Ion–dipole collisions can facilitate the formation of atmospheric aerosol particles and play an important role in their detection in chemical ionization mass spectrometers. Conventionally, analytical models, or simple parametrizations, have been used to calculate the rate coefficients of ion–dipole collisions in the gas phase. Such models, however, neglect the atomistic structure and charge distribution of the collision partners. To determine the accuracy and applicability of these approaches under atmospheric conditions, we calculated collision cross sections and rate coefficients from all-atom molecular dynamics collision trajectories, sampling the relevant range of impact parameters and relative velocities, and from a central field model using an effective attractive interaction fitted to the long-range potential of mean force between the collision partners. We considered collisions between various atmospherically relevant molecular ions and dipoles and charged and neutral dipolar clusters. Based on the good agreement between collision cross sections and rate coefficients obtained from molecular dynamics trajectories and a generalized central field model, we conclude that the effective interactions between the collision partners are isotropic to a high degree, and the model is able to capture the relevant physicochemical properties of the systems. In addition, when the potential of mean force is recalculated at the respective temperatures, the central field model exhibits the correct temperature dependence of the collision process. The classical parametrization by Su and Chesnavich (1982), which combines a central field model with simplified trajectory simulations, is able to predict the collision rate coefficients and their temperature dependence quite well for molecular systems, but the agreement worsens for systems containing clusters. Based on our results, we propose the combination of potential of mean force calculation and a central field model as a viable and elegant alternative to the brute force sampling of individual collision trajectories over a large range of impact parameters and relative velocities.

In the atmosphere, gas-phase molecules can aggregate to form molecular clusters and subsequently grow into larger-sized atmospheric aerosol particles in a process called new particle formation (NPF)

The first stage of NPF is the gas-phase collision between single molecules or ions to form a dimer. For a theoretical description of NPF, it is therefore crucial to properly characterize the thermodynamics and kinetics of these initial collisions. In current NPF models, the cluster thermodynamics (e.g., cluster binding free energies and therefrom derived fragmentation rate coefficients) are treated with high-level quantum chemical calculations

The theoretical prediction of collision kinetics is a longstanding topic throughout physics and chemistry (e.g., atmospheric chemistry, subatomic physics, and mass spectrometry), and thus, several theoretical and computational methods have been developed. Collision rate coefficients generally depend on both the relative velocity between the collision partners and the fluid density regime

Neglecting the attractive forces between the collision partners can result in significant discrepancies with experiments, especially for systems with strong intermolecular interactions, such as systems containing ions. In 1905,

In addition, various statistical models (often referred to as variational transition state theories), with quantized energy levels, exist for collision processes

Although the aforementioned theoretical approaches are flexible and readily applicable, they often rely on simplified characterizations of the studied collision system and the intermolecular interactions. This can potentially lead to significant inaccuracies in the predicted collision rate coefficients. As mentioned earlier, an ion–dipole complex is often reduced to a point-like charge and a polar rod. However, especially for larger molecules (or clusters), the non-symmetric molecular structure and dynamic partial charge distribution should be considered to determine the actual strength of the interaction. Recently,

In this study, we examine collisions between one charged and one neutral dipolar collision partner. While there are typically significantly fewer ions present compared to neutral molecules, the ion–neutral collision rate coefficients are higher than for neutral–neutral collisions due to relatively strong long-range interactions. Such collisions usually do not involve a significant electronic activation energy barrier. However, the collision process does involve a centrifugal barrier due to the conservation of the system's angular momentum, which can lead to interesting, non-standard, temperature dependencies

Here, as test systems, we considered collisions of the atmospherically relevant molecular dipole sulfuric acid (

We carried out all-atom MD trajectory simulations of the collisions to determine the collision rate coefficient directly from the collision probabilities in relevant ranges of the impact parameter and relative velocity. Additionally, we calculated the potentials of mean force (PMF) between collision partners from well-tempered metadynamics simulations to determine the effective potential, arising from the same underlying atomistic interactions, at finite temperature. Attractive interactions fitted to the tail of the PMFs were used to predict collision cross sections and canonical rate coefficients using a central field model. Last, we compared the analytical Langevin–Gioumousis–Stevenson

The remainder of this paper is organized as follows: in Sect.

The formation of a molecular cluster through collisions requires asymptotic attractive intermolecular interaction potentials which can be ideally modeled as a function of the distance

In the central field model, one of the collision partners is set to be stationary, while the other approaches from infinitely far away with some initial velocity

A schematic diagram of the central field approach with the corresponding potential energy profile. The molecule on the left is at rest, and its center of mass (c.o.m.; center of the gray circle) designates the center of the field, while the molecule on the right initially moves along a trajectory set by a velocity vector,

When inserting Eq. (

The presented central field model is essentially adiabatic; it is assumed that there is neither an exchange of energy between rotational and vibrational modes of the collision partners nor an exchange of angular momentum between the rotations of the collision partners and the orbiting motion of the system as a whole

The expressions for both the collision cross section and rate coefficient are derived for a general, well-behaving, asymptotic attractive interaction given by Eq. (

When the interaction exponent

The main contribution to the intermolecular interactions for collisions between an ion and neutral particle is the ion–induced dipole interaction, as follows:

For collisions between an ion and polar neutral compound, angle-dependent ion–permanent dipole interactions should also be considered. In the most extreme case, the orientation of the dipole can be locked in so that the strength of the interaction is maximized. While thermal rotations of the collision partners will often prevent the dipole from locking in, the ion–permanent dipole interaction is not necessarily averaged over all angles.

We studied a total of eight ion–dipole collision systems. Systems with only molecular ions and dipoles were studied, together with systems with either a dipolar or charged dimer or both. For five of the systems, sulfuric acid (

Stick-and-ball representations of the studied dipoles and ions, including

To describe the test systems, we employed a force field fitted according to the OPLS (optimized potentials for liquid simulations) all-atom procedure

The intermolecular interactions, and intramolecular interactions between atoms separated by more than three covalent bonds, are described by Lennard–Jones potentials between atoms

The OPLS force field parameters used in this study were obtained from

Temperature-dependent long-range attractive interactions and binding free energies of the ion–dipole systems can be obtained from the potential of mean force (PMF) as a function of the distance

We ran well-tempered metadynamics simulations using the LAMMPS code

The location of the potential minimum

To obtain ion–dipole collision cross sections and rate coefficients from MD simulations, we determined the collision probability over a relevant range of impact parameters and relative velocities. All collision simulations were carried out with the LAMMPS code

Collisions were determined based on the minimum center-of-mass distance between the collision partners during the trajectory. All collisions were simulated in the NVE (microcanonical) ensemble, with an initial thermal energy corresponding to a temperature of 300 K achieved during equilibration. In addition, we studied the temperature dependence of the collision probability for the

Figure

The potential of mean force (PMF) along the center-of-mass distance between the collision partners for the systems of

The fitted attractive interactions reveal interesting differences between the systems. The interactions between sulfuric acid (

Figures

Heat maps of the collision probabilities from molecular dynamics (MD) for molecular ion–dipole systems as a function of initial relative velocity

Unlike in the PMF calculations, no constraints were imposed on center-of-mass distances within the dimer structures [

Heat maps of the collision probabilities from molecular dynamics (MD) for ion–dipole systems, including at least one dimer as a function of initial relative velocity

The collision probability heat maps all exhibit similar dependencies on

The dynamic collision cross section is a measure for the velocity-dependent collision probability over all impact parameters considered in the MD simulations and can be calculated as follows:

This agreement indicates that the framework of the central field model adequately captures the underlying atomistic collision dynamics of the studied ion–dipole systems. As the model assumes point-like, structureless collision partners, it disregards any energy transfer between the partners' respective internal degrees of freedom and the orbiting motion of the system as a whole. As a result, the centrifugal barrier is solely determined by the system's initial orbital angular momentum. Based on the demonstrated predictive power of Eq. (

Collision rate coefficients obtained from the molecular dynamics collision simulations

The canonical collision rate coefficient can be calculated from the effective collision cross section of Eq. (

Overall, we find collision rate coefficients of similar magnitudes for the systems containing a molecular dipole. In comparison, systems containing a dipolar dimer exhibit larger collision rate coefficients and more variation in magnitude between the systems. We find excellent agreement within 10 % across all systems for the collision rate coefficients obtained from MD and the central field model using the interactions obtained from the PMF calculation. This indicates that collision dynamics are indeed well captured by an adiabatic model and isotropic interactions.

The Langevin–Gioumousis–Stevenson

Due to the Maxwell–Boltzmann distribution of velocities, the collision rate coefficients given by the central field model and the Su and Chesnavich (1982) parametrization are proportional to

Collision rate coefficient

To study the temperature dependence of the collision kinetics, and the extent to which this is captured by the theoretical models, we performed additional MD trajectory simulations and PMF calculations, for the

The potential of mean force (PMF) along the center-of-mass distance between the collision partners for the

Figure

Due to the temperature dependence of the PMFs, correct collision rate coefficients cannot be computed by simply temperature-scaling

The collision cross section

A proper theoretical treatment of bimolecular reactions requires an accurate assessment of the intermolecular potential. In the context of atmospheric clusters and their formation, high-level ab initio calculations are necessary for assessing the clusters' stability in equilibrium

The demand for accurate theoretical modeling of collisions between atmospheric molecules and clusters (neutral or charged) arises from several recent observations and considerations.

At polluted sites, new particle formation (NPF) is controlled predominantly by collisions due to high vapor concentrations and extremely stable dimers

Due to the immense improvement in ab initio calculations

In chemical ionization mass spectrometry, collisions between a studied atmospheric cluster and a charging ion lead to the formation of a detectable charged cluster in the ionization chamber, while non-sticking collisions between the charged cluster and residual carrier gas in the atmospheric pressure interface can lead to cluster fragmentation, causing systematic errors in the mass spectra

One reason that ion-induced NPF is sometimes disregarded in global aerosol models is the lack of accurate rate coefficients for charged clusters

While simple analytical and parameterized models exist for the calculation of collision rate coefficients of ion–dipole systems, these models do not directly account for the complexity encountered in molecular ions and dipoles, let alone dipolar or charged clusters. Thus, in this study we considered two fundamentally different modeling approaches to calculate ion–dipole collision rates of atmospheric molecules and clusters in the free molecular regime, namely molecular dynamics trajectory simulations and a central field model. Since accurate experimental data are currently missing for the investigated systems, the presented MD simulations serve as reference as each collision trajectory evolves under the full set of atomistic interactions defined by the force field. In contrast, the motion of colliding particles in the central field model follows from the assumption that the effective interaction potential is close to isotropic and adiabatic and thus provides an adequate analytical solution for a certain group of systems at specific conditions.

In this work, we studied the collisions of eight atmospherically relevant ion–dipole systems, described by an atomistic OPLS-based force field.
To achieve comparability between the MD simulations and the central field model, the attractive interaction in the central field model was fitted to the potentials of mean force between the collision partners obtained from well-tempered metadynamics calculations at the respective temperature.
The velocity-dependent collision cross sections from the central field model and the molecular dynamics simulations were found to be in excellent agreement, supporting the assumption that the process can be described by isotropic and adiabatic intermolecular dynamics. Furthermore, the cross sections are very similar for systems with the same neutral dipole, despite the differences in the underlying interaction potentials. Thus, we concluded that, for the studied systems, the canonical collision rate coefficients depend mostly on the dipole's properties, while the ion affects only the velocity distribution through its mass. This finding and the collision rate coefficients calculated from atomistic simulations are in good agreement with the widely used parametrization by

We have demonstrated that the combination of a PMF calculation and central field model is a viable and elegant alternative to the brute force sampling of collision trajectories over a large range of impact parameters and relative velocities, in particular for systems with long-range attractive interactions, such as those between ions and dipoles in the gas phase.
The presented approaches will be used in the future to obtain the collision rate coefficients of a large group of molecules and clusters. The resulting data will allow the assessment of the relative importance of particle growth pathways involving ions in the initial stages of atmospheric new particle formation

Ab initio values for the dipole moment and polarizability of

For the force field model, the dipole moments

To determine the average polarizability of the compounds

Dipole moments

Conversion between cgs and SI units of polarizability is as follows: 1 Å

We benchmarked the accuracy of the all-atom OPLS force field in describing the structures and energies of stable clusters formed upon ion–dipole collision against ab initio calculations.

Ab initio single-point energies of all collision partners, and clusters formed after collision, were obtained from the Atmospheric Cluster Database of

Binding energies of collision systems obtained from ab initio calculations,

For consistency with the single-point energy values obtained from the database, we followed a similar procedure for

We used the global minimum energy configurations obtained from the ab initio calculations and performed an energy minimization with the all-atom OPLS force field in LAMMPS. For all compounds and clusters, the differences between the re-optimized geometry and the ab initio reference structure were quite small, even for the larger clusters.

The cluster binding energies from ab initio and using the OPLS force field reported in Table

This study has been carried out using several computer codes, as explained and referenced in the paper. All these codes are open source, except for GAUSSIAN16, for which open-source alternatives can be found.

The data generated in this study are fully represented in the figures and tables shown in the paper. Input files for simulations are available from the authors upon reasonable request.

IN, HV, and BR planned the study. IN, RH, and BR designed the simulation framework. IN carried out the ab initio calculations, and IN and BR carried out the force-field-based molecular dynamics collision and well-tempered metadynamics simulations. RH provided the theoretical framework. IN, RH, and BR analyzed the simulation data and wrote the first draft. All authors contributed to writing the final paper.

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Computational resources were provided by the CSC–IT Center for Science Ltd., Finland. The authors thank the Finnish Grid and Cloud Infrastructure (FGCI) for supporting this project with computational and data storage resources. The authors thank Valtteri Tikkanen and Huan Yang, for stimulating discussions.

This research has been supported by the European Research Council (project no. 692891 DAMOCLES), the Academy of Finland flagship programme (grant no. 337549) and Centres of Excellence programme (CoE VILMA), and the University of Helsinki, Faculty of Science ATMATH project.Open-access funding was provided by the Helsinki University Library.

This paper was edited by Fangqun Yu and reviewed by Kai Leonhard and one anonymous referee.