Kinetics of collision-sticking processes between vapor molecules and clusters of low-volatility compounds govern the initial steps of atmospheric new particle formation.
Conventional non-interacting hard-sphere models underestimate the collision rate by neglecting long-range attractive forces, and the commonly adopted assumption that every collision leads to the formation of a stable cluster (unit mass accommodation coefficient) is questionable for small clusters, especially at elevated temperatures.
Here, we present a generally applicable analytical interacting hard-sphere model for evaluating collision rates between molecules and clusters, accounting for long-range attractive forces.
In the model, the collision cross section is calculated based on an effective molecule–cluster potential, derived using Hamaker's approach.
Applied to collisions of sulfuric acid or dimethylamine with neutral bisulfate–dimethylammonium clusters composed of 1–32 dimers, our new model predicts collision rates 2–3 times higher than the non-interacting model for small clusters, while decaying asymptotically to the non-interacting limit as cluster size increases, in excellent agreement with a collision-rate-theory atomistic molecular dynamics simulation approach.
Additionally, we calculated sticking rates and mass accommodation coefficients (MACs) using atomistic molecular dynamics collision simulations.
For sulfuric acid, a MAC

In the first steps of atmospheric new particle formation (NPF), condensed-phase clusters

Precise determination of the collision and sticking rate coefficients is non-trivial, as they are sensitive to multiple variables including cluster size, structure, composition, system temperature, and vapor concentration.
Conventional models, like the commonly used classical nucleation theory (CNT;

Molecular dynamics (MD) simulations allow us to study the time evolution of collision systems, where all intra- and intermolecular interactions are described through force fields. Recently, we have developed several atomistic molecular dynamics simulation frameworks

Rate coefficients can also be obtained through analytical models that consider the interactions between the collision partners.
These models usually require a fraction of the computational cost of MD simulations but generally still rely on significant approximations.
In the central field approach, the collision partners are approximated by point particles interacting through an isotropic attractive potential.
Along this line,

We have previously shown that the central field approach yields very similar results to those from sampling collision trajectories using MD, provided that the attractive interaction used in the central field model is fitted to the tail of the potential of mean force (PMF) between the collision partners

In this study, we present two extensions to the central field approach to efficiently predict collision rate coefficients between monomers and clusters of arbitrary sizes.

We consider an interacting hard-sphere model, in which the collision partners are still treated as point particles, interacting through an effective long-ranged isotropic attractive potential, but a collision is considered to have occurred if the center of mass distance between the collision partners is less than the sum of their hard-sphere radii to account for their extended structures.

We integrate the monomer–monomer attractive potential following the approach of Hamaker

As test systems, we considered collisions between the monomers of sulfuric acid (

In the central field model, both collision partners are represented by point particles.
We consider the initial condition where one particle remains at rest while the other particle approaches from infinitely far away with some initial relative velocity

Central field model. The collision partners are described by point particles. In the frame of reference of the target particle, the collision geometry is fully described by the incoming particle's initial velocity

Reducing the two-dimensional single-body orbital motion to a one-dimensional motion along the center-to-center direction introduces a centrifugal energy barrier (the repulsive part of the effective potential) between the point particles.
In the central field model, a “collision” is defined by zero distance between the point particles. Crossing the centrifugal energy barrier in the center-to-center direction is, therefore, a necessary and sufficient condition for a collision. This condition is equivalent to the kinetic energy in the center-to-center direction

Depending on the attractive potential

If

If

If

Shape of

In the interacting hard-sphere model, the aforementioned situations are accounted for by using a revised collision criterion; i.e., if the distance between the collision partners is at any point smaller than the sum of their hard-sphere radii,

To determine the critical impact parameter

Vapors relevant to atmospheric cluster formation are mostly composed of polar and polarizable molecules interacting through a van der Waals potential

Following the approach of Hamaker

Computer simulations were used to

compute the PMF between monomer pairs to obtain the

validate the analytical model by comparison with atomistic and point particle molecular dynamics (MD) collision simulations;

calculate predictions for the sticking probability, sticking rate coefficient, and mass accommodation coefficient by analyzing the atomistic MD collision simulations over a range of relative velocities and impact parameters.

We considered the atmospherically relevant sulfuric acid (

In the PMF and atomistic MD simulations, these systems are described by a force field fitted according to the OPLS (optimized potentials for liquid simulations) all-atom procedure

The intermolecular interactions, as well as 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

We have validated these force field parameters for MD collision simulations in our previous work

To determine the thermally averaged interaction potential between the collision partners, we calculated the PMF as a function of their center of mass distance from well-tempered meta-dynamics simulations

We note that, though in the current paper the monomer–monomer interacting parameters were obtained from the PMF calculated by atomistic simulations, they could in principle be taken directly from literature values if available.

