molecular complex. In many theoretical approaches, the statistics of such collisions are simply taken from kinetic gas theory, i.e. the molecules are considered to be noninteracting hard spheres, and a collision occurs when the impact parameter, i.e. the perpendicular distance between the spheres' trajectories, is smaller than the sum of the hard spheres' radii. The hard-sphere collision cross section is independent of the relative velocity of the colliding bodies, and the collision rate coefficient for hard spheres of identical radii is customarily expressed as

It is well known that acid–base clusters, in particular clusters containing sulfuric acid and ammonia, or amines, are very relevant in nucleation and growth of particles that can serve as cloud condensation nuclei

The effect of long-range interactions between neutral polar molecules on the capture rate constant has been studied by classical trajectory integration

In the present work, we use atomistic molecular dynamics simulations to study the statistics of collisions between sulfuric acid molecules in a vacuum, determine the collision rate coefficient, and calculate the enhancement factor over kinetic gas theory. Here, we focus on “reactive” collisions, defined by the formation of one or more hydrogen bonds between the molecules.
Detailed modelling of e.g. proton transfer processes related to hydrogen bond formation in such reactive collisions would require first-principle simulations

In Sect.

We considered two force fields to describe the sulfuric acid molecules in the present study. The first choice was the force field by

To validate the force fields, we compare the structures and energies of four stable configurations of the sulfuric acid dimer illustrated in Fig.

Four minimum energy structures for the sulfuric acid dimer

Relative energies

Energy unit conversion: 1 eV

However, the vibrational spectra, calculated from the Fourier transform of the velocity autocorrelation functions of an isolated

Vibrational spectra of the sulfuric acid molecule obtained with OPLS

We first calculated the binding free energy of two sulfuric acid molecules in a vacuum as described by the force fields of

Potential of mean force between two

Molecular dynamics simulations were performed with the

Three example MD trajectories with a relative velocity of 350 m s

The results from the atomistic simulations will be compared to different theoretical models described in the following.

As the collision rate in the context of atomistic simulations is defined as the reaction rate of hydrogen bonding, the related theoretical models are often based on the assumption that if the trajectory of the colliding molecules is able to surmount a centrifugal barrier the reaction is certain. This is known as the capture approximation; to emphasize this conceptual difference between simulations and theoretical models, we use the word

The interaction between two identical polar molecules is usually written as

In the study by

For the attractive potential described by Eq. (

It should be noted that the model of Brownian coagulation does not describe the correct transport physics of collisions of molecules in the gas phase. For a discussion on the transition from the free molecular (ballistic) regime to the continuum (diffusive) regime, see e.g.

The statistics of the collision probabilities as a function of the impact parameter and relative velocity,

Heat map of the collision probability of sulfuric acid molecules plotted as a function of impact parameter

The dynamical collision cross section, obtained from the integral over the collision probability functions,

Ratio between the collision cross section

The discrepancy between

The canonical collision rate coefficient can be calculated in a similar fashion to Eq. (

Collision rate coefficient

For the Langevin model (Eq.

In addition to the coefficients obtained by different approaches, we examine the enhancement factor

We find that

The enhancement factor obtained by atomistic simulations is in very good agreement with the kinetic modelling on the recent experimental results of the formation of atmospheric sulfuric acid dimers

In summary, we have benchmarked two classical force fields against experimental and ab initio data and determined that the OPLS force field by

In the future, the atomistic collision modelling approach presented in this work can be applied to other atmospherically relevant molecules, clusters, or ions, exhibiting dipoles of varying magnitudes – and in some cases several times larger than that of the sulfuric acid molecule – to help understand the effect of long-range interactions in cluster formation rates.
However, before we can quantitatively assess the influence of collision rate enhancement on atmospherical new particle formation rates obtained from cluster dynamics models (for example, Atmospheric Cluster Dynamics Code;

Simulation data and input files can be made available upon request from the corresponding author.

As in Eq. (

As all different long-range interactions are included in the attractive part of the potential of mean force between two sulfuric acid molecules (Eq.

Enhancement factor

As we were unable to distinguish the actual dipole–dipole interaction from the total attractive potential, we estimated the interaction using the Keesom equation (see Fig.

The attractive potential parameters

The canonical collision rate coefficient can be calculated from the collision probabilities obtained from atomistic simulation at arbitrary temperatures by shifting the Maxwell–Boltzmann relative velocity distribution, provided that changes in the internal motion of the molecules do not affect the collision probabilities. We tested the effect of the different rotational and vibrational motion on the collision statistics in an atmospherically relevant temperature range by carrying out MD collision simulations for a subset of impact parameters

Collision probabilities of sulfuric acid molecules, as a function of the impact parameter squared, for different values of the relative velocity, obtained from molecular dynamics simulations at 300 K (solid coloured lines), at 250 K (coloured dots), and at 400 K (coloured crosses). The step-like collision probabilities for a hard-sphere model (

In order to vary temperature in calculating the thermal collision rate coefficient using the Langevin approach, the potential of mean force between two sulfuric acid molecules was calculated at 250, 300, and 400 K, and the parameters describing the attractive intermolecular interaction (Eq.

As shown in Figs.

Time evolution of the intermolecular energy (red line) and the sulfur–sulfur distance (solid black line) in one MD trajectory with relative velocity

This process is illustrated in Fig.

RH and BR planned the simulation set-up and performed benchmark calculations. BR carried out collision and metadynamics simulations and wrote the first draft of the paper. RH and EZ provided the theoretical framework. RH analysed the simulation data. TK and HV helped plan the project. All authors contributed to writing the final paper.

The authors declare that they have no conflict of interest.

Computational resources were provided by the CSC–IT Center for Science Ltd., Finland.

This research has been supported by the European Research Council (DAMOCLES; grant no. 692891), the Academy of Finland (grant nos. 266388 and 285067), and the University of Helsinki, Faculty of Science (ATMATH project).Open access funding was provided by Helsinki University Library.

This paper was edited by Fangqun Yu and reviewed by two anonymous referees.