Atomistic simulations were performed with the LAMMPS code

For the collision systems with clusters containing 1, 2, 4, 8, 16, and 32 [

In the point particle molecular dynamics (PP MD) simulations, collision partners were treated as point masses interacting through effective pair potentials. The purpose of performing such simulations was to verify the critical impact parameters derived from the analytical interacting hard-sphere model. The collision simulation setup in this case is similar to that in the atomistic MD simulation. First, the point particles were placed 200 Å apart on the

The MD collision rate coefficient

The mass accommodation coefficient is defined as

To validate our analytical interacting hard-sphere model, we first compared the analytical critical impact parameter

Excellent agreement is observed in both cases (note that the dependent variable

Critical impact parameter. Critical impact parameters

In Fig.

In general,

For both systems, the analytical

Collision probability and critical impact parameter. Collision probability

Having validated the analytical interacting hard-sphere model by comparison with MD simulations, we now use the model to predict enhancement factors

Collision rate enhancement factor. Monomer–cluster collision rate enhancement factors gradually decay to 1 with increasing cluster radii, indicating a transition to hard-sphere-like behavior. The main plot shows the data with logarithmic radius axis and the inset with linear radius axis.

In this work, we use classical, atomistic molecular dynamics simulations to determine parameters used in the analytical model and to validate the analytical model by comparing them to collision MD simulations. It is therefore important to recall the limitations of classical force fields regarding the physico-chemical processes occurring in gas-phase collisions and the timescales on which these different processes occur.

Whether or not a collision will occur is typically determined at a relatively large intermolecular distance, where the collision partners can accurately be modeled using classical force field descriptions in a molecular dynamics simulation. Modeling the probability of the collision partners forming a stable product using MD simulations is, however, significantly more complex, as the “sticking” probabilities are influenced by the possibility of bond formation between the collision partners, which is not captured by non-reactive classical force fields. The acid–base clusters considered in this study can form hydrogen bonds upon collision, which is captured by the classical force field employed in the atomistic simulation; however, proton transfer between acids and bases cannot happen.

After a collision, the newly formed cluster will generally have some excess energy due to these bond formations. In the atmosphere, this energy will be removed from the cluster through collisions with surrounding gas molecules. In the collision MD simulations, however, no additional carrier gas is modeled. As such, the excess energy after a collision cannot be released.

Lastly, it is likely that the cluster dissociation process is unphysically enhanced due to the fact that in the classical atomistic model employed, each vibrational mode possesses

The mentioned limitations of MD simulations in modeling the sticking probability all enhance the dissociation of the formed cluster. As such, the sticking probability obtained from MD simulations can be seen as a limiting value. If the collision partners are able to stick together for a certain time after collision in the MD simulations, then it is likely that the same collision partners would be able to stick together for the same amount of time in the atmosphere. MD simulations are, however, limited in the timescale that they can model due to the computational cost. We, therefore, discuss some characteristic timescales in atmospheric cluster formation after collision.

We first differentiate the two types of monomer–cluster interactions influencing the cluster formation process:
(1) the vibration and diffusion of a condensable vapor monomer on the cluster surface immediately after a collision, which, if successful, drives the cluster formation while storing excess energy due to bond formation (hydrogen bonds in this study),
and (2) collisions between background carrier gas and the cluster, dissipating the excess energy and equilibrating/thermalizing the nascent cluster.
We consider the following scenario: after a condensable vapor–cluster collision, the vapor monomer experiences a few rounds of unsteady vibrations on the cluster surface before forming stable hydrogen bonds (i.e., sticking) or rebounding from the cluster surface.
In the former case, a stable cluster with excess thermal energy, distributed over its degrees of freedom, is produced.
At this point, the captured monomer can only escape from the cluster through thermal fluctuation (i.e., evaporation), as the monomer has become thermally indistinguishable from other molecules in the cluster.
At typical atmospheric conditions, the timescale

For example, for

The timescale analysis strongly indicates that a stable cluster is very likely to form if the impinging condensable vapor monomer can survive a very short window of unsteady vibration right after a collision.

Events of interest in atmospheric cluster growth. The schematic shows events of interest in the acid–base cluster growth process and the corresponding timescales at atmospherically relevant pressures and temperatures.

To calculate collision and sticking probabilities in atomistic molecular dynamics simulations we need simple, but robust, criteria for when two collision partners have collided and when they are sticking together, as the large number of individual trajectories makes a manual inspection unfeasible.
Collision and sticking criteria can be based on geometric and/or energetic properties of the system

Classification of MD collision trajectories. Trajectories can be classified as “flyby” (no collision), “rebounding collision” (no formation of a metastable complex), collisions leading to the formation of a metastable complex that will re-dissociate on a timescale

In analyzing the MD collision simulations, we broadly encounter four different kinds of trajectories:
flyby, rebounding collision, collision with metastable product, and collision with stable product.
An example of each of these types of trajectories is shown in Fig.

A distance criterion that is too strict runs the risk of miscounting collisions with relatively large center of mass distances arising from certain relative orientations as flybys.
Conversely, a distance criterion that is too loose results in certain close flybys being counted as rebounding collisions. Generally, a good distance criterion would lie just above the largest amplitudes of the oscillating sticking trajectories in Fig.

Based on our analyses, a collision is considered to have occurred if the center of mass distance between the collision partners is less than the sum of their hard-sphere radii in at least one time frame of the trajectory.
Based on our timescale analyses in Sect.

Based on the above analysis, we monitored the center of mass distance between the impinging monomer and cluster during the atomistic MD simulations.
Collision probabilities and sticking probabilities 25 ps or 50 ps after collision are shown in Fig.

Collision and sticking probability. Collision probability

The sticking rate coefficients and rate enhancement factors for different clusters sizes and different temperatures are shown in Fig.

Rate coefficient and enhancement factor vs. cluster size (

It is important to remember the limitations of MD simulations using a classical force field description and the timescale analysis performed in Sect.

Precise rate coefficients are essential in modeling the gas-phase collisions which are the initial step in atmospheric new particle formation (NPF). The non-interacting hard-sphere model commonly adopted for neutral molecules and clusters may significantly underestimate collision rate coefficients due to neglecting long-range attractive forces. On the other hand, the assumption of a unit mass accommodation coefficient can overestimate the number of stable clusters formed, in particular for small clusters at elevated temperatures. It is important to note, however, that these two errors will not necessarily cancel each other out.

Simulation approaches based on atomistic modeling of the collision partners have recently been developed and shown to give reasonable estimates for collision rates and enhancement factors for collisions of individual molecules or ions. To systematically calculate rate coefficients for collisions between molecules and clusters of atmospherically relevant acid–base systems, we have developed a new analytical interacting hard-sphere model. The molecule–cluster interactions are obtained from Hamaker's approach by integrating monomer–monomer interactions over the volume of the cluster. Here, the underlying monomer–monomer interacting parameters were obtained from fitting Lennard-Jones potential to the monomer–monomer potential of mean force calculated from atomistic simulations, but we note that the monomer–monomer interacting parameters could also be obtained by other methods or taken directly from literature values if available. The critical impact parameter is determined using the sum of hard-sphere radii or the minimum distance between the point-like collision partners which cannot be crossed due to the centrifugal barrier in the central field model. The accuracy of the analytical model was validated against molecular dynamics collision simulations using the full atomistic model of the collision partners. The analytical model has an accuracy comparable to atomistic molecular dynamics simulations but can be applied efficiently for the systematic calculation of molecule–cluster collision rates required in atmospheric NPF models. The same approach can also be applied to calculate effective cluster–cluster interactions.

For collisions of sulfuric acid (

The analytical interacting hard-sphere model cannot give the probability of sticking or rebounding after collisions. To assess the fraction of collisions leading to a stable product cluster, we analyzed the large data set of collisions obtained during the validation of the analytical interacting hard-sphere model, taking into account the limitations of the classical force fields employed, as well as the timescales of equilibration processes in the atmosphere, not simulated explicitly in the binary collision setup.

Our analysis shows that if a monomer can stay on the cluster surface for more than a characteristic time (typically tens of picoseconds) after a collision, then a stable product cluster is formed. With this criterion, the sticking probability, sticking rate coefficient, and mass accommodation coefficient can be calculated by analyzing a set of molecular dynamics collision trajectories where impact parameters and relative speeds are properly sampled.

The mass accommodation coefficient decreases with the increasing temperature and decreasing cluster size. For collisions between

Future research directions include the investigation of systems where the transition between the two regimes defining the critical impact parameter occurs at lower and more relevant relative initial velocities, as well as the application of the analytical interacting hard-sphere model to cluster–cluster collisions. The model presented in this paper is expected to provide a useful tool for the atmospheric NPF community due to its relative simplicity, demonstrated good accuracy, and widespread applicability.

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

The supplement related to this article is available online at:

HY conceived the research and derived the theoretical framework of the model. IN and HY performed the molecular dynamics collision simulations. IN provided theoretical background for the central field approach. BR and VT performed the metadynamics simulations. VT and JK helped to prepare the clusters used in the simulation. TK and HV helped to plan the project. HY, IN, and BR analyzed the simulation data and wrote the manuscript with comments from all other authors.

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 wish to thank the Finnish Computing Competence Infrastructure (FCCI) for supporting this project with computational and data storage resources.

This research has been supported by the European Research Council (project no. 692891 DAMOCLES), the Academy of Finland Flagship Program (grant no. 337549 ACCC) and Centers of Excellence Program (grant no. 346368 VILMA). Open-access funding was provided by the Helsinki University Library.

This paper was edited by Hinrich Grothe and reviewed by two anonymous referees